%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP075^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n026.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 2.68s 1.59s
% Output : Refutation 2.68s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP075^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n026.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 08:25:25 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.45 ---- Original DTF file ---
% 0.20/0.45 thf(spec,logic,$$dhol).
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % File : ITP075^2 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.45 % Domain : Interactive Theorem Proving
% 0.20/0.45 % Problem : Sledgehammer HF problem prob_754__5337380_1
% 0.20/0.45 % Version : Especial.
% 0.20/0.45 % English :
% 0.20/0.45
% 0.20/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.45 % Source : [Des21]
% 0.20/0.45 % Names : HF/prob_754__5337380_1 [Des21]
% 0.20/0.45
% 0.20/0.45 % Status : Theorem
% 0.20/0.45 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.20/0.45 % Syntax : Number of formulae : 364 ( 179 unt; 54 typ; 0 def)
% 0.20/0.45 % Number of atoms : 663 ( 295 equ; 0 cnn)
% 0.20/0.45 % Maximal formula atoms : 4 ( 2 avg)
% 0.20/0.45 % Number of connectives : 2460 ( 83 ~; 5 |; 42 &;2097 @)
% 0.20/0.45 % ( 0 <=>; 233 =>; 0 <=; 0 <~>)
% 0.20/0.45 % Maximal formula depth : 15 ( 5 avg)
% 0.20/0.45 % Number of types : 4 ( 3 usr)
% 0.20/0.45 % Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% 0.20/0.45 % Number of symbols : 54 ( 51 usr; 4 con; 0-4 aty)
% 0.20/0.45 % Number of variables : 835 ( 114 ^; 661 !; 29 ?; 835 :)
% 0.20/0.45 % ( 31 !>; 0 ?*; 0 @-; 0 @+)
% 0.20/0.45 % SPC : TH1_THM_EQU_NAR
% 0.20/0.45
% 0.20/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:23:06.328
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % Could-be-implicit typings (4)
% 0.20/0.45 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 0.20/0.45 hF_Mirabelle_hf: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_t_Set_Oset,type,
% 0.20/0.45 set: $tType > $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_t_Nat_Onat,type,
% 0.20/0.45 nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_t_Int_Oint,type,
% 0.20/0.45 int: $tType ).
% 0.20/0.45
% 0.20/0.45 % Explicit typings (50)
% 0.20/0.45 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.20/0.45 bounded_lattice:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_HOL_Otype,type,
% 0.20/0.45 type:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ozero,type,
% 0.20/0.45 zero:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ominus,type,
% 0.20/0.45 minus:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.20/0.45 bot:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ogroup__add,type,
% 0.20/0.45 group_add:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Lattices_Olattice,type,
% 0.20/0.45 lattice:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Rings_Osemiring__1,type,
% 0.20/0.45 semiring_1:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.20/0.45 semiring_char_0:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.20/0.45 order_bot:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.20/0.45 comm_monoid_diff:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.20/0.45 semilattice_inf:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.20/0.45 ordered_ab_group_add:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.20/0.45 bounded_lattice_bot:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 0.20/0.45 cancel1352612707id_add:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.20/0.45 cancel146912293up_add:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.20/0.45 linord1659791738miring:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.20/0.45 canoni770627133id_add:
% 0.20/0.45 !>[A: $tType] : $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.20/0.45 minus_minus:
% 0.20/0.45 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.20/0.45 zero_zero:
% 0.20/0.45 !>[A: $tType] : A ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBall,type,
% 0.20/0.45 hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBex,type,
% 0.20/0.45 hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 0.20/0.45 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 0.20/0.45 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 0.20/0.46 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 0.20/0.46 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 0.20/0.46 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 0.20/0.46 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 0.20/0.46 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard,type,
% 0.20/0.46 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 0.20/0.46 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 0.20/0.46 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 0.20/0.46 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 0.20/0.46 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 0.20/0.46 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 0.20/0.46 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 0.20/0.46 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 0.20/0.46 hF_Mirabelle_hsplit:
% 0.20/0.46 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_HOL_OThe,type,
% 0.20/0.46 the:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_If,type,
% 0.20/0.46 if:
% 0.20/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.20/0.46 inf_inf:
% 0.20/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.20/0.46 semiring_1_of_nat:
% 0.20/0.46 !>[A: $tType] : ( nat > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.20/0.46 bot_bot:
% 0.20/0.46 !>[A: $tType] : A ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord_OLeast,type,
% 0.20/0.46 least:
% 0.20/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.20/0.46 ord_less:
% 0.20/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect,type,
% 0.20/0.46 collect:
% 0.20/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.20/0.46 is_empty:
% 0.20/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member,type,
% 0.20/0.46 member:
% 0.20/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_A,type,
% 0.20/0.46 a: hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_P,type,
% 0.20/0.46 p: $o ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (254)
% 0.20/0.46 thf(fact_0_hf__equalityI,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.20/0.46 = ( hF_Mirabelle_hmem @ X @ B ) )
% 0.20/0.46 => ( A2 = B ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_equalityI
% 0.20/0.46 thf(fact_1_hbexI,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ X2 )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 0.20/0.46 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hbexI
% 0.20/0.46 thf(fact_2_hbexE,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_HBex @ A3 @ P )
% 0.20/0.46 => ~ ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ A3 )
% 0.20/0.46 => ~ ( P @ X ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hbexE
% 0.20/0.46 thf(fact_3_hf__ext,axiom,
% 0.20/0.46 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.20/0.46 = ( hF_Mirabelle_hmem @ X3 @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_ext
% 0.20/0.46 thf(fact_4_hmem__ne,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.20/0.46 => ( X2 != Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_ne
% 0.20/0.46 thf(fact_5_HBex__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_HBex
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ? [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X3 @ A5 )
% 0.20/0.46 & ( P2 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HBex_def
% 0.20/0.46 thf(fact_6_hbex__cong,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( A3 = A6 )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ A6 )
% 0.20/0.46 => ( ( P @ X )
% 0.20/0.46 = ( P3 @ X ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_HBex @ A3 @ P )
% 0.20/0.46 = ( hF_Mirabelle_HBex @ A6 @ P3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hbex_cong
% 0.20/0.46 thf(fact_7_rev__hbexI,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 0.20/0.46 => ( ( P @ X2 )
% 0.20/0.46 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % rev_hbexI
% 0.20/0.46 thf(fact_8_replacement,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U @ X2 )
% 0.20/0.46 => ( ( R @ U @ V )
% 0.20/0.46 => ( ( R @ U @ V2 )
% 0.20/0.46 => ( V2 = V ) ) ) )
% 0.20/0.46 => ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [V3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.20/0.46 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % replacement
% 0.20/0.46 thf(fact_9_binary__union,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.20/0.46 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % binary_union
% 0.20/0.46 thf(fact_10_hmem__not__sym,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ Y2 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_not_sym
% 0.20/0.46 thf(fact_11_union__of__set,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ X2 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % union_of_set
% 0.20/0.46 thf(fact_12_comprehension,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.20/0.46 & ( P @ U3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % comprehension
% 0.20/0.46 thf(fact_13_hmem__not__refl,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ X2 @ X2 ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_not_refl
% 0.20/0.46 thf(fact_14_replacement__fun,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ? [Z2: hF_Mirabelle_hf] :
% 0.20/0.46 ! [V3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.20/0.46 & ( V3
% 0.20/0.46 = ( F @ U2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % replacement_fun
% 0.20/0.46 thf(fact_15_HF__Mirabelle__fsbjehakzm_ObexCI,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_HBall @ A3
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( P @ X3 ) )
% 0.20/0.46 => ( P @ A2 ) )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 0.20/0.46 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HF_Mirabelle_fsbjehakzm.bexCI
% 0.20/0.46 thf(fact_16_hballI,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ A3 )
% 0.20/0.46 => ( P @ X ) )
% 0.20/0.46 => ( hF_Mirabelle_HBall @ A3 @ P ) ) ).
% 0.20/0.46
% 0.20/0.46 % hballI
% 0.20/0.46 thf(fact_17_hball__triv,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: $o] :
% 0.20/0.46 ( ( hF_Mirabelle_HBall @ A3
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] : P )
% 0.20/0.46 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.20/0.46 => P ) ) ).
% 0.20/0.46
% 0.20/0.46 % hball_triv
% 0.20/0.46 thf(fact_18_PrimReplace__iff,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U @ A3 )
% 0.20/0.46 => ( ( R @ U @ V )
% 0.20/0.46 => ( ( R @ U @ V2 )
% 0.20/0.46 => ( V2 = V ) ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir569462966eplace @ A3 @ R ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 0.20/0.46 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % PrimReplace_iff
% 0.20/0.46 thf(fact_19_HCollect__iff,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mir1687042746ollect @ P @ A3 ) )
% 0.20/0.46 = ( ( P @ X2 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ X2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollect_iff
% 0.20/0.46 thf(fact_20_Replace__iff,axiom,
% 0.20/0.46 ! [V4: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A3 @ R ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 0.20/0.46 & ( R @ U2 @ V4 )
% 0.20/0.46 & ! [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( R @ U2 @ Y3 )
% 0.20/0.46 => ( Y3 = V4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_iff
% 0.20/0.46 thf(fact_21_HUnion__iff,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HUnion @ A3 ) )
% 0.20/0.46 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A3 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_iff
% 0.20/0.46 thf(fact_22_hballE,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_HBall @ A3 @ P )
% 0.20/0.46 => ( ~ ( P @ X2 )
% 0.20/0.46 => ~ ( hF_Mirabelle_hmem @ X2 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hballE
% 0.20/0.46 thf(fact_23_hbspec,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_HBall @ A3 @ P )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 0.20/0.46 => ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hbspec
% 0.20/0.46 thf(fact_24_HCollectE,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mir1687042746ollect @ P @ A3 ) )
% 0.20/0.46 => ~ ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 0.20/0.46 => ~ ( P @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollectE
% 0.20/0.46 thf(fact_25_HCollectI,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 0.20/0.46 => ( ( P @ A2 )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ A2 @ ( hF_Mir1687042746ollect @ P @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollectI
% 0.20/0.46 thf(fact_26_hball__cong,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( A3 = A6 )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ A6 )
% 0.20/0.46 => ( ( P @ X )
% 0.20/0.46 = ( P3 @ X ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_HBall @ A3 @ P )
% 0.20/0.46 = ( hF_Mirabelle_HBall @ A6 @ P3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hball_cong
% 0.20/0.46 thf(fact_27_hmem__Sup__ne,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.20/0.46 => ( ( hF_Mirabelle_HUnion @ X2 )
% 0.20/0.46 != Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_Sup_ne
% 0.20/0.46 thf(fact_28_Replace__cong,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( A3 = B3 )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.20/0.46 => ( ( P @ X @ Y4 )
% 0.20/0.46 = ( Q @ X @ Y4 ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_Replace @ A3 @ P )
% 0.20/0.46 = ( hF_Mirabelle_Replace @ B3 @ Q ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_cong
% 0.20/0.46 thf(fact_29_HBall__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_HBall
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ! [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X3 @ A5 )
% 0.20/0.46 => ( P2 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HBall_def
% 0.20/0.46 thf(fact_30_Replace__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_Replace
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( hF_Mir569462966eplace @ A5
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ? [Z3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( R2 @ X3 @ Z3 )
% 0.20/0.46 & ! [Aa: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( R2 @ X3 @ Aa )
% 0.20/0.46 => ( Aa = Z3 ) ) )
% 0.20/0.46 & ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_def
% 0.20/0.46 thf(fact_31_RepFun__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_RepFun
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( hF_Mirabelle_Replace @ A5
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( Y3
% 0.20/0.46 = ( F2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_def
% 0.20/0.46 thf(fact_32_minus__hf__def,axiom,
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf )
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.20/0.46 ( hF_Mir1687042746ollect
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.20/0.46 @ A5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % minus_hf_def
% 0.20/0.46 thf(fact_33_inf__hf__def,axiom,
% 0.20/0.46 ( ( inf_inf @ hF_Mirabelle_hf )
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.20/0.46 ( hF_Mir1687042746ollect
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ B2 )
% 0.20/0.46 @ A4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_hf_def
% 0.20/0.46 thf(fact_34_HInter__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_HInter
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( hF_Mir1687042746ollect
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ! [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ X3 @ Y3 ) )
% 0.20/0.46 @ ( hF_Mirabelle_HUnion @ A5 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_def
% 0.20/0.46 thf(fact_35_RepFun__iff,axiom,
% 0.20/0.46 ! [V4: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A3 @ F ) )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 0.20/0.46 & ( V4
% 0.20/0.46 = ( F @ U2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_iff
% 0.20/0.46 thf(fact_36_HCollect__hempty,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollect_hempty
% 0.20/0.46 thf(fact_37_Replace__0,axiom,
% 0.20/0.46 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % Replace_0
% 0.20/0.46 thf(fact_38_hmem__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_hmem
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A4 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_def
% 0.20/0.46 thf(fact_39_triv__RepFun,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ A3
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] : X3 )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % triv_RepFun
% 0.20/0.46 thf(fact_40_hinter__iff,axiom,
% 0.20/0.46 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_inf @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_iff
% 0.20/0.46 thf(fact_41_hdiff__iff,axiom,
% 0.20/0.46 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U4 @ ( minus_minus @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.20/0.46 & ~ ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_iff
% 0.20/0.46 thf(fact_42_hinter__hempty__right,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( inf_inf @ hF_Mirabelle_hf @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hempty_right
% 0.20/0.46 thf(fact_43_hinter__hempty__left,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A3 )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hempty_left
% 0.20/0.46 thf(fact_44_zero__hdiff,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X2 )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_hdiff
% 0.20/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.20/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.20/0.46 = ( P @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % mem_Collect_eq
% 0.20/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_mem_eq
% 0.20/0.46 thf(fact_47_Collect__cong,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 = ( Q @ X ) )
% 0.20/0.46 => ( ( collect @ A @ P )
% 0.20/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_cong
% 0.20/0.46 thf(fact_48_ext,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType,F: A > B5,G: A > B5] :
% 0.20/0.46 ( ! [X: A] :
% 0.20/0.46 ( ( F @ X )
% 0.20/0.46 = ( G @ X ) )
% 0.20/0.46 => ( F = G ) ) ).
% 0.20/0.46
% 0.20/0.46 % ext
% 0.20/0.46 thf(fact_49_hdiff__zero,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ X2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = X2 ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_zero
% 0.20/0.46 thf(fact_50_RepFun__0,axiom,
% 0.20/0.46 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_0
% 0.20/0.46 thf(fact_51_HUnion__hempty,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_hempty
% 0.20/0.46 thf(fact_52_HInter__hempty,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_hempty
% 0.20/0.46 thf(fact_53_HInter__iff,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HInter @ A3 ) )
% 0.20/0.46 = ( ! [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A3 )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_iff
% 0.20/0.46 thf(fact_54_foundation,axiom,
% 0.20/0.46 ! [Z4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z4
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ? [W: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ W @ Z4 )
% 0.20/0.46 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z4 )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % foundation
% 0.20/0.46 thf(fact_55_hemptyE,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hemptyE
% 0.20/0.46 thf(fact_56_hempty__iff,axiom,
% 0.20/0.46 ! [Z4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z4
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( ! [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ X3 @ Z4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hempty_iff
% 0.20/0.46 thf(fact_57_hmem__hempty,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_hempty
% 0.20/0.46 thf(fact_58_RepFun__cong,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A3 = B3 )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( hF_Mirabelle_RepFun @ A3 @ F )
% 0.20/0.46 = ( hF_Mirabelle_RepFun @ B3 @ G ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_cong
% 0.20/0.46 thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel1352612707id_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.20/0.46 thf(fact_60_diff__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel1352612707id_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_zero
% 0.20/0.46 thf(fact_61_zero__diff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( comm_monoid_diff @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_diff
% 0.20/0.46 thf(fact_62_diff__0__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_0_right
% 0.20/0.46 thf(fact_63_diff__self,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A2 @ A2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_self
% 0.20/0.46 thf(fact_64_inf__right__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X2: A,Y2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 )
% 0.20/0.46 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_right_idem
% 0.20/0.46 thf(fact_65_inf_Oright__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ B )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.right_idem
% 0.20/0.46 thf(fact_66_inf__left__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X2: A,Y2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
% 0.20/0.46 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_left_idem
% 0.20/0.46 thf(fact_67_minus__apply,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType] :
% 0.20/0.46 ( ( minus @ B5 )
% 0.20/0.46 => ( ( minus_minus @ ( A > B5 ) )
% 0.20/0.46 = ( ^ [A5: A > B5,B4: A > B5,X3: A] : ( minus_minus @ B5 @ ( A5 @ X3 ) @ ( B4 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % minus_apply
% 0.20/0.46 thf(fact_68_inf__apply,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ B5 )
% 0.20/0.46 => ( ( inf_inf @ ( A > B5 ) )
% 0.20/0.46 = ( ^ [F2: A > B5,G2: A > B5,X3: A] : ( inf_inf @ B5 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_apply
% 0.20/0.46 thf(fact_69_inf_Oidem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ A2 @ A2 )
% 0.20/0.46 = A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.idem
% 0.20/0.46 thf(fact_70_inf__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ X2 )
% 0.20/0.46 = X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_idem
% 0.20/0.46 thf(fact_71_inf_Oleft__idem,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B ) )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.left_idem
% 0.20/0.46 thf(fact_72_hfset__hdiff,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hfset @ ( minus_minus @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hfset @ X2 ) @ ( hF_Mirabelle_hfset @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hfset_hdiff
% 0.20/0.46 thf(fact_73_zero__reorient,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( zero @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( ( zero_zero @ A )
% 0.20/0.46 = X2 )
% 0.20/0.46 = ( X2
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_reorient
% 0.20/0.46 thf(fact_74_fun__diff__def,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType] :
% 0.20/0.46 ( ( minus @ B5 )
% 0.20/0.46 => ( ( minus_minus @ ( A > B5 ) )
% 0.20/0.46 = ( ^ [A5: A > B5,B4: A > B5,X3: A] : ( minus_minus @ B5 @ ( A5 @ X3 ) @ ( B4 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fun_diff_def
% 0.20/0.46 thf(fact_75_diff__eq__diff__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.20/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.20/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.20/0.46 => ( ( A2 = B )
% 0.20/0.46 = ( C = D ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_eq_diff_eq
% 0.20/0.46 thf(fact_76_diff__right__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( cancel146912293up_add @ A )
% 0.20/0.46 => ! [A2: A,C: A,B: A] :
% 0.20/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 0.20/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_right_commute
% 0.20/0.46 thf(fact_77_inf__sup__aci_I4_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X2: A,Y2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
% 0.20/0.46 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(4)
% 0.20/0.46 thf(fact_78_inf__sup__aci_I3_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X2: A,Y2: A,Z4: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) )
% 0.20/0.46 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(3)
% 0.20/0.46 thf(fact_79_inf__sup__aci_I2_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ! [X2: A,Y2: A,Z4: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z4 )
% 0.20/0.46 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(2)
% 0.20/0.46 thf(fact_80_inf__sup__aci_I1_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( lattice @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_sup_aci(1)
% 0.20/0.46 thf(fact_81_inf__fun__def,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ B5 )
% 0.20/0.46 => ( ( inf_inf @ ( A > B5 ) )
% 0.20/0.46 = ( ^ [F2: A > B5,G2: A > B5,X3: A] : ( inf_inf @ B5 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_fun_def
% 0.20/0.46 thf(fact_82_boolean__algebra__cancel_Oinf1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A3: A,K: A,A2: A,B: A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( inf_inf @ A @ K @ A2 ) )
% 0.20/0.46 => ( ( inf_inf @ A @ A3 @ B )
% 0.20/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.inf1
% 0.20/0.46 thf(fact_83_boolean__algebra__cancel_Oinf2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B3: A,K: A,B: A,A2: A] :
% 0.20/0.46 ( ( B3
% 0.20/0.46 = ( inf_inf @ A @ K @ B ) )
% 0.20/0.46 => ( ( inf_inf @ A @ A2 @ B3 )
% 0.20/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % boolean_algebra_cancel.inf2
% 0.20/0.46 thf(fact_84_inf_Oassoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ C )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.assoc
% 0.20/0.46 thf(fact_85_inf__assoc,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X2: A,Y2: A,Z4: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z4 )
% 0.20/0.46 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_assoc
% 0.20/0.46 thf(fact_86_inf_Ocommute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [A4: A,B2: A] : ( inf_inf @ A @ B2 @ A4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.commute
% 0.20/0.46 thf(fact_87_inf__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ( ( inf_inf @ A )
% 0.20/0.46 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_commute
% 0.20/0.46 thf(fact_88_inf_Oleft__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B: A,A2: A,C: A] :
% 0.20/0.46 ( ( inf_inf @ A @ B @ ( inf_inf @ A @ A2 @ C ) )
% 0.20/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.left_commute
% 0.20/0.46 thf(fact_89_inf__left__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [X2: A,Y2: A,Z4: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) )
% 0.20/0.46 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_left_commute
% 0.20/0.46 thf(fact_90_eq__iff__diff__eq__0,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( group_add @ A )
% 0.20/0.46 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.20/0.46 = ( ^ [A4: A,B2: A] :
% 0.20/0.46 ( ( minus_minus @ A @ A4 @ B2 )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % eq_iff_diff_eq_0
% 0.20/0.46 thf(fact_91_HInter__hinsert,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A3 ) )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HInter @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HInter_hinsert
% 0.20/0.46 thf(fact_92_fun__cong__unused__0,axiom,
% 0.20/0.46 ! [A: $tType,B5: $tType,C2: $tType] :
% 0.20/0.46 ( ( zero @ B5 )
% 0.20/0.46 => ! [F: ( A > B5 ) > C2,G: C2] :
% 0.20/0.46 ( ( F
% 0.20/0.46 = ( ^ [X3: A > B5] : G ) )
% 0.20/0.46 => ( ( F
% 0.20/0.46 @ ^ [X3: A] : ( zero_zero @ B5 ) )
% 0.20/0.46 = G ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % fun_cong_unused_0
% 0.20/0.46 thf(fact_93_HCollect__def,axiom,
% 0.20/0.46 ( hF_Mir1687042746ollect
% 0.20/0.46 = ( ^ [P2: hF_Mirabelle_hf > $o,A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( the @ hF_Mirabelle_hf
% 0.20/0.46 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.20/0.46 = ( ( P2 @ U2 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U2 @ A5 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HCollect_def
% 0.20/0.46 thf(fact_94_HF__hfset,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % HF_hfset
% 0.20/0.46 thf(fact_95_zero__notin__hpair,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ~ ( hF_Mirabelle_hmem @ ( zero_zero @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_notin_hpair
% 0.20/0.46 thf(fact_96_Abs__hf__0,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % Abs_hf_0
% 0.20/0.46 thf(fact_97_hfset__0,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_hfset @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hfset_0
% 0.20/0.46 thf(fact_98_hpair__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 = ( hF_Mirabelle_hpair @ A7 @ B6 ) )
% 0.20/0.46 = ( ( A2 = A7 )
% 0.20/0.46 & ( B = B6 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_iff
% 0.20/0.46 thf(fact_99_inf__bot__right,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bounded_lattice_bot @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ X2 @ ( bot_bot @ A ) )
% 0.20/0.46 = ( bot_bot @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_bot_right
% 0.20/0.46 thf(fact_100_inf__bot__left,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bounded_lattice_bot @ A )
% 0.20/0.46 => ! [X2: A] :
% 0.20/0.46 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X2 )
% 0.20/0.46 = ( bot_bot @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_bot_left
% 0.20/0.46 thf(fact_101_hmem__hinsert,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.20/0.46 = ( ( A2 = B )
% 0.20/0.46 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hmem_hinsert
% 0.20/0.46 thf(fact_102_singleton__eq__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.20/0.46 = ( A2 = B ) ) ).
% 0.20/0.46
% 0.20/0.46 % singleton_eq_iff
% 0.20/0.46 thf(fact_103_RepFun__hinsert,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % RepFun_hinsert
% 0.20/0.46 thf(fact_104_hpair__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_hpair
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_def
% 0.20/0.46 thf(fact_105_hpair__def_H,axiom,
% 0.20/0.46 ( hF_Mirabelle_hpair
% 0.20/0.46 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_def'
% 0.20/0.46 thf(fact_106_Zero__hf__def,axiom,
% 0.20/0.46 ( ( zero_zero @ hF_Mirabelle_hf )
% 0.20/0.46 = ( hF_Mirabelle_HF @ ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Zero_hf_def
% 0.20/0.46 thf(fact_107_hpair__neq__snd,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 != B ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_neq_snd
% 0.20/0.46 thf(fact_108_hpair__neq__fst,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 != A2 ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_neq_fst
% 0.20/0.46 thf(fact_109_hpair__inject,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 0.20/0.46 = ( hF_Mirabelle_hpair @ A7 @ B6 ) )
% 0.20/0.46 => ~ ( ( A2 = A7 )
% 0.20/0.46 => ( B != B6 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_inject
% 0.20/0.46 thf(fact_110_hinsert__commute,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hinsert @ X2 @ ( hF_Mirabelle_hinsert @ Y2 @ Z4 ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ Y2 @ ( hF_Mirabelle_hinsert @ X2 @ Z4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_commute
% 0.20/0.46 thf(fact_111_hinsert__iff,axiom,
% 0.20/0.46 ! [Z4: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Z4
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ Y2 @ X2 ) )
% 0.20/0.46 = ( ! [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ Z4 )
% 0.20/0.46 = ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.20/0.46 | ( U2 = Y2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_iff
% 0.20/0.46 thf(fact_112_hinsert__nonempty,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hinsert @ A2 @ A3 )
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_nonempty
% 0.20/0.46 thf(fact_113_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 0.20/0.46 = ( ( ( A2 = C )
% 0.20/0.46 & ( B = D ) )
% 0.20/0.46 | ( ( A2 = D )
% 0.20/0.46 & ( B = C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HF_Mirabelle_fsbjehakzm.doubleton_eq_iff
% 0.20/0.46 thf(fact_114_hf__induct__ax,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ! [Y4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ Y4 )
% 0.20/0.46 => ( P @ ( hF_Mirabelle_hinsert @ Y4 @ X ) ) ) )
% 0.20/0.46 => ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_induct_ax
% 0.20/0.46 thf(fact_115_hpair__nonzero,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hpair @ X2 @ Y2 )
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % hpair_nonzero
% 0.20/0.46 thf(fact_116_hf__cases,axiom,
% 0.20/0.46 ! [Y2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Y2
% 0.20/0.46 != ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ~ ! [A8: hF_Mirabelle_hf,B7: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( Y2
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ A8 @ B7 ) )
% 0.20/0.46 => ( hF_Mirabelle_hmem @ A8 @ B7 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_cases
% 0.20/0.46 thf(fact_117_hf__induct,axiom,
% 0.20/0.46 ! [P: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( ( P @ Y4 )
% 0.20/0.46 => ( ~ ( hF_Mirabelle_hmem @ X @ Y4 )
% 0.20/0.46 => ( P @ ( hF_Mirabelle_hinsert @ X @ Y4 ) ) ) ) )
% 0.20/0.46 => ( P @ Z4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hf_induct
% 0.20/0.46 thf(fact_118_hinter__hinsert__right,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_inf @ hF_Mirabelle_hf @ B3 @ A3 ) ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ B3 @ A3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hinsert_right
% 0.20/0.46 thf(fact_119_hinter__hinsert__left,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_inf @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( inf_inf @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinter_hinsert_left
% 0.20/0.46 thf(fact_120_hinsert__hdiff__if,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) ) )
% 0.20/0.46 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.20/0.46 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 0.20/0.46 = ( hF_Mirabelle_hinsert @ X2 @ ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hinsert_hdiff_if
% 0.20/0.46 thf(fact_121_hdiff__insert,axiom,
% 0.20/0.46 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( minus_minus @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_hinsert @ A2 @ B3 ) )
% 0.20/0.46 = ( minus_minus @ hF_Mirabelle_hf @ ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) @ ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hdiff_insert
% 0.20/0.46 thf(fact_122_hsplit,axiom,
% 0.20/0.46 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = ( C @ A2 @ B ) ) ).
% 0.20/0.46
% 0.20/0.46 % hsplit
% 0.20/0.46 thf(fact_123_Diff__disjoint,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_disjoint
% 0.20/0.46 thf(fact_124_Diff__cancel,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_cancel
% 0.20/0.46 thf(fact_125_empty__Diff,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_Diff
% 0.20/0.46 thf(fact_126_Diff__empty,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_empty
% 0.20/0.46 thf(fact_127_empty__iff,axiom,
% 0.20/0.46 ! [A: $tType,C: A] :
% 0.20/0.46 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_iff
% 0.20/0.46 thf(fact_128_all__not__in__conv,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( ! [X3: A] :
% 0.20/0.46 ~ ( member @ A @ X3 @ A3 ) )
% 0.20/0.46 = ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % all_not_in_conv
% 0.20/0.46 thf(fact_129_Collect__empty__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( collect @ A @ P )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( ! [X3: A] :
% 0.20/0.46 ~ ( P @ X3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_empty_eq
% 0.20/0.46 thf(fact_130_empty__Collect__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A @ P ) )
% 0.20/0.46 = ( ! [X3: A] :
% 0.20/0.46 ~ ( P @ X3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_Collect_eq
% 0.20/0.46 thf(fact_131_Int__iff,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( ( member @ A @ C @ A3 )
% 0.20/0.46 & ( member @ A @ C @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_iff
% 0.20/0.46 thf(fact_132_IntI,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ A3 )
% 0.20/0.46 => ( ( member @ A @ C @ B3 )
% 0.20/0.46 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % IntI
% 0.20/0.46 thf(fact_133_DiffI,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ A3 )
% 0.20/0.46 => ( ~ ( member @ A @ C @ B3 )
% 0.20/0.46 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffI
% 0.20/0.46 thf(fact_134_Diff__iff,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( ( member @ A @ C @ A3 )
% 0.20/0.46 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_iff
% 0.20/0.46 thf(fact_135_Diff__idemp,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_idemp
% 0.20/0.46 thf(fact_136_Int__left__commute,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_left_commute
% 0.20/0.46 thf(fact_137_Int__left__absorb,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_left_absorb
% 0.20/0.46 thf(fact_138_Collect__conj__eq,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ( collect @ A
% 0.20/0.46 @ ^ [X3: A] :
% 0.20/0.46 ( ( P @ X3 )
% 0.20/0.46 & ( Q @ X3 ) ) )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_conj_eq
% 0.20/0.46 thf(fact_139_inf__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ( inf_inf @ ( A > $o )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_set_def
% 0.20/0.46 thf(fact_140_Int__commute,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_commute
% 0.20/0.46 thf(fact_141_Int__Collect,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,A3: set @ A,P: A > $o] :
% 0.20/0.46 ( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
% 0.20/0.46 = ( ( member @ A @ X2 @ A3 )
% 0.20/0.46 & ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_Collect
% 0.20/0.46 thf(fact_142_Int__absorb,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 0.20/0.46 = A3 ) ).
% 0.20/0.46
% 0.20/0.46 % Int_absorb
% 0.20/0.46 thf(fact_143_Int__assoc,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_assoc
% 0.20/0.46 thf(fact_144_Int__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ^ [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A5 )
% 0.20/0.46 & ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_def
% 0.20/0.46 thf(fact_145_IntD2,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ( member @ A @ C @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % IntD2
% 0.20/0.46 thf(fact_146_IntD1,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ( member @ A @ C @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % IntD1
% 0.20/0.46 thf(fact_147_IntE,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ~ ( ( member @ A @ C @ A3 )
% 0.20/0.46 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % IntE
% 0.20/0.46 thf(fact_148_bot__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_set_def
% 0.20/0.46 thf(fact_149_bot__hf__def,axiom,
% 0.20/0.46 ( ( bot_bot @ hF_Mirabelle_hf )
% 0.20/0.46 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_hf_def
% 0.20/0.46 thf(fact_150_emptyE,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] :
% 0.20/0.46 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % emptyE
% 0.20/0.46 thf(fact_151_equals0D,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,A2: A] :
% 0.20/0.46 ( ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % equals0D
% 0.20/0.46 thf(fact_152_equals0I,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ! [Y4: A] :
% 0.20/0.46 ~ ( member @ A @ Y4 @ A3 )
% 0.20/0.46 => ( A3
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % equals0I
% 0.20/0.46 thf(fact_153_ex__in__conv,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.20/0.46 = ( A3
% 0.20/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_in_conv
% 0.20/0.46 thf(fact_154_Int__emptyI,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ! [X: A] :
% 0.20/0.46 ( ( member @ A @ X @ A3 )
% 0.20/0.46 => ~ ( member @ A @ X @ B3 ) )
% 0.20/0.46 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_emptyI
% 0.20/0.46 thf(fact_155_disjoint__iff,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A3 )
% 0.20/0.46 => ~ ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % disjoint_iff
% 0.20/0.46 thf(fact_156_Int__empty__left,axiom,
% 0.20/0.46 ! [A: $tType,B3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_empty_left
% 0.20/0.46 thf(fact_157_Int__empty__right,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_empty_right
% 0.20/0.46 thf(fact_158_disjoint__iff__not__equal,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( ! [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A3 )
% 0.20/0.46 => ! [Y3: A] :
% 0.20/0.46 ( ( member @ A @ Y3 @ B3 )
% 0.20/0.46 => ( X3 != Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % disjoint_iff_not_equal
% 0.20/0.46 thf(fact_159_DiffE,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ~ ( ( member @ A @ C @ A3 )
% 0.20/0.46 => ( member @ A @ C @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffE
% 0.20/0.46 thf(fact_160_DiffD1,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ( member @ A @ C @ A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffD1
% 0.20/0.46 thf(fact_161_DiffD2,axiom,
% 0.20/0.46 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 => ~ ( member @ A @ C @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % DiffD2
% 0.20/0.46 thf(fact_162_Int__Diff,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Int_Diff
% 0.20/0.46 thf(fact_163_Diff__Int2,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_Int2
% 0.20/0.46 thf(fact_164_Diff__Diff__Int,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_Diff_Int
% 0.20/0.46 thf(fact_165_Diff__Int__distrib,axiom,
% 0.20/0.46 ! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_Int_distrib
% 0.20/0.46 thf(fact_166_Diff__Int__distrib2,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 0.20/0.46 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_Int_distrib2
% 0.20/0.46 thf(fact_167_empty__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.20/0.46 = ( collect @ A
% 0.20/0.46 @ ^ [X3: A] : $false ) ) ).
% 0.20/0.46
% 0.20/0.46 % empty_def
% 0.20/0.46 thf(fact_168_set__diff__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ^ [X3: A] :
% 0.20/0.46 ( ( member @ A @ X3 @ A5 )
% 0.20/0.46 & ~ ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % set_diff_eq
% 0.20/0.46 thf(fact_169_minus__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( minus_minus @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( collect @ A
% 0.20/0.46 @ ( minus_minus @ ( A > $o )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % minus_set_def
% 0.20/0.46 thf(fact_170_Diff__triv,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 => ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 = A3 ) ) ).
% 0.20/0.46
% 0.20/0.46 % Diff_triv
% 0.20/0.46 thf(fact_171_hfst__conv,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % hfst_conv
% 0.20/0.46 thf(fact_172_hsnd__conv,axiom,
% 0.20/0.46 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 0.20/0.46 = B ) ).
% 0.20/0.46
% 0.20/0.46 % hsnd_conv
% 0.20/0.46 thf(fact_173_the__equality,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.46 ( ( P @ A2 )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( X = A2 ) )
% 0.20/0.46 => ( ( the @ A @ P )
% 0.20/0.46 = A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % the_equality
% 0.20/0.46 thf(fact_174_the__eq__trivial,axiom,
% 0.20/0.46 ! [A: $tType,A2: A] :
% 0.20/0.46 ( ( the @ A
% 0.20/0.46 @ ^ [X3: A] : ( X3 = A2 ) )
% 0.20/0.46 = A2 ) ).
% 0.20/0.46
% 0.20/0.46 % the_eq_trivial
% 0.20/0.46 thf(fact_175_inf1I,axiom,
% 0.20/0.46 ! [A: $tType,A3: A > $o,X2: A,B3: A > $o] :
% 0.20/0.46 ( ( A3 @ X2 )
% 0.20/0.46 => ( ( B3 @ X2 )
% 0.20/0.46 => ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf1I
% 0.20/0.46 thf(fact_176_the__sym__eq__trivial,axiom,
% 0.20/0.46 ! [A: $tType,X2: A] :
% 0.20/0.46 ( ( the @ A
% 0.20/0.46 @ ( ^ [Y: A,Z: A] : ( Y = Z )
% 0.20/0.46 @ X2 ) )
% 0.20/0.46 = X2 ) ).
% 0.20/0.46
% 0.20/0.46 % the_sym_eq_trivial
% 0.20/0.46 thf(fact_177_inf1D2,axiom,
% 0.20/0.46 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 0.20/0.46 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 0.20/0.46 => ( B3 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf1D2
% 0.20/0.46 thf(fact_178_inf1D1,axiom,
% 0.20/0.46 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 0.20/0.46 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 0.20/0.46 => ( A3 @ X2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf1D1
% 0.20/0.46 thf(fact_179_inf1E,axiom,
% 0.20/0.46 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 0.20/0.46 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 0.20/0.46 => ~ ( ( A3 @ X2 )
% 0.20/0.46 => ~ ( B3 @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf1E
% 0.20/0.46 thf(fact_180_the1__equality,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.46 ( ? [X4: A] :
% 0.20/0.46 ( ( P @ X4 )
% 0.20/0.46 & ! [Y4: A] :
% 0.20/0.46 ( ( P @ Y4 )
% 0.20/0.46 => ( Y4 = X4 ) ) )
% 0.20/0.46 => ( ( P @ A2 )
% 0.20/0.46 => ( ( the @ A @ P )
% 0.20/0.46 = A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % the1_equality
% 0.20/0.46 thf(fact_181_the1I2,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.20/0.46 ( ? [X4: A] :
% 0.20/0.46 ( ( P @ X4 )
% 0.20/0.46 & ! [Y4: A] :
% 0.20/0.46 ( ( P @ Y4 )
% 0.20/0.46 => ( Y4 = X4 ) ) )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( Q @ X ) )
% 0.20/0.46 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % the1I2
% 0.20/0.46 thf(fact_182_If__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( if @ A )
% 0.20/0.46 = ( ^ [P2: $o,X3: A,Y3: A] :
% 0.20/0.46 ( the @ A
% 0.20/0.46 @ ^ [Z3: A] :
% 0.20/0.46 ( ( P2
% 0.20/0.46 => ( Z3 = X3 ) )
% 0.20/0.46 & ( ~ P2
% 0.20/0.46 => ( Z3 = Y3 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % If_def
% 0.20/0.46 thf(fact_183_theI2,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.20/0.46 ( ( P @ A2 )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( X = A2 ) )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( Q @ X ) )
% 0.20/0.46 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % theI2
% 0.20/0.46 thf(fact_184_theI_H,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ? [X4: A] :
% 0.20/0.46 ( ( P @ X4 )
% 0.20/0.46 & ! [Y4: A] :
% 0.20/0.46 ( ( P @ Y4 )
% 0.20/0.46 => ( Y4 = X4 ) ) )
% 0.20/0.46 => ( P @ ( the @ A @ P ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % theI'
% 0.20/0.46 thf(fact_185_theI,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o,A2: A] :
% 0.20/0.46 ( ( P @ A2 )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( P @ X )
% 0.20/0.46 => ( X = A2 ) )
% 0.20/0.46 => ( P @ ( the @ A @ P ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % theI
% 0.20/0.46 thf(fact_186_bot__apply,axiom,
% 0.20/0.46 ! [C2: $tType,D2: $tType] :
% 0.20/0.46 ( ( bot @ C2 )
% 0.20/0.46 => ( ( bot_bot @ ( D2 > C2 ) )
% 0.20/0.46 = ( ^ [X3: D2] : ( bot_bot @ C2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_apply
% 0.20/0.46 thf(fact_187_inf__Int__eq,axiom,
% 0.20/0.46 ! [A: $tType,R: set @ A,S: set @ A] :
% 0.20/0.46 ( ( inf_inf @ ( A > $o )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 0.20/0.46 = ( ^ [X3: A] : ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf_Int_eq
% 0.20/0.46 thf(fact_188_diff__self__eq__0,axiom,
% 0.20/0.46 ! [M: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ M @ M )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_self_eq_0
% 0.20/0.46 thf(fact_189_diff__0__eq__0,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_0_eq_0
% 0.20/0.46 thf(fact_190_diff__commute,axiom,
% 0.20/0.46 ! [I: nat,J: nat,K: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.20/0.46 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_commute
% 0.20/0.46 thf(fact_191_bot__nat__def,axiom,
% 0.20/0.46 ( ( bot_bot @ nat )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_nat_def
% 0.20/0.46 thf(fact_192_diffs0__imp__equal,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ( minus_minus @ nat @ M @ N )
% 0.20/0.46 = ( zero_zero @ nat ) )
% 0.20/0.46 => ( ( ( minus_minus @ nat @ N @ M )
% 0.20/0.46 = ( zero_zero @ nat ) )
% 0.20/0.46 => ( M = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diffs0_imp_equal
% 0.20/0.46 thf(fact_193_minus__nat_Odiff__0,axiom,
% 0.20/0.46 ! [M: nat] :
% 0.20/0.46 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 0.20/0.46 = M ) ).
% 0.20/0.46
% 0.20/0.46 % minus_nat.diff_0
% 0.20/0.46 thf(fact_194_bot__fun__def,axiom,
% 0.20/0.46 ! [B5: $tType,A: $tType] :
% 0.20/0.46 ( ( bot @ B5 )
% 0.20/0.46 => ( ( bot_bot @ ( A > B5 ) )
% 0.20/0.46 = ( ^ [X3: A] : ( bot_bot @ B5 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_fun_def
% 0.20/0.46 thf(fact_195_bot__empty__eq,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.20/0.46 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_empty_eq
% 0.20/0.46 thf(fact_196_Collect__empty__eq__bot,axiom,
% 0.20/0.46 ! [A: $tType,P: A > $o] :
% 0.20/0.46 ( ( ( collect @ A @ P )
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.20/0.46 = ( P
% 0.20/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Collect_empty_eq_bot
% 0.20/0.46 thf(fact_197_ord_OLeast__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( least @ A )
% 0.20/0.46 = ( ^ [Less_eq: A > A > $o,P2: A > $o] :
% 0.20/0.46 ( the @ A
% 0.20/0.46 @ ^ [X3: A] :
% 0.20/0.46 ( ( P2 @ X3 )
% 0.20/0.46 & ! [Y3: A] :
% 0.20/0.46 ( ( P2 @ Y3 )
% 0.20/0.46 => ( Less_eq @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ord.Least_def
% 0.20/0.46 thf(fact_198_PrimReplace__def,axiom,
% 0.20/0.46 ( hF_Mir569462966eplace
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.20/0.46 ( the @ hF_Mirabelle_hf
% 0.20/0.46 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.20/0.46 ! [V5: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ V5 @ Z3 )
% 0.20/0.46 = ( ? [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 0.20/0.46 & ( R2 @ U2 @ V5 ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % PrimReplace_def
% 0.20/0.46 thf(fact_199_hcard__0,axiom,
% 0.20/0.46 ( ( hF_Mirabelle_hcard @ ( zero_zero @ hF_Mirabelle_hf ) )
% 0.20/0.46 = ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % hcard_0
% 0.20/0.46 thf(fact_200_hfst__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_hfst
% 0.20/0.46 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.20/0.46 ( the @ hF_Mirabelle_hf
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( P4
% 0.20/0.46 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hfst_def
% 0.20/0.46 thf(fact_201_hsnd__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_hsnd
% 0.20/0.46 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.20/0.46 ( the @ hF_Mirabelle_hf
% 0.20/0.46 @ ^ [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ? [X3: hF_Mirabelle_hf] :
% 0.20/0.46 ( P4
% 0.20/0.46 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hsnd_def
% 0.20/0.46 thf(fact_202_HUnion__def,axiom,
% 0.20/0.46 ( hF_Mirabelle_HUnion
% 0.20/0.46 = ( ^ [A5: hF_Mirabelle_hf] :
% 0.20/0.46 ( the @ hF_Mirabelle_hf
% 0.20/0.46 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.20/0.46 ! [U2: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.20/0.46 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 0.20/0.46 & ( hF_Mirabelle_hmem @ U2 @ Y3 ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % HUnion_def
% 0.20/0.46 thf(fact_203_hcard__hdiff1__less,axiom,
% 0.20/0.46 ! [X2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.20/0.46 ( ( hF_Mirabelle_hmem @ X2 @ Z4 )
% 0.20/0.46 => ( ord_less @ nat @ ( hF_Mirabelle_hcard @ ( minus_minus @ hF_Mirabelle_hf @ Z4 @ ( hF_Mirabelle_hinsert @ X2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) @ ( hF_Mirabelle_hcard @ Z4 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % hcard_hdiff1_less
% 0.20/0.46 thf(fact_204_Set_Ois__empty__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( is_empty @ A )
% 0.20/0.46 = ( ^ [A5: set @ A] :
% 0.20/0.46 ( A5
% 0.20/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % Set.is_empty_def
% 0.20/0.46 thf(fact_205_not__gr__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [N: A] :
% 0.20/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.20/0.46 = ( N
% 0.20/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_gr_zero
% 0.20/0.46 thf(fact_206_bot__nat__0_Onot__eq__extremum,axiom,
% 0.20/0.46 ! [A2: nat] :
% 0.20/0.46 ( ( A2
% 0.20/0.46 != ( zero_zero @ nat ) )
% 0.20/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_nat_0.not_eq_extremum
% 0.20/0.46 thf(fact_207_less__nat__zero__code,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_nat_zero_code
% 0.20/0.46 thf(fact_208_neq0__conv,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( N
% 0.20/0.46 != ( zero_zero @ nat ) )
% 0.20/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % neq0_conv
% 0.20/0.46 thf(fact_209_diff__gt__0__iff__gt,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A] :
% 0.20/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 0.20/0.46 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_gt_0_iff_gt
% 0.20/0.46 thf(fact_210_zero__less__diff,axiom,
% 0.20/0.46 ! [N: nat,M: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 0.20/0.46 = ( ord_less @ nat @ M @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_less_diff
% 0.20/0.46 thf(fact_211_less__imp__diff__less,axiom,
% 0.20/0.46 ! [J: nat,K: nat,N: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ J @ K )
% 0.20/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_imp_diff_less
% 0.20/0.46 thf(fact_212_diff__less__mono2,axiom,
% 0.20/0.46 ! [M: nat,N: nat,L: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ M @ N )
% 0.20/0.46 => ( ( ord_less @ nat @ M @ L )
% 0.20/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_less_mono2
% 0.20/0.46 thf(fact_213_diff__less,axiom,
% 0.20/0.46 ! [N: nat,M: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.20/0.46 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.20/0.46 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_less
% 0.20/0.46 thf(fact_214_diff__strict__right__mono,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A] :
% 0.20/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.20/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_strict_right_mono
% 0.20/0.46 thf(fact_215_diff__strict__left__mono,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ! [B: A,A2: A,C: A] :
% 0.20/0.46 ( ( ord_less @ A @ B @ A2 )
% 0.20/0.46 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_strict_left_mono
% 0.20/0.46 thf(fact_216_diff__eq__diff__less,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A,D: A] :
% 0.20/0.46 ( ( ( minus_minus @ A @ A2 @ B )
% 0.20/0.46 = ( minus_minus @ A @ C @ D ) )
% 0.20/0.46 => ( ( ord_less @ A @ A2 @ B )
% 0.20/0.46 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_eq_diff_less
% 0.20/0.46 thf(fact_217_diff__strict__mono,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ! [A2: A,B: A,D: A,C: A] :
% 0.20/0.46 ( ( ord_less @ A @ A2 @ B )
% 0.20/0.46 => ( ( ord_less @ A @ D @ C )
% 0.20/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % diff_strict_mono
% 0.20/0.46 thf(fact_218_infinite__descent0__measure,axiom,
% 0.20/0.46 ! [A: $tType,V6: A > nat,P: A > $o,X2: A] :
% 0.20/0.46 ( ! [X: A] :
% 0.20/0.46 ( ( ( V6 @ X )
% 0.20/0.46 = ( zero_zero @ nat ) )
% 0.20/0.46 => ( P @ X ) )
% 0.20/0.46 => ( ! [X: A] :
% 0.20/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V6 @ X ) )
% 0.20/0.46 => ( ~ ( P @ X )
% 0.20/0.46 => ? [Y5: A] :
% 0.20/0.46 ( ( ord_less @ nat @ ( V6 @ Y5 ) @ ( V6 @ X ) )
% 0.20/0.46 & ~ ( P @ Y5 ) ) ) )
% 0.20/0.46 => ( P @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_descent0_measure
% 0.20/0.46 thf(fact_219_bot__nat__0_Oextremum__strict,axiom,
% 0.20/0.46 ! [A2: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot_nat_0.extremum_strict
% 0.20/0.46 thf(fact_220_infinite__descent0,axiom,
% 0.20/0.46 ! [P: nat > $o,N: nat] :
% 0.20/0.46 ( ( P @ ( zero_zero @ nat ) )
% 0.20/0.46 => ( ! [N2: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.46 => ( ~ ( P @ N2 )
% 0.20/0.46 => ? [M2: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.20/0.46 & ~ ( P @ M2 ) ) ) )
% 0.20/0.46 => ( P @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % infinite_descent0
% 0.20/0.46 thf(fact_221_gr__implies__not0,axiom,
% 0.20/0.46 ! [M: nat,N: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ M @ N )
% 0.20/0.46 => ( N
% 0.20/0.46 != ( zero_zero @ nat ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % gr_implies_not0
% 0.20/0.46 thf(fact_222_less__zeroE,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_zeroE
% 0.20/0.46 thf(fact_223_not__less0,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_less0
% 0.20/0.46 thf(fact_224_not__gr0,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.20/0.46 = ( N
% 0.20/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_gr0
% 0.20/0.46 thf(fact_225_gr0I,axiom,
% 0.20/0.46 ! [N: nat] :
% 0.20/0.46 ( ( N
% 0.20/0.46 != ( zero_zero @ nat ) )
% 0.20/0.46 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.20/0.46
% 0.20/0.46 % gr0I
% 0.20/0.46 thf(fact_226_less__iff__diff__less__0,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ordered_ab_group_add @ A )
% 0.20/0.46 => ( ( ord_less @ A )
% 0.20/0.46 = ( ^ [A4: A,B2: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_iff_diff_less_0
% 0.20/0.46 thf(fact_227_zero__less__iff__neq__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [N: A] :
% 0.20/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.20/0.46 = ( N
% 0.20/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_less_iff_neq_zero
% 0.20/0.46 thf(fact_228_gr__implies__not__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [M: A,N: A] :
% 0.20/0.46 ( ( ord_less @ A @ M @ N )
% 0.20/0.46 => ( N
% 0.20/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % gr_implies_not_zero
% 0.20/0.46 thf(fact_229_not__less__zero,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [N: A] :
% 0.20/0.46 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_less_zero
% 0.20/0.46 thf(fact_230_gr__zeroI,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( canoni770627133id_add @ A )
% 0.20/0.46 => ! [N: A] :
% 0.20/0.46 ( ( N
% 0.20/0.46 != ( zero_zero @ A ) )
% 0.20/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % gr_zeroI
% 0.20/0.46 thf(fact_231_less__infI1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,X2: A,B: A] :
% 0.20/0.46 ( ( ord_less @ A @ A2 @ X2 )
% 0.20/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_infI1
% 0.20/0.46 thf(fact_232_less__infI2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B: A,X2: A,A2: A] :
% 0.20/0.46 ( ( ord_less @ A @ B @ X2 )
% 0.20/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ X2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_infI2
% 0.20/0.46 thf(fact_233_inf_Ostrict__boundedE,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,B: A,C: A] :
% 0.20/0.46 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 0.20/0.46 => ~ ( ( ord_less @ A @ A2 @ B )
% 0.20/0.46 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.strict_boundedE
% 0.20/0.46 thf(fact_234_inf_Ostrict__order__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ( ( ord_less @ A )
% 0.20/0.46 = ( ^ [A4: A,B2: A] :
% 0.20/0.46 ( ( A4
% 0.20/0.46 = ( inf_inf @ A @ A4 @ B2 ) )
% 0.20/0.46 & ( A4 != B2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.strict_order_iff
% 0.20/0.46 thf(fact_235_inf_Ostrict__coboundedI1,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [A2: A,C: A,B: A] :
% 0.20/0.46 ( ( ord_less @ A @ A2 @ C )
% 0.20/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.strict_coboundedI1
% 0.20/0.46 thf(fact_236_inf_Ostrict__coboundedI2,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A )
% 0.20/0.46 => ! [B: A,C: A,A2: A] :
% 0.20/0.46 ( ( ord_less @ A @ B @ C )
% 0.20/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inf.strict_coboundedI2
% 0.20/0.46 thf(fact_237_bot_Oextremum__strict,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order_bot @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot.extremum_strict
% 0.20/0.46 thf(fact_238_bot_Onot__eq__extremum,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( order_bot @ A )
% 0.20/0.46 => ! [A2: A] :
% 0.20/0.46 ( ( A2
% 0.20/0.46 != ( bot_bot @ A ) )
% 0.20/0.46 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % bot.not_eq_extremum
% 0.20/0.46 thf(fact_239_less__numeral__extra_I3_J,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linord1659791738miring @ A )
% 0.20/0.46 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_numeral_extra(3)
% 0.20/0.46 thf(fact_240_of__nat__0__less__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linord1659791738miring @ A )
% 0.20/0.46 => ! [N: nat] :
% 0.20/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.46 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % of_nat_0_less_iff
% 0.20/0.46 thf(fact_241_of__nat__0,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semiring_1 @ A )
% 0.20/0.46 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.20/0.46 = ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % of_nat_0
% 0.20/0.46 thf(fact_242_of__nat__0__eq__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semiring_char_0 @ A )
% 0.20/0.46 => ! [N: nat] :
% 0.20/0.46 ( ( ( zero_zero @ A )
% 0.20/0.46 = ( semiring_1_of_nat @ A @ N ) )
% 0.20/0.46 = ( ( zero_zero @ nat )
% 0.20/0.46 = N ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % of_nat_0_eq_iff
% 0.20/0.46 thf(fact_243_of__nat__eq__0__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( semiring_char_0 @ A )
% 0.20/0.46 => ! [M: nat] :
% 0.20/0.46 ( ( ( semiring_1_of_nat @ A @ M )
% 0.20/0.46 = ( zero_zero @ A ) )
% 0.20/0.46 = ( M
% 0.20/0.46 = ( zero_zero @ nat ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % of_nat_eq_0_iff
% 0.20/0.46 thf(fact_244_psubset__imp__ex__mem,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.20/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 => ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % psubset_imp_ex_mem
% 0.20/0.46 thf(fact_245_of__nat__less__0__iff,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( linord1659791738miring @ A )
% 0.20/0.46 => ! [M: nat] :
% 0.20/0.46 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % of_nat_less_0_iff
% 0.20/0.46 thf(fact_246_not__psubset__empty,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A] :
% 0.20/0.46 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % not_psubset_empty
% 0.20/0.46 thf(fact_247_pos__int__cases,axiom,
% 0.20/0.46 ! [K: int] :
% 0.20/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.20/0.46 => ~ ! [N2: nat] :
% 0.20/0.46 ( ( K
% 0.20/0.46 = ( semiring_1_of_nat @ int @ N2 ) )
% 0.20/0.46 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % pos_int_cases
% 0.20/0.46 thf(fact_248_zero__less__imp__eq__int,axiom,
% 0.20/0.46 ! [K: int] :
% 0.20/0.46 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.20/0.46 => ? [N2: nat] :
% 0.20/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.20/0.46 & ( K
% 0.20/0.46 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % zero_less_imp_eq_int
% 0.20/0.46 thf(fact_249_psubsetD,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
% 0.20/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 => ( ( member @ A @ C @ A3 )
% 0.20/0.46 => ( member @ A @ C @ B3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % psubsetD
% 0.20/0.46 thf(fact_250_less__set__def,axiom,
% 0.20/0.46 ! [A: $tType] :
% 0.20/0.46 ( ( ord_less @ ( set @ A ) )
% 0.20/0.46 = ( ^ [A5: set @ A,B4: set @ A] :
% 0.20/0.46 ( ord_less @ ( A > $o )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 0.20/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % less_set_def
% 0.20/0.46 thf(fact_251_psubset__trans,axiom,
% 0.20/0.46 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.20/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.20/0.46 => ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
% 0.20/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % psubset_trans
% 0.20/0.46 thf(fact_252_int__diff__cases,axiom,
% 0.20/0.46 ! [Z4: int] :
% 0.20/0.46 ~ ! [M3: nat,N2: nat] :
% 0.20/0.46 ( Z4
% 0.20/0.46 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % int_diff_cases
% 0.20/0.46 thf(fact_253_minus__int__code_I1_J,axiom,
% 0.20/0.46 ! [K: int] :
% 0.20/0.46 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 0.20/0.46 = K ) ).
% 0.20/0.46
% 0.20/0.46 % minus_int_code(1)
% 0.20/0.46
% 0.20/0.46 % Type constructors (52)
% 0.20/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.20/0.46 bounded_lattice @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.20/0.46 ! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( bounded_lattice @ A10 )
% 0.20/0.46 => ( bounded_lattice @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( bounded_lattice @ A10 )
% 0.20/0.46 => ( bounded_lattice_bot @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( semilattice_inf @ A10 )
% 0.20/0.46 => ( semilattice_inf @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( order_bot @ A10 )
% 0.20/0.46 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( lattice @ A10 )
% 0.20/0.46 => ( lattice @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( bot @ A10 )
% 0.20/0.46 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_fun___Groups_Ominus,axiom,
% 0.20/0.46 ! [A9: $tType,A10: $tType] :
% 0.20/0.46 ( ( minus @ A10 )
% 0.20/0.46 => ( minus @ ( A9 > A10 ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 0.20/0.46 linord1659791738miring @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 0.20/0.46 cancel146912293up_add @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 0.20/0.46 cancel1352612707id_add @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 0.20/0.46 ordered_ab_group_add @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
% 0.20/0.46 semilattice_inf @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.20/0.46 semiring_char_0 @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.20/0.46 semiring_1 @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Lattices_Olattice_4,axiom,
% 0.20/0.46 lattice @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 0.20/0.46 group_add @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Ominus_5,axiom,
% 0.20/0.46 minus @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.20/0.46 zero @ int ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.20/0.46 canoni770627133id_add @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6,axiom,
% 0.20/0.46 linord1659791738miring @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 0.20/0.46 cancel146912293up_add @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 0.20/0.46 cancel1352612707id_add @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_9,axiom,
% 0.20/0.46 semilattice_inf @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 0.20/0.46 comm_monoid_diff @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_10,axiom,
% 0.20/0.46 order_bot @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 0.20/0.46 semiring_char_0 @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_12,axiom,
% 0.20/0.46 semiring_1 @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Lattices_Olattice_13,axiom,
% 0.20/0.46 lattice @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Orderings_Obot_14,axiom,
% 0.20/0.46 bot @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ominus_15,axiom,
% 0.20/0.46 minus @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Nat_Onat___Groups_Ozero_16,axiom,
% 0.20/0.46 zero @ nat ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_17,axiom,
% 0.20/0.46 ! [A9: $tType] : ( bounded_lattice_bot @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_18,axiom,
% 0.20/0.46 ! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_19,axiom,
% 0.20/0.46 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Lattices_Olattice_20,axiom,
% 0.20/0.46 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Orderings_Obot_21,axiom,
% 0.20/0.46 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_Set_Oset___Groups_Ominus_22,axiom,
% 0.20/0.46 ! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_23,axiom,
% 0.20/0.46 bounded_lattice_bot @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_24,axiom,
% 0.20/0.46 semilattice_inf @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_25,axiom,
% 0.20/0.46 order_bot @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_26,axiom,
% 0.20/0.46 lattice @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Orderings_Obot_27,axiom,
% 0.20/0.46 bot @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HOL_Obool___Groups_Ominus_28,axiom,
% 0.20/0.46 minus @ $o ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Obounded__lattice__bot_29,axiom,
% 0.20/0.46 bounded_lattice_bot @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_30,axiom,
% 0.20/0.46 semilattice_inf @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder__bot_31,axiom,
% 0.20/0.46 order_bot @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_32,axiom,
% 0.20/0.46 lattice @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Obot_33,axiom,
% 0.20/0.46 bot @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_34,axiom,
% 0.20/0.46 minus @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_35,axiom,
% 0.20/0.46 zero @ hF_Mirabelle_hf ).
% 0.20/0.46
% 0.20/0.46 % Helper facts (3)
% 0.20/0.46 thf(help_If_3_1_T,axiom,
% 0.20/0.46 ! [P: $o] :
% 0.20/0.46 ( ( P = $true )
% 0.20/0.46 | ( P = $false ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(help_If_2_1_T,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,Y2: A] :
% 0.20/0.46 ( ( if @ A @ $false @ X2 @ Y2 )
% 0.20/0.46 = Y2 ) ).
% 0.20/0.46
% 0.20/0.46 thf(help_If_1_1_T,axiom,
% 0.20/0.46 ! [A: $tType,X2: A,Y2: A] :
% 0.20/0.46 ( ( if @ A @ $true @ X2 @ Y2 )
% 0.20/0.46 = X2 ) ).
% 0.20/0.46
% 0.20/0.46 % Conjectures (1)
% 0.20/0.46 thf(conj_0,conjecture,
% 0.20/0.46 ( ( hF_Mirabelle_HBex @ a
% 0.20/0.46 @ ^ [X3: hF_Mirabelle_hf] : p )
% 0.20/0.46 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ a )
% 0.20/0.46 & p ) ) ).
% 0.20/0.46
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 ------------------------
% 2.56/1.43 ---- Embedded in THF ---
% 2.56/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.56/1.43 %%% Generated on Tue Feb 24 08:25:26 EST 2026
% 2.56/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.56/1.43 %%% Logic specification used:
% 2.56/1.43 %%% thf(spec, logic, $$dhol).
% 2.56/1.43
% 2.56/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.yW9if3OHxl/DTF2DTF25592.p
% 2.56/1.43 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf, type, hF_Mirabelle_hf: $tType).
% 2.56/1.43 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.56/1.43 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.56/1.43 thf(ty_t_Int_Oint, type, int: $tType).
% 2.56/1.43 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1352612707id_add: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.56/1.43 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.56/1.43 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBall, type, hF_Mirabelle_HBall: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > $o) > $o))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBex, type, hF_Mirabelle_HBex: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > $o) > $o))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect, type, hF_Mir1687042746ollect: ((hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF, type, hF_Mirabelle_HF: ((set @ hF_Mirabelle_hf) > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter, type, hF_Mirabelle_HInter: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion, type, hF_Mirabelle_HUnion: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace, type, hF_Mir569462966eplace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun, type, hF_Mirabelle_RepFun: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace, type, hF_Mirabelle_Replace: (hF_Mirabelle_hf > ((hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)) > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard, type, hF_Mirabelle_hcard: (hF_Mirabelle_hf > nat)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf, type, hF_Mirabelle_Abs_hf: (nat > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset, type, hF_Mirabelle_hfset: (hF_Mirabelle_hf > (set @ hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst, type, hF_Mirabelle_hfst: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert, type, hF_Mirabelle_hinsert: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem, type, hF_Mirabelle_hmem: (hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair, type, hF_Mirabelle_hpair: (hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf))).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd, type, hF_Mirabelle_hsnd: (hF_Mirabelle_hf > hF_Mirabelle_hf)).
% 2.56/1.43 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit, type, hF_Mirabelle_hsplit: (!> [A:$tType]: (((hF_Mirabelle_hf > (hF_Mirabelle_hf > A)) > (hF_Mirabelle_hf > A))))).
% 2.56/1.43 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 2.56/1.43 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.56/1.43 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.56/1.43 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.56/1.43 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.56/1.43 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 2.56/1.43 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.56/1.43 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.56/1.43 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.56/1.43 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.56/1.43 thf(sy_v_A, type, a: hF_Mirabelle_hf).
% 2.56/1.43 thf(sy_v_P, type, p: $o).
% 2.56/1.43 thf(fact_0_hf__equalityI, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A2) = ((hF_Mirabelle_hmem @ X) @ B)))) => (A2 = B))))).
% 2.56/1.43 thf(fact_1_hbexI, axiom, (! [P:(hF_Mirabelle_hf > $o),X2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: (((P @ X2) => (((hF_Mirabelle_hmem @ X2) @ A3) => ((hF_Mirabelle_HBex @ A3) @ P)))))).
% 2.56/1.43 thf(fact_2_hbexE, axiom, (! [A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((((hF_Mirabelle_HBex @ A3) @ P) => (~ ((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A3) => (~ ((P @ X)))))))))))).
% 2.56/1.43 thf(fact_3_hf__ext, axiom, ((^ [Y:hF_Mirabelle_hf,Z:hF_Mirabelle_hf]: ((Y = Z))) = (^ [A4:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A4) = ((hF_Mirabelle_hmem @ X3) @ B2)))))))).
% 2.56/1.43 thf(fact_4_hmem__ne, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ Y2) => (X2 != Y2))))).
% 2.56/1.43 thf(fact_5_HBex__def, axiom, (hF_Mirabelle_HBex = (^ [A5:hF_Mirabelle_hf,P2:(hF_Mirabelle_hf > $o)]: ((? [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A5) & (P2 @ X3)))))))).
% 2.56/1.43 thf(fact_6_hbex__cong, axiom, (! [A3:hF_Mirabelle_hf,A6:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),P3:(hF_Mirabelle_hf > $o)]: (((A3 = A6) => ((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A6) => ((P @ X) = (P3 @ X))))) => (((hF_Mirabelle_HBex @ A3) @ P) = ((hF_Mirabelle_HBex @ A6) @ P3))))))).
% 2.56/1.43 thf(fact_7_rev__hbexI, axiom, (! [X2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((((hF_Mirabelle_hmem @ X2) @ A3) => ((P @ X2) => ((hF_Mirabelle_HBex @ A3) @ P)))))).
% 2.56/1.43 thf(fact_8_replacement, axiom, (! [X2:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((! [U:hF_Mirabelle_hf,V:hF_Mirabelle_hf,V2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U) @ X2) => (((R @ U) @ V) => (((R @ U) @ V2) => (V2 = V)))))) => (? [Z2:hF_Mirabelle_hf]: ((! [V3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V3) @ Z2) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X2) & ((R @ U2) @ V3))))))))))))).
% 2.56/1.43 thf(fact_9_binary__union, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (((hF_Mirabelle_hmem @ U3) @ X2) | ((hF_Mirabelle_hmem @ U3) @ Y2)))))))))).
% 2.56/1.43 thf(fact_10_hmem__not__sym, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((~ ((((hF_Mirabelle_hmem @ X2) @ Y2) & ((hF_Mirabelle_hmem @ Y2) @ X2))))))).
% 2.56/1.43 thf(fact_11_union__of__set, axiom, (! [X2:hF_Mirabelle_hf]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (? [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ X2) & ((hF_Mirabelle_hmem @ U3) @ Y3)))))))))))).
% 2.56/1.43 thf(fact_12_comprehension, axiom, (! [X2:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((? [Z2:hF_Mirabelle_hf]: ((! [U3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U3) @ Z2) = (((hF_Mirabelle_hmem @ U3) @ X2) & (P @ U3)))))))))).
% 2.56/1.43 thf(fact_13_hmem__not__refl, axiom, (! [X2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X2) @ X2)))))).
% 2.56/1.43 thf(fact_14_replacement__fun, axiom, (! [X2:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((? [Z2:hF_Mirabelle_hf]: ((! [V3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V3) @ Z2) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ X2) & (V3 = (F @ U2))))))))))))).
% 2.56/1.43 thf(fact_15_HF__Mirabelle__fsbjehakzm_ObexCI, axiom, (! [A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),A2:hF_Mirabelle_hf]: (((((hF_Mirabelle_HBall @ A3) @ (^ [X3:hF_Mirabelle_hf]: ((~ ((P @ X3)))))) => (P @ A2)) => (((hF_Mirabelle_hmem @ A2) @ A3) => ((hF_Mirabelle_HBex @ A3) @ P)))))).
% 2.56/1.43 thf(fact_16_hballI, axiom, (! [A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: (((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A3) => (P @ X)))) => ((hF_Mirabelle_HBall @ A3) @ P))))).
% 2.56/1.43 thf(fact_17_hball__triv, axiom, (! [A3:hF_Mirabelle_hf,P:$o]: ((((hF_Mirabelle_HBall @ A3) @ (^ [X3:hF_Mirabelle_hf]: (P))) = ((? [X3:hF_Mirabelle_hf]: (((hF_Mirabelle_hmem @ X3) @ A3))) => P))))).
% 2.56/1.43 thf(fact_18_PrimReplace__iff, axiom, (! [A3:hF_Mirabelle_hf,R:(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) @ A3) => (((R @ U) @ V) => (((R @ U) @ V2) => (V2 = V)))))) => (((hF_Mirabelle_hmem @ V4) @ ((hF_Mir569462966eplace @ A3) @ R)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A3) & ((R @ U2) @ V4))))))))).
% 2.56/1.43 thf(fact_19_HCollect__iff, axiom, (! [X2:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),A3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ ((hF_Mir1687042746ollect @ P) @ A3)) = ((P @ X2) & ((hF_Mirabelle_hmem @ X2) @ A3)))))).
% 2.56/1.43 thf(fact_20_Replace__iff, axiom, (! [V4:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_hmem @ V4) @ ((hF_Mirabelle_Replace @ A3) @ R)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A3) & (((R @ U2) @ V4) & (! [Y3:hF_Mirabelle_hf]: ((((R @ U2) @ Y3) => (Y3 = V4)))))))))))).
% 2.56/1.43 thf(fact_21_HUnion__iff, axiom, (! [X2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ (hF_Mirabelle_HUnion @ A3)) = (? [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A3) & ((hF_Mirabelle_hmem @ X2) @ Y3)))))))).
% 2.56/1.43 thf(fact_22_hballE, axiom, (! [A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_HBall @ A3) @ P) => ((~ ((P @ X2))) => (~ (((hF_Mirabelle_hmem @ X2) @ A3)))))))).
% 2.56/1.43 thf(fact_23_hbspec, axiom, (! [A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),X2:hF_Mirabelle_hf]: ((((hF_Mirabelle_HBall @ A3) @ P) => (((hF_Mirabelle_hmem @ X2) @ A3) => (P @ X2)))))).
% 2.56/1.43 thf(fact_24_HCollectE, axiom, (! [A2:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),A3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ A2) @ ((hF_Mir1687042746ollect @ P) @ A3)) => (~ ((((hF_Mirabelle_hmem @ A2) @ A3) => (~ ((P @ A2)))))))))).
% 2.56/1.43 thf(fact_25_HCollectI, axiom, (! [A2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o)]: ((((hF_Mirabelle_hmem @ A2) @ A3) => ((P @ A2) => ((hF_Mirabelle_hmem @ A2) @ ((hF_Mir1687042746ollect @ P) @ A3))))))).
% 2.56/1.43 thf(fact_26_hball__cong, axiom, (! [A3:hF_Mirabelle_hf,A6:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > $o),P3:(hF_Mirabelle_hf > $o)]: (((A3 = A6) => ((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ A6) => ((P @ X) = (P3 @ X))))) => (((hF_Mirabelle_HBall @ A3) @ P) = ((hF_Mirabelle_HBall @ A6) @ P3))))))).
% 2.56/1.43 thf(fact_27_hmem__Sup__ne, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ Y2) => ((hF_Mirabelle_HUnion @ X2) != Y2))))).
% 2.56/1.43 thf(fact_28_Replace__cong, axiom, (! [A3:hF_Mirabelle_hf,B3:hF_Mirabelle_hf,P:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o)),Q:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((A3 = B3) => ((! [X:hF_Mirabelle_hf,Y4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ B3) => (((P @ X) @ Y4) = ((Q @ X) @ Y4))))) => (((hF_Mirabelle_Replace @ A3) @ P) = ((hF_Mirabelle_Replace @ B3) @ Q))))))).
% 2.56/1.43 thf(fact_29_HBall__def, axiom, (hF_Mirabelle_HBall = (^ [A5:hF_Mirabelle_hf,P2:(hF_Mirabelle_hf > $o)]: ((! [X3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X3) @ A5) => (P2 @ X3)))))))).
% 2.56/1.43 thf(fact_30_Replace__def, axiom, (hF_Mirabelle_Replace = (^ [A5:hF_Mirabelle_hf,R2:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((hF_Mir569462966eplace @ A5) @ (^ [X3:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: (((? [Z3:hF_Mirabelle_hf]: ((((R2 @ X3) @ Z3) & (! [Aa:hF_Mirabelle_hf]: ((((R2 @ X3) @ Aa) => (Aa = Z3))))))) & ((R2 @ X3) @ Y3))))))))).
% 2.56/1.43 thf(fact_31_RepFun__def, axiom, (hF_Mirabelle_RepFun = (^ [A5:hF_Mirabelle_hf,F2:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: (((hF_Mirabelle_Replace @ A5) @ (^ [X3:hF_Mirabelle_hf,Y3:hF_Mirabelle_hf]: ((Y3 = (F2 @ X3))))))))).
% 2.56/1.43 thf(fact_32_minus__hf__def, axiom, ((minus_minus @ hF_Mirabelle_hf) = (^ [A5:hF_Mirabelle_hf,B4:hF_Mirabelle_hf]: (((hF_Mir1687042746ollect @ (^ [X3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X3) @ B4)))))) @ A5))))).
% 2.56/1.43 thf(fact_33_inf__hf__def, axiom, ((inf_inf @ hF_Mirabelle_hf) = (^ [A4:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mir1687042746ollect @ (^ [X3:hF_Mirabelle_hf]: (((hF_Mirabelle_hmem @ X3) @ B2)))) @ A4))))).
% 2.56/1.43 thf(fact_34_HInter__def, axiom, (hF_Mirabelle_HInter = (^ [A5:hF_Mirabelle_hf]: (((hF_Mir1687042746ollect @ (^ [X3:hF_Mirabelle_hf]: ((! [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A5) => ((hF_Mirabelle_hmem @ X3) @ Y3))))))) @ (hF_Mirabelle_HUnion @ A5)))))).
% 2.56/1.43 thf(fact_35_RepFun__iff, axiom, (! [V4:hF_Mirabelle_hf,A3:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_hmem @ V4) @ ((hF_Mirabelle_RepFun @ A3) @ F)) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A3) & (V4 = (F @ U2))))))))).
% 2.56/1.43 thf(fact_36_HCollect__hempty, axiom, (! [P:(hF_Mirabelle_hf > $o)]: ((((hF_Mir1687042746ollect @ P) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.43 thf(fact_37_Replace__0, axiom, (! [R:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: ((((hF_Mirabelle_Replace @ (zero_zero @ hF_Mirabelle_hf)) @ R) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.43 thf(fact_38_hmem__def, axiom, (hF_Mirabelle_hmem = (^ [A4:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: ((((member @ hF_Mirabelle_hf) @ A4) @ (hF_Mirabelle_hfset @ B2)))))).
% 2.56/1.43 thf(fact_39_triv__RepFun, axiom, (! [A3:hF_Mirabelle_hf]: ((((hF_Mirabelle_RepFun @ A3) @ (^ [X3:hF_Mirabelle_hf]: (X3))) = A3)))).
% 2.56/1.43 thf(fact_40_hinter__iff, axiom, (! [U4:hF_Mirabelle_hf,X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U4) @ (((inf_inf @ hF_Mirabelle_hf) @ X2) @ Y2)) = (((hF_Mirabelle_hmem @ U4) @ X2) & ((hF_Mirabelle_hmem @ U4) @ Y2)))))).
% 2.56/1.43 thf(fact_41_hdiff__iff, axiom, (! [U4:hF_Mirabelle_hf,X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U4) @ (((minus_minus @ hF_Mirabelle_hf) @ X2) @ Y2)) = (((hF_Mirabelle_hmem @ U4) @ X2) & (~ (((hF_Mirabelle_hmem @ U4) @ Y2)))))))).
% 2.56/1.43 thf(fact_42_hinter__hempty__right, axiom, (! [A3:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ A3) @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.43 thf(fact_43_hinter__hempty__left, axiom, (! [A3:hF_Mirabelle_hf]: (((((inf_inf @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ A3) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.43 thf(fact_44_zero__hdiff, axiom, (! [X2:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ (zero_zero @ hF_Mirabelle_hf)) @ X2) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.44 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.56/1.44 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 2.56/1.44 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.56/1.44 thf(fact_48_ext, axiom, (! [B5:$tType,A:$tType,F:(A > B5),G:(A > B5)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.56/1.44 thf(fact_49_hdiff__zero, axiom, (! [X2:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ X2) @ (zero_zero @ hF_Mirabelle_hf)) = X2)))).
% 2.56/1.44 thf(fact_50_RepFun__0, axiom, (! [F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ (zero_zero @ hF_Mirabelle_hf)) @ F) = (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.44 thf(fact_51_HUnion__hempty, axiom, ((hF_Mirabelle_HUnion @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.56/1.44 thf(fact_52_HInter__hempty, axiom, ((hF_Mirabelle_HInter @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.56/1.44 thf(fact_53_HInter__iff, axiom, (! [A3:hF_Mirabelle_hf,X2:hF_Mirabelle_hf]: (((A3 != (zero_zero @ hF_Mirabelle_hf)) => (((hF_Mirabelle_hmem @ X2) @ (hF_Mirabelle_HInter @ A3)) = (! [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A3) => ((hF_Mirabelle_hmem @ X2) @ Y3))))))))).
% 2.56/1.44 thf(fact_54_foundation, axiom, (! [Z4:hF_Mirabelle_hf]: (((Z4 != (zero_zero @ hF_Mirabelle_hf)) => (? [W:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ W) @ Z4) & ((((inf_inf @ hF_Mirabelle_hf) @ W) @ Z4) = (zero_zero @ hF_Mirabelle_hf))))))))).
% 2.56/1.44 thf(fact_55_hemptyE, axiom, (! [A2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.56/1.44 thf(fact_56_hempty__iff, axiom, (! [Z4:hF_Mirabelle_hf]: (((Z4 = (zero_zero @ hF_Mirabelle_hf)) = (! [X3:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ X3) @ Z4))))))))).
% 2.56/1.44 thf(fact_57_hmem__hempty, axiom, (! [A2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.56/1.44 thf(fact_58_RepFun__cong, axiom, (! [A3:hF_Mirabelle_hf,B3:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf),G:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: (((A3 = B3) => ((! [X:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X) @ B3) => ((F @ X) = (G @ X))))) => (((hF_Mirabelle_RepFun @ A3) @ F) = ((hF_Mirabelle_RepFun @ B3) @ G))))))).
% 2.56/1.44 thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.56/1.44 thf(fact_60_diff__zero, axiom, (! [A:$tType]: (((cancel1352612707id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.56/1.44 thf(fact_61_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.56/1.44 thf(fact_62_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))).
% 2.56/1.44 thf(fact_63_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))).
% 2.56/1.44 thf(fact_64_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y2)) @ Y2) = (((inf_inf @ A) @ X2) @ Y2)))))))).
% 2.56/1.44 thf(fact_65_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B)) @ B) = (((inf_inf @ A) @ A2) @ B)))))))).
% 2.56/1.44 thf(fact_66_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y2:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ X2) @ Y2)) = (((inf_inf @ A) @ X2) @ Y2)))))))).
% 2.56/1.44 thf(fact_67_minus__apply, axiom, (! [B5:$tType,A:$tType]: (((minus @ B5) => ((minus_minus @ (A > B5)) = (^ [A5:(A > B5),B4:(A > B5),X3:A]: ((((minus_minus @ B5) @ (A5 @ X3)) @ (B4 @ X3))))))))).
% 2.56/1.44 thf(fact_68_inf__apply, axiom, (! [B5:$tType,A:$tType]: (((semilattice_inf @ B5) => ((inf_inf @ (A > B5)) = (^ [F2:(A > B5),G2:(A > B5),X3:A]: ((((inf_inf @ B5) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.56/1.44 thf(fact_69_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.56/1.44 thf(fact_70_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A]: (((((inf_inf @ A) @ X2) @ X2) = X2))))))).
% 2.56/1.44 thf(fact_71_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B)) = (((inf_inf @ A) @ A2) @ B)))))))).
% 2.56/1.44 thf(fact_72_hfset__hdiff, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: (((hF_Mirabelle_hfset @ (((minus_minus @ hF_Mirabelle_hf) @ X2) @ Y2)) = (((minus_minus @ (set @ hF_Mirabelle_hf)) @ (hF_Mirabelle_hfset @ X2)) @ (hF_Mirabelle_hfset @ Y2)))))).
% 2.56/1.44 thf(fact_73_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_74_fun__diff__def, axiom, (! [B5:$tType,A:$tType]: (((minus @ B5) => ((minus_minus @ (A > B5)) = (^ [A5:(A > B5),B4:(A > B5),X3:A]: ((((minus_minus @ B5) @ (A5 @ X3)) @ (B4 @ X3))))))))).
% 2.56/1.44 thf(fact_75_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((A2 = B) = (C = D))))))))).
% 2.56/1.44 thf(fact_76_diff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A2:A,C:A,B:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C)) @ B) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B)) @ C)))))))).
% 2.56/1.44 thf(fact_77_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y2:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ X2) @ Y2)) = (((inf_inf @ A) @ X2) @ Y2)))))))).
% 2.56/1.44 thf(fact_78_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y2:A,Z4:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y2) @ Z4)) = (((inf_inf @ A) @ Y2) @ (((inf_inf @ A) @ X2) @ Z4))))))))).
% 2.56/1.44 thf(fact_79_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y2:A,Z4:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y2)) @ Z4) = (((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y2) @ Z4))))))))).
% 2.56/1.44 thf(fact_80_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X3:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X3)))))))).
% 2.56/1.44 thf(fact_81_inf__fun__def, axiom, (! [B5:$tType,A:$tType]: (((semilattice_inf @ B5) => ((inf_inf @ (A > B5)) = (^ [F2:(A > B5),G2:(A > B5),X3:A]: ((((inf_inf @ B5) @ (F2 @ X3)) @ (G2 @ X3))))))))).
% 2.56/1.44 thf(fact_82_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:A,K:A,A2:A,B:A]: (((A3 = (((inf_inf @ A) @ K) @ A2)) => ((((inf_inf @ A) @ A3) @ B) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B)))))))))).
% 2.56/1.44 thf(fact_83_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B3:A,K:A,B:A,A2:A]: (((B3 = (((inf_inf @ A) @ K) @ B)) => ((((inf_inf @ A) @ A2) @ B3) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B)))))))))).
% 2.56/1.44 thf(fact_84_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B) @ C))))))))).
% 2.56/1.44 thf(fact_85_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y2:A,Z4:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y2)) @ Z4) = (((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y2) @ Z4))))))))).
% 2.56/1.44 thf(fact_86_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A4:A,B2:A]: ((((inf_inf @ A) @ B2) @ A4)))))))).
% 2.56/1.44 thf(fact_87_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X3:A,Y3:A]: ((((inf_inf @ A) @ Y3) @ X3)))))))).
% 2.56/1.44 thf(fact_88_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,A2:A,C:A]: (((((inf_inf @ A) @ B) @ (((inf_inf @ A) @ A2) @ C)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B) @ C))))))))).
% 2.56/1.44 thf(fact_89_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y2:A,Z4:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y2) @ Z4)) = (((inf_inf @ A) @ Y2) @ (((inf_inf @ A) @ X2) @ Z4))))))))).
% 2.56/1.44 thf(fact_90_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [A4:A,B2:A]: (((((minus_minus @ A) @ A4) @ B2) = (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_91_HInter__hinsert, axiom, (! [A3:hF_Mirabelle_hf,A2:hF_Mirabelle_hf]: (((A3 != (zero_zero @ hF_Mirabelle_hf)) => ((hF_Mirabelle_HInter @ ((hF_Mirabelle_hinsert @ A2) @ A3)) = (((inf_inf @ hF_Mirabelle_hf) @ A2) @ (hF_Mirabelle_HInter @ A3))))))).
% 2.56/1.44 thf(fact_92_fun__cong__unused__0, axiom, (! [A:$tType,B5:$tType,C2:$tType]: (((zero @ B5) => (! [F:((A > B5) > C2),G:C2]: (((F = (^ [X3:(A > B5)]: (G))) => ((F @ (^ [X3:A]: ((zero_zero @ B5)))) = G)))))))).
% 2.56/1.44 thf(fact_93_HCollect__def, axiom, (hF_Mir1687042746ollect = (^ [P2:(hF_Mirabelle_hf > $o),A5:hF_Mirabelle_hf]: (((the @ hF_Mirabelle_hf) @ (^ [Z3:hF_Mirabelle_hf]: ((! [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ Z3) = ((P2 @ U2) & ((hF_Mirabelle_hmem @ U2) @ A5)))))))))))).
% 2.56/1.44 thf(fact_94_HF__hfset, axiom, (! [A2:hF_Mirabelle_hf]: (((hF_Mirabelle_HF @ (hF_Mirabelle_hfset @ A2)) = A2)))).
% 2.56/1.44 thf(fact_95_zero__notin__hpair, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((~ (((hF_Mirabelle_hmem @ (zero_zero @ hF_Mirabelle_hf)) @ ((hF_Mirabelle_hpair @ X2) @ Y2))))))).
% 2.56/1.44 thf(fact_96_Abs__hf__0, axiom, ((hF_Mirabelle_Abs_hf @ (zero_zero @ nat)) = (zero_zero @ hF_Mirabelle_hf))).
% 2.56/1.44 thf(fact_97_hfset__0, axiom, ((hF_Mirabelle_hfset @ (zero_zero @ hF_Mirabelle_hf)) = (bot_bot @ (set @ hF_Mirabelle_hf)))).
% 2.56/1.44 thf(fact_98_hpair__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,A7:hF_Mirabelle_hf,B6:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A2) @ B) = ((hF_Mirabelle_hpair @ A7) @ B6)) = ((A2 = A7) & (B = B6)))))).
% 2.56/1.44 thf(fact_99_inf__bot__right, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X2:A]: (((((inf_inf @ A) @ X2) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 2.56/1.44 thf(fact_100_inf__bot__left, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X2:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X2) = (bot_bot @ A)))))))).
% 2.56/1.44 thf(fact_101_hmem__hinsert, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,C:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ A2) @ ((hF_Mirabelle_hinsert @ B) @ C)) = ((A2 = B) | ((hF_Mirabelle_hmem @ A2) @ C)))))).
% 2.56/1.44 thf(fact_102_singleton__eq__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A2) @ (zero_zero @ hF_Mirabelle_hf)) = ((hF_Mirabelle_hinsert @ B) @ (zero_zero @ hF_Mirabelle_hf))) = (A2 = B))))).
% 2.56/1.44 thf(fact_103_RepFun__hinsert, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,F:(hF_Mirabelle_hf > hF_Mirabelle_hf)]: ((((hF_Mirabelle_RepFun @ ((hF_Mirabelle_hinsert @ A2) @ B)) @ F) = ((hF_Mirabelle_hinsert @ (F @ A2)) @ ((hF_Mirabelle_RepFun @ B) @ F)))))).
% 2.56/1.44 thf(fact_104_hpair__def, axiom, (hF_Mirabelle_hpair = (^ [A4:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ (zero_zero @ hF_Mirabelle_hf))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ B2) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.56/1.44 thf(fact_105_hpair__def_H, axiom, (hF_Mirabelle_hpair = (^ [A4:hF_Mirabelle_hf,B2:hF_Mirabelle_hf]: (((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ A4) @ (zero_zero @ hF_Mirabelle_hf)))) @ ((hF_Mirabelle_hinsert @ ((hF_Mirabelle_hinsert @ A4) @ ((hF_Mirabelle_hinsert @ B2) @ (zero_zero @ hF_Mirabelle_hf)))) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.56/1.44 thf(fact_106_Zero__hf__def, axiom, ((zero_zero @ hF_Mirabelle_hf) = (hF_Mirabelle_HF @ (bot_bot @ (set @ hF_Mirabelle_hf))))).
% 2.56/1.44 thf(fact_107_hpair__neq__snd, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A2) @ B) != B)))).
% 2.56/1.44 thf(fact_108_hpair__neq__fst, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ A2) @ B) != A2)))).
% 2.56/1.44 thf(fact_109_hpair__inject, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,A7:hF_Mirabelle_hf,B6:hF_Mirabelle_hf]: (((((hF_Mirabelle_hpair @ A2) @ B) = ((hF_Mirabelle_hpair @ A7) @ B6)) => (~ (((A2 = A7) => (B != B6)))))))).
% 2.56/1.44 thf(fact_110_hinsert__commute, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf,Z4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ X2) @ ((hF_Mirabelle_hinsert @ Y2) @ Z4)) = ((hF_Mirabelle_hinsert @ Y2) @ ((hF_Mirabelle_hinsert @ X2) @ Z4)))))).
% 2.56/1.44 thf(fact_111_hinsert__iff, axiom, (! [Z4:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf,X2:hF_Mirabelle_hf]: (((Z4 = ((hF_Mirabelle_hinsert @ Y2) @ X2)) = (! [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ Z4) = (((hF_Mirabelle_hmem @ U2) @ X2) | (U2 = Y2))))))))).
% 2.56/1.44 thf(fact_112_hinsert__nonempty, axiom, (! [A2:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hinsert @ A2) @ A3) != (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.44 thf(fact_113_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf,C:hF_Mirabelle_hf,D:hF_Mirabelle_hf]: (((((hF_Mirabelle_hinsert @ A2) @ ((hF_Mirabelle_hinsert @ B) @ (zero_zero @ hF_Mirabelle_hf))) = ((hF_Mirabelle_hinsert @ C) @ ((hF_Mirabelle_hinsert @ D) @ (zero_zero @ hF_Mirabelle_hf)))) = (((A2 = C) & (B = D)) | ((A2 = D) & (B = C))))))).
% 2.56/1.44 thf(fact_114_hf__induct__ax, axiom, (! [P:(hF_Mirabelle_hf > $o),X2:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X:hF_Mirabelle_hf]: (((P @ X) => (! [Y4:hF_Mirabelle_hf]: (((P @ Y4) => (P @ ((hF_Mirabelle_hinsert @ Y4) @ X)))))))) => (P @ X2)))))).
% 2.56/1.44 thf(fact_115_hpair__nonzero, axiom, (! [X2:hF_Mirabelle_hf,Y2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hpair @ X2) @ Y2) != (zero_zero @ hF_Mirabelle_hf))))).
% 2.56/1.44 thf(fact_116_hf__cases, axiom, (! [Y2:hF_Mirabelle_hf]: (((Y2 != (zero_zero @ hF_Mirabelle_hf)) => (~ ((! [A8:hF_Mirabelle_hf,B7:hF_Mirabelle_hf]: (((Y2 = ((hF_Mirabelle_hinsert @ A8) @ B7)) => ((hF_Mirabelle_hmem @ A8) @ B7)))))))))).
% 2.56/1.44 thf(fact_117_hf__induct, axiom, (! [P:(hF_Mirabelle_hf > $o),Z4:hF_Mirabelle_hf]: (((P @ (zero_zero @ hF_Mirabelle_hf)) => ((! [X:hF_Mirabelle_hf,Y4:hF_Mirabelle_hf]: (((P @ X) => ((P @ Y4) => ((~ (((hF_Mirabelle_hmem @ X) @ Y4))) => (P @ ((hF_Mirabelle_hinsert @ X) @ Y4))))))) => (P @ Z4)))))).
% 2.56/1.44 thf(fact_118_hinter__hinsert__right, axiom, (! [X2:hF_Mirabelle_hf,B3:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X2) @ B3) => ((((inf_inf @ hF_Mirabelle_hf) @ B3) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) = ((hF_Mirabelle_hinsert @ X2) @ (((inf_inf @ hF_Mirabelle_hf) @ B3) @ A3)))) & ((~ (((hF_Mirabelle_hmem @ X2) @ B3))) => ((((inf_inf @ hF_Mirabelle_hf) @ B3) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) = (((inf_inf @ hF_Mirabelle_hf) @ B3) @ A3))))))).
% 2.56/1.44 thf(fact_119_hinter__hinsert__left, axiom, (! [X2:hF_Mirabelle_hf,B3:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X2) @ B3) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) @ B3) = ((hF_Mirabelle_hinsert @ X2) @ (((inf_inf @ hF_Mirabelle_hf) @ A3) @ B3)))) & ((~ (((hF_Mirabelle_hmem @ X2) @ B3))) => ((((inf_inf @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) @ B3) = (((inf_inf @ hF_Mirabelle_hf) @ A3) @ B3))))))).
% 2.56/1.44 thf(fact_120_hinsert__hdiff__if, axiom, (! [X2:hF_Mirabelle_hf,B3:hF_Mirabelle_hf,A3:hF_Mirabelle_hf]: (((((hF_Mirabelle_hmem @ X2) @ B3) => ((((minus_minus @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) @ B3) = (((minus_minus @ hF_Mirabelle_hf) @ A3) @ B3))) & ((~ (((hF_Mirabelle_hmem @ X2) @ B3))) => ((((minus_minus @ hF_Mirabelle_hf) @ ((hF_Mirabelle_hinsert @ X2) @ A3)) @ B3) = ((hF_Mirabelle_hinsert @ X2) @ (((minus_minus @ hF_Mirabelle_hf) @ A3) @ B3)))))))).
% 2.56/1.44 thf(fact_121_hdiff__insert, axiom, (! [A3:hF_Mirabelle_hf,A2:hF_Mirabelle_hf,B3:hF_Mirabelle_hf]: (((((minus_minus @ hF_Mirabelle_hf) @ A3) @ ((hF_Mirabelle_hinsert @ A2) @ B3)) = (((minus_minus @ hF_Mirabelle_hf) @ (((minus_minus @ hF_Mirabelle_hf) @ A3) @ B3)) @ ((hF_Mirabelle_hinsert @ A2) @ (zero_zero @ hF_Mirabelle_hf))))))).
% 2.56/1.44 thf(fact_122_hsplit, axiom, (! [A:$tType,C:(hF_Mirabelle_hf > (hF_Mirabelle_hf > A)),A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((((hF_Mirabelle_hsplit @ A) @ C) @ ((hF_Mirabelle_hpair @ A2) @ B)) = ((C @ A2) @ B))))).
% 2.56/1.44 thf(fact_123_Diff__disjoint, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) = (bot_bot @ (set @ A)))))).
% 2.56/1.44 thf(fact_124_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 2.56/1.44 thf(fact_125_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 2.56/1.44 thf(fact_126_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.56/1.44 thf(fact_127_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.56/1.44 thf(fact_128_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.56/1.44 thf(fact_129_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.56/1.44 thf(fact_130_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X3:A]: ((~ ((P @ X3))))))))).
% 2.56/1.44 thf(fact_131_Int__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C) @ A3) & (((member @ A) @ C) @ B3)))))).
% 2.56/1.44 thf(fact_132_IntI, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A3) => ((((member @ A) @ C) @ B3) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))).
% 2.56/1.44 thf(fact_133_DiffI, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A3) => ((~ ((((member @ A) @ C) @ B3))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B3))))))).
% 2.56/1.44 thf(fact_134_Diff__iff, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C) @ A3) & (~ ((((member @ A) @ C) @ B3)))))))).
% 2.56/1.44 thf(fact_135_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))))).
% 2.56/1.44 thf(fact_136_Int__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C3)) = (((inf_inf @ (set @ A)) @ B3) @ (((inf_inf @ (set @ A)) @ A3) @ C3)))))).
% 2.56/1.44 thf(fact_137_Int__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))).
% 2.56/1.44 thf(fact_138_Collect__conj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3))))) = (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.56/1.44 thf(fact_139_inf__set__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (((inf_inf @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B4))))))))))).
% 2.56/1.44 thf(fact_140_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: ((((inf_inf @ (set @ A)) @ B4) @ A5))))))).
% 2.56/1.44 thf(fact_141_Int__Collect, axiom, (! [A:$tType,X2:A,A3:(set @ A),P:(A > $o)]: (((((member @ A) @ X2) @ (((inf_inf @ (set @ A)) @ A3) @ ((collect @ A) @ P))) = ((((member @ A) @ X2) @ A3) & (P @ X2)))))).
% 2.56/1.44 thf(fact_142_Int__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ A3) = A3)))).
% 2.56/1.44 thf(fact_143_Int__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C3) = (((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C3)))))).
% 2.56/1.44 thf(fact_144_Int__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A5) & (((member @ A) @ X3) @ B4))))))))))).
% 2.56/1.44 thf(fact_145_IntD2, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C) @ B3))))).
% 2.56/1.44 thf(fact_146_IntD1, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C) @ A3))))).
% 2.56/1.44 thf(fact_147_IntE, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C) @ A3) => (~ ((((member @ A) @ C) @ B3)))))))))).
% 2.56/1.44 thf(fact_148_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.56/1.44 thf(fact_149_bot__hf__def, axiom, ((bot_bot @ hF_Mirabelle_hf) = (zero_zero @ hF_Mirabelle_hf))).
% 2.56/1.44 thf(fact_150_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.56/1.44 thf(fact_151_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.56/1.44 thf(fact_152_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.56/1.44 thf(fact_153_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.56/1.44 thf(fact_154_Int__emptyI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A3) => (~ ((((member @ A) @ X) @ B3)))))) => ((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))))))).
% 2.56/1.44 thf(fact_155_disjoint__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (! [X3:A]: (((((member @ A) @ X3) @ A3) => (~ ((((member @ A) @ X3) @ B3)))))))))).
% 2.56/1.44 thf(fact_156_Int__empty__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (bot_bot @ (set @ A))) @ B3) = (bot_bot @ (set @ A)))))).
% 2.56/1.44 thf(fact_157_Int__empty__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.56/1.44 thf(fact_158_disjoint__iff__not__equal, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (! [X3:A]: (((((member @ A) @ X3) @ A3) => (! [Y3:A]: (((((member @ A) @ Y3) @ B3) => (X3 != Y3))))))))))).
% 2.56/1.44 thf(fact_159_DiffE, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B3)))))))).
% 2.56/1.44 thf(fact_160_DiffD1, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C) @ A3))))).
% 2.56/1.44 thf(fact_161_DiffD2, axiom, (! [A:$tType,C:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ ((((member @ A) @ C) @ B3))))))).
% 2.56/1.44 thf(fact_162_Int__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C3) = (((inf_inf @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ C3)))))).
% 2.56/1.44 thf(fact_163_Diff__Int2, axiom, (! [A:$tType,A3:(set @ A),C3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C3)) @ (((inf_inf @ (set @ A)) @ B3) @ C3)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C3)) @ B3))))).
% 2.56/1.44 thf(fact_164_Diff__Diff__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))).
% 2.56/1.44 thf(fact_165_Diff__Int__distrib, axiom, (! [A:$tType,C3:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ C3) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ C3) @ A3)) @ (((inf_inf @ (set @ A)) @ C3) @ B3)))))).
% 2.56/1.44 thf(fact_166_Diff__Int__distrib2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ C3) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C3)) @ (((inf_inf @ (set @ A)) @ B3) @ C3)))))).
% 2.56/1.44 thf(fact_167_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X3:A]: ($false))))))).
% 2.56/1.44 thf(fact_168_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A5) & (~ ((((member @ A) @ X3) @ B4))))))))))))).
% 2.56/1.44 thf(fact_169_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B4))))))))))).
% 2.56/1.44 thf(fact_170_Diff__triv, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) => ((((minus_minus @ (set @ A)) @ A3) @ B3) = A3))))).
% 2.56/1.44 thf(fact_171_hfst__conv, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((hF_Mirabelle_hfst @ ((hF_Mirabelle_hpair @ A2) @ B)) = A2)))).
% 2.56/1.44 thf(fact_172_hsnd__conv, axiom, (! [A2:hF_Mirabelle_hf,B:hF_Mirabelle_hf]: (((hF_Mirabelle_hsnd @ ((hF_Mirabelle_hpair @ A2) @ B)) = B)))).
% 2.56/1.44 thf(fact_173_the__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X:A]: (((P @ X) => (X = A2)))) => (((the @ A) @ P) = A2)))))).
% 2.56/1.44 thf(fact_174_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X3:A]: ((X3 = A2)))) = A2)))).
% 2.56/1.44 thf(fact_175_inf1I, axiom, (! [A:$tType,A3:(A > $o),X2:A,B3:(A > $o)]: (((A3 @ X2) => ((B3 @ X2) => ((((inf_inf @ (A > $o)) @ A3) @ B3) @ X2)))))).
% 2.56/1.44 thf(fact_176_the__sym__eq__trivial, axiom, (! [A:$tType,X2:A]: ((((the @ A) @ ((^ [Y:A,Z:A]: ((Y = Z))) @ X2)) = X2)))).
% 2.56/1.44 thf(fact_177_inf1D2, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X2:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X2) => (B3 @ X2))))).
% 2.56/1.44 thf(fact_178_inf1D1, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X2:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X2) => (A3 @ X2))))).
% 2.56/1.44 thf(fact_179_inf1E, axiom, (! [A:$tType,A3:(A > $o),B3:(A > $o),X2:A]: ((((((inf_inf @ (A > $o)) @ A3) @ B3) @ X2) => (~ (((A3 @ X2) => (~ ((B3 @ X2)))))))))).
% 2.56/1.44 thf(fact_180_the1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X4:A]: (((P @ X4) & (! [Y4:A]: (((P @ Y4) => (Y4 = X4))))))) => ((P @ A2) => (((the @ A) @ P) = A2)))))).
% 2.56/1.44 thf(fact_181_the1I2, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X4:A]: (((P @ X4) & (! [Y4:A]: (((P @ Y4) => (Y4 = X4))))))) => ((! [X:A]: (((P @ X) => (Q @ X)))) => (Q @ ((the @ A) @ P))))))).
% 2.56/1.44 thf(fact_182_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P2:$o,X3:A,Y3:A]: (((the @ A) @ (^ [Z3:A]: (((P2 => (Z3 = X3)) & ((~ (P2)) => (Z3 = Y3)))))))))))).
% 2.56/1.44 thf(fact_183_theI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X:A]: (((P @ X) => (X = A2)))) => ((! [X:A]: (((P @ X) => (Q @ X)))) => (Q @ ((the @ A) @ P)))))))).
% 2.56/1.44 thf(fact_184_theI_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X4:A]: (((P @ X4) & (! [Y4:A]: (((P @ Y4) => (Y4 = X4))))))) => (P @ ((the @ A) @ P)))))).
% 2.56/1.44 thf(fact_185_theI, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X:A]: (((P @ X) => (X = A2)))) => (P @ ((the @ A) @ P))))))).
% 2.56/1.44 thf(fact_186_bot__apply, axiom, (! [C2:$tType,D2:$tType]: (((bot @ C2) => ((bot_bot @ (D2 > C2)) = (^ [X3:D2]: ((bot_bot @ C2)))))))).
% 2.56/1.44 thf(fact_187_inf__Int__eq, axiom, (! [A:$tType,R:(set @ A),S:(set @ A)]: (((((inf_inf @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S)))) = (^ [X3:A]: ((((member @ A) @ X3) @ (((inf_inf @ (set @ A)) @ R) @ S)))))))).
% 2.56/1.44 thf(fact_188_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))).
% 2.56/1.44 thf(fact_189_diff__0__eq__0, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.56/1.44 thf(fact_190_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))).
% 2.56/1.44 thf(fact_191_bot__nat__def, axiom, ((bot_bot @ nat) = (zero_zero @ nat))).
% 2.56/1.44 thf(fact_192_diffs0__imp__equal, axiom, (! [M:nat,N:nat]: ((((((minus_minus @ nat) @ M) @ N) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N) @ M) = (zero_zero @ nat)) => (M = N)))))).
% 2.56/1.44 thf(fact_193_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))).
% 2.56/1.44 thf(fact_194_bot__fun__def, axiom, (! [B5:$tType,A:$tType]: (((bot @ B5) => ((bot_bot @ (A > B5)) = (^ [X3:A]: ((bot_bot @ B5)))))))).
% 2.56/1.44 thf(fact_195_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.56/1.44 thf(fact_196_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.56/1.44 thf(fact_197_ord_OLeast__def, axiom, (! [A:$tType]: (((least @ A) = (^ [Less_eq:(A > (A > $o)),P2:(A > $o)]: (((the @ A) @ (^ [X3:A]: (((P2 @ X3) & (! [Y3:A]: (((P2 @ Y3) => ((Less_eq @ X3) @ Y3)))))))))))))).
% 2.56/1.44 thf(fact_198_PrimReplace__def, axiom, (hF_Mir569462966eplace = (^ [A5:hF_Mirabelle_hf,R2:(hF_Mirabelle_hf > (hF_Mirabelle_hf > $o))]: (((the @ hF_Mirabelle_hf) @ (^ [Z3:hF_Mirabelle_hf]: ((! [V5:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ V5) @ Z3) = (? [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ A5) & ((R2 @ U2) @ V5)))))))))))))).
% 2.56/1.44 thf(fact_199_hcard__0, axiom, ((hF_Mirabelle_hcard @ (zero_zero @ hF_Mirabelle_hf)) = (zero_zero @ nat))).
% 2.56/1.44 thf(fact_200_hfst__def, axiom, (hF_Mirabelle_hfst = (^ [P4:hF_Mirabelle_hf]: (((the @ hF_Mirabelle_hf) @ (^ [X3:hF_Mirabelle_hf]: ((? [Y3:hF_Mirabelle_hf]: ((P4 = ((hF_Mirabelle_hpair @ X3) @ Y3))))))))))).
% 2.56/1.44 thf(fact_201_hsnd__def, axiom, (hF_Mirabelle_hsnd = (^ [P4:hF_Mirabelle_hf]: (((the @ hF_Mirabelle_hf) @ (^ [Y3:hF_Mirabelle_hf]: ((? [X3:hF_Mirabelle_hf]: ((P4 = ((hF_Mirabelle_hpair @ X3) @ Y3))))))))))).
% 2.56/1.44 thf(fact_202_HUnion__def, axiom, (hF_Mirabelle_HUnion = (^ [A5:hF_Mirabelle_hf]: (((the @ hF_Mirabelle_hf) @ (^ [Z3:hF_Mirabelle_hf]: ((! [U2:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ U2) @ Z3) = (? [Y3:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ Y3) @ A5) & ((hF_Mirabelle_hmem @ U2) @ Y3)))))))))))))).
% 2.56/1.44 thf(fact_203_hcard__hdiff1__less, axiom, (! [X2:hF_Mirabelle_hf,Z4:hF_Mirabelle_hf]: ((((hF_Mirabelle_hmem @ X2) @ Z4) => (((ord_less @ nat) @ (hF_Mirabelle_hcard @ (((minus_minus @ hF_Mirabelle_hf) @ Z4) @ ((hF_Mirabelle_hinsert @ X2) @ (zero_zero @ hF_Mirabelle_hf))))) @ (hF_Mirabelle_hcard @ Z4)))))).
% 2.56/1.44 thf(fact_204_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.56/1.44 thf(fact_205_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_206_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.56/1.44 thf(fact_207_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.56/1.44 thf(fact_208_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.56/1.44 thf(fact_209_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B)) = (((ord_less @ A) @ B) @ A2)))))))).
% 2.56/1.44 thf(fact_210_zero__less__diff, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M)) = (((ord_less @ nat) @ M) @ N))))).
% 2.56/1.44 thf(fact_211_less__imp__diff__less, axiom, (! [J:nat,K:nat,N:nat]: (((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ N)) @ K))))).
% 2.56/1.44 thf(fact_212_diff__less__mono2, axiom, (! [M:nat,N:nat,L:nat]: (((((ord_less @ nat) @ M) @ N) => ((((ord_less @ nat) @ M) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N)) @ (((minus_minus @ nat) @ L) @ M))))))).
% 2.56/1.44 thf(fact_213_diff__less, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N)) @ M)))))).
% 2.56/1.44 thf(fact_214_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ C))))))))).
% 2.56/1.44 thf(fact_215_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C) @ A2)) @ (((minus_minus @ A) @ C) @ B))))))))).
% 2.56/1.44 thf(fact_216_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,C:A,D:A]: ((((((minus_minus @ A) @ A2) @ B) = (((minus_minus @ A) @ C) @ D)) => ((((ord_less @ A) @ A2) @ B) = (((ord_less @ A) @ C) @ D))))))))).
% 2.56/1.44 thf(fact_217_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B:A,D:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ D) @ C) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C)) @ (((minus_minus @ A) @ B) @ D)))))))))).
% 2.56/1.44 thf(fact_218_infinite__descent0__measure, axiom, (! [A:$tType,V6:(A > nat),P:(A > $o),X2:A]: (((! [X:A]: ((((V6 @ X) = (zero_zero @ nat)) => (P @ X)))) => ((! [X:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V6 @ X)) => ((~ ((P @ X))) => (? [Y5:A]: (((((ord_less @ nat) @ (V6 @ Y5)) @ (V6 @ X)) & (~ ((P @ Y5)))))))))) => (P @ X2)))))).
% 2.56/1.44 thf(fact_219_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.56/1.44 thf(fact_220_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 2.56/1.44 thf(fact_221_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.56/1.44 thf(fact_222_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.56/1.44 thf(fact_223_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.56/1.44 thf(fact_224_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.56/1.44 thf(fact_225_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.56/1.44 thf(fact_226_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A4:A,B2:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A4) @ B2)) @ (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_227_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_228_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.56/1.44 thf(fact_229_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.56/1.44 thf(fact_230_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.56/1.44 thf(fact_231_less__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X2:A,B:A]: (((((ord_less @ A) @ A2) @ X2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B)) @ X2)))))))).
% 2.56/1.44 thf(fact_232_less__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,X2:A,A2:A]: (((((ord_less @ A) @ B) @ X2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B)) @ X2)))))))).
% 2.56/1.44 thf(fact_233_inf_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ (((inf_inf @ A) @ B) @ C)) => (~ (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ A2) @ C))))))))))))).
% 2.56/1.44 thf(fact_234_inf_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less @ A) = (^ [A4:A,B2:A]: (((A4 = (((inf_inf @ A) @ A4) @ B2)) & (A4 != B2))))))))).
% 2.56/1.44 thf(fact_235_inf_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ A2) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C)))))))).
% 2.56/1.44 thf(fact_236_inf_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B:A,C:A,A2:A]: (((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B)) @ C)))))))).
% 2.56/1.44 thf(fact_237_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.56/1.44 thf(fact_238_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.56/1.44 thf(fact_239_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.56/1.44 thf(fact_240_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 2.56/1.44 thf(fact_241_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.56/1.44 thf(fact_242_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.56/1.44 thf(fact_243_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.56/1.44 thf(fact_244_psubset__imp__ex__mem, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (? [B7:A]: ((((member @ A) @ B7) @ (((minus_minus @ (set @ A)) @ B3) @ A3)))))))).
% 2.56/1.44 thf(fact_245_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 2.56/1.44 thf(fact_246_not__psubset__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.56/1.44 thf(fact_247_pos__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N2:nat]: (((K = ((semiring_1_of_nat @ int) @ N2)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))))))))).
% 2.56/1.44 thf(fact_248_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (K = ((semiring_1_of_nat @ int) @ N2))))))))).
% 2.56/1.44 thf(fact_249_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B3)))))).
% 2.56/1.44 thf(fact_250_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B4:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ A5)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ B4)))))))))).
% 2.56/1.44 thf(fact_251_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.56/1.44 thf(fact_252_int__diff__cases, axiom, (! [Z4:int]: ((~ ((! [M3:nat,N2:nat]: ((Z4 != (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ M3)) @ ((semiring_1_of_nat @ int) @ N2)))))))))).
% 2.56/1.44 thf(fact_253_minus__int__code_I1_J, axiom, (! [K:int]: (((((minus_minus @ int) @ K) @ (zero_zero @ int)) = K)))).
% 2.56/1.44 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.56/1.44 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A9:$tType]: ((bounded_lattice @ (set @ A9))))).
% 2.56/1.44 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A9:$tType,A10:$tType]: (((bounded_lattice @ A10) => (bounded_lattice @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A9:$tType,A10:$tType]: (((bounded_lattice @ A10) => (bounded_lattice_bot @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_inf @ A10) => (semilattice_inf @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: (((order_bot @ A10) => (order_bot @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Lattices_Olattice, axiom, (! [A9:$tType,A10:$tType]: (((lattice @ A10) => (lattice @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_fun___Groups_Ominus, axiom, (! [A9:$tType,A10:$tType]: (((minus @ A10) => (minus @ (A9 > A10)))))).
% 2.56/1.44 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel146912293up_add @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1352612707id_add @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3, axiom, (semilattice_inf @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Lattices_Olattice_4, axiom, (lattice @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Ominus_5, axiom, (minus @ int)).
% 2.56/1.44 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6, axiom, (linord1659791738miring @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7, axiom, (cancel146912293up_add @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8, axiom, (cancel1352612707id_add @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_9, axiom, (semilattice_inf @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Orderings_Oorder__bot_10, axiom, (order_bot @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11, axiom, (semiring_char_0 @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Rings_Osemiring__1_12, axiom, (semiring_1 @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Lattices_Olattice_13, axiom, (lattice @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Orderings_Obot_14, axiom, (bot @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ominus_15, axiom, (minus @ nat)).
% 2.56/1.44 thf(tcon_Nat_Onat___Groups_Ozero_16, axiom, (zero @ nat)).
% 2.56/1.44 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_17, axiom, (! [A9:$tType]: ((bounded_lattice_bot @ (set @ A9))))).
% 2.56/1.44 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_18, axiom, (! [A9:$tType]: ((semilattice_inf @ (set @ A9))))).
% 2.56/1.44 thf(tcon_Set_Oset___Orderings_Oorder__bot_19, axiom, (! [A9:$tType]: ((order_bot @ (set @ A9))))).
% 2.56/1.44 thf(tcon_Set_Oset___Lattices_Olattice_20, axiom, (! [A9:$tType]: ((lattice @ (set @ A9))))).
% 2.56/1.44 thf(tcon_Set_Oset___Orderings_Obot_21, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 2.56/1.44 thf(tcon_Set_Oset___Groups_Ominus_22, axiom, (! [A9:$tType]: ((minus @ (set @ A9))))).
% 2.56/1.44 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_23, axiom, (bounded_lattice_bot @ $o)).
% 2.56/1.44 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_24, axiom, (semilattice_inf @ $o)).
% 2.56/1.44 thf(tcon_HOL_Obool___Orderings_Oorder__bot_25, axiom, (order_bot @ $o)).
% 2.56/1.44 thf(tcon_HOL_Obool___Lattices_Olattice_26, axiom, (lattice @ $o)).
% 2.56/1.44 thf(tcon_HOL_Obool___Orderings_Obot_27, axiom, (bot @ $o)).
% 2.56/1.44 thf(tcon_HOL_Obool___Groups_Ominus_28, axiom, (minus @ $o)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Obounded__lattice__bot_29, axiom, (bounded_lattice_bot @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_30, axiom, (semilattice_inf @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder__bot_31, axiom, (order_bot @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_32, axiom, (lattice @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Obot_33, axiom, (bot @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_34, axiom, (minus @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_35, axiom, (zero @ hF_Mirabelle_hf)).
% 2.56/1.44 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.56/1.44 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y2:A]: ((((((if @ A) @ $false) @ X2) @ Y2) = Y2)))).
% 2.56/1.44 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y2:A]: ((((((if @ A) @ $true) @ X2) @ Y2) = X2)))).
% 2.56/1.44 thf(conj_0, conjecture, (((hF_Mirabelle_HBex @ a) @ (^ [X3:hF_Mirabelle_hf]: (p))) = ((? [X3:hF_Mirabelle_hf]: (((hF_Mirabelle_hmem @ X3) @ a))) & p))).
% 2.56/1.44 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.yW9if3OHxl/DTF2DTF25592.p
% 2.56/1.44 ------------------------
% 2.56/1.47 ---- Cleaned THF ---
% 2.56/1.48 thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
% 2.56/1.48 hF_Mirabelle_hf: $tType ).
% 2.56/1.48
% 2.56/1.48 thf(ty_t_Set_Oset,type,
% 2.56/1.48 set: $tType > $tType ).
% 2.56/1.48
% 2.56/1.48 thf(ty_t_Nat_Onat,type,
% 2.56/1.48 nat: $tType ).
% 2.56/1.48
% 2.56/1.48 thf(ty_t_Int_Oint,type,
% 2.56/1.48 int: $tType ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.56/1.48 bounded_lattice:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_HOL_Otype,type,
% 2.56/1.48 type:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ozero,type,
% 2.56/1.48 zero:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ominus,type,
% 2.56/1.48 minus:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Orderings_Obot,type,
% 2.56/1.48 bot:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ogroup__add,type,
% 2.56/1.48 group_add:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Lattices_Olattice,type,
% 2.56/1.48 lattice:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Rings_Osemiring__1,type,
% 2.56/1.48 semiring_1:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.56/1.48 semiring_char_0:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.56/1.48 order_bot:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 2.56/1.48 comm_monoid_diff:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.56/1.48 semilattice_inf:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.56/1.48 ordered_ab_group_add:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.56/1.48 bounded_lattice_bot:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 2.56/1.48 cancel1352612707id_add:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.56/1.48 cancel146912293up_add:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.56/1.48 linord1659791738miring:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.56/1.48 canoni770627133id_add:
% 2.56/1.48 !>[A: $tType] : $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.56/1.48 minus_minus:
% 2.56/1.48 !>[A: $tType] : ( A > A > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.56/1.48 zero_zero:
% 2.56/1.48 !>[A: $tType] : A ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBall,type,
% 2.56/1.48 hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHBex,type,
% 2.56/1.48 hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
% 2.56/1.48 hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
% 2.56/1.48 hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHInter,type,
% 2.56/1.48 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
% 2.56/1.48 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
% 2.56/1.48 hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
% 2.56/1.48 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
% 2.56/1.48 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohcard,type,
% 2.56/1.48 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohf_OAbs__hf,type,
% 2.56/1.48 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
% 2.56/1.48 hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfst,type,
% 2.56/1.48 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohinsert,type,
% 2.56/1.48 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
% 2.56/1.48 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohpair,type,
% 2.56/1.48 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsnd,type,
% 2.56/1.48 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohsplit,type,
% 2.56/1.48 hF_Mirabelle_hsplit:
% 2.56/1.48 !>[A: $tType] : ( ( hF_Mirabelle_hf > hF_Mirabelle_hf > A ) > hF_Mirabelle_hf > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_HOL_OThe,type,
% 2.56/1.48 the:
% 2.56/1.48 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_If,type,
% 2.56/1.48 if:
% 2.56/1.48 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.56/1.48 inf_inf:
% 2.56/1.48 !>[A: $tType] : ( A > A > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.56/1.48 semiring_1_of_nat:
% 2.56/1.48 !>[A: $tType] : ( nat > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.56/1.48 bot_bot:
% 2.56/1.48 !>[A: $tType] : A ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Orderings_Oord_OLeast,type,
% 2.56/1.48 least:
% 2.56/1.48 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.56/1.48 ord_less:
% 2.56/1.48 !>[A: $tType] : ( A > A > $o ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Set_OCollect,type,
% 2.56/1.48 collect:
% 2.56/1.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_Set_Ois__empty,type,
% 2.56/1.48 is_empty:
% 2.56/1.48 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_c_member,type,
% 2.56/1.48 member:
% 2.56/1.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.56/1.48
% 2.56/1.48 thf(sy_v_A,type,
% 2.56/1.48 a: hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(sy_v_P,type,
% 2.56/1.48 p: $o ).
% 2.56/1.48
% 2.56/1.48 thf(fact_0_hf__equalityI,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 2.56/1.48 = ( hF_Mirabelle_hmem @ X @ B ) )
% 2.56/1.48 => ( A2 = B ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_1_hbexI,axiom,
% 2.56/1.48 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ X2 )
% 2.56/1.48 => ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 2.56/1.48 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_2_hbexE,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mirabelle_HBex @ A3 @ P )
% 2.56/1.48 => ~ ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ A3 )
% 2.56/1.48 => ~ ( P @ X ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_3_hf__ext,axiom,
% 2.56/1.48 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 2.56/1.48 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 2.56/1.48 = ( hF_Mirabelle_hmem @ X3 @ B2 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_4_hmem__ne,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 2.56/1.48 => ( X2 != Y2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_5_HBex__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_HBex
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ? [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X3 @ A5 )
% 2.56/1.48 & ( P2 @ X3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_6_hbex__cong,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( A3 = A6 )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ A6 )
% 2.56/1.48 => ( ( P @ X )
% 2.56/1.48 = ( P3 @ X ) ) )
% 2.56/1.48 => ( ( hF_Mirabelle_HBex @ A3 @ P )
% 2.56/1.48 = ( hF_Mirabelle_HBex @ A6 @ P3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_7_rev__hbexI,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 2.56/1.48 => ( ( P @ X2 )
% 2.56/1.48 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_8_replacement,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U @ X2 )
% 2.56/1.48 => ( ( R @ U @ V )
% 2.56/1.48 => ( ( R @ U @ V2 )
% 2.56/1.48 => ( V2 = V ) ) ) )
% 2.56/1.48 => ? [Z2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [V3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 2.56/1.48 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_9_binary__union,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ? [Z2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [U3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.56/1.48 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 2.56/1.48 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_10_hmem__not__sym,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ Y2 @ X2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_11_union__of__set,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf] :
% 2.56/1.48 ? [Z2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [U3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.56/1.48 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ Y3 @ X2 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_12_comprehension,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ? [Z2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [U3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 2.56/1.48 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 2.56/1.48 & ( P @ U3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_13_hmem__not__refl,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ X2 @ X2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_14_replacement__fun,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ? [Z2: hF_Mirabelle_hf] :
% 2.56/1.48 ! [V3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 2.56/1.48 & ( V3
% 2.56/1.48 = ( F @ U2 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_15_HF__Mirabelle__fsbjehakzm_ObexCI,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_HBall @ A3
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( P @ X3 ) )
% 2.56/1.48 => ( P @ A2 ) )
% 2.56/1.48 => ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 2.56/1.48 => ( hF_Mirabelle_HBex @ A3 @ P ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_16_hballI,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ A3 )
% 2.56/1.48 => ( P @ X ) )
% 2.56/1.48 => ( hF_Mirabelle_HBall @ A3 @ P ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_17_hball__triv,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: $o] :
% 2.56/1.48 ( ( hF_Mirabelle_HBall @ A3
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] : P )
% 2.56/1.48 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ A3 )
% 2.56/1.48 => P ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_18_PrimReplace__iff,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U @ A3 )
% 2.56/1.48 => ( ( R @ U @ V )
% 2.56/1.48 => ( ( R @ U @ V2 )
% 2.56/1.48 => ( V2 = V ) ) ) )
% 2.56/1.48 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir569462966eplace @ A3 @ R ) )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 2.56/1.48 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_19_HCollect__iff,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mir1687042746ollect @ P @ A3 ) )
% 2.56/1.48 = ( ( P @ X2 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ X2 @ A3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_20_Replace__iff,axiom,
% 2.56/1.48 ! [V4: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A3 @ R ) )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 2.56/1.48 & ( R @ U2 @ V4 )
% 2.56/1.48 & ! [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( R @ U2 @ Y3 )
% 2.56/1.48 => ( Y3 = V4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_21_HUnion__iff,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HUnion @ A3 ) )
% 2.56/1.48 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ Y3 @ A3 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_22_hballE,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_HBall @ A3 @ P )
% 2.56/1.48 => ( ~ ( P @ X2 )
% 2.56/1.48 => ~ ( hF_Mirabelle_hmem @ X2 @ A3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_23_hbspec,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_HBall @ A3 @ P )
% 2.56/1.48 => ( ( hF_Mirabelle_hmem @ X2 @ A3 )
% 2.56/1.48 => ( P @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_24_HCollectE,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mir1687042746ollect @ P @ A3 ) )
% 2.56/1.48 => ~ ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 2.56/1.48 => ~ ( P @ A2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_25_HCollectI,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ A2 @ A3 )
% 2.56/1.48 => ( ( P @ A2 )
% 2.56/1.48 => ( hF_Mirabelle_hmem @ A2 @ ( hF_Mir1687042746ollect @ P @ A3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_26_hball__cong,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( A3 = A6 )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ A6 )
% 2.56/1.48 => ( ( P @ X )
% 2.56/1.48 = ( P3 @ X ) ) )
% 2.56/1.48 => ( ( hF_Mirabelle_HBall @ A3 @ P )
% 2.56/1.48 = ( hF_Mirabelle_HBall @ A6 @ P3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_27_hmem__Sup__ne,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 2.56/1.48 => ( ( hF_Mirabelle_HUnion @ X2 )
% 2.56/1.48 != Y2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_28_Replace__cong,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( A3 = B3 )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 2.56/1.48 => ( ( P @ X @ Y4 )
% 2.56/1.48 = ( Q @ X @ Y4 ) ) )
% 2.56/1.48 => ( ( hF_Mirabelle_Replace @ A3 @ P )
% 2.56/1.48 = ( hF_Mirabelle_Replace @ B3 @ Q ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_29_HBall__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_HBall
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ! [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X3 @ A5 )
% 2.56/1.48 => ( P2 @ X3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_30_Replace__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_Replace
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( hF_Mir569462966eplace @ A5
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ? [Z3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( R2 @ X3 @ Z3 )
% 2.56/1.48 & ! [Aa: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( R2 @ X3 @ Aa )
% 2.56/1.48 => ( Aa = Z3 ) ) )
% 2.56/1.48 & ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_31_RepFun__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_RepFun
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ( hF_Mirabelle_Replace @ A5
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( Y3
% 2.56/1.48 = ( F2 @ X3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_32_minus__hf__def,axiom,
% 2.56/1.48 ( ( minus_minus @ hF_Mirabelle_hf )
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 2.56/1.48 ( hF_Mir1687042746ollect
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ X3 @ B4 )
% 2.56/1.48 @ A5 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_33_inf__hf__def,axiom,
% 2.56/1.48 ( ( inf_inf @ hF_Mirabelle_hf )
% 2.56/1.48 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 2.56/1.48 ( hF_Mir1687042746ollect
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ B2 )
% 2.56/1.48 @ A4 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_34_HInter__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_HInter
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf] :
% 2.56/1.48 ( hF_Mir1687042746ollect
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ! [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 2.56/1.48 => ( hF_Mirabelle_hmem @ X3 @ Y3 ) )
% 2.56/1.48 @ ( hF_Mirabelle_HUnion @ A5 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_35_RepFun__iff,axiom,
% 2.56/1.48 ! [V4: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A3 @ F ) )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ A3 )
% 2.56/1.48 & ( V4
% 2.56/1.48 = ( F @ U2 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_36_HCollect__hempty,axiom,
% 2.56/1.48 ! [P: hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mir1687042746ollect @ P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_37_Replace__0,axiom,
% 2.56/1.48 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_38_hmem__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_hmem
% 2.56/1.48 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A4 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_39_triv__RepFun,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_RepFun @ A3
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] : X3 )
% 2.56/1.48 = A3 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_40_hinter__iff,axiom,
% 2.56/1.48 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_inf @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 2.56/1.48 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_41_hdiff__iff,axiom,
% 2.56/1.48 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U4 @ ( minus_minus @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 2.56/1.48 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 2.56/1.48 & ~ ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_42_hinter__hempty__right,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( inf_inf @ hF_Mirabelle_hf @ A3 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_43_hinter__hempty__left,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( inf_inf @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ A3 )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_44_zero__hdiff,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( minus_minus @ hF_Mirabelle_hf @ ( zero_zero @ hF_Mirabelle_hf ) @ X2 )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_45_mem__Collect__eq,axiom,
% 2.56/1.48 ! [A: $tType,A2: A,P: A > $o] :
% 2.56/1.48 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.56/1.48 = ( P @ A2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_46_Collect__mem__eq,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( collect @ A
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 2.56/1.48 = A3 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_47_Collect__cong,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.56/1.48 ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 = ( Q @ X ) )
% 2.56/1.48 => ( ( collect @ A @ P )
% 2.56/1.48 = ( collect @ A @ Q ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_48_ext,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType,F: A > B5,G: A > B5] :
% 2.56/1.48 ( ! [X: A] :
% 2.56/1.48 ( ( F @ X )
% 2.56/1.48 = ( G @ X ) )
% 2.56/1.48 => ( F = G ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_49_hdiff__zero,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( minus_minus @ hF_Mirabelle_hf @ X2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = X2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_50_RepFun__0,axiom,
% 2.56/1.48 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_51_HUnion__hempty,axiom,
% 2.56/1.48 ( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_52_HInter__hempty,axiom,
% 2.56/1.48 ( ( hF_Mirabelle_HInter @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_53_HInter__iff,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( A3
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HInter @ A3 ) )
% 2.56/1.48 = ( ! [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ Y3 @ A3 )
% 2.56/1.48 => ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_54_foundation,axiom,
% 2.56/1.48 ! [Z4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( Z4
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ? [W: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ W @ Z4 )
% 2.56/1.48 & ( ( inf_inf @ hF_Mirabelle_hf @ W @ Z4 )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_55_hemptyE,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_56_hempty__iff,axiom,
% 2.56/1.48 ! [Z4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( Z4
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( ! [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ X3 @ Z4 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_57_hmem__hempty,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_58_RepFun__cong,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ( ( A3 = B3 )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 2.56/1.48 => ( ( F @ X )
% 2.56/1.48 = ( G @ X ) ) )
% 2.56/1.48 => ( ( hF_Mirabelle_RepFun @ A3 @ F )
% 2.56/1.48 = ( hF_Mirabelle_RepFun @ B3 @ G ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( cancel1352612707id_add @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ A2 @ A2 )
% 2.56/1.48 = ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_60_diff__zero,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( cancel1352612707id_add @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.56/1.48 = A2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_61_zero__diff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( comm_monoid_diff @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
% 2.56/1.48 = ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_62_diff__0__right,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( group_add @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
% 2.56/1.48 = A2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_63_diff__self,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( group_add @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ A2 @ A2 )
% 2.56/1.48 = ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_64_inf__right__idem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [X2: A,Y2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 )
% 2.56/1.48 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_65_inf_Oright__idem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,B: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ B )
% 2.56/1.48 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_66_inf__left__idem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [X2: A,Y2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
% 2.56/1.48 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_67_minus__apply,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType] :
% 2.56/1.48 ( ( minus @ B5 )
% 2.56/1.48 => ( ( minus_minus @ ( A > B5 ) )
% 2.56/1.48 = ( ^ [A5: A > B5,B4: A > B5,X3: A] : ( minus_minus @ B5 @ ( A5 @ X3 ) @ ( B4 @ X3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_68_inf__apply,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ B5 )
% 2.56/1.48 => ( ( inf_inf @ ( A > B5 ) )
% 2.56/1.48 = ( ^ [F2: A > B5,G2: A > B5,X3: A] : ( inf_inf @ B5 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_69_inf_Oidem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ A2 @ A2 )
% 2.56/1.48 = A2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_70_inf__idem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [X2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ X2 )
% 2.56/1.48 = X2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_71_inf_Oleft__idem,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,B: A] :
% 2.56/1.48 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B ) )
% 2.56/1.48 = ( inf_inf @ A @ A2 @ B ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_72_hfset__hdiff,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hfset @ ( minus_minus @ hF_Mirabelle_hf @ X2 @ Y2 ) )
% 2.56/1.48 = ( minus_minus @ ( set @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hfset @ X2 ) @ ( hF_Mirabelle_hfset @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_73_zero__reorient,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( zero @ A )
% 2.56/1.48 => ! [X2: A] :
% 2.56/1.48 ( ( ( zero_zero @ A )
% 2.56/1.48 = X2 )
% 2.56/1.48 = ( X2
% 2.56/1.48 = ( zero_zero @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_74_fun__diff__def,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType] :
% 2.56/1.48 ( ( minus @ B5 )
% 2.56/1.48 => ( ( minus_minus @ ( A > B5 ) )
% 2.56/1.48 = ( ^ [A5: A > B5,B4: A > B5,X3: A] : ( minus_minus @ B5 @ ( A5 @ X3 ) @ ( B4 @ X3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_75_diff__eq__diff__eq,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( group_add @ A )
% 2.56/1.48 => ! [A2: A,B: A,C: A,D: A] :
% 2.56/1.48 ( ( ( minus_minus @ A @ A2 @ B )
% 2.56/1.48 = ( minus_minus @ A @ C @ D ) )
% 2.56/1.48 => ( ( A2 = B )
% 2.56/1.48 = ( C = D ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_76_diff__right__commute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( cancel146912293up_add @ A )
% 2.56/1.48 => ! [A2: A,C: A,B: A] :
% 2.56/1.48 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B )
% 2.56/1.48 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_77_inf__sup__aci_I4_J,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( lattice @ A )
% 2.56/1.48 => ! [X2: A,Y2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
% 2.56/1.48 = ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_78_inf__sup__aci_I3_J,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( lattice @ A )
% 2.56/1.48 => ! [X2: A,Y2: A,Z4: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) )
% 2.56/1.48 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_79_inf__sup__aci_I2_J,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( lattice @ A )
% 2.56/1.48 => ! [X2: A,Y2: A,Z4: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z4 )
% 2.56/1.48 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_80_inf__sup__aci_I1_J,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( lattice @ A )
% 2.56/1.48 => ( ( inf_inf @ A )
% 2.56/1.48 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_81_inf__fun__def,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ B5 )
% 2.56/1.48 => ( ( inf_inf @ ( A > B5 ) )
% 2.56/1.48 = ( ^ [F2: A > B5,G2: A > B5,X3: A] : ( inf_inf @ B5 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_82_boolean__algebra__cancel_Oinf1,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A3: A,K: A,A2: A,B: A] :
% 2.56/1.48 ( ( A3
% 2.56/1.48 = ( inf_inf @ A @ K @ A2 ) )
% 2.56/1.48 => ( ( inf_inf @ A @ A3 @ B )
% 2.56/1.48 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_83_boolean__algebra__cancel_Oinf2,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [B3: A,K: A,B: A,A2: A] :
% 2.56/1.48 ( ( B3
% 2.56/1.48 = ( inf_inf @ A @ K @ B ) )
% 2.56/1.48 => ( ( inf_inf @ A @ A2 @ B3 )
% 2.56/1.48 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_84_inf_Oassoc,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,B: A,C: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B ) @ C )
% 2.56/1.48 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_85_inf__assoc,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [X2: A,Y2: A,Z4: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z4 )
% 2.56/1.48 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_86_inf_Ocommute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ( ( inf_inf @ A )
% 2.56/1.48 = ( ^ [A4: A,B2: A] : ( inf_inf @ A @ B2 @ A4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_87_inf__commute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ( ( inf_inf @ A )
% 2.56/1.48 = ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_88_inf_Oleft__commute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [B: A,A2: A,C: A] :
% 2.56/1.48 ( ( inf_inf @ A @ B @ ( inf_inf @ A @ A2 @ C ) )
% 2.56/1.48 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B @ C ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_89_inf__left__commute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [X2: A,Y2: A,Z4: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) )
% 2.56/1.48 = ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_90_eq__iff__diff__eq__0,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( group_add @ A )
% 2.56/1.48 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.56/1.48 = ( ^ [A4: A,B2: A] :
% 2.56/1.48 ( ( minus_minus @ A @ A4 @ B2 )
% 2.56/1.48 = ( zero_zero @ A ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_91_HInter__hinsert,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( A3
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A3 ) )
% 2.56/1.48 = ( inf_inf @ hF_Mirabelle_hf @ A2 @ ( hF_Mirabelle_HInter @ A3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_92_fun__cong__unused__0,axiom,
% 2.56/1.48 ! [A: $tType,B5: $tType,C2: $tType] :
% 2.56/1.48 ( ( zero @ B5 )
% 2.56/1.48 => ! [F: ( A > B5 ) > C2,G: C2] :
% 2.56/1.48 ( ( F
% 2.56/1.48 = ( ^ [X3: A > B5] : G ) )
% 2.56/1.48 => ( ( F
% 2.56/1.48 @ ^ [X3: A] : ( zero_zero @ B5 ) )
% 2.56/1.48 = G ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_93_HCollect__def,axiom,
% 2.56/1.48 ( hF_Mir1687042746ollect
% 2.56/1.48 = ( ^ [P2: hF_Mirabelle_hf > $o,A5: hF_Mirabelle_hf] :
% 2.56/1.48 ( the @ hF_Mirabelle_hf
% 2.56/1.48 @ ^ [Z3: hF_Mirabelle_hf] :
% 2.56/1.48 ! [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 2.56/1.48 = ( ( P2 @ U2 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ U2 @ A5 ) ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_94_HF__hfset,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
% 2.56/1.48 = A2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_95_zero__notin__hpair,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ~ ( hF_Mirabelle_hmem @ ( zero_zero @ hF_Mirabelle_hf ) @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_96_Abs__hf__0,axiom,
% 2.56/1.48 ( ( hF_Mirabelle_Abs_hf @ ( zero_zero @ nat ) )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_97_hfset__0,axiom,
% 2.56/1.48 ( ( hF_Mirabelle_hfset @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_98_hpair__iff,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.56/1.48 = ( hF_Mirabelle_hpair @ A7 @ B6 ) )
% 2.56/1.48 = ( ( A2 = A7 )
% 2.56/1.48 & ( B = B6 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_99_inf__bot__right,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( bounded_lattice_bot @ A )
% 2.56/1.48 => ! [X2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ X2 @ ( bot_bot @ A ) )
% 2.56/1.48 = ( bot_bot @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_100_inf__bot__left,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( bounded_lattice_bot @ A )
% 2.56/1.48 => ! [X2: A] :
% 2.56/1.48 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X2 )
% 2.56/1.48 = ( bot_bot @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_101_hmem__hinsert,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 2.56/1.48 = ( ( A2 = B )
% 2.56/1.48 | ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_102_singleton__eq__iff,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 2.56/1.48 = ( A2 = B ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_103_RepFun__hinsert,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_104_hpair__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_hpair
% 2.56/1.48 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_105_hpair__def_H,axiom,
% 2.56/1.48 ( hF_Mirabelle_hpair
% 2.56/1.48 = ( ^ [A4: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_106_Zero__hf__def,axiom,
% 2.56/1.48 ( ( zero_zero @ hF_Mirabelle_hf )
% 2.56/1.48 = ( hF_Mirabelle_HF @ ( bot_bot @ ( set @ hF_Mirabelle_hf ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_107_hpair__neq__snd,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.56/1.48 != B ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_108_hpair__neq__fst,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.56/1.48 != A2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_109_hpair__inject,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hpair @ A2 @ B )
% 2.56/1.48 = ( hF_Mirabelle_hpair @ A7 @ B6 ) )
% 2.56/1.48 => ~ ( ( A2 = A7 )
% 2.56/1.48 => ( B != B6 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_110_hinsert__commute,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hinsert @ X2 @ ( hF_Mirabelle_hinsert @ Y2 @ Z4 ) )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ Y2 @ ( hF_Mirabelle_hinsert @ X2 @ Z4 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_111_hinsert__iff,axiom,
% 2.56/1.48 ! [Z4: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( Z4
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ Y2 @ X2 ) )
% 2.56/1.48 = ( ! [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ Z4 )
% 2.56/1.48 = ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 2.56/1.48 | ( U2 = Y2 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_112_hinsert__nonempty,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hinsert @ A2 @ A3 )
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_113_HF__Mirabelle__fsbjehakzm_Odoubleton__eq__iff,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ ( zero_zero @ hF_Mirabelle_hf ) ) )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ ( zero_zero @ hF_Mirabelle_hf ) ) ) )
% 2.56/1.48 = ( ( ( A2 = C )
% 2.56/1.48 & ( B = D ) )
% 2.56/1.48 | ( ( A2 = D )
% 2.56/1.48 & ( B = C ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_114_hf__induct__ax,axiom,
% 2.56/1.48 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ! [Y4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ Y4 )
% 2.56/1.48 => ( P @ ( hF_Mirabelle_hinsert @ Y4 @ X ) ) ) )
% 2.56/1.48 => ( P @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_115_hpair__nonzero,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hpair @ X2 @ Y2 )
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_116_hf__cases,axiom,
% 2.56/1.48 ! [Y2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( Y2
% 2.56/1.48 != ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ~ ! [A8: hF_Mirabelle_hf,B7: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( Y2
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ A8 @ B7 ) )
% 2.56/1.48 => ( hF_Mirabelle_hmem @ A8 @ B7 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_117_hf__induct,axiom,
% 2.56/1.48 ! [P: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( ( P @ Y4 )
% 2.56/1.48 => ( ~ ( hF_Mirabelle_hmem @ X @ Y4 )
% 2.56/1.48 => ( P @ ( hF_Mirabelle_hinsert @ X @ Y4 ) ) ) ) )
% 2.56/1.48 => ( P @ Z4 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_118_hinter__hinsert__right,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( inf_inf @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_inf @ hF_Mirabelle_hf @ B3 @ A3 ) ) ) )
% 2.56/1.48 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( inf_inf @ hF_Mirabelle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) )
% 2.56/1.48 = ( inf_inf @ hF_Mirabelle_hf @ B3 @ A3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_119_hinter__hinsert__left,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_inf @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) )
% 2.56/1.48 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( inf_inf @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 2.56/1.48 = ( inf_inf @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_120_hinsert__hdiff__if,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 2.56/1.48 = ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) ) )
% 2.56/1.48 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 2.56/1.48 => ( ( minus_minus @ hF_Mirabelle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A3 ) @ B3 )
% 2.56/1.48 = ( hF_Mirabelle_hinsert @ X2 @ ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_121_hdiff__insert,axiom,
% 2.56/1.48 ! [A3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( minus_minus @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_hinsert @ A2 @ B3 ) )
% 2.56/1.48 = ( minus_minus @ hF_Mirabelle_hf @ ( minus_minus @ hF_Mirabelle_hf @ A3 @ B3 ) @ ( hF_Mirabelle_hinsert @ A2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_122_hsplit,axiom,
% 2.56/1.48 ! [A: $tType,C: hF_Mirabelle_hf > hF_Mirabelle_hf > A,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hsplit @ A @ C @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.56/1.48 = ( C @ A2 @ B ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_123_Diff__disjoint,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_124_Diff__cancel,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_125_empty__Diff,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_126_Diff__empty,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = A3 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_127_empty__iff,axiom,
% 2.56/1.48 ! [A: $tType,C: A] :
% 2.56/1.48 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_128_all__not__in__conv,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( ! [X3: A] :
% 2.56/1.48 ~ ( member @ A @ X3 @ A3 ) )
% 2.56/1.48 = ( A3
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_129_Collect__empty__eq,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o] :
% 2.56/1.48 ( ( ( collect @ A @ P )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = ( ! [X3: A] :
% 2.56/1.48 ~ ( P @ X3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_130_empty__Collect__eq,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o] :
% 2.56/1.48 ( ( ( bot_bot @ ( set @ A ) )
% 2.56/1.48 = ( collect @ A @ P ) )
% 2.56/1.48 = ( ! [X3: A] :
% 2.56/1.48 ~ ( P @ X3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_131_Int__iff,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 = ( ( member @ A @ C @ A3 )
% 2.56/1.48 & ( member @ A @ C @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_132_IntI,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ A3 )
% 2.56/1.48 => ( ( member @ A @ C @ B3 )
% 2.56/1.48 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_133_DiffI,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ A3 )
% 2.56/1.48 => ( ~ ( member @ A @ C @ B3 )
% 2.56/1.48 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_134_Diff__iff,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 = ( ( member @ A @ C @ A3 )
% 2.56/1.48 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_135_Diff__idemp,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 2.56/1.48 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_136_Int__left__commute,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_137_Int__left__absorb,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_138_Collect__conj__eq,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.56/1.48 ( ( collect @ A
% 2.56/1.48 @ ^ [X3: A] :
% 2.56/1.48 ( ( P @ X3 )
% 2.56/1.48 & ( Q @ X3 ) ) )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_139_inf__set__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.56/1.48 ( collect @ A
% 2.56/1.48 @ ( inf_inf @ ( A > $o )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_140_Int__commute,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_141_Int__Collect,axiom,
% 2.56/1.48 ! [A: $tType,X2: A,A3: set @ A,P: A > $o] :
% 2.56/1.48 ( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
% 2.56/1.48 = ( ( member @ A @ X2 @ A3 )
% 2.56/1.48 & ( P @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_142_Int__absorb,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
% 2.56/1.48 = A3 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_143_Int__assoc,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_144_Int__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.56/1.48 ( collect @ A
% 2.56/1.48 @ ^ [X3: A] :
% 2.56/1.48 ( ( member @ A @ X3 @ A5 )
% 2.56/1.48 & ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_145_IntD2,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ( member @ A @ C @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_146_IntD1,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ( member @ A @ C @ A3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_147_IntE,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ~ ( ( member @ A @ C @ A3 )
% 2.56/1.48 => ~ ( member @ A @ C @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_148_bot__set__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( bot_bot @ ( set @ A ) )
% 2.56/1.48 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_149_bot__hf__def,axiom,
% 2.56/1.48 ( ( bot_bot @ hF_Mirabelle_hf )
% 2.56/1.48 = ( zero_zero @ hF_Mirabelle_hf ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_150_emptyE,axiom,
% 2.56/1.48 ! [A: $tType,A2: A] :
% 2.56/1.48 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_151_equals0D,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,A2: A] :
% 2.56/1.48 ( ( A3
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_152_equals0I,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ! [Y4: A] :
% 2.56/1.48 ~ ( member @ A @ Y4 @ A3 )
% 2.56/1.48 => ( A3
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_153_ex__in__conv,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
% 2.56/1.48 = ( A3
% 2.56/1.48 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_154_Int__emptyI,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ! [X: A] :
% 2.56/1.48 ( ( member @ A @ X @ A3 )
% 2.56/1.48 => ~ ( member @ A @ X @ B3 ) )
% 2.56/1.48 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_155_disjoint__iff,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = ( ! [X3: A] :
% 2.56/1.48 ( ( member @ A @ X3 @ A3 )
% 2.56/1.48 => ~ ( member @ A @ X3 @ B3 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_156_Int__empty__left,axiom,
% 2.56/1.48 ! [A: $tType,B3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_157_Int__empty__right,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_158_disjoint__iff__not__equal,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = ( ! [X3: A] :
% 2.56/1.48 ( ( member @ A @ X3 @ A3 )
% 2.56/1.48 => ! [Y3: A] :
% 2.56/1.48 ( ( member @ A @ Y3 @ B3 )
% 2.56/1.48 => ( X3 != Y3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_159_DiffE,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ~ ( ( member @ A @ C @ A3 )
% 2.56/1.48 => ( member @ A @ C @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_160_DiffD1,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ( member @ A @ C @ A3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_161_DiffD2,axiom,
% 2.56/1.48 ! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 => ~ ( member @ A @ C @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_162_Int__Diff,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_163_Diff__Int2,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
% 2.56/1.48 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_164_Diff__Diff__Int,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_165_Diff__Int__distrib,axiom,
% 2.56/1.48 ! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.56/1.48 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_166_Diff__Int__distrib2,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 2.56/1.48 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_167_empty__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( bot_bot @ ( set @ A ) )
% 2.56/1.48 = ( collect @ A
% 2.56/1.48 @ ^ [X3: A] : $false ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_168_set__diff__eq,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.56/1.48 ( collect @ A
% 2.56/1.48 @ ^ [X3: A] :
% 2.56/1.48 ( ( member @ A @ X3 @ A5 )
% 2.56/1.48 & ~ ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_169_minus__set__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( minus_minus @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.56/1.48 ( collect @ A
% 2.56/1.48 @ ( minus_minus @ ( A > $o )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_170_Diff__triv,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 => ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 = A3 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_171_hfst__conv,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.56/1.48 = A2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_172_hsnd__conv,axiom,
% 2.56/1.48 ! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
% 2.56/1.48 = B ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_173_the__equality,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,A2: A] :
% 2.56/1.48 ( ( P @ A2 )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( X = A2 ) )
% 2.56/1.48 => ( ( the @ A @ P )
% 2.56/1.48 = A2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_174_the__eq__trivial,axiom,
% 2.56/1.48 ! [A: $tType,A2: A] :
% 2.56/1.48 ( ( the @ A
% 2.56/1.48 @ ^ [X3: A] : ( X3 = A2 ) )
% 2.56/1.48 = A2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_175_inf1I,axiom,
% 2.56/1.48 ! [A: $tType,A3: A > $o,X2: A,B3: A > $o] :
% 2.56/1.48 ( ( A3 @ X2 )
% 2.56/1.48 => ( ( B3 @ X2 )
% 2.56/1.48 => ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_176_the__sym__eq__trivial,axiom,
% 2.56/1.48 ! [A: $tType,X2: A] :
% 2.56/1.48 ( ( the @ A
% 2.56/1.48 @ ( ^ [Y: A,Z: A] : ( Y = Z )
% 2.56/1.48 @ X2 ) )
% 2.56/1.48 = X2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_177_inf1D2,axiom,
% 2.56/1.48 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 2.56/1.48 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 2.56/1.48 => ( B3 @ X2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_178_inf1D1,axiom,
% 2.56/1.48 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 2.56/1.48 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 2.56/1.48 => ( A3 @ X2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_179_inf1E,axiom,
% 2.56/1.48 ! [A: $tType,A3: A > $o,B3: A > $o,X2: A] :
% 2.56/1.48 ( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X2 )
% 2.56/1.48 => ~ ( ( A3 @ X2 )
% 2.56/1.48 => ~ ( B3 @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_180_the1__equality,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,A2: A] :
% 2.56/1.48 ( ? [X4: A] :
% 2.56/1.48 ( ( P @ X4 )
% 2.56/1.48 & ! [Y4: A] :
% 2.56/1.48 ( ( P @ Y4 )
% 2.56/1.48 => ( Y4 = X4 ) ) )
% 2.56/1.48 => ( ( P @ A2 )
% 2.56/1.48 => ( ( the @ A @ P )
% 2.56/1.48 = A2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_181_the1I2,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.56/1.48 ( ? [X4: A] :
% 2.56/1.48 ( ( P @ X4 )
% 2.56/1.48 & ! [Y4: A] :
% 2.56/1.48 ( ( P @ Y4 )
% 2.56/1.48 => ( Y4 = X4 ) ) )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( Q @ X ) )
% 2.56/1.48 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_182_If__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( if @ A )
% 2.56/1.48 = ( ^ [P2: $o,X3: A,Y3: A] :
% 2.56/1.48 ( the @ A
% 2.56/1.48 @ ^ [Z3: A] :
% 2.56/1.48 ( ( P2
% 2.56/1.48 => ( Z3 = X3 ) )
% 2.56/1.48 & ( ~ P2
% 2.56/1.48 => ( Z3 = Y3 ) ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_183_theI2,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 2.56/1.48 ( ( P @ A2 )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( X = A2 ) )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( Q @ X ) )
% 2.56/1.48 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_184_theI_H,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o] :
% 2.56/1.48 ( ? [X4: A] :
% 2.56/1.48 ( ( P @ X4 )
% 2.56/1.48 & ! [Y4: A] :
% 2.56/1.48 ( ( P @ Y4 )
% 2.56/1.48 => ( Y4 = X4 ) ) )
% 2.56/1.48 => ( P @ ( the @ A @ P ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_185_theI,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o,A2: A] :
% 2.56/1.48 ( ( P @ A2 )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( P @ X )
% 2.56/1.48 => ( X = A2 ) )
% 2.56/1.48 => ( P @ ( the @ A @ P ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_186_bot__apply,axiom,
% 2.56/1.48 ! [C2: $tType,D2: $tType] :
% 2.56/1.48 ( ( bot @ C2 )
% 2.56/1.48 => ( ( bot_bot @ ( D2 > C2 ) )
% 2.56/1.48 = ( ^ [X3: D2] : ( bot_bot @ C2 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_187_inf__Int__eq,axiom,
% 2.56/1.48 ! [A: $tType,R: set @ A,S: set @ A] :
% 2.56/1.48 ( ( inf_inf @ ( A > $o )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ R )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ S ) )
% 2.56/1.48 = ( ^ [X3: A] : ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_188_diff__self__eq__0,axiom,
% 2.56/1.48 ! [M: nat] :
% 2.56/1.48 ( ( minus_minus @ nat @ M @ M )
% 2.56/1.48 = ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_189_diff__0__eq__0,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
% 2.56/1.48 = ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_190_diff__commute,axiom,
% 2.56/1.48 ! [I: nat,J: nat,K: nat] :
% 2.56/1.48 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 2.56/1.48 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_191_bot__nat__def,axiom,
% 2.56/1.48 ( ( bot_bot @ nat )
% 2.56/1.48 = ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_192_diffs0__imp__equal,axiom,
% 2.56/1.48 ! [M: nat,N: nat] :
% 2.56/1.48 ( ( ( minus_minus @ nat @ M @ N )
% 2.56/1.48 = ( zero_zero @ nat ) )
% 2.56/1.48 => ( ( ( minus_minus @ nat @ N @ M )
% 2.56/1.48 = ( zero_zero @ nat ) )
% 2.56/1.48 => ( M = N ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_193_minus__nat_Odiff__0,axiom,
% 2.56/1.48 ! [M: nat] :
% 2.56/1.48 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
% 2.56/1.48 = M ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_194_bot__fun__def,axiom,
% 2.56/1.48 ! [B5: $tType,A: $tType] :
% 2.56/1.48 ( ( bot @ B5 )
% 2.56/1.48 => ( ( bot_bot @ ( A > B5 ) )
% 2.56/1.48 = ( ^ [X3: A] : ( bot_bot @ B5 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_195_bot__empty__eq,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( bot_bot @ ( A > $o ) )
% 2.56/1.48 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_196_Collect__empty__eq__bot,axiom,
% 2.56/1.48 ! [A: $tType,P: A > $o] :
% 2.56/1.48 ( ( ( collect @ A @ P )
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) )
% 2.56/1.48 = ( P
% 2.56/1.48 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_197_ord_OLeast__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( least @ A )
% 2.56/1.48 = ( ^ [Less_eq: A > A > $o,P2: A > $o] :
% 2.56/1.48 ( the @ A
% 2.56/1.48 @ ^ [X3: A] :
% 2.56/1.48 ( ( P2 @ X3 )
% 2.56/1.48 & ! [Y3: A] :
% 2.56/1.48 ( ( P2 @ Y3 )
% 2.56/1.48 => ( Less_eq @ X3 @ Y3 ) ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_198_PrimReplace__def,axiom,
% 2.56/1.48 ( hF_Mir569462966eplace
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 2.56/1.48 ( the @ hF_Mirabelle_hf
% 2.56/1.48 @ ^ [Z3: hF_Mirabelle_hf] :
% 2.56/1.48 ! [V5: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ V5 @ Z3 )
% 2.56/1.48 = ( ? [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ A5 )
% 2.56/1.48 & ( R2 @ U2 @ V5 ) ) ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_199_hcard__0,axiom,
% 2.56/1.48 ( ( hF_Mirabelle_hcard @ ( zero_zero @ hF_Mirabelle_hf ) )
% 2.56/1.48 = ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_200_hfst__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_hfst
% 2.56/1.48 = ( ^ [P4: hF_Mirabelle_hf] :
% 2.56/1.48 ( the @ hF_Mirabelle_hf
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ? [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( P4
% 2.56/1.48 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_201_hsnd__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_hsnd
% 2.56/1.48 = ( ^ [P4: hF_Mirabelle_hf] :
% 2.56/1.48 ( the @ hF_Mirabelle_hf
% 2.56/1.48 @ ^ [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ? [X3: hF_Mirabelle_hf] :
% 2.56/1.48 ( P4
% 2.56/1.48 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_202_HUnion__def,axiom,
% 2.56/1.48 ( hF_Mirabelle_HUnion
% 2.56/1.48 = ( ^ [A5: hF_Mirabelle_hf] :
% 2.56/1.48 ( the @ hF_Mirabelle_hf
% 2.56/1.48 @ ^ [Z3: hF_Mirabelle_hf] :
% 2.56/1.48 ! [U2: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 2.56/1.48 = ( ? [Y3: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ Y3 @ A5 )
% 2.56/1.48 & ( hF_Mirabelle_hmem @ U2 @ Y3 ) ) ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_203_hcard__hdiff1__less,axiom,
% 2.56/1.48 ! [X2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 2.56/1.48 ( ( hF_Mirabelle_hmem @ X2 @ Z4 )
% 2.56/1.48 => ( ord_less @ nat @ ( hF_Mirabelle_hcard @ ( minus_minus @ hF_Mirabelle_hf @ Z4 @ ( hF_Mirabelle_hinsert @ X2 @ ( zero_zero @ hF_Mirabelle_hf ) ) ) ) @ ( hF_Mirabelle_hcard @ Z4 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_204_Set_Ois__empty__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( is_empty @ A )
% 2.56/1.48 = ( ^ [A5: set @ A] :
% 2.56/1.48 ( A5
% 2.56/1.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_205_not__gr__zero,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( canoni770627133id_add @ A )
% 2.56/1.48 => ! [N: A] :
% 2.56/1.48 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.56/1.48 = ( N
% 2.56/1.48 = ( zero_zero @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_206_bot__nat__0_Onot__eq__extremum,axiom,
% 2.56/1.48 ! [A2: nat] :
% 2.56/1.48 ( ( A2
% 2.56/1.48 != ( zero_zero @ nat ) )
% 2.56/1.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_207_less__nat__zero__code,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_208_neq0__conv,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ( ( N
% 2.56/1.48 != ( zero_zero @ nat ) )
% 2.56/1.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_209_diff__gt__0__iff__gt,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ! [A2: A,B: A] :
% 2.56/1.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B ) )
% 2.56/1.48 = ( ord_less @ A @ B @ A2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_210_zero__less__diff,axiom,
% 2.56/1.48 ! [N: nat,M: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
% 2.56/1.48 = ( ord_less @ nat @ M @ N ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_211_less__imp__diff__less,axiom,
% 2.56/1.48 ! [J: nat,K: nat,N: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ J @ K )
% 2.56/1.48 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_212_diff__less__mono2,axiom,
% 2.56/1.48 ! [M: nat,N: nat,L: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ M @ N )
% 2.56/1.48 => ( ( ord_less @ nat @ M @ L )
% 2.56/1.48 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_213_diff__less,axiom,
% 2.56/1.48 ! [N: nat,M: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 2.56/1.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.56/1.48 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_214_diff__strict__right__mono,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ! [A2: A,B: A,C: A] :
% 2.56/1.48 ( ( ord_less @ A @ A2 @ B )
% 2.56/1.48 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ C ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_215_diff__strict__left__mono,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ! [B: A,A2: A,C: A] :
% 2.56/1.48 ( ( ord_less @ A @ B @ A2 )
% 2.56/1.48 => ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_216_diff__eq__diff__less,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ! [A2: A,B: A,C: A,D: A] :
% 2.56/1.48 ( ( ( minus_minus @ A @ A2 @ B )
% 2.56/1.48 = ( minus_minus @ A @ C @ D ) )
% 2.56/1.48 => ( ( ord_less @ A @ A2 @ B )
% 2.56/1.48 = ( ord_less @ A @ C @ D ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_217_diff__strict__mono,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ! [A2: A,B: A,D: A,C: A] :
% 2.56/1.48 ( ( ord_less @ A @ A2 @ B )
% 2.56/1.48 => ( ( ord_less @ A @ D @ C )
% 2.56/1.48 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B @ D ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_218_infinite__descent0__measure,axiom,
% 2.56/1.48 ! [A: $tType,V6: A > nat,P: A > $o,X2: A] :
% 2.56/1.48 ( ! [X: A] :
% 2.56/1.48 ( ( ( V6 @ X )
% 2.56/1.48 = ( zero_zero @ nat ) )
% 2.56/1.48 => ( P @ X ) )
% 2.56/1.48 => ( ! [X: A] :
% 2.56/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V6 @ X ) )
% 2.56/1.48 => ( ~ ( P @ X )
% 2.56/1.48 => ? [Y5: A] :
% 2.56/1.48 ( ( ord_less @ nat @ ( V6 @ Y5 ) @ ( V6 @ X ) )
% 2.56/1.48 & ~ ( P @ Y5 ) ) ) )
% 2.56/1.48 => ( P @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_219_bot__nat__0_Oextremum__strict,axiom,
% 2.56/1.48 ! [A2: nat] :
% 2.56/1.48 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_220_infinite__descent0,axiom,
% 2.56/1.48 ! [P: nat > $o,N: nat] :
% 2.56/1.48 ( ( P @ ( zero_zero @ nat ) )
% 2.56/1.48 => ( ! [N2: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.56/1.48 => ( ~ ( P @ N2 )
% 2.56/1.48 => ? [M2: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ M2 @ N2 )
% 2.56/1.48 & ~ ( P @ M2 ) ) ) )
% 2.56/1.48 => ( P @ N ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_221_gr__implies__not0,axiom,
% 2.56/1.48 ! [M: nat,N: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ M @ N )
% 2.56/1.48 => ( N
% 2.56/1.48 != ( zero_zero @ nat ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_222_less__zeroE,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_223_not__less0,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_224_not__gr0,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.56/1.48 = ( N
% 2.56/1.48 = ( zero_zero @ nat ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_225_gr0I,axiom,
% 2.56/1.48 ! [N: nat] :
% 2.56/1.48 ( ( N
% 2.56/1.48 != ( zero_zero @ nat ) )
% 2.56/1.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_226_less__iff__diff__less__0,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ordered_ab_group_add @ A )
% 2.56/1.48 => ( ( ord_less @ A )
% 2.56/1.48 = ( ^ [A4: A,B2: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_227_zero__less__iff__neq__zero,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( canoni770627133id_add @ A )
% 2.56/1.48 => ! [N: A] :
% 2.56/1.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.56/1.48 = ( N
% 2.56/1.48 != ( zero_zero @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_228_gr__implies__not__zero,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( canoni770627133id_add @ A )
% 2.56/1.48 => ! [M: A,N: A] :
% 2.56/1.48 ( ( ord_less @ A @ M @ N )
% 2.56/1.48 => ( N
% 2.56/1.48 != ( zero_zero @ A ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_229_not__less__zero,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( canoni770627133id_add @ A )
% 2.56/1.48 => ! [N: A] :
% 2.56/1.48 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_230_gr__zeroI,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( canoni770627133id_add @ A )
% 2.56/1.48 => ! [N: A] :
% 2.56/1.48 ( ( N
% 2.56/1.48 != ( zero_zero @ A ) )
% 2.56/1.48 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_231_less__infI1,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,X2: A,B: A] :
% 2.56/1.48 ( ( ord_less @ A @ A2 @ X2 )
% 2.56/1.48 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_232_less__infI2,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [B: A,X2: A,A2: A] :
% 2.56/1.48 ( ( ord_less @ A @ B @ X2 )
% 2.56/1.48 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ X2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_233_inf_Ostrict__boundedE,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,B: A,C: A] :
% 2.56/1.48 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B @ C ) )
% 2.56/1.48 => ~ ( ( ord_less @ A @ A2 @ B )
% 2.56/1.48 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_234_inf_Ostrict__order__iff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ( ( ord_less @ A )
% 2.56/1.48 = ( ^ [A4: A,B2: A] :
% 2.56/1.48 ( ( A4
% 2.56/1.48 = ( inf_inf @ A @ A4 @ B2 ) )
% 2.56/1.48 & ( A4 != B2 ) ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_235_inf_Ostrict__coboundedI1,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [A2: A,C: A,B: A] :
% 2.56/1.48 ( ( ord_less @ A @ A2 @ C )
% 2.56/1.48 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_236_inf_Ostrict__coboundedI2,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A )
% 2.56/1.48 => ! [B: A,C: A,A2: A] :
% 2.56/1.48 ( ( ord_less @ A @ B @ C )
% 2.56/1.48 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B ) @ C ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_237_bot_Oextremum__strict,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( order_bot @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_238_bot_Onot__eq__extremum,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( order_bot @ A )
% 2.56/1.48 => ! [A2: A] :
% 2.56/1.48 ( ( A2
% 2.56/1.48 != ( bot_bot @ A ) )
% 2.56/1.48 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_239_less__numeral__extra_I3_J,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( linord1659791738miring @ A )
% 2.56/1.48 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_240_of__nat__0__less__iff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( linord1659791738miring @ A )
% 2.56/1.48 => ! [N: nat] :
% 2.56/1.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.56/1.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_241_of__nat__0,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semiring_1 @ A )
% 2.56/1.48 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.56/1.48 = ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_242_of__nat__0__eq__iff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semiring_char_0 @ A )
% 2.56/1.48 => ! [N: nat] :
% 2.56/1.48 ( ( ( zero_zero @ A )
% 2.56/1.48 = ( semiring_1_of_nat @ A @ N ) )
% 2.56/1.48 = ( ( zero_zero @ nat )
% 2.56/1.48 = N ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_243_of__nat__eq__0__iff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( semiring_char_0 @ A )
% 2.56/1.48 => ! [M: nat] :
% 2.56/1.48 ( ( ( semiring_1_of_nat @ A @ M )
% 2.56/1.48 = ( zero_zero @ A ) )
% 2.56/1.48 = ( M
% 2.56/1.48 = ( zero_zero @ nat ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_244_psubset__imp__ex__mem,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.56/1.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 => ? [B7: A] : ( member @ A @ B7 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_245_of__nat__less__0__iff,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( linord1659791738miring @ A )
% 2.56/1.48 => ! [M: nat] :
% 2.56/1.48 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_246_not__psubset__empty,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A] :
% 2.56/1.48 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_247_pos__int__cases,axiom,
% 2.56/1.48 ! [K: int] :
% 2.56/1.48 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.56/1.48 => ~ ! [N2: nat] :
% 2.56/1.48 ( ( K
% 2.56/1.48 = ( semiring_1_of_nat @ int @ N2 ) )
% 2.56/1.48 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_248_zero__less__imp__eq__int,axiom,
% 2.56/1.48 ! [K: int] :
% 2.56/1.48 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.56/1.48 => ? [N2: nat] :
% 2.56/1.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.56/1.48 & ( K
% 2.56/1.48 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_249_psubsetD,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
% 2.56/1.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 => ( ( member @ A @ C @ A3 )
% 2.56/1.48 => ( member @ A @ C @ B3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_250_less__set__def,axiom,
% 2.56/1.48 ! [A: $tType] :
% 2.56/1.48 ( ( ord_less @ ( set @ A ) )
% 2.56/1.48 = ( ^ [A5: set @ A,B4: set @ A] :
% 2.56/1.48 ( ord_less @ ( A > $o )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ A5 )
% 2.56/1.48 @ ^ [X3: A] : ( member @ A @ X3 @ B4 ) ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_251_psubset__trans,axiom,
% 2.56/1.48 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.56/1.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.56/1.48 => ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
% 2.56/1.48 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_252_int__diff__cases,axiom,
% 2.56/1.48 ! [Z4: int] :
% 2.56/1.48 ~ ! [M3: nat,N2: nat] :
% 2.56/1.48 ( Z4
% 2.56/1.48 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(fact_253_minus__int__code_I1_J,axiom,
% 2.56/1.48 ! [K: int] :
% 2.56/1.48 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
% 2.56/1.48 = K ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.56/1.48 bounded_lattice @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.56/1.48 ! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( bounded_lattice @ A10 )
% 2.56/1.48 => ( bounded_lattice @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( bounded_lattice @ A10 )
% 2.56/1.48 => ( bounded_lattice_bot @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( semilattice_inf @ A10 )
% 2.56/1.48 => ( semilattice_inf @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( order_bot @ A10 )
% 2.56/1.48 => ( order_bot @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( lattice @ A10 )
% 2.56/1.48 => ( lattice @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Orderings_Obot,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( bot @ A10 )
% 2.56/1.48 => ( bot @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_fun___Groups_Ominus,axiom,
% 2.56/1.48 ! [A9: $tType,A10: $tType] :
% 2.56/1.48 ( ( minus @ A10 )
% 2.56/1.48 => ( minus @ ( A9 > A10 ) ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 2.56/1.48 linord1659791738miring @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 2.56/1.48 cancel146912293up_add @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 2.56/1.48 cancel1352612707id_add @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 2.56/1.48 ordered_ab_group_add @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
% 2.56/1.48 semilattice_inf @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.56/1.48 semiring_char_0 @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.56/1.48 semiring_1 @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Lattices_Olattice_4,axiom,
% 2.56/1.48 lattice @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 2.56/1.48 group_add @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Ominus_5,axiom,
% 2.56/1.48 minus @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.56/1.48 zero @ int ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.56/1.48 canoni770627133id_add @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6,axiom,
% 2.56/1.48 linord1659791738miring @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7,axiom,
% 2.56/1.48 cancel146912293up_add @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8,axiom,
% 2.56/1.48 cancel1352612707id_add @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_9,axiom,
% 2.56/1.48 semilattice_inf @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 2.56/1.48 comm_monoid_diff @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Orderings_Oorder__bot_10,axiom,
% 2.56/1.48 order_bot @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_11,axiom,
% 2.56/1.48 semiring_char_0 @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Rings_Osemiring__1_12,axiom,
% 2.56/1.48 semiring_1 @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Lattices_Olattice_13,axiom,
% 2.56/1.48 lattice @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Orderings_Obot_14,axiom,
% 2.56/1.48 bot @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ominus_15,axiom,
% 2.56/1.48 minus @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Nat_Onat___Groups_Ozero_16,axiom,
% 2.56/1.48 zero @ nat ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_17,axiom,
% 2.56/1.48 ! [A9: $tType] : ( bounded_lattice_bot @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_18,axiom,
% 2.56/1.48 ! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Orderings_Oorder__bot_19,axiom,
% 2.56/1.48 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Lattices_Olattice_20,axiom,
% 2.56/1.48 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Orderings_Obot_21,axiom,
% 2.56/1.48 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_Set_Oset___Groups_Ominus_22,axiom,
% 2.56/1.48 ! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_23,axiom,
% 2.56/1.48 bounded_lattice_bot @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_24,axiom,
% 2.56/1.48 semilattice_inf @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Orderings_Oorder__bot_25,axiom,
% 2.56/1.48 order_bot @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Lattices_Olattice_26,axiom,
% 2.56/1.48 lattice @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Orderings_Obot_27,axiom,
% 2.56/1.48 bot @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HOL_Obool___Groups_Ominus_28,axiom,
% 2.56/1.48 minus @ $o ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Obounded__lattice__bot_29,axiom,
% 2.56/1.48 bounded_lattice_bot @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Osemilattice__inf_30,axiom,
% 2.56/1.48 semilattice_inf @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder__bot_31,axiom,
% 2.56/1.48 order_bot @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Lattices_Olattice_32,axiom,
% 2.56/1.48 lattice @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Obot_33,axiom,
% 2.56/1.48 bot @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ominus_34,axiom,
% 2.56/1.48 minus @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Groups_Ozero_35,axiom,
% 2.56/1.48 zero @ hF_Mirabelle_hf ).
% 2.56/1.48
% 2.56/1.48 thf(help_If_3_1_T,axiom,
% 2.56/1.48 ! [P: $o] :
% 2.56/1.48 ( ( P = $true )
% 2.56/1.48 | ( P = $false ) ) ).
% 2.56/1.48
% 2.56/1.48 thf(help_If_2_1_T,axiom,
% 2.56/1.48 ! [A: $tType,X2: A,Y2: A] :
% 2.56/1.48 ( ( if @ A @ $false @ X2 @ Y2 )
% 2.56/1.48 = Y2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(help_If_1_1_T,axiom,
% 2.56/1.48 ! [A: $tType,X2: A,Y2: A] :
% 2.56/1.48 ( ( if @ A @ $true @ X2 @ Y2 )
% 2.56/1.48 = X2 ) ).
% 2.56/1.48
% 2.56/1.48 thf(conj_0,conjecture,
% 2.56/1.48 ( ( hF_Mirabelle_HBex @ a
% 2.56/1.48 @ ^ [X3: hF_Mirabelle_hf] : p )
% 2.56/1.48 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ a )
% 2.56/1.48 & p ) ) ).
% 2.56/1.48 ------------------------
% 2.68/1.51 % (25786)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_25592 for (2999ds/27Mi)
% 2.68/1.51 % (25788)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_25592 for (2999ds/2Mi)
% 2.68/1.51 % (25788)Instruction limit reached!
% 2.68/1.51 % (25788)------------------------------
% 2.68/1.51 % (25788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.51 % (25788)Termination reason: Unknown
% 2.68/1.51 % (25788)Termination phase: shuffling
% 2.68/1.51
% 2.68/1.51 % (25788)Memory used [KB]: 1279
% 2.68/1.51 % (25788)Time elapsed: 0.002 s
% 2.68/1.51 % (25788)Instructions burned: 2 (million)
% 2.68/1.51 % (25788)------------------------------
% 2.68/1.51 % (25788)------------------------------
% 2.68/1.51 % (25789)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_25592 for (2999ds/275Mi)
% 2.68/1.52 % (25787)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/2Mi)
% 2.68/1.52 % (25787)Instruction limit reached!
% 2.68/1.52 % (25787)------------------------------
% 2.68/1.52 % (25787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.52 % (25787)Termination reason: Unknown
% 2.68/1.52 % (25787)Termination phase: shuffling
% 2.68/1.52
% 2.68/1.52 % (25787)Memory used [KB]: 1279
% 2.68/1.52 % (25787)Time elapsed: 0.002 s
% 2.68/1.52 % (25787)Instructions burned: 3 (million)
% 2.68/1.52 % (25787)------------------------------
% 2.68/1.52 % (25787)------------------------------
% 2.68/1.52 % (25786)Instruction limit reached!
% 2.68/1.52 % (25786)------------------------------
% 2.68/1.52 % (25786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.52 % (25786)Termination reason: Unknown
% 2.68/1.52 % (25786)Termination phase: shuffling
% 2.68/1.52
% 2.68/1.52 % (25786)Memory used [KB]: 1663
% 2.68/1.52 % (25786)Time elapsed: 0.012 s
% 2.68/1.52 % (25786)Instructions burned: 28 (million)
% 2.68/1.52 % (25786)------------------------------
% 2.68/1.52 % (25786)------------------------------
% 2.68/1.52 % (25785)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_25592 for (2999ds/4Mi)
% 2.68/1.52 % (25791)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/3Mi)
% 2.68/1.52 % (25791)Instruction limit reached!
% 2.68/1.52 % (25791)------------------------------
% 2.68/1.52 % (25791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.52 % (25791)Termination reason: Unknown
% 2.68/1.52 % (25791)Termination phase: shuffling
% 2.68/1.52
% 2.68/1.52 % (25791)Memory used [KB]: 1279
% 2.68/1.52 % (25791)Time elapsed: 0.002 s
% 2.68/1.52 % (25791)Instructions burned: 3 (million)
% 2.68/1.52 % (25791)------------------------------
% 2.68/1.52 % (25791)------------------------------
% 2.68/1.52 % (25785)Instruction limit reached!
% 2.68/1.52 % (25785)------------------------------
% 2.68/1.52 % (25785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.52 % (25785)Termination reason: Unknown
% 2.68/1.52 % (25785)Termination phase: shuffling
% 2.68/1.52
% 2.68/1.52 % (25785)Memory used [KB]: 1407
% 2.68/1.52 % (25785)Time elapsed: 0.003 s
% 2.68/1.52 % (25785)Instructions burned: 5 (million)
% 2.68/1.52 % (25785)------------------------------
% 2.68/1.52 % (25785)------------------------------
% 2.68/1.52 % (25790)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/18Mi)
% 2.68/1.53 % (25784)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/183Mi)
% 2.68/1.53 % (25792)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_25592 for (2999ds/37Mi)
% 2.68/1.53 % (25793)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_25592 for (2999ds/15Mi)
% 2.68/1.53 % (25794)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_25592 for (2999ds/3Mi)
% 2.68/1.53 % (25795)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_25592 for (2999ds/1041Mi)
% 2.68/1.53 % (25794)Instruction limit reached!
% 2.68/1.53 % (25794)------------------------------
% 2.68/1.53 % (25794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.53 % (25794)Termination reason: Unknown
% 2.68/1.53 % (25794)Termination phase: shuffling
% 2.68/1.53
% 2.68/1.53 % (25794)Memory used [KB]: 1407
% 2.68/1.53 % (25794)Time elapsed: 0.002 s
% 2.68/1.53 % (25794)Instructions burned: 3 (million)
% 2.68/1.53 % (25794)------------------------------
% 2.68/1.53 % (25794)------------------------------
% 2.68/1.53 % (25790)Instruction limit reached!
% 2.68/1.53 % (25790)------------------------------
% 2.68/1.53 % (25790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.53 % (25790)Termination reason: Unknown
% 2.68/1.53 % (25790)Termination phase: Property scanning
% 2.68/1.53
% 2.68/1.53 % (25790)Memory used [KB]: 1663
% 2.68/1.53 % (25790)Time elapsed: 0.013 s
% 2.68/1.53 % (25790)Instructions burned: 21 (million)
% 2.68/1.53 % (25790)------------------------------
% 2.68/1.53 % (25790)------------------------------
% 2.68/1.54 % (25793)Instruction limit reached!
% 2.68/1.54 % (25793)------------------------------
% 2.68/1.54 % (25793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.54 % (25793)Termination reason: Unknown
% 2.68/1.54 % (25793)Termination phase: shuffling
% 2.68/1.54
% 2.68/1.54 % (25793)Memory used [KB]: 1663
% 2.68/1.54 % (25793)Time elapsed: 0.008 s
% 2.68/1.54 % (25793)Instructions burned: 15 (million)
% 2.68/1.54 % (25793)------------------------------
% 2.68/1.54 % (25793)------------------------------
% 2.68/1.54 % (25792)Instruction limit reached!
% 2.68/1.54 % (25792)------------------------------
% 2.68/1.54 % (25792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.54 % (25792)Termination reason: Unknown
% 2.68/1.54 % (25792)Termination phase: Property scanning
% 2.68/1.54
% 2.68/1.54 % (25792)Memory used [KB]: 2046
% 2.68/1.54 % (25792)Time elapsed: 0.013 s
% 2.68/1.54 % (25792)Instructions burned: 37 (million)
% 2.68/1.54 % (25792)------------------------------
% 2.68/1.54 % (25792)------------------------------
% 2.68/1.54 % (25796)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/7Mi)
% 2.68/1.55 % (25797)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/16Mi)
% 2.68/1.55 % (25796)Instruction limit reached!
% 2.68/1.55 % (25796)------------------------------
% 2.68/1.55 % (25796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.55 % (25796)Termination reason: Unknown
% 2.68/1.55 % (25796)Termination phase: shuffling
% 2.68/1.55
% 2.68/1.55 % (25796)Memory used [KB]: 1535
% 2.68/1.55 % (25796)Time elapsed: 0.004 s
% 2.68/1.55 % (25796)Instructions burned: 9 (million)
% 2.68/1.55 % (25796)------------------------------
% 2.68/1.55 % (25796)------------------------------
% 2.68/1.55 % (25798)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_25592 for (2999ds/3Mi)
% 2.68/1.55 % (25798)Instruction limit reached!
% 2.68/1.55 % (25798)------------------------------
% 2.68/1.55 % (25798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.55 % (25798)Termination reason: Unknown
% 2.68/1.55 % (25798)Termination phase: shuffling
% 2.68/1.55
% 2.68/1.55 % (25798)Memory used [KB]: 1407
% 2.68/1.55 % (25798)Time elapsed: 0.002 s
% 2.68/1.55 % (25798)Instructions burned: 3 (million)
% 2.68/1.55 % (25798)------------------------------
% 2.68/1.55 % (25798)------------------------------
% 2.68/1.55 % (25799)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_25592 for (2999ds/3Mi)
% 2.68/1.55 % (25799)Instruction limit reached!
% 2.68/1.55 % (25799)------------------------------
% 2.68/1.55 % (25799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.55 % (25799)Termination reason: Unknown
% 2.68/1.55 % (25799)Termination phase: shuffling
% 2.68/1.55
% 2.68/1.55 % (25799)Memory used [KB]: 1407
% 2.68/1.55 % (25799)Time elapsed: 0.003 s
% 2.68/1.55 % (25799)Instructions burned: 5 (million)
% 2.68/1.55 % (25799)------------------------------
% 2.68/1.55 % (25799)------------------------------
% 2.68/1.55 % (25797)Instruction limit reached!
% 2.68/1.55 % (25797)------------------------------
% 2.68/1.55 % (25797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.55 % (25797)Termination reason: Unknown
% 2.68/1.55 % (25797)Termination phase: Property scanning
% 2.68/1.55
% 2.68/1.55 % (25797)Memory used [KB]: 1663
% 2.68/1.55 % (25797)Time elapsed: 0.007 s
% 2.68/1.55 % (25797)Instructions burned: 17 (million)
% 2.68/1.55 % (25797)------------------------------
% 2.68/1.55 % (25797)------------------------------
% 2.68/1.56 % (25800)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/7Mi)
% 2.68/1.56 % (25800)Instruction limit reached!
% 2.68/1.56 % (25800)------------------------------
% 2.68/1.56 % (25800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.56 % (25800)Termination reason: Unknown
% 2.68/1.56 % (25800)Termination phase: shuffling
% 2.68/1.56
% 2.68/1.56 % (25800)Memory used [KB]: 1407
% 2.68/1.56 % (25800)Time elapsed: 0.004 s
% 2.68/1.56 % (25800)Instructions burned: 9 (million)
% 2.68/1.56 % (25800)------------------------------
% 2.68/1.56 % (25800)------------------------------
% 2.68/1.56 % (25801)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/3Mi)
% 2.68/1.56 % (25801)Instruction limit reached!
% 2.68/1.56 % (25801)------------------------------
% 2.68/1.56 % (25801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.56 % (25801)Termination reason: Unknown
% 2.68/1.56 % (25801)Termination phase: shuffling
% 2.68/1.56
% 2.68/1.56 % (25801)Memory used [KB]: 1407
% 2.68/1.56 % (25801)Time elapsed: 0.003 s
% 2.68/1.56 % (25801)Instructions burned: 4 (million)
% 2.68/1.56 % (25801)------------------------------
% 2.68/1.56 % (25801)------------------------------
% 2.68/1.56 % (25803)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/18Mi)
% 2.68/1.56 % (25802)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/4Mi)
% 2.68/1.57 % (25802)Instruction limit reached!
% 2.68/1.57 % (25802)------------------------------
% 2.68/1.57 % (25802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (25802)Termination reason: Unknown
% 2.68/1.57 % (25802)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (25802)Memory used [KB]: 1407
% 2.68/1.57 % (25802)Time elapsed: 0.004 s
% 2.68/1.57 % (25802)Instructions burned: 4 (million)
% 2.68/1.57 % (25802)------------------------------
% 2.68/1.57 % (25802)------------------------------
% 2.68/1.57 % (25809)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/710Mi)
% 2.68/1.57 % (25803)Instruction limit reached!
% 2.68/1.57 % (25803)------------------------------
% 2.68/1.57 % (25803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (25803)Termination reason: Unknown
% 2.68/1.57 % (25803)Termination phase: Property scanning
% 2.68/1.57
% 2.68/1.57 % (25803)Memory used [KB]: 1663
% 2.68/1.57 % (25803)Time elapsed: 0.008 s
% 2.68/1.57 % (25803)Instructions burned: 20 (million)
% 2.68/1.57 % (25803)------------------------------
% 2.68/1.57 % (25803)------------------------------
% 2.68/1.57 % (25810)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/6Mi)
% 2.68/1.57 % (25810)Instruction limit reached!
% 2.68/1.57 % (25810)------------------------------
% 2.68/1.57 % (25810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.57 % (25810)Termination reason: Unknown
% 2.68/1.57 % (25810)Termination phase: shuffling
% 2.68/1.57
% 2.68/1.57 % (25810)Memory used [KB]: 1407
% 2.68/1.57 % (25810)Time elapsed: 0.004 s
% 2.68/1.57 % (25810)Instructions burned: 7 (million)
% 2.68/1.57 % (25810)------------------------------
% 2.68/1.57 % (25810)------------------------------
% 2.68/1.58 % (25812)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/21Mi)
% 2.68/1.58 % (25813)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/5Mi)
% 2.68/1.58 % (25811)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_25592 for (2999ds/902Mi)
% 2.68/1.58 % (25813)Instruction limit reached!
% 2.68/1.58 % (25813)------------------------------
% 2.68/1.58 % (25813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (25813)Termination reason: Unknown
% 2.68/1.58 % (25813)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (25813)Memory used [KB]: 1407
% 2.68/1.58 % (25813)Time elapsed: 0.004 s
% 2.68/1.58 % (25813)Instructions burned: 7 (million)
% 2.68/1.58 % (25813)------------------------------
% 2.68/1.58 % (25813)------------------------------
% 2.68/1.58 % (25809)First to succeed.
% 2.68/1.59 % (25809)Refutation found. Thanks to Tanya!
% 2.68/1.59 % SZS status Theorem for DTF2THF_25592
% 2.68/1.59 % SZS output start Proof for DTF2THF_25592
% 2.68/1.59 thf(type_def_6, type, hF_Mirabelle_hf: $tType).
% 2.68/1.59 thf(type_def_7, type, set: ($tType) > $tType).
% 2.68/1.59 thf(type_def_8, type, nat: $tType).
% 2.68/1.59 thf(type_def_9, type, int: $tType).
% 2.68/1.59 thf(func_def_0, type, hF_Mirabelle_hf: $tType).
% 2.68/1.59 thf(func_def_1, type, set: ($tType) > $tType).
% 2.68/1.59 thf(func_def_2, type, nat: $tType).
% 2.68/1.59 thf(func_def_3, type, int: $tType).
% 2.68/1.59 thf(func_def_4, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_6, type, zero: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_7, type, minus: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_8, type, bot: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_9, type, group_add: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_10, type, lattice: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_11, type, semiring_1: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_12, type, semiring_char_0: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_13, type, order_bot: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_14, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_15, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_16, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_17, type, bounded_lattice_bot: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_18, type, cancel1352612707id_add: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_19, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_20, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_21, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.68/1.59 thf(func_def_22, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.68/1.59 thf(func_def_23, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.68/1.59 thf(func_def_24, type, hF_Mirabelle_HBall: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > $o).
% 2.68/1.59 thf(func_def_25, type, hF_Mirabelle_HBex: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > $o).
% 2.68/1.59 thf(func_def_26, type, hF_Mir1687042746ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_27, type, hF_Mirabelle_HF: set @ hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_28, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_29, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_30, type, hF_Mir569462966eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_31, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_32, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_33, type, hF_Mirabelle_hcard: hF_Mirabelle_hf > nat).
% 2.68/1.59 thf(func_def_34, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_35, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set @ hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_36, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_37, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_38, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 2.68/1.59 thf(func_def_39, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_40, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_41, type, hF_Mirabelle_hsplit: !>[X0: $tType]:((hF_Mirabelle_hf > hF_Mirabelle_hf > X0) > hF_Mirabelle_hf > X0)).
% 2.68/1.59 thf(func_def_42, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.68/1.59 thf(func_def_43, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.68/1.59 thf(func_def_44, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.68/1.59 thf(func_def_45, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.68/1.59 thf(func_def_46, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.68/1.59 thf(func_def_47, type, least: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 2.68/1.59 thf(func_def_48, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.68/1.59 thf(func_def_49, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.68/1.59 thf(func_def_50, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.68/1.59 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.68/1.59 thf(func_def_52, type, a: hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_69, type, sP0: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > $o).
% 2.68/1.59 thf(func_def_70, type, sK1: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_71, type, sK2: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_72, type, sK3: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_73, type, sK4: hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_74, type, sK5: (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_75, type, sK6: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_76, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.68/1.59 thf(func_def_77, type, sK8: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_78, type, sK9: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_79, type, sK10: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_80, type, sK11: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_81, type, sK12: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_82, type, sK13: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_83, type, sK14: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_84, type, sK15: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_85, type, sK16: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_86, type, sK17: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_87, type, sK18: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_89, type, sK20: hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_90, type, sK21: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_91, type, sK22: hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_92, type, sK23: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_93, type, sK24: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_94, type, sK25: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_95, type, sK26: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_96, type, sK27: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_97, type, sK28: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_98, type, sK29: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_99, type, sK30: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 2.68/1.59 thf(func_def_100, type, sK31: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 2.68/1.59 thf(f1112,plain,(
% 2.68/1.59 $false),
% 2.68/1.59 inference(avatar_sat_refutation,[status(thm)],[f1078,f1084,f1093,f1097,f1101,f1103,f1106,f1110])).
% 2.68/1.59 thf(f1110,plain,(
% 2.68/1.59 ~spl19_1 | ~spl19_6),
% 2.68/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1109])).
% 2.68/1.59 thf(f1109,plain,(
% 2.68/1.59 $false | (~spl19_1 | ~spl19_6)),
% 2.68/1.59 inference(trivial_inequality_removal,[status(thm)],[f1108])).
% 2.68/1.59 thf(f1108,plain,(
% 2.68/1.59 ($true = $false) | (~spl19_1 | ~spl19_6)),
% 2.68/1.59 inference(forward_demodulation,[status(thm)],[f1088,f1066])).
% 2.68/1.59 thf(f1066,plain,(
% 2.68/1.59 ( ! [X1 : hF_Mirabelle_hf] : (((hF_Mirabelle_hmem @ X1 @ a) = $false)) ) | ~spl19_1),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1065])).
% 2.68/1.59 thf(f1065,definition,(
% 2.68/1.59 spl19_1 <=> ! [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ a) = $false)),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_1])],[avatar_definition])).
% 2.68/1.59 thf(f1088,plain,(
% 2.68/1.59 ((hF_Mirabelle_hmem @ sK4 @ a) = $true) | ~spl19_6),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1086])).
% 2.68/1.59 thf(f1086,definition,(
% 2.68/1.59 spl19_6 <=> ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_6])],[avatar_definition])).
% 2.68/1.59 thf(f1106,plain,(
% 2.68/1.59 ~spl19_1 | ~spl19_7),
% 2.68/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1105])).
% 2.68/1.59 thf(f1105,plain,(
% 2.68/1.59 $false | (~spl19_1 | ~spl19_7)),
% 2.68/1.59 inference(trivial_inequality_removal,[status(thm)],[f1104])).
% 2.68/1.59 thf(f1104,plain,(
% 2.68/1.59 ($true = $false) | (~spl19_1 | ~spl19_7)),
% 2.68/1.59 inference(forward_demodulation,[status(thm)],[f1092,f1066])).
% 2.68/1.59 thf(f1092,plain,(
% 2.68/1.59 ($true = (hF_Mirabelle_hmem @ sK20 @ a)) | ~spl19_7),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1090])).
% 2.68/1.59 thf(f1090,definition,(
% 2.68/1.59 spl19_7 <=> ($true = (hF_Mirabelle_hmem @ sK20 @ a))),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_7])],[avatar_definition])).
% 2.68/1.59 thf(f1103,plain,(
% 2.68/1.59 ~spl19_4 | ~spl19_7),
% 2.68/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1102])).
% 2.68/1.59 thf(f1102,plain,(
% 2.68/1.59 $false | (~spl19_4 | ~spl19_7)),
% 2.68/1.59 inference(subsumption_resolution,[status(thm)],[f1092,f1077])).
% 2.68/1.59 thf(f1077,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a))) ) | ~spl19_4),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1076])).
% 2.68/1.59 thf(f1076,definition,(
% 2.68/1.59 spl19_4 <=> ! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a))),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_4])],[avatar_definition])).
% 2.68/1.59 thf(f1101,plain,(
% 2.68/1.59 ~spl19_4 | ~spl19_6),
% 2.68/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1100])).
% 2.68/1.59 thf(f1100,plain,(
% 2.68/1.59 $false | (~spl19_4 | ~spl19_6)),
% 2.68/1.59 inference(trivial_inequality_removal,[status(thm)],[f1099])).
% 2.68/1.59 thf(f1099,plain,(
% 2.68/1.59 ($true != $true) | (~spl19_4 | ~spl19_6)),
% 2.68/1.59 inference(superposition,[status(thm)],[f1077,f1088])).
% 2.68/1.59 thf(f1097,plain,(
% 2.68/1.59 ~spl19_2 | ~spl19_3),
% 2.68/1.59 inference(avatar_contradiction_clause,[status(thm)],[f1096])).
% 2.68/1.59 thf(f1096,plain,(
% 2.68/1.59 $false | (~spl19_2 | ~spl19_3)),
% 2.68/1.59 inference(trivial_inequality_removal,[status(thm)],[f1095])).
% 2.68/1.59 thf(f1095,plain,(
% 2.68/1.59 ($true = $false) | (~spl19_2 | ~spl19_3)),
% 2.68/1.59 inference(backward_demodulation,[status(thm)],[f1073,f1070])).
% 2.68/1.59 thf(f1070,plain,(
% 2.68/1.59 (p = $false) | ~spl19_2),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1068])).
% 2.68/1.59 thf(f1068,definition,(
% 2.68/1.59 spl19_2 <=> (p = $false)),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_2])],[avatar_definition])).
% 2.68/1.59 thf(f1073,plain,(
% 2.68/1.59 (p = $true) | ~spl19_3),
% 2.68/1.59 inference(avatar_component_clause,[status(thm)],[f1072])).
% 2.68/1.59 thf(f1072,definition,(
% 2.68/1.59 spl19_3 <=> (p = $true)),
% 2.68/1.59 introduced(definition,[new_symbols(naming,[spl19_3])],[avatar_definition])).
% 2.68/1.59 thf(f1093,plain,(
% 2.68/1.59 spl19_6 | spl19_7),
% 2.68/1.59 inference(avatar_split_clause,[status(thm)],[f958,f1090,f1086])).
% 2.68/1.59 thf(f958,plain,(
% 2.68/1.59 ($true = (hF_Mirabelle_hmem @ sK20 @ a)) | ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 inference(binary_proxy_clausification,[status(thm)],[f957])).
% 2.68/1.59 thf(f957,plain,(
% 2.68/1.59 ($true = (p & (hF_Mirabelle_hmem @ sK20 @ a))) | ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f956])).
% 2.68/1.59 thf(f956,plain,(
% 2.68/1.59 ($true = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ sK20)) | ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 inference(sigma_clausification,[status(thm)],[f955])).
% 2.68/1.59 thf(f955,plain,(
% 2.68/1.59 ($true = (?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))))) | ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f941])).
% 2.68/1.59 thf(f941,plain,(
% 2.68/1.59 ($true = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf > $o]: (?? @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((Y1 @ Y2) & (hF_Mirabelle_hmem @ Y2 @ Y0))))))) @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) | ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 inference(definition_unfolding,[status(thm)],[f906,f893])).
% 2.68/1.59 thf(f893,plain,(
% 2.68/1.59 (hF_Mirabelle_HBex = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf > $o]: (?? @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((Y1 @ Y2) & (hF_Mirabelle_hmem @ Y2 @ Y0))))))))),
% 2.68/1.59 inference(cnf_transformation,[status(thm)],[f328])).
% 2.68/1.59 thf(f328,plain,(
% 2.68/1.59 (hF_Mirabelle_HBex = (^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf > $o]: (?? @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((Y1 @ Y2) & (hF_Mirabelle_hmem @ Y2 @ Y0))))))))),
% 2.68/1.59 inference(fool_elimination,[status(thm)],[f327])).
% 2.68/1.59 thf(f327,plain,(
% 2.68/1.59 (hF_Mirabelle_HBex = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf > $o] : (? [X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X2 @ X0) & (X1 @ X2)))))),
% 2.68/1.59 inference(rectify,[status(thm)],[f6])).
% 2.68/1.59 thf(f6,axiom,(
% 2.68/1.59 (hF_Mirabelle_HBex = (^[X13 : hF_Mirabelle_hf, X14 : hF_Mirabelle_hf > $o] : (? [X11 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X11 @ X13) & (X14 @ X11)))))),
% 2.68/1.59 file('/export/starexec/sandbox/tmp/tmp.yW9if3OHxl/DTF2THF_25592.p',fact_5_HBex__def)).
% 2.68/1.59 thf(f906,plain,(
% 2.68/1.59 ((hF_Mirabelle_hmem @ sK4 @ a) = $true) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))),
% 2.68/1.59 inference(cnf_transformation,[status(thm)],[f858])).
% 2.68/1.59 thf(f858,plain,(
% 2.68/1.59 (! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true) | ($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))) & ((((hF_Mirabelle_hmem @ sK4 @ a) = $true) & (p = $true)) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))))),
% 2.68/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4])],[f856,f857])).
% 2.68/1.59 thf(f857,plain,(
% 2.68/1.59 ? [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ a) = $true) => ((hF_Mirabelle_hmem @ sK4 @ a) = $true)),
% 2.68/1.59 introduced(definition,[],[choice_axiom])).
% 2.68/1.59 thf(f856,plain,(
% 2.68/1.59 (! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true) | ($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))) & ((? [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ a) = $true) & (p = $true)) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))))),
% 2.68/1.59 inference(rectify,[status(thm)],[f855])).
% 2.68/1.59 thf(f855,plain,(
% 2.68/1.59 (! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true) | ($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))) & ((? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))))),
% 2.68/1.59 inference(flattening,[status(thm)],[f854])).
% 2.68/1.59 thf(f854,plain,(
% 2.68/1.59 ((! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true)) | ($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))) & ((? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))))),
% 2.68/1.59 inference(nnf_transformation,[status(thm)],[f826])).
% 2.68/1.59 thf(f826,plain,(
% 2.68/1.59 ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) <~> (? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true))),
% 2.68/1.59 inference(ennf_transformation,[status(thm)],[f815])).
% 2.68/1.59 thf(f815,plain,(
% 2.68/1.59 ~((? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)) <=> ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))))),
% 2.68/1.59 inference(rectify,[status(thm)],[f764])).
% 2.68/1.59 thf(f764,plain,(
% 2.68/1.59 ~(($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) <=> ((p = $true) & ? [X1 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X1 @ a) = $true)))),
% 2.68/1.59 inference(fool_elimination,[status(thm)],[f763])).
% 2.68/1.59 thf(f763,plain,(
% 2.68/1.59 ~((hF_Mirabelle_HBex @ a @ (^[X0 : hF_Mirabelle_hf] : (p))) = p & ? [X1 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X1 @ a))),
% 2.68/1.59 inference(rectify,[status(thm)],[f311])).
% 2.68/1.59 thf(f311,negated_conjecture,(
% 2.68/1.59 ~((hF_Mirabelle_HBex @ a @ (^[X11 : hF_Mirabelle_hf] : (p))) = p & ? [X11 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X11 @ a))),
% 2.68/1.59 inference(negated_conjecture,[status(cth)],[f310])).
% 2.68/1.59 thf(f310,conjecture,(
% 2.68/1.59 ((hF_Mirabelle_HBex @ a @ (^[X11 : hF_Mirabelle_hf] : (p))) = p & ? [X11 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X11 @ a))),
% 2.68/1.59 file('/export/starexec/sandbox/tmp/tmp.yW9if3OHxl/DTF2THF_25592.p',conj_0)).
% 2.68/1.59 thf(f1084,plain,(
% 2.68/1.59 spl19_3),
% 2.68/1.59 inference(avatar_split_clause,[status(thm)],[f976,f1072])).
% 2.68/1.59 thf(f976,plain,(
% 2.68/1.59 (p = $true)),
% 2.68/1.59 inference(duplicate_literal_removal,[status(thm)],[f975])).
% 2.68/1.59 thf(f975,plain,(
% 2.68/1.59 (p = $true) | (p = $true)),
% 2.68/1.59 inference(binary_proxy_clausification,[status(thm)],[f973])).
% 2.68/1.59 thf(f973,plain,(
% 2.68/1.59 ($true = (p & (hF_Mirabelle_hmem @ sK22 @ a))) | (p = $true)),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f972])).
% 2.68/1.59 thf(f972,plain,(
% 2.68/1.59 ($true = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ sK22)) | (p = $true)),
% 2.68/1.59 inference(sigma_clausification,[status(thm)],[f971])).
% 2.68/1.59 thf(f971,plain,(
% 2.68/1.59 (p = $true) | ($true = (?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a)))))),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f942])).
% 2.68/1.59 thf(f942,plain,(
% 2.68/1.59 (p = $true) | ($true = ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf > $o]: (?? @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((Y1 @ Y2) & (hF_Mirabelle_hmem @ Y2 @ Y0))))))) @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))),
% 2.68/1.59 inference(definition_unfolding,[status(thm)],[f905,f893])).
% 2.68/1.59 thf(f905,plain,(
% 2.68/1.59 (p = $true) | ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))),
% 2.68/1.59 inference(cnf_transformation,[status(thm)],[f858])).
% 2.68/1.59 thf(f1078,plain,(
% 2.68/1.59 spl19_1 | spl19_2 | ~spl19_3 | spl19_4),
% 2.68/1.59 inference(avatar_split_clause,[status(thm)],[f985,f1076,f1072,f1068,f1065])).
% 2.68/1.59 thf(f985,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((p != $true) | ($true != (hF_Mirabelle_hmem @ X0 @ a)) | ((hF_Mirabelle_hmem @ X1 @ a) = $false) | (p = $false)) )),
% 2.68/1.59 inference(binary_proxy_clausification,[status(thm)],[f984])).
% 2.68/1.59 thf(f984,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((p != $true) | ($false = (p & (hF_Mirabelle_hmem @ X1 @ a))) | ($true != (hF_Mirabelle_hmem @ X0 @ a))) )),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f983])).
% 2.68/1.59 thf(f983,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true) | ($false = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ X1))) )),
% 2.68/1.59 inference(pi_clausification,[status(thm)],[f982])).
% 2.68/1.59 thf(f982,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | ($true != (?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))))) | (p != $true)) )),
% 2.68/1.59 inference(beta_eta_normalization,[status(thm)],[f940])).
% 2.68/1.59 thf(f940,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf] : (($true != ((^[Y0 : hF_Mirabelle_hf]: ((^[Y1 : hF_Mirabelle_hf > $o]: (?? @ hF_Mirabelle_hf @ (^[Y2 : hF_Mirabelle_hf]: ((Y1 @ Y2) & (hF_Mirabelle_hmem @ Y2 @ Y0))))))) @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) | ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true)) )),
% 2.68/1.59 inference(definition_unfolding,[status(thm)],[f907,f893])).
% 2.68/1.59 thf(f907,plain,(
% 2.68/1.59 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true) | ($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))) )),
% 2.68/1.59 inference(cnf_transformation,[status(thm)],[f858])).
% 2.68/1.59 % SZS output end Proof for DTF2THF_25592
% 2.68/1.59 % (25809)------------------------------
% 2.68/1.59 % (25809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (25809)Termination reason: Refutation
% 2.68/1.59
% 2.68/1.59 % (25809)Memory used [KB]: 6524
% 2.68/1.59 % (25809)Time elapsed: 0.019 s
% 2.68/1.59 % (25809)Instructions burned: 55 (million)
% 2.68/1.59 % (25809)------------------------------
% 2.68/1.59 % (25809)------------------------------
% 2.68/1.59 % (25783)Success in time 0.105 s
%------------------------------------------------------------------------------