%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP075^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n016.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 08:55:20 AM UTC 2026
% Result : Theorem 3.04s 1.77s
% Output : Refutation 3.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 9
% Syntax : Number of formulae : 64 ( 7 unt; 0 typ; 6 def)
% Number of atoms : 394 ( 100 equ; 0 cnn)
% Maximal formula atoms : 6 ( 6 avg)
% Number of connectives : 359 ( 71 ~; 72 |; 29 &; 170 @)
% ( 7 <=>; 1 =>; 0 <=; 1 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 89 ( 85 usr; 20 con; 0-3 aty)
% ( 0 !!; 8 ??; 0 @@+; 0 @@-)
% Number of variables : 72 ( 44 ^; 17 !; 11 ?; 72 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
hF_Mirabelle_hf: $tType ).
thf(type_def_7,type,
int: $tType ).
thf(type_def_8,type,
nat: $tType ).
thf(type_def_9,type,
set_HF_Mirabelle_hf: $tType ).
thf(func_def_0,type,
set_HF_Mirabelle_hf: $tType ).
thf(func_def_1,type,
hF_Mirabelle_hf: $tType ).
thf(func_def_2,type,
nat: $tType ).
thf(func_def_3,type,
int: $tType ).
thf(func_def_4,type,
minus_1668263787e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(func_def_5,type,
minus_minus_o: $o > $o > $o ).
thf(func_def_6,type,
minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_7,type,
minus_minus_int: int > int > int ).
thf(func_def_8,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_9,type,
minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_10,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(func_def_11,type,
zero_zero_int: int ).
thf(func_def_12,type,
zero_zero_nat: nat ).
thf(func_def_13,type,
hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
thf(func_def_14,type,
hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
thf(func_def_15,type,
hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_16,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_17,type,
hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_18,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_19,type,
hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_20,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_21,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_22,type,
hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
thf(func_def_23,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(func_def_24,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_25,type,
hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_26,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_27,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_28,type,
hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_29,type,
hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_30,type,
the_HF_Mirabelle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_31,type,
if_HF_Mirabelle_hf: $o > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_32,type,
inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(func_def_33,type,
inf_inf_o: $o > $o > $o ).
thf(func_def_34,type,
inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_35,type,
inf_inf_int: int > int > int ).
thf(func_def_36,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_37,type,
inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(func_def_38,type,
semiri2019852685at_int: nat > int ).
thf(func_def_39,type,
semiri1382578993at_nat: nat > nat ).
thf(func_def_40,type,
bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
thf(func_def_42,type,
bot_bo1001194783lle_hf: hF_Mirabelle_hf ).
thf(func_def_43,type,
bot_bot_nat: nat ).
thf(func_def_44,type,
bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
thf(func_def_45,type,
least_917110455lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_46,type,
ord_le616625840e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > $o ).
thf(func_def_47,type,
ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(func_def_48,type,
ord_less_int: int > int > $o ).
thf(func_def_49,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_50,type,
ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_51,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(func_def_52,type,
is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
thf(func_def_53,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(func_def_54,type,
a: hF_Mirabelle_hf ).
thf(func_def_71,type,
sP0: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > $o ).
thf(func_def_72,type,
sK1: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_73,type,
sK2: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_74,type,
sK3: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_75,type,
sK4: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_76,type,
sK5: hF_Mirabelle_hf > hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_77,type,
sK6: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_78,type,
sK7: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_79,type,
sK8: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_80,type,
sK9: hF_Mirabelle_hf > hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(func_def_81,type,
sK10: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_82,type,
sK11: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_83,type,
sK12: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_84,type,
sK13: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_85,type,
sK14: hF_Mirabelle_hf ).
thf(func_def_86,type,
sK15: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_88,type,
sK17: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_89,type,
sK18: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_90,type,
sK19: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_91,type,
sK20: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_92,type,
sK21: hF_Mirabelle_hf ).
thf(func_def_93,type,
sK22: hF_Mirabelle_hf ).
thf(func_def_94,type,
sK23: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_95,type,
sK24: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_96,type,
sK25: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_97,type,
sK26: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(func_def_98,type,
sK27: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(func_def_99,type,
sK28: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(f1025,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f981,f992,f1003,f1006,f1011,f1015,f1018,f1023]) ).
thf(f1023,plain,
( ~ spl16_1
| ~ spl16_6 ),
inference(avatar_contradiction_clause,[status(thm)],[f1022]) ).
thf(f1022,plain,
( $false
| ~ spl16_1
| ~ spl16_6 ),
inference(trivial_inequality_removal,[status(thm)],[f1020]) ).
thf(f1020,plain,
( ( $true != $true )
| ~ spl16_1
| ~ spl16_6 ),
inference(superposition,[status(thm)],[f999,f976]) ).
thf(f976,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK22 @ a ) )
| ~ spl16_1 ),
inference(avatar_component_clause,[status(thm)],[f974]) ).
thf(f974,definition,
( spl16_1
<=> ( $true
= ( hF_Mirabelle_hmem @ sK22 @ a ) ) ),
introduced(definition,[new_symbols(naming,[spl16_1])],[avatar_definition]) ).
thf(f999,plain,
( ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ~ spl16_6 ),
inference(avatar_component_clause,[status(thm)],[f998]) ).
thf(f998,definition,
( spl16_6
<=> ! [X0: hF_Mirabelle_hf] :
( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) ) ),
introduced(definition,[new_symbols(naming,[spl16_6])],[avatar_definition]) ).
thf(f1018,plain,
( ~ spl16_1
| ~ spl16_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f1017]) ).
thf(f1017,plain,
( $false
| ~ spl16_1
| ~ spl16_7 ),
inference(trivial_inequality_removal,[status(thm)],[f1016]) ).
thf(f1016,plain,
( ( $true = $false )
| ~ spl16_1
| ~ spl16_7 ),
inference(forward_demodulation,[status(thm)],[f976,f1002]) ).
thf(f1002,plain,
( ! [X1: hF_Mirabelle_hf] :
( $false
= ( hF_Mirabelle_hmem @ X1 @ a ) )
| ~ spl16_7 ),
inference(avatar_component_clause,[status(thm)],[f1001]) ).
thf(f1001,definition,
( spl16_7
<=> ! [X1: hF_Mirabelle_hf] :
( $false
= ( hF_Mirabelle_hmem @ X1 @ a ) ) ),
introduced(definition,[new_symbols(naming,[spl16_7])],[avatar_definition]) ).
thf(f1015,plain,
( ~ spl16_2
| ~ spl16_7 ),
inference(avatar_contradiction_clause,[status(thm)],[f1014]) ).
thf(f1014,plain,
( $false
| ~ spl16_2
| ~ spl16_7 ),
inference(trivial_inequality_removal,[status(thm)],[f1013]) ).
thf(f1013,plain,
( ( $true = $false )
| ~ spl16_2
| ~ spl16_7 ),
inference(backward_demodulation,[status(thm)],[f980,f1002]) ).
thf(f980,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) )
| ~ spl16_2 ),
inference(avatar_component_clause,[status(thm)],[f978]) ).
thf(f978,definition,
( spl16_2
<=> ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) ) ),
introduced(definition,[new_symbols(naming,[spl16_2])],[avatar_definition]) ).
thf(f1011,plain,
( ~ spl16_2
| ~ spl16_6 ),
inference(avatar_contradiction_clause,[status(thm)],[f1010]) ).
thf(f1010,plain,
( $false
| ~ spl16_2
| ~ spl16_6 ),
inference(trivial_inequality_removal,[status(thm)],[f1008]) ).
thf(f1008,plain,
( ( $true != $true )
| ~ spl16_2
| ~ spl16_6 ),
inference(superposition,[status(thm)],[f999,f980]) ).
thf(f1006,plain,
( ~ spl16_3
| ~ spl16_5 ),
inference(avatar_contradiction_clause,[status(thm)],[f1005]) ).
thf(f1005,plain,
( $false
| ~ spl16_3
| ~ spl16_5 ),
inference(trivial_inequality_removal,[status(thm)],[f1004]) ).
thf(f1004,plain,
( ( $true = $false )
| ~ spl16_3
| ~ spl16_5 ),
inference(forward_demodulation,[status(thm)],[f985,f996]) ).
thf(f996,plain,
( ( p = $false )
| ~ spl16_5 ),
inference(avatar_component_clause,[status(thm)],[f994]) ).
thf(f994,definition,
( spl16_5
<=> ( p = $false ) ),
introduced(definition,[new_symbols(naming,[spl16_5])],[avatar_definition]) ).
thf(f985,plain,
( ( p = $true )
| ~ spl16_3 ),
inference(avatar_component_clause,[status(thm)],[f983]) ).
thf(f983,definition,
( spl16_3
<=> ( p = $true ) ),
introduced(definition,[new_symbols(naming,[spl16_3])],[avatar_definition]) ).
thf(f1003,plain,
( spl16_5
| spl16_6
| ~ spl16_3
| spl16_7 ),
inference(avatar_split_clause,[status(thm)],[f908,f1001,f983,f998,f994]) ).
thf(f908,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( p = $false )
| ( $false
= ( hF_Mirabelle_hmem @ X1 @ a ) )
| ( p != $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f907]) ).
thf(f907,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( p != $true )
| ( $false
= ( p
& ( hF_Mirabelle_hmem @ X1 @ a ) ) )
| ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) ) ),
inference(beta_eta_normalization,[status(thm)],[f906]) ).
thf(f906,plain,
! [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf] :
( ( p != $true )
| ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( $false
= ( ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) )
@ X1 ) ) ),
inference(pi_clausification,[status(thm)],[f905]) ).
thf(f905,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( ( ?? @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) ) )
!= $true )
| ( p != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f859]) ).
thf(f859,plain,
! [X0: hF_Mirabelle_hf] :
( ( ( ^ [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 )
| ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( p != $true ) ),
inference(definition_unfolding,[status(thm)],[f849,f832]) ).
thf(f832,plain,
( 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 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f405]) ).
thf(f405,plain,
( 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 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f404]) ).
thf(f404,plain,
( hF_Mirabelle_HBex
= ( ^ [X0: hF_Mirabelle_hf,X1: hF_Mirabelle_hf > $o] :
? [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ X0 )
& ( X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( hF_Mirabelle_HBex
= ( ^ [X12: hF_Mirabelle_hf,X13: hF_Mirabelle_hf > $o] :
? [X10: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X10 @ X12 )
& ( X13 @ X10 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.mKzX9y748V/DTF2THF_32593.p',fact_5_HBex__def) ).
thf(f849,plain,
! [X0: hF_Mirabelle_hf] :
( ( $true
!= ( hF_Mirabelle_HBex @ a
@ ^ [Y0: hF_Mirabelle_hf] : p ) )
| ( $true
!= ( hF_Mirabelle_hmem @ X0 @ a ) )
| ( p != $true ) ),
inference(cnf_transformation,[status(thm)],[f814]) ).
thf(f814,plain,
( ( ( $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 ) )
| ( ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) )
& ( p = $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK14])],[f812,f813]) ).
thf(f813,plain,
( ? [X1: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X1 @ a ) )
=> ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f812,plain,
( ( ( $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 ) )
| ( ? [X1: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X1 @ a ) )
& ( p = $true ) ) ) ),
inference(rectify,[status(thm)],[f811]) ).
thf(f811,plain,
( ( ( $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 ) )
| ( ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ a ) )
& ( p = $true ) ) ) ),
inference(flattening,[status(thm)],[f810]) ).
thf(f810,plain,
( ( ( $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 ) )
| ( ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ a ) )
& ( p = $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f777]) ).
thf(f777,plain,
( ( ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ a ) )
& ( p = $true ) )
<~> ( $true
= ( hF_Mirabelle_HBex @ a
@ ^ [Y0: hF_Mirabelle_hf] : p ) ) ),
inference(ennf_transformation,[status(thm)],[f657]) ).
thf(f657,plain,
~ ( ( $true
= ( hF_Mirabelle_HBex @ a
@ ^ [Y0: hF_Mirabelle_hf] : p ) )
<=> ( ? [X0: hF_Mirabelle_hf] :
( $true
= ( hF_Mirabelle_hmem @ X0 @ a ) )
& ( p = $true ) ) ),
inference(fool_elimination,[status(thm)],[f656]) ).
thf(f656,plain,
~ ( p
& ? [X0: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X0 @ a )
= ( hF_Mirabelle_HBex @ a
@ ^ [X1: hF_Mirabelle_hf] : p ) ) ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
~ ( p
& ? [X10: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X10 @ a )
= ( hF_Mirabelle_HBex @ a
@ ^ [X10: hF_Mirabelle_hf] : p ) ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
( p
& ? [X10: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X10 @ a )
= ( hF_Mirabelle_HBex @ a
@ ^ [X10: hF_Mirabelle_hf] : p ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.mKzX9y748V/DTF2THF_32593.p',conj_0) ).
thf(f992,plain,
spl16_3,
inference(avatar_split_clause,[status(thm)],[f914,f983]) ).
thf(f914,plain,
p = $true,
inference(duplicate_literal_removal,[status(thm)],[f913]) ).
thf(f913,plain,
( ( p = $true )
| ( p = $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f911]) ).
thf(f911,plain,
( ( p = $true )
| ( $true
= ( p
& ( hF_Mirabelle_hmem @ sK21 @ a ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f910]) ).
thf(f910,plain,
( ( $true
= ( ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) )
@ sK21 ) )
| ( p = $true ) ),
inference(sigma_clausification,[status(thm)],[f909]) ).
thf(f909,plain,
( ( p = $true )
| ( ( ?? @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f861]) ).
thf(f861,plain,
( ( p = $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 ) ),
inference(definition_unfolding,[status(thm)],[f847,f832]) ).
thf(f847,plain,
( ( $true
= ( hF_Mirabelle_HBex @ a
@ ^ [Y0: hF_Mirabelle_hf] : p ) )
| ( p = $true ) ),
inference(cnf_transformation,[status(thm)],[f814]) ).
thf(f981,plain,
( spl16_1
| spl16_2 ),
inference(avatar_split_clause,[status(thm)],[f920,f978,f974]) ).
thf(f920,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK22 @ a ) )
| ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f919]) ).
thf(f919,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) )
| ( $true
= ( p
& ( hF_Mirabelle_hmem @ sK22 @ a ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f918]) ).
thf(f918,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) )
| ( $true
= ( ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) )
@ sK22 ) ) ),
inference(sigma_clausification,[status(thm)],[f917]) ).
thf(f917,plain,
( ( ( ?? @ hF_Mirabelle_hf
@ ^ [Y0: hF_Mirabelle_hf] :
( p
& ( hF_Mirabelle_hmem @ Y0 @ a ) ) )
= $true )
| ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) ) ),
inference(beta_eta_normalization,[status(thm)],[f860]) ).
thf(f860,plain,
( ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) )
| ( ( ^ [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 ) ),
inference(definition_unfolding,[status(thm)],[f848,f832]) ).
thf(f848,plain,
( ( $true
= ( hF_Mirabelle_HBex @ a
@ ^ [Y0: hF_Mirabelle_hf] : p ) )
| ( $true
= ( hF_Mirabelle_hmem @ sK14 @ a ) ) ),
inference(cnf_transformation,[status(thm)],[f814]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP075^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n016.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 08:25:36 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.41 ---- Original DTF file ---
% 0.15/0.41 thf(spec,logic,$$dhol).
% 0.15/0.41 %------------------------------------------------------------------------------
% 0.15/0.41 % File : ITP075^1 : TPTP v9.2.1. Released v7.5.0.
% 0.15/0.41 % Domain : Interactive Theorem Proving
% 0.15/0.41 % Problem : Sledgehammer HF problem prob_754__5337380_1
% 0.15/0.41 % Version : Especial.
% 0.15/0.41 % English :
% 0.15/0.41
% 0.15/0.41 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.15/0.41 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.15/0.41 % Source : [Des21]
% 0.15/0.41 % Names : HF/prob_754__5337380_1 [Des21]
% 0.15/0.41
% 0.15/0.41 % Status : Theorem
% 0.15/0.41 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.15/0.41 % Syntax : Number of formulae : 411 ( 267 unt; 56 typ; 0 def)
% 0.15/0.41 % Number of atoms : 755 ( 399 equ; 0 cnn)
% 0.15/0.41 % Maximal formula atoms : 4 ( 2 avg)
% 0.15/0.41 % Number of connectives : 2036 ( 93 ~; 5 |; 43 &;1719 @)
% 0.15/0.41 % ( 0 <=>; 176 =>; 0 <=; 0 <~>)
% 0.15/0.41 % Maximal formula depth : 13 ( 4 avg)
% 0.15/0.41 % Number of types : 5 ( 4 usr)
% 0.15/0.41 % Number of type conns : 221 ( 221 >; 0 *; 0 +; 0 <<)
% 0.15/0.41 % Number of symbols : 55 ( 52 usr; 11 con; 0-3 aty)
% 0.15/0.41 % Number of variables : 844 ( 139 ^; 677 !; 28 ?; 844 :)
% 0.15/0.41 % SPC : TH0_THM_EQU_NAR
% 0.15/0.41
% 0.15/0.41 % Comments : This file was generated by Sledgehammer 2021-02-23 15:38:10.866
% 0.15/0.41 %------------------------------------------------------------------------------
% 0.15/0.41 % Could-be-implicit typings (4)
% 0.15/0.41 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.15/0.41 set_HF_Mirabelle_hf: $tType ).
% 0.15/0.41
% 0.15/0.41 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 hF_Mirabelle_hf: $tType ).
% 0.15/0.41
% 0.15/0.41 thf(ty_n_t__Nat__Onat,type,
% 0.15/0.41 nat: $tType ).
% 0.15/0.41
% 0.15/0.41 thf(ty_n_t__Int__Oint,type,
% 0.15/0.41 int: $tType ).
% 0.15/0.41
% 0.15/0.41 % Explicit typings (52)
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.15/0.41 minus_1668263787e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
% 0.15/0.41 minus_minus_o: $o > $o > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.15/0.41 minus_minus_int: int > int > int ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.15/0.41 minus_minus_nat: nat > nat > nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.15/0.41 minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.15/0.41 zero_zero_int: int ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.15/0.41 zero_zero_nat: nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHBall,type,
% 0.15/0.41 hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHBex,type,
% 0.15/0.41 hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.15/0.41 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.15/0.41 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 0.15/0.41 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.15/0.41 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.15/0.41 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.15/0.41 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.15/0.41 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohcard,type,
% 0.15/0.41 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.15/0.41 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.15/0.41 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 0.15/0.41 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.15/0.41 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.15/0.41 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 0.15/0.41 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 0.15/0.41 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_HOL_OThe_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 the_HF_Mirabelle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_If_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 if_HF_Mirabelle_hf: $o > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.15/0.41 inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
% 0.15/0.41 inf_inf_o: $o > $o > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
% 0.15/0.41 inf_inf_int: int > int > int ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.15/0.41 inf_inf_nat: nat > nat > nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.15/0.41 inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.15/0.41 semiri2019852685at_int: nat > int ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.15/0.41 semiri1382578993at_nat: nat > nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.15/0.41 bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.15/0.41 bot_bot_o: $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Obot__class_Obot_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 bot_bo1001194783lle_hf: hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.15/0.41 bot_bot_nat: nat ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.15/0.41 bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord_OLeast_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 least_917110455lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.15/0.41 ord_le616625840e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.15/0.41 ord_less_int: int > int > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.15/0.41 ord_less_nat: nat > nat > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.15/0.41 ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.15/0.41 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.15/0.41
% 0.15/0.41 thf(sy_v_A,type,
% 0.15/0.41 a: hF_Mirabelle_hf ).
% 0.15/0.41
% 0.15/0.41 thf(sy_v_P,type,
% 0.15/0.41 p: $o ).
% 0.15/0.41
% 0.15/0.41 % Relevant facts (351)
% 0.15/0.41 thf(fact_0_hf__equalityI,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.41 ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ A )
% 0.15/0.41 = ( hF_Mirabelle_hmem @ X @ B ) )
% 0.15/0.41 => ( A = B ) ) ).
% 0.15/0.41
% 0.15/0.41 % hf_equalityI
% 0.15/0.41 thf(fact_1_hbexI,axiom,
% 0.15/0.41 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( P @ X2 )
% 0.15/0.41 => ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.15/0.41 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hbexI
% 0.15/0.41 thf(fact_2_hbexE,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mirabelle_HBex @ A2 @ P )
% 0.15/0.41 => ~ ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.15/0.41 => ~ ( P @ X ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hbexE
% 0.15/0.41 thf(fact_3_hf__ext,axiom,
% 0.15/0.41 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.15/0.41 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.15/0.41 = ( hF_Mirabelle_hmem @ X3 @ B2 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hf_ext
% 0.15/0.41 thf(fact_4_hmem__ne,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.15/0.41 => ( X2 != Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_ne
% 0.15/0.41 thf(fact_5_HBex__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_HBex
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ? [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.15/0.41 & ( P2 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HBex_def
% 0.15/0.41 thf(fact_6_hbex__cong,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( A2 = A5 )
% 0.15/0.41 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ A5 )
% 0.15/0.41 => ( ( P @ X )
% 0.15/0.41 = ( P3 @ X ) ) )
% 0.15/0.41 => ( ( hF_Mirabelle_HBex @ A2 @ P )
% 0.15/0.41 = ( hF_Mirabelle_HBex @ A5 @ P3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hbex_cong
% 0.15/0.41 thf(fact_7_rev__hbexI,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.15/0.41 => ( ( P @ X2 )
% 0.15/0.41 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % rev_hbexI
% 0.15/0.41 thf(fact_8_replacement,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U @ X2 )
% 0.15/0.41 => ( ( R @ U @ V )
% 0.15/0.41 => ( ( R @ U @ V2 )
% 0.15/0.41 => ( V2 = V ) ) ) )
% 0.15/0.41 => ? [Z2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [V3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.15/0.41 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.15/0.41 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % replacement
% 0.15/0.41 thf(fact_9_binary__union,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ? [Z2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [U3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.15/0.41 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.15/0.41 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % binary_union
% 0.15/0.41 thf(fact_10_hmem__not__sym,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.15/0.41 & ( hF_Mirabelle_hmem @ Y2 @ X2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_not_sym
% 0.15/0.41 thf(fact_11_union__of__set,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ? [Z2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [U3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.15/0.41 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ Y3 @ X2 )
% 0.15/0.41 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % union_of_set
% 0.15/0.41 thf(fact_12_comprehension,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ? [Z2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [U3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.15/0.41 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.15/0.41 & ( P @ U3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % comprehension
% 0.15/0.41 thf(fact_13_hmem__not__refl,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( hF_Mirabelle_hmem @ X2 @ X2 ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_not_refl
% 0.15/0.41 thf(fact_14_replacement__fun,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.41 ? [Z2: hF_Mirabelle_hf] :
% 0.15/0.41 ! [V3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.15/0.41 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.15/0.41 & ( V3
% 0.15/0.41 = ( F @ U2 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % replacement_fun
% 0.15/0.41 thf(fact_15_HF__Mirabelle__glliljednj_ObexCI,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( ( hF_Mirabelle_HBall @ A2
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( P @ X3 ) )
% 0.15/0.41 => ( P @ A ) )
% 0.15/0.41 => ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.15/0.41 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HF_Mirabelle_glliljednj.bexCI
% 0.15/0.41 thf(fact_16_hballI,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.15/0.41 => ( P @ X ) )
% 0.15/0.41 => ( hF_Mirabelle_HBall @ A2 @ P ) ) ).
% 0.15/0.41
% 0.15/0.41 % hballI
% 0.15/0.41 thf(fact_17_hball__triv,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: $o] :
% 0.15/0.41 ( ( hF_Mirabelle_HBall @ A2
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] : P )
% 0.15/0.41 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ A2 )
% 0.15/0.41 => P ) ) ).
% 0.15/0.41
% 0.15/0.41 % hball_triv
% 0.15/0.41 thf(fact_18_PrimReplace__iff,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U @ A2 )
% 0.15/0.41 => ( ( R @ U @ V )
% 0.15/0.41 => ( ( R @ U @ V2 )
% 0.15/0.41 => ( V2 = V ) ) ) )
% 0.15/0.41 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A2 @ R ) )
% 0.15/0.41 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.15/0.41 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % PrimReplace_iff
% 0.15/0.41 thf(fact_19_HCollect__iff,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mir818139703ollect @ P @ A2 ) )
% 0.15/0.41 = ( ( P @ X2 )
% 0.15/0.41 & ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HCollect_iff
% 0.15/0.41 thf(fact_20_Replace__iff,axiom,
% 0.15/0.41 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A2 @ R ) )
% 0.15/0.41 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.15/0.41 & ( R @ U2 @ V4 )
% 0.15/0.41 & ! [Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( R @ U2 @ Y3 )
% 0.15/0.41 => ( Y3 = V4 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % Replace_iff
% 0.15/0.41 thf(fact_21_HUnion__iff,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HUnion @ A2 ) )
% 0.15/0.41 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ Y3 @ A2 )
% 0.15/0.41 & ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HUnion_iff
% 0.15/0.41 thf(fact_22_hballE,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.15/0.41 => ( ~ ( P @ X2 )
% 0.15/0.41 => ~ ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hballE
% 0.15/0.41 thf(fact_23_hbspec,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.15/0.41 => ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.15/0.41 => ( P @ X2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hbspec
% 0.15/0.41 thf(fact_24_HCollectE,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) )
% 0.15/0.41 => ~ ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.15/0.41 => ~ ( P @ A ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HCollectE
% 0.15/0.41 thf(fact_25_HCollectI,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.15/0.41 => ( ( P @ A )
% 0.15/0.41 => ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HCollectI
% 0.15/0.41 thf(fact_26_hball__cong,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( A2 = A5 )
% 0.15/0.41 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ A5 )
% 0.15/0.41 => ( ( P @ X )
% 0.15/0.41 = ( P3 @ X ) ) )
% 0.15/0.41 => ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.15/0.41 = ( hF_Mirabelle_HBall @ A5 @ P3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hball_cong
% 0.15/0.41 thf(fact_27_hmem__Sup__ne,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.15/0.41 => ( ( hF_Mirabelle_HUnion @ X2 )
% 0.15/0.41 != Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_Sup_ne
% 0.15/0.41 thf(fact_28_Replace__cong,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( A2 = B3 )
% 0.15/0.41 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.15/0.41 => ( ( P @ X @ Y4 )
% 0.15/0.41 = ( Q @ X @ Y4 ) ) )
% 0.15/0.41 => ( ( hF_Mirabelle_Replace @ A2 @ P )
% 0.15/0.41 = ( hF_Mirabelle_Replace @ B3 @ Q ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % Replace_cong
% 0.15/0.41 thf(fact_29_HBall__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_HBall
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ! [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.15/0.41 => ( P2 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HBall_def
% 0.15/0.41 thf(fact_30_Replace__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_Replace
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( hF_Mir1248913145eplace @ A4
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ? [Z3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( R2 @ X3 @ Z3 )
% 0.15/0.41 & ! [Aa: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( R2 @ X3 @ Aa )
% 0.15/0.41 => ( Aa = Z3 ) ) )
% 0.15/0.41 & ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % Replace_def
% 0.15/0.41 thf(fact_31_RepFun__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_RepFun
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf,F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.41 ( hF_Mirabelle_Replace @ A4
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( Y3
% 0.15/0.41 = ( F2 @ X3 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % RepFun_def
% 0.15/0.41 thf(fact_32_minus__hf__def,axiom,
% 0.15/0.41 ( minus_1232880740lle_hf
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.15/0.41 ( hF_Mir818139703ollect
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.15/0.41 @ A4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % minus_hf_def
% 0.15/0.41 thf(fact_33_inf__hf__def,axiom,
% 0.15/0.41 ( inf_in956532509lle_hf
% 0.15/0.41 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.15/0.41 ( hF_Mir818139703ollect
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ B2 )
% 0.15/0.41 @ A3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_hf_def
% 0.15/0.41 thf(fact_34_HInter__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_HInter
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf] :
% 0.15/0.41 ( hF_Mir818139703ollect
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ! [Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.15/0.41 => ( hF_Mirabelle_hmem @ X3 @ Y3 ) )
% 0.15/0.41 @ ( hF_Mirabelle_HUnion @ A4 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HInter_def
% 0.15/0.41 thf(fact_35_RepFun__iff,axiom,
% 0.15/0.41 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A2 @ F ) )
% 0.15/0.41 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.15/0.41 & ( V4
% 0.15/0.41 = ( F @ U2 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % RepFun_iff
% 0.15/0.41 thf(fact_36_HCollect__hempty,axiom,
% 0.15/0.41 ! [P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % HCollect_hempty
% 0.15/0.41 thf(fact_37_Replace__0,axiom,
% 0.15/0.41 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % Replace_0
% 0.15/0.41 thf(fact_38_hmem__def,axiom,
% 0.15/0.41 ( hF_Mirabelle_hmem
% 0.15/0.41 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_def
% 0.15/0.41 thf(fact_39_triv__RepFun,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_RepFun @ A2
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] : X3 )
% 0.15/0.41 = A2 ) ).
% 0.15/0.41
% 0.15/0.41 % triv_RepFun
% 0.15/0.41 thf(fact_40_hinter__iff,axiom,
% 0.15/0.41 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.15/0.41 & ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hinter_iff
% 0.15/0.41 thf(fact_41_hdiff__iff,axiom,
% 0.15/0.41 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ U4 @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.15/0.41 & ~ ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hdiff_iff
% 0.15/0.41 thf(fact_42_hinter__hempty__right,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ A2 @ zero_z189798548lle_hf )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % hinter_hempty_right
% 0.15/0.41 thf(fact_43_hinter__hempty__left,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A2 )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % hinter_hempty_left
% 0.15/0.41 thf(fact_44_zero__hdiff,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X2 )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % zero_hdiff
% 0.15/0.41 thf(fact_45_mem__Collect__eq,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 0.15/0.41 = ( P @ A ) ) ).
% 0.15/0.41
% 0.15/0.41 % mem_Collect_eq
% 0.15/0.41 thf(fact_46_Collect__mem__eq,axiom,
% 0.15/0.41 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( collec2046588256lle_hf
% 0.15/0.41 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.15/0.41 = A2 ) ).
% 0.15/0.41
% 0.15/0.41 % Collect_mem_eq
% 0.15/0.41 thf(fact_47_Collect__cong,axiom,
% 0.15/0.41 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( P @ X )
% 0.15/0.41 = ( Q @ X ) )
% 0.15/0.41 => ( ( collec2046588256lle_hf @ P )
% 0.15/0.41 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % Collect_cong
% 0.15/0.41 thf(fact_48_hdiff__zero,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( minus_1232880740lle_hf @ X2 @ zero_z189798548lle_hf )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % hdiff_zero
% 0.15/0.41 thf(fact_49_RepFun__0,axiom,
% 0.15/0.41 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % RepFun_0
% 0.15/0.41 thf(fact_50_HUnion__hempty,axiom,
% 0.15/0.41 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % HUnion_hempty
% 0.15/0.41 thf(fact_51_HInter__hempty,axiom,
% 0.15/0.41 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 0.15/0.41 = zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % HInter_hempty
% 0.15/0.41 thf(fact_52_HInter__iff,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( A2 != zero_z189798548lle_hf )
% 0.15/0.41 => ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HInter @ A2 ) )
% 0.15/0.41 = ( ! [Y3: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ Y3 @ A2 )
% 0.15/0.41 => ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % HInter_iff
% 0.15/0.41 thf(fact_53_foundation,axiom,
% 0.15/0.41 ! [Z4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( Z4 != zero_z189798548lle_hf )
% 0.15/0.41 => ? [W: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ W @ Z4 )
% 0.15/0.41 & ( ( inf_in956532509lle_hf @ W @ Z4 )
% 0.15/0.41 = zero_z189798548lle_hf ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % foundation
% 0.15/0.41 thf(fact_54_hemptyE,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % hemptyE
% 0.15/0.41 thf(fact_55_hempty__iff,axiom,
% 0.15/0.41 ! [Z4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( Z4 = zero_z189798548lle_hf )
% 0.15/0.41 = ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( hF_Mirabelle_hmem @ X3 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hempty_iff
% 0.15/0.41 thf(fact_56_hmem__hempty,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf] :
% 0.15/0.41 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.15/0.41
% 0.15/0.41 % hmem_hempty
% 0.15/0.41 thf(fact_57_RepFun__cong,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.41 ( ( A2 = B3 )
% 0.15/0.41 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.15/0.41 => ( ( F @ X )
% 0.15/0.41 = ( G @ X ) ) )
% 0.15/0.41 => ( ( hF_Mirabelle_RepFun @ A2 @ F )
% 0.15/0.41 = ( hF_Mirabelle_RepFun @ B3 @ G ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % RepFun_cong
% 0.15/0.41 thf(fact_58_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.15/0.41 ! [A: nat] :
% 0.15/0.41 ( ( minus_minus_nat @ A @ A )
% 0.15/0.41 = zero_zero_nat ) ).
% 0.15/0.41
% 0.15/0.41 % cancel_comm_monoid_add_class.diff_cancel
% 0.15/0.41 thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.15/0.41 ! [A: int] :
% 0.15/0.41 ( ( minus_minus_int @ A @ A )
% 0.15/0.41 = zero_zero_int ) ).
% 0.15/0.41
% 0.15/0.41 % cancel_comm_monoid_add_class.diff_cancel
% 0.15/0.41 thf(fact_60_diff__zero,axiom,
% 0.15/0.41 ! [A: nat] :
% 0.15/0.41 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % diff_zero
% 0.15/0.41 thf(fact_61_diff__zero,axiom,
% 0.15/0.41 ! [A: int] :
% 0.15/0.41 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % diff_zero
% 0.15/0.41 thf(fact_62_zero__diff,axiom,
% 0.15/0.41 ! [A: nat] :
% 0.15/0.41 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.15/0.41 = zero_zero_nat ) ).
% 0.15/0.41
% 0.15/0.41 % zero_diff
% 0.15/0.41 thf(fact_63_diff__0__right,axiom,
% 0.15/0.41 ! [A: int] :
% 0.15/0.41 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % diff_0_right
% 0.15/0.41 thf(fact_64_diff__self,axiom,
% 0.15/0.41 ! [A: int] :
% 0.15/0.41 ( ( minus_minus_int @ A @ A )
% 0.15/0.41 = zero_zero_int ) ).
% 0.15/0.41
% 0.15/0.41 % diff_self
% 0.15/0.41 thf(fact_65_inf__right__idem,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Y2 )
% 0.15/0.41 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_right_idem
% 0.15/0.41 thf(fact_66_inf__right__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Y2 )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_right_idem
% 0.15/0.41 thf(fact_67_inf__right__idem,axiom,
% 0.15/0.41 ! [X2: int,Y2: int] :
% 0.15/0.41 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Y2 )
% 0.15/0.41 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_right_idem
% 0.15/0.41 thf(fact_68_inf__right__idem,axiom,
% 0.15/0.41 ! [X2: nat,Y2: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 )
% 0.15/0.41 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_right_idem
% 0.15/0.41 thf(fact_69_inf__right__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Y2 )
% 0.15/0.41 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_right_idem
% 0.15/0.41 thf(fact_70_inf_Oright__idem,axiom,
% 0.15/0.41 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ B )
% 0.15/0.41 = ( inf_in923488851lle_hf @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.right_idem
% 0.15/0.41 thf(fact_71_inf_Oright__idem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ B )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.right_idem
% 0.15/0.41 thf(fact_72_inf_Oright__idem,axiom,
% 0.15/0.41 ! [A: int,B: int] :
% 0.15/0.41 ( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ B )
% 0.15/0.41 = ( inf_inf_int @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.right_idem
% 0.15/0.41 thf(fact_73_inf_Oright__idem,axiom,
% 0.15/0.41 ! [A: nat,B: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
% 0.15/0.41 = ( inf_inf_nat @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.right_idem
% 0.15/0.41 thf(fact_74_inf_Oright__idem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B )
% 0.15/0.41 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.right_idem
% 0.15/0.41 thf(fact_75_inf__left__idem,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_left_idem
% 0.15/0.41 thf(fact_76_inf__left__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_left_idem
% 0.15/0.41 thf(fact_77_inf__left__idem,axiom,
% 0.15/0.41 ! [X2: int,Y2: int] :
% 0.15/0.41 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_left_idem
% 0.15/0.41 thf(fact_78_inf__left__idem,axiom,
% 0.15/0.41 ! [X2: nat,Y2: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_left_idem
% 0.15/0.41 thf(fact_79_inf__left__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_left_idem
% 0.15/0.41 thf(fact_80_minus__apply,axiom,
% 0.15/0.41 ( minus_1668263787e_hf_o
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % minus_apply
% 0.15/0.41 thf(fact_81_inf__apply,axiom,
% 0.15/0.41 ( inf_in307783154e_hf_o
% 0.15/0.41 = ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_apply
% 0.15/0.41 thf(fact_82_inf_Oidem,axiom,
% 0.15/0.41 ! [A: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ A @ A )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % inf.idem
% 0.15/0.41 thf(fact_83_inf_Oidem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ A @ A )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % inf.idem
% 0.15/0.41 thf(fact_84_inf_Oidem,axiom,
% 0.15/0.41 ! [A: int] :
% 0.15/0.41 ( ( inf_inf_int @ A @ A )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % inf.idem
% 0.15/0.41 thf(fact_85_inf_Oidem,axiom,
% 0.15/0.41 ! [A: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ A @ A )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % inf.idem
% 0.15/0.41 thf(fact_86_inf_Oidem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ A @ A )
% 0.15/0.41 = A ) ).
% 0.15/0.41
% 0.15/0.41 % inf.idem
% 0.15/0.41 thf(fact_87_inf__idem,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ X2 @ X2 )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % inf_idem
% 0.15/0.41 thf(fact_88_inf__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ X2 @ X2 )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % inf_idem
% 0.15/0.41 thf(fact_89_inf__idem,axiom,
% 0.15/0.41 ! [X2: int] :
% 0.15/0.41 ( ( inf_inf_int @ X2 @ X2 )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % inf_idem
% 0.15/0.41 thf(fact_90_inf__idem,axiom,
% 0.15/0.41 ! [X2: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ X2 @ X2 )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % inf_idem
% 0.15/0.41 thf(fact_91_inf__idem,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ X2 @ X2 )
% 0.15/0.41 = X2 ) ).
% 0.15/0.41
% 0.15/0.41 % inf_idem
% 0.15/0.41 thf(fact_92_inf_Oleft__idem,axiom,
% 0.15/0.41 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ A @ B ) )
% 0.15/0.41 = ( inf_in923488851lle_hf @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.left_idem
% 0.15/0.41 thf(fact_93_inf_Oleft__idem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ A @ B ) )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.left_idem
% 0.15/0.41 thf(fact_94_inf_Oleft__idem,axiom,
% 0.15/0.41 ! [A: int,B: int] :
% 0.15/0.41 ( ( inf_inf_int @ A @ ( inf_inf_int @ A @ B ) )
% 0.15/0.41 = ( inf_inf_int @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.left_idem
% 0.15/0.41 thf(fact_95_inf_Oleft__idem,axiom,
% 0.15/0.41 ! [A: nat,B: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
% 0.15/0.41 = ( inf_inf_nat @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.left_idem
% 0.15/0.41 thf(fact_96_inf_Oleft__idem,axiom,
% 0.15/0.41 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ A @ B ) )
% 0.15/0.41 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf.left_idem
% 0.15/0.41 thf(fact_97_hfset__hdiff,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( hF_Mirabelle_hfset @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( minus_1450406810lle_hf @ ( hF_Mirabelle_hfset @ X2 ) @ ( hF_Mirabelle_hfset @ Y2 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % hfset_hdiff
% 0.15/0.41 thf(fact_98_zero__reorient,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( zero_z189798548lle_hf = X2 )
% 0.15/0.41 = ( X2 = zero_z189798548lle_hf ) ) ).
% 0.15/0.41
% 0.15/0.41 % zero_reorient
% 0.15/0.41 thf(fact_99_zero__reorient,axiom,
% 0.15/0.41 ! [X2: nat] :
% 0.15/0.41 ( ( zero_zero_nat = X2 )
% 0.15/0.41 = ( X2 = zero_zero_nat ) ) ).
% 0.15/0.41
% 0.15/0.41 % zero_reorient
% 0.15/0.41 thf(fact_100_zero__reorient,axiom,
% 0.15/0.41 ! [X2: int] :
% 0.15/0.41 ( ( zero_zero_int = X2 )
% 0.15/0.41 = ( X2 = zero_zero_int ) ) ).
% 0.15/0.41
% 0.15/0.41 % zero_reorient
% 0.15/0.41 thf(fact_101_fun__diff__def,axiom,
% 0.15/0.41 ( minus_1668263787e_hf_o
% 0.15/0.41 = ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % fun_diff_def
% 0.15/0.41 thf(fact_102_diff__eq__diff__eq,axiom,
% 0.15/0.41 ! [A: int,B: int,C: int,D: int] :
% 0.15/0.41 ( ( ( minus_minus_int @ A @ B )
% 0.15/0.41 = ( minus_minus_int @ C @ D ) )
% 0.15/0.41 => ( ( A = B )
% 0.15/0.41 = ( C = D ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % diff_eq_diff_eq
% 0.15/0.41 thf(fact_103_diff__right__commute,axiom,
% 0.15/0.41 ! [A: nat,C: nat,B: nat] :
% 0.15/0.41 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.15/0.41 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.15/0.41
% 0.15/0.41 % diff_right_commute
% 0.15/0.41 thf(fact_104_diff__right__commute,axiom,
% 0.15/0.41 ! [A: int,C: int,B: int] :
% 0.15/0.41 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.15/0.41 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.15/0.41
% 0.15/0.41 % diff_right_commute
% 0.15/0.41 thf(fact_105_inf__sup__aci_I4_J,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(4)
% 0.15/0.41 thf(fact_106_inf__sup__aci_I4_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(4)
% 0.15/0.41 thf(fact_107_inf__sup__aci_I4_J,axiom,
% 0.15/0.41 ! [X2: int,Y2: int] :
% 0.15/0.41 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(4)
% 0.15/0.41 thf(fact_108_inf__sup__aci_I4_J,axiom,
% 0.15/0.41 ! [X2: nat,Y2: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(4)
% 0.15/0.41 thf(fact_109_inf__sup__aci_I4_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.15/0.41 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(4)
% 0.15/0.41 thf(fact_110_inf__sup__aci_I3_J,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
% 0.15/0.41 = ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(3)
% 0.15/0.41 thf(fact_111_inf__sup__aci_I3_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(3)
% 0.15/0.41 thf(fact_112_inf__sup__aci_I3_J,axiom,
% 0.15/0.41 ! [X2: int,Y2: int,Z4: int] :
% 0.15/0.41 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
% 0.15/0.41 = ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(3)
% 0.15/0.41 thf(fact_113_inf__sup__aci_I3_J,axiom,
% 0.15/0.41 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
% 0.15/0.41 = ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(3)
% 0.15/0.41 thf(fact_114_inf__sup__aci_I3_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
% 0.15/0.41 = ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(3)
% 0.15/0.41 thf(fact_115_inf__sup__aci_I2_J,axiom,
% 0.15/0.41 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.15/0.41 = ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(2)
% 0.15/0.41 thf(fact_116_inf__sup__aci_I2_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(2)
% 0.15/0.41 thf(fact_117_inf__sup__aci_I2_J,axiom,
% 0.15/0.41 ! [X2: int,Y2: int,Z4: int] :
% 0.15/0.41 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
% 0.15/0.41 = ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(2)
% 0.15/0.41 thf(fact_118_inf__sup__aci_I2_J,axiom,
% 0.15/0.41 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.15/0.41 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
% 0.15/0.41 = ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(2)
% 0.15/0.41 thf(fact_119_inf__sup__aci_I2_J,axiom,
% 0.15/0.41 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.15/0.41 = ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(2)
% 0.15/0.41 thf(fact_120_inf__sup__aci_I1_J,axiom,
% 0.15/0.41 ( inf_in923488851lle_hf
% 0.15/0.41 = ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(1)
% 0.15/0.41 thf(fact_121_inf__sup__aci_I1_J,axiom,
% 0.15/0.41 ( inf_in307783154e_hf_o
% 0.15/0.41 = ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(1)
% 0.15/0.41 thf(fact_122_inf__sup__aci_I1_J,axiom,
% 0.15/0.41 ( inf_inf_int
% 0.15/0.41 = ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(1)
% 0.15/0.41 thf(fact_123_inf__sup__aci_I1_J,axiom,
% 0.15/0.41 ( inf_inf_nat
% 0.15/0.41 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(1)
% 0.15/0.41 thf(fact_124_inf__sup__aci_I1_J,axiom,
% 0.15/0.41 ( inf_in956532509lle_hf
% 0.15/0.41 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_sup_aci(1)
% 0.15/0.41 thf(fact_125_inf__fun__def,axiom,
% 0.15/0.41 ( inf_in307783154e_hf_o
% 0.15/0.41 = ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % inf_fun_def
% 0.15/0.41 thf(fact_126_boolean__algebra__cancel_Oinf1,axiom,
% 0.15/0.41 ! [A2: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( A2
% 0.15/0.41 = ( inf_in923488851lle_hf @ K @ A ) )
% 0.15/0.41 => ( ( inf_in923488851lle_hf @ A2 @ B )
% 0.15/0.41 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf1
% 0.15/0.41 thf(fact_127_boolean__algebra__cancel_Oinf1,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( A2
% 0.15/0.41 = ( inf_in307783154e_hf_o @ K @ A ) )
% 0.15/0.41 => ( ( inf_in307783154e_hf_o @ A2 @ B )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf1
% 0.15/0.41 thf(fact_128_boolean__algebra__cancel_Oinf1,axiom,
% 0.15/0.41 ! [A2: int,K: int,A: int,B: int] :
% 0.15/0.41 ( ( A2
% 0.15/0.41 = ( inf_inf_int @ K @ A ) )
% 0.15/0.41 => ( ( inf_inf_int @ A2 @ B )
% 0.15/0.41 = ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf1
% 0.15/0.41 thf(fact_129_boolean__algebra__cancel_Oinf1,axiom,
% 0.15/0.41 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.15/0.41 ( ( A2
% 0.15/0.41 = ( inf_inf_nat @ K @ A ) )
% 0.15/0.41 => ( ( inf_inf_nat @ A2 @ B )
% 0.15/0.41 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf1
% 0.15/0.41 thf(fact_130_boolean__algebra__cancel_Oinf1,axiom,
% 0.15/0.41 ! [A2: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.41 ( ( A2
% 0.15/0.41 = ( inf_in956532509lle_hf @ K @ A ) )
% 0.15/0.41 => ( ( inf_in956532509lle_hf @ A2 @ B )
% 0.15/0.41 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf1
% 0.15/0.41 thf(fact_131_boolean__algebra__cancel_Oinf2,axiom,
% 0.15/0.41 ! [B3: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.15/0.41 ( ( B3
% 0.15/0.41 = ( inf_in923488851lle_hf @ K @ B ) )
% 0.15/0.41 => ( ( inf_in923488851lle_hf @ A @ B3 )
% 0.15/0.41 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf2
% 0.15/0.41 thf(fact_132_boolean__algebra__cancel_Oinf2,axiom,
% 0.15/0.41 ! [B3: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o] :
% 0.15/0.41 ( ( B3
% 0.15/0.41 = ( inf_in307783154e_hf_o @ K @ B ) )
% 0.15/0.41 => ( ( inf_in307783154e_hf_o @ A @ B3 )
% 0.15/0.41 = ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf2
% 0.15/0.41 thf(fact_133_boolean__algebra__cancel_Oinf2,axiom,
% 0.15/0.41 ! [B3: int,K: int,B: int,A: int] :
% 0.15/0.41 ( ( B3
% 0.15/0.41 = ( inf_inf_int @ K @ B ) )
% 0.15/0.41 => ( ( inf_inf_int @ A @ B3 )
% 0.15/0.41 = ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf2
% 0.15/0.41 thf(fact_134_boolean__algebra__cancel_Oinf2,axiom,
% 0.15/0.41 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.15/0.41 ( ( B3
% 0.15/0.41 = ( inf_inf_nat @ K @ B ) )
% 0.15/0.41 => ( ( inf_inf_nat @ A @ B3 )
% 0.15/0.41 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.15/0.41
% 0.15/0.41 % boolean_algebra_cancel.inf2
% 0.15/0.41 thf(fact_135_boolean__algebra__cancel_Oinf2,axiom,
% 0.15/0.42 ! [B3: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( B3
% 0.15/0.42 = ( inf_in956532509lle_hf @ K @ B ) )
% 0.15/0.42 => ( ( inf_in956532509lle_hf @ A @ B3 )
% 0.15/0.42 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % boolean_algebra_cancel.inf2
% 0.15/0.42 thf(fact_136_inf_Oassoc,axiom,
% 0.15/0.42 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ C )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.assoc
% 0.15/0.42 thf(fact_137_inf_Oassoc,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ C )
% 0.15/0.42 = ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.assoc
% 0.15/0.42 thf(fact_138_inf_Oassoc,axiom,
% 0.15/0.42 ! [A: int,B: int,C: int] :
% 0.15/0.42 ( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ C )
% 0.15/0.42 = ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.assoc
% 0.15/0.42 thf(fact_139_inf_Oassoc,axiom,
% 0.15/0.42 ! [A: nat,B: nat,C: nat] :
% 0.15/0.42 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
% 0.15/0.42 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.assoc
% 0.15/0.42 thf(fact_140_inf_Oassoc,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C )
% 0.15/0.42 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.assoc
% 0.15/0.42 thf(fact_141_inf__assoc,axiom,
% 0.15/0.42 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.15/0.42 = ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_assoc
% 0.15/0.42 thf(fact_142_inf__assoc,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
% 0.15/0.42 = ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_assoc
% 0.15/0.42 thf(fact_143_inf__assoc,axiom,
% 0.15/0.42 ! [X2: int,Y2: int,Z4: int] :
% 0.15/0.42 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
% 0.15/0.42 = ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_assoc
% 0.15/0.42 thf(fact_144_inf__assoc,axiom,
% 0.15/0.42 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.15/0.42 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
% 0.15/0.42 = ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_assoc
% 0.15/0.42 thf(fact_145_inf__assoc,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.15/0.42 = ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_assoc
% 0.15/0.42 thf(fact_146_inf_Ocommute,axiom,
% 0.15/0.42 ( inf_in923488851lle_hf
% 0.15/0.42 = ( ^ [A3: set_HF_Mirabelle_hf,B2: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B2 @ A3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.commute
% 0.15/0.42 thf(fact_147_inf_Ocommute,axiom,
% 0.15/0.42 ( inf_in307783154e_hf_o
% 0.15/0.42 = ( ^ [A3: hF_Mirabelle_hf > $o,B2: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ B2 @ A3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.commute
% 0.15/0.42 thf(fact_148_inf_Ocommute,axiom,
% 0.15/0.42 ( inf_inf_int
% 0.15/0.42 = ( ^ [A3: int,B2: int] : ( inf_inf_int @ B2 @ A3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.commute
% 0.15/0.42 thf(fact_149_inf_Ocommute,axiom,
% 0.15/0.42 ( inf_inf_nat
% 0.15/0.42 = ( ^ [A3: nat,B2: nat] : ( inf_inf_nat @ B2 @ A3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.commute
% 0.15/0.42 thf(fact_150_inf_Ocommute,axiom,
% 0.15/0.42 ( inf_in956532509lle_hf
% 0.15/0.42 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B2 @ A3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.commute
% 0.15/0.42 thf(fact_151_inf__commute,axiom,
% 0.15/0.42 ( inf_in923488851lle_hf
% 0.15/0.42 = ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_commute
% 0.15/0.42 thf(fact_152_inf__commute,axiom,
% 0.15/0.42 ( inf_in307783154e_hf_o
% 0.15/0.42 = ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_commute
% 0.15/0.42 thf(fact_153_inf__commute,axiom,
% 0.15/0.42 ( inf_inf_int
% 0.15/0.42 = ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_commute
% 0.15/0.42 thf(fact_154_inf__commute,axiom,
% 0.15/0.42 ( inf_inf_nat
% 0.15/0.42 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_commute
% 0.15/0.42 thf(fact_155_inf__commute,axiom,
% 0.15/0.42 ( inf_in956532509lle_hf
% 0.15/0.42 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_commute
% 0.15/0.42 thf(fact_156_inf_Oleft__commute,axiom,
% 0.15/0.42 ! [B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A @ C ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.left_commute
% 0.15/0.42 thf(fact_157_inf_Oleft__commute,axiom,
% 0.15/0.42 ! [B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ B @ ( inf_in307783154e_hf_o @ A @ C ) )
% 0.15/0.42 = ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.left_commute
% 0.15/0.42 thf(fact_158_inf_Oleft__commute,axiom,
% 0.15/0.42 ! [B: int,A: int,C: int] :
% 0.15/0.42 ( ( inf_inf_int @ B @ ( inf_inf_int @ A @ C ) )
% 0.15/0.42 = ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.left_commute
% 0.15/0.42 thf(fact_159_inf_Oleft__commute,axiom,
% 0.15/0.42 ! [B: nat,A: nat,C: nat] :
% 0.15/0.42 ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
% 0.15/0.42 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.left_commute
% 0.15/0.42 thf(fact_160_inf_Oleft__commute,axiom,
% 0.15/0.42 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A @ C ) )
% 0.15/0.42 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.left_commute
% 0.15/0.42 thf(fact_161_inf__left__commute,axiom,
% 0.15/0.42 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_left_commute
% 0.15/0.42 thf(fact_162_inf__left__commute,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
% 0.15/0.42 = ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_left_commute
% 0.15/0.42 thf(fact_163_inf__left__commute,axiom,
% 0.15/0.42 ! [X2: int,Y2: int,Z4: int] :
% 0.15/0.42 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
% 0.15/0.42 = ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_left_commute
% 0.15/0.42 thf(fact_164_inf__left__commute,axiom,
% 0.15/0.42 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.15/0.42 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
% 0.15/0.42 = ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_left_commute
% 0.15/0.42 thf(fact_165_inf__left__commute,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
% 0.15/0.42 = ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_left_commute
% 0.15/0.42 thf(fact_166_eq__iff__diff__eq__0,axiom,
% 0.15/0.42 ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
% 0.15/0.42 = ( ^ [A3: int,B2: int] :
% 0.15/0.42 ( ( minus_minus_int @ A3 @ B2 )
% 0.15/0.42 = zero_zero_int ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % eq_iff_diff_eq_0
% 0.15/0.42 thf(fact_167_HInter__hinsert,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( A2 != zero_z189798548lle_hf )
% 0.15/0.42 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A2 ) )
% 0.15/0.42 = ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % HInter_hinsert
% 0.15/0.42 thf(fact_168_HCollect__def,axiom,
% 0.15/0.42 ( hF_Mir818139703ollect
% 0.15/0.42 = ( ^ [P2: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.15/0.42 ! [U2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.15/0.42 = ( ( P2 @ U2 )
% 0.15/0.42 & ( hF_Mirabelle_hmem @ U2 @ A4 ) ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % HCollect_def
% 0.15/0.42 thf(fact_169_HF__hfset,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 0.15/0.42 = A ) ).
% 0.15/0.42
% 0.15/0.42 % HF_hfset
% 0.15/0.42 thf(fact_170_zero__notin__hpair,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % zero_notin_hpair
% 0.15/0.42 thf(fact_171_Abs__hf__0,axiom,
% 0.15/0.42 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.15/0.42 = zero_z189798548lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % Abs_hf_0
% 0.15/0.42 thf(fact_172_hfset__0,axiom,
% 0.15/0.42 ( ( hF_Mirabelle_hfset @ zero_z189798548lle_hf )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % hfset_0
% 0.15/0.42 thf(fact_173_hpair__iff,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.15/0.42 = ( hF_Mirabelle_hpair @ A6 @ B5 ) )
% 0.15/0.42 = ( ( A = A6 )
% 0.15/0.42 & ( B = B5 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_iff
% 0.15/0.42 thf(fact_174_inf__bot__right,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ X2 @ bot_bo1263054448e_hf_o )
% 0.15/0.42 = bot_bo1263054448e_hf_o ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_right
% 0.15/0.42 thf(fact_175_inf__bot__right,axiom,
% 0.15/0.42 ! [X2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ X2 @ bot_bo53200981lle_hf )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_right
% 0.15/0.42 thf(fact_176_inf__bot__right,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ X2 @ bot_bo1001194783lle_hf )
% 0.15/0.42 = bot_bo1001194783lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_right
% 0.15/0.42 thf(fact_177_inf__bot__left,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ bot_bo1263054448e_hf_o @ X2 )
% 0.15/0.42 = bot_bo1263054448e_hf_o ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_left
% 0.15/0.42 thf(fact_178_inf__bot__left,axiom,
% 0.15/0.42 ! [X2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ X2 )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_left
% 0.15/0.42 thf(fact_179_inf__bot__left,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in956532509lle_hf @ bot_bo1001194783lle_hf @ X2 )
% 0.15/0.42 = bot_bo1001194783lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % inf_bot_left
% 0.15/0.42 thf(fact_180_hmem__hinsert,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.15/0.42 = ( ( A = B )
% 0.15/0.42 | ( hF_Mirabelle_hmem @ A @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hmem_hinsert
% 0.15/0.42 thf(fact_181_singleton__eq__iff,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.15/0.42 = ( A = B ) ) ).
% 0.15/0.42
% 0.15/0.42 % singleton_eq_iff
% 0.15/0.42 thf(fact_182_RepFun__hinsert,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A @ B ) @ F )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ ( F @ A ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % RepFun_hinsert
% 0.15/0.42 thf(fact_183_hpair__def,axiom,
% 0.15/0.42 ( hF_Mirabelle_hpair
% 0.15/0.42 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_def
% 0.15/0.42 thf(fact_184_hpair__def_H,axiom,
% 0.15/0.42 ( hF_Mirabelle_hpair
% 0.15/0.42 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_def'
% 0.15/0.42 thf(fact_185_Zero__hf__def,axiom,
% 0.15/0.42 ( zero_z189798548lle_hf
% 0.15/0.42 = ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% 0.15/0.42
% 0.15/0.42 % Zero_hf_def
% 0.15/0.42 thf(fact_186_hpair__neq__snd,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.15/0.42 != B ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_neq_snd
% 0.15/0.42 thf(fact_187_hpair__neq__fst,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.15/0.42 != A ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_neq_fst
% 0.15/0.42 thf(fact_188_hpair__inject,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.15/0.42 = ( hF_Mirabelle_hpair @ A6 @ B5 ) )
% 0.15/0.42 => ~ ( ( A = A6 )
% 0.15/0.42 => ( B != B5 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_inject
% 0.15/0.42 thf(fact_189_hinsert__commute,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hinsert @ X2 @ ( hF_Mirabelle_hinsert @ Y2 @ Z4 ) )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ Y2 @ ( hF_Mirabelle_hinsert @ X2 @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hinsert_commute
% 0.15/0.42 thf(fact_190_hinsert__iff,axiom,
% 0.15/0.42 ! [Z4: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( Z4
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ Y2 @ X2 ) )
% 0.15/0.42 = ( ! [U2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ U2 @ Z4 )
% 0.15/0.42 = ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.15/0.42 | ( U2 = Y2 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hinsert_iff
% 0.15/0.42 thf(fact_191_hinsert__nonempty,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hinsert @ A @ A2 )
% 0.15/0.42 != zero_z189798548lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % hinsert_nonempty
% 0.15/0.42 thf(fact_192_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hinsert @ A @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 0.15/0.42 = ( ( ( A = C )
% 0.15/0.42 & ( B = D ) )
% 0.15/0.42 | ( ( A = D )
% 0.15/0.42 & ( B = C ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % HF_Mirabelle_glliljednj.doubleton_eq_iff
% 0.15/0.42 thf(fact_193_hf__induct__ax,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ zero_z189798548lle_hf )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ! [Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ Y4 )
% 0.15/0.42 => ( P @ ( hF_Mirabelle_hinsert @ Y4 @ X ) ) ) )
% 0.15/0.42 => ( P @ X2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hf_induct_ax
% 0.15/0.42 thf(fact_194_hpair__nonzero,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hpair @ X2 @ Y2 )
% 0.15/0.42 != zero_z189798548lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % hpair_nonzero
% 0.15/0.42 thf(fact_195_hf__cases,axiom,
% 0.15/0.42 ! [Y2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( Y2 != zero_z189798548lle_hf )
% 0.15/0.42 => ~ ! [A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( Y2
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ A7 @ B6 ) )
% 0.15/0.42 => ( hF_Mirabelle_hmem @ A7 @ B6 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hf_cases
% 0.15/0.42 thf(fact_196_hf__induct,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ zero_z189798548lle_hf )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( ( P @ Y4 )
% 0.15/0.42 => ( ~ ( hF_Mirabelle_hmem @ X @ Y4 )
% 0.15/0.42 => ( P @ ( hF_Mirabelle_hinsert @ X @ Y4 ) ) ) ) )
% 0.15/0.42 => ( P @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hf_induct
% 0.15/0.42 thf(fact_197_hinter__hinsert__right,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) )
% 0.15/0.42 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
% 0.15/0.42 = ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hinter_hinsert_right
% 0.15/0.42 thf(fact_198_hinter__hinsert__left,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) )
% 0.15/0.42 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.15/0.42 = ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hinter_hinsert_left
% 0.15/0.42 thf(fact_199_hinsert__hdiff__if,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.15/0.42 = ( minus_1232880740lle_hf @ A2 @ B3 ) ) )
% 0.15/0.42 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.15/0.42 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.15/0.42 = ( hF_Mirabelle_hinsert @ X2 @ ( minus_1232880740lle_hf @ A2 @ B3 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hinsert_hdiff_if
% 0.15/0.42 thf(fact_200_hdiff__insert,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1232880740lle_hf @ A2 @ ( hF_Mirabelle_hinsert @ A @ B3 ) )
% 0.15/0.42 = ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A2 @ B3 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hdiff_insert
% 0.15/0.42 thf(fact_201_Diff__disjoint,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ A2 ) )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_disjoint
% 0.15/0.42 thf(fact_202_Diff__cancel,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ A2 @ A2 )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_cancel
% 0.15/0.42 thf(fact_203_empty__Diff,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A2 )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % empty_Diff
% 0.15/0.42 thf(fact_204_Diff__empty,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.15/0.42 = A2 ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_empty
% 0.15/0.42 thf(fact_205_empty__iff,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % empty_iff
% 0.15/0.42 thf(fact_206_all__not__in__conv,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.15/0.42 = ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.15/0.42
% 0.15/0.42 % all_not_in_conv
% 0.15/0.42 thf(fact_207_Collect__empty__eq,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( ( collec2046588256lle_hf @ P )
% 0.15/0.42 = bot_bo53200981lle_hf )
% 0.15/0.42 = ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( P @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Collect_empty_eq
% 0.15/0.42 thf(fact_208_empty__Collect__eq,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( bot_bo53200981lle_hf
% 0.15/0.42 = ( collec2046588256lle_hf @ P ) )
% 0.15/0.42 = ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( P @ X3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % empty_Collect_eq
% 0.15/0.42 thf(fact_209_Int__iff,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.15/0.42 = ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 & ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_iff
% 0.15/0.42 thf(fact_210_IntI,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 => ( ( member1367349282lle_hf @ C @ B3 )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % IntI
% 0.15/0.42 thf(fact_211_DiffI,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 => ( ~ ( member1367349282lle_hf @ C @ B3 )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % DiffI
% 0.15/0.42 thf(fact_212_Diff__iff,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 = ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 & ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_iff
% 0.15/0.42 thf(fact_213_Diff__idemp,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ B3 )
% 0.15/0.42 = ( minus_1450406810lle_hf @ A2 @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_idemp
% 0.15/0.42 thf(fact_214_Int__left__commute,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ B3 @ ( inf_in923488851lle_hf @ A2 @ C2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_left_commute
% 0.15/0.42 thf(fact_215_Int__left__absorb,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_left_absorb
% 0.15/0.42 thf(fact_216_Collect__conj__eq,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( collec2046588256lle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X3 )
% 0.15/0.42 & ( Q @ X3 ) ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ ( collec2046588256lle_hf @ P ) @ ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Collect_conj_eq
% 0.15/0.42 thf(fact_217_inf__set__def,axiom,
% 0.15/0.42 ( inf_in923488851lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( collec2046588256lle_hf
% 0.15/0.42 @ ( inf_in307783154e_hf_o
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_set_def
% 0.15/0.42 thf(fact_218_Int__commute,axiom,
% 0.15/0.42 ( inf_in923488851lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B4 @ A4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_commute
% 0.15/0.42 thf(fact_219_Int__Collect,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X2 @ ( inf_in923488851lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) ) )
% 0.15/0.42 = ( ( member1367349282lle_hf @ X2 @ A2 )
% 0.15/0.42 & ( P @ X2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_Collect
% 0.15/0.42 thf(fact_220_Int__absorb,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ A2 @ A2 )
% 0.15/0.42 = A2 ) ).
% 0.15/0.42
% 0.15/0.42 % Int_absorb
% 0.15/0.42 thf(fact_221_Int__assoc,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_assoc
% 0.15/0.42 thf(fact_222_Int__def,axiom,
% 0.15/0.42 ( inf_in923488851lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( collec2046588256lle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.15/0.42 & ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_def
% 0.15/0.42 thf(fact_223_IntD2,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % IntD2
% 0.15/0.42 thf(fact_224_IntD1,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ A2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % IntD1
% 0.15/0.42 thf(fact_225_IntE,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ~ ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 => ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % IntE
% 0.15/0.42 thf(fact_226_bot__set__def,axiom,
% 0.15/0.42 ( bot_bo53200981lle_hf
% 0.15/0.42 = ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot_set_def
% 0.15/0.42 thf(fact_227_bot__hf__def,axiom,
% 0.15/0.42 bot_bo1001194783lle_hf = zero_z189798548lle_hf ).
% 0.15/0.42
% 0.15/0.42 % bot_hf_def
% 0.15/0.42 thf(fact_228_emptyE,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( member1367349282lle_hf @ A @ bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % emptyE
% 0.15/0.42 thf(fact_229_equals0D,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( A2 = bot_bo53200981lle_hf )
% 0.15/0.42 => ~ ( member1367349282lle_hf @ A @ A2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % equals0D
% 0.15/0.42 thf(fact_230_equals0I,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ! [Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( member1367349282lle_hf @ Y4 @ A2 )
% 0.15/0.42 => ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.15/0.42
% 0.15/0.42 % equals0I
% 0.15/0.42 thf(fact_231_ex__in__conv,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.15/0.42 = ( A2 != bot_bo53200981lle_hf ) ) ).
% 0.15/0.42
% 0.15/0.42 % ex_in_conv
% 0.15/0.42 thf(fact_232_Int__emptyI,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X @ A2 )
% 0.15/0.42 => ~ ( member1367349282lle_hf @ X @ B3 ) )
% 0.15/0.42 => ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.15/0.42 = bot_bo53200981lle_hf ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_emptyI
% 0.15/0.42 thf(fact_233_disjoint__iff,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.15/0.42 = bot_bo53200981lle_hf )
% 0.15/0.42 = ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X3 @ A2 )
% 0.15/0.42 => ~ ( member1367349282lle_hf @ X3 @ B3 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % disjoint_iff
% 0.15/0.42 thf(fact_234_Int__empty__left,axiom,
% 0.15/0.42 ! [B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ B3 )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % Int_empty_left
% 0.15/0.42 thf(fact_235_Int__empty__right,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.15/0.42 = bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % Int_empty_right
% 0.15/0.42 thf(fact_236_disjoint__iff__not__equal,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.15/0.42 = bot_bo53200981lle_hf )
% 0.15/0.42 = ( ! [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X3 @ A2 )
% 0.15/0.42 => ! [Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ Y3 @ B3 )
% 0.15/0.42 => ( X3 != Y3 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % disjoint_iff_not_equal
% 0.15/0.42 thf(fact_237_DiffE,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ~ ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % DiffE
% 0.15/0.42 thf(fact_238_DiffD1,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ A2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % DiffD1
% 0.15/0.42 thf(fact_239_DiffD2,axiom,
% 0.15/0.42 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 => ~ ( member1367349282lle_hf @ C @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % DiffD2
% 0.15/0.42 thf(fact_240_Int__Diff,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ C2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Int_Diff
% 0.15/0.42 thf(fact_241_Diff__Int2,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
% 0.15/0.42 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_Int2
% 0.15/0.42 thf(fact_242_Diff__Diff__Int,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( minus_1450406810lle_hf @ A2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 = ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_Diff_Int
% 0.15/0.42 thf(fact_243_Diff__Int__distrib,axiom,
% 0.15/0.42 ! [C2: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ C2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.15/0.42 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ C2 @ A2 ) @ ( inf_in923488851lle_hf @ C2 @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_Int_distrib
% 0.15/0.42 thf(fact_244_Diff__Int__distrib2,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in923488851lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ C2 )
% 0.15/0.42 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_Int_distrib2
% 0.15/0.42 thf(fact_245_empty__def,axiom,
% 0.15/0.42 ( bot_bo53200981lle_hf
% 0.15/0.42 = ( collec2046588256lle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : $false ) ) ).
% 0.15/0.42
% 0.15/0.42 % empty_def
% 0.15/0.42 thf(fact_246_set__diff__eq,axiom,
% 0.15/0.42 ( minus_1450406810lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( collec2046588256lle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.15/0.42 & ~ ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % set_diff_eq
% 0.15/0.42 thf(fact_247_minus__set__def,axiom,
% 0.15/0.42 ( minus_1450406810lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( collec2046588256lle_hf
% 0.15/0.42 @ ( minus_1668263787e_hf_o
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % minus_set_def
% 0.15/0.42 thf(fact_248_Diff__triv,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.15/0.42 = bot_bo53200981lle_hf )
% 0.15/0.42 => ( ( minus_1450406810lle_hf @ A2 @ B3 )
% 0.15/0.42 = A2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % Diff_triv
% 0.15/0.42 thf(fact_249_hfst__conv,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.15/0.42 = A ) ).
% 0.15/0.42
% 0.15/0.42 % hfst_conv
% 0.15/0.42 thf(fact_250_hsnd__conv,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.15/0.42 = B ) ).
% 0.15/0.42
% 0.15/0.42 % hsnd_conv
% 0.15/0.42 thf(fact_251_the__equality,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ A )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( X = A ) )
% 0.15/0.42 => ( ( the_HF_Mirabelle_hf @ P )
% 0.15/0.42 = A ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % the_equality
% 0.15/0.42 thf(fact_252_the__eq__trivial,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( X3 = A ) )
% 0.15/0.42 = A ) ).
% 0.15/0.42
% 0.15/0.42 % the_eq_trivial
% 0.15/0.42 thf(fact_253_inf1I,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( A2 @ X2 )
% 0.15/0.42 => ( ( B3 @ X2 )
% 0.15/0.42 => ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf1I
% 0.15/0.42 thf(fact_254_the__sym__eq__trivial,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z )
% 0.15/0.42 @ X2 ) )
% 0.15/0.42 = X2 ) ).
% 0.15/0.42
% 0.15/0.42 % the_sym_eq_trivial
% 0.15/0.42 thf(fact_255_inf1D2,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.15/0.42 => ( B3 @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf1D2
% 0.15/0.42 thf(fact_256_inf1D1,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.15/0.42 => ( A2 @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf1D1
% 0.15/0.42 thf(fact_257_inf1E,axiom,
% 0.15/0.42 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.15/0.42 => ~ ( ( A2 @ X2 )
% 0.15/0.42 => ~ ( B3 @ X2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf1E
% 0.15/0.42 thf(fact_258_the1__equality,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ? [X4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X4 )
% 0.15/0.42 & ! [Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ Y4 )
% 0.15/0.42 => ( Y4 = X4 ) ) )
% 0.15/0.42 => ( ( P @ A )
% 0.15/0.42 => ( ( the_HF_Mirabelle_hf @ P )
% 0.15/0.42 = A ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % the1_equality
% 0.15/0.42 thf(fact_259_the1I2,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ? [X4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X4 )
% 0.15/0.42 & ! [Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ Y4 )
% 0.15/0.42 => ( Y4 = X4 ) ) )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( Q @ X ) )
% 0.15/0.42 => ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % the1I2
% 0.15/0.42 thf(fact_260_If__def,axiom,
% 0.15/0.42 ( if_HF_Mirabelle_hf
% 0.15/0.42 = ( ^ [P2: $o,X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P2
% 0.15/0.42 => ( Z3 = X3 ) )
% 0.15/0.42 & ( ~ P2
% 0.15/0.42 => ( Z3 = Y3 ) ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % If_def
% 0.15/0.42 thf(fact_261_theI2,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( P @ A )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( X = A ) )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( Q @ X ) )
% 0.15/0.42 => ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % theI2
% 0.15/0.42 thf(fact_262_theI_H,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ? [X4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X4 )
% 0.15/0.42 & ! [Y4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ Y4 )
% 0.15/0.42 => ( Y4 = X4 ) ) )
% 0.15/0.42 => ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % theI'
% 0.15/0.42 thf(fact_263_theI,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ A )
% 0.15/0.42 => ( ! [X: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P @ X )
% 0.15/0.42 => ( X = A ) )
% 0.15/0.42 => ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % theI
% 0.15/0.42 thf(fact_264_bot__apply,axiom,
% 0.15/0.42 ( bot_bo1263054448e_hf_o
% 0.15/0.42 = ( ^ [X3: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot_apply
% 0.15/0.42 thf(fact_265_inf__Int__eq,axiom,
% 0.15/0.42 ! [R: set_HF_Mirabelle_hf,S: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( inf_in307783154e_hf_o
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ R )
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ S ) )
% 0.15/0.42 = ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ ( inf_in923488851lle_hf @ R @ S ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf_Int_eq
% 0.15/0.42 thf(fact_266_diff__self__eq__0,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ( ( minus_minus_nat @ M @ M )
% 0.15/0.42 = zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % diff_self_eq_0
% 0.15/0.42 thf(fact_267_diff__0__eq__0,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.15/0.42 = zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % diff_0_eq_0
% 0.15/0.42 thf(fact_268_diff__commute,axiom,
% 0.15/0.42 ! [I: nat,J: nat,K: nat] :
% 0.15/0.42 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.15/0.42 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_commute
% 0.15/0.42 thf(fact_269_bot__nat__def,axiom,
% 0.15/0.42 bot_bot_nat = zero_zero_nat ).
% 0.15/0.42
% 0.15/0.42 % bot_nat_def
% 0.15/0.42 thf(fact_270_diffs0__imp__equal,axiom,
% 0.15/0.42 ! [M: nat,N: nat] :
% 0.15/0.42 ( ( ( minus_minus_nat @ M @ N )
% 0.15/0.42 = zero_zero_nat )
% 0.15/0.42 => ( ( ( minus_minus_nat @ N @ M )
% 0.15/0.42 = zero_zero_nat )
% 0.15/0.42 => ( M = N ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diffs0_imp_equal
% 0.15/0.42 thf(fact_271_minus__nat_Odiff__0,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.15/0.42 = M ) ).
% 0.15/0.42
% 0.15/0.42 % minus_nat.diff_0
% 0.15/0.42 thf(fact_272_bot__empty__eq,axiom,
% 0.15/0.42 ( bot_bo1263054448e_hf_o
% 0.15/0.42 = ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot_empty_eq
% 0.15/0.42 thf(fact_273_Collect__empty__eq__bot,axiom,
% 0.15/0.42 ! [P: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( ( ( collec2046588256lle_hf @ P )
% 0.15/0.42 = bot_bo53200981lle_hf )
% 0.15/0.42 = ( P = bot_bo1263054448e_hf_o ) ) ).
% 0.15/0.42
% 0.15/0.42 % Collect_empty_eq_bot
% 0.15/0.42 thf(fact_274_ord_OLeast__def,axiom,
% 0.15/0.42 ( least_917110455lle_hf
% 0.15/0.42 = ( ^ [Less_eq: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,P2: hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P2 @ X3 )
% 0.15/0.42 & ! [Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( P2 @ Y3 )
% 0.15/0.42 => ( Less_eq @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % ord.Least_def
% 0.15/0.42 thf(fact_275_PrimReplace__def,axiom,
% 0.15/0.42 ( hF_Mir1248913145eplace
% 0.15/0.42 = ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.15/0.42 ! [V5: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ V5 @ Z3 )
% 0.15/0.42 = ( ? [U2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.15/0.42 & ( R2 @ U2 @ V5 ) ) ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % PrimReplace_def
% 0.15/0.42 thf(fact_276_hcard__0,axiom,
% 0.15/0.42 ( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
% 0.15/0.42 = zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % hcard_0
% 0.15/0.42 thf(fact_277_hfst__def,axiom,
% 0.15/0.42 ( hF_Mirabelle_hfst
% 0.15/0.42 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ? [Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ( P4
% 0.15/0.42 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hfst_def
% 0.15/0.42 thf(fact_278_hsnd__def,axiom,
% 0.15/0.42 ( hF_Mirabelle_hsnd
% 0.15/0.42 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ? [X3: hF_Mirabelle_hf] :
% 0.15/0.42 ( P4
% 0.15/0.42 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hsnd_def
% 0.15/0.42 thf(fact_279_HUnion__def,axiom,
% 0.15/0.42 ( hF_Mirabelle_HUnion
% 0.15/0.42 = ( ^ [A4: hF_Mirabelle_hf] :
% 0.15/0.42 ( the_HF_Mirabelle_hf
% 0.15/0.42 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.15/0.42 ! [U2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.15/0.42 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.15/0.42 & ( hF_Mirabelle_hmem @ U2 @ Y3 ) ) ) ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % HUnion_def
% 0.15/0.42 thf(fact_280_hcard__hdiff1__less,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( hF_Mirabelle_hmem @ X2 @ Z4 )
% 0.15/0.42 => ( ord_less_nat @ ( hF_Mirabelle_hcard @ ( minus_1232880740lle_hf @ Z4 @ ( hF_Mirabelle_hinsert @ X2 @ zero_z189798548lle_hf ) ) ) @ ( hF_Mirabelle_hcard @ Z4 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % hcard_hdiff1_less
% 0.15/0.42 thf(fact_281_Set_Ois__empty__def,axiom,
% 0.15/0.42 ( is_emp566801209lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf] : ( A4 = bot_bo53200981lle_hf ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % Set.is_empty_def
% 0.15/0.42 thf(fact_282_not__gr__zero,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.15/0.42 = ( N = zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % not_gr_zero
% 0.15/0.42 thf(fact_283_bot__nat__0_Onot__eq__extremum,axiom,
% 0.15/0.42 ! [A: nat] :
% 0.15/0.42 ( ( A != zero_zero_nat )
% 0.15/0.42 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot_nat_0.not_eq_extremum
% 0.15/0.42 thf(fact_284_less__nat__zero__code,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % less_nat_zero_code
% 0.15/0.42 thf(fact_285_neq0__conv,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( N != zero_zero_nat )
% 0.15/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % neq0_conv
% 0.15/0.42 thf(fact_286_diff__gt__0__iff__gt,axiom,
% 0.15/0.42 ! [A: int,B: int] :
% 0.15/0.42 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.15/0.42 = ( ord_less_int @ B @ A ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_gt_0_iff_gt
% 0.15/0.42 thf(fact_287_zero__less__diff,axiom,
% 0.15/0.42 ! [N: nat,M: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.15/0.42 = ( ord_less_nat @ M @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % zero_less_diff
% 0.15/0.42 thf(fact_288_less__imp__diff__less,axiom,
% 0.15/0.42 ! [J: nat,K: nat,N: nat] :
% 0.15/0.42 ( ( ord_less_nat @ J @ K )
% 0.15/0.42 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_imp_diff_less
% 0.15/0.42 thf(fact_289_diff__less__mono2,axiom,
% 0.15/0.42 ! [M: nat,N: nat,L: nat] :
% 0.15/0.42 ( ( ord_less_nat @ M @ N )
% 0.15/0.42 => ( ( ord_less_nat @ M @ L )
% 0.15/0.42 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_less_mono2
% 0.15/0.42 thf(fact_290_diff__less,axiom,
% 0.15/0.42 ! [N: nat,M: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.15/0.42 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.15/0.42 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_less
% 0.15/0.42 thf(fact_291_diff__strict__right__mono,axiom,
% 0.15/0.42 ! [A: int,B: int,C: int] :
% 0.15/0.42 ( ( ord_less_int @ A @ B )
% 0.15/0.42 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_strict_right_mono
% 0.15/0.42 thf(fact_292_diff__strict__left__mono,axiom,
% 0.15/0.42 ! [B: int,A: int,C: int] :
% 0.15/0.42 ( ( ord_less_int @ B @ A )
% 0.15/0.42 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_strict_left_mono
% 0.15/0.42 thf(fact_293_diff__eq__diff__less,axiom,
% 0.15/0.42 ! [A: int,B: int,C: int,D: int] :
% 0.15/0.42 ( ( ( minus_minus_int @ A @ B )
% 0.15/0.42 = ( minus_minus_int @ C @ D ) )
% 0.15/0.42 => ( ( ord_less_int @ A @ B )
% 0.15/0.42 = ( ord_less_int @ C @ D ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_eq_diff_less
% 0.15/0.42 thf(fact_294_diff__strict__mono,axiom,
% 0.15/0.42 ! [A: int,B: int,D: int,C: int] :
% 0.15/0.42 ( ( ord_less_int @ A @ B )
% 0.15/0.42 => ( ( ord_less_int @ D @ C )
% 0.15/0.42 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % diff_strict_mono
% 0.15/0.42 thf(fact_295_bot__nat__0_Oextremum__strict,axiom,
% 0.15/0.42 ! [A: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % bot_nat_0.extremum_strict
% 0.15/0.42 thf(fact_296_infinite__descent0,axiom,
% 0.15/0.42 ! [P: nat > $o,N: nat] :
% 0.15/0.42 ( ( P @ zero_zero_nat )
% 0.15/0.42 => ( ! [N2: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.15/0.42 => ( ~ ( P @ N2 )
% 0.15/0.42 => ? [M2: nat] :
% 0.15/0.42 ( ( ord_less_nat @ M2 @ N2 )
% 0.15/0.42 & ~ ( P @ M2 ) ) ) )
% 0.15/0.42 => ( P @ N ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % infinite_descent0
% 0.15/0.42 thf(fact_297_gr__implies__not0,axiom,
% 0.15/0.42 ! [M: nat,N: nat] :
% 0.15/0.42 ( ( ord_less_nat @ M @ N )
% 0.15/0.42 => ( N != zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % gr_implies_not0
% 0.15/0.42 thf(fact_298_less__zeroE,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % less_zeroE
% 0.15/0.42 thf(fact_299_not__less0,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % not_less0
% 0.15/0.42 thf(fact_300_not__gr0,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.15/0.42 = ( N = zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % not_gr0
% 0.15/0.42 thf(fact_301_gr0I,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( N != zero_zero_nat )
% 0.15/0.42 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % gr0I
% 0.15/0.42 thf(fact_302_less__iff__diff__less__0,axiom,
% 0.15/0.42 ( ord_less_int
% 0.15/0.42 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_iff_diff_less_0
% 0.15/0.42 thf(fact_303_zero__less__iff__neq__zero,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.15/0.42 = ( N != zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % zero_less_iff_neq_zero
% 0.15/0.42 thf(fact_304_gr__implies__not__zero,axiom,
% 0.15/0.42 ! [M: nat,N: nat] :
% 0.15/0.42 ( ( ord_less_nat @ M @ N )
% 0.15/0.42 => ( N != zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % gr_implies_not_zero
% 0.15/0.42 thf(fact_305_not__less__zero,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % not_less_zero
% 0.15/0.42 thf(fact_306_gr__zeroI,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( N != zero_zero_nat )
% 0.15/0.42 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % gr_zeroI
% 0.15/0.42 thf(fact_307_less__infI1,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1310584031lle_hf @ A @ X2 )
% 0.15/0.42 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI1
% 0.15/0.42 thf(fact_308_less__infI1,axiom,
% 0.15/0.42 ! [A: nat,X2: nat,B: nat] :
% 0.15/0.42 ( ( ord_less_nat @ A @ X2 )
% 0.15/0.42 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI1
% 0.15/0.42 thf(fact_309_less__infI1,axiom,
% 0.15/0.42 ! [A: int,X2: int,B: int] :
% 0.15/0.42 ( ( ord_less_int @ A @ X2 )
% 0.15/0.42 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI1
% 0.15/0.42 thf(fact_310_less__infI2,axiom,
% 0.15/0.42 ! [B: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1310584031lle_hf @ B @ X2 )
% 0.15/0.42 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI2
% 0.15/0.42 thf(fact_311_less__infI2,axiom,
% 0.15/0.42 ! [B: nat,X2: nat,A: nat] :
% 0.15/0.42 ( ( ord_less_nat @ B @ X2 )
% 0.15/0.42 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI2
% 0.15/0.42 thf(fact_312_less__infI2,axiom,
% 0.15/0.42 ! [B: int,X2: int,A: int] :
% 0.15/0.42 ( ( ord_less_int @ B @ X2 )
% 0.15/0.42 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_infI2
% 0.15/0.42 thf(fact_313_inf_Ostrict__boundedE,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1310584031lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 0.15/0.42 => ~ ( ( ord_le1310584031lle_hf @ A @ B )
% 0.15/0.42 => ~ ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_boundedE
% 0.15/0.42 thf(fact_314_inf_Ostrict__boundedE,axiom,
% 0.15/0.42 ! [A: nat,B: nat,C: nat] :
% 0.15/0.42 ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 0.15/0.42 => ~ ( ( ord_less_nat @ A @ B )
% 0.15/0.42 => ~ ( ord_less_nat @ A @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_boundedE
% 0.15/0.42 thf(fact_315_inf_Ostrict__boundedE,axiom,
% 0.15/0.42 ! [A: int,B: int,C: int] :
% 0.15/0.42 ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
% 0.15/0.42 => ~ ( ( ord_less_int @ A @ B )
% 0.15/0.42 => ~ ( ord_less_int @ A @ C ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_boundedE
% 0.15/0.42 thf(fact_316_inf_Ostrict__order__iff,axiom,
% 0.15/0.42 ( ord_le1310584031lle_hf
% 0.15/0.42 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( A3
% 0.15/0.42 = ( inf_in956532509lle_hf @ A3 @ B2 ) )
% 0.15/0.42 & ( A3 != B2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_order_iff
% 0.15/0.42 thf(fact_317_inf_Ostrict__order__iff,axiom,
% 0.15/0.42 ( ord_less_nat
% 0.15/0.42 = ( ^ [A3: nat,B2: nat] :
% 0.15/0.42 ( ( A3
% 0.15/0.42 = ( inf_inf_nat @ A3 @ B2 ) )
% 0.15/0.42 & ( A3 != B2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_order_iff
% 0.15/0.42 thf(fact_318_inf_Ostrict__order__iff,axiom,
% 0.15/0.42 ( ord_less_int
% 0.15/0.42 = ( ^ [A3: int,B2: int] :
% 0.15/0.42 ( ( A3
% 0.15/0.42 = ( inf_inf_int @ A3 @ B2 ) )
% 0.15/0.42 & ( A3 != B2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_order_iff
% 0.15/0.42 thf(fact_319_inf_Ostrict__coboundedI1,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1310584031lle_hf @ A @ C )
% 0.15/0.42 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI1
% 0.15/0.42 thf(fact_320_inf_Ostrict__coboundedI1,axiom,
% 0.15/0.42 ! [A: nat,C: nat,B: nat] :
% 0.15/0.42 ( ( ord_less_nat @ A @ C )
% 0.15/0.42 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI1
% 0.15/0.42 thf(fact_321_inf_Ostrict__coboundedI1,axiom,
% 0.15/0.42 ! [A: int,C: int,B: int] :
% 0.15/0.42 ( ( ord_less_int @ A @ C )
% 0.15/0.42 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI1
% 0.15/0.42 thf(fact_322_inf_Ostrict__coboundedI2,axiom,
% 0.15/0.42 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1310584031lle_hf @ B @ C )
% 0.15/0.42 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI2
% 0.15/0.42 thf(fact_323_inf_Ostrict__coboundedI2,axiom,
% 0.15/0.42 ! [B: nat,C: nat,A: nat] :
% 0.15/0.42 ( ( ord_less_nat @ B @ C )
% 0.15/0.42 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI2
% 0.15/0.42 thf(fact_324_inf_Ostrict__coboundedI2,axiom,
% 0.15/0.42 ! [B: int,C: int,A: int] :
% 0.15/0.42 ( ( ord_less_int @ B @ C )
% 0.15/0.42 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% 0.15/0.42
% 0.15/0.42 % inf.strict_coboundedI2
% 0.15/0.42 thf(fact_325_bot_Oextremum__strict,axiom,
% 0.15/0.42 ! [A: set_HF_Mirabelle_hf] :
% 0.15/0.42 ~ ( ord_le1344122901lle_hf @ A @ bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % bot.extremum_strict
% 0.15/0.42 thf(fact_326_bot_Oextremum__strict,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf] :
% 0.15/0.42 ~ ( ord_le1310584031lle_hf @ A @ bot_bo1001194783lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % bot.extremum_strict
% 0.15/0.42 thf(fact_327_bot_Oextremum__strict,axiom,
% 0.15/0.42 ! [A: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.15/0.42
% 0.15/0.42 % bot.extremum_strict
% 0.15/0.42 thf(fact_328_bot_Onot__eq__extremum,axiom,
% 0.15/0.42 ! [A: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( A != bot_bo53200981lle_hf )
% 0.15/0.42 = ( ord_le1344122901lle_hf @ bot_bo53200981lle_hf @ A ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot.not_eq_extremum
% 0.15/0.42 thf(fact_329_bot_Onot__eq__extremum,axiom,
% 0.15/0.42 ! [A: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( A != bot_bo1001194783lle_hf )
% 0.15/0.42 = ( ord_le1310584031lle_hf @ bot_bo1001194783lle_hf @ A ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot.not_eq_extremum
% 0.15/0.42 thf(fact_330_bot_Onot__eq__extremum,axiom,
% 0.15/0.42 ! [A: nat] :
% 0.15/0.42 ( ( A != bot_bot_nat )
% 0.15/0.42 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.15/0.42
% 0.15/0.42 % bot.not_eq_extremum
% 0.15/0.42 thf(fact_331_less__numeral__extra_I3_J,axiom,
% 0.15/0.42 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % less_numeral_extra(3)
% 0.15/0.42 thf(fact_332_less__numeral__extra_I3_J,axiom,
% 0.15/0.42 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.15/0.42
% 0.15/0.42 % less_numeral_extra(3)
% 0.15/0.42 thf(fact_333_of__nat__0__less__iff,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 0.15/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0_less_iff
% 0.15/0.42 thf(fact_334_of__nat__0__less__iff,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 0.15/0.42 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0_less_iff
% 0.15/0.42 thf(fact_335_of__nat__0,axiom,
% 0.15/0.42 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.15/0.42 = zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0
% 0.15/0.42 thf(fact_336_of__nat__0,axiom,
% 0.15/0.42 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.15/0.42 = zero_zero_int ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0
% 0.15/0.42 thf(fact_337_of__nat__0__eq__iff,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( zero_zero_nat
% 0.15/0.42 = ( semiri1382578993at_nat @ N ) )
% 0.15/0.42 = ( zero_zero_nat = N ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0_eq_iff
% 0.15/0.42 thf(fact_338_of__nat__0__eq__iff,axiom,
% 0.15/0.42 ! [N: nat] :
% 0.15/0.42 ( ( zero_zero_int
% 0.15/0.42 = ( semiri2019852685at_int @ N ) )
% 0.15/0.42 = ( zero_zero_nat = N ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_0_eq_iff
% 0.15/0.42 thf(fact_339_of__nat__eq__0__iff,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ( ( ( semiri1382578993at_nat @ M )
% 0.15/0.42 = zero_zero_nat )
% 0.15/0.42 = ( M = zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_eq_0_iff
% 0.15/0.42 thf(fact_340_of__nat__eq__0__iff,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ( ( ( semiri2019852685at_int @ M )
% 0.15/0.42 = zero_zero_int )
% 0.15/0.42 = ( M = zero_zero_nat ) ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_eq_0_iff
% 0.15/0.42 thf(fact_341_psubset__imp__ex__mem,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1344122901lle_hf @ A2 @ B3 )
% 0.15/0.42 => ? [B6: hF_Mirabelle_hf] : ( member1367349282lle_hf @ B6 @ ( minus_1450406810lle_hf @ B3 @ A2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % psubset_imp_ex_mem
% 0.15/0.42 thf(fact_342_of__nat__less__0__iff,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_less_0_iff
% 0.15/0.42 thf(fact_343_of__nat__less__0__iff,axiom,
% 0.15/0.42 ! [M: nat] :
% 0.15/0.42 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 0.15/0.42
% 0.15/0.42 % of_nat_less_0_iff
% 0.15/0.42 thf(fact_344_not__psubset__empty,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf] :
% 0.15/0.42 ~ ( ord_le1344122901lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% 0.15/0.42
% 0.15/0.42 % not_psubset_empty
% 0.15/0.42 thf(fact_345_pos__int__cases,axiom,
% 0.15/0.42 ! [K: int] :
% 0.15/0.42 ( ( ord_less_int @ zero_zero_int @ K )
% 0.15/0.42 => ~ ! [N2: nat] :
% 0.15/0.42 ( ( K
% 0.15/0.42 = ( semiri2019852685at_int @ N2 ) )
% 0.15/0.42 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % pos_int_cases
% 0.15/0.42 thf(fact_346_zero__less__imp__eq__int,axiom,
% 0.15/0.42 ! [K: int] :
% 0.15/0.42 ( ( ord_less_int @ zero_zero_int @ K )
% 0.15/0.42 => ? [N2: nat] :
% 0.15/0.42 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.15/0.42 & ( K
% 0.15/0.42 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % zero_less_imp_eq_int
% 0.15/0.42 thf(fact_347_psubsetD,axiom,
% 0.15/0.42 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( ord_le1344122901lle_hf @ A2 @ B3 )
% 0.15/0.42 => ( ( member1367349282lle_hf @ C @ A2 )
% 0.15/0.42 => ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % psubsetD
% 0.15/0.42 thf(fact_348_less__set__def,axiom,
% 0.15/0.42 ( ord_le1344122901lle_hf
% 0.15/0.42 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.15/0.42 ( ord_le616625840e_hf_o
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % less_set_def
% 0.15/0.42 thf(fact_349_int__diff__cases,axiom,
% 0.15/0.42 ! [Z4: int] :
% 0.15/0.42 ~ ! [M3: nat,N2: nat] :
% 0.15/0.42 ( Z4
% 0.15/0.42 != ( minus_minus_int @ ( semiri2019852685at_int @ M3 ) @ ( semiri2019852685at_int @ N2 ) ) ) ).
% 0.15/0.42
% 0.15/0.42 % int_diff_cases
% 0.15/0.42 thf(fact_350_minus__int__code_I1_J,axiom,
% 0.15/0.42 ! [K: int] :
% 0.15/0.42 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.15/0.42 = K ) ).
% 0.15/0.42
% 0.15/0.42 % minus_int_code(1)
% 0.15/0.42
% 0.15/0.42 % Helper facts (3)
% 0.15/0.42 thf(help_If_3_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.15/0.42 ! [P: $o] :
% 0.15/0.42 ( ( P = $true )
% 0.15/0.42 | ( P = $false ) ) ).
% 0.15/0.42
% 0.15/0.42 thf(help_If_2_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( if_HF_Mirabelle_hf @ $false @ X2 @ Y2 )
% 0.15/0.42 = Y2 ) ).
% 0.15/0.42
% 0.15/0.42 thf(help_If_1_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.15/0.42 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.15/0.42 ( ( if_HF_Mirabelle_hf @ $true @ X2 @ Y2 )
% 0.15/0.42 = X2 ) ).
% 0.15/0.42
% 0.15/0.42 % Conjectures (1)
% 0.15/0.42 thf(conj_0,conjecture,
% 0.15/0.42 ( ( hF_Mirabelle_HBex @ a
% 0.15/0.42 @ ^ [X3: hF_Mirabelle_hf] : p )
% 0.15/0.42 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ a )
% 0.15/0.42 & p ) ) ).
% 0.15/0.42
% 0.15/0.42 %------------------------------------------------------------------------------
% 0.15/0.42 ------------------------
% 0.22/1.60 ---- Embedded in THF ---
% 0.22/1.60 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.22/1.60 %%% Generated on Tue Feb 24 08:25:37 EST 2026
% 0.22/1.60 %%% using '$$dhol' embedding, version 1.3.0.
% 0.22/1.60 %%% Logic specification used:
% 0.22/1.60 %%% thf(spec, logic, $$dhol).
% 0.22/1.60
% 0.22/1.60 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.mKzX9y748V/DTF2DTF32593.p
% See solution above
% 0.22/1.60 ------------------------
% 0.22/1.64 ---- Cleaned THF ---
% 0.22/1.64 thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.22/1.64 set_HF_Mirabelle_hf: $tType ).
% 0.22/1.64
% 0.22/1.64 thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 hF_Mirabelle_hf: $tType ).
% 0.22/1.64
% 0.22/1.64 thf(ty_n_t__Nat__Onat,type,
% 0.22/1.64 nat: $tType ).
% 0.22/1.64
% 0.22/1.64 thf(ty_n_t__Int__Oint,type,
% 0.22/1.64 int: $tType ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.22/1.64 minus_1668263787e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
% 0.22/1.64 minus_minus_o: $o > $o > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.22/1.64 minus_minus_int: int > int > int ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.22/1.64 minus_minus_nat: nat > nat > nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.22/1.64 minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 zero_z189798548lle_hf: hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
% 0.22/1.64 zero_zero_int: int ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.22/1.64 zero_zero_nat: nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHBall,type,
% 0.22/1.64 hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHBex,type,
% 0.22/1.64 hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
% 0.22/1.64 hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
% 0.22/1.64 hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
% 0.22/1.64 hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
% 0.22/1.64 hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
% 0.22/1.64 hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
% 0.22/1.64 hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
% 0.22/1.64 hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohcard,type,
% 0.22/1.64 hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
% 0.22/1.64 hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
% 0.22/1.64 hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
% 0.22/1.64 hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
% 0.22/1.64 hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
% 0.22/1.64 hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
% 0.22/1.64 hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
% 0.22/1.64 hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_HOL_OThe_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 the_HF_Mirabelle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_If_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 if_HF_Mirabelle_hf: $o > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.22/1.64 inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
% 0.22/1.64 inf_inf_o: $o > $o > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
% 0.22/1.64 inf_inf_int: int > int > int ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.22/1.64 inf_inf_nat: nat > nat > nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.22/1.64 inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.22/1.64 semiri2019852685at_int: nat > int ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.22/1.64 semiri1382578993at_nat: nat > nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.22/1.64 bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
% 0.22/1.64 bot_bot_o: $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Obot__class_Obot_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 bot_bo1001194783lle_hf: hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.22/1.64 bot_bot_nat: nat ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.22/1.64 bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord_OLeast_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 least_917110455lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
% 0.22/1.64 ord_le616625840e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 0.22/1.64 ord_less_int: int > int > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.22/1.64 ord_less_nat: nat > nat > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
% 0.22/1.64 ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
% 0.22/1.64 member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
% 0.22/1.64
% 0.22/1.64 thf(sy_v_A,type,
% 0.22/1.64 a: hF_Mirabelle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(sy_v_P,type,
% 0.22/1.64 p: $o ).
% 0.22/1.64
% 0.22/1.64 thf(fact_0_hf__equalityI,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ A )
% 0.22/1.64 = ( hF_Mirabelle_hmem @ X @ B ) )
% 0.22/1.64 => ( A = B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_1_hbexI,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X2 )
% 0.22/1.64 => ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.22/1.64 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_2_hbexE,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mirabelle_HBex @ A2 @ P )
% 0.22/1.64 => ~ ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.22/1.64 => ~ ( P @ X ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_3_hf__ext,axiom,
% 0.22/1.64 ( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z ) )
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X3 @ A3 )
% 0.22/1.64 = ( hF_Mirabelle_hmem @ X3 @ B2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_4_hmem__ne,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.22/1.64 => ( X2 != Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_5_HBex__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_HBex
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ? [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.22/1.64 & ( P2 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_6_hbex__cong,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( A2 = A5 )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ A5 )
% 0.22/1.64 => ( ( P @ X )
% 0.22/1.64 = ( P3 @ X ) ) )
% 0.22/1.64 => ( ( hF_Mirabelle_HBex @ A2 @ P )
% 0.22/1.64 = ( hF_Mirabelle_HBex @ A5 @ P3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_7_rev__hbexI,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.22/1.64 => ( ( P @ X2 )
% 0.22/1.64 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_8_replacement,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U @ X2 )
% 0.22/1.64 => ( ( R @ U @ V )
% 0.22/1.64 => ( ( R @ U @ V2 )
% 0.22/1.64 => ( V2 = V ) ) ) )
% 0.22/1.64 => ? [Z2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [V3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.22/1.64 & ( R @ U2 @ V3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_9_binary__union,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ? [Z2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [U3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.22/1.64 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.22/1.64 | ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_10_hmem__not__sym,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ Y2 @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_11_union__of__set,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ? [Z2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [U3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.22/1.64 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ Y3 @ X2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_12_comprehension,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ? [Z2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [U3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U3 @ Z2 )
% 0.22/1.64 = ( ( hF_Mirabelle_hmem @ U3 @ X2 )
% 0.22/1.64 & ( P @ U3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_13_hmem__not__refl,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ X2 @ X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_14_replacement__fun,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ? [Z2: hF_Mirabelle_hf] :
% 0.22/1.64 ! [V3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ V3 @ Z2 )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.22/1.64 & ( V3
% 0.22/1.64 = ( F @ U2 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_15_HF__Mirabelle__glliljednj_ObexCI,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_HBall @ A2
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( P @ X3 ) )
% 0.22/1.64 => ( P @ A ) )
% 0.22/1.64 => ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.22/1.64 => ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_16_hballI,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ A2 )
% 0.22/1.64 => ( P @ X ) )
% 0.22/1.64 => ( hF_Mirabelle_HBall @ A2 @ P ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_17_hball__triv,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: $o] :
% 0.22/1.64 ( ( hF_Mirabelle_HBall @ A2
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : P )
% 0.22/1.64 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ A2 )
% 0.22/1.64 => P ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_18_PrimReplace__iff,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U @ A2 )
% 0.22/1.64 => ( ( R @ U @ V )
% 0.22/1.64 => ( ( R @ U @ V2 )
% 0.22/1.64 => ( V2 = V ) ) ) )
% 0.22/1.64 => ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A2 @ R ) )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.22/1.64 & ( R @ U2 @ V4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_19_HCollect__iff,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mir818139703ollect @ P @ A2 ) )
% 0.22/1.64 = ( ( P @ X2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_20_Replace__iff,axiom,
% 0.22/1.64 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A2 @ R ) )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.22/1.64 & ( R @ U2 @ V4 )
% 0.22/1.64 & ! [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( R @ U2 @ Y3 )
% 0.22/1.64 => ( Y3 = V4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_21_HUnion__iff,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HUnion @ A2 ) )
% 0.22/1.64 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ Y3 @ A2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_22_hballE,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.22/1.64 => ( ~ ( P @ X2 )
% 0.22/1.64 => ~ ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_23_hbspec,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.22/1.64 => ( ( hF_Mirabelle_hmem @ X2 @ A2 )
% 0.22/1.64 => ( P @ X2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_24_HCollectE,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) )
% 0.22/1.64 => ~ ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.22/1.64 => ~ ( P @ A ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_25_HCollectI,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ A @ A2 )
% 0.22/1.64 => ( ( P @ A )
% 0.22/1.64 => ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_26_hball__cong,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( A2 = A5 )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ A5 )
% 0.22/1.64 => ( ( P @ X )
% 0.22/1.64 = ( P3 @ X ) ) )
% 0.22/1.64 => ( ( hF_Mirabelle_HBall @ A2 @ P )
% 0.22/1.64 = ( hF_Mirabelle_HBall @ A5 @ P3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_27_hmem__Sup__ne,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
% 0.22/1.64 => ( ( hF_Mirabelle_HUnion @ X2 )
% 0.22/1.64 != Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_28_Replace__cong,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( A2 = B3 )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.22/1.64 => ( ( P @ X @ Y4 )
% 0.22/1.64 = ( Q @ X @ Y4 ) ) )
% 0.22/1.64 => ( ( hF_Mirabelle_Replace @ A2 @ P )
% 0.22/1.64 = ( hF_Mirabelle_Replace @ B3 @ Q ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_29_HBall__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_HBall
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X3 @ A4 )
% 0.22/1.64 => ( P2 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_30_Replace__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_Replace
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( hF_Mir1248913145eplace @ A4
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ? [Z3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( R2 @ X3 @ Z3 )
% 0.22/1.64 & ! [Aa: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( R2 @ X3 @ Aa )
% 0.22/1.64 => ( Aa = Z3 ) ) )
% 0.22/1.64 & ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_31_RepFun__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_RepFun
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ( hF_Mirabelle_Replace @ A4
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( Y3
% 0.22/1.64 = ( F2 @ X3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_32_minus__hf__def,axiom,
% 0.22/1.64 ( minus_1232880740lle_hf
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
% 0.22/1.64 ( hF_Mir818139703ollect
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ X3 @ B4 )
% 0.22/1.64 @ A4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_33_inf__hf__def,axiom,
% 0.22/1.64 ( inf_in956532509lle_hf
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.22/1.64 ( hF_Mir818139703ollect
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ B2 )
% 0.22/1.64 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_34_HInter__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_HInter
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf] :
% 0.22/1.64 ( hF_Mir818139703ollect
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ! [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.22/1.64 => ( hF_Mirabelle_hmem @ X3 @ Y3 ) )
% 0.22/1.64 @ ( hF_Mirabelle_HUnion @ A4 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_35_RepFun__iff,axiom,
% 0.22/1.64 ! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A2 @ F ) )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ A2 )
% 0.22/1.64 & ( V4
% 0.22/1.64 = ( F @ U2 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_36_HCollect__hempty,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_37_Replace__0,axiom,
% 0.22/1.64 ! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_38_hmem__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_hmem
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_39_triv__RepFun,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_RepFun @ A2
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : X3 )
% 0.22/1.64 = A2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_40_hinter__iff,axiom,
% 0.22/1.64 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_41_hdiff__iff,axiom,
% 0.22/1.64 ! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U4 @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( ( hF_Mirabelle_hmem @ U4 @ X2 )
% 0.22/1.64 & ~ ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_42_hinter__hempty__right,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ A2 @ zero_z189798548lle_hf )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_43_hinter__hempty__left,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A2 )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_44_zero__hdiff,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X2 )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_45_mem__Collect__eq,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
% 0.22/1.64 = ( P @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_46_Collect__mem__eq,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( collec2046588256lle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.22/1.64 = A2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_47_Collect__cong,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 = ( Q @ X ) )
% 0.22/1.64 => ( ( collec2046588256lle_hf @ P )
% 0.22/1.64 = ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_48_hdiff__zero,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1232880740lle_hf @ X2 @ zero_z189798548lle_hf )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_49_RepFun__0,axiom,
% 0.22/1.64 ! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_50_HUnion__hempty,axiom,
% 0.22/1.64 ( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_51_HInter__hempty,axiom,
% 0.22/1.64 ( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_52_HInter__iff,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2 != zero_z189798548lle_hf )
% 0.22/1.64 => ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HInter @ A2 ) )
% 0.22/1.64 = ( ! [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ Y3 @ A2 )
% 0.22/1.64 => ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_53_foundation,axiom,
% 0.22/1.64 ! [Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( Z4 != zero_z189798548lle_hf )
% 0.22/1.64 => ? [W: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ W @ Z4 )
% 0.22/1.64 & ( ( inf_in956532509lle_hf @ W @ Z4 )
% 0.22/1.64 = zero_z189798548lle_hf ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_54_hemptyE,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_55_hempty__iff,axiom,
% 0.22/1.64 ! [Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( Z4 = zero_z189798548lle_hf )
% 0.22/1.64 = ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ X3 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_56_hmem__hempty,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_57_RepFun__cong,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2 = B3 )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X @ B3 )
% 0.22/1.64 => ( ( F @ X )
% 0.22/1.64 = ( G @ X ) ) )
% 0.22/1.64 => ( ( hF_Mirabelle_RepFun @ A2 @ F )
% 0.22/1.64 = ( hF_Mirabelle_RepFun @ B3 @ G ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_58_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ A @ A )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.22/1.64 ! [A: int] :
% 0.22/1.64 ( ( minus_minus_int @ A @ A )
% 0.22/1.64 = zero_zero_int ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_60_diff__zero,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_61_diff__zero,axiom,
% 0.22/1.64 ! [A: int] :
% 0.22/1.64 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_62_zero__diff,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_63_diff__0__right,axiom,
% 0.22/1.64 ! [A: int] :
% 0.22/1.64 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_64_diff__self,axiom,
% 0.22/1.64 ! [A: int] :
% 0.22/1.64 ( ( minus_minus_int @ A @ A )
% 0.22/1.64 = zero_zero_int ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_65_inf__right__idem,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Y2 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_66_inf__right__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Y2 )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_67_inf__right__idem,axiom,
% 0.22/1.64 ! [X2: int,Y2: int] :
% 0.22/1.64 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Y2 )
% 0.22/1.64 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_68_inf__right__idem,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 )
% 0.22/1.64 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_69_inf__right__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Y2 )
% 0.22/1.64 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_70_inf_Oright__idem,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ B )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_71_inf_Oright__idem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ B )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_72_inf_Oright__idem,axiom,
% 0.22/1.64 ! [A: int,B: int] :
% 0.22/1.64 ( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ B )
% 0.22/1.64 = ( inf_inf_int @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_73_inf_Oright__idem,axiom,
% 0.22/1.64 ! [A: nat,B: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
% 0.22/1.64 = ( inf_inf_nat @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_74_inf_Oright__idem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_75_inf__left__idem,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_76_inf__left__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_77_inf__left__idem,axiom,
% 0.22/1.64 ! [X2: int,Y2: int] :
% 0.22/1.64 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_78_inf__left__idem,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_79_inf__left__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_80_minus__apply,axiom,
% 0.22/1.64 ( minus_1668263787e_hf_o
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_81_inf__apply,axiom,
% 0.22/1.64 ( inf_in307783154e_hf_o
% 0.22/1.64 = ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_82_inf_Oidem,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A @ A )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_83_inf_Oidem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ A @ A )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_84_inf_Oidem,axiom,
% 0.22/1.64 ! [A: int] :
% 0.22/1.64 ( ( inf_inf_int @ A @ A )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_85_inf_Oidem,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ A @ A )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_86_inf_Oidem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ A @ A )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_87_inf__idem,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ X2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_88_inf__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ X2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_89_inf__idem,axiom,
% 0.22/1.64 ! [X2: int] :
% 0.22/1.64 ( ( inf_inf_int @ X2 @ X2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_90_inf__idem,axiom,
% 0.22/1.64 ! [X2: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ X2 @ X2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_91_inf__idem,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ X2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_92_inf_Oleft__idem,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ A @ B ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_93_inf_Oleft__idem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ A @ B ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_94_inf_Oleft__idem,axiom,
% 0.22/1.64 ! [A: int,B: int] :
% 0.22/1.64 ( ( inf_inf_int @ A @ ( inf_inf_int @ A @ B ) )
% 0.22/1.64 = ( inf_inf_int @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_95_inf_Oleft__idem,axiom,
% 0.22/1.64 ! [A: nat,B: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
% 0.22/1.64 = ( inf_inf_nat @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_96_inf_Oleft__idem,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ A @ B ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A @ B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_97_hfset__hdiff,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hfset @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( minus_1450406810lle_hf @ ( hF_Mirabelle_hfset @ X2 ) @ ( hF_Mirabelle_hfset @ Y2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_98_zero__reorient,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( zero_z189798548lle_hf = X2 )
% 0.22/1.64 = ( X2 = zero_z189798548lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_99_zero__reorient,axiom,
% 0.22/1.64 ! [X2: nat] :
% 0.22/1.64 ( ( zero_zero_nat = X2 )
% 0.22/1.64 = ( X2 = zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_100_zero__reorient,axiom,
% 0.22/1.64 ! [X2: int] :
% 0.22/1.64 ( ( zero_zero_int = X2 )
% 0.22/1.64 = ( X2 = zero_zero_int ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_101_fun__diff__def,axiom,
% 0.22/1.64 ( minus_1668263787e_hf_o
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_102_diff__eq__diff__eq,axiom,
% 0.22/1.64 ! [A: int,B: int,C: int,D: int] :
% 0.22/1.64 ( ( ( minus_minus_int @ A @ B )
% 0.22/1.64 = ( minus_minus_int @ C @ D ) )
% 0.22/1.64 => ( ( A = B )
% 0.22/1.64 = ( C = D ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_103_diff__right__commute,axiom,
% 0.22/1.64 ! [A: nat,C: nat,B: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.22/1.64 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_104_diff__right__commute,axiom,
% 0.22/1.64 ! [A: int,C: int,B: int] :
% 0.22/1.64 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.22/1.64 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_105_inf__sup__aci_I4_J,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_106_inf__sup__aci_I4_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_107_inf__sup__aci_I4_J,axiom,
% 0.22/1.64 ! [X2: int,Y2: int] :
% 0.22/1.64 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_inf_int @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_108_inf__sup__aci_I4_J,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_inf_nat @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_109_inf__sup__aci_I4_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_110_inf__sup__aci_I3_J,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_111_inf__sup__aci_I3_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_112_inf__sup__aci_I3_J,axiom,
% 0.22/1.64 ! [X2: int,Y2: int,Z4: int] :
% 0.22/1.64 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_113_inf__sup__aci_I3_J,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_114_inf__sup__aci_I3_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_115_inf__sup__aci_I2_J,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_116_inf__sup__aci_I2_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_117_inf__sup__aci_I2_J,axiom,
% 0.22/1.64 ! [X2: int,Y2: int,Z4: int] :
% 0.22/1.64 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_118_inf__sup__aci_I2_J,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_119_inf__sup__aci_I2_J,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_120_inf__sup__aci_I1_J,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_121_inf__sup__aci_I1_J,axiom,
% 0.22/1.64 ( inf_in307783154e_hf_o
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_122_inf__sup__aci_I1_J,axiom,
% 0.22/1.64 ( inf_inf_int
% 0.22/1.64 = ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_123_inf__sup__aci_I1_J,axiom,
% 0.22/1.64 ( inf_inf_nat
% 0.22/1.64 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_124_inf__sup__aci_I1_J,axiom,
% 0.22/1.64 ( inf_in956532509lle_hf
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_125_inf__fun__def,axiom,
% 0.22/1.64 ( inf_in307783154e_hf_o
% 0.22/1.64 = ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_126_boolean__algebra__cancel_Oinf1,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2
% 0.22/1.64 = ( inf_in923488851lle_hf @ K @ A ) )
% 0.22/1.64 => ( ( inf_in923488851lle_hf @ A2 @ B )
% 0.22/1.64 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_127_boolean__algebra__cancel_Oinf1,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( A2
% 0.22/1.64 = ( inf_in307783154e_hf_o @ K @ A ) )
% 0.22/1.64 => ( ( inf_in307783154e_hf_o @ A2 @ B )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_128_boolean__algebra__cancel_Oinf1,axiom,
% 0.22/1.64 ! [A2: int,K: int,A: int,B: int] :
% 0.22/1.64 ( ( A2
% 0.22/1.64 = ( inf_inf_int @ K @ A ) )
% 0.22/1.64 => ( ( inf_inf_int @ A2 @ B )
% 0.22/1.64 = ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_129_boolean__algebra__cancel_Oinf1,axiom,
% 0.22/1.64 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.22/1.64 ( ( A2
% 0.22/1.64 = ( inf_inf_nat @ K @ A ) )
% 0.22/1.64 => ( ( inf_inf_nat @ A2 @ B )
% 0.22/1.64 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_130_boolean__algebra__cancel_Oinf1,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2
% 0.22/1.64 = ( inf_in956532509lle_hf @ K @ A ) )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ A2 @ B )
% 0.22/1.64 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_131_boolean__algebra__cancel_Oinf2,axiom,
% 0.22/1.64 ! [B3: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( B3
% 0.22/1.64 = ( inf_in923488851lle_hf @ K @ B ) )
% 0.22/1.64 => ( ( inf_in923488851lle_hf @ A @ B3 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_132_boolean__algebra__cancel_Oinf2,axiom,
% 0.22/1.64 ! [B3: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( B3
% 0.22/1.64 = ( inf_in307783154e_hf_o @ K @ B ) )
% 0.22/1.64 => ( ( inf_in307783154e_hf_o @ A @ B3 )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_133_boolean__algebra__cancel_Oinf2,axiom,
% 0.22/1.64 ! [B3: int,K: int,B: int,A: int] :
% 0.22/1.64 ( ( B3
% 0.22/1.64 = ( inf_inf_int @ K @ B ) )
% 0.22/1.64 => ( ( inf_inf_int @ A @ B3 )
% 0.22/1.64 = ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_134_boolean__algebra__cancel_Oinf2,axiom,
% 0.22/1.64 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.22/1.64 ( ( B3
% 0.22/1.64 = ( inf_inf_nat @ K @ B ) )
% 0.22/1.64 => ( ( inf_inf_nat @ A @ B3 )
% 0.22/1.64 = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_135_boolean__algebra__cancel_Oinf2,axiom,
% 0.22/1.64 ! [B3: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( B3
% 0.22/1.64 = ( inf_in956532509lle_hf @ K @ B ) )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ A @ B3 )
% 0.22/1.64 = ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_136_inf_Oassoc,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ C )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_137_inf_Oassoc,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ C )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_138_inf_Oassoc,axiom,
% 0.22/1.64 ! [A: int,B: int,C: int] :
% 0.22/1.64 ( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ C )
% 0.22/1.64 = ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_139_inf_Oassoc,axiom,
% 0.22/1.64 ! [A: nat,B: nat,C: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
% 0.22/1.64 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_140_inf_Oassoc,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_141_inf__assoc,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_142_inf__assoc,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_143_inf__assoc,axiom,
% 0.22/1.64 ! [X2: int,Y2: int,Z4: int] :
% 0.22/1.64 ( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_144_inf__assoc,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_145_inf__assoc,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
% 0.22/1.64 = ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_146_inf_Ocommute,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [A3: set_HF_Mirabelle_hf,B2: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B2 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_147_inf_Ocommute,axiom,
% 0.22/1.64 ( inf_in307783154e_hf_o
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf > $o,B2: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ B2 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_148_inf_Ocommute,axiom,
% 0.22/1.64 ( inf_inf_int
% 0.22/1.64 = ( ^ [A3: int,B2: int] : ( inf_inf_int @ B2 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_149_inf_Ocommute,axiom,
% 0.22/1.64 ( inf_inf_nat
% 0.22/1.64 = ( ^ [A3: nat,B2: nat] : ( inf_inf_nat @ B2 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_150_inf_Ocommute,axiom,
% 0.22/1.64 ( inf_in956532509lle_hf
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B2 @ A3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_151_inf__commute,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_152_inf__commute,axiom,
% 0.22/1.64 ( inf_in307783154e_hf_o
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_153_inf__commute,axiom,
% 0.22/1.64 ( inf_inf_int
% 0.22/1.64 = ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_154_inf__commute,axiom,
% 0.22/1.64 ( inf_inf_nat
% 0.22/1.64 = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_155_inf__commute,axiom,
% 0.22/1.64 ( inf_in956532509lle_hf
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_156_inf_Oleft__commute,axiom,
% 0.22/1.64 ! [B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A @ C ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_157_inf_Oleft__commute,axiom,
% 0.22/1.64 ! [B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ B @ ( inf_in307783154e_hf_o @ A @ C ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_158_inf_Oleft__commute,axiom,
% 0.22/1.64 ! [B: int,A: int,C: int] :
% 0.22/1.64 ( ( inf_inf_int @ B @ ( inf_inf_int @ A @ C ) )
% 0.22/1.64 = ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_159_inf_Oleft__commute,axiom,
% 0.22/1.64 ! [B: nat,A: nat,C: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
% 0.22/1.64 = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_160_inf_Oleft__commute,axiom,
% 0.22/1.64 ! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A @ C ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_161_inf__left__commute,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_162_inf__left__commute,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_163_inf__left__commute,axiom,
% 0.22/1.64 ! [X2: int,Y2: int,Z4: int] :
% 0.22/1.64 ( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_164_inf__left__commute,axiom,
% 0.22/1.64 ! [X2: nat,Y2: nat,Z4: nat] :
% 0.22/1.64 ( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_165_inf__left__commute,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_166_eq__iff__diff__eq__0,axiom,
% 0.22/1.64 ( ( ^ [Y: int,Z: int] : ( Y = Z ) )
% 0.22/1.64 = ( ^ [A3: int,B2: int] :
% 0.22/1.64 ( ( minus_minus_int @ A3 @ B2 )
% 0.22/1.64 = zero_zero_int ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_167_HInter__hinsert,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2 != zero_z189798548lle_hf )
% 0.22/1.64 => ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A2 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_168_HCollect__def,axiom,
% 0.22/1.64 ( hF_Mir818139703ollect
% 0.22/1.64 = ( ^ [P2: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.22/1.64 ! [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.22/1.64 = ( ( P2 @ U2 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ U2 @ A4 ) ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_169_HF__hfset,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_170_zero__notin__hpair,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_171_Abs__hf__0,axiom,
% 0.22/1.64 ( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
% 0.22/1.64 = zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_172_hfset__0,axiom,
% 0.22/1.64 ( ( hF_Mirabelle_hfset @ zero_z189798548lle_hf )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_173_hpair__iff,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.22/1.64 = ( hF_Mirabelle_hpair @ A6 @ B5 ) )
% 0.22/1.64 = ( ( A = A6 )
% 0.22/1.64 & ( B = B5 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_174_inf__bot__right,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ X2 @ bot_bo1263054448e_hf_o )
% 0.22/1.64 = bot_bo1263054448e_hf_o ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_175_inf__bot__right,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ X2 @ bot_bo53200981lle_hf )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_176_inf__bot__right,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ X2 @ bot_bo1001194783lle_hf )
% 0.22/1.64 = bot_bo1001194783lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_177_inf__bot__left,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ bot_bo1263054448e_hf_o @ X2 )
% 0.22/1.64 = bot_bo1263054448e_hf_o ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_178_inf__bot__left,axiom,
% 0.22/1.64 ! [X2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ X2 )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_179_inf__bot__left,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in956532509lle_hf @ bot_bo1001194783lle_hf @ X2 )
% 0.22/1.64 = bot_bo1001194783lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_180_hmem__hinsert,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ A @ ( hF_Mirabelle_hinsert @ B @ C ) )
% 0.22/1.64 = ( ( A = B )
% 0.22/1.64 | ( hF_Mirabelle_hmem @ A @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_181_singleton__eq__iff,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.22/1.64 = ( A = B ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_182_RepFun__hinsert,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A @ B ) @ F )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ ( F @ A ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_183_hpair__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_hpair
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_184_hpair__def_H,axiom,
% 0.22/1.64 ( hF_Mirabelle_hpair
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B2 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_185_Zero__hf__def,axiom,
% 0.22/1.64 ( zero_z189798548lle_hf
% 0.22/1.64 = ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_186_hpair__neq__snd,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.22/1.64 != B ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_187_hpair__neq__fst,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hpair @ A @ B )
% 0.22/1.64 != A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_188_hpair__inject,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hpair @ A @ B )
% 0.22/1.64 = ( hF_Mirabelle_hpair @ A6 @ B5 ) )
% 0.22/1.64 => ~ ( ( A = A6 )
% 0.22/1.64 => ( B != B5 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_189_hinsert__commute,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hinsert @ X2 @ ( hF_Mirabelle_hinsert @ Y2 @ Z4 ) )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ Y2 @ ( hF_Mirabelle_hinsert @ X2 @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_190_hinsert__iff,axiom,
% 0.22/1.64 ! [Z4: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( Z4
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ Y2 @ X2 ) )
% 0.22/1.64 = ( ! [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ Z4 )
% 0.22/1.64 = ( ( hF_Mirabelle_hmem @ U2 @ X2 )
% 0.22/1.64 | ( U2 = Y2 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_191_hinsert__nonempty,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hinsert @ A @ A2 )
% 0.22/1.64 != zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_192_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hinsert @ A @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
% 0.22/1.64 = ( ( ( A = C )
% 0.22/1.64 & ( B = D ) )
% 0.22/1.64 | ( ( A = D )
% 0.22/1.64 & ( B = C ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_193_hf__induct__ax,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ zero_z189798548lle_hf )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ! [Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ Y4 )
% 0.22/1.64 => ( P @ ( hF_Mirabelle_hinsert @ Y4 @ X ) ) ) )
% 0.22/1.64 => ( P @ X2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_194_hpair__nonzero,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hpair @ X2 @ Y2 )
% 0.22/1.64 != zero_z189798548lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_195_hf__cases,axiom,
% 0.22/1.64 ! [Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( Y2 != zero_z189798548lle_hf )
% 0.22/1.64 => ~ ! [A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( Y2
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ A7 @ B6 ) )
% 0.22/1.64 => ( hF_Mirabelle_hmem @ A7 @ B6 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_196_hf__induct,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ zero_z189798548lle_hf )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( ( P @ Y4 )
% 0.22/1.64 => ( ~ ( hF_Mirabelle_hmem @ X @ Y4 )
% 0.22/1.64 => ( P @ ( hF_Mirabelle_hinsert @ X @ Y4 ) ) ) ) )
% 0.22/1.64 => ( P @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_197_hinter__hinsert__right,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) )
% 0.22/1.64 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
% 0.22/1.64 = ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_198_hinter__hinsert__left,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) )
% 0.22/1.64 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.22/1.64 = ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_199_hinsert__hdiff__if,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.22/1.64 = ( minus_1232880740lle_hf @ A2 @ B3 ) ) )
% 0.22/1.64 & ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
% 0.22/1.64 => ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
% 0.22/1.64 = ( hF_Mirabelle_hinsert @ X2 @ ( minus_1232880740lle_hf @ A2 @ B3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_200_hdiff__insert,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1232880740lle_hf @ A2 @ ( hF_Mirabelle_hinsert @ A @ B3 ) )
% 0.22/1.64 = ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A2 @ B3 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_201_Diff__disjoint,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ A2 ) )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_202_Diff__cancel,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ A2 @ A2 )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_203_empty__Diff,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A2 )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_204_Diff__empty,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.22/1.64 = A2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_205_empty__iff,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_206_all__not__in__conv,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.22/1.64 = ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_207_Collect__empty__eq,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( ( collec2046588256lle_hf @ P )
% 0.22/1.64 = bot_bo53200981lle_hf )
% 0.22/1.64 = ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( P @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_208_empty__Collect__eq,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( bot_bo53200981lle_hf
% 0.22/1.64 = ( collec2046588256lle_hf @ P ) )
% 0.22/1.64 = ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( P @ X3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_209_Int__iff,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.22/1.64 = ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 & ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_210_IntI,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 => ( ( member1367349282lle_hf @ C @ B3 )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_211_DiffI,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 => ( ~ ( member1367349282lle_hf @ C @ B3 )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_212_Diff__iff,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 = ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 & ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_213_Diff__idemp,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ B3 )
% 0.22/1.64 = ( minus_1450406810lle_hf @ A2 @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_214_Int__left__commute,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ B3 @ ( inf_in923488851lle_hf @ A2 @ C2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_215_Int__left__absorb,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_216_Collect__conj__eq,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( collec2046588256lle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X3 )
% 0.22/1.64 & ( Q @ X3 ) ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ ( collec2046588256lle_hf @ P ) @ ( collec2046588256lle_hf @ Q ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_217_inf__set__def,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( collec2046588256lle_hf
% 0.22/1.64 @ ( inf_in307783154e_hf_o
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_218_Int__commute,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B4 @ A4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_219_Int__Collect,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X2 @ ( inf_in923488851lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) ) )
% 0.22/1.64 = ( ( member1367349282lle_hf @ X2 @ A2 )
% 0.22/1.64 & ( P @ X2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_220_Int__absorb,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A2 @ A2 )
% 0.22/1.64 = A2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_221_Int__assoc,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_222_Int__def,axiom,
% 0.22/1.64 ( inf_in923488851lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( collec2046588256lle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.22/1.64 & ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_223_IntD2,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_224_IntD1,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ A2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_225_IntE,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ~ ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 => ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_226_bot__set__def,axiom,
% 0.22/1.64 ( bot_bo53200981lle_hf
% 0.22/1.64 = ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_227_bot__hf__def,axiom,
% 0.22/1.64 bot_bo1001194783lle_hf = zero_z189798548lle_hf ).
% 0.22/1.64
% 0.22/1.64 thf(fact_228_emptyE,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( member1367349282lle_hf @ A @ bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_229_equals0D,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A2 = bot_bo53200981lle_hf )
% 0.22/1.64 => ~ ( member1367349282lle_hf @ A @ A2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_230_equals0I,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ! [Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( member1367349282lle_hf @ Y4 @ A2 )
% 0.22/1.64 => ( A2 = bot_bo53200981lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_231_ex__in__conv,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
% 0.22/1.64 = ( A2 != bot_bo53200981lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_232_Int__emptyI,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X @ A2 )
% 0.22/1.64 => ~ ( member1367349282lle_hf @ X @ B3 ) )
% 0.22/1.64 => ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.22/1.64 = bot_bo53200981lle_hf ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_233_disjoint__iff,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.22/1.64 = bot_bo53200981lle_hf )
% 0.22/1.64 = ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X3 @ A2 )
% 0.22/1.64 => ~ ( member1367349282lle_hf @ X3 @ B3 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_234_Int__empty__left,axiom,
% 0.22/1.64 ! [B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ B3 )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_235_Int__empty__right,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ A2 @ bot_bo53200981lle_hf )
% 0.22/1.64 = bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_236_disjoint__iff__not__equal,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.22/1.64 = bot_bo53200981lle_hf )
% 0.22/1.64 = ( ! [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X3 @ A2 )
% 0.22/1.64 => ! [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ Y3 @ B3 )
% 0.22/1.64 => ( X3 != Y3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_237_DiffE,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ~ ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_238_DiffD1,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ A2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_239_DiffD2,axiom,
% 0.22/1.64 ! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 => ~ ( member1367349282lle_hf @ C @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_240_Int__Diff,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ C2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_241_Diff__Int2,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
% 0.22/1.64 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_242_Diff__Diff__Int,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( minus_1450406810lle_hf @ A2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 = ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_243_Diff__Int__distrib,axiom,
% 0.22/1.64 ! [C2: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ C2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
% 0.22/1.64 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ C2 @ A2 ) @ ( inf_in923488851lle_hf @ C2 @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_244_Diff__Int__distrib2,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in923488851lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ C2 )
% 0.22/1.64 = ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_245_empty__def,axiom,
% 0.22/1.64 ( bot_bo53200981lle_hf
% 0.22/1.64 = ( collec2046588256lle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : $false ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_246_set__diff__eq,axiom,
% 0.22/1.64 ( minus_1450406810lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( collec2046588256lle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( member1367349282lle_hf @ X3 @ A4 )
% 0.22/1.64 & ~ ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_247_minus__set__def,axiom,
% 0.22/1.64 ( minus_1450406810lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( collec2046588256lle_hf
% 0.22/1.64 @ ( minus_1668263787e_hf_o
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_248_Diff__triv,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ( inf_in923488851lle_hf @ A2 @ B3 )
% 0.22/1.64 = bot_bo53200981lle_hf )
% 0.22/1.64 => ( ( minus_1450406810lle_hf @ A2 @ B3 )
% 0.22/1.64 = A2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_249_hfst__conv,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_250_hsnd__conv,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
% 0.22/1.64 = B ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_251_the__equality,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ A )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( X = A ) )
% 0.22/1.64 => ( ( the_HF_Mirabelle_hf @ P )
% 0.22/1.64 = A ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_252_the__eq__trivial,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( X3 = A ) )
% 0.22/1.64 = A ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_253_inf1I,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( A2 @ X2 )
% 0.22/1.64 => ( ( B3 @ X2 )
% 0.22/1.64 => ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_254_the__sym__eq__trivial,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( Y = Z )
% 0.22/1.64 @ X2 ) )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_255_inf1D2,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.22/1.64 => ( B3 @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_256_inf1D1,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.22/1.64 => ( A2 @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_257_inf1E,axiom,
% 0.22/1.64 ! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
% 0.22/1.64 => ~ ( ( A2 @ X2 )
% 0.22/1.64 => ~ ( B3 @ X2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_258_the1__equality,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ? [X4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X4 )
% 0.22/1.64 & ! [Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ Y4 )
% 0.22/1.64 => ( Y4 = X4 ) ) )
% 0.22/1.64 => ( ( P @ A )
% 0.22/1.64 => ( ( the_HF_Mirabelle_hf @ P )
% 0.22/1.64 = A ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_259_the1I2,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ? [X4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X4 )
% 0.22/1.64 & ! [Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ Y4 )
% 0.22/1.64 => ( Y4 = X4 ) ) )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( Q @ X ) )
% 0.22/1.64 => ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_260_If__def,axiom,
% 0.22/1.64 ( if_HF_Mirabelle_hf
% 0.22/1.64 = ( ^ [P2: $o,X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P2
% 0.22/1.64 => ( Z3 = X3 ) )
% 0.22/1.64 & ( ~ P2
% 0.22/1.64 => ( Z3 = Y3 ) ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_261_theI2,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( P @ A )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( X = A ) )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( Q @ X ) )
% 0.22/1.64 => ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_262_theI_H,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ? [X4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X4 )
% 0.22/1.64 & ! [Y4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ Y4 )
% 0.22/1.64 => ( Y4 = X4 ) ) )
% 0.22/1.64 => ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_263_theI,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ A )
% 0.22/1.64 => ( ! [X: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P @ X )
% 0.22/1.64 => ( X = A ) )
% 0.22/1.64 => ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_264_bot__apply,axiom,
% 0.22/1.64 ( bot_bo1263054448e_hf_o
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_265_inf__Int__eq,axiom,
% 0.22/1.64 ! [R: set_HF_Mirabelle_hf,S: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( inf_in307783154e_hf_o
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ R )
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ S ) )
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ ( inf_in923488851lle_hf @ R @ S ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_266_diff__self__eq__0,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ M @ M )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_267_diff__0__eq__0,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_268_diff__commute,axiom,
% 0.22/1.64 ! [I: nat,J: nat,K: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.22/1.64 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_269_bot__nat__def,axiom,
% 0.22/1.64 bot_bot_nat = zero_zero_nat ).
% 0.22/1.64
% 0.22/1.64 thf(fact_270_diffs0__imp__equal,axiom,
% 0.22/1.64 ! [M: nat,N: nat] :
% 0.22/1.64 ( ( ( minus_minus_nat @ M @ N )
% 0.22/1.64 = zero_zero_nat )
% 0.22/1.64 => ( ( ( minus_minus_nat @ N @ M )
% 0.22/1.64 = zero_zero_nat )
% 0.22/1.64 => ( M = N ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_271_minus__nat_Odiff__0,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.22/1.64 = M ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_272_bot__empty__eq,axiom,
% 0.22/1.64 ( bot_bo1263054448e_hf_o
% 0.22/1.64 = ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_273_Collect__empty__eq__bot,axiom,
% 0.22/1.64 ! [P: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( ( ( collec2046588256lle_hf @ P )
% 0.22/1.64 = bot_bo53200981lle_hf )
% 0.22/1.64 = ( P = bot_bo1263054448e_hf_o ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_274_ord_OLeast__def,axiom,
% 0.22/1.64 ( least_917110455lle_hf
% 0.22/1.64 = ( ^ [Less_eq: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,P2: hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P2 @ X3 )
% 0.22/1.64 & ! [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( P2 @ Y3 )
% 0.22/1.64 => ( Less_eq @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_275_PrimReplace__def,axiom,
% 0.22/1.64 ( hF_Mir1248913145eplace
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.22/1.64 ! [V5: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ V5 @ Z3 )
% 0.22/1.64 = ( ? [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ A4 )
% 0.22/1.64 & ( R2 @ U2 @ V5 ) ) ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_276_hcard__0,axiom,
% 0.22/1.64 ( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_277_hfst__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_hfst
% 0.22/1.64 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ? [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( P4
% 0.22/1.64 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_278_hsnd__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_hsnd
% 0.22/1.64 = ( ^ [P4: hF_Mirabelle_hf] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ? [X3: hF_Mirabelle_hf] :
% 0.22/1.64 ( P4
% 0.22/1.64 = ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_279_HUnion__def,axiom,
% 0.22/1.64 ( hF_Mirabelle_HUnion
% 0.22/1.64 = ( ^ [A4: hF_Mirabelle_hf] :
% 0.22/1.64 ( the_HF_Mirabelle_hf
% 0.22/1.64 @ ^ [Z3: hF_Mirabelle_hf] :
% 0.22/1.64 ! [U2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ U2 @ Z3 )
% 0.22/1.64 = ( ? [Y3: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ Y3 @ A4 )
% 0.22/1.64 & ( hF_Mirabelle_hmem @ U2 @ Y3 ) ) ) ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_280_hcard__hdiff1__less,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( hF_Mirabelle_hmem @ X2 @ Z4 )
% 0.22/1.64 => ( ord_less_nat @ ( hF_Mirabelle_hcard @ ( minus_1232880740lle_hf @ Z4 @ ( hF_Mirabelle_hinsert @ X2 @ zero_z189798548lle_hf ) ) ) @ ( hF_Mirabelle_hcard @ Z4 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_281_Set_Ois__empty__def,axiom,
% 0.22/1.64 ( is_emp566801209lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf] : ( A4 = bot_bo53200981lle_hf ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_282_not__gr__zero,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.22/1.64 = ( N = zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_283_bot__nat__0_Onot__eq__extremum,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( A != zero_zero_nat )
% 0.22/1.64 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_284_less__nat__zero__code,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_285_neq0__conv,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( N != zero_zero_nat )
% 0.22/1.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_286_diff__gt__0__iff__gt,axiom,
% 0.22/1.64 ! [A: int,B: int] :
% 0.22/1.64 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.22/1.64 = ( ord_less_int @ B @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_287_zero__less__diff,axiom,
% 0.22/1.64 ! [N: nat,M: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.22/1.64 = ( ord_less_nat @ M @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_288_less__imp__diff__less,axiom,
% 0.22/1.64 ! [J: nat,K: nat,N: nat] :
% 0.22/1.64 ( ( ord_less_nat @ J @ K )
% 0.22/1.64 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_289_diff__less__mono2,axiom,
% 0.22/1.64 ! [M: nat,N: nat,L: nat] :
% 0.22/1.64 ( ( ord_less_nat @ M @ N )
% 0.22/1.64 => ( ( ord_less_nat @ M @ L )
% 0.22/1.64 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_290_diff__less,axiom,
% 0.22/1.64 ! [N: nat,M: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/1.64 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.22/1.64 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_291_diff__strict__right__mono,axiom,
% 0.22/1.64 ! [A: int,B: int,C: int] :
% 0.22/1.64 ( ( ord_less_int @ A @ B )
% 0.22/1.64 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_292_diff__strict__left__mono,axiom,
% 0.22/1.64 ! [B: int,A: int,C: int] :
% 0.22/1.64 ( ( ord_less_int @ B @ A )
% 0.22/1.64 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_293_diff__eq__diff__less,axiom,
% 0.22/1.64 ! [A: int,B: int,C: int,D: int] :
% 0.22/1.64 ( ( ( minus_minus_int @ A @ B )
% 0.22/1.64 = ( minus_minus_int @ C @ D ) )
% 0.22/1.64 => ( ( ord_less_int @ A @ B )
% 0.22/1.64 = ( ord_less_int @ C @ D ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_294_diff__strict__mono,axiom,
% 0.22/1.64 ! [A: int,B: int,D: int,C: int] :
% 0.22/1.64 ( ( ord_less_int @ A @ B )
% 0.22/1.64 => ( ( ord_less_int @ D @ C )
% 0.22/1.64 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_295_bot__nat__0_Oextremum__strict,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_296_infinite__descent0,axiom,
% 0.22/1.64 ! [P: nat > $o,N: nat] :
% 0.22/1.64 ( ( P @ zero_zero_nat )
% 0.22/1.64 => ( ! [N2: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/1.64 => ( ~ ( P @ N2 )
% 0.22/1.64 => ? [M2: nat] :
% 0.22/1.64 ( ( ord_less_nat @ M2 @ N2 )
% 0.22/1.64 & ~ ( P @ M2 ) ) ) )
% 0.22/1.64 => ( P @ N ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_297_gr__implies__not0,axiom,
% 0.22/1.64 ! [M: nat,N: nat] :
% 0.22/1.64 ( ( ord_less_nat @ M @ N )
% 0.22/1.64 => ( N != zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_298_less__zeroE,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_299_not__less0,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_300_not__gr0,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.22/1.64 = ( N = zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_301_gr0I,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( N != zero_zero_nat )
% 0.22/1.64 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_302_less__iff__diff__less__0,axiom,
% 0.22/1.64 ( ord_less_int
% 0.22/1.64 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_303_zero__less__iff__neq__zero,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.22/1.64 = ( N != zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_304_gr__implies__not__zero,axiom,
% 0.22/1.64 ! [M: nat,N: nat] :
% 0.22/1.64 ( ( ord_less_nat @ M @ N )
% 0.22/1.64 => ( N != zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_305_not__less__zero,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_306_gr__zeroI,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( N != zero_zero_nat )
% 0.22/1.64 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_307_less__infI1,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1310584031lle_hf @ A @ X2 )
% 0.22/1.64 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_308_less__infI1,axiom,
% 0.22/1.64 ! [A: nat,X2: nat,B: nat] :
% 0.22/1.64 ( ( ord_less_nat @ A @ X2 )
% 0.22/1.64 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_309_less__infI1,axiom,
% 0.22/1.64 ! [A: int,X2: int,B: int] :
% 0.22/1.64 ( ( ord_less_int @ A @ X2 )
% 0.22/1.64 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_310_less__infI2,axiom,
% 0.22/1.64 ! [B: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1310584031lle_hf @ B @ X2 )
% 0.22/1.64 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_311_less__infI2,axiom,
% 0.22/1.64 ! [B: nat,X2: nat,A: nat] :
% 0.22/1.64 ( ( ord_less_nat @ B @ X2 )
% 0.22/1.64 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_312_less__infI2,axiom,
% 0.22/1.64 ! [B: int,X2: int,A: int] :
% 0.22/1.64 ( ( ord_less_int @ B @ X2 )
% 0.22/1.64 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_313_inf_Ostrict__boundedE,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1310584031lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
% 0.22/1.64 => ~ ( ( ord_le1310584031lle_hf @ A @ B )
% 0.22/1.64 => ~ ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_314_inf_Ostrict__boundedE,axiom,
% 0.22/1.64 ! [A: nat,B: nat,C: nat] :
% 0.22/1.64 ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
% 0.22/1.64 => ~ ( ( ord_less_nat @ A @ B )
% 0.22/1.64 => ~ ( ord_less_nat @ A @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_315_inf_Ostrict__boundedE,axiom,
% 0.22/1.64 ! [A: int,B: int,C: int] :
% 0.22/1.64 ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
% 0.22/1.64 => ~ ( ( ord_less_int @ A @ B )
% 0.22/1.64 => ~ ( ord_less_int @ A @ C ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_316_inf_Ostrict__order__iff,axiom,
% 0.22/1.64 ( ord_le1310584031lle_hf
% 0.22/1.64 = ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A3
% 0.22/1.64 = ( inf_in956532509lle_hf @ A3 @ B2 ) )
% 0.22/1.64 & ( A3 != B2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_317_inf_Ostrict__order__iff,axiom,
% 0.22/1.64 ( ord_less_nat
% 0.22/1.64 = ( ^ [A3: nat,B2: nat] :
% 0.22/1.64 ( ( A3
% 0.22/1.64 = ( inf_inf_nat @ A3 @ B2 ) )
% 0.22/1.64 & ( A3 != B2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_318_inf_Ostrict__order__iff,axiom,
% 0.22/1.64 ( ord_less_int
% 0.22/1.64 = ( ^ [A3: int,B2: int] :
% 0.22/1.64 ( ( A3
% 0.22/1.64 = ( inf_inf_int @ A3 @ B2 ) )
% 0.22/1.64 & ( A3 != B2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_319_inf_Ostrict__coboundedI1,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1310584031lle_hf @ A @ C )
% 0.22/1.64 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_320_inf_Ostrict__coboundedI1,axiom,
% 0.22/1.64 ! [A: nat,C: nat,B: nat] :
% 0.22/1.64 ( ( ord_less_nat @ A @ C )
% 0.22/1.64 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_321_inf_Ostrict__coboundedI1,axiom,
% 0.22/1.64 ! [A: int,C: int,B: int] :
% 0.22/1.64 ( ( ord_less_int @ A @ C )
% 0.22/1.64 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_322_inf_Ostrict__coboundedI2,axiom,
% 0.22/1.64 ! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1310584031lle_hf @ B @ C )
% 0.22/1.64 => ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_323_inf_Ostrict__coboundedI2,axiom,
% 0.22/1.64 ! [B: nat,C: nat,A: nat] :
% 0.22/1.64 ( ( ord_less_nat @ B @ C )
% 0.22/1.64 => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_324_inf_Ostrict__coboundedI2,axiom,
% 0.22/1.64 ! [B: int,C: int,A: int] :
% 0.22/1.64 ( ( ord_less_int @ B @ C )
% 0.22/1.64 => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_325_bot_Oextremum__strict,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf] :
% 0.22/1.64 ~ ( ord_le1344122901lle_hf @ A @ bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_326_bot_Oextremum__strict,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ~ ( ord_le1310584031lle_hf @ A @ bot_bo1001194783lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_327_bot_Oextremum__strict,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_328_bot_Onot__eq__extremum,axiom,
% 0.22/1.64 ! [A: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( A != bot_bo53200981lle_hf )
% 0.22/1.64 = ( ord_le1344122901lle_hf @ bot_bo53200981lle_hf @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_329_bot_Onot__eq__extremum,axiom,
% 0.22/1.64 ! [A: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( A != bot_bo1001194783lle_hf )
% 0.22/1.64 = ( ord_le1310584031lle_hf @ bot_bo1001194783lle_hf @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_330_bot_Onot__eq__extremum,axiom,
% 0.22/1.64 ! [A: nat] :
% 0.22/1.64 ( ( A != bot_bot_nat )
% 0.22/1.64 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_331_less__numeral__extra_I3_J,axiom,
% 0.22/1.64 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_332_less__numeral__extra_I3_J,axiom,
% 0.22/1.64 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_333_of__nat__0__less__iff,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
% 0.22/1.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_334_of__nat__0__less__iff,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
% 0.22/1.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_335_of__nat__0,axiom,
% 0.22/1.64 ( ( semiri1382578993at_nat @ zero_zero_nat )
% 0.22/1.64 = zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_336_of__nat__0,axiom,
% 0.22/1.64 ( ( semiri2019852685at_int @ zero_zero_nat )
% 0.22/1.64 = zero_zero_int ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_337_of__nat__0__eq__iff,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( zero_zero_nat
% 0.22/1.64 = ( semiri1382578993at_nat @ N ) )
% 0.22/1.64 = ( zero_zero_nat = N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_338_of__nat__0__eq__iff,axiom,
% 0.22/1.64 ! [N: nat] :
% 0.22/1.64 ( ( zero_zero_int
% 0.22/1.64 = ( semiri2019852685at_int @ N ) )
% 0.22/1.64 = ( zero_zero_nat = N ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_339_of__nat__eq__0__iff,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ( ( ( semiri1382578993at_nat @ M )
% 0.22/1.64 = zero_zero_nat )
% 0.22/1.64 = ( M = zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_340_of__nat__eq__0__iff,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ( ( ( semiri2019852685at_int @ M )
% 0.22/1.64 = zero_zero_int )
% 0.22/1.64 = ( M = zero_zero_nat ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_341_psubset__imp__ex__mem,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1344122901lle_hf @ A2 @ B3 )
% 0.22/1.64 => ? [B6: hF_Mirabelle_hf] : ( member1367349282lle_hf @ B6 @ ( minus_1450406810lle_hf @ B3 @ A2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_342_of__nat__less__0__iff,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_343_of__nat__less__0__iff,axiom,
% 0.22/1.64 ! [M: nat] :
% 0.22/1.64 ~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_344_not__psubset__empty,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf] :
% 0.22/1.64 ~ ( ord_le1344122901lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_345_pos__int__cases,axiom,
% 0.22/1.64 ! [K: int] :
% 0.22/1.64 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/1.64 => ~ ! [N2: nat] :
% 0.22/1.64 ( ( K
% 0.22/1.64 = ( semiri2019852685at_int @ N2 ) )
% 0.22/1.64 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_346_zero__less__imp__eq__int,axiom,
% 0.22/1.64 ! [K: int] :
% 0.22/1.64 ( ( ord_less_int @ zero_zero_int @ K )
% 0.22/1.64 => ? [N2: nat] :
% 0.22/1.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.22/1.64 & ( K
% 0.22/1.64 = ( semiri2019852685at_int @ N2 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_347_psubsetD,axiom,
% 0.22/1.64 ! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( ord_le1344122901lle_hf @ A2 @ B3 )
% 0.22/1.64 => ( ( member1367349282lle_hf @ C @ A2 )
% 0.22/1.64 => ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_348_less__set__def,axiom,
% 0.22/1.64 ( ord_le1344122901lle_hf
% 0.22/1.64 = ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
% 0.22/1.64 ( ord_le616625840e_hf_o
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_349_int__diff__cases,axiom,
% 0.22/1.64 ! [Z4: int] :
% 0.22/1.64 ~ ! [M3: nat,N2: nat] :
% 0.22/1.64 ( Z4
% 0.22/1.64 != ( minus_minus_int @ ( semiri2019852685at_int @ M3 ) @ ( semiri2019852685at_int @ N2 ) ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(fact_350_minus__int__code_I1_J,axiom,
% 0.22/1.64 ! [K: int] :
% 0.22/1.64 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.22/1.64 = K ) ).
% 0.22/1.64
% 0.22/1.64 thf(help_If_3_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.22/1.64 ! [P: $o] :
% 0.22/1.64 ( ( P = $true )
% 0.22/1.64 | ( P = $false ) ) ).
% 0.22/1.64
% 0.22/1.64 thf(help_If_2_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( if_HF_Mirabelle_hf @ $false @ X2 @ Y2 )
% 0.22/1.64 = Y2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(help_If_1_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
% 0.22/1.64 ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
% 0.22/1.64 ( ( if_HF_Mirabelle_hf @ $true @ X2 @ Y2 )
% 0.22/1.64 = X2 ) ).
% 0.22/1.64
% 0.22/1.64 thf(conj_0,conjecture,
% 0.22/1.64 ( ( hF_Mirabelle_HBex @ a
% 0.22/1.64 @ ^ [X3: hF_Mirabelle_hf] : p )
% 0.22/1.64 = ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ a )
% 0.22/1.64 & p ) ) ).
% 0.22/1.64 ------------------------
% 0.22/1.67 % (32694)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/183Mi)
% 0.22/1.67 % (32696)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_32593 for (2999ds/27Mi)
% 0.22/1.67 % (32697)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/2Mi)
% 0.22/1.67 % (32695)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_32593 for (2999ds/4Mi)
% 0.22/1.67 % (32700)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/18Mi)
% 0.22/1.67 % (32698)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_32593 for (2999ds/2Mi)
% 0.22/1.68 % (32699)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_32593 for (2999ds/275Mi)
% 0.22/1.68 % (32697)Instruction limit reached!
% 0.22/1.68 % (32697)------------------------------
% 0.22/1.68 % (32697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.68 % (32697)Termination reason: Unknown
% 0.22/1.68 % (32697)Termination phase: shuffling
% 0.22/1.68 % (32698)Instruction limit reached!
% 0.22/1.68 % (32698)------------------------------
% 0.22/1.68 % (32698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.68 % (32698)Termination reason: Unknown
% 0.22/1.68 % (32698)Termination phase: shuffling
% 0.22/1.68
% 0.22/1.68 % (32698)Memory used [KB]: 1407
% 0.22/1.68 % (32698)Time elapsed: 0.003 s
% 0.22/1.68 % (32698)Instructions burned: 3 (million)
% 0.22/1.68 % (32698)------------------------------
% 0.22/1.68 % (32698)------------------------------
% 0.22/1.68
% 0.22/1.68 % (32697)Memory used [KB]: 1407
% 0.22/1.68 % (32697)Time elapsed: 0.003 s
% 0.22/1.68 % (32697)Instructions burned: 3 (million)
% 0.22/1.68 % (32697)------------------------------
% 0.22/1.68 % (32697)------------------------------
% 0.22/1.68 % (32695)Instruction limit reached!
% 0.22/1.68 % (32695)------------------------------
% 0.22/1.68 % (32695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.68 % (32695)Termination reason: Unknown
% 0.22/1.68 % (32695)Termination phase: shuffling
% 0.22/1.68
% 0.22/1.68 % (32695)Memory used [KB]: 1407
% 0.22/1.68 % (32695)Time elapsed: 0.004 s
% 0.22/1.68 % (32695)Instructions burned: 5 (million)
% 0.22/1.68 % (32695)------------------------------
% 0.22/1.68 % (32695)------------------------------
% 0.22/1.68 % (32700)Instruction limit reached!
% 0.22/1.68 % (32700)------------------------------
% 0.22/1.68 % (32700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.68 % (32700)Termination reason: Unknown
% 0.22/1.68 % (32700)Termination phase: Property scanning
% 0.22/1.68
% 0.22/1.68 % (32700)Memory used [KB]: 1535
% 0.22/1.68 % (32700)Time elapsed: 0.010 s
% 0.22/1.68 % (32700)Instructions burned: 18 (million)
% 0.22/1.68 % (32700)------------------------------
% 0.22/1.68 % (32700)------------------------------
% 0.22/1.69 % (32696)Instruction limit reached!
% 0.22/1.69 % (32696)------------------------------
% 0.22/1.69 % (32696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.69 % (32696)Termination reason: Unknown
% 0.22/1.69 % (32696)Termination phase: Preprocessing 1
% 0.22/1.69
% 0.22/1.69 % (32696)Memory used [KB]: 1535
% 0.22/1.69 % (32696)Time elapsed: 0.015 s
% 0.22/1.69 % (32696)Instructions burned: 28 (million)
% 0.22/1.69 % (32696)------------------------------
% 0.22/1.69 % (32696)------------------------------
% 0.22/1.69 % (32702)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_32593 for (2999ds/37Mi)
% 0.22/1.69 % (32703)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_32593 for (2999ds/15Mi)
% 3.04/1.70 % (32704)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_32593 for (2999ds/3Mi)
% 3.04/1.70 % (32701)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/3Mi)
% 3.04/1.70 % (32703)Instruction limit reached!
% 3.04/1.70 % (32703)------------------------------
% 3.04/1.70 % (32703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (32703)Termination reason: Unknown
% 3.04/1.70 % (32703)Termination phase: Property scanning
% 3.04/1.70
% 3.04/1.70 % (32703)Memory used [KB]: 1535
% 3.04/1.70 % (32703)Time elapsed: 0.009 s
% 3.04/1.70 % (32703)Instructions burned: 15 (million)
% 3.04/1.70 % (32703)------------------------------
% 3.04/1.70 % (32703)------------------------------
% 3.04/1.70 % (32704)Instruction limit reached!
% 3.04/1.70 % (32704)------------------------------
% 3.04/1.70 % (32704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (32704)Termination reason: Unknown
% 3.04/1.70 % (32704)Termination phase: shuffling
% 3.04/1.70
% 3.04/1.70 % (32704)Memory used [KB]: 1407
% 3.04/1.70 % (32704)Time elapsed: 0.003 s
% 3.04/1.70 % (32704)Instructions burned: 3 (million)
% 3.04/1.70 % (32704)------------------------------
% 3.04/1.70 % (32704)------------------------------
% 3.04/1.70 % (32701)Instruction limit reached!
% 3.04/1.70 % (32701)------------------------------
% 3.04/1.70 % (32701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (32701)Termination reason: Unknown
% 3.04/1.70 % (32701)Termination phase: shuffling
% 3.04/1.70
% 3.04/1.70 % (32701)Memory used [KB]: 1407
% 3.04/1.70 % (32701)Time elapsed: 0.003 s
% 3.04/1.70 % (32701)Instructions burned: 3 (million)
% 3.04/1.70 % (32701)------------------------------
% 3.04/1.70 % (32701)------------------------------
% 3.04/1.70 % (32705)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_32593 for (2999ds/1041Mi)
% 3.04/1.71 % (32702)Instruction limit reached!
% 3.04/1.71 % (32702)------------------------------
% 3.04/1.71 % (32702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.71 % (32702)Termination reason: Unknown
% 3.04/1.71 % (32702)Termination phase: shuffling
% 3.04/1.71
% 3.04/1.71 % (32702)Memory used [KB]: 2046
% 3.04/1.71 % (32702)Time elapsed: 0.018 s
% 3.04/1.71 % (32702)Instructions burned: 37 (million)
% 3.04/1.71 % (32702)------------------------------
% 3.04/1.71 % (32702)------------------------------
% 3.04/1.72 % (32706)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/7Mi)
% 3.04/1.72 % (32707)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/16Mi)
% 3.04/1.72 % (32708)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_32593 for (2999ds/3Mi)
% 3.04/1.72 % (32708)Instruction limit reached!
% 3.04/1.72 % (32708)------------------------------
% 3.04/1.72 % (32708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.72 % (32708)Termination reason: Unknown
% 3.04/1.72 % (32708)Termination phase: shuffling
% 3.04/1.72
% 3.04/1.72 % (32708)Memory used [KB]: 1407
% 3.04/1.72 % (32708)Time elapsed: 0.003 s
% 3.04/1.72 % (32708)Instructions burned: 4 (million)
% 3.04/1.72 % (32708)------------------------------
% 3.04/1.72 % (32708)------------------------------
% 3.04/1.72 % (32706)Instruction limit reached!
% 3.04/1.72 % (32706)------------------------------
% 3.04/1.72 % (32706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.72 % (32706)Termination reason: Unknown
% 3.04/1.72 % (32706)Termination phase: shuffling
% 3.04/1.72
% 3.04/1.72 % (32706)Memory used [KB]: 1407
% 3.04/1.72 % (32706)Time elapsed: 0.005 s
% 3.04/1.72 % (32706)Instructions burned: 7 (million)
% 3.04/1.72 % (32706)------------------------------
% 3.04/1.72 % (32706)------------------------------
% 3.04/1.72 % (32707)Instruction limit reached!
% 3.04/1.72 % (32707)------------------------------
% 3.04/1.72 % (32707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.72 % (32707)Termination reason: Unknown
% 3.04/1.72 % (32707)Termination phase: Property scanning
% 3.04/1.72
% 3.04/1.72 % (32707)Memory used [KB]: 1535
% 3.04/1.72 % (32707)Time elapsed: 0.009 s
% 3.04/1.72 % (32707)Instructions burned: 16 (million)
% 3.04/1.72 % (32707)------------------------------
% 3.04/1.72 % (32707)------------------------------
% 3.04/1.73 % (32694)Instruction limit reached!
% 3.04/1.73 % (32694)------------------------------
% 3.04/1.73 % (32694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.73 % (32694)Termination reason: Unknown
% 3.04/1.73 % (32694)Termination phase: Saturation
% 3.04/1.73
% 3.04/1.73 % (32694)Memory used [KB]: 6908
% 3.04/1.73 % (32694)Time elapsed: 0.058 s
% 3.04/1.73 % (32694)Instructions burned: 184 (million)
% 3.04/1.73 % (32694)------------------------------
% 3.04/1.73 % (32694)------------------------------
% 3.04/1.73 % (32710)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_32593 for (2999ds/3Mi)
% 3.04/1.73 % (32710)Instruction limit reached!
% 3.04/1.73 % (32710)------------------------------
% 3.04/1.73 % (32710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.73 % (32710)Termination reason: Unknown
% 3.04/1.73 % (32710)Termination phase: shuffling
% 3.04/1.73
% 3.04/1.73 % (32710)Memory used [KB]: 1407
% 3.04/1.73 % (32710)Time elapsed: 0.003 s
% 3.04/1.73 % (32710)Instructions burned: 4 (million)
% 3.04/1.73 % (32710)------------------------------
% 3.04/1.73 % (32710)------------------------------
% 3.04/1.74 % (32711)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/7Mi)
% 3.04/1.74 % (32712)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/3Mi)
% 3.04/1.74 % (32712)Instruction limit reached!
% 3.04/1.74 % (32712)------------------------------
% 3.04/1.74 % (32712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.74 % (32712)Termination reason: Unknown
% 3.04/1.74 % (32712)Termination phase: shuffling
% 3.04/1.74
% 3.04/1.74 % (32712)Memory used [KB]: 1407
% 3.04/1.74 % (32712)Time elapsed: 0.003 s
% 3.04/1.74 % (32712)Instructions burned: 4 (million)
% 3.04/1.74 % (32712)------------------------------
% 3.04/1.74 % (32712)------------------------------
% 3.04/1.74 % (32711)Instruction limit reached!
% 3.04/1.74 % (32711)------------------------------
% 3.04/1.74 % (32711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.74 % (32711)Termination reason: Unknown
% 3.04/1.74 % (32711)Termination phase: shuffling
% 3.04/1.74
% 3.04/1.74 % (32711)Memory used [KB]: 1407
% 3.04/1.74 % (32711)Time elapsed: 0.005 s
% 3.04/1.74 % (32711)Instructions burned: 7 (million)
% 3.04/1.74 % (32711)------------------------------
% 3.04/1.74 % (32711)------------------------------
% 3.04/1.74 % (32713)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/4Mi)
% 3.04/1.74 % (32714)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/18Mi)
% 3.04/1.74 % (32713)Instruction limit reached!
% 3.04/1.74 % (32713)------------------------------
% 3.04/1.74 % (32713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.74 % (32713)Termination reason: Unknown
% 3.04/1.74 % (32713)Termination phase: shuffling
% 3.04/1.74
% 3.04/1.74 % (32713)Memory used [KB]: 1407
% 3.04/1.74 % (32713)Time elapsed: 0.003 s
% 3.04/1.74 % (32713)Instructions burned: 4 (million)
% 3.04/1.74 % (32713)------------------------------
% 3.04/1.74 % (32713)------------------------------
% 3.04/1.74 % (32715)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/710Mi)
% 3.04/1.75 % (32714)Instruction limit reached!
% 3.04/1.75 % (32714)------------------------------
% 3.04/1.75 % (32714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.75 % (32714)Termination reason: Unknown
% 3.04/1.75 % (32714)Termination phase: Property scanning
% 3.04/1.75
% 3.04/1.75 % (32714)Memory used [KB]: 1535
% 3.04/1.75 % (32714)Time elapsed: 0.010 s
% 3.04/1.75 % (32714)Instructions burned: 18 (million)
% 3.04/1.75 % (32714)------------------------------
% 3.04/1.75 % (32714)------------------------------
% 3.04/1.76 % (32717)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_32593 for (2999ds/902Mi)
% 3.04/1.76 % (32716)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/6Mi)
% 3.04/1.76 % (32718)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_32593 for (2999ds/21Mi)
% 3.04/1.76 % (32716)Instruction limit reached!
% 3.04/1.76 % (32716)------------------------------
% 3.04/1.76 % (32716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.76 % (32716)Termination reason: Unknown
% 3.04/1.76 % (32716)Termination phase: shuffling
% 3.04/1.76
% 3.04/1.76 % (32716)Memory used [KB]: 1407
% 3.04/1.76 % (32716)Time elapsed: 0.004 s
% 3.04/1.76 % (32716)Instructions burned: 6 (million)
% 3.04/1.76 % (32716)------------------------------
% 3.04/1.76 % (32716)------------------------------
% 3.04/1.77 % (32719)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32593 for (2999ds/5Mi)
% 3.04/1.77 % (32719)Instruction limit reached!
% 3.04/1.77 % (32719)------------------------------
% 3.04/1.77 % (32719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.77 % (32719)Termination reason: Unknown
% 3.04/1.77 % (32719)Termination phase: shuffling
% 3.04/1.77
% 3.04/1.77 % (32719)Memory used [KB]: 1407
% 3.04/1.77 % (32719)Time elapsed: 0.004 s
% 3.04/1.77 % (32719)Instructions burned: 6 (million)
% 3.04/1.77 % (32719)------------------------------
% 3.04/1.77 % (32719)------------------------------
% 3.04/1.77 % (32718)Instruction limit reached!
% 3.04/1.77 % (32718)------------------------------
% 3.04/1.77 % (32718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.77 % (32718)Termination reason: Unknown
% 3.04/1.77 % (32718)Termination phase: shuffling
% 3.04/1.77
% 3.04/1.77 % (32718)Memory used [KB]: 1663
% 3.04/1.77 % (32718)Time elapsed: 0.012 s
% 3.04/1.77 % (32718)Instructions burned: 22 (million)
% 3.04/1.77 % (32718)------------------------------
% 3.04/1.77 % (32718)------------------------------
% 3.04/1.77 % (32715)First to succeed.
% 3.04/1.77 % (32715)Refutation found. Thanks to Tanya!
% 3.04/1.77 % SZS status Theorem for DTF2THF_32593
% 3.04/1.77 % SZS output start Proof for DTF2THF_32593
% 3.04/1.77 thf(type_def_5, type, hF_Mirabelle_hf: $tType).
% 3.04/1.77 thf(type_def_7, type, int: $tType).
% 3.04/1.77 thf(type_def_8, type, nat: $tType).
% 3.04/1.77 thf(type_def_9, type, set_HF_Mirabelle_hf: $tType).
% 3.04/1.77 thf(func_def_0, type, set_HF_Mirabelle_hf: $tType).
% 3.04/1.77 thf(func_def_1, type, hF_Mirabelle_hf: $tType).
% 3.04/1.77 thf(func_def_2, type, nat: $tType).
% 3.04/1.77 thf(func_def_3, type, int: $tType).
% 3.04/1.77 thf(func_def_4, type, minus_1668263787e_hf_o: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_5, type, minus_minus_o: $o > $o > $o).
% 3.04/1.77 thf(func_def_6, type, minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_7, type, minus_minus_int: int > int > int).
% 3.04/1.77 thf(func_def_8, type, minus_minus_nat: nat > nat > nat).
% 3.04/1.77 thf(func_def_9, type, minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_10, type, zero_z189798548lle_hf: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_11, type, zero_zero_int: int).
% 3.04/1.77 thf(func_def_12, type, zero_zero_nat: nat).
% 3.04/1.77 thf(func_def_13, type, hF_Mirabelle_HBall: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > $o).
% 3.04/1.77 thf(func_def_14, type, hF_Mirabelle_HBex: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > $o).
% 3.04/1.77 thf(func_def_15, type, hF_Mir818139703ollect: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_16, type, hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_17, type, hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_18, type, hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_19, type, hF_Mir1248913145eplace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_20, type, hF_Mirabelle_RepFun: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_21, type, hF_Mirabelle_Replace: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_22, type, hF_Mirabelle_hcard: hF_Mirabelle_hf > nat).
% 3.04/1.77 thf(func_def_23, type, hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_24, type, hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_25, type, hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_26, type, hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_27, type, hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_28, type, hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_29, type, hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_30, type, the_HF_Mirabelle_hf: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_31, type, if_HF_Mirabelle_hf: $o > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_32, type, inf_in307783154e_hf_o: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_33, type, inf_inf_o: $o > $o > $o).
% 3.04/1.77 thf(func_def_34, type, inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_35, type, inf_inf_int: int > int > int).
% 3.04/1.77 thf(func_def_36, type, inf_inf_nat: nat > nat > nat).
% 3.04/1.77 thf(func_def_37, type, inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_38, type, semiri2019852685at_int: nat > int).
% 3.04/1.77 thf(func_def_39, type, semiri1382578993at_nat: nat > nat).
% 3.04/1.77 thf(func_def_40, type, bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_42, type, bot_bo1001194783lle_hf: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_43, type, bot_bot_nat: nat).
% 3.04/1.77 thf(func_def_44, type, bot_bo53200981lle_hf: set_HF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_45, type, least_917110455lle_hf: (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_46, type, ord_le616625840e_hf_o: (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > $o).
% 3.04/1.77 thf(func_def_47, type, ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_48, type, ord_less_int: int > int > $o).
% 3.04/1.77 thf(func_def_49, type, ord_less_nat: nat > nat > $o).
% 3.04/1.77 thf(func_def_50, type, ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_51, type, collec2046588256lle_hf: (hF_Mirabelle_hf > $o) > set_HF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_52, type, is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_53, type, member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o).
% 3.04/1.77 thf(func_def_54, type, a: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_71, type, sP0: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > $o).
% 3.04/1.77 thf(func_def_72, type, sK1: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_73, type, sK2: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_74, type, sK3: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_75, type, sK4: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_76, type, sK5: hF_Mirabelle_hf > hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_77, type, sK6: hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_78, type, sK7: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_79, type, sK8: hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_80, type, sK9: hF_Mirabelle_hf > hF_Mirabelle_hf > (hF_Mirabelle_hf > hF_Mirabelle_hf) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_81, type, sK10: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_82, type, sK11: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_83, type, sK12: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_84, type, sK13: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_85, type, sK14: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_86, type, sK15: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_88, type, sK17: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_89, type, sK18: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_90, type, sK19: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_91, type, sK20: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_92, type, sK21: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_93, type, sK22: hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_94, type, sK23: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_95, type, sK24: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_96, type, sK25: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_97, type, sK26: (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_98, type, sK27: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(func_def_99, type, sK28: hF_Mirabelle_hf > (hF_Mirabelle_hf > $o) > hF_Mirabelle_hf).
% 3.04/1.77 thf(f1025,plain,(
% 3.04/1.77 $false),
% 3.04/1.77 inference(avatar_sat_refutation,[status(thm)],[f981,f992,f1003,f1006,f1011,f1015,f1018,f1023])).
% 3.04/1.77 thf(f1023,plain,(
% 3.04/1.77 ~spl16_1 | ~spl16_6),
% 3.04/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1022])).
% 3.04/1.77 thf(f1022,plain,(
% 3.04/1.77 $false | (~spl16_1 | ~spl16_6)),
% 3.04/1.77 inference(trivial_inequality_removal,[status(thm)],[f1020])).
% 3.04/1.77 thf(f1020,plain,(
% 3.04/1.77 ($true != $true) | (~spl16_1 | ~spl16_6)),
% 3.04/1.77 inference(superposition,[status(thm)],[f999,f976])).
% 3.04/1.77 thf(f976,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK22 @ a)) | ~spl16_1),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f974])).
% 3.04/1.77 thf(f974,definition,(
% 3.04/1.77 spl16_1 <=> ($true = (hF_Mirabelle_hmem @ sK22 @ a))),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_1])],[avatar_definition])).
% 3.04/1.77 thf(f999,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a))) ) | ~spl16_6),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f998])).
% 3.04/1.77 thf(f998,definition,(
% 3.04/1.77 spl16_6 <=> ! [X0 : hF_Mirabelle_hf] : ($true != (hF_Mirabelle_hmem @ X0 @ a))),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_6])],[avatar_definition])).
% 3.04/1.77 thf(f1018,plain,(
% 3.04/1.77 ~spl16_1 | ~spl16_7),
% 3.04/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1017])).
% 3.04/1.77 thf(f1017,plain,(
% 3.04/1.77 $false | (~spl16_1 | ~spl16_7)),
% 3.04/1.77 inference(trivial_inequality_removal,[status(thm)],[f1016])).
% 3.04/1.77 thf(f1016,plain,(
% 3.04/1.77 ($true = $false) | (~spl16_1 | ~spl16_7)),
% 3.04/1.77 inference(forward_demodulation,[status(thm)],[f976,f1002])).
% 3.04/1.77 thf(f1002,plain,(
% 3.04/1.77 ( ! [X1 : hF_Mirabelle_hf] : (($false = (hF_Mirabelle_hmem @ X1 @ a))) ) | ~spl16_7),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f1001])).
% 3.04/1.77 thf(f1001,definition,(
% 3.04/1.77 spl16_7 <=> ! [X1 : hF_Mirabelle_hf] : ($false = (hF_Mirabelle_hmem @ X1 @ a))),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_7])],[avatar_definition])).
% 3.04/1.77 thf(f1015,plain,(
% 3.04/1.77 ~spl16_2 | ~spl16_7),
% 3.04/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1014])).
% 3.04/1.77 thf(f1014,plain,(
% 3.04/1.77 $false | (~spl16_2 | ~spl16_7)),
% 3.04/1.77 inference(trivial_inequality_removal,[status(thm)],[f1013])).
% 3.04/1.77 thf(f1013,plain,(
% 3.04/1.77 ($true = $false) | (~spl16_2 | ~spl16_7)),
% 3.04/1.77 inference(backward_demodulation,[status(thm)],[f980,f1002])).
% 3.04/1.77 thf(f980,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK14 @ a)) | ~spl16_2),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f978])).
% 3.04/1.77 thf(f978,definition,(
% 3.04/1.77 spl16_2 <=> ($true = (hF_Mirabelle_hmem @ sK14 @ a))),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_2])],[avatar_definition])).
% 3.04/1.77 thf(f1011,plain,(
% 3.04/1.77 ~spl16_2 | ~spl16_6),
% 3.04/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1010])).
% 3.04/1.77 thf(f1010,plain,(
% 3.04/1.77 $false | (~spl16_2 | ~spl16_6)),
% 3.04/1.77 inference(trivial_inequality_removal,[status(thm)],[f1008])).
% 3.04/1.77 thf(f1008,plain,(
% 3.04/1.77 ($true != $true) | (~spl16_2 | ~spl16_6)),
% 3.04/1.77 inference(superposition,[status(thm)],[f999,f980])).
% 3.04/1.77 thf(f1006,plain,(
% 3.04/1.77 ~spl16_3 | ~spl16_5),
% 3.04/1.77 inference(avatar_contradiction_clause,[status(thm)],[f1005])).
% 3.04/1.77 thf(f1005,plain,(
% 3.04/1.77 $false | (~spl16_3 | ~spl16_5)),
% 3.04/1.77 inference(trivial_inequality_removal,[status(thm)],[f1004])).
% 3.04/1.77 thf(f1004,plain,(
% 3.04/1.77 ($true = $false) | (~spl16_3 | ~spl16_5)),
% 3.04/1.77 inference(forward_demodulation,[status(thm)],[f985,f996])).
% 3.04/1.77 thf(f996,plain,(
% 3.04/1.77 (p = $false) | ~spl16_5),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f994])).
% 3.04/1.77 thf(f994,definition,(
% 3.04/1.77 spl16_5 <=> (p = $false)),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_5])],[avatar_definition])).
% 3.04/1.77 thf(f985,plain,(
% 3.04/1.77 (p = $true) | ~spl16_3),
% 3.04/1.77 inference(avatar_component_clause,[status(thm)],[f983])).
% 3.04/1.77 thf(f983,definition,(
% 3.04/1.77 spl16_3 <=> (p = $true)),
% 3.04/1.77 introduced(definition,[new_symbols(naming,[spl16_3])],[avatar_definition])).
% 3.04/1.77 thf(f1003,plain,(
% 3.04/1.77 spl16_5 | spl16_6 | ~spl16_3 | spl16_7),
% 3.04/1.77 inference(avatar_split_clause,[status(thm)],[f908,f1001,f983,f998,f994])).
% 3.04/1.77 thf(f908,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p = $false) | ($false = (hF_Mirabelle_hmem @ X1 @ a)) | (p != $true)) )),
% 3.04/1.77 inference(binary_proxy_clausification,[status(thm)],[f907])).
% 3.04/1.77 thf(f907,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((p != $true) | ($false = (p & (hF_Mirabelle_hmem @ X1 @ a))) | ($true != (hF_Mirabelle_hmem @ X0 @ a))) )),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f906])).
% 3.04/1.77 thf(f906,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf,X1 : hF_Mirabelle_hf] : ((p != $true) | ($true != (hF_Mirabelle_hmem @ X0 @ a)) | ($false = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ X1))) )),
% 3.04/1.77 inference(pi_clausification,[status(thm)],[f905])).
% 3.04/1.77 thf(f905,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_hmem @ X0 @ a)) | ((?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a)))) != $true) | (p != $true)) )),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f859])).
% 3.04/1.77 thf(f859,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf] : ((((^[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))) != $true) | ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true)) )),
% 3.04/1.77 inference(definition_unfolding,[status(thm)],[f849,f832])).
% 3.04/1.77 thf(f832,plain,(
% 3.04/1.77 (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))))))))),
% 3.04/1.77 inference(cnf_transformation,[status(thm)],[f405])).
% 3.04/1.77 thf(f405,plain,(
% 3.04/1.77 (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))))))))),
% 3.04/1.77 inference(fool_elimination,[status(thm)],[f404])).
% 3.04/1.77 thf(f404,plain,(
% 3.04/1.77 (hF_Mirabelle_HBex = (^[X0 : hF_Mirabelle_hf, X1 : hF_Mirabelle_hf > $o] : (? [X2 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X2 @ X0) & (X1 @ X2)))))),
% 3.04/1.77 inference(rectify,[status(thm)],[f6])).
% 3.04/1.77 thf(f6,axiom,(
% 3.04/1.77 (hF_Mirabelle_HBex = (^[X12 : hF_Mirabelle_hf, X13 : hF_Mirabelle_hf > $o] : (? [X10 : hF_Mirabelle_hf] : ((hF_Mirabelle_hmem @ X10 @ X12) & (X13 @ X10)))))),
% 3.04/1.77 file('/export/starexec/sandbox2/tmp/tmp.mKzX9y748V/DTF2THF_32593.p',fact_5_HBex__def)).
% 3.04/1.77 thf(f849,plain,(
% 3.04/1.77 ( ! [X0 : hF_Mirabelle_hf] : (($true != (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) | ($true != (hF_Mirabelle_hmem @ X0 @ a)) | (p != $true)) )),
% 3.04/1.77 inference(cnf_transformation,[status(thm)],[f814])).
% 3.04/1.77 thf(f814,plain,(
% 3.04/1.77 (($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)))) | (($true = (hF_Mirabelle_hmem @ sK14 @ a)) & (p = $true)))),
% 3.04/1.77 inference(skolemisation,[status(esa),new_symbols(skolem,[sK14])],[f812,f813])).
% 3.04/1.77 thf(f813,plain,(
% 3.04/1.77 ? [X1 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X1 @ a)) => ($true = (hF_Mirabelle_hmem @ sK14 @ a))),
% 3.04/1.77 introduced(definition,[],[choice_axiom])).
% 3.04/1.77 thf(f812,plain,(
% 3.04/1.77 (($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)))) | (? [X1 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X1 @ a)) & (p = $true)))),
% 3.04/1.77 inference(rectify,[status(thm)],[f811])).
% 3.04/1.77 thf(f811,plain,(
% 3.04/1.77 (($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)))) | (? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)))),
% 3.04/1.77 inference(flattening,[status(thm)],[f810])).
% 3.04/1.77 thf(f810,plain,(
% 3.04/1.77 (($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)))) | (? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)))),
% 3.04/1.77 inference(nnf_transformation,[status(thm)],[f777])).
% 3.04/1.77 thf(f777,plain,(
% 3.04/1.77 (? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)) <~> ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p))))),
% 3.04/1.77 inference(ennf_transformation,[status(thm)],[f657])).
% 3.04/1.77 thf(f657,plain,(
% 3.04/1.77 ~(($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) <=> (? [X0 : hF_Mirabelle_hf] : ($true = (hF_Mirabelle_hmem @ X0 @ a)) & (p = $true)))),
% 3.04/1.77 inference(fool_elimination,[status(thm)],[f656])).
% 3.04/1.77 thf(f656,plain,(
% 3.04/1.77 ~(p & ? [X0 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X0 @ a) = (hF_Mirabelle_HBex @ a @ (^[X1 : hF_Mirabelle_hf] : (p))))),
% 3.04/1.77 inference(rectify,[status(thm)],[f356])).
% 3.04/1.77 thf(f356,negated_conjecture,(
% 3.04/1.77 ~(p & ? [X10 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X10 @ a) = (hF_Mirabelle_HBex @ a @ (^[X10 : hF_Mirabelle_hf] : (p))))),
% 3.04/1.77 inference(negated_conjecture,[status(cth)],[f355])).
% 3.04/1.77 thf(f355,conjecture,(
% 3.04/1.77 (p & ? [X10 : hF_Mirabelle_hf] : (hF_Mirabelle_hmem @ X10 @ a) = (hF_Mirabelle_HBex @ a @ (^[X10 : hF_Mirabelle_hf] : (p))))),
% 3.04/1.77 file('/export/starexec/sandbox2/tmp/tmp.mKzX9y748V/DTF2THF_32593.p',conj_0)).
% 3.04/1.77 thf(f992,plain,(
% 3.04/1.77 spl16_3),
% 3.04/1.77 inference(avatar_split_clause,[status(thm)],[f914,f983])).
% 3.04/1.77 thf(f914,plain,(
% 3.04/1.77 (p = $true)),
% 3.04/1.77 inference(duplicate_literal_removal,[status(thm)],[f913])).
% 3.04/1.77 thf(f913,plain,(
% 3.04/1.77 (p = $true) | (p = $true)),
% 3.04/1.77 inference(binary_proxy_clausification,[status(thm)],[f911])).
% 3.04/1.77 thf(f911,plain,(
% 3.04/1.77 (p = $true) | ($true = (p & (hF_Mirabelle_hmem @ sK21 @ a)))),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f910])).
% 3.04/1.77 thf(f910,plain,(
% 3.04/1.77 ($true = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ sK21)) | (p = $true)),
% 3.04/1.77 inference(sigma_clausification,[status(thm)],[f909])).
% 3.04/1.77 thf(f909,plain,(
% 3.04/1.77 (p = $true) | ((?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a)))) = $true)),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f861])).
% 3.04/1.77 thf(f861,plain,(
% 3.04/1.77 (p = $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))) = $true)),
% 3.04/1.77 inference(definition_unfolding,[status(thm)],[f847,f832])).
% 3.04/1.77 thf(f847,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) | (p = $true)),
% 3.04/1.77 inference(cnf_transformation,[status(thm)],[f814])).
% 3.04/1.77 thf(f981,plain,(
% 3.04/1.77 spl16_1 | spl16_2),
% 3.04/1.77 inference(avatar_split_clause,[status(thm)],[f920,f978,f974])).
% 3.04/1.77 thf(f920,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK22 @ a)) | ($true = (hF_Mirabelle_hmem @ sK14 @ a))),
% 3.04/1.77 inference(binary_proxy_clausification,[status(thm)],[f919])).
% 3.04/1.77 thf(f919,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK14 @ a)) | ($true = (p & (hF_Mirabelle_hmem @ sK22 @ a)))),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f918])).
% 3.04/1.77 thf(f918,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK14 @ a)) | ($true = ((^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a))) @ sK22))),
% 3.04/1.77 inference(sigma_clausification,[status(thm)],[f917])).
% 3.04/1.77 thf(f917,plain,(
% 3.04/1.77 ((?? @ hF_Mirabelle_hf @ (^[Y0 : hF_Mirabelle_hf]: (p & (hF_Mirabelle_hmem @ Y0 @ a)))) = $true) | ($true = (hF_Mirabelle_hmem @ sK14 @ a))),
% 3.04/1.77 inference(beta_eta_normalization,[status(thm)],[f860])).
% 3.04/1.77 thf(f860,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_hmem @ sK14 @ a)) | (((^[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))) = $true)),
% 3.04/1.77 inference(definition_unfolding,[status(thm)],[f848,f832])).
% 3.04/1.77 thf(f848,plain,(
% 3.04/1.77 ($true = (hF_Mirabelle_HBex @ a @ (^[Y0 : hF_Mirabelle_hf]: (p)))) | ($true = (hF_Mirabelle_hmem @ sK14 @ a))),
% 3.04/1.77 inference(cnf_transformation,[status(thm)],[f814])).
% 3.04/1.77 % SZS output end Proof for DTF2THF_32593
% 3.04/1.77 % (32715)------------------------------
% 3.04/1.77 % (32715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.77 % (32715)Termination reason: Refutation
% 3.04/1.77
% 3.04/1.77 % (32715)Memory used [KB]: 6396
% 3.04/1.77 % (32715)Time elapsed: 0.030 s
% 3.04/1.77 % (32715)Instructions burned: 57 (million)
% 3.04/1.77 % (32715)------------------------------
% 3.04/1.77 % (32715)------------------------------
% 3.04/1.77 % (32693)Success in time 0.117 s
%------------------------------------------------------------------------------