%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP076^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : 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 08:55:21 AM UTC 2026
% Result : Theorem 2.65s 1.63s
% Output : Refutation 2.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP076^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/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 08:25:12 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.46 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : ITP076^2 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.47 % Domain : Interactive Theorem Proving
% 0.26/0.47 % Problem : Sledgehammer HF problem prob_850__5339526_1
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English :
% 0.26/0.47
% 0.26/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.47 % Source : [Des21]
% 0.26/0.47 % Names : HF/prob_850__5339526_1 [Des21]
% 0.26/0.47
% 0.26/0.47 % Status : Theorem
% 0.26/0.47 % Rating : 0.00 v7.5.0
% 0.26/0.47 % Syntax : Number of formulae : 355 ( 155 unt; 56 typ; 0 def)
% 0.26/0.47 % Number of atoms : 648 ( 339 equ; 0 cnn)
% 0.26/0.47 % Maximal formula atoms : 5 ( 2 avg)
% 0.26/0.47 % Number of connectives : 2796 ( 52 ~; 8 |; 46 &;2477 @)
% 0.26/0.47 % ( 0 <=>; 213 =>; 0 <=; 0 <~>)
% 0.26/0.47 % Maximal formula depth : 15 ( 5 avg)
% 0.26/0.47 % Number of types : 3 ( 2 usr)
% 0.26/0.47 % Number of type conns : 149 ( 149 >; 0 *; 0 +; 0 <<)
% 0.26/0.47 % Number of symbols : 55 ( 54 usr; 2 con; 0-5 aty)
% 0.26/0.47 % Number of variables : 901 ( 51 ^; 794 !; 25 ?; 901 :)
% 0.26/0.47 % ( 31 !>; 0 ?*; 0 @-; 0 @+)
% 0.26/0.47 % SPC : TH1_THM_EQU_NAR
% 0.26/0.47
% 0.26/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:23:17.517
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % Could-be-implicit typings (3)
% 0.26/0.47 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 0.26/0.47 hF_Mirabelle_hf: $tType ).
% 0.26/0.47
% 0.26/0.47 thf(ty_t_Set_Oset,type,
% 0.26/0.47 set: $tType > $tType ).
% 0.26/0.47
% 0.26/0.47 thf(ty_t_Nat_Onat,type,
% 0.26/0.47 nat: $tType ).
% 0.26/0.47
% 0.26/0.47 % Explicit typings (53)
% 0.26/0.47 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.26/0.47 bounded_lattice:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_HOL_Otype,type,
% 0.26/0.47 type:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.26/0.47 zero:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Orderings_Otop,type,
% 0.26/0.47 top:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ogroup__add,type,
% 0.26/0.47 group_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Olattice,type,
% 0.26/0.47 lattice:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Omonoid__add,type,
% 0.26/0.47 monoid_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.26/0.47 semigroup_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.26/0.47 comm_monoid_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.26/0.47 ab_semigroup_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.26/0.47 distrib_lattice:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.26/0.47 semilattice_inf:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.26/0.47 semilattice_sup:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.26/0.47 cancel_semigroup_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.26/0.47 bounded_lattice_top:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.26/0.47 cancel1352612707id_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.26/0.47 linord219039673up_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.26/0.47 ordere779506340up_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 0.26/0.47 bounde1561333602nf_top:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.26/0.47 canoni770627133id_add:
% 0.26/0.47 !>[A: $tType] : $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.26/0.47 plus_plus:
% 0.26/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.26/0.47 zero_zero:
% 0.26/0.47 !>[A: $tType] : A ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 0.26/0.47 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 0.26/0.47 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 0.26/0.47 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 0.26/0.47 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 0.26/0.47 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 0.26/0.47 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 0.26/0.47 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard,type,
% 0.26/0.47 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 0.26/0.47 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_ORep__hf,type,
% 0.26/0.47 hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 0.26/0.47 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 0.26/0.47 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfunction,type,
% 0.26/0.47 hF_Mir859708136nction: hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 0.26/0.47 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 0.26/0.47 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 0.26/0.47 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrelation,type,
% 0.26/0.47 hF_Mir1093797708lation: hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrestrict,type,
% 0.26/0.47 hF_Mir165288108strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 0.26/0.47 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 0.26/0.47 hF_Mirabelle_hsplit:
% 0.26/0.47 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ois__hpair,type,
% 0.26/0.47 hF_Mir796903520_hpair: hF_Mirabelle_hf > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.26/0.47 inf_inf:
% 0.26/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.26/0.47 sup_sup:
% 0.26/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Nat_OSuc,type,
% 0.26/0.47 suc: nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.26/0.47 top_top:
% 0.26/0.47 !>[A: $tType] : A ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect,type,
% 0.26/0.47 collect:
% 0.26/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oinsert,type,
% 0.26/0.47 insert:
% 0.26/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Typedef_Otype__definition,type,
% 0.26/0.47 type_definition:
% 0.26/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member,type,
% 0.26/0.47 member:
% 0.26/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_r,type,
% 0.26/0.47 r: hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_x,type,
% 0.26/0.47 x: hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 % Relevant facts (255)
% 0.26/0.47 thf(fact_0_hrelation__restr,axiom,
% 0.26/0.47 ! [R: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( hF_Mir1093797708lation @ ( hF_Mir165288108strict @ R @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % hrelation_restr
% 0.26/0.47 thf(fact_1_hfunction__def,axiom,
% 0.26/0.47 ( hF_Mir859708136nction
% 0.26/0.47 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.26/0.47 ! [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y ) @ R2 )
% 0.26/0.47 => ! [Y2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) @ R2 )
% 0.26/0.47 => ( Y = Y2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hfunction_def
% 0.26/0.47 thf(fact_2_hrestrict__iff,axiom,
% 0.26/0.47 ! [Z: hF_Mirabelle_hf,R: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ Z @ ( hF_Mir165288108strict @ R @ A2 ) )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ Z @ R )
% 0.26/0.47 & ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Z
% 0.26/0.47 = ( hF_Mirabelle_hpair @ X2 @ Y ) )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hrestrict_iff
% 0.26/0.47 thf(fact_3_hpair__iff,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.26/0.47 = ( hF_Mirabelle_hpair @ A4 @ B3 ) )
% 0.26/0.47 = ( ( A3 = A4 )
% 0.26/0.47 & ( B2 = B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_iff
% 0.26/0.47 thf(fact_4_Rep__hf__inject,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_Rep_hf @ X )
% 0.26/0.47 = ( hF_Mirabelle_Rep_hf @ Y3 ) )
% 0.26/0.47 = ( X = Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Rep_hf_inject
% 0.26/0.47 thf(fact_5_hf__equalityI,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ! [X3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.26/0.47 = ( hF_Mirabelle_hmem @ X3 @ B2 ) )
% 0.26/0.47 => ( A3 = B2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % hf_equalityI
% 0.26/0.47 thf(fact_6_hf__ext,axiom,
% 0.26/0.47 ( ( ^ [Y4: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] : ( Y4 = Z2 ) )
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.26/0.47 ! [X2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X2 @ A5 )
% 0.26/0.47 = ( hF_Mirabelle_hmem @ X2 @ B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hf_ext
% 0.26/0.47 thf(fact_7_hmem__ne,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.26/0.47 => ( X != Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_ne
% 0.26/0.47 thf(fact_8_replacement,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U @ X )
% 0.26/0.47 => ( ( R3 @ U @ V )
% 0.26/0.47 => ( ( R3 @ U @ V2 )
% 0.26/0.47 => ( V2 = V ) ) ) )
% 0.26/0.47 => ? [Z3: hF_Mirabelle_hf] :
% 0.26/0.47 ! [V3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.26/0.47 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.47 & ( R3 @ U2 @ V3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % replacement
% 0.26/0.47 thf(fact_9_binary__union,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ? [Z3: hF_Mirabelle_hf] :
% 0.26/0.47 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.26/0.47 | ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % binary_union
% 0.26/0.47 thf(fact_10_hmem__not__sym,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ Y3 @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_not_sym
% 0.26/0.47 thf(fact_11_hpair__inject,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.26/0.47 = ( hF_Mirabelle_hpair @ A4 @ B3 ) )
% 0.26/0.47 => ~ ( ( A3 = A4 )
% 0.26/0.47 => ( B2 != B3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_inject
% 0.26/0.47 thf(fact_12_union__of__set,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] :
% 0.26/0.47 ? [Z3: hF_Mirabelle_hf] :
% 0.26/0.47 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.26/0.47 = ( ? [Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ Y @ X )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % union_of_set
% 0.26/0.47 thf(fact_13_comprehension,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.26/0.47 ? [Z3: hF_Mirabelle_hf] :
% 0.26/0.47 ! [U3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.26/0.47 & ( P @ U3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % comprehension
% 0.26/0.47 thf(fact_14_hmem__not__refl,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_not_refl
% 0.26/0.47 thf(fact_15_hpair__neq__fst,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.26/0.47 != A3 ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_neq_fst
% 0.26/0.47 thf(fact_16_hpair__neq__snd,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.26/0.47 != B2 ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_neq_snd
% 0.26/0.47 thf(fact_17_replacement__fun,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ? [Z3: hF_Mirabelle_hf] :
% 0.26/0.47 ! [V3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.26/0.47 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.47 & ( V3
% 0.26/0.47 = ( F @ U2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % replacement_fun
% 0.26/0.47 thf(fact_18_hsplit,axiom,
% 0.26/0.47 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.26/0.47 = ( C @ A3 @ B2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % hsplit
% 0.26/0.47 thf(fact_19_hfst__conv,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % hfst_conv
% 0.26/0.47 thf(fact_20_hsnd__conv,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.26/0.47 = B2 ) ).
% 0.26/0.47
% 0.26/0.47 % hsnd_conv
% 0.26/0.47 thf(fact_21_hrelation__def,axiom,
% 0.26/0.47 ( hF_Mir1093797708lation
% 0.26/0.47 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.26/0.47 ! [Z4: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ Z4 @ R2 )
% 0.26/0.47 => ( hF_Mir796903520_hpair @ Z4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hrelation_def
% 0.26/0.47 thf(fact_22_is__hpair__def,axiom,
% 0.26/0.47 ( hF_Mir796903520_hpair
% 0.26/0.47 = ( ^ [Z4: hF_Mirabelle_hf] :
% 0.26/0.47 ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( Z4
% 0.26/0.47 = ( hF_Mirabelle_hpair @ X2 @ Y ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % is_hpair_def
% 0.26/0.47 thf(fact_23_PrimReplace__iff,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.26/0.47 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U @ A2 )
% 0.26/0.47 => ( ( R3 @ U @ V )
% 0.26/0.47 => ( ( R3 @ U @ V2 )
% 0.26/0.47 => ( V2 = V ) ) ) )
% 0.26/0.47 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir569462966eplace @ A2 @ R3 ) )
% 0.26/0.47 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.26/0.47 & ( R3 @ U2 @ V4 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % PrimReplace_iff
% 0.26/0.47 thf(fact_24_HUnion__iff,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A2 ) )
% 0.26/0.47 = ( ? [Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ Y @ A2 )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HUnion_iff
% 0.26/0.47 thf(fact_25_Replace__iff,axiom,
% 0.26/0.47 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A2 @ R3 ) )
% 0.26/0.47 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.26/0.47 & ( R3 @ U2 @ V4 )
% 0.26/0.47 & ! [Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( R3 @ U2 @ Y )
% 0.26/0.47 => ( Y = V4 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Replace_iff
% 0.26/0.47 thf(fact_26_HCollect__iff,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir1687042746ollect @ P @ A2 ) )
% 0.26/0.47 = ( ( P @ X )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ X @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HCollect_iff
% 0.26/0.47 thf(fact_27_RepFun__iff,axiom,
% 0.26/0.47 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A2 @ F ) )
% 0.26/0.47 = ( ? [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.26/0.47 & ( V4
% 0.26/0.47 = ( F @ U2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_iff
% 0.26/0.47 thf(fact_28_Rep__hf__inverse,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_Abs_hf @ ( hF_Mirabelle_Rep_hf @ X ) )
% 0.26/0.47 = X ) ).
% 0.26/0.47
% 0.26/0.47 % Rep_hf_inverse
% 0.26/0.47 thf(fact_29_RepFun__cong,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( A2 = B5 )
% 0.26/0.47 => ( ! [X3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X3 @ B5 )
% 0.26/0.47 => ( ( F @ X3 )
% 0.26/0.47 = ( G @ X3 ) ) )
% 0.26/0.47 => ( ( hF_Mirabelle_RepFun @ A2 @ F )
% 0.26/0.47 = ( hF_Mirabelle_RepFun @ B5 @ G ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_cong
% 0.26/0.47 thf(fact_30_Replace__cong,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( A2 = B5 )
% 0.26/0.47 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X3 @ B5 )
% 0.26/0.47 => ( ( P @ X3 @ Y5 )
% 0.26/0.47 = ( Q @ X3 @ Y5 ) ) )
% 0.26/0.47 => ( ( hF_Mirabelle_Replace @ A2 @ P )
% 0.26/0.47 = ( hF_Mirabelle_Replace @ B5 @ Q ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Replace_cong
% 0.26/0.47 thf(fact_31_hmem__Sup__ne,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.26/0.47 => ( ( hF_Mirabelle_HUnion @ X )
% 0.26/0.47 != Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_Sup_ne
% 0.26/0.47 thf(fact_32_Abs__hf__inverse,axiom,
% 0.26/0.47 ! [Y3: nat] :
% 0.26/0.47 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ( ( hF_Mirabelle_Rep_hf @ ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.26/0.47 = Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Abs_hf_inverse
% 0.26/0.47 thf(fact_33_zero__notin__hpair,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( hF_Mirabelle_hmem @ ( zero_zero @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hpair @ X @ Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_notin_hpair
% 0.26/0.47 thf(fact_34_hmem__def,axiom,
% 0.26/0.47 ( hF_Mirabelle_hmem
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hfset @ B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_def
% 0.26/0.47 thf(fact_35_HCollect__hempty,axiom,
% 0.26/0.47 ! [P: hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % HCollect_hempty
% 0.26/0.47 thf(fact_36_HUnion__hempty,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % HUnion_hempty
% 0.26/0.47 thf(fact_37_Replace__0,axiom,
% 0.26/0.47 ! [R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R3 )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % Replace_0
% 0.26/0.47 thf(fact_38_RepFun__0,axiom,
% 0.26/0.47 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_0
% 0.26/0.47 thf(fact_39_RepFun__hinsert,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A3 @ B2 ) @ F )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ ( F @ A3 ) @ ( hF_Mirabelle_RepFun @ B2 @ F ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_hinsert
% 0.26/0.47 thf(fact_40_hrelation__hunion,axiom,
% 0.26/0.47 ! [F: hF_Mirabelle_hf,G: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mir1093797708lation @ ( sup_sup @ hF_Mirabelle_hf @ F @ G ) )
% 0.26/0.47 = ( ( hF_Mir1093797708lation @ F )
% 0.26/0.47 & ( hF_Mir1093797708lation @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hrelation_hunion
% 0.26/0.47 thf(fact_41_hmem__hinsert,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ C ) )
% 0.26/0.47 = ( ( A3 = B2 )
% 0.26/0.47 | ( hF_Mirabelle_hmem @ A3 @ C ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_hinsert
% 0.26/0.47 thf(fact_42_singleton__eq__iff,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hinsert @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.26/0.47 = ( A3 = B2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_eq_iff
% 0.26/0.47 thf(fact_43_hunion__iff,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ X @ ( sup_sup @ hF_Mirabelle_hf @ A3 @ B2 ) )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ X @ A3 )
% 0.26/0.47 | ( hF_Mirabelle_hmem @ X @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hunion_iff
% 0.26/0.47 thf(fact_44_hunion__hempty__left,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( sup_sup @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A2 )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % hunion_hempty_left
% 0.26/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,P: A > $o] :
% 0.26/0.47 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.26/0.47 = ( P @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_Collect_eq
% 0.26/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ( collect @ A
% 0.26/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mem_eq
% 0.26/0.47 thf(fact_47_Collect__cong,axiom,
% 0.26/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.26/0.47 ( ! [X3: A] :
% 0.26/0.47 ( ( P @ X3 )
% 0.26/0.47 = ( Q @ X3 ) )
% 0.26/0.47 => ( ( collect @ A @ P )
% 0.26/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_cong
% 0.26/0.47 thf(fact_48_ext,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.26/0.47 ( ! [X3: A] :
% 0.26/0.47 ( ( F @ X3 )
% 0.26/0.47 = ( G @ X3 ) )
% 0.26/0.47 => ( F = G ) ) ).
% 0.26/0.47
% 0.26/0.47 % ext
% 0.26/0.47 thf(fact_49_hunion__hempty__right,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( sup_sup @ hF_Mirabelle_hf @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % hunion_hempty_right
% 0.26/0.47 thf(fact_50_RepFun__hunion,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_RepFun @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) @ F )
% 0.26/0.47 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_RepFun @ A2 @ F ) @ ( hF_Mirabelle_RepFun @ B5 @ F ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % RepFun_hunion
% 0.26/0.47 thf(fact_51_HUnion__hunion,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_HUnion @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) )
% 0.26/0.47 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_HUnion @ A2 ) @ ( hF_Mirabelle_HUnion @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HUnion_hunion
% 0.26/0.47 thf(fact_52_Replace__hunion,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.26/0.47 ( ( hF_Mirabelle_Replace @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) @ R3 )
% 0.26/0.47 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_Replace @ A2 @ R3 ) @ ( hF_Mirabelle_Replace @ B5 @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Replace_hunion
% 0.26/0.47 thf(fact_53_HUnion__hinsert,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A3 @ A2 ) )
% 0.26/0.47 = ( sup_sup @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_HUnion @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HUnion_hinsert
% 0.26/0.47 thf(fact_54_hf__induct__ax,axiom,
% 0.26/0.47 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ( ! [X3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( P @ X3 )
% 0.26/0.47 => ! [Y5: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( P @ Y5 )
% 0.26/0.47 => ( P @ ( hF_Mirabelle_hinsert @ Y5 @ X3 ) ) ) )
% 0.26/0.47 => ( P @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hf_induct_ax
% 0.26/0.47 thf(fact_55_hinsert__eq__sup,axiom,
% 0.26/0.47 ( hF_Mirabelle_hinsert
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( sup_sup @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_eq_sup
% 0.26/0.47 thf(fact_56_hinsert__commute,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y3 @ Z ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ Y3 @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_commute
% 0.26/0.47 thf(fact_57_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 0.26/0.47 = ( ( ( A3 = C )
% 0.26/0.47 & ( B2 = D ) )
% 0.26/0.47 | ( ( A3 = D )
% 0.26/0.47 & ( B2 = C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HF_Mirabelle_fsbjehakzm.doubleton_eq_iff
% 0.26/0.47 thf(fact_58_hinsert__nonempty,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hinsert @ A3 @ A2 )
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_nonempty
% 0.26/0.47 thf(fact_59_hunion__hinsert__left,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hunion_hinsert_left
% 0.26/0.47 thf(fact_60_hunion__hinsert__right,axiom,
% 0.26/0.47 ! [B5: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( sup_sup @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hunion_hinsert_right
% 0.26/0.47 thf(fact_61_hf__cases,axiom,
% 0.26/0.47 ! [Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Y3
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ~ ! [A6: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Y3
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ A6 @ B6 ) )
% 0.26/0.47 => ( hF_Mirabelle_hmem @ A6 @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hf_cases
% 0.26/0.47 thf(fact_62_hf__induct,axiom,
% 0.26/0.47 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( P @ X3 )
% 0.26/0.47 => ( ( P @ Y5 )
% 0.26/0.47 => ( ~ ( hF_Mirabelle_hmem @ X3 @ Y5 )
% 0.26/0.47 => ( P @ ( hF_Mirabelle_hinsert @ X3 @ Y5 ) ) ) ) )
% 0.26/0.47 => ( P @ Z ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hf_induct
% 0.26/0.47 thf(fact_63_hpair__def,axiom,
% 0.26/0.47 ( hF_Mirabelle_hpair
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ B4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_def
% 0.26/0.47 thf(fact_64_hpair__def_H,axiom,
% 0.26/0.47 ( hF_Mirabelle_hpair
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ B4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_def'
% 0.26/0.47 thf(fact_65_hinsert__iff,axiom,
% 0.26/0.47 ! [Z: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Z
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ Y3 @ X ) )
% 0.26/0.47 = ( ! [U2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.26/0.47 | ( U2 = Y3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_iff
% 0.26/0.47 thf(fact_66_hemptyE,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hemptyE
% 0.26/0.47 thf(fact_67_hempty__iff,axiom,
% 0.26/0.47 ! [Z: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Z
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( ! [X2: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( hF_Mirabelle_hmem @ X2 @ Z ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hempty_iff
% 0.26/0.47 thf(fact_68_hmem__hempty,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hmem_hempty
% 0.26/0.47 thf(fact_69_hpair__nonzero,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hpair @ X @ Y3 )
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hpair_nonzero
% 0.26/0.47 thf(fact_70_Rep__hf,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] : ( member @ nat @ ( hF_Mirabelle_Rep_hf @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Rep_hf
% 0.26/0.47 thf(fact_71_Rep__hf__cases,axiom,
% 0.26/0.47 ! [Y3: nat] :
% 0.26/0.47 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.26/0.47 ( Y3
% 0.26/0.47 != ( hF_Mirabelle_Rep_hf @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Rep_hf_cases
% 0.26/0.47 thf(fact_72_Rep__hf__induct,axiom,
% 0.26/0.47 ! [Y3: nat,P: nat > $o] :
% 0.26/0.47 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( hF_Mirabelle_Rep_hf @ X3 ) )
% 0.26/0.47 => ( P @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Rep_hf_induct
% 0.26/0.47 thf(fact_73_Abs__hf__cases,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf] :
% 0.26/0.47 ~ ! [Y5: nat] :
% 0.26/0.47 ( ( X
% 0.26/0.47 = ( hF_Mirabelle_Abs_hf @ Y5 ) )
% 0.26/0.47 => ~ ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Abs_hf_cases
% 0.26/0.47 thf(fact_74_Abs__hf__induct,axiom,
% 0.26/0.47 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ! [Y5: nat] :
% 0.26/0.47 ( ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ( P @ ( hF_Mirabelle_Abs_hf @ Y5 ) ) )
% 0.26/0.47 => ( P @ X ) ) ).
% 0.26/0.47
% 0.26/0.47 % Abs_hf_induct
% 0.26/0.47 thf(fact_75_Abs__hf__inject,axiom,
% 0.26/0.47 ! [X: nat,Y3: nat] :
% 0.26/0.47 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.26/0.47 => ( ( ( hF_Mirabelle_Abs_hf @ X )
% 0.26/0.47 = ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.26/0.47 = ( X = Y3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Abs_hf_inject
% 0.26/0.47 thf(fact_76_hrelation__0,axiom,
% 0.26/0.47 hF_Mir1093797708lation @ ( zero_zero @ hF_Mirabelle_hf ) ).
% 0.26/0.47
% 0.26/0.47 % hrelation_0
% 0.26/0.47 thf(fact_77_sup__top__left,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounded_lattice_top @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.26/0.47 = ( top_top @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_top_left
% 0.26/0.47 thf(fact_78_sup__top__right,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounded_lattice_top @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.26/0.47 = ( top_top @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_top_right
% 0.26/0.47 thf(fact_79_HInter__iff,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A2 ) )
% 0.26/0.47 = ( ! [Y: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ Y @ A2 )
% 0.26/0.47 => ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HInter_iff
% 0.26/0.47 thf(fact_80_sup__apply,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ B )
% 0.26/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.26/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_apply
% 0.26/0.47 thf(fact_81_sup_Oidem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( sup_sup @ A @ A3 @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.idem
% 0.26/0.47 thf(fact_82_sup__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ X )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_idem
% 0.26/0.47 thf(fact_83_sup_Oleft__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
% 0.26/0.47 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.left_idem
% 0.26/0.47 thf(fact_84_sup__left__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.26/0.47 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_left_idem
% 0.26/0.47 thf(fact_85_sup_Oright__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
% 0.26/0.47 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.right_idem
% 0.26/0.47 thf(fact_86_HInter__hempty,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % HInter_hempty
% 0.26/0.47 thf(fact_87_sup__left__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_left_commute
% 0.26/0.47 thf(fact_88_sup_Oleft__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [B2: A,A3: A,C: A] :
% 0.26/0.47 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C ) )
% 0.26/0.47 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.left_commute
% 0.26/0.47 thf(fact_89_sup__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ( ( sup_sup @ A )
% 0.26/0.47 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_commute
% 0.26/0.47 thf(fact_90_sup_Ocommute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ( ( sup_sup @ A )
% 0.26/0.47 = ( ^ [A5: A,B4: A] : ( sup_sup @ A @ B4 @ A5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.commute
% 0.26/0.47 thf(fact_91_sup__assoc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
% 0.26/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_assoc
% 0.26/0.47 thf(fact_92_sup_Oassoc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C )
% 0.26/0.47 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup.assoc
% 0.26/0.47 thf(fact_93_boolean__algebra__cancel_Osup2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.26/0.47 ( ( B5
% 0.26/0.47 = ( sup_sup @ A @ K @ B2 ) )
% 0.26/0.47 => ( ( sup_sup @ A @ A3 @ B5 )
% 0.26/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % boolean_algebra_cancel.sup2
% 0.26/0.47 thf(fact_94_boolean__algebra__cancel_Osup1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A )
% 0.26/0.47 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( sup_sup @ A @ K @ A3 ) )
% 0.26/0.47 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.26/0.47 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % boolean_algebra_cancel.sup1
% 0.26/0.47 thf(fact_95_sup__fun__def,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ B )
% 0.26/0.47 => ( ( sup_sup @ ( A > B ) )
% 0.26/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_fun_def
% 0.26/0.47 thf(fact_96_inf__sup__aci_I5_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ( ( sup_sup @ A )
% 0.26/0.47 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(5)
% 0.26/0.47 thf(fact_97_inf__sup__aci_I6_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
% 0.26/0.47 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(6)
% 0.26/0.47 thf(fact_98_inf__sup__aci_I7_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(7)
% 0.26/0.47 thf(fact_99_inf__sup__aci_I8_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.26/0.47 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(8)
% 0.26/0.47 thf(fact_100_iso__tuple__UNIV__I,axiom,
% 0.26/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % iso_tuple_UNIV_I
% 0.26/0.47 thf(fact_101_UNIV__I,axiom,
% 0.26/0.47 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UNIV_I
% 0.26/0.47 thf(fact_102_top__apply,axiom,
% 0.26/0.47 ! [C2: $tType,D2: $tType] :
% 0.26/0.47 ( ( top @ C2 )
% 0.26/0.47 => ( ( top_top @ ( D2 > C2 ) )
% 0.26/0.47 = ( ^ [X2: D2] : ( top_top @ C2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % top_apply
% 0.26/0.47 thf(fact_103_HInter__hinsert,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A3 @ A2 ) )
% 0.26/0.47 = ( inf_inf @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_HInter @ A2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % HInter_hinsert
% 0.26/0.47 thf(fact_104_type__definition__hf,axiom,
% 0.26/0.47 type_definition @ hF_Mirabelle_hf @ nat @ hF_Mirabelle_Rep_hf @ hF_Mirabelle_Abs_hf @ ( top_top @ ( set @ nat ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition_hf
% 0.26/0.47 thf(fact_105_inf__right__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 )
% 0.26/0.47 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_right_idem
% 0.26/0.47 thf(fact_106_inf_Oright__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
% 0.26/0.47 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.right_idem
% 0.26/0.47 thf(fact_107_inf__left__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.26/0.47 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_left_idem
% 0.26/0.47 thf(fact_108_inf_Oleft__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
% 0.26/0.47 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.left_idem
% 0.26/0.47 thf(fact_109_inf__idem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ X )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_idem
% 0.26/0.47 thf(fact_110_inf_Oidem,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ A3 @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.idem
% 0.26/0.47 thf(fact_111_inf__apply,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ B )
% 0.26/0.47 => ( ( inf_inf @ ( A > B ) )
% 0.26/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_apply
% 0.26/0.47 thf(fact_112_inf__top_Oright__neutral,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top.right_neutral
% 0.26/0.47 thf(fact_113_inf__top_Oneutr__eq__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( ( top_top @ A )
% 0.26/0.47 = ( inf_inf @ A @ A3 @ B2 ) )
% 0.26/0.47 = ( ( A3
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 & ( B2
% 0.26/0.47 = ( top_top @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top.neutr_eq_iff
% 0.26/0.47 thf(fact_114_inf__top_Oleft__neutral,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top.left_neutral
% 0.26/0.47 thf(fact_115_inf__top_Oeq__neutr__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( ( inf_inf @ A @ A3 @ B2 )
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 = ( ( A3
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 & ( B2
% 0.26/0.47 = ( top_top @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top.eq_neutr_iff
% 0.26/0.47 thf(fact_116_top__eq__inf__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( ( top_top @ A )
% 0.26/0.47 = ( inf_inf @ A @ X @ Y3 ) )
% 0.26/0.47 = ( ( X
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 & ( Y3
% 0.26/0.47 = ( top_top @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % top_eq_inf_iff
% 0.26/0.47 thf(fact_117_inf__eq__top__iff,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( ( inf_inf @ A @ X @ Y3 )
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 = ( ( X
% 0.26/0.47 = ( top_top @ A ) )
% 0.26/0.47 & ( Y3
% 0.26/0.47 = ( top_top @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_eq_top_iff
% 0.26/0.47 thf(fact_118_inf__top__right,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top_right
% 0.26/0.47 thf(fact_119_inf__top__left,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( bounde1561333602nf_top @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_top_left
% 0.26/0.47 thf(fact_120_inf__sup__absorb,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_absorb
% 0.26/0.47 thf(fact_121_sup__inf__absorb,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_inf_absorb
% 0.26/0.47 thf(fact_122_hinter__iff,axiom,
% 0.26/0.47 ! [U4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y3 ) )
% 0.26/0.47 = ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.26/0.47 & ( hF_Mirabelle_hmem @ U4 @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinter_iff
% 0.26/0.47 thf(fact_123_hinter__hempty__left,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A2 )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinter_hempty_left
% 0.26/0.47 thf(fact_124_hinter__hempty__right,axiom,
% 0.26/0.47 ! [A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinter_hempty_right
% 0.26/0.47 thf(fact_125_Abs__hf__0,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.26/0.47
% 0.26/0.47 % Abs_hf_0
% 0.26/0.47 thf(fact_126_inf__left__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_left_commute
% 0.26/0.47 thf(fact_127_inf_Oleft__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [B2: A,A3: A,C: A] :
% 0.26/0.47 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C ) )
% 0.26/0.47 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.left_commute
% 0.26/0.47 thf(fact_128_inf__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ( ( inf_inf @ A )
% 0.26/0.47 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_commute
% 0.26/0.47 thf(fact_129_inf_Ocommute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ( ( inf_inf @ A )
% 0.26/0.47 = ( ^ [A5: A,B4: A] : ( inf_inf @ A @ B4 @ A5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.commute
% 0.26/0.47 thf(fact_130_inf__assoc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z )
% 0.26/0.47 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_assoc
% 0.26/0.47 thf(fact_131_inf_Oassoc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C )
% 0.26/0.47 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf.assoc
% 0.26/0.47 thf(fact_132_boolean__algebra__cancel_Oinf2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.26/0.47 ( ( B5
% 0.26/0.47 = ( inf_inf @ A @ K @ B2 ) )
% 0.26/0.47 => ( ( inf_inf @ A @ A3 @ B5 )
% 0.26/0.47 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % boolean_algebra_cancel.inf2
% 0.26/0.47 thf(fact_133_boolean__algebra__cancel_Oinf1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A )
% 0.26/0.47 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( inf_inf @ A @ K @ A3 ) )
% 0.26/0.47 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.26/0.47 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % boolean_algebra_cancel.inf1
% 0.26/0.47 thf(fact_134_inf__fun__def,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ B )
% 0.26/0.47 => ( ( inf_inf @ ( A > B ) )
% 0.26/0.47 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_fun_def
% 0.26/0.47 thf(fact_135_inf__sup__aci_I1_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ( ( inf_inf @ A )
% 0.26/0.47 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(1)
% 0.26/0.47 thf(fact_136_inf__sup__aci_I2_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z )
% 0.26/0.47 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(2)
% 0.26/0.47 thf(fact_137_inf__sup__aci_I3_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(3)
% 0.26/0.47 thf(fact_138_inf__sup__aci_I4_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.26/0.47 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_aci(4)
% 0.26/0.47 thf(fact_139_sup__inf__distrib2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( distrib_lattice @ A )
% 0.26/0.47 => ! [Y3: A,Z: A,X: A] :
% 0.26/0.47 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z ) @ X )
% 0.26/0.47 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_inf_distrib2
% 0.26/0.47 thf(fact_140_sup__inf__distrib1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( distrib_lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % sup_inf_distrib1
% 0.26/0.47 thf(fact_141_inf__sup__distrib2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( distrib_lattice @ A )
% 0.26/0.47 => ! [Y3: A,Z: A,X: A] :
% 0.26/0.47 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z ) @ X )
% 0.26/0.47 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_distrib2
% 0.26/0.47 thf(fact_142_inf__sup__distrib1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( distrib_lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inf_sup_distrib1
% 0.26/0.47 thf(fact_143_distrib__imp2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ! [X3: A,Y5: A,Z3: A] :
% 0.26/0.47 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y5 @ Z3 ) )
% 0.26/0.47 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( sup_sup @ A @ X3 @ Z3 ) ) )
% 0.26/0.47 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % distrib_imp2
% 0.26/0.47 thf(fact_144_distrib__imp1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( lattice @ A )
% 0.26/0.47 => ! [X: A,Y3: A,Z: A] :
% 0.26/0.47 ( ! [X3: A,Y5: A,Z3: A] :
% 0.26/0.47 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y5 @ Z3 ) )
% 0.26/0.47 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( inf_inf @ A @ X3 @ Z3 ) ) )
% 0.26/0.47 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.26/0.47 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % distrib_imp1
% 0.26/0.47 thf(fact_145_foundation,axiom,
% 0.26/0.47 ! [Z: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( Z
% 0.26/0.47 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 => ? [W: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.26/0.47 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z )
% 0.26/0.47 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % foundation
% 0.26/0.47 thf(fact_146_hinter__hinsert__right,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hmem @ X @ B5 )
% 0.26/0.47 => ( ( inf_inf @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) )
% 0.26/0.47 & ( ~ ( hF_Mirabelle_hmem @ X @ B5 )
% 0.26/0.47 => ( ( inf_inf @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.26/0.47 = ( inf_inf @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinter_hinsert_right
% 0.26/0.47 thf(fact_147_hinter__hinsert__left,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hmem @ X @ B5 )
% 0.26/0.47 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.26/0.47 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) )
% 0.26/0.47 & ( ~ ( hF_Mirabelle_hmem @ X @ B5 )
% 0.26/0.47 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.26/0.47 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinter_hinsert_left
% 0.26/0.47 thf(fact_148_zero__reorient,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( zero @ A )
% 0.26/0.47 => ! [X: A] :
% 0.26/0.47 ( ( ( zero_zero @ A )
% 0.26/0.47 = X )
% 0.26/0.47 = ( X
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_reorient
% 0.26/0.47 thf(fact_149_UNIV__eq__I,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.26/0.47 => ( ( top_top @ ( set @ A ) )
% 0.26/0.47 = A2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % UNIV_eq_I
% 0.26/0.47 thf(fact_150_UNIV__witness,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UNIV_witness
% 0.26/0.47 thf(fact_151_Un__UNIV__right,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.26/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_UNIV_right
% 0.26/0.47 thf(fact_152_Un__UNIV__left,axiom,
% 0.26/0.47 ! [A: $tType,B5: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
% 0.26/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_UNIV_left
% 0.26/0.47 thf(fact_153_top__set__def,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( top_top @ ( set @ A ) )
% 0.26/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % top_set_def
% 0.26/0.47 thf(fact_154_type__copy__ex__RepI,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F3: B > $o] :
% 0.26/0.47 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.26/0.47 => ( ( ? [X4: B] : ( F3 @ X4 ) )
% 0.26/0.47 = ( ? [B4: A] : ( F3 @ ( Rep @ B4 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_copy_ex_RepI
% 0.26/0.47 thf(fact_155_type__copy__obj__one__point__absE,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
% 0.26/0.47 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.26/0.47 => ~ ! [X3: B] :
% 0.26/0.47 ( S
% 0.26/0.47 != ( Abs @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_copy_obj_one_point_absE
% 0.26/0.47 thf(fact_156_HF__hfset,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A3 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % HF_hfset
% 0.26/0.47 thf(fact_157_type__definition_ORep,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( member @ A @ ( Rep @ X ) @ A2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Rep
% 0.26/0.47 thf(fact_158_type__definition__def,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType] :
% 0.26/0.47 ( ( type_definition @ B @ A )
% 0.26/0.47 = ( ^ [Rep2: B > A,Abs2: A > B,A7: set @ A] :
% 0.26/0.47 ( ! [X2: B] : ( member @ A @ ( Rep2 @ X2 ) @ A7 )
% 0.26/0.47 & ! [X2: B] :
% 0.26/0.47 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.26/0.47 = X2 )
% 0.26/0.47 & ! [Y: A] :
% 0.26/0.47 ( ( member @ A @ Y @ A7 )
% 0.26/0.47 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.26/0.47 = Y ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition_def
% 0.26/0.47 thf(fact_159_Int__UNIV,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B5 )
% 0.26/0.47 = ( top_top @ ( set @ A ) ) )
% 0.26/0.47 = ( ( A2
% 0.26/0.47 = ( top_top @ ( set @ A ) ) )
% 0.26/0.47 & ( B5
% 0.26/0.47 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_UNIV
% 0.26/0.47 thf(fact_160_Int__Un__eq_I4_J,axiom,
% 0.26/0.47 ! [A: $tType,T: set @ A,S2: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ T @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.26/0.47 = T ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_eq(4)
% 0.26/0.47 thf(fact_161_Int__Un__eq_I3_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ S2 @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.26/0.47 = S2 ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_eq(3)
% 0.26/0.47 thf(fact_162_Int__Un__eq_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) @ T )
% 0.26/0.47 = T ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_eq(2)
% 0.26/0.47 thf(fact_163_Int__Un__eq_I1_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) @ S2 )
% 0.26/0.47 = S2 ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_eq(1)
% 0.26/0.47 thf(fact_164_Un__Int__eq_I4_J,axiom,
% 0.26/0.47 ! [A: $tType,T: set @ A,S2: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ T @ ( sup_sup @ ( set @ A ) @ S2 @ T ) )
% 0.26/0.47 = T ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_eq(4)
% 0.26/0.47 thf(fact_165_Un__Int__eq_I3_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ S2 @ ( sup_sup @ ( set @ A ) @ S2 @ T ) )
% 0.26/0.47 = S2 ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_eq(3)
% 0.26/0.47 thf(fact_166_Un__Int__eq_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T ) @ T )
% 0.26/0.47 = T ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_eq(2)
% 0.26/0.47 thf(fact_167_Un__Int__eq_I1_J,axiom,
% 0.26/0.47 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T ) @ S2 )
% 0.26/0.47 = S2 ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_eq(1)
% 0.26/0.47 thf(fact_168_UnCI,axiom,
% 0.26/0.47 ! [A: $tType,C: A,B5: set @ A,A2: set @ A] :
% 0.26/0.47 ( ( ~ ( member @ A @ C @ B5 )
% 0.26/0.47 => ( member @ A @ C @ A2 ) )
% 0.26/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UnCI
% 0.26/0.47 thf(fact_169_Un__iff,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 = ( ( member @ A @ C @ A2 )
% 0.26/0.47 | ( member @ A @ C @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_iff
% 0.26/0.47 thf(fact_170_top1I,axiom,
% 0.26/0.47 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.26/0.47
% 0.26/0.47 % top1I
% 0.26/0.47 thf(fact_171_top__conj_I2_J,axiom,
% 0.26/0.47 ! [A: $tType,P: $o,X: A] :
% 0.26/0.47 ( ( P
% 0.26/0.47 & ( top_top @ ( A > $o ) @ X ) )
% 0.26/0.47 = P ) ).
% 0.26/0.47
% 0.26/0.47 % top_conj(2)
% 0.26/0.47 thf(fact_172_top__conj_I1_J,axiom,
% 0.26/0.47 ! [A: $tType,X: A,P: $o] :
% 0.26/0.47 ( ( ( top_top @ ( A > $o ) @ X )
% 0.26/0.47 & P )
% 0.26/0.47 = P ) ).
% 0.26/0.47
% 0.26/0.47 % top_conj(1)
% 0.26/0.47 thf(fact_173_UnE,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 => ( ~ ( member @ A @ C @ A2 )
% 0.26/0.47 => ( member @ A @ C @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UnE
% 0.26/0.47 thf(fact_174_UnI1,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ A2 )
% 0.26/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UnI1
% 0.26/0.47 thf(fact_175_UnI2,axiom,
% 0.26/0.47 ! [A: $tType,C: A,B5: set @ A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ B5 )
% 0.26/0.47 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % UnI2
% 0.26/0.47 thf(fact_176_bex__Un,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
% 0.26/0.47 ( ( ? [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 & ( P @ X2 ) ) )
% 0.26/0.47 = ( ? [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ A2 )
% 0.26/0.47 & ( P @ X2 ) )
% 0.26/0.47 | ? [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ B5 )
% 0.26/0.47 & ( P @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % bex_Un
% 0.26/0.47 thf(fact_177_ball__Un,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
% 0.26/0.47 ( ( ! [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 => ( P @ X2 ) ) )
% 0.26/0.47 = ( ! [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ A2 )
% 0.26/0.47 => ( P @ X2 ) )
% 0.26/0.47 & ! [X2: A] :
% 0.26/0.47 ( ( member @ A @ X2 @ B5 )
% 0.26/0.47 => ( P @ X2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ball_Un
% 0.26/0.47 thf(fact_178_Un__assoc,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ C3 )
% 0.26/0.47 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_assoc
% 0.26/0.47 thf(fact_179_Un__absorb,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % Un_absorb
% 0.26/0.47 thf(fact_180_Un__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.26/0.47 = ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_commute
% 0.26/0.47 thf(fact_181_Un__Int__crazy,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) @ ( inf_inf @ ( set @ A ) @ C3 @ A2 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) @ ( sup_sup @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_crazy
% 0.26/0.47 thf(fact_182_Int__UNIV__left,axiom,
% 0.26/0.47 ! [A: $tType,B5: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
% 0.26/0.47 = B5 ) ).
% 0.26/0.47
% 0.26/0.47 % Int_UNIV_left
% 0.26/0.47 thf(fact_183_Int__UNIV__right,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % Int_UNIV_right
% 0.26/0.47 thf(fact_184_Int__Un__distrib,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.26/0.47 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_distrib
% 0.26/0.47 thf(fact_185_Un__Int__distrib,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_distrib
% 0.26/0.47 thf(fact_186_Un__left__absorb,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 = ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_left_absorb
% 0.26/0.47 thf(fact_187_Int__Un__distrib2,axiom,
% 0.26/0.47 ! [A: $tType,B5: set @ A,C3: set @ A,A2: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) @ A2 )
% 0.26/0.47 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_Un_distrib2
% 0.26/0.47 thf(fact_188_Un__Int__distrib2,axiom,
% 0.26/0.47 ! [A: $tType,B5: set @ A,C3: set @ A,A2: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) @ A2 )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ A2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_Int_distrib2
% 0.26/0.47 thf(fact_189_Un__left__commute,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.26/0.47 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_left_commute
% 0.26/0.47 thf(fact_190_type__definition_ORep__inverse,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( Abs @ ( Rep @ X ) )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Rep_inverse
% 0.26/0.47 thf(fact_191_type__definition_OAbs__inverse,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( member @ A @ Y3 @ A2 )
% 0.26/0.47 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.26/0.47 = Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Abs_inverse
% 0.26/0.47 thf(fact_192_type__definition_ORep__inject,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B,Y3: B] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( ( Rep @ X )
% 0.26/0.47 = ( Rep @ Y3 ) )
% 0.26/0.47 = ( X = Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Rep_inject
% 0.26/0.47 thf(fact_193_type__definition_ORep__induct,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A,P: A > $o] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( member @ A @ Y3 @ A2 )
% 0.26/0.47 => ( ! [X3: B] : ( P @ ( Rep @ X3 ) )
% 0.26/0.47 => ( P @ Y3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Rep_induct
% 0.26/0.47 thf(fact_194_type__definition_OAbs__inject,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: A,Y3: A] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( member @ A @ X @ A2 )
% 0.26/0.47 => ( ( member @ A @ Y3 @ A2 )
% 0.26/0.47 => ( ( ( Abs @ X )
% 0.26/0.47 = ( Abs @ Y3 ) )
% 0.26/0.47 = ( X = Y3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Abs_inject
% 0.26/0.47 thf(fact_195_type__definition_OAbs__induct,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,P: B > $o,X: B] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ! [Y5: A] :
% 0.26/0.47 ( ( member @ A @ Y5 @ A2 )
% 0.26/0.47 => ( P @ ( Abs @ Y5 ) ) )
% 0.26/0.47 => ( P @ X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Abs_induct
% 0.26/0.47 thf(fact_196_type__definition_ORep__cases,axiom,
% 0.26/0.47 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ( ( member @ A @ Y3 @ A2 )
% 0.26/0.47 => ~ ! [X3: B] :
% 0.26/0.47 ( Y3
% 0.26/0.47 != ( Rep @ X3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Rep_cases
% 0.26/0.47 thf(fact_197_type__definition_OAbs__cases,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.26/0.47 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.26/0.47 => ~ ! [Y5: A] :
% 0.26/0.47 ( ( X
% 0.26/0.47 = ( Abs @ Y5 ) )
% 0.26/0.47 => ~ ( member @ A @ Y5 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.Abs_cases
% 0.26/0.47 thf(fact_198_type__definition_Ointro,axiom,
% 0.26/0.47 ! [B: $tType,A: $tType,Rep: B > A,A2: set @ A,Abs: A > B] :
% 0.26/0.47 ( ! [X3: B] : ( member @ A @ ( Rep @ X3 ) @ A2 )
% 0.26/0.47 => ( ! [X3: B] :
% 0.26/0.47 ( ( Abs @ ( Rep @ X3 ) )
% 0.26/0.47 = X3 )
% 0.26/0.47 => ( ! [Y5: A] :
% 0.26/0.47 ( ( member @ A @ Y5 @ A2 )
% 0.26/0.47 => ( ( Rep @ ( Abs @ Y5 ) )
% 0.26/0.47 = Y5 ) )
% 0.26/0.47 => ( type_definition @ B @ A @ Rep @ Abs @ A2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % type_definition.intro
% 0.26/0.47 thf(fact_199_top__empty__eq,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( top_top @ ( A > $o ) )
% 0.26/0.47 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % top_empty_eq
% 0.26/0.47 thf(fact_200_hcard__0,axiom,
% 0.26/0.47 ( ( hF_Mirabelle_hcard @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.26/0.47 = ( zero_zero @ nat ) ) ).
% 0.26/0.47
% 0.26/0.47 % hcard_0
% 0.26/0.47 thf(fact_201_hinsert__def,axiom,
% 0.26/0.47 ( hF_Mirabelle_hinsert
% 0.26/0.47 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hfset @ B4 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hinsert_def
% 0.26/0.47 thf(fact_202_insert__absorb2,axiom,
% 0.26/0.47 ! [A: $tType,X: A,A2: set @ A] :
% 0.26/0.47 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 0.26/0.47 = ( insert @ A @ X @ A2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb2
% 0.26/0.47 thf(fact_203_insert__iff,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.26/0.47 = ( ( A3 = B2 )
% 0.26/0.47 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_iff
% 0.26/0.47 thf(fact_204_insertCI,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B5: set @ A,B2: A] :
% 0.26/0.47 ( ( ~ ( member @ A @ A3 @ B5 )
% 0.26/0.47 => ( A3 = B2 ) )
% 0.26/0.47 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertCI
% 0.26/0.47 thf(fact_205_IntI,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ A2 )
% 0.26/0.47 => ( ( member @ A @ C @ B5 )
% 0.26/0.47 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % IntI
% 0.26/0.47 thf(fact_206_Int__iff,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 = ( ( member @ A @ C @ A2 )
% 0.26/0.47 & ( member @ A @ C @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_iff
% 0.26/0.47 thf(fact_207_Int__insert__left__if0,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.26/0.47 ( ~ ( member @ A @ A3 @ C3 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_left_if0
% 0.26/0.47 thf(fact_208_Int__insert__left__if1,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ C3 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.26/0.47 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_left_if1
% 0.26/0.47 thf(fact_209_insert__inter__insert,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_inter_insert
% 0.26/0.47 thf(fact_210_Int__insert__right__if0,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ~ ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_right_if0
% 0.26/0.47 thf(fact_211_Int__insert__right__if1,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_right_if1
% 0.26/0.47 thf(fact_212_Un__insert__left,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.26/0.47 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_insert_left
% 0.26/0.47 thf(fact_213_Un__insert__right,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,A3: A,B5: set @ A] :
% 0.26/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Un_insert_right
% 0.26/0.47 thf(fact_214_insert__UNIV,axiom,
% 0.26/0.47 ! [A: $tType,X: A] :
% 0.26/0.47 ( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
% 0.26/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_UNIV
% 0.26/0.47 thf(fact_215_IntE,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 => ~ ( ( member @ A @ C @ A2 )
% 0.26/0.47 => ~ ( member @ A @ C @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % IntE
% 0.26/0.47 thf(fact_216_IntD1,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 => ( member @ A @ C @ A2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % IntD1
% 0.26/0.47 thf(fact_217_IntD2,axiom,
% 0.26/0.47 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 => ( member @ A @ C @ B5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % IntD2
% 0.26/0.47 thf(fact_218_Int__assoc,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ C3 )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_assoc
% 0.26/0.47 thf(fact_219_Int__absorb,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
% 0.26/0.47 = A2 ) ).
% 0.26/0.47
% 0.26/0.47 % Int_absorb
% 0.26/0.47 thf(fact_220_Int__commute,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) )
% 0.26/0.47 = ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_commute
% 0.26/0.47 thf(fact_221_Int__insert__left,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( ( member @ A @ A3 @ C3 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.26/0.47 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) )
% 0.26/0.47 & ( ~ ( member @ A @ A3 @ C3 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_left
% 0.26/0.47 thf(fact_222_Int__left__absorb,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_left_absorb
% 0.26/0.47 thf(fact_223_Int__insert__right,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ( ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) )
% 0.26/0.47 & ( ~ ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_insert_right
% 0.26/0.47 thf(fact_224_Int__left__commute,axiom,
% 0.26/0.47 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.26/0.47 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) )
% 0.26/0.47 = ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Int_left_commute
% 0.26/0.47 thf(fact_225_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ? [B8: set @ A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( insert @ A @ A3 @ B8 ) )
% 0.26/0.47 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mk_disjoint_insert
% 0.26/0.47 thf(fact_226_insert__commute,axiom,
% 0.26/0.47 ! [A: $tType,X: A,Y3: A,A2: set @ A] :
% 0.26/0.47 ( ( insert @ A @ X @ ( insert @ A @ Y3 @ A2 ) )
% 0.26/0.47 = ( insert @ A @ Y3 @ ( insert @ A @ X @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_commute
% 0.26/0.47 thf(fact_227_insert__eq__iff,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A,B2: A,B5: set @ A] :
% 0.26/0.47 ( ~ ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ~ ( member @ A @ B2 @ B5 )
% 0.26/0.47 => ( ( ( insert @ A @ A3 @ A2 )
% 0.26/0.47 = ( insert @ A @ B2 @ B5 ) )
% 0.26/0.47 = ( ( ( A3 = B2 )
% 0.26/0.47 => ( A2 = B5 ) )
% 0.26/0.47 & ( ( A3 != B2 )
% 0.26/0.47 => ? [C4: set @ A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( insert @ A @ B2 @ C4 ) )
% 0.26/0.47 & ~ ( member @ A @ B2 @ C4 )
% 0.26/0.47 & ( B5
% 0.26/0.47 = ( insert @ A @ A3 @ C4 ) )
% 0.26/0.47 & ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_eq_iff
% 0.26/0.47 thf(fact_228_insert__absorb,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ A2 )
% 0.26/0.47 => ( ( insert @ A @ A3 @ A2 )
% 0.26/0.47 = A2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb
% 0.26/0.47 thf(fact_229_insert__ident,axiom,
% 0.26/0.47 ! [A: $tType,X: A,A2: set @ A,B5: set @ A] :
% 0.26/0.47 ( ~ ( member @ A @ X @ A2 )
% 0.26/0.47 => ( ~ ( member @ A @ X @ B5 )
% 0.26/0.47 => ( ( ( insert @ A @ X @ A2 )
% 0.26/0.47 = ( insert @ A @ X @ B5 ) )
% 0.26/0.47 = ( A2 = B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_ident
% 0.26/0.47 thf(fact_230_Set_Oset__insert,axiom,
% 0.26/0.47 ! [A: $tType,X: A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ X @ A2 )
% 0.26/0.47 => ~ ! [B8: set @ A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( insert @ A @ X @ B8 ) )
% 0.26/0.47 => ( member @ A @ X @ B8 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Set.set_insert
% 0.26/0.47 thf(fact_231_insertI2,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B5: set @ A,B2: A] :
% 0.26/0.47 ( ( member @ A @ A3 @ B5 )
% 0.26/0.47 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertI2
% 0.26/0.47 thf(fact_232_insertI1,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertI1
% 0.26/0.47 thf(fact_233_insertE,axiom,
% 0.26/0.47 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.26/0.47 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.26/0.47 => ( ( A3 != B2 )
% 0.26/0.47 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertE
% 0.26/0.47 thf(fact_234_hfset__hinsert,axiom,
% 0.26/0.47 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ A3 @ B2 ) )
% 0.26/0.47 = ( insert @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hfset_hinsert
% 0.26/0.47 thf(fact_235_hcard__union__inter,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( plus_plus @ nat @ ( hF_Mirabelle_hcard @ ( sup_sup @ hF_Mirabelle_hf @ X @ Y3 ) ) @ ( hF_Mirabelle_hcard @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y3 ) ) )
% 0.26/0.47 = ( plus_plus @ nat @ ( hF_Mirabelle_hcard @ X ) @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hcard_union_inter
% 0.26/0.47 thf(fact_236_hcard__hinsert__if,axiom,
% 0.26/0.47 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.26/0.47 ( ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.26/0.47 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.26/0.47 = ( hF_Mirabelle_hcard @ Y3 ) ) )
% 0.26/0.47 & ( ~ ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.26/0.47 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.26/0.47 = ( suc @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % hcard_hinsert_if
% 0.26/0.47 thf(fact_237_add__left__cancel,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel_semigroup_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.26/0.47 = ( plus_plus @ A @ A3 @ C ) )
% 0.26/0.47 = ( B2 = C ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_left_cancel
% 0.26/0.47 thf(fact_238_add__right__cancel,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel_semigroup_add @ A )
% 0.26/0.47 => ! [B2: A,A3: A,C: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.26/0.47 = ( plus_plus @ A @ C @ A3 ) )
% 0.26/0.47 = ( B2 = C ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_right_cancel
% 0.26/0.47 thf(fact_239_zero__eq__add__iff__both__eq__0,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( canoni770627133id_add @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( ( zero_zero @ A )
% 0.26/0.47 = ( plus_plus @ A @ X @ Y3 ) )
% 0.26/0.47 = ( ( X
% 0.26/0.47 = ( zero_zero @ A ) )
% 0.26/0.47 & ( Y3
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_eq_add_iff_both_eq_0
% 0.26/0.47 thf(fact_240_add__eq__0__iff__both__eq__0,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( canoni770627133id_add @ A )
% 0.26/0.47 => ! [X: A,Y3: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ X @ Y3 )
% 0.26/0.47 = ( zero_zero @ A ) )
% 0.26/0.47 = ( ( X
% 0.26/0.47 = ( zero_zero @ A ) )
% 0.26/0.47 & ( Y3
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_eq_0_iff_both_eq_0
% 0.26/0.47 thf(fact_241_add__cancel__right__right,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel1352612707id_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( plus_plus @ A @ A3 @ B2 ) )
% 0.26/0.47 = ( B2
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_cancel_right_right
% 0.26/0.47 thf(fact_242_add__cancel__right__left,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel1352612707id_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.26/0.47 = ( B2
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_cancel_right_left
% 0.26/0.47 thf(fact_243_add__cancel__left__right,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel1352612707id_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.26/0.47 = A3 )
% 0.26/0.47 = ( B2
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_cancel_left_right
% 0.26/0.47 thf(fact_244_add__cancel__left__left,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( cancel1352612707id_add @ A )
% 0.26/0.47 => ! [B2: A,A3: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.26/0.47 = A3 )
% 0.26/0.47 = ( B2
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_cancel_left_left
% 0.26/0.47 thf(fact_245_double__zero__sym,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linord219039673up_add @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( ( zero_zero @ A )
% 0.26/0.47 = ( plus_plus @ A @ A3 @ A3 ) )
% 0.26/0.47 = ( A3
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % double_zero_sym
% 0.26/0.47 thf(fact_246_double__zero,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( linord219039673up_add @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.26/0.47 = ( zero_zero @ A ) )
% 0.26/0.47 = ( A3
% 0.26/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % double_zero
% 0.26/0.47 thf(fact_247_add_Oright__neutral,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( monoid_add @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.right_neutral
% 0.26/0.47 thf(fact_248_add_Oleft__neutral,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( monoid_add @ A )
% 0.26/0.47 => ! [A3: A] :
% 0.26/0.47 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.left_neutral
% 0.26/0.47 thf(fact_249_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ab_semigroup_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C )
% 0.26/0.47 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ab_semigroup_add_class.add_ac(1)
% 0.26/0.47 thf(fact_250_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( ordere779506340up_add @ A )
% 0.26/0.47 => ! [I: A,J: A,K: A,L: A] :
% 0.26/0.47 ( ( ( I = J )
% 0.26/0.47 & ( K = L ) )
% 0.26/0.47 => ( ( plus_plus @ A @ I @ K )
% 0.26/0.47 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_mono_thms_linordered_semiring(4)
% 0.26/0.47 thf(fact_251_group__cancel_Oadd1,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( comm_monoid_add @ A )
% 0.26/0.47 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.26/0.47 ( ( A2
% 0.26/0.47 = ( plus_plus @ A @ K @ A3 ) )
% 0.26/0.47 => ( ( plus_plus @ A @ A2 @ B2 )
% 0.26/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group_cancel.add1
% 0.26/0.47 thf(fact_252_group__cancel_Oadd2,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( comm_monoid_add @ A )
% 0.26/0.47 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.26/0.47 ( ( B5
% 0.26/0.47 = ( plus_plus @ A @ K @ B2 ) )
% 0.26/0.47 => ( ( plus_plus @ A @ A3 @ B5 )
% 0.26/0.47 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group_cancel.add2
% 0.26/0.47 thf(fact_253_add_Oassoc,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( semigroup_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C )
% 0.26/0.47 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.assoc
% 0.26/0.47 thf(fact_254_add_Oleft__cancel,axiom,
% 0.26/0.47 ! [A: $tType] :
% 0.26/0.47 ( ( group_add @ A )
% 0.26/0.47 => ! [A3: A,B2: A,C: A] :
% 0.26/0.47 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.26/0.47 = ( plus_plus @ A @ A3 @ C ) )
% 0.26/0.47 = ( B2 = C ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.left_cancel
% 0.26/0.47
% 0.26/0.47 % Type constructors (42)
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.26/0.47 bounded_lattice @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.26/0.47 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( bounded_lattice @ A9 )
% 0.26/0.47 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( bounded_lattice @ A9 )
% 0.26/0.47 => ( bounde1561333602nf_top @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( bounded_lattice @ A9 )
% 0.26/0.47 => ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( semilattice_sup @ A9 )
% 0.26/0.47 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( semilattice_inf @ A9 )
% 0.26/0.47 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( distrib_lattice @ A9 )
% 0.26/0.47 => ( distrib_lattice @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( lattice @ A9 )
% 0.26/0.47 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.26/0.47 ! [A8: $tType,A9: $tType] :
% 0.26/0.47 ( ( top @ A9 )
% 0.26/0.47 => ( top @ ( A8 > A9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.26/0.47 canoni770627133id_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.26/0.47 ordere779506340up_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.26/0.47 cancel1352612707id_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.26/0.47 cancel_semigroup_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
% 0.26/0.47 semilattice_sup @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_4,axiom,
% 0.26/0.47 semilattice_inf @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_5,axiom,
% 0.26/0.47 distrib_lattice @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.26/0.47 ab_semigroup_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.26/0.47 comm_monoid_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.26/0.47 semigroup_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.26/0.47 monoid_add @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Lattices_Olattice_6,axiom,
% 0.26/0.47 lattice @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.26/0.47 zero @ nat ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_7,axiom,
% 0.26/0.47 ! [A8: $tType] : ( bounde1561333602nf_top @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_8,axiom,
% 0.26/0.47 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
% 0.26/0.47 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_10,axiom,
% 0.26/0.47 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_11,axiom,
% 0.26/0.47 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Lattices_Olattice_12,axiom,
% 0.26/0.47 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_Set_Oset___Orderings_Otop_13,axiom,
% 0.26/0.47 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_14,axiom,
% 0.26/0.47 bounde1561333602nf_top @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_15,axiom,
% 0.26/0.47 bounded_lattice_top @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16,axiom,
% 0.26/0.47 semilattice_sup @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_17,axiom,
% 0.26/0.47 semilattice_inf @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_18,axiom,
% 0.26/0.47 distrib_lattice @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 0.26/0.47 lattice @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HOL_Obool___Orderings_Otop_20,axiom,
% 0.26/0.47 top @ $o ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_21,axiom,
% 0.26/0.47 semilattice_sup @ hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_22,axiom,
% 0.26/0.47 semilattice_inf @ hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_23,axiom,
% 0.26/0.47 distrib_lattice @ hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_24,axiom,
% 0.26/0.47 lattice @ hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_25,axiom,
% 0.26/0.47 zero @ hF_Mirabelle_hf ).
% 0.26/0.47
% 0.26/0.47 % Conjectures (2)
% 0.26/0.47 thf(conj_0,hypothesis,
% 0.26/0.47 hF_Mir859708136nction @ r ).
% 0.26/0.47
% 0.26/0.47 thf(conj_1,conjecture,
% 0.26/0.47 hF_Mir859708136nction @ ( hF_Mir165288108strict @ r @ x ) ).
% 0.26/0.47
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 ------------------------
% 0.31/1.50 ---- Embedded in THF ---
% 0.31/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.50 %%% Generated on Tue Feb 24 08:25:13 EST 2026
% 0.31/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.50 %%% Logic specification used:
% 0.31/1.50 %%% thf(spec, logic, $$dhol).
% 0.31/1.50
% 0.31/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2DTF504.p
% 0.31/1.50 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf, type, hF_Mirabelle_hf: $tType).
% 0.31/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.31/1.50 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.31/1.50 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Lattices_Obounded__semilattice__inf__top, type, bounde1561333602nf_top: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 0.31/1.50 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 0.31/1.50 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect, type, hF_Mir1687042746ollect: ((hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF, type, hF_Mirabelle_HF: ((set @ hF_Mirabelle_hf) > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter, type, hF_Mirabelle_HInter: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion, type, hF_Mirabelle_HUnion: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace, type, hF_Mir569462966eplace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun, type, hF_Mirabelle_RepFun: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace, type, hF_Mirabelle_Replace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard, type, hF_Mirabelle_hcard: (hF_Mirabelle_hf > nat)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf, type, hF_Mirabelle_Abs_hf: (nat > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_ORep__hf, type, hF_Mirabelle_Rep_hf: (hF_Mirabelle_hf > nat)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset, type, hF_Mirabelle_hfset: (hF_Mirabelle_hf > (set @ hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst, type, hF_Mirabelle_hfst: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfunction, type, hF_Mir859708136nction: (hF_Mirabelle_hf > $o)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert, type, hF_Mirabelle_hinsert: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem, type, hF_Mirabelle_hmem: (hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair, type, hF_Mirabelle_hpair: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrelation, type, hF_Mir1093797708lation: (hF_Mirabelle_hf > $o)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrestrict, type, hF_Mir165288108strict: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd, type, hF_Mirabelle_hsnd: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit, type, hF_Mirabelle_hsplit: (!> [A:$tType]: (((hF_Mirabelle_hf > (hF_Mirabelle_hf > A)) > (hF_Mirabelle_hf > A))))).
% 0.31/1.50 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ois__hpair, type, hF_Mir796903520_hpair: (hF_Mirabelle_hf > $o)).
% 0.31/1.50 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 0.31/1.50 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 0.31/1.50 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 0.31/1.50 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 0.31/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.31/1.50 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 0.31/1.50 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 0.31/1.50 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.31/1.50 thf(sy_v_r, type, r: hF_Mirabelle_hf).
% 0.31/1.50 thf(sy_v_x, type, x: hF_Mirabelle_hf).
% 0.31/1.50 thf(fact_0_hrelation__restr, axiom, (! [R:hF_Mirabelle_hf,X:hF_Mirabelle_hf]: ((hF_Mir1093797708lation @ ((hF_Mir165288108strict @ R) @ X))))).
% 0.31/1.50 thf(fact_1_hfunction__def, axiom, (hF_Mir859708136nction = (^ [R2:hF_Mirabelle_hf]: ((! [X2:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ ((hF_Mirabelle_hpair @ X2) @ Y)) @ R2) => (! [Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ ((hF_Mirabelle_hpair @ X2) @ Y2)) @ R2) => (Y = Y2))))))))))).
% 0.31/1.50 thf(fact_2_hrestrict__iff, axiom, (! [Z:hF_Mirabelle_hf,R:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Z) @ ((hF_Mir165288108strict @ R) @ A2)) = (((hF_Mirabelle_hmem @ Z) @ R) & (? [X2:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: (((Z = ((hF_Mirabelle_hpair @ X2) @ Y)) & ((hF_Mirabelle_hmem @ X2) @ A2))))))))).
% 0.31/1.50 thf(fact_3_hpair__iff, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf,A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A3) @ B2) = ((hF_Mirabelle_hpair @ A4) @ B3)) = ((A3 = A4) & (B2 = B3)))))).
% 0.31/1.50 thf(fact_4_Rep__hf__inject, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_Rep_hf @ X) = (hF_Mirabelle_Rep_hf @ Y3)) = (X = Y3))))).
% 0.31/1.50 thf(fact_5_hf__equalityI, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A3) = ((hF_Mirabelle_hmem @ X3) @ B2)))) => (A3 = B2))))).
% 0.31/1.50 thf(fact_6_hf__ext, axiom, ((^ [Y4:hF_Mirabelle_hf,Z2:hF_Mirabelle_hf]: ((Y4 = Z2))) = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: ((! [X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ A5) = ((hF_Mirabelle_hmem @ X2) @ B4)))))))).
% 0.31/1.50 thf(fact_7_hmem__ne, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ Y3) => (X != Y3))))).
% 0.31/1.50 thf(fact_8_replacement, axiom, (! [X:hF_Mirabelle_hf,R3:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((! [U:hF_Mirabelle_hf,V:hF_Mirabelle_hf,V2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U) @ X) => (((R3 @ U) @ V) => (((R3 @ U) @ V2) => (V2 = V)))))) => (? [Z3:hF_Mirabelle_hf]: ((! [V3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V3) @ Z3) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X) & ((R3 @ U2) @ V3))))))))))))).
% 0.31/1.50 thf(fact_9_binary__union, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((? [Z3:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z3) = (((hF_Mirabelle_hmem @ U3) @ X) | ((hF_Mirabelle_hmem @ U3) @ Y3)))))))))).
% 0.31/1.50 thf(fact_10_hmem__not__sym, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((~ ((((hF_Mirabelle_hmem @ X) @ Y3) & ((hF_Mirabelle_hmem @ Y3) @ X))))))).
% 0.31/1.50 thf(fact_11_hpair__inject, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf,A4:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A3) @ B2) = ((hF_Mirabelle_hpair @ A4) @ B3)) => (~ (((A3 = A4) => (B2 != B3)))))))).
% 0.31/1.50 thf(fact_12_union__of__set, axiom, (! [X:hF_Mirabelle_hf]: ((? [Z3:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z3) = (? [Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y) @ X) & ((hF_Mirabelle_hmem @ U3) @ Y)))))))))))).
% 0.31/1.50 thf(fact_13_comprehension, axiom, (! [X:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((? [Z3:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z3) = (((hF_Mirabelle_hmem @ U3) @ X) & (P @ U3)))))))))).
% 0.31/1.50 thf(fact_14_hmem__not__refl, axiom, (! [X:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X) @ X)))))).
% 0.31/1.50 thf(fact_15_hpair__neq__fst, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A3) @ B2) != A3)))).
% 0.31/1.50 thf(fact_16_hpair__neq__snd, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A3) @ B2) != B2)))).
% 0.31/1.50 thf(fact_17_replacement__fun, axiom, (! [X:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((? [Z3:hF_Mirabelle_hf]: ((! [V3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V3) @ Z3) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X) & (V3 = (F @ U2))))))))))))).
% 0.31/1.50 thf(fact_18_hsplit, axiom, (! [A:$tType,C:(hF_Mirabelle_hf > (hF_Mirabelle_hf > A)),A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hsplit @ A) @ C) @ ((hF_Mirabelle_hpair @ A3) @ B2)) = ((C @ A3) @ B2))))).
% 0.31/1.50 thf(fact_19_hfst__conv, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mirabelle_hfst @ ((hF_Mirabelle_hpair @ A3) @ B2)) = A3)))).
% 0.31/1.50 thf(fact_20_hsnd__conv, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mirabelle_hsnd @ ((hF_Mirabelle_hpair @ A3) @ B2)) = B2)))).
% 0.31/1.50 thf(fact_21_hrelation__def, axiom, (hF_Mir1093797708lation = (^ [R2:hF_Mirabelle_hf]: ((! [Z4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Z4) @ R2) => (hF_Mir796903520_hpair @ Z4)))))))).
% 0.31/1.50 thf(fact_22_is__hpair__def, axiom, (hF_Mir796903520_hpair = (^ [Z4:hF_Mirabelle_hf]: ((? [X2:hF_Mirabelle_hf,Y:hF_Mirabelle_hf]: ((Z4 = ((hF_Mirabelle_hpair @ X2) @ Y)))))))).
% 0.31/1.50 thf(fact_23_PrimReplace__iff, axiom, (! [A2:hF_Mirabelle_hf,R3:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)),V4:hF_Mirabelle_hf]: (((! [U:hF_Mirabelle_hf,V:hF_Mirabelle_hf,V2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U) @ A2) => (((R3 @ U) @ V) => (((R3 @ U) @ V2) => (V2 = V)))))) => (((hF_Mirabelle_hmem @ V4) @ ((hF_Mir569462966eplace @ A2) @ R3)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A2) & ((R3 @ U2) @ V4))))))))).
% 0.31/1.50 thf(fact_24_HUnion__iff, axiom, (! [X:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ (hF_Mirabelle_HUnion @ A2)) = (? [Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y) @ A2) & ((hF_Mirabelle_hmem @ X) @ Y)))))))).
% 0.31/1.50 thf(fact_25_Replace__iff, axiom, (! [V4:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,R3:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_hmem @ V4) @ ((hF_Mirabelle_Replace @ A2) @ R3)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A2) & (((R3 @ U2) @ V4) & (! [Y:hF_Mirabelle_hf]: ((((R3 @ U2) @ Y) => (Y = V4)))))))))))).
% 0.31/1.50 thf(fact_26_HCollect__iff, axiom, (! [X:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),A2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ ((hF_Mir1687042746ollect @ P) @ A2)) = ((P @ X) & ((hF_Mirabelle_hmem @ X) @ A2)))))).
% 0.31/1.50 thf(fact_27_RepFun__iff, axiom, (! [V4:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_hmem @ V4) @ ((hF_Mirabelle_RepFun @ A2) @ F)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A2) & (V4 = (F @ U2))))))))).
% 0.31/1.50 thf(fact_28_Rep__hf__inverse, axiom, (! [X:hF_Mirabelle_hf]: (((hF_Mirabelle_Abs_hf @ (hF_Mirabelle_Rep_hf @ X)) = X)))).
% 0.31/1.50 thf(fact_29_RepFun__cong, axiom, (! [A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf),G:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: (((A2 = B5) => ((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ B5) => ((F @ X3) = (G @ X3))))) => (((hF_Mirabelle_RepFun @ A2) @ F) = ((hF_Mirabelle_RepFun @ B5) @ G))))))).
% 0.31/1.50 thf(fact_30_Replace__cong, axiom, (! [A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)),Q:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((A2 = B5) => ((! [X3:hF_Mirabelle_hf,Y5:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ B5) => (((P @ X3) @ Y5) = ((Q @ X3) @ Y5))))) => (((hF_Mirabelle_Replace @ A2) @ P) = ((hF_Mirabelle_Replace @ B5) @ Q))))))).
% 0.31/1.50 thf(fact_31_hmem__Sup__ne, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ Y3) => ((hF_Mirabelle_HUnion @ X) != Y3))))).
% 0.31/1.50 thf(fact_32_Abs__hf__inverse, axiom, (! [Y3:nat]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => ((hF_Mirabelle_Rep_hf @ (hF_Mirabelle_Abs_hf @ Y3)) = Y3))))).
% 0.31/1.50 thf(fact_33_zero__notin__hpair, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ (zero_zero @ hF_Mirabelle_hf)) @ ((hF_Mirabelle_hpair @ X) @ Y3))))))).
% 0.31/1.50 thf(fact_34_hmem__def, axiom, (hF_Mirabelle_hmem = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: ((((member @ hF_Mirabelle_hf) @ A5) @ (hF_Mirabelle_hfset @ B4)))))).
% 0.31/1.50 thf(fact_35_HCollect__hempty, axiom, (! [P:(hF_Mirabelle_hf > $o)]: ((((hF_Mir1687042746ollect @ P) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_36_HUnion__hempty, axiom, ((hF_Mirabelle_HUnion @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 0.31/1.50 thf(fact_37_Replace__0, axiom, (! [R3:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_Replace @ (zero_zero @ hF_Mirabelle_hf)) @ R3) = (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_38_RepFun__0, axiom, (! [F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ (zero_zero @ hF_Mirabelle_hf)) @ F) = (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_39_RepFun__hinsert, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ ((hF_Mirabelle_hinsert @ A3) @ B2)) @ F) = ((hF_Mirabelle_hinsert @ (F @ A3)) @ ((hF_Mirabelle_RepFun @ B2) @ F)))))).
% 0.31/1.50 thf(fact_40_hrelation__hunion, axiom, (! [F:hF_Mirabelle_hf,G:hF_Mirabelle_hf]: (((hF_Mir1093797708lation @ (((sup_sup @ hF_Mirabelle_hf) @ F) @ G)) = ((hF_Mir1093797708lation @ F) & (hF_Mir1093797708lation @ G)))))).
% 0.31/1.50 thf(fact_41_hmem__hinsert, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ A3) @ ((hF_Mirabelle_hinsert @ B2) @ C)) = ((A3 = B2) | ((hF_Mirabelle_hmem @ A3) @ C)))))).
% 0.31/1.50 thf(fact_42_singleton__eq__iff, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A3) @ (zero_zero @ hF_Mirabelle_hf)) = ((hF_Mirabelle_hinsert @ B2) @ (zero_zero @ hF_Mirabelle_hf))) = (A3 = B2))))).
% 0.31/1.50 thf(fact_43_hunion__iff, axiom, (! [X:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ A3) @ B2)) = (((hF_Mirabelle_hmem @ X) @ A3) | ((hF_Mirabelle_hmem @ X) @ B2)))))).
% 0.31/1.50 thf(fact_44_hunion__hempty__left, axiom, (! [A2:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ A2) = A2)))).
% 0.31/1.50 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 0.31/1.50 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 0.31/1.50 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.31/1.50 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 0.31/1.50 thf(fact_49_hunion__hempty__right, axiom, (! [A2:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ A2) @ (zero_zero @ hF_Mirabelle_hf)) = A2)))).
% 0.31/1.50 thf(fact_50_RepFun__hunion, axiom, (! [A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ (((sup_sup @ hF_Mirabelle_hf) @ A2) @ B5)) @ F) = (((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_RepFun @ A2) @ F)) @ ((hF_Mirabelle_RepFun @ B5) @ F)))))).
% 0.31/1.50 thf(fact_51_HUnion__hunion, axiom, (! [A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf]: (((hF_Mirabelle_HUnion @ (((sup_sup @ hF_Mirabelle_hf) @ A2) @ B5)) = (((sup_sup @ hF_Mirabelle_hf) @ (hF_Mirabelle_HUnion @ A2)) @ (hF_Mirabelle_HUnion @ B5)))))).
% 0.31/1.50 thf(fact_52_Replace__hunion, axiom, (! [A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,R3:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_Replace @ (((sup_sup @ hF_Mirabelle_hf) @ A2) @ B5)) @ R3) = (((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_Replace @ A2) @ R3)) @ ((hF_Mirabelle_Replace @ B5) @ R3)))))).
% 0.31/1.50 thf(fact_53_HUnion__hinsert, axiom, (! [A3:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((hF_Mirabelle_HUnion @ ((hF_Mirabelle_hinsert @ A3) @ A2)) = (((sup_sup @ hF_Mirabelle_hf) @ A3) @ (hF_Mirabelle_HUnion @ A2)))))).
% 0.31/1.50 thf(fact_54_hf__induct__ax, axiom, (! [P:(hF_Mirabelle_hf > $o),X:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X3:hF_Mirabelle_hf]: (((P @ X3) => (! [Y5:hF_Mirabelle_hf]: (((P @ Y5) => (P @ ((hF_Mirabelle_hinsert @ Y5) @ X3)))))))) => (P @ X)))))).
% 0.31/1.50 thf(fact_55_hinsert__eq__sup, axiom, (hF_Mirabelle_hinsert = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: ((((sup_sup @ hF_Mirabelle_hf) @ B4) @ ((hF_Mirabelle_hinsert @ A5) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 0.31/1.50 thf(fact_56_hinsert__commute, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf,Z:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ X) @ ((hF_Mirabelle_hinsert @ Y3) @ Z)) = ((hF_Mirabelle_hinsert @ Y3) @ ((hF_Mirabelle_hinsert @ X) @ Z)))))).
% 0.31/1.50 thf(fact_57_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf,C:hF_Mirabelle_hf,D:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A3) @ ((hF_Mirabelle_hinsert @ B2) @ (zero_zero @ hF_Mirabelle_hf))) = ((hF_Mirabelle_hinsert @ C) @ ((hF_Mirabelle_hinsert @ D) @ (zero_zero @ hF_Mirabelle_hf)))) = (((A3 = C) & (B2 = D)) | ((A3 = D) & (B2 = C))))))).
% 0.31/1.50 thf(fact_58_hinsert__nonempty, axiom, (! [A3:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ A3) @ A2) != (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_59_hunion__hinsert__left, axiom, (! [X:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,B5:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A2)) @ B5) = ((hF_Mirabelle_hinsert @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_60_hunion__hinsert__right, axiom, (! [B5:hF_Mirabelle_hf,X:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((((sup_sup @ hF_Mirabelle_hf) @ B5) @ ((hF_Mirabelle_hinsert @ X) @ A2)) = ((hF_Mirabelle_hinsert @ X) @ (((sup_sup @ hF_Mirabelle_hf) @ B5) @ A2)))))).
% 0.31/1.50 thf(fact_61_hf__cases, axiom, (! [Y3:hF_Mirabelle_hf]: (((Y3 != (zero_zero @ hF_Mirabelle_hf)) => (~ ((! [A6:hF_Mirabelle_hf,B6:hF_Mirabelle_hf]: (((Y3 = ((hF_Mirabelle_hinsert @ A6) @ B6)) => ((hF_Mirabelle_hmem @ A6) @ B6)))))))))).
% 0.31/1.50 thf(fact_62_hf__induct, axiom, (! [P:(hF_Mirabelle_hf > $o),Z:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X3:hF_Mirabelle_hf,Y5:hF_Mirabelle_hf]: (((P @ X3) => ((P @ Y5) => ((~ (((hF_Mirabelle_hmem @ X3) @ Y5))) => (P @ ((hF_Mirabelle_hinsert @ X3) @ Y5))))))) => (P @ Z)))))).
% 0.31/1.50 thf(fact_63_hpair__def, axiom, (hF_Mirabelle_hpair = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A5) @ (zero_zero @ hF_Mirabelle_hf))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A5) @ ((hF_Mirabelle_hinsert @ B4) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 0.31/1.50 thf(fact_64_hpair__def_H, axiom, (hF_Mirabelle_hpair = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A5) @ ((hF_Mirabelle_hinsert @ A5) @ (zero_zero @ hF_Mirabelle_hf)))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A5) @ ((hF_Mirabelle_hinsert @ B4) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 0.31/1.50 thf(fact_65_hinsert__iff, axiom, (! [Z:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf,X:hF_Mirabelle_hf]: (((Z = ((hF_Mirabelle_hinsert @ Y3) @ X)) = (! [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ Z) = (((hF_Mirabelle_hmem @ U2) @ X) | (U2 = Y3))))))))).
% 0.31/1.50 thf(fact_66_hemptyE, axiom, (! [A3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A3) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 0.31/1.50 thf(fact_67_hempty__iff, axiom, (! [Z:hF_Mirabelle_hf]: (((Z = (zero_zero @ hF_Mirabelle_hf)) = (! [X2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X2) @ Z))))))))).
% 0.31/1.50 thf(fact_68_hmem__hempty, axiom, (! [A3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A3) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 0.31/1.50 thf(fact_69_hpair__nonzero, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ X) @ Y3) != (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_70_Rep__hf, axiom, (! [X:hF_Mirabelle_hf]: ((((member @ nat) @ (hF_Mirabelle_Rep_hf @ X)) @ (top_top @ (set @ nat)))))).
% 0.31/1.50 thf(fact_71_Rep__hf__cases, axiom, (! [Y3:nat]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => (~ ((! [X3:hF_Mirabelle_hf]: ((Y3 != (hF_Mirabelle_Rep_hf @ X3)))))))))).
% 0.31/1.50 thf(fact_72_Rep__hf__induct, axiom, (! [Y3:nat,P:(nat > $o)]: (((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => ((! [X3:hF_Mirabelle_hf]: ((P @ (hF_Mirabelle_Rep_hf @ X3)))) => (P @ Y3)))))).
% 0.31/1.50 thf(fact_73_Abs__hf__cases, axiom, (! [X:hF_Mirabelle_hf]: ((~ ((! [Y5:nat]: (((X = (hF_Mirabelle_Abs_hf @ Y5)) => (~ ((((member @ nat) @ Y5) @ (top_top @ (set @ nat))))))))))))).
% 0.31/1.50 thf(fact_74_Abs__hf__induct, axiom, (! [P:(hF_Mirabelle_hf > $o),X:hF_Mirabelle_hf]: (((! [Y5:nat]: (((((member @ nat) @ Y5) @ (top_top @ (set @ nat))) => (P @ (hF_Mirabelle_Abs_hf @ Y5))))) => (P @ X))))).
% 0.31/1.50 thf(fact_75_Abs__hf__inject, axiom, (! [X:nat,Y3:nat]: (((((member @ nat) @ X) @ (top_top @ (set @ nat))) => ((((member @ nat) @ Y3) @ (top_top @ (set @ nat))) => (((hF_Mirabelle_Abs_hf @ X) = (hF_Mirabelle_Abs_hf @ Y3)) = (X = Y3))))))).
% 0.31/1.50 thf(fact_76_hrelation__0, axiom, (hF_Mir1093797708lation @ (zero_zero @ hF_Mirabelle_hf))).
% 0.31/1.50 thf(fact_77_sup__top__left, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 0.31/1.50 thf(fact_78_sup__top__right, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 0.31/1.50 thf(fact_79_HInter__iff, axiom, (! [A2:hF_Mirabelle_hf,X:hF_Mirabelle_hf]: (((A2 != (zero_zero @ hF_Mirabelle_hf)) => (((hF_Mirabelle_hmem @ X) @ (hF_Mirabelle_HInter @ A2)) = (! [Y:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y) @ A2) => ((hF_Mirabelle_hmem @ X) @ Y))))))))).
% 0.31/1.50 thf(fact_80_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.31/1.50 thf(fact_81_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 0.31/1.50 thf(fact_82_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 0.31/1.50 thf(fact_83_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B2)) = (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.31/1.50 thf(fact_84_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.31/1.50 thf(fact_85_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ B2) = (((sup_sup @ A) @ A3) @ B2)))))))).
% 0.31/1.50 thf(fact_86_HInter__hempty, axiom, ((hF_Mirabelle_HInter @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 0.31/1.50 thf(fact_87_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z)) = (((sup_sup @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_88_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A3:A,C:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A3) @ C)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_89_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 0.31/1.50 thf(fact_90_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A5:A,B4:A]: ((((sup_sup @ A) @ B4) @ A5)))))))).
% 0.31/1.50 thf(fact_91_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X:A,Y3:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z))))))))).
% 0.31/1.50 thf(fact_92_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B2:A,C:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B2)) @ C) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_93_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B5:A,K:A,B2:A,A3:A]: (((B5 = (((sup_sup @ A) @ K) @ B2)) => ((((sup_sup @ A) @ A3) @ B5) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_94_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,K:A,A3:A,B2:A]: (((A2 = (((sup_sup @ A) @ K) @ A3)) => ((((sup_sup @ A) @ A2) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_95_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.31/1.50 thf(fact_96_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 0.31/1.50 thf(fact_97_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ Z) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z))))))))).
% 0.31/1.50 thf(fact_98_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z)) = (((sup_sup @ A) @ Y3) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_99_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = (((sup_sup @ A) @ X) @ Y3)))))))).
% 0.31/1.50 thf(fact_100_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 0.31/1.50 thf(fact_101_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 0.31/1.50 thf(fact_102_top__apply, axiom, (! [C2:$tType,D2:$tType]: (((top @ C2) => ((top_top @ (D2 > C2)) = (^ [X2:D2]: ((top_top @ C2)))))))).
% 0.31/1.50 thf(fact_103_HInter__hinsert, axiom, (! [A2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: (((A2 != (zero_zero @ hF_Mirabelle_hf)) => ((hF_Mirabelle_HInter @ ((hF_Mirabelle_hinsert @ A3) @ A2)) = (((inf_inf @ hF_Mirabelle_hf) @ A3) @ (hF_Mirabelle_HInter @ A2))))))).
% 0.31/1.50 thf(fact_104_type__definition__hf, axiom, (((((type_definition @ hF_Mirabelle_hf) @ nat) @ hF_Mirabelle_Rep_hf) @ hF_Mirabelle_Abs_hf) @ (top_top @ (set @ nat)))).
% 0.31/1.50 thf(fact_105_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Y3) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.31/1.50 thf(fact_106_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ B2) = (((inf_inf @ A) @ A3) @ B2)))))))).
% 0.31/1.50 thf(fact_107_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.31/1.50 thf(fact_108_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A]: (((((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ A3) @ B2)) = (((inf_inf @ A) @ A3) @ B2)))))))).
% 0.31/1.50 thf(fact_109_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ X) = X))))))).
% 0.31/1.50 thf(fact_110_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A]: (((((inf_inf @ A) @ A3) @ A3) = A3))))))).
% 0.31/1.50 thf(fact_111_inf__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.31/1.50 thf(fact_112_inf__top_Oright__neutral, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [A3:A]: (((((inf_inf @ A) @ A3) @ (top_top @ A)) = A3))))))).
% 0.31/1.50 thf(fact_113_inf__top_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [A3:A,B2:A]: ((((top_top @ A) = (((inf_inf @ A) @ A3) @ B2)) = ((A3 = (top_top @ A)) & (B2 = (top_top @ A)))))))))).
% 0.31/1.50 thf(fact_114_inf__top_Oleft__neutral, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [A3:A]: (((((inf_inf @ A) @ (top_top @ A)) @ A3) = A3))))))).
% 0.31/1.50 thf(fact_115_inf__top_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [A3:A,B2:A]: ((((((inf_inf @ A) @ A3) @ B2) = (top_top @ A)) = ((A3 = (top_top @ A)) & (B2 = (top_top @ A)))))))))).
% 0.31/1.50 thf(fact_116_top__eq__inf__iff, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [X:A,Y3:A]: ((((top_top @ A) = (((inf_inf @ A) @ X) @ Y3)) = ((X = (top_top @ A)) & (Y3 = (top_top @ A)))))))))).
% 0.31/1.50 thf(fact_117_inf__eq__top__iff, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [X:A,Y3:A]: ((((((inf_inf @ A) @ X) @ Y3) = (top_top @ A)) = ((X = (top_top @ A)) & (Y3 = (top_top @ A)))))))))).
% 0.31/1.50 thf(fact_118_inf__top__right, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [X:A]: (((((inf_inf @ A) @ X) @ (top_top @ A)) = X))))))).
% 0.31/1.50 thf(fact_119_inf__top__left, axiom, (! [A:$tType]: (((bounde1561333602nf_top @ A) => (! [X:A]: (((((inf_inf @ A) @ (top_top @ A)) @ X) = X))))))).
% 0.31/1.50 thf(fact_120_inf__sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y3)) = X))))))).
% 0.31/1.50 thf(fact_121_sup__inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = X))))))).
% 0.31/1.50 thf(fact_122_hinter__iff, axiom, (! [U4:hF_Mirabelle_hf,X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U4) @ (((inf_inf @ hF_Mirabelle_hf) @ X) @ Y3)) = (((hF_Mirabelle_hmem @ U4) @ X) & ((hF_Mirabelle_hmem @ U4) @ Y3)))))).
% 0.31/1.50 thf(fact_123_hinter__hempty__left, axiom, (! [A2:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ A2) = (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_124_hinter__hempty__right, axiom, (! [A2:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ A2) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 0.31/1.50 thf(fact_125_Abs__hf__0, axiom, ((hF_Mirabelle_Abs_hf @ (zero_zero @ nat)) = (zero_zero @ hF_Mirabelle_hf))).
% 0.31/1.50 thf(fact_126_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z)) = (((inf_inf @ A) @ Y3) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_127_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A3:A,C:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A3) @ C)) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_128_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X2:A,Y:A]: ((((inf_inf @ A) @ Y) @ X2)))))))).
% 0.31/1.50 thf(fact_129_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A5:A,B4:A]: ((((inf_inf @ A) @ B4) @ A5)))))))).
% 0.31/1.50 thf(fact_130_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X:A,Y3:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z))))))))).
% 0.31/1.50 thf(fact_131_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,B2:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A3) @ B2)) @ C) = (((inf_inf @ A) @ A3) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_132_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B5:A,K:A,B2:A,A3:A]: (((B5 = (((inf_inf @ A) @ K) @ B2)) => ((((inf_inf @ A) @ A3) @ B5) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_133_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,K:A,A3:A,B2:A]: (((A2 = (((inf_inf @ A) @ K) @ A3)) => ((((inf_inf @ A) @ A2) @ B2) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_134_inf__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F2:(A > B),G2:(A > B),X2:A]: ((((inf_inf @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 0.31/1.50 thf(fact_135_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X2:A,Y:A]: ((((inf_inf @ A) @ Y) @ X2)))))))).
% 0.31/1.50 thf(fact_136_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ Z) = (((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z))))))))).
% 0.31/1.50 thf(fact_137_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z)) = (((inf_inf @ A) @ Y3) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_138_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A]: (((((inf_inf @ A) @ X) @ (((inf_inf @ A) @ X) @ Y3)) = (((inf_inf @ A) @ X) @ Y3)))))))).
% 0.31/1.50 thf(fact_139_sup__inf__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y3:A,Z:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ Z)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ X)) @ (((sup_sup @ A) @ Z) @ X))))))))).
% 0.31/1.50 thf(fact_140_sup__inf__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_141_inf__sup__distrib2, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [Y3:A,Z:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y3) @ Z)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y3) @ X)) @ (((inf_inf @ A) @ Z) @ X))))))))).
% 0.31/1.50 thf(fact_142_inf__sup__distrib1, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [X:A,Y3:A,Z:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z))))))))).
% 0.31/1.50 thf(fact_143_distrib__imp2, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((! [X3:A,Y5:A,Z3:A]: (((((sup_sup @ A) @ X3) @ (((inf_inf @ A) @ Y5) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X3) @ Y5)) @ (((sup_sup @ A) @ X3) @ Z3))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y3) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y3)) @ (((inf_inf @ A) @ X) @ Z)))))))))).
% 0.31/1.50 thf(fact_144_distrib__imp1, axiom, (! [A:$tType]: (((lattice @ A) => (! [X:A,Y3:A,Z:A]: (((! [X3:A,Y5:A,Z3:A]: (((((inf_inf @ A) @ X3) @ (((sup_sup @ A) @ Y5) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X3) @ Y5)) @ (((inf_inf @ A) @ X3) @ Z3))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y3) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y3)) @ (((sup_sup @ A) @ X) @ Z)))))))))).
% 0.31/1.50 thf(fact_145_foundation, axiom, (! [Z:hF_Mirabelle_hf]: (((Z != (zero_zero @ hF_Mirabelle_hf)) => (? [W:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ W) @ Z) & ((((inf_inf @ hF_Mirabelle_hf) @ W) @ Z) = (zero_zero @ hF_Mirabelle_hf))))))))).
% 0.31/1.50 thf(fact_146_hinter__hinsert__right, axiom, (! [X:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ B5) => ((((inf_inf @ hF_Mirabelle_hf) @ B5) @ ((hF_Mirabelle_hinsert @ X) @ A2)) = ((hF_Mirabelle_hinsert @ X) @ (((inf_inf @ hF_Mirabelle_hf) @ B5) @ A2)))) & ((~ (((hF_Mirabelle_hmem @ X) @ B5))) => ((((inf_inf @ hF_Mirabelle_hf) @ B5) @ ((hF_Mirabelle_hinsert @ X) @ A2)) = (((inf_inf @ hF_Mirabelle_hf) @ B5) @ A2))))))).
% 0.31/1.50 thf(fact_147_hinter__hinsert__left, axiom, (! [X:hF_Mirabelle_hf,B5:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ B5) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A2)) @ B5) = ((hF_Mirabelle_hinsert @ X) @ (((inf_inf @ hF_Mirabelle_hf) @ A2) @ B5)))) & ((~ (((hF_Mirabelle_hmem @ X) @ B5))) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X) @ A2)) @ B5) = (((inf_inf @ hF_Mirabelle_hf) @ A2) @ B5))))))).
% 0.31/1.50 thf(fact_148_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_149_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 0.31/1.50 thf(fact_150_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 0.31/1.50 thf(fact_151_Un__UNIV__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 0.31/1.50 thf(fact_152_Un__UNIV__left, axiom, (! [A:$tType,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B5) = (top_top @ (set @ A)))))).
% 0.31/1.50 thf(fact_153_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 0.31/1.50 thf(fact_154_type__copy__ex__RepI, axiom, (! [B:$tType,A:$tType,Rep:(A > B),Abs:(B > A),F3:(B > $o)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((? [X4:B]: ((F3 @ X4))) = (? [B4:A]: ((F3 @ (Rep @ B4))))))))).
% 0.31/1.50 thf(fact_155_type__copy__obj__one__point__absE, axiom, (! [A:$tType,B:$tType,Rep:(A > B),Abs:(B > A),S:A]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (~ ((! [X3:B]: ((S != (Abs @ X3)))))))))).
% 0.31/1.50 thf(fact_156_HF__hfset, axiom, (! [A3:hF_Mirabelle_hf]: (((hF_Mirabelle_HF @ (hF_Mirabelle_hfset @ A3)) = A3)))).
% 0.31/1.50 thf(fact_157_type__definition_ORep, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((member @ A) @ (Rep @ X)) @ A2))))).
% 0.31/1.50 thf(fact_158_type__definition__def, axiom, (! [A:$tType,B:$tType]: ((((type_definition @ B) @ A) = (^ [Rep2:(B > A),Abs2:(A > B),A7:(set @ A)]: (((! [X2:B]: ((((member @ A) @ (Rep2 @ X2)) @ A7))) & ((! [X2:B]: (((Abs2 @ (Rep2 @ X2)) = X2))) & (! [Y:A]: (((((member @ A) @ Y) @ A7) => ((Rep2 @ (Abs2 @ Y)) = Y)))))))))))).
% 0.31/1.50 thf(fact_159_Int__UNIV, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A2) @ B5) = (top_top @ (set @ A))) = ((A2 = (top_top @ (set @ A))) & (B5 = (top_top @ (set @ A)))))))).
% 0.31/1.50 thf(fact_160_Int__Un__eq_I4_J, axiom, (! [A:$tType,T:(set @ A),S2:(set @ A)]: (((((sup_sup @ (set @ A)) @ T) @ (((inf_inf @ (set @ A)) @ S2) @ T)) = T)))).
% 0.31/1.50 thf(fact_161_Int__Un__eq_I3_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((sup_sup @ (set @ A)) @ S2) @ (((inf_inf @ (set @ A)) @ S2) @ T)) = S2)))).
% 0.31/1.50 thf(fact_162_Int__Un__eq_I2_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ S2) @ T)) @ T) = T)))).
% 0.31/1.50 thf(fact_163_Int__Un__eq_I1_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ S2) @ T)) @ S2) = S2)))).
% 0.31/1.50 thf(fact_164_Un__Int__eq_I4_J, axiom, (! [A:$tType,T:(set @ A),S2:(set @ A)]: (((((inf_inf @ (set @ A)) @ T) @ (((sup_sup @ (set @ A)) @ S2) @ T)) = T)))).
% 0.31/1.50 thf(fact_165_Un__Int__eq_I3_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((inf_inf @ (set @ A)) @ S2) @ (((sup_sup @ (set @ A)) @ S2) @ T)) = S2)))).
% 0.31/1.50 thf(fact_166_Un__Int__eq_I2_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ S2) @ T)) @ T) = T)))).
% 0.31/1.50 thf(fact_167_Un__Int__eq_I1_J, axiom, (! [A:$tType,S2:(set @ A),T:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ S2) @ T)) @ S2) = S2)))).
% 0.31/1.50 thf(fact_168_UnCI, axiom, (! [A:$tType,C:A,B5:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C) @ B5))) => (((member @ A) @ C) @ A2)) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_169_Un__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) = ((((member @ A) @ C) @ A2) | (((member @ A) @ C) @ B5)))))).
% 0.31/1.50 thf(fact_170_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 0.31/1.50 thf(fact_171_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 0.31/1.50 thf(fact_172_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 0.31/1.50 thf(fact_173_UnE, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) => ((~ ((((member @ A) @ C) @ A2))) => (((member @ A) @ C) @ B5)))))).
% 0.31/1.50 thf(fact_174_UnI1, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_175_UnI2, axiom, (! [A:$tType,C:A,B5:(set @ A),A2:(set @ A)]: (((((member @ A) @ C) @ B5) => (((member @ A) @ C) @ (((sup_sup @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_176_bex__Un, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A2) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B5) & (P @ X2))))))))).
% 0.31/1.50 thf(fact_177_ball__Un, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A2) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B5) => (P @ X2))))))))).
% 0.31/1.50 thf(fact_178_Un__assoc, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) @ C3) = (((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B5) @ C3)))))).
% 0.31/1.50 thf(fact_179_Un__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ A2) = A2)))).
% 0.31/1.50 thf(fact_180_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((((sup_sup @ (set @ A)) @ B7) @ A7))))))).
% 0.31/1.50 thf(fact_181_Un__Int__crazy, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) @ (((inf_inf @ (set @ A)) @ B5) @ C3))) @ (((inf_inf @ (set @ A)) @ C3) @ A2)) = (((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) @ (((sup_sup @ (set @ A)) @ B5) @ C3))) @ (((sup_sup @ (set @ A)) @ C3) @ A2)))))).
% 0.31/1.50 thf(fact_182_Int__UNIV__left, axiom, (! [A:$tType,B5:(set @ A)]: (((((inf_inf @ (set @ A)) @ (top_top @ (set @ A))) @ B5) = B5)))).
% 0.31/1.50 thf(fact_183_Int__UNIV__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (top_top @ (set @ A))) = A2)))).
% 0.31/1.50 thf(fact_184_Int__Un__distrib, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B5) @ C3)) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) @ (((inf_inf @ (set @ A)) @ A2) @ C3)))))).
% 0.31/1.50 thf(fact_185_Un__Int__distrib, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ B5) @ C3)) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) @ (((sup_sup @ (set @ A)) @ A2) @ C3)))))).
% 0.31/1.50 thf(fact_186_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B5)) = (((sup_sup @ (set @ A)) @ A2) @ B5))))).
% 0.31/1.50 thf(fact_187_Int__Un__distrib2, axiom, (! [A:$tType,B5:(set @ A),C3:(set @ A),A2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B5) @ C3)) @ A2) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B5) @ A2)) @ (((inf_inf @ (set @ A)) @ C3) @ A2)))))).
% 0.31/1.50 thf(fact_188_Un__Int__distrib2, axiom, (! [A:$tType,B5:(set @ A),C3:(set @ A),A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B5) @ C3)) @ A2) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B5) @ A2)) @ (((sup_sup @ (set @ A)) @ C3) @ A2)))))).
% 0.31/1.50 thf(fact_189_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B5) @ C3)) = (((sup_sup @ (set @ A)) @ B5) @ (((sup_sup @ (set @ A)) @ A2) @ C3)))))).
% 0.31/1.50 thf(fact_190_type__definition_ORep__inverse, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((Abs @ (Rep @ X)) = X))))).
% 0.31/1.50 thf(fact_191_type__definition_OAbs__inverse, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y3:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y3) @ A2) => ((Rep @ (Abs @ Y3)) = Y3)))))).
% 0.31/1.50 thf(fact_192_type__definition_ORep__inject, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B,Y3:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((Rep @ X) = (Rep @ Y3)) = (X = Y3)))))).
% 0.31/1.50 thf(fact_193_type__definition_ORep__induct, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y3:A,P:(A > $o)]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y3) @ A2) => ((! [X3:B]: ((P @ (Rep @ X3)))) => (P @ Y3))))))).
% 0.31/1.50 thf(fact_194_type__definition_OAbs__inject, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:A,Y3:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ X) @ A2) => ((((member @ A) @ Y3) @ A2) => (((Abs @ X) = (Abs @ Y3)) = (X = Y3)))))))).
% 0.31/1.50 thf(fact_195_type__definition_OAbs__induct, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),P:(B > $o),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((! [Y5:A]: (((((member @ A) @ Y5) @ A2) => (P @ (Abs @ Y5))))) => (P @ X)))))).
% 0.31/1.50 thf(fact_196_type__definition_ORep__cases, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),Y3:A]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((((member @ A) @ Y3) @ A2) => (~ ((! [X3:B]: ((Y3 != (Rep @ X3))))))))))).
% 0.31/1.50 thf(fact_197_type__definition_OAbs__cases, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),X:B]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (~ ((! [Y5:A]: (((X = (Abs @ Y5)) => (~ ((((member @ A) @ Y5) @ A2)))))))))))).
% 0.31/1.50 thf(fact_198_type__definition_Ointro, axiom, (! [B:$tType,A:$tType,Rep:(B > A),A2:(set @ A),Abs:(A > B)]: (((! [X3:B]: ((((member @ A) @ (Rep @ X3)) @ A2))) => ((! [X3:B]: (((Abs @ (Rep @ X3)) = X3))) => ((! [Y5:A]: (((((member @ A) @ Y5) @ A2) => ((Rep @ (Abs @ Y5)) = Y5)))) => (((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2))))))).
% 0.31/1.50 thf(fact_199_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 0.31/1.50 thf(fact_200_hcard__0, axiom, ((hF_Mirabelle_hcard @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ nat))).
% 0.31/1.50 thf(fact_201_hinsert__def, axiom, (hF_Mirabelle_hinsert = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: ((hF_Mirabelle_HF @ (((insert @ hF_Mirabelle_hf) @ A5) @ (hF_Mirabelle_hfset @ B4))))))).
% 0.31/1.50 thf(fact_202_insert__absorb2, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A2)) = (((insert @ A) @ X) @ A2))))).
% 0.31/1.50 thf(fact_203_insert__iff, axiom, (! [A:$tType,A3:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B2) @ A2)) = ((A3 = B2) | (((member @ A) @ A3) @ A2)))))).
% 0.31/1.50 thf(fact_204_insertCI, axiom, (! [A:$tType,A3:A,B5:(set @ A),B2:A]: ((((~ ((((member @ A) @ A3) @ B5))) => (A3 = B2)) => (((member @ A) @ A3) @ (((insert @ A) @ B2) @ B5)))))).
% 0.31/1.50 thf(fact_205_IntI, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ A2) => ((((member @ A) @ C) @ B5) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B5))))))).
% 0.31/1.50 thf(fact_206_Int__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) = ((((member @ A) @ C) @ A2) & (((member @ A) @ C) @ B5)))))).
% 0.31/1.50 thf(fact_207_Int__insert__left__if0, axiom, (! [A:$tType,A3:A,C3:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ A3) @ C3))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((inf_inf @ (set @ A)) @ B5) @ C3)))))).
% 0.31/1.50 thf(fact_208_Int__insert__left__if1, axiom, (! [A:$tType,A3:A,C3:(set @ A),B5:(set @ A)]: (((((member @ A) @ A3) @ C3) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ B5) @ C3))))))).
% 0.31/1.50 thf(fact_209_insert__inter__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A),B5:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ A2)) @ (((insert @ A) @ A3) @ B5)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_210_Int__insert__right__if0, axiom, (! [A:$tType,A3:A,A2:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B5)) = (((inf_inf @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_211_Int__insert__right__if1, axiom, (! [A:$tType,A3:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B5)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B5))))))).
% 0.31/1.50 thf(fact_212_Un__insert__left, axiom, (! [A:$tType,A3:A,B5:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ B5) @ C3)))))).
% 0.31/1.50 thf(fact_213_Un__insert__right, axiom, (! [A:$tType,A2:(set @ A),A3:A,B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B5)) = (((insert @ A) @ A3) @ (((sup_sup @ (set @ A)) @ A2) @ B5)))))).
% 0.31/1.50 thf(fact_214_insert__UNIV, axiom, (! [A:$tType,X:A]: (((((insert @ A) @ X) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 0.31/1.50 thf(fact_215_IntE, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) => (~ (((((member @ A) @ C) @ A2) => (~ ((((member @ A) @ C) @ B5)))))))))).
% 0.31/1.50 thf(fact_216_IntD1, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) => (((member @ A) @ C) @ A2))))).
% 0.31/1.50 thf(fact_217_IntD2, axiom, (! [A:$tType,C:A,A2:(set @ A),B5:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) => (((member @ A) @ C) @ B5))))).
% 0.31/1.50 thf(fact_218_Int__assoc, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) @ C3) = (((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ B5) @ C3)))))).
% 0.31/1.50 thf(fact_219_Int__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ A2) = A2)))).
% 0.31/1.50 thf(fact_220_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((((inf_inf @ (set @ A)) @ B7) @ A7))))))).
% 0.31/1.50 thf(fact_221_Int__insert__left, axiom, (! [A:$tType,A3:A,C3:(set @ A),B5:(set @ A)]: ((((((member @ A) @ A3) @ C3) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ B5) @ C3)))) & ((~ ((((member @ A) @ A3) @ C3))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A3) @ B5)) @ C3) = (((inf_inf @ (set @ A)) @ B5) @ C3))))))).
% 0.31/1.50 thf(fact_222_Int__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ A2) @ B5)) = (((inf_inf @ (set @ A)) @ A2) @ B5))))).
% 0.31/1.50 thf(fact_223_Int__insert__right, axiom, (! [A:$tType,A3:A,A2:(set @ A),B5:(set @ A)]: ((((((member @ A) @ A3) @ A2) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B5)) = (((insert @ A) @ A3) @ (((inf_inf @ (set @ A)) @ A2) @ B5)))) & ((~ ((((member @ A) @ A3) @ A2))) => ((((inf_inf @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B5)) = (((inf_inf @ (set @ A)) @ A2) @ B5))))))).
% 0.31/1.50 thf(fact_224_Int__left__commute, axiom, (! [A:$tType,A2:(set @ A),B5:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A2) @ (((inf_inf @ (set @ A)) @ B5) @ C3)) = (((inf_inf @ (set @ A)) @ B5) @ (((inf_inf @ (set @ A)) @ A2) @ C3)))))).
% 0.31/1.50 thf(fact_225_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (? [B8:(set @ A)]: (((A2 = (((insert @ A) @ A3) @ B8)) & (~ ((((member @ A) @ A3) @ B8)))))))))).
% 0.31/1.50 thf(fact_226_insert__commute, axiom, (! [A:$tType,X:A,Y3:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y3) @ A2)) = (((insert @ A) @ Y3) @ (((insert @ A) @ X) @ A2)))))).
% 0.31/1.50 thf(fact_227_insert__eq__iff, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:A,B5:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((~ ((((member @ A) @ B2) @ B5))) => (((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B2) @ B5)) = (((A3 = B2) => (A2 = B5)) & ((A3 != B2) => (? [C4:(set @ A)]: (((A2 = (((insert @ A) @ B2) @ C4)) & ((~ ((((member @ A) @ B2) @ C4))) & ((B5 = (((insert @ A) @ A3) @ C4)) & (~ ((((member @ A) @ A3) @ C4)))))))))))))))).
% 0.31/1.50 thf(fact_228_insert__absorb, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ A2) = A2))))).
% 0.31/1.50 thf(fact_229_insert__ident, axiom, (! [A:$tType,X:A,A2:(set @ A),B5:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((~ ((((member @ A) @ X) @ B5))) => (((((insert @ A) @ X) @ A2) = (((insert @ A) @ X) @ B5)) = (A2 = B5))))))).
% 0.31/1.50 thf(fact_230_Set_Oset__insert, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((member @ A) @ X) @ A2) => (~ ((! [B8:(set @ A)]: (((A2 = (((insert @ A) @ X) @ B8)) => (((member @ A) @ X) @ B8)))))))))).
% 0.31/1.50 thf(fact_231_insertI2, axiom, (! [A:$tType,A3:A,B5:(set @ A),B2:A]: (((((member @ A) @ A3) @ B5) => (((member @ A) @ A3) @ (((insert @ A) @ B2) @ B5)))))).
% 0.31/1.50 thf(fact_232_insertI1, axiom, (! [A:$tType,A3:A,B5:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B5))))).
% 0.31/1.50 thf(fact_233_insertE, axiom, (! [A:$tType,A3:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B2) @ A2)) => ((A3 != B2) => (((member @ A) @ A3) @ A2)))))).
% 0.31/1.50 thf(fact_234_hfset__hinsert, axiom, (! [A3:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mirabelle_hfset @ ((hF_Mirabelle_hinsert @ A3) @ B2)) = (((insert @ hF_Mirabelle_hf) @ A3) @ (hF_Mirabelle_hfset @ B2)))))).
% 0.31/1.50 thf(fact_235_hcard__union__inter, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: (((((plus_plus @ nat) @ (hF_Mirabelle_hcard @ (((sup_sup @ hF_Mirabelle_hf) @ X) @ Y3))) @ (hF_Mirabelle_hcard @ (((inf_inf @ hF_Mirabelle_hf) @ X) @ Y3))) = (((plus_plus @ nat) @ (hF_Mirabelle_hcard @ X)) @ (hF_Mirabelle_hcard @ Y3)))))).
% 0.31/1.50 thf(fact_236_hcard__hinsert__if, axiom, (! [X:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X) @ Y3) => ((hF_Mirabelle_hcard @ ((hF_Mirabelle_hinsert @ X) @ Y3)) = (hF_Mirabelle_hcard @ Y3))) & ((~ (((hF_Mirabelle_hmem @ X) @ Y3))) => ((hF_Mirabelle_hcard @ ((hF_Mirabelle_hinsert @ X) @ Y3)) = (suc @ (hF_Mirabelle_hcard @ Y3)))))))).
% 0.31/1.50 thf(fact_237_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B2:A,C:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C)) = (B2 = C)))))))).
% 0.31/1.50 thf(fact_238_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A3:A,C:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C) @ A3)) = (B2 = C)))))))).
% 0.31/1.50 thf(fact_239_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X) @ Y3)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.31/1.50 thf(fact_240_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X:A,Y3:A]: ((((((plus_plus @ A) @ X) @ Y3) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y3 = (zero_zero @ A)))))))))).
% 0.31/1.50 thf(fact_241_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ A3) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_242_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ B2) @ A3)) = (B2 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_243_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = A3) = (B2 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_244_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [B2:A,A3:A]: ((((((plus_plus @ A) @ B2) @ A3) = A3) = (B2 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_245_double__zero__sym, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A3) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_246_double__zero, axiom, (! [A:$tType]: (((linord219039673up_add @ A) => (! [A3:A]: ((((((plus_plus @ A) @ A3) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 0.31/1.50 thf(fact_247_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 0.31/1.50 thf(fact_248_add_Oleft__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 0.31/1.50 thf(fact_249_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A3:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_250_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 0.31/1.50 thf(fact_251_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A,K:A,A3:A,B2:A]: (((A2 = (((plus_plus @ A) @ K) @ A3)) => ((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_252_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B5:A,K:A,B2:A,A3:A]: (((B5 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A3) @ B5) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 0.31/1.50 thf(fact_253_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A3:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 0.31/1.50 thf(fact_254_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C)) = (B2 = C)))))))).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A8:$tType]: ((bounded_lattice @ (set @ A8))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounde1561333602nf_top @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice_top @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_sup @ A9) => (semilattice_sup @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_inf @ A9) => (semilattice_inf @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A8:$tType,A9:$tType]: (((distrib_lattice @ A9) => (distrib_lattice @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3, axiom, (semilattice_sup @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_4, axiom, (semilattice_inf @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_5, axiom, (distrib_lattice @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Omonoid__add, axiom, (monoid_add @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Lattices_Olattice_6, axiom, (lattice @ nat)).
% 0.31/1.50 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_7, axiom, (! [A8:$tType]: ((bounde1561333602nf_top @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_8, axiom, (! [A8:$tType]: ((bounded_lattice_top @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9, axiom, (! [A8:$tType]: ((semilattice_sup @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_10, axiom, (! [A8:$tType]: ((semilattice_inf @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_11, axiom, (! [A8:$tType]: ((distrib_lattice @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Lattices_Olattice_12, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 0.31/1.50 thf(tcon_Set_Oset___Orderings_Otop_13, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_14, axiom, (bounde1561333602nf_top @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_15, axiom, (bounded_lattice_top @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16, axiom, (semilattice_sup @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_17, axiom, (semilattice_inf @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_18, axiom, (distrib_lattice @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Lattices_Olattice_19, axiom, (lattice @ $o)).
% 0.31/1.50 thf(tcon_HOL_Obool___Orderings_Otop_20, axiom, (top @ $o)).
% 0.31/1.50 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_21, axiom, (semilattice_sup @ hF_Mirabelle_hf)).
% 0.31/1.50 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_22, axiom, (semilattice_inf @ hF_Mirabelle_hf)).
% 0.31/1.50 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_23, axiom, (distrib_lattice @ hF_Mirabelle_hf)).
% 0.31/1.50 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_24, axiom, (lattice @ hF_Mirabelle_hf)).
% 0.31/1.50 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_25, axiom, (zero @ hF_Mirabelle_hf)).
% 0.31/1.50 thf(conj_0, hypothesis, (hF_Mir859708136nction @ r)).
% 0.31/1.50 thf(conj_1, conjecture, (hF_Mir859708136nction @ ((hF_Mir165288108strict @ r) @ x))).
% 0.31/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2DTF504.p
% 0.31/1.50 ------------------------
% 0.31/1.53 ---- Cleaned THF ---
% 0.31/1.53 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 0.31/1.53 hF_Mirabelle_hf: $tType ).
% 0.31/1.53
% 0.31/1.53 thf(ty_t_Set_Oset,type,
% 0.31/1.53 set: $tType > $tType ).
% 0.31/1.53
% 0.31/1.53 thf(ty_t_Nat_Onat,type,
% 0.31/1.53 nat: $tType ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.31/1.53 bounded_lattice:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_HOL_Otype,type,
% 0.31/1.53 type:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ozero,type,
% 0.31/1.53 zero:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Orderings_Otop,type,
% 0.31/1.53 top:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ogroup__add,type,
% 0.31/1.53 group_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Olattice,type,
% 0.31/1.53 lattice:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Omonoid__add,type,
% 0.31/1.53 monoid_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.31/1.53 semigroup_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.31/1.53 comm_monoid_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.31/1.53 ab_semigroup_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.31/1.53 distrib_lattice:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.31/1.53 semilattice_inf:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.31/1.53 semilattice_sup:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.31/1.53 cancel_semigroup_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.31/1.53 bounded_lattice_top:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.31/1.53 cancel1352612707id_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.31/1.53 linord219039673up_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.31/1.53 ordere779506340up_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
% 0.31/1.53 bounde1561333602nf_top:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.31/1.53 canoni770627133id_add:
% 0.31/1.53 !>[A: $tType] : $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.31/1.53 plus_plus:
% 0.31/1.53 !>[A: $tType] : ( A > A > A ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.31/1.53 zero_zero:
% 0.31/1.53 !>[A: $tType] : A ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 0.31/1.53 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 0.31/1.53 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 0.31/1.53 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 0.31/1.53 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 0.31/1.53 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 0.31/1.53 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 0.31/1.53 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard,type,
% 0.31/1.53 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 0.31/1.53 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_ORep__hf,type,
% 0.31/1.53 hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 0.31/1.53 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 0.31/1.53 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfunction,type,
% 0.31/1.53 hF_Mir859708136nction: hF_Mirabelle_hf > $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 0.31/1.53 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 0.31/1.53 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 0.31/1.53 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrelation,type,
% 0.31/1.53 hF_Mir1093797708lation: hF_Mirabelle_hf > $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohrestrict,type,
% 0.31/1.53 hF_Mir165288108strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 0.31/1.53 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 0.31/1.53 hF_Mirabelle_hsplit:
% 0.31/1.53 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ois__hpair,type,
% 0.31/1.53 hF_Mir796903520_hpair: hF_Mirabelle_hf > $o ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.31/1.53 inf_inf:
% 0.31/1.53 !>[A: $tType] : ( A > A > A ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.31/1.53 sup_sup:
% 0.31/1.53 !>[A: $tType] : ( A > A > A ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Nat_OSuc,type,
% 0.31/1.53 suc: nat > nat ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.31/1.53 top_top:
% 0.31/1.53 !>[A: $tType] : A ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Set_OCollect,type,
% 0.31/1.53 collect:
% 0.31/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Set_Oinsert,type,
% 0.31/1.53 insert:
% 0.31/1.53 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_Typedef_Otype__definition,type,
% 0.31/1.53 type_definition:
% 0.31/1.53 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_c_member,type,
% 0.31/1.53 member:
% 0.31/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/1.53
% 0.31/1.53 thf(sy_v_r,type,
% 0.31/1.53 r: hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(sy_v_x,type,
% 0.31/1.53 x: hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(fact_0_hrelation__restr,axiom,
% 0.31/1.53 ! [R: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( hF_Mir1093797708lation @ ( hF_Mir165288108strict @ R @ X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_1_hfunction__def,axiom,
% 0.31/1.53 ( hF_Mir859708136nction
% 0.31/1.53 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.31/1.53 ! [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y ) @ R2 )
% 0.31/1.53 => ! [Y2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) @ R2 )
% 0.31/1.53 => ( Y = Y2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_2_hrestrict__iff,axiom,
% 0.31/1.53 ! [Z: hF_Mirabelle_hf,R: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ Z @ ( hF_Mir165288108strict @ R @ A2 ) )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ Z @ R )
% 0.31/1.53 & ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Z
% 0.31/1.53 = ( hF_Mirabelle_hpair @ X2 @ Y ) )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_3_hpair__iff,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.31/1.53 = ( hF_Mirabelle_hpair @ A4 @ B3 ) )
% 0.31/1.53 = ( ( A3 = A4 )
% 0.31/1.53 & ( B2 = B3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_4_Rep__hf__inject,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_Rep_hf @ X )
% 0.31/1.53 = ( hF_Mirabelle_Rep_hf @ Y3 ) )
% 0.31/1.53 = ( X = Y3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_5_hf__equalityI,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ! [X3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.31/1.53 = ( hF_Mirabelle_hmem @ X3 @ B2 ) )
% 0.31/1.53 => ( A3 = B2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_6_hf__ext,axiom,
% 0.31/1.53 ( ( ^ [Y4: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] : ( Y4 = Z2 ) )
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.31/1.53 ! [X2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X2 @ A5 )
% 0.31/1.53 = ( hF_Mirabelle_hmem @ X2 @ B4 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_7_hmem__ne,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.31/1.53 => ( X != Y3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_8_replacement,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U @ X )
% 0.31/1.53 => ( ( R3 @ U @ V )
% 0.31/1.53 => ( ( R3 @ U @ V2 )
% 0.31/1.53 => ( V2 = V ) ) ) )
% 0.31/1.53 => ? [Z3: hF_Mirabelle_hf] :
% 0.31/1.53 ! [V3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.31/1.53 = ( ? [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.31/1.53 & ( R3 @ U2 @ V3 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_9_binary__union,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ? [Z3: hF_Mirabelle_hf] :
% 0.31/1.53 ! [U3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.31/1.53 | ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_10_hmem__not__sym,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ Y3 @ X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_11_hpair__inject,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.31/1.53 = ( hF_Mirabelle_hpair @ A4 @ B3 ) )
% 0.31/1.53 => ~ ( ( A3 = A4 )
% 0.31/1.53 => ( B2 != B3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_12_union__of__set,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf] :
% 0.31/1.53 ? [Z3: hF_Mirabelle_hf] :
% 0.31/1.53 ! [U3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.31/1.53 = ( ? [Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ Y @ X )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_13_comprehension,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.31/1.53 ? [Z3: hF_Mirabelle_hf] :
% 0.31/1.53 ! [U3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U3 @ Z3 )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ U3 @ X )
% 0.31/1.53 & ( P @ U3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_14_hmem__not__refl,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( hF_Mirabelle_hmem @ X @ X ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_15_hpair__neq__fst,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.31/1.53 != A3 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_16_hpair__neq__snd,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hpair @ A3 @ B2 )
% 0.31/1.53 != B2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_17_replacement__fun,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ? [Z3: hF_Mirabelle_hf] :
% 0.31/1.53 ! [V3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ V3 @ Z3 )
% 0.31/1.53 = ( ? [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.31/1.53 & ( V3
% 0.31/1.53 = ( F @ U2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_18_hsplit,axiom,
% 0.31/1.53 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.31/1.53 = ( C @ A3 @ B2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_19_hfst__conv,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.31/1.53 = A3 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_20_hsnd__conv,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A3 @ B2 ) )
% 0.31/1.53 = B2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_21_hrelation__def,axiom,
% 0.31/1.53 ( hF_Mir1093797708lation
% 0.31/1.53 = ( ^ [R2: hF_Mirabelle_hf] :
% 0.31/1.53 ! [Z4: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ Z4 @ R2 )
% 0.31/1.53 => ( hF_Mir796903520_hpair @ Z4 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_22_is__hpair__def,axiom,
% 0.31/1.53 ( hF_Mir796903520_hpair
% 0.31/1.53 = ( ^ [Z4: hF_Mirabelle_hf] :
% 0.31/1.53 ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( Z4
% 0.31/1.53 = ( hF_Mirabelle_hpair @ X2 @ Y ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_23_PrimReplace__iff,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.31/1.53 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U @ A2 )
% 0.31/1.53 => ( ( R3 @ U @ V )
% 0.31/1.53 => ( ( R3 @ U @ V2 )
% 0.31/1.53 => ( V2 = V ) ) ) )
% 0.31/1.53 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir569462966eplace @ A2 @ R3 ) )
% 0.31/1.53 = ( ? [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.31/1.53 & ( R3 @ U2 @ V4 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_24_HUnion__iff,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A2 ) )
% 0.31/1.53 = ( ? [Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ Y @ A2 )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_25_Replace__iff,axiom,
% 0.31/1.53 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A2 @ R3 ) )
% 0.31/1.53 = ( ? [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.31/1.53 & ( R3 @ U2 @ V4 )
% 0.31/1.53 & ! [Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( R3 @ U2 @ Y )
% 0.31/1.53 => ( Y = V4 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_26_HCollect__iff,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X @ ( hF_Mir1687042746ollect @ P @ A2 ) )
% 0.31/1.53 = ( ( P @ X )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ X @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_27_RepFun__iff,axiom,
% 0.31/1.53 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A2 @ F ) )
% 0.31/1.53 = ( ? [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.31/1.53 & ( V4
% 0.31/1.53 = ( F @ U2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_28_Rep__hf__inverse,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_Abs_hf @ ( hF_Mirabelle_Rep_hf @ X ) )
% 0.31/1.53 = X ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_29_RepFun__cong,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ( ( A2 = B5 )
% 0.31/1.53 => ( ! [X3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X3 @ B5 )
% 0.31/1.53 => ( ( F @ X3 )
% 0.31/1.53 = ( G @ X3 ) ) )
% 0.31/1.53 => ( ( hF_Mirabelle_RepFun @ A2 @ F )
% 0.31/1.53 = ( hF_Mirabelle_RepFun @ B5 @ G ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_30_Replace__cong,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ( A2 = B5 )
% 0.31/1.53 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X3 @ B5 )
% 0.31/1.53 => ( ( P @ X3 @ Y5 )
% 0.31/1.53 = ( Q @ X3 @ Y5 ) ) )
% 0.31/1.53 => ( ( hF_Mirabelle_Replace @ A2 @ P )
% 0.31/1.53 = ( hF_Mirabelle_Replace @ B5 @ Q ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_31_hmem__Sup__ne,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.31/1.53 => ( ( hF_Mirabelle_HUnion @ X )
% 0.31/1.53 != Y3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_32_Abs__hf__inverse,axiom,
% 0.31/1.53 ! [Y3: nat] :
% 0.31/1.53 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ( ( hF_Mirabelle_Rep_hf @ ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.31/1.53 = Y3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_33_zero__notin__hpair,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( hF_Mirabelle_hmem @ ( zero_zero @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hpair @ X @ Y3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_34_hmem__def,axiom,
% 0.31/1.53 ( hF_Mirabelle_hmem
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hfset @ B4 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_35_HCollect__hempty,axiom,
% 0.31/1.53 ! [P: hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_36_HUnion__hempty,axiom,
% 0.31/1.53 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_37_Replace__0,axiom,
% 0.31/1.53 ! [R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R3 )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_38_RepFun__0,axiom,
% 0.31/1.53 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_39_RepFun__hinsert,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A3 @ B2 ) @ F )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ ( F @ A3 ) @ ( hF_Mirabelle_RepFun @ B2 @ F ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_40_hrelation__hunion,axiom,
% 0.31/1.53 ! [F: hF_Mirabelle_hf,G: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mir1093797708lation @ ( sup_sup @ hF_Mirabelle_hf @ F @ G ) )
% 0.31/1.53 = ( ( hF_Mir1093797708lation @ F )
% 0.31/1.53 & ( hF_Mir1093797708lation @ G ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_41_hmem__hinsert,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ C ) )
% 0.31/1.53 = ( ( A3 = B2 )
% 0.31/1.53 | ( hF_Mirabelle_hmem @ A3 @ C ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_42_singleton__eq__iff,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hinsert @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.31/1.53 = ( A3 = B2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_43_hunion__iff,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ X @ ( sup_sup @ hF_Mirabelle_hf @ A3 @ B2 ) )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ X @ A3 )
% 0.31/1.53 | ( hF_Mirabelle_hmem @ X @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_44_hunion__hempty__left,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( sup_sup @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A2 )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,P: A > $o] :
% 0.31/1.53 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.31/1.53 = ( P @ A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ( collect @ A
% 0.31/1.53 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_47_Collect__cong,axiom,
% 0.31/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/1.53 ( ! [X3: A] :
% 0.31/1.53 ( ( P @ X3 )
% 0.31/1.53 = ( Q @ X3 ) )
% 0.31/1.53 => ( ( collect @ A @ P )
% 0.31/1.53 = ( collect @ A @ Q ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_48_ext,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/1.53 ( ! [X3: A] :
% 0.31/1.53 ( ( F @ X3 )
% 0.31/1.53 = ( G @ X3 ) )
% 0.31/1.53 => ( F = G ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_49_hunion__hempty__right,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( sup_sup @ hF_Mirabelle_hf @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_50_RepFun__hunion,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_RepFun @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) @ F )
% 0.31/1.53 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_RepFun @ A2 @ F ) @ ( hF_Mirabelle_RepFun @ B5 @ F ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_51_HUnion__hunion,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_HUnion @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) )
% 0.31/1.53 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_HUnion @ A2 ) @ ( hF_Mirabelle_HUnion @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_52_Replace__hunion,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.31/1.53 ( ( hF_Mirabelle_Replace @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) @ R3 )
% 0.31/1.53 = ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_Replace @ A2 @ R3 ) @ ( hF_Mirabelle_Replace @ B5 @ R3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_53_HUnion__hinsert,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A3 @ A2 ) )
% 0.31/1.53 = ( sup_sup @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_HUnion @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_54_hf__induct__ax,axiom,
% 0.31/1.53 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ( ! [X3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( P @ X3 )
% 0.31/1.53 => ! [Y5: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( P @ Y5 )
% 0.31/1.53 => ( P @ ( hF_Mirabelle_hinsert @ Y5 @ X3 ) ) ) )
% 0.31/1.53 => ( P @ X ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_55_hinsert__eq__sup,axiom,
% 0.31/1.53 ( hF_Mirabelle_hinsert
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( sup_sup @ hF_Mirabelle_hf @ B4 @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_56_hinsert__commute,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y3 @ Z ) )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ Y3 @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_57_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 0.31/1.53 = ( ( ( A3 = C )
% 0.31/1.53 & ( B2 = D ) )
% 0.31/1.53 | ( ( A3 = D )
% 0.31/1.53 & ( B2 = C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_58_hinsert__nonempty,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hinsert @ A3 @ A2 )
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_59_hunion__hinsert__left,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( sup_sup @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_60_hunion__hinsert__right,axiom,
% 0.31/1.53 ! [B5: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( sup_sup @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ X @ ( sup_sup @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_61_hf__cases,axiom,
% 0.31/1.53 ! [Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Y3
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ~ ! [A6: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Y3
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ A6 @ B6 ) )
% 0.31/1.53 => ( hF_Mirabelle_hmem @ A6 @ B6 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_62_hf__induct,axiom,
% 0.31/1.53 ! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( P @ X3 )
% 0.31/1.53 => ( ( P @ Y5 )
% 0.31/1.53 => ( ~ ( hF_Mirabelle_hmem @ X3 @ Y5 )
% 0.31/1.53 => ( P @ ( hF_Mirabelle_hinsert @ X3 @ Y5 ) ) ) ) )
% 0.31/1.53 => ( P @ Z ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_63_hpair__def,axiom,
% 0.31/1.53 ( hF_Mirabelle_hpair
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ B4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_64_hpair__def_H,axiom,
% 0.31/1.53 ( hF_Mirabelle_hpair
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ A5 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A5 @ ( hF_Mirabelle_hinsert @ B4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_65_hinsert__iff,axiom,
% 0.31/1.53 ! [Z: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Z
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ Y3 @ X ) )
% 0.31/1.53 = ( ! [U2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U2 @ Z )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ U2 @ X )
% 0.31/1.53 | ( U2 = Y3 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_66_hemptyE,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_67_hempty__iff,axiom,
% 0.31/1.53 ! [Z: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Z
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( ! [X2: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( hF_Mirabelle_hmem @ X2 @ Z ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_68_hmem__hempty,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ( hF_Mirabelle_hmem @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_69_hpair__nonzero,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hpair @ X @ Y3 )
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_70_Rep__hf,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf] : ( member @ nat @ ( hF_Mirabelle_Rep_hf @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_71_Rep__hf__cases,axiom,
% 0.31/1.53 ! [Y3: nat] :
% 0.31/1.53 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ~ ! [X3: hF_Mirabelle_hf] :
% 0.31/1.53 ( Y3
% 0.31/1.53 != ( hF_Mirabelle_Rep_hf @ X3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_72_Rep__hf__induct,axiom,
% 0.31/1.53 ! [Y3: nat,P: nat > $o] :
% 0.31/1.53 ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ( ! [X3: hF_Mirabelle_hf] : ( P @ ( hF_Mirabelle_Rep_hf @ X3 ) )
% 0.31/1.53 => ( P @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_73_Abs__hf__cases,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf] :
% 0.31/1.53 ~ ! [Y5: nat] :
% 0.31/1.53 ( ( X
% 0.31/1.53 = ( hF_Mirabelle_Abs_hf @ Y5 ) )
% 0.31/1.53 => ~ ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_74_Abs__hf__induct,axiom,
% 0.31/1.53 ! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
% 0.31/1.53 ( ! [Y5: nat] :
% 0.31/1.53 ( ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ( P @ ( hF_Mirabelle_Abs_hf @ Y5 ) ) )
% 0.31/1.53 => ( P @ X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_75_Abs__hf__inject,axiom,
% 0.31/1.53 ! [X: nat,Y3: nat] :
% 0.31/1.53 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
% 0.31/1.53 => ( ( ( hF_Mirabelle_Abs_hf @ X )
% 0.31/1.53 = ( hF_Mirabelle_Abs_hf @ Y3 ) )
% 0.31/1.53 = ( X = Y3 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_76_hrelation__0,axiom,
% 0.31/1.53 hF_Mir1093797708lation @ ( zero_zero @ hF_Mirabelle_hf ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_77_sup__top__left,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounded_lattice_top @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.31/1.53 = ( top_top @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_78_sup__top__right,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounded_lattice_top @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.31/1.53 = ( top_top @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_79_HInter__iff,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A2 ) )
% 0.31/1.53 = ( ! [Y: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ Y @ A2 )
% 0.31/1.53 => ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_80_sup__apply,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ B )
% 0.31/1.53 => ( ( sup_sup @ ( A > B ) )
% 0.31/1.53 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_81_sup_Oidem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( sup_sup @ A @ A3 @ A3 )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_82_sup__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ X )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_83_sup_Oleft__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
% 0.31/1.53 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_84_sup__left__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.31/1.53 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_85_sup_Oright__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
% 0.31/1.53 = ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_86_HInter__hempty,axiom,
% 0.31/1.53 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_87_sup__left__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_88_sup_Oleft__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [B2: A,A3: A,C: A] :
% 0.31/1.53 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C ) )
% 0.31/1.53 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_89_sup__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ( ( sup_sup @ A )
% 0.31/1.53 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_90_sup_Ocommute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ( ( sup_sup @ A )
% 0.31/1.53 = ( ^ [A5: A,B4: A] : ( sup_sup @ A @ B4 @ A5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_91_sup__assoc,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
% 0.31/1.53 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_92_sup_Oassoc,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C )
% 0.31/1.53 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_93_boolean__algebra__cancel_Osup2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.31/1.53 ( ( B5
% 0.31/1.53 = ( sup_sup @ A @ K @ B2 ) )
% 0.31/1.53 => ( ( sup_sup @ A @ A3 @ B5 )
% 0.31/1.53 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_94_boolean__algebra__cancel_Osup1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A )
% 0.31/1.53 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( sup_sup @ A @ K @ A3 ) )
% 0.31/1.53 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.31/1.53 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_95_sup__fun__def,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ B )
% 0.31/1.53 => ( ( sup_sup @ ( A > B ) )
% 0.31/1.53 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_96_inf__sup__aci_I5_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ( ( sup_sup @ A )
% 0.31/1.53 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_97_inf__sup__aci_I6_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z )
% 0.31/1.53 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_98_inf__sup__aci_I7_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_99_inf__sup__aci_I8_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.31/1.53 = ( sup_sup @ A @ X @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_100_iso__tuple__UNIV__I,axiom,
% 0.31/1.53 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_101_UNIV__I,axiom,
% 0.31/1.53 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_102_top__apply,axiom,
% 0.31/1.53 ! [C2: $tType,D2: $tType] :
% 0.31/1.53 ( ( top @ C2 )
% 0.31/1.53 => ( ( top_top @ ( D2 > C2 ) )
% 0.31/1.53 = ( ^ [X2: D2] : ( top_top @ C2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_103_HInter__hinsert,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A3 @ A2 ) )
% 0.31/1.53 = ( inf_inf @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_HInter @ A2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_104_type__definition__hf,axiom,
% 0.31/1.53 type_definition @ hF_Mirabelle_hf @ nat @ hF_Mirabelle_Rep_hf @ hF_Mirabelle_Abs_hf @ ( top_top @ ( set @ nat ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_105_inf__right__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 )
% 0.31/1.53 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_106_inf_Oright__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
% 0.31/1.53 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_107_inf__left__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.31/1.53 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_108_inf_Oleft__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
% 0.31/1.53 = ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_109_inf__idem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ X )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_110_inf_Oidem,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ A3 @ A3 )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_111_inf__apply,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ B )
% 0.31/1.53 => ( ( inf_inf @ ( A > B ) )
% 0.31/1.53 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_112_inf__top_Oright__neutral,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_113_inf__top_Oneutr__eq__iff,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( ( top_top @ A )
% 0.31/1.53 = ( inf_inf @ A @ A3 @ B2 ) )
% 0.31/1.53 = ( ( A3
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 & ( B2
% 0.31/1.53 = ( top_top @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_114_inf__top_Oleft__neutral,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_115_inf__top_Oeq__neutr__iff,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( ( inf_inf @ A @ A3 @ B2 )
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 = ( ( A3
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 & ( B2
% 0.31/1.53 = ( top_top @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_116_top__eq__inf__iff,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( ( top_top @ A )
% 0.31/1.53 = ( inf_inf @ A @ X @ Y3 ) )
% 0.31/1.53 = ( ( X
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 & ( Y3
% 0.31/1.53 = ( top_top @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_117_inf__eq__top__iff,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( ( inf_inf @ A @ X @ Y3 )
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 = ( ( X
% 0.31/1.53 = ( top_top @ A ) )
% 0.31/1.53 & ( Y3
% 0.31/1.53 = ( top_top @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_118_inf__top__right,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( top_top @ A ) )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_119_inf__top__left,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( bounde1561333602nf_top @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( top_top @ A ) @ X )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_120_inf__sup__absorb,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_121_sup__inf__absorb,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_122_hinter__iff,axiom,
% 0.31/1.53 ! [U4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y3 ) )
% 0.31/1.53 = ( ( hF_Mirabelle_hmem @ U4 @ X )
% 0.31/1.53 & ( hF_Mirabelle_hmem @ U4 @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_123_hinter__hempty__left,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A2 )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_124_hinter__hempty__right,axiom,
% 0.31/1.53 ! [A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_125_Abs__hf__0,axiom,
% 0.31/1.53 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_126_inf__left__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_127_inf_Oleft__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [B2: A,A3: A,C: A] :
% 0.31/1.53 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C ) )
% 0.31/1.53 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_128_inf__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ( ( inf_inf @ A )
% 0.31/1.53 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_129_inf_Ocommute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ( ( inf_inf @ A )
% 0.31/1.53 = ( ^ [A5: A,B4: A] : ( inf_inf @ A @ B4 @ A5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_130_inf__assoc,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z )
% 0.31/1.53 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_131_inf_Oassoc,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C )
% 0.31/1.53 = ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_132_boolean__algebra__cancel_Oinf2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.31/1.53 ( ( B5
% 0.31/1.53 = ( inf_inf @ A @ K @ B2 ) )
% 0.31/1.53 => ( ( inf_inf @ A @ A3 @ B5 )
% 0.31/1.53 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_133_boolean__algebra__cancel_Oinf1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A )
% 0.31/1.53 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( inf_inf @ A @ K @ A3 ) )
% 0.31/1.53 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.31/1.53 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_134_inf__fun__def,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ B )
% 0.31/1.53 => ( ( inf_inf @ ( A > B ) )
% 0.31/1.53 = ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_135_inf__sup__aci_I1_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ( ( inf_inf @ A )
% 0.31/1.53 = ( ^ [X2: A,Y: A] : ( inf_inf @ A @ Y @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_136_inf__sup__aci_I2_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z )
% 0.31/1.53 = ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_137_inf__sup__aci_I3_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_138_inf__sup__aci_I4_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
% 0.31/1.53 = ( inf_inf @ A @ X @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_139_sup__inf__distrib2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( distrib_lattice @ A )
% 0.31/1.53 => ! [Y3: A,Z: A,X: A] :
% 0.31/1.53 ( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z ) @ X )
% 0.31/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_140_sup__inf__distrib1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( distrib_lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_141_inf__sup__distrib2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( distrib_lattice @ A )
% 0.31/1.53 => ! [Y3: A,Z: A,X: A] :
% 0.31/1.53 ( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z ) @ X )
% 0.31/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_142_inf__sup__distrib1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( distrib_lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_143_distrib__imp2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ! [X3: A,Y5: A,Z3: A] :
% 0.31/1.53 ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y5 @ Z3 ) )
% 0.31/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( sup_sup @ A @ X3 @ Z3 ) ) )
% 0.31/1.53 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_144_distrib__imp1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( lattice @ A )
% 0.31/1.53 => ! [X: A,Y3: A,Z: A] :
% 0.31/1.53 ( ! [X3: A,Y5: A,Z3: A] :
% 0.31/1.53 ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y5 @ Z3 ) )
% 0.31/1.53 = ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( inf_inf @ A @ X3 @ Z3 ) ) )
% 0.31/1.53 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z ) )
% 0.31/1.53 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_145_foundation,axiom,
% 0.31/1.53 ! [Z: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( Z
% 0.31/1.53 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 => ? [W: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hmem @ W @ Z )
% 0.31/1.53 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z )
% 0.31/1.53 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_146_hinter__hinsert__right,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hmem @ X @ B5 )
% 0.31/1.53 => ( ( inf_inf @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) )
% 0.31/1.53 & ( ~ ( hF_Mirabelle_hmem @ X @ B5 )
% 0.31/1.53 => ( ( inf_inf @ hF_Mirabelle_hf @ B5 @ ( hF_Mirabelle_hinsert @ X @ A2 ) )
% 0.31/1.53 = ( inf_inf @ hF_Mirabelle_hf @ B5 @ A2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_147_hinter__hinsert__left,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hmem @ X @ B5 )
% 0.31/1.53 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.31/1.53 = ( hF_Mirabelle_hinsert @ X @ ( inf_inf @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) )
% 0.31/1.53 & ( ~ ( hF_Mirabelle_hmem @ X @ B5 )
% 0.31/1.53 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X @ A2 ) @ B5 )
% 0.31/1.53 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ B5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_148_zero__reorient,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( zero @ A )
% 0.31/1.53 => ! [X: A] :
% 0.31/1.53 ( ( ( zero_zero @ A )
% 0.31/1.53 = X )
% 0.31/1.53 = ( X
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_149_UNIV__eq__I,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.31/1.53 => ( ( top_top @ ( set @ A ) )
% 0.31/1.53 = A2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_150_UNIV__witness,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_151_Un__UNIV__right,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.31/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_152_Un__UNIV__left,axiom,
% 0.31/1.53 ! [A: $tType,B5: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
% 0.31/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_153_top__set__def,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( top_top @ ( set @ A ) )
% 0.31/1.53 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_154_type__copy__ex__RepI,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F3: B > $o] :
% 0.31/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.31/1.53 => ( ( ? [X4: B] : ( F3 @ X4 ) )
% 0.31/1.53 = ( ? [B4: A] : ( F3 @ ( Rep @ B4 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_155_type__copy__obj__one__point__absE,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
% 0.31/1.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.31/1.53 => ~ ! [X3: B] :
% 0.31/1.53 ( S
% 0.31/1.53 != ( Abs @ X3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_156_HF__hfset,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A3 ) )
% 0.31/1.53 = A3 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_157_type__definition_ORep,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( member @ A @ ( Rep @ X ) @ A2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_158_type__definition__def,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType] :
% 0.31/1.53 ( ( type_definition @ B @ A )
% 0.31/1.53 = ( ^ [Rep2: B > A,Abs2: A > B,A7: set @ A] :
% 0.31/1.53 ( ! [X2: B] : ( member @ A @ ( Rep2 @ X2 ) @ A7 )
% 0.31/1.53 & ! [X2: B] :
% 0.31/1.53 ( ( Abs2 @ ( Rep2 @ X2 ) )
% 0.31/1.53 = X2 )
% 0.31/1.53 & ! [Y: A] :
% 0.31/1.53 ( ( member @ A @ Y @ A7 )
% 0.31/1.53 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.31/1.53 = Y ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_159_Int__UNIV,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( ( inf_inf @ ( set @ A ) @ A2 @ B5 )
% 0.31/1.53 = ( top_top @ ( set @ A ) ) )
% 0.31/1.53 = ( ( A2
% 0.31/1.53 = ( top_top @ ( set @ A ) ) )
% 0.31/1.53 & ( B5
% 0.31/1.53 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_160_Int__Un__eq_I4_J,axiom,
% 0.31/1.53 ! [A: $tType,T: set @ A,S2: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ T @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.31/1.53 = T ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_161_Int__Un__eq_I3_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ S2 @ ( inf_inf @ ( set @ A ) @ S2 @ T ) )
% 0.31/1.53 = S2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_162_Int__Un__eq_I2_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) @ T )
% 0.31/1.53 = T ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_163_Int__Un__eq_I1_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T ) @ S2 )
% 0.31/1.53 = S2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_164_Un__Int__eq_I4_J,axiom,
% 0.31/1.53 ! [A: $tType,T: set @ A,S2: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ T @ ( sup_sup @ ( set @ A ) @ S2 @ T ) )
% 0.31/1.53 = T ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_165_Un__Int__eq_I3_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ S2 @ ( sup_sup @ ( set @ A ) @ S2 @ T ) )
% 0.31/1.53 = S2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_166_Un__Int__eq_I2_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T ) @ T )
% 0.31/1.53 = T ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_167_Un__Int__eq_I1_J,axiom,
% 0.31/1.53 ! [A: $tType,S2: set @ A,T: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T ) @ S2 )
% 0.31/1.53 = S2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_168_UnCI,axiom,
% 0.31/1.53 ! [A: $tType,C: A,B5: set @ A,A2: set @ A] :
% 0.31/1.53 ( ( ~ ( member @ A @ C @ B5 )
% 0.31/1.53 => ( member @ A @ C @ A2 ) )
% 0.31/1.53 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_169_Un__iff,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 = ( ( member @ A @ C @ A2 )
% 0.31/1.53 | ( member @ A @ C @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_170_top1I,axiom,
% 0.31/1.53 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_171_top__conj_I2_J,axiom,
% 0.31/1.53 ! [A: $tType,P: $o,X: A] :
% 0.31/1.53 ( ( P
% 0.31/1.53 & ( top_top @ ( A > $o ) @ X ) )
% 0.31/1.53 = P ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_172_top__conj_I1_J,axiom,
% 0.31/1.53 ! [A: $tType,X: A,P: $o] :
% 0.31/1.53 ( ( ( top_top @ ( A > $o ) @ X )
% 0.31/1.53 & P )
% 0.31/1.53 = P ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_173_UnE,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 => ( ~ ( member @ A @ C @ A2 )
% 0.31/1.53 => ( member @ A @ C @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_174_UnI1,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ A2 )
% 0.31/1.53 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_175_UnI2,axiom,
% 0.31/1.53 ! [A: $tType,C: A,B5: set @ A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ B5 )
% 0.31/1.53 => ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_176_bex__Un,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
% 0.31/1.53 ( ( ? [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 & ( P @ X2 ) ) )
% 0.31/1.53 = ( ? [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ A2 )
% 0.31/1.53 & ( P @ X2 ) )
% 0.31/1.53 | ? [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ B5 )
% 0.31/1.53 & ( P @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_177_ball__Un,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
% 0.31/1.53 ( ( ! [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 => ( P @ X2 ) ) )
% 0.31/1.53 = ( ! [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ A2 )
% 0.31/1.53 => ( P @ X2 ) )
% 0.31/1.53 & ! [X2: A] :
% 0.31/1.53 ( ( member @ A @ X2 @ B5 )
% 0.31/1.53 => ( P @ X2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_178_Un__assoc,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ C3 )
% 0.31/1.53 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_179_Un__absorb,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_180_Un__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) )
% 0.31/1.53 = ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_181_Un__Int__crazy,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) @ ( inf_inf @ ( set @ A ) @ C3 @ A2 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) @ ( sup_sup @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_182_Int__UNIV__left,axiom,
% 0.31/1.53 ! [A: $tType,B5: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
% 0.31/1.53 = B5 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_183_Int__UNIV__right,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_184_Int__Un__distrib,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.31/1.53 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ ( inf_inf @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_185_Un__Int__distrib,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_186_Un__left__absorb,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 = ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_187_Int__Un__distrib2,axiom,
% 0.31/1.53 ! [A: $tType,B5: set @ A,C3: set @ A,A2: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) @ A2 )
% 0.31/1.53 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ A2 ) @ ( inf_inf @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_188_Un__Int__distrib2,axiom,
% 0.31/1.53 ! [A: $tType,B5: set @ A,C3: set @ A,A2: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) @ A2 )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B5 @ A2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_189_Un__left__commute,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
% 0.31/1.53 = ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_190_type__definition_ORep__inverse,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( Abs @ ( Rep @ X ) )
% 0.31/1.53 = X ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_191_type__definition_OAbs__inverse,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( member @ A @ Y3 @ A2 )
% 0.31/1.53 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.31/1.53 = Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_192_type__definition_ORep__inject,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B,Y3: B] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( ( Rep @ X )
% 0.31/1.53 = ( Rep @ Y3 ) )
% 0.31/1.53 = ( X = Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_193_type__definition_ORep__induct,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A,P: A > $o] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( member @ A @ Y3 @ A2 )
% 0.31/1.53 => ( ! [X3: B] : ( P @ ( Rep @ X3 ) )
% 0.31/1.53 => ( P @ Y3 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_194_type__definition_OAbs__inject,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: A,Y3: A] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( member @ A @ X @ A2 )
% 0.31/1.53 => ( ( member @ A @ Y3 @ A2 )
% 0.31/1.53 => ( ( ( Abs @ X )
% 0.31/1.53 = ( Abs @ Y3 ) )
% 0.31/1.53 = ( X = Y3 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_195_type__definition_OAbs__induct,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,P: B > $o,X: B] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ! [Y5: A] :
% 0.31/1.53 ( ( member @ A @ Y5 @ A2 )
% 0.31/1.53 => ( P @ ( Abs @ Y5 ) ) )
% 0.31/1.53 => ( P @ X ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_196_type__definition_ORep__cases,axiom,
% 0.31/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y3: A] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ( ( member @ A @ Y3 @ A2 )
% 0.31/1.53 => ~ ! [X3: B] :
% 0.31/1.53 ( Y3
% 0.31/1.53 != ( Rep @ X3 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_197_type__definition_OAbs__cases,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
% 0.31/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.31/1.53 => ~ ! [Y5: A] :
% 0.31/1.53 ( ( X
% 0.31/1.53 = ( Abs @ Y5 ) )
% 0.31/1.53 => ~ ( member @ A @ Y5 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_198_type__definition_Ointro,axiom,
% 0.31/1.53 ! [B: $tType,A: $tType,Rep: B > A,A2: set @ A,Abs: A > B] :
% 0.31/1.53 ( ! [X3: B] : ( member @ A @ ( Rep @ X3 ) @ A2 )
% 0.31/1.53 => ( ! [X3: B] :
% 0.31/1.53 ( ( Abs @ ( Rep @ X3 ) )
% 0.31/1.53 = X3 )
% 0.31/1.53 => ( ! [Y5: A] :
% 0.31/1.53 ( ( member @ A @ Y5 @ A2 )
% 0.31/1.53 => ( ( Rep @ ( Abs @ Y5 ) )
% 0.31/1.53 = Y5 ) )
% 0.31/1.53 => ( type_definition @ B @ A @ Rep @ Abs @ A2 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_199_top__empty__eq,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( top_top @ ( A > $o ) )
% 0.31/1.53 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_200_hcard__0,axiom,
% 0.31/1.53 ( ( hF_Mirabelle_hcard @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.31/1.53 = ( zero_zero @ nat ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_201_hinsert__def,axiom,
% 0.31/1.53 ( hF_Mirabelle_hinsert
% 0.31/1.53 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert @ hF_Mirabelle_hf @ A5 @ ( hF_Mirabelle_hfset @ B4 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_202_insert__absorb2,axiom,
% 0.31/1.53 ! [A: $tType,X: A,A2: set @ A] :
% 0.31/1.53 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 0.31/1.53 = ( insert @ A @ X @ A2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_203_insert__iff,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.31/1.53 = ( ( A3 = B2 )
% 0.31/1.53 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_204_insertCI,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B5: set @ A,B2: A] :
% 0.31/1.53 ( ( ~ ( member @ A @ A3 @ B5 )
% 0.31/1.53 => ( A3 = B2 ) )
% 0.31/1.53 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_205_IntI,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ A2 )
% 0.31/1.53 => ( ( member @ A @ C @ B5 )
% 0.31/1.53 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_206_Int__iff,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 = ( ( member @ A @ C @ A2 )
% 0.31/1.53 & ( member @ A @ C @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_207_Int__insert__left__if0,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.31/1.53 ( ~ ( member @ A @ A3 @ C3 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_208_Int__insert__left__if1,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ C3 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.31/1.53 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_209_insert__inter__insert,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A2 ) @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_210_Int__insert__right__if0,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ~ ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_211_Int__insert__right__if1,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_212_Un__insert__left,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.31/1.53 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_213_Un__insert__right,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,A3: A,B5: set @ A] :
% 0.31/1.53 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_214_insert__UNIV,axiom,
% 0.31/1.53 ! [A: $tType,X: A] :
% 0.31/1.53 ( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
% 0.31/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_215_IntE,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 => ~ ( ( member @ A @ C @ A2 )
% 0.31/1.53 => ~ ( member @ A @ C @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_216_IntD1,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 => ( member @ A @ C @ A2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_217_IntD2,axiom,
% 0.31/1.53 ! [A: $tType,C: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 => ( member @ A @ C @ B5 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_218_Int__assoc,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) @ C3 )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_219_Int__absorb,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ A2 @ A2 )
% 0.31/1.53 = A2 ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_220_Int__commute,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) )
% 0.31/1.53 = ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_221_Int__insert__left,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,C3: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( ( member @ A @ A3 @ C3 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.31/1.53 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) )
% 0.31/1.53 & ( ~ ( member @ A @ A3 @ C3 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B5 ) @ C3 )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_222_Int__left__absorb,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_223_Int__insert__right,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ( ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) )
% 0.31/1.53 & ( ~ ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ( inf_inf @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B5 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ A2 @ B5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_224_Int__left__commute,axiom,
% 0.31/1.53 ! [A: $tType,A2: set @ A,B5: set @ A,C3: set @ A] :
% 0.31/1.53 ( ( inf_inf @ ( set @ A ) @ A2 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) )
% 0.31/1.53 = ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_225_mk__disjoint__insert,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ? [B8: set @ A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( insert @ A @ A3 @ B8 ) )
% 0.31/1.53 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_226_insert__commute,axiom,
% 0.31/1.53 ! [A: $tType,X: A,Y3: A,A2: set @ A] :
% 0.31/1.53 ( ( insert @ A @ X @ ( insert @ A @ Y3 @ A2 ) )
% 0.31/1.53 = ( insert @ A @ Y3 @ ( insert @ A @ X @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_227_insert__eq__iff,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A,B2: A,B5: set @ A] :
% 0.31/1.53 ( ~ ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ~ ( member @ A @ B2 @ B5 )
% 0.31/1.53 => ( ( ( insert @ A @ A3 @ A2 )
% 0.31/1.53 = ( insert @ A @ B2 @ B5 ) )
% 0.31/1.53 = ( ( ( A3 = B2 )
% 0.31/1.53 => ( A2 = B5 ) )
% 0.31/1.53 & ( ( A3 != B2 )
% 0.31/1.53 => ? [C4: set @ A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( insert @ A @ B2 @ C4 ) )
% 0.31/1.53 & ~ ( member @ A @ B2 @ C4 )
% 0.31/1.53 & ( B5
% 0.31/1.53 = ( insert @ A @ A3 @ C4 ) )
% 0.31/1.53 & ~ ( member @ A @ A3 @ C4 ) ) ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_228_insert__absorb,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ A2 )
% 0.31/1.53 => ( ( insert @ A @ A3 @ A2 )
% 0.31/1.53 = A2 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_229_insert__ident,axiom,
% 0.31/1.53 ! [A: $tType,X: A,A2: set @ A,B5: set @ A] :
% 0.31/1.53 ( ~ ( member @ A @ X @ A2 )
% 0.31/1.53 => ( ~ ( member @ A @ X @ B5 )
% 0.31/1.53 => ( ( ( insert @ A @ X @ A2 )
% 0.31/1.53 = ( insert @ A @ X @ B5 ) )
% 0.31/1.53 = ( A2 = B5 ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_230_Set_Oset__insert,axiom,
% 0.31/1.53 ! [A: $tType,X: A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ X @ A2 )
% 0.31/1.53 => ~ ! [B8: set @ A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( insert @ A @ X @ B8 ) )
% 0.31/1.53 => ( member @ A @ X @ B8 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_231_insertI2,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B5: set @ A,B2: A] :
% 0.31/1.53 ( ( member @ A @ A3 @ B5 )
% 0.31/1.53 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_232_insertI1,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B5: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B5 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_233_insertE,axiom,
% 0.31/1.53 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.31/1.53 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.31/1.53 => ( ( A3 != B2 )
% 0.31/1.53 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_234_hfset__hinsert,axiom,
% 0.31/1.53 ! [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ A3 @ B2 ) )
% 0.31/1.53 = ( insert @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_235_hcard__union__inter,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( plus_plus @ nat @ ( hF_Mirabelle_hcard @ ( sup_sup @ hF_Mirabelle_hf @ X @ Y3 ) ) @ ( hF_Mirabelle_hcard @ ( inf_inf @ hF_Mirabelle_hf @ X @ Y3 ) ) )
% 0.31/1.53 = ( plus_plus @ nat @ ( hF_Mirabelle_hcard @ X ) @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_236_hcard__hinsert__if,axiom,
% 0.31/1.53 ! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.31/1.53 ( ( ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.31/1.53 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.31/1.53 = ( hF_Mirabelle_hcard @ Y3 ) ) )
% 0.31/1.53 & ( ~ ( hF_Mirabelle_hmem @ X @ Y3 )
% 0.31/1.53 => ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
% 0.31/1.53 = ( suc @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_237_add__left__cancel,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel_semigroup_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.31/1.53 = ( plus_plus @ A @ A3 @ C ) )
% 0.31/1.53 = ( B2 = C ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_238_add__right__cancel,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel_semigroup_add @ A )
% 0.31/1.53 => ! [B2: A,A3: A,C: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.31/1.53 = ( plus_plus @ A @ C @ A3 ) )
% 0.31/1.53 = ( B2 = C ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_239_zero__eq__add__iff__both__eq__0,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( canoni770627133id_add @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( ( zero_zero @ A )
% 0.31/1.53 = ( plus_plus @ A @ X @ Y3 ) )
% 0.31/1.53 = ( ( X
% 0.31/1.53 = ( zero_zero @ A ) )
% 0.31/1.53 & ( Y3
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_240_add__eq__0__iff__both__eq__0,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( canoni770627133id_add @ A )
% 0.31/1.53 => ! [X: A,Y3: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ X @ Y3 )
% 0.31/1.53 = ( zero_zero @ A ) )
% 0.31/1.53 = ( ( X
% 0.31/1.53 = ( zero_zero @ A ) )
% 0.31/1.53 & ( Y3
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_241_add__cancel__right__right,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel1352612707id_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( A3
% 0.31/1.53 = ( plus_plus @ A @ A3 @ B2 ) )
% 0.31/1.53 = ( B2
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_242_add__cancel__right__left,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel1352612707id_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( A3
% 0.31/1.53 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.31/1.53 = ( B2
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_243_add__cancel__left__right,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel1352612707id_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.31/1.53 = A3 )
% 0.31/1.53 = ( B2
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_244_add__cancel__left__left,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( cancel1352612707id_add @ A )
% 0.31/1.53 => ! [B2: A,A3: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.31/1.53 = A3 )
% 0.31/1.53 = ( B2
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_245_double__zero__sym,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( linord219039673up_add @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( ( zero_zero @ A )
% 0.31/1.53 = ( plus_plus @ A @ A3 @ A3 ) )
% 0.31/1.53 = ( A3
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_246_double__zero,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( linord219039673up_add @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.31/1.53 = ( zero_zero @ A ) )
% 0.31/1.53 = ( A3
% 0.31/1.53 = ( zero_zero @ A ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_247_add_Oright__neutral,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( monoid_add @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_248_add_Oleft__neutral,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( monoid_add @ A )
% 0.31/1.53 => ! [A3: A] :
% 0.31/1.53 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.31/1.53 = A3 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_249_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( ab_semigroup_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C )
% 0.31/1.53 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_250_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( ordere779506340up_add @ A )
% 0.31/1.53 => ! [I: A,J: A,K: A,L: A] :
% 0.31/1.53 ( ( ( I = J )
% 0.31/1.53 & ( K = L ) )
% 0.31/1.53 => ( ( plus_plus @ A @ I @ K )
% 0.31/1.53 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_251_group__cancel_Oadd1,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( comm_monoid_add @ A )
% 0.31/1.53 => ! [A2: A,K: A,A3: A,B2: A] :
% 0.31/1.53 ( ( A2
% 0.31/1.53 = ( plus_plus @ A @ K @ A3 ) )
% 0.31/1.53 => ( ( plus_plus @ A @ A2 @ B2 )
% 0.31/1.53 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_252_group__cancel_Oadd2,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( comm_monoid_add @ A )
% 0.31/1.53 => ! [B5: A,K: A,B2: A,A3: A] :
% 0.31/1.53 ( ( B5
% 0.31/1.53 = ( plus_plus @ A @ K @ B2 ) )
% 0.31/1.53 => ( ( plus_plus @ A @ A3 @ B5 )
% 0.31/1.53 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_253_add_Oassoc,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( semigroup_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C )
% 0.31/1.53 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(fact_254_add_Oleft__cancel,axiom,
% 0.31/1.53 ! [A: $tType] :
% 0.31/1.53 ( ( group_add @ A )
% 0.31/1.53 => ! [A3: A,B2: A,C: A] :
% 0.31/1.53 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.31/1.53 = ( plus_plus @ A @ A3 @ C ) )
% 0.31/1.53 = ( B2 = C ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.31/1.53 bounded_lattice @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.31/1.53 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( bounded_lattice @ A9 )
% 0.31/1.53 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( bounded_lattice @ A9 )
% 0.31/1.53 => ( bounde1561333602nf_top @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( bounded_lattice @ A9 )
% 0.31/1.53 => ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( semilattice_sup @ A9 )
% 0.31/1.53 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( semilattice_inf @ A9 )
% 0.31/1.53 => ( semilattice_inf @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( distrib_lattice @ A9 )
% 0.31/1.53 => ( distrib_lattice @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( lattice @ A9 )
% 0.31/1.53 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/1.53 ! [A8: $tType,A9: $tType] :
% 0.31/1.53 ( ( top @ A9 )
% 0.31/1.53 => ( top @ ( A8 > A9 ) ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/1.53 canoni770627133id_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.31/1.53 ordere779506340up_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
% 0.31/1.53 cancel1352612707id_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.31/1.53 cancel_semigroup_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
% 0.31/1.53 semilattice_sup @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_4,axiom,
% 0.31/1.53 semilattice_inf @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_5,axiom,
% 0.31/1.53 distrib_lattice @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.31/1.53 ab_semigroup_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.31/1.53 comm_monoid_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.31/1.53 semigroup_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
% 0.31/1.53 monoid_add @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Lattices_Olattice_6,axiom,
% 0.31/1.53 lattice @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.31/1.53 zero @ nat ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_7,axiom,
% 0.31/1.53 ! [A8: $tType] : ( bounde1561333602nf_top @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_8,axiom,
% 0.31/1.53 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
% 0.31/1.53 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_10,axiom,
% 0.31/1.53 ! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_11,axiom,
% 0.31/1.53 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Lattices_Olattice_12,axiom,
% 0.31/1.53 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_Set_Oset___Orderings_Otop_13,axiom,
% 0.31/1.53 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_14,axiom,
% 0.31/1.53 bounde1561333602nf_top @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_15,axiom,
% 0.31/1.53 bounded_lattice_top @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_16,axiom,
% 0.31/1.53 semilattice_sup @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_17,axiom,
% 0.31/1.53 semilattice_inf @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_18,axiom,
% 0.31/1.53 distrib_lattice @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Lattices_Olattice_19,axiom,
% 0.31/1.53 lattice @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HOL_Obool___Orderings_Otop_20,axiom,
% 0.31/1.53 top @ $o ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__sup_21,axiom,
% 0.31/1.53 semilattice_sup @ hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_22,axiom,
% 0.31/1.53 semilattice_inf @ hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Odistrib__lattice_23,axiom,
% 0.31/1.53 distrib_lattice @ hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_24,axiom,
% 0.31/1.53 lattice @ hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_25,axiom,
% 0.31/1.53 zero @ hF_Mirabelle_hf ).
% 0.31/1.53
% 0.31/1.53 thf(conj_0,hypothesis,
% 0.31/1.53 hF_Mir859708136nction @ r ).
% 0.31/1.53
% 0.31/1.53 thf(conj_1,conjecture,
% 0.31/1.53 hF_Mir859708136nction @ ( hF_Mir165288108strict @ r @ x ) ).
% 0.31/1.53 ------------------------
% 0.31/1.56 % (816)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/183Mi)
% 0.31/1.56 % (817)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_504 for (2999ds/4Mi)
% 0.31/1.56 % (821)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_504 for (2999ds/275Mi)
% 0.31/1.56 % (817)Instruction limit reached!
% 0.31/1.56 % (817)------------------------------
% 0.31/1.56 % (817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.56 % (817)Termination reason: Unknown
% 0.31/1.56 % (817)Termination phase: shuffling
% 0.31/1.56
% 0.31/1.56 % (817)Memory used [KB]: 1407
% 0.31/1.56 % (817)Time elapsed: 0.003 s
% 0.31/1.56 % (817)Instructions burned: 6 (million)
% 0.31/1.56 % (817)------------------------------
% 0.31/1.56 % (817)------------------------------
% 0.31/1.56 % (818)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_504 for (2999ds/27Mi)
% 0.31/1.56 % (819)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/2Mi)
% 0.31/1.56 % (820)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_504 for (2999ds/2Mi)
% 0.31/1.56 % (822)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/18Mi)
% 0.31/1.57 % (819)Instruction limit reached!
% 0.31/1.57 % (819)------------------------------
% 0.31/1.57 % (819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.57 % (819)Termination reason: Unknown
% 0.31/1.57 % (819)Termination phase: shuffling
% 0.31/1.57
% 0.31/1.57 % (819)Memory used [KB]: 1279
% 0.31/1.57 % (819)Time elapsed: 0.004 s
% 0.31/1.57 % (819)Instructions burned: 3 (million)
% 0.31/1.57 % (819)------------------------------
% 0.31/1.57 % (819)------------------------------
% 0.31/1.57 % (820)Instruction limit reached!
% 0.31/1.57 % (820)------------------------------
% 0.31/1.57 % (820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.57 % (820)Termination reason: Unknown
% 0.31/1.57 % (820)Termination phase: shuffling
% 0.31/1.57
% 0.31/1.57 % (820)Memory used [KB]: 1279
% 0.31/1.57 % (820)Time elapsed: 0.004 s
% 0.31/1.57 % (820)Instructions burned: 3 (million)
% 0.31/1.57 % (820)------------------------------
% 0.31/1.57 % (820)------------------------------
% 0.31/1.57 % (823)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/3Mi)
% 0.31/1.57 % (823)Instruction limit reached!
% 0.31/1.57 % (823)------------------------------
% 0.31/1.57 % (823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.57 % (823)Termination reason: Unknown
% 0.31/1.57 % (823)Termination phase: shuffling
% 0.31/1.57
% 0.31/1.57 % (823)Memory used [KB]: 1407
% 0.31/1.57 % (823)Time elapsed: 0.002 s
% 0.31/1.57 % (823)Instructions burned: 3 (million)
% 0.31/1.57 % (823)------------------------------
% 0.31/1.57 % (823)------------------------------
% 0.31/1.57 % (821)Refutation not found, incomplete strategy
% 0.31/1.57 % (821)------------------------------
% 0.31/1.57 % (821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.57 % (821)Termination reason: Refutation not found, incomplete strategy
% 0.31/1.57
% 0.31/1.57
% 0.31/1.57 % (821)Memory used [KB]: 6268
% 0.31/1.57 % (821)Time elapsed: 0.014 s
% 0.31/1.57 % (821)Instructions burned: 40 (million)
% 0.31/1.57 % (821)------------------------------
% 0.31/1.57 % (821)------------------------------
% 0.31/1.58 % (822)Instruction limit reached!
% 0.31/1.58 % (822)------------------------------
% 0.31/1.58 % (822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.58 % (822)Termination reason: Unknown
% 0.31/1.58 % (822)Termination phase: Property scanning
% 0.31/1.58
% 0.31/1.58 % (822)Memory used [KB]: 1663
% 0.31/1.58 % (822)Time elapsed: 0.014 s
% 0.31/1.58 % (822)Instructions burned: 20 (million)
% 0.31/1.58 % (822)------------------------------
% 0.31/1.58 % (822)------------------------------
% 0.31/1.58 % (826)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_504 for (2999ds/3Mi)
% 0.31/1.58 % (826)Instruction limit reached!
% 0.31/1.58 % (826)------------------------------
% 0.31/1.58 % (826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.58 % (826)Termination reason: Unknown
% 0.31/1.58 % (826)Termination phase: shuffling
% 0.31/1.58
% 0.31/1.58 % (826)Memory used [KB]: 1407
% 0.31/1.58 % (826)Time elapsed: 0.002 s
% 0.31/1.58 % (826)Instructions burned: 3 (million)
% 0.31/1.58 % (826)------------------------------
% 0.31/1.58 % (826)------------------------------
% 0.31/1.58 % (827)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_504 for (2999ds/1041Mi)
% 0.31/1.58 % (818)Instruction limit reached!
% 0.31/1.58 % (818)------------------------------
% 0.31/1.58 % (818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.58 % (818)Termination reason: Unknown
% 0.31/1.58 % (818)Termination phase: Preprocessing 1
% 0.31/1.58
% 0.31/1.58 % (818)Memory used [KB]: 1663
% 0.31/1.58 % (818)Time elapsed: 0.018 s
% 0.31/1.58 % (818)Instructions burned: 28 (million)
% 0.31/1.58 % (818)------------------------------
% 0.31/1.58 % (818)------------------------------
% 0.31/1.58 % (824)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_504 for (2999ds/37Mi)
% 0.31/1.58 % (825)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_504 for (2999ds/15Mi)
% 0.31/1.59 % (829)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/16Mi)
% 0.31/1.59 % (825)Instruction limit reached!
% 0.31/1.59 % (825)------------------------------
% 0.31/1.59 % (825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.59 % (825)Termination reason: Unknown
% 0.31/1.59 % (825)Termination phase: shuffling
% 0.31/1.59
% 0.31/1.59 % (825)Memory used [KB]: 1663
% 0.31/1.59 % (825)Time elapsed: 0.012 s
% 0.31/1.59 % (825)Instructions burned: 15 (million)
% 0.31/1.59 % (825)------------------------------
% 0.31/1.59 % (825)------------------------------
% 0.31/1.59 % (829)Instruction limit reached!
% 0.31/1.59 % (829)------------------------------
% 0.31/1.59 % (829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.59 % (829)Termination reason: Unknown
% 0.31/1.59 % (829)Termination phase: Property scanning
% 0.31/1.59
% 0.31/1.59 % (829)Memory used [KB]: 1663
% 0.31/1.59 % (829)Time elapsed: 0.007 s
% 0.31/1.59 % (829)Instructions burned: 18 (million)
% 0.31/1.59 % (829)------------------------------
% 0.31/1.59 % (829)------------------------------
% 0.31/1.59 % (828)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/7Mi)
% 2.65/1.60 % (830)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/3Mi)
% 2.65/1.60 % (828)Instruction limit reached!
% 2.65/1.60 % (828)------------------------------
% 2.65/1.60 % (828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (828)Termination reason: Unknown
% 2.65/1.60 % (828)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (828)Memory used [KB]: 1407
% 2.65/1.60 % (828)Time elapsed: 0.007 s
% 2.65/1.60 % (828)Instructions burned: 8 (million)
% 2.65/1.60 % (828)------------------------------
% 2.65/1.60 % (828)------------------------------
% 2.65/1.60 % (830)Instruction limit reached!
% 2.65/1.60 % (830)------------------------------
% 2.65/1.60 % (830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (830)Termination reason: Unknown
% 2.65/1.60 % (830)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (830)Memory used [KB]: 1407
% 2.65/1.60 % (830)Time elapsed: 0.004 s
% 2.65/1.60 % (830)Instructions burned: 3 (million)
% 2.65/1.60 % (830)------------------------------
% 2.65/1.60 % (830)------------------------------
% 2.65/1.60 % (824)Instruction limit reached!
% 2.65/1.60 % (824)------------------------------
% 2.65/1.60 % (824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (824)Termination reason: Unknown
% 2.65/1.60 % (824)Termination phase: Property scanning
% 2.65/1.60
% 2.65/1.60 % (824)Memory used [KB]: 2046
% 2.65/1.60 % (824)Time elapsed: 0.021 s
% 2.65/1.60 % (824)Instructions burned: 39 (million)
% 2.65/1.60 % (824)------------------------------
% 2.65/1.60 % (824)------------------------------
% 2.65/1.60 % (831)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/3Mi)
% 2.65/1.60 % (831)Instruction limit reached!
% 2.65/1.60 % (831)------------------------------
% 2.65/1.60 % (831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (831)Termination reason: Unknown
% 2.65/1.60 % (831)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (831)Memory used [KB]: 1279
% 2.65/1.60 % (831)Time elapsed: 0.002 s
% 2.65/1.60 % (831)Instructions burned: 3 (million)
% 2.65/1.60 % (831)------------------------------
% 2.65/1.60 % (831)------------------------------
% 2.65/1.61 % (816)Instruction limit reached!
% 2.65/1.61 % (816)------------------------------
% 2.65/1.61 % (816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (816)Termination reason: Unknown
% 2.65/1.61 % (816)Termination phase: Saturation
% 2.65/1.61
% 2.65/1.61 % (816)Memory used [KB]: 7164
% 2.65/1.61 % (816)Time elapsed: 0.054 s
% 2.65/1.61 % (816)Instructions burned: 183 (million)
% 2.65/1.61 % (816)------------------------------
% 2.65/1.61 % (816)------------------------------
% 2.65/1.61 % (832)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/7Mi)
% 2.65/1.61 % (836)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/710Mi)
% 2.65/1.62 % (835)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/18Mi)
% 2.65/1.62 % (833)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/3Mi)
% 2.65/1.62 % (832)Instruction limit reached!
% 2.65/1.62 % (832)------------------------------
% 2.65/1.62 % (832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (832)Termination reason: Unknown
% 2.65/1.62 % (832)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (832)Memory used [KB]: 1407
% 2.65/1.62 % (832)Time elapsed: 0.007 s
% 2.65/1.62 % (832)Instructions burned: 8 (million)
% 2.65/1.62 % (832)------------------------------
% 2.65/1.62 % (832)------------------------------
% 2.65/1.62 % (834)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/4Mi)
% 2.65/1.62 % (833)Instruction limit reached!
% 2.65/1.62 % (833)------------------------------
% 2.65/1.62 % (833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (833)Termination reason: Unknown
% 2.65/1.62 % (833)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (833)Memory used [KB]: 1407
% 2.65/1.62 % (833)Time elapsed: 0.004 s
% 2.65/1.62 % (833)Instructions burned: 5 (million)
% 2.65/1.62 % (833)------------------------------
% 2.65/1.62 % (833)------------------------------
% 2.65/1.62 % (837)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_504 for (2999ds/6Mi)
% 2.65/1.62 % (834)Instruction limit reached!
% 2.65/1.62 % (834)------------------------------
% 2.65/1.62 % (834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (834)Termination reason: Unknown
% 2.65/1.62 % (834)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (834)Memory used [KB]: 1407
% 2.65/1.62 % (834)Time elapsed: 0.005 s
% 2.65/1.62 % (834)Instructions burned: 5 (million)
% 2.65/1.62 % (834)------------------------------
% 2.65/1.62 % (834)------------------------------
% 2.65/1.62 % (837)Instruction limit reached!
% 2.65/1.62 % (837)------------------------------
% 2.65/1.62 % (837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (837)Termination reason: Unknown
% 2.65/1.62 % (837)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (837)Memory used [KB]: 1407
% 2.65/1.62 % (837)Time elapsed: 0.003 s
% 2.65/1.62 % (837)Instructions burned: 6 (million)
% 2.65/1.62 % (837)------------------------------
% 2.65/1.62 % (837)------------------------------
% 2.65/1.62 % (835)Instruction limit reached!
% 2.65/1.62 % (835)------------------------------
% 2.65/1.62 % (835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (835)Termination reason: Unknown
% 2.65/1.62 % (835)Termination phase: Property scanning
% 2.65/1.62
% 2.65/1.62 % (835)Memory used [KB]: 1663
% 2.65/1.62 % (835)Time elapsed: 0.008 s
% 2.65/1.62 % (835)Instructions burned: 21 (million)
% 2.65/1.62 % (835)------------------------------
% 2.65/1.62 % (835)------------------------------
% 2.65/1.63 % (836)First to succeed.
% 2.65/1.63 % (836)Refutation found. Thanks to Tanya!
% 2.65/1.63 % SZS status Theorem for DTF2THF_504
% 2.65/1.63 % SZS output start Proof for DTF2THF_504
% 2.65/1.63 thf(type_def_6, type, hF_Mirabelle_hf: $tType).
% 2.65/1.63 thf(type_def_7, type, set: ($tType) > $tType).
% 2.65/1.63 thf(type_def_8, type, nat: $tType).
% 2.65/1.63 thf(func_def_0, type, hF_Mirabelle_hf: $tType).
% 2.65/1.63 thf(func_def_1, type, set: ($tType) > $tType).
% 2.65/1.63 thf(func_def_2, type, nat: $tType).
% 2.65/1.63 thf(func_def_3, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_4, type, type: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_5, type, zero: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_6, type, top: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_7, type, group_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_8, type, lattice: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_9, type, monoid_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_10, type, semigroup_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_11, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_12, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_13, type, distrib_lattice: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_14, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_15, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_16, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_17, type, bounded_lattice_top: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_18, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_19, type, linord219039673up_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_20, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_21, type, bounde1561333602nf_top: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_22, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.65/1.63 thf(func_def_23, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.65/1.63 thf(func_def_24, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.65/1.63 thf(func_def_25, type, hF_Mir1687042746ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_26, type, hF_Mirabelle_HF: set @ hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_27, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_28, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_29, type, hF_Mir569462966eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_30, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_31, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_32, type, hF_Mirabelle_hcard: hF_Mirabelle_hf > nat).
% 2.65/1.63 thf(func_def_33, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_34, type, hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat).
% 2.65/1.63 thf(func_def_35, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set @ hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_36, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_37, type, hF_Mir859708136nction: hF_Mirabelle_hf > $o).
% 2.65/1.63 thf(func_def_38, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_39, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.65/1.63 thf(func_def_40, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_41, type, hF_Mir1093797708lation: hF_Mirabelle_hf > $o).
% 2.65/1.63 thf(func_def_42, type, hF_Mir165288108strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_43, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_44, type, hF_Mirabelle_hsplit: !>[X0: $tType]:((hF_Mirabelle_hf > hF_Mirabelle_hf > X0) > hF_Mirabelle_hf > X0)).
% 2.65/1.63 thf(func_def_45, type, hF_Mir796903520_hpair: hF_Mirabelle_hf > $o).
% 2.65/1.63 thf(func_def_46, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.65/1.63 thf(func_def_47, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.65/1.63 thf(func_def_48, type, suc: nat > nat).
% 2.65/1.63 thf(func_def_49, type, top_top: !>[X0: $tType]:(X0)).
% 2.65/1.63 thf(func_def_50, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.65/1.63 thf(func_def_51, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.65/1.63 thf(func_def_52, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 2.65/1.63 thf(func_def_53, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.65/1.63 thf(func_def_54, type, r: hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_55, type, x: hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_69, type, sK0: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_70, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_71, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.65/1.63 thf(func_def_73, type, sK4: hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_74, type, sK5: hF_Mirabelle_hf).
% 2.65/1.63 thf(func_def_75, type, sK6: hF_Mirabelle_hf).
% 2.65/1.63 thf(f841,plain,(
% 2.65/1.63 $false),
% 2.65/1.63 inference(subsumption_resolution,[status(thm)],[f840,f807])).
% 2.65/1.63 thf(f807,plain,(
% 2.65/1.63 (sK4 != sK6)),
% 2.65/1.63 inference(equality_proxy_clausification,[status(thm)],[f805])).
% 2.65/1.63 thf(f805,plain,(
% 2.65/1.63 ((sK4 = sK6) = $false)),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f804])).
% 2.65/1.63 thf(f804,plain,(
% 2.65/1.63 ($false = ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK6) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = sK6)))),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f803])).
% 2.65/1.63 thf(f803,plain,(
% 2.65/1.63 (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ Y0) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = Y0))) @ sK6) = $false)),
% 2.65/1.63 inference(sigma_clausification,[status(thm)],[f801])).
% 2.65/1.63 thf(f801,plain,(
% 2.65/1.63 ((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ Y0) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = Y0)))) = $false)),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f800])).
% 2.65/1.63 thf(f800,plain,(
% 2.65/1.63 (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK4) @ (hF_Mir165288108strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ Y0) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = Y0))))) = $false)),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f799])).
% 2.65/1.63 thf(f799,plain,(
% 2.65/1.63 (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ sK4) @ (hF_Mir165288108strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = Y1)))))) @ sK5) = $false)),
% 2.65/1.63 inference(sigma_clausification,[status(thm)],[f798])).
% 2.65/1.63 thf(f798,plain,(
% 2.65/1.63 ((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ sK4) @ (hF_Mir165288108strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ (hF_Mir165288108strict @ r @ x)) => (sK4 = Y1))))))) = $false)),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f797])).
% 2.65/1.63 thf(f797,plain,(
% 2.65/1.63 ($false = ((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y0) @ (hF_Mir165288108strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ (hF_Mir165288108strict @ r @ x)) => (Y0 = Y2)))))))) @ sK4))),
% 2.65/1.63 inference(sigma_clausification,[status(thm)],[f796])).
% 2.65/1.63 thf(f796,plain,(
% 2.65/1.63 ((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y0) @ (hF_Mir165288108strict @ r @ x)) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ (hF_Mir165288108strict @ r @ x)) => (Y0 = Y2))))))))) != $true)),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f783])).
% 2.65/1.63 thf(f783,plain,(
% 2.65/1.63 ($true != ((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y1) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y3) @ Y0) => (Y1 = Y3)))))))))) @ (hF_Mir165288108strict @ r @ x)))),
% 2.65/1.63 inference(definition_unfolding,[status(thm)],[f780,f779])).
% 2.65/1.63 thf(f779,plain,(
% 2.65/1.63 (hF_Mir859708136nction = (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y1) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y3) @ Y0) => (Y1 = Y3)))))))))))),
% 2.65/1.63 inference(cnf_transformation,[status(thm)],[f381])).
% 2.65/1.63 thf(f381,plain,(
% 2.65/1.63 (hF_Mir859708136nction = (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y1) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y3) @ Y0) => (Y1 = Y3)))))))))))),
% 2.65/1.63 inference(fool_elimination,[status(thm)],[f380])).
% 2.65/1.63 thf(f380,plain,(
% 2.65/1.63 (hF_Mir859708136nction = (^[X0 : hF_Mirabelle_hf] : (! [X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X2) @ X0) => ! [X3 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X1 @ X3) @ X0) => (X2 = X3))))))),
% 2.65/1.63 inference(rectify,[status(thm)],[f2])).
% 2.65/1.63 thf(f2,axiom,(
% 2.65/1.63 (hF_Mir859708136nction = (^[X4 : hF_Mirabelle_hf] : (! [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X5 @ X6) @ X4) => ! [X7 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X5 @ X7) @ X4) => (X6 = X7))))))),
% 2.65/1.63 file('/export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2THF_504.p',fact_1_hfunction__def)).
% 2.65/1.63 thf(f780,plain,(
% 2.65/1.63 ((hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x)) != $true)),
% 2.65/1.63 inference(cnf_transformation,[status(thm)],[f764])).
% 2.65/1.63 thf(f764,plain,(
% 2.65/1.63 ((hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x)) != $true)),
% 2.65/1.63 inference(flattening,[status(thm)],[f549])).
% 2.65/1.63 thf(f549,plain,(
% 2.65/1.63 ~((hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x)) = $true)),
% 2.65/1.63 inference(fool_elimination,[status(thm)],[f548])).
% 2.65/1.63 thf(f548,plain,(
% 2.65/1.63 ~(hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x))),
% 2.65/1.63 inference(rectify,[status(thm)],[f300])).
% 2.65/1.63 thf(f300,negated_conjecture,(
% 2.65/1.63 ~(hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x))),
% 2.65/1.63 inference(negated_conjecture,[status(cth)],[f299])).
% 2.65/1.63 thf(f299,conjecture,(
% 2.65/1.63 (hF_Mir859708136nction @ (hF_Mir165288108strict @ r @ x))),
% 2.65/1.63 file('/export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2THF_504.p',conj_1)).
% 2.65/1.63 thf(f840,plain,(
% 2.65/1.63 (sK4 = sK6)),
% 2.65/1.63 inference(trivial_inequality_removal,[status(thm)],[f836])).
% 2.65/1.63 thf(f836,plain,(
% 2.65/1.63 ($false = $true) | (sK4 = sK6)),
% 2.65/1.63 inference(superposition,[status(thm)],[f825,f814])).
% 2.65/1.63 thf(f814,plain,(
% 2.65/1.63 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK6) @ r) = $true)),
% 2.65/1.63 inference(trivial_inequality_removal,[status(thm)],[f813])).
% 2.65/1.63 thf(f813,plain,(
% 2.65/1.63 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK6) @ r) = $true) | ($true != $true)),
% 2.65/1.63 inference(superposition,[status(thm)],[f776,f806])).
% 2.65/1.63 thf(f806,plain,(
% 2.65/1.63 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK6) @ (hF_Mir165288108strict @ r @ x)) = $true)),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f804])).
% 2.65/1.63 thf(f776,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))) | ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) )),
% 2.65/1.63 inference(cnf_transformation,[status(thm)],[f771])).
% 2.65/1.63 thf(f771,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((((($true = (hF_Mirabelle_hmem @ (sK0 @ X2 @ X0) @ X2)) & ((hF_Mirabelle_hpair @ (sK0 @ X2 @ X0) @ (sK1 @ X2 @ X0)) = X0)) & ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))) | ! [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X5 @ X2) != $true) | ((hF_Mirabelle_hpair @ X5 @ X6) != X0)) | ((hF_Mirabelle_hmem @ X0 @ X1) != $true)))),
% 2.65/1.63 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f769,f770])).
% 2.65/1.63 thf(f770,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X3 @ X2)) & ((hF_Mirabelle_hpair @ X3 @ X4) = X0)) => (($true = (hF_Mirabelle_hmem @ (sK0 @ X2 @ X0) @ X2)) & ((hF_Mirabelle_hpair @ (sK0 @ X2 @ X0) @ (sK1 @ X2 @ X0)) = X0)))),
% 2.65/1.63 introduced(definition,[],[choice_axiom])).
% 2.65/1.63 thf(f769,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X3 @ X2)) & ((hF_Mirabelle_hpair @ X3 @ X4) = X0)) & ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))) | ! [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X5 @ X2) != $true) | ((hF_Mirabelle_hpair @ X5 @ X6) != X0)) | ((hF_Mirabelle_hmem @ X0 @ X1) != $true)))),
% 2.65/1.63 inference(rectify,[status(thm)],[f768])).
% 2.65/1.63 thf(f768,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X3 @ X2)) & ((hF_Mirabelle_hpair @ X3 @ X4) = X0)) & ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))) | ! [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X3 @ X2)) | ((hF_Mirabelle_hpair @ X3 @ X4) != X0)) | ((hF_Mirabelle_hmem @ X0 @ X1) != $true)))),
% 2.65/1.63 inference(flattening,[status(thm)],[f767])).
% 2.65/1.63 thf(f767,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X3 @ X2)) & ((hF_Mirabelle_hpair @ X3 @ X4) = X0)) & ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) | ($true != (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2)))) & (($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))) | (! [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X3 @ X2)) | ((hF_Mirabelle_hpair @ X3 @ X4) != X0)) | ((hF_Mirabelle_hmem @ X0 @ X1) != $true))))),
% 2.65/1.63 inference(nnf_transformation,[status(thm)],[f749])).
% 2.65/1.63 thf(f749,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : ((? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (($true = (hF_Mirabelle_hmem @ X3 @ X2)) & ((hF_Mirabelle_hpair @ X3 @ X4) = X0)) & ((hF_Mirabelle_hmem @ X0 @ X1) = $true)) <=> ($true = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2))))),
% 2.65/1.63 inference(fool_elimination,[status(thm)],[f748])).
% 2.65/1.63 thf(f748,plain,(
% 2.65/1.63 ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf] : (? [X3 : hF_Mirabelle_hf,X4 : hF_Mirabelle_hf] : (((hF_Mirabelle_hpair @ X3 @ X4) = X0) & (hF_Mirabelle_hmem @ X3 @ X2)) & (hF_Mirabelle_hmem @ X0 @ X1) = (hF_Mirabelle_hmem @ X0 @ (hF_Mir165288108strict @ X1 @ X2)))),
% 2.65/1.63 inference(rectify,[status(thm)],[f3])).
% 2.65/1.63 thf(f3,axiom,(
% 2.65/1.63 ! [X8 : hF_Mirabelle_hf,X2 : hF_Mirabelle_hf,X9 : hF_Mirabelle_hf] : (? [X5 : hF_Mirabelle_hf,X6 : hF_Mirabelle_hf] : (((hF_Mirabelle_hpair @ X5 @ X6) = X8) & (hF_Mirabelle_hmem @ X5 @ X9)) & (hF_Mirabelle_hmem @ X8 @ X2) = (hF_Mirabelle_hmem @ X8 @ (hF_Mir165288108strict @ X2 @ X9)))),
% 2.65/1.63 file('/export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2THF_504.p',fact_2_hrestrict__iff)).
% 2.65/1.63 thf(f825,plain,(
% 2.65/1.63 ( ! [X0 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ X0) @ r) = $false) | (sK4 = X0)) )),
% 2.65/1.63 inference(trivial_inequality_removal,[status(thm)],[f817])).
% 2.65/1.63 thf(f817,plain,(
% 2.65/1.63 ( ! [X0 : hF_Mirabelle_hf] : (($false = $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ X0) @ r) = $false) | (sK4 = X0)) )),
% 2.65/1.63 inference(superposition,[status(thm)],[f795,f809])).
% 2.65/1.63 thf(f809,plain,(
% 2.65/1.63 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK4) @ r) = $true)),
% 2.65/1.63 inference(trivial_inequality_removal,[status(thm)],[f808])).
% 2.65/1.63 thf(f808,plain,(
% 2.65/1.63 ($true != $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK4) @ r) = $true)),
% 2.65/1.63 inference(superposition,[status(thm)],[f776,f802])).
% 2.65/1.63 thf(f802,plain,(
% 2.65/1.63 ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ sK5 @ sK4) @ (hF_Mir165288108strict @ r @ x)) = $true)),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f800])).
% 2.65/1.63 thf(f795,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) = $false) | (X1 = X3) | ($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X3) @ r))) )),
% 2.65/1.63 inference(equality_proxy_clausification,[status(thm)],[f794])).
% 2.65/1.63 thf(f794,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((X1 = X3) = $true) | ($false = (hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X3) @ r)) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) = $false)) )),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f793])).
% 2.65/1.63 thf(f793,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X3) @ r) => (X1 = X3)) = $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) = $false)) )),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f792])).
% 2.65/1.63 thf(f792,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X3 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) = $false) | (((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ Y0) @ r) => (X1 = Y0))) @ X3) = $true)) )),
% 2.65/1.63 inference(pi_clausification,[status(thm)],[f791])).
% 2.65/1.63 thf(f791,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ Y0) @ r) => (X1 = Y0)))) = $true) | ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) = $false)) )),
% 2.65/1.63 inference(binary_proxy_clausification,[status(thm)],[f790])).
% 2.65/1.63 thf(f790,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ X1) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ X2 @ Y0) @ r) => (X1 = Y0))))) = $true)) )),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f789])).
% 2.65/1.63 thf(f789,plain,(
% 2.65/1.63 ( ! [X2 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ X1) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ r) => (X1 = Y1)))))) @ X2) = $true)) )),
% 2.65/1.63 inference(pi_clausification,[status(thm)],[f788])).
% 2.65/1.63 thf(f788,plain,(
% 2.65/1.63 ( ! [X1 : hF_Mirabelle_hf] : (((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ X1) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y0 @ Y1) @ r) => (X1 = Y1))))))) = $true)) )),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f787])).
% 2.65/1.63 thf(f787,plain,(
% 2.65/1.63 ( ! [X1 : hF_Mirabelle_hf] : ((((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y0) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ r) => (Y0 = Y2)))))))) @ X1) = $true)) )),
% 2.65/1.63 inference(pi_clausification,[status(thm)],[f786])).
% 2.65/1.63 thf(f786,plain,(
% 2.65/1.63 ((!! @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y0) @ r) => (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y1 @ Y2) @ r) => (Y0 = Y2))))))))) = $true)),
% 2.65/1.63 inference(beta_eta_normalization,[status(thm)],[f784])).
% 2.65/1.63 thf(f784,plain,(
% 2.65/1.63 (((^[Y0 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y1 : hF_Mirabelle_hf]: (!! @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y1) @ Y0) => (!! @ hF_Mirabelle_hf @ (^[Y3 : hF_Mirabelle_hf]: ((hF_Mirabelle_hmem @ (hF_Mirabelle_hpair @ Y2 @ Y3) @ Y0) => (Y1 = Y3)))))))))) @ r) = $true)),
% 2.65/1.63 inference(definition_unfolding,[status(thm)],[f781,f779])).
% 2.65/1.63 thf(f781,plain,(
% 2.65/1.63 ((hF_Mir859708136nction @ r) = $true)),
% 2.65/1.63 inference(cnf_transformation,[status(thm)],[f313])).
% 2.65/1.63 thf(f313,plain,(
% 2.65/1.63 ((hF_Mir859708136nction @ r) = $true)),
% 2.65/1.63 inference(fool_elimination,[status(thm)],[f312])).
% 2.65/1.63 thf(f312,plain,(
% 2.65/1.63 (hF_Mir859708136nction @ r)),
% 2.65/1.63 inference(rectify,[status(thm)],[f298])).
% 2.65/1.63 thf(f298,axiom,(
% 2.65/1.63 (hF_Mir859708136nction @ r)),
% 2.65/1.63 file('/export/starexec/sandbox2/tmp/tmp.JrhNd7SFCY/DTF2THF_504.p',conj_0)).
% 2.65/1.63 % SZS output end Proof for DTF2THF_504
% 2.65/1.63 % (836)------------------------------
% 2.65/1.63 % (836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.63 % (836)Termination reason: Refutation
% 2.65/1.63
% 2.65/1.63 % (836)Memory used [KB]: 6268
% 2.65/1.63 % (836)Time elapsed: 0.014 s
% 2.65/1.63 % (836)Instructions burned: 37 (million)
% 2.65/1.63 % (836)------------------------------
% 2.65/1.63 % (836)------------------------------
% 2.65/1.63 % (815)Success in time 0.085 s
%------------------------------------------------------------------------------