TPTP Problem File: ITP075^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP075^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_754__5337380_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : HF/prob_754__5337380_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 411 ( 267 unt; 56 typ; 0 def)
% Number of atoms : 755 ( 399 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 2036 ( 93 ~; 5 |; 43 &;1719 @)
% ( 0 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 221 ( 221 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 11 con; 0-3 aty)
% Number of variables : 844 ( 139 ^; 677 !; 28 ?; 844 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:10.866
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
set_HF_Mirabelle_hf: $tType ).
thf(ty_n_t__HF____Mirabelle____glliljednj__Ohf,type,
hF_Mirabelle_hf: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (52)
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
minus_1668263787e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__HF____Mirabelle____glliljednj__Ohf,type,
minus_1232880740lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
minus_1450406810lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__HF____Mirabelle____glliljednj__Ohf,type,
zero_z189798548lle_hf: hF_Mirabelle_hf ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HF__Mirabelle__glliljednj_OHBall,type,
hF_Mirabelle_HBall: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
thf(sy_c_HF__Mirabelle__glliljednj_OHBex,type,
hF_Mirabelle_HBex: hF_Mirabelle_hf > ( hF_Mirabelle_hf > $o ) > $o ).
thf(sy_c_HF__Mirabelle__glliljednj_OHCollect,type,
hF_Mir818139703ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OHF,type,
hF_Mirabelle_HF: set_HF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OHInter,type,
hF_Mirabelle_HInter: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OHUnion,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OPrimReplace,type,
hF_Mir1248913145eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_ORepFun,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_OReplace,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohcard,type,
hF_Mirabelle_hcard: hF_Mirabelle_hf > nat ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohf_OAbs__hf,type,
hF_Mirabelle_Abs_hf: nat > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohfset,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohfst,type,
hF_Mirabelle_hfst: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohinsert,type,
hF_Mirabelle_hinsert: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohpair,type,
hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohsnd,type,
hF_Mirabelle_hsnd: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HOL_OThe_001t__HF____Mirabelle____glliljednj__Ohf,type,
the_HF_Mirabelle_hf: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_If_001t__HF____Mirabelle____glliljednj__Ohf,type,
if_HF_Mirabelle_hf: $o > hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
inf_in307783154e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__HF____Mirabelle____glliljednj__Ohf,type,
inf_in956532509lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
inf_in923488851lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri2019852685at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1382578993at_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
bot_bo1263054448e_hf_o: hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__HF____Mirabelle____glliljednj__Ohf,type,
bot_bo1001194783lle_hf: hF_Mirabelle_hf ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
bot_bo53200981lle_hf: set_HF_Mirabelle_hf ).
thf(sy_c_Orderings_Oord_OLeast_001t__HF____Mirabelle____glliljednj__Ohf,type,
least_917110455lle_hf: ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
ord_le616625840e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__HF____Mirabelle____glliljednj__Ohf,type,
ord_le1310584031lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
ord_le1344122901lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_c_Set_OCollect_001t__HF____Mirabelle____glliljednj__Ohf,type,
collec2046588256lle_hf: ( hF_Mirabelle_hf > $o ) > set_HF_Mirabelle_hf ).
thf(sy_c_Set_Ois__empty_001t__HF____Mirabelle____glliljednj__Ohf,type,
is_emp566801209lle_hf: set_HF_Mirabelle_hf > $o ).
thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_v_A,type,
a: hF_Mirabelle_hf ).
thf(sy_v_P,type,
p: $o ).
% Relevant facts (351)
thf(fact_0_hf__equalityI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A )
= ( hF_Mirabelle_hmem @ X @ B ) )
=> ( A = B ) ) ).
% hf_equalityI
thf(fact_1_hbexI,axiom,
! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( P @ X2 )
=> ( ( hF_Mirabelle_hmem @ X2 @ A2 )
=> ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% hbexI
thf(fact_2_hbexE,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_HBex @ A2 @ P )
=> ~ ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A2 )
=> ~ ( P @ X ) ) ) ).
% hbexE
thf(fact_3_hf__ext,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A3 )
= ( hF_Mirabelle_hmem @ X3 @ B2 ) ) ) ) ).
% hf_ext
thf(fact_4_hmem__ne,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% hmem_ne
thf(fact_5_HBex__def,axiom,
( hF_Mirabelle_HBex
= ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
? [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A4 )
& ( P2 @ X3 ) ) ) ) ).
% HBex_def
thf(fact_6_hbex__cong,axiom,
! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
( ( A2 = A5 )
=> ( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A5 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ( hF_Mirabelle_HBex @ A2 @ P )
= ( hF_Mirabelle_HBex @ A5 @ P3 ) ) ) ) ).
% hbex_cong
thf(fact_7_rev__hbexI,axiom,
! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% rev_hbexI
thf(fact_8_replacement,axiom,
! [X2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ X2 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X2 )
& ( R @ U2 @ V3 ) ) ) ) ) ).
% replacement
thf(fact_9_binary__union,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X2 )
| ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% binary_union
thf(fact_10_hmem__not__sym,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
~ ( ( hF_Mirabelle_hmem @ X2 @ Y2 )
& ( hF_Mirabelle_hmem @ Y2 @ X2 ) ) ).
% hmem_not_sym
thf(fact_11_union__of__set,axiom,
! [X2: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ X2 )
& ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% union_of_set
thf(fact_12_comprehension,axiom,
! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X2 )
& ( P @ U3 ) ) ) ).
% comprehension
thf(fact_13_hmem__not__refl,axiom,
! [X2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X2 @ X2 ) ).
% hmem_not_refl
thf(fact_14_replacement__fun,axiom,
! [X2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X2 )
& ( V3
= ( F @ U2 ) ) ) ) ) ).
% replacement_fun
thf(fact_15_HF__Mirabelle__glliljednj_ObexCI,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_HBall @ A2
@ ^ [X3: hF_Mirabelle_hf] :
~ ( P @ X3 ) )
=> ( P @ A ) )
=> ( ( hF_Mirabelle_hmem @ A @ A2 )
=> ( hF_Mirabelle_HBex @ A2 @ P ) ) ) ).
% HF_Mirabelle_glliljednj.bexCI
thf(fact_16_hballI,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A2 )
=> ( P @ X ) )
=> ( hF_Mirabelle_HBall @ A2 @ P ) ) ).
% hballI
thf(fact_17_hball__triv,axiom,
! [A2: hF_Mirabelle_hf,P: $o] :
( ( hF_Mirabelle_HBall @ A2
@ ^ [X3: hF_Mirabelle_hf] : P )
= ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ A2 )
=> P ) ) ).
% hball_triv
thf(fact_18_PrimReplace__iff,axiom,
! [A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ A2 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A2 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A2 )
& ( R @ U2 @ V4 ) ) ) ) ) ).
% PrimReplace_iff
thf(fact_19_HCollect__iff,axiom,
! [X2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mir818139703ollect @ P @ A2 ) )
= ( ( P @ X2 )
& ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% HCollect_iff
thf(fact_20_Replace__iff,axiom,
! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A2 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A2 )
& ( R @ U2 @ V4 )
& ! [Y3: hF_Mirabelle_hf] :
( ( R @ U2 @ Y3 )
=> ( Y3 = V4 ) ) ) ) ) ).
% Replace_iff
thf(fact_21_HUnion__iff,axiom,
! [X2: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HUnion @ A2 ) )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A2 )
& ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ).
% HUnion_iff
thf(fact_22_hballE,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HBall @ A2 @ P )
=> ( ~ ( P @ X2 )
=> ~ ( hF_Mirabelle_hmem @ X2 @ A2 ) ) ) ).
% hballE
thf(fact_23_hbspec,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HBall @ A2 @ P )
=> ( ( hF_Mirabelle_hmem @ X2 @ A2 )
=> ( P @ X2 ) ) ) ).
% hbspec
thf(fact_24_HCollectE,axiom,
! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) )
=> ~ ( ( hF_Mirabelle_hmem @ A @ A2 )
=> ~ ( P @ A ) ) ) ).
% HCollectE
thf(fact_25_HCollectI,axiom,
! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ A @ A2 )
=> ( ( P @ A )
=> ( hF_Mirabelle_hmem @ A @ ( hF_Mir818139703ollect @ P @ A2 ) ) ) ) ).
% HCollectI
thf(fact_26_hball__cong,axiom,
! [A2: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,P3: hF_Mirabelle_hf > $o] :
( ( A2 = A5 )
=> ( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A5 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ( hF_Mirabelle_HBall @ A2 @ P )
= ( hF_Mirabelle_HBall @ A5 @ P3 ) ) ) ) ).
% hball_cong
thf(fact_27_hmem__Sup__ne,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ Y2 )
=> ( ( hF_Mirabelle_HUnion @ X2 )
!= Y2 ) ) ).
% hmem_Sup_ne
thf(fact_28_Replace__cong,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( A2 = B3 )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B3 )
=> ( ( P @ X @ Y4 )
= ( Q @ X @ Y4 ) ) )
=> ( ( hF_Mirabelle_Replace @ A2 @ P )
= ( hF_Mirabelle_Replace @ B3 @ Q ) ) ) ) ).
% Replace_cong
thf(fact_29_HBall__def,axiom,
( hF_Mirabelle_HBall
= ( ^ [A4: hF_Mirabelle_hf,P2: hF_Mirabelle_hf > $o] :
! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A4 )
=> ( P2 @ X3 ) ) ) ) ).
% HBall_def
thf(fact_30_Replace__def,axiom,
( hF_Mirabelle_Replace
= ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( hF_Mir1248913145eplace @ A4
@ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ? [Z3: hF_Mirabelle_hf] :
( ( R2 @ X3 @ Z3 )
& ! [Aa: hF_Mirabelle_hf] :
( ( R2 @ X3 @ Aa )
=> ( Aa = Z3 ) ) )
& ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% Replace_def
thf(fact_31_RepFun__def,axiom,
( hF_Mirabelle_RepFun
= ( ^ [A4: hF_Mirabelle_hf,F2: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( hF_Mirabelle_Replace @ A4
@ ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( Y3
= ( F2 @ X3 ) ) ) ) ) ).
% RepFun_def
thf(fact_32_minus__hf__def,axiom,
( minus_1232880740lle_hf
= ( ^ [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( hF_Mir818139703ollect
@ ^ [X3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X3 @ B4 )
@ A4 ) ) ) ).
% minus_hf_def
thf(fact_33_inf__hf__def,axiom,
( inf_in956532509lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( hF_Mir818139703ollect
@ ^ [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ B2 )
@ A3 ) ) ) ).
% inf_hf_def
thf(fact_34_HInter__def,axiom,
( hF_Mirabelle_HInter
= ( ^ [A4: hF_Mirabelle_hf] :
( hF_Mir818139703ollect
@ ^ [X3: hF_Mirabelle_hf] :
! [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A4 )
=> ( hF_Mirabelle_hmem @ X3 @ Y3 ) )
@ ( hF_Mirabelle_HUnion @ A4 ) ) ) ) ).
% HInter_def
thf(fact_35_RepFun__iff,axiom,
! [V4: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A2 @ F ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A2 )
& ( V4
= ( F @ U2 ) ) ) ) ) ).
% RepFun_iff
thf(fact_36_HCollect__hempty,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HCollect_hempty
thf(fact_37_Replace__0,axiom,
! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R )
= zero_z189798548lle_hf ) ).
% Replace_0
thf(fact_38_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B2 ) ) ) ) ).
% hmem_def
thf(fact_39_triv__RepFun,axiom,
! [A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ A2
@ ^ [X3: hF_Mirabelle_hf] : X3 )
= A2 ) ).
% triv_RepFun
thf(fact_40_hinter__iff,axiom,
! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
= ( ( hF_Mirabelle_hmem @ U4 @ X2 )
& ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% hinter_iff
thf(fact_41_hdiff__iff,axiom,
! [U4: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U4 @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
= ( ( hF_Mirabelle_hmem @ U4 @ X2 )
& ~ ( hF_Mirabelle_hmem @ U4 @ Y2 ) ) ) ).
% hdiff_iff
thf(fact_42_hinter__hempty__right,axiom,
! [A2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A2 @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% hinter_hempty_right
thf(fact_43_hinter__hempty__left,axiom,
! [A2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A2 )
= zero_z189798548lle_hf ) ).
% hinter_hempty_left
thf(fact_44_zero__hdiff,axiom,
! [X2: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X2 )
= zero_z189798548lle_hf ) ).
% zero_hdiff
thf(fact_45_mem__Collect__eq,axiom,
! [A: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( member1367349282lle_hf @ A @ ( collec2046588256lle_hf @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec2046588256lle_hf @ P )
= ( collec2046588256lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_48_hdiff__zero,axiom,
! [X2: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ X2 @ zero_z189798548lle_hf )
= X2 ) ).
% hdiff_zero
thf(fact_49_RepFun__0,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
= zero_z189798548lle_hf ) ).
% RepFun_0
thf(fact_50_HUnion__hempty,axiom,
( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HUnion_hempty
thf(fact_51_HInter__hempty,axiom,
( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HInter_hempty
thf(fact_52_HInter__iff,axiom,
! [A2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( A2 != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_hmem @ X2 @ ( hF_Mirabelle_HInter @ A2 ) )
= ( ! [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A2 )
=> ( hF_Mirabelle_hmem @ X2 @ Y3 ) ) ) ) ) ).
% HInter_iff
thf(fact_53_foundation,axiom,
! [Z4: hF_Mirabelle_hf] :
( ( Z4 != zero_z189798548lle_hf )
=> ? [W: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ W @ Z4 )
& ( ( inf_in956532509lle_hf @ W @ Z4 )
= zero_z189798548lle_hf ) ) ) ).
% foundation
thf(fact_54_hemptyE,axiom,
! [A: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% hemptyE
thf(fact_55_hempty__iff,axiom,
! [Z4: hF_Mirabelle_hf] :
( ( Z4 = zero_z189798548lle_hf )
= ( ! [X3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X3 @ Z4 ) ) ) ).
% hempty_iff
thf(fact_56_hmem__hempty,axiom,
! [A: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% hmem_hempty
thf(fact_57_RepFun__cong,axiom,
! [A2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( A2 = B3 )
=> ( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hF_Mirabelle_RepFun @ A2 @ F )
= ( hF_Mirabelle_RepFun @ B3 @ G ) ) ) ) ).
% RepFun_cong
thf(fact_58_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_59_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_60_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_61_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_62_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_63_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_64_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_65_inf__right__idem,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Y2 )
= ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_66_inf__right__idem,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Y2 )
= ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_67_inf__right__idem,axiom,
! [X2: int,Y2: int] :
( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_int @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_68_inf__right__idem,axiom,
! [X2: nat,Y2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_nat @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_69_inf__right__idem,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Y2 )
= ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_70_inf_Oright__idem,axiom,
! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ B )
= ( inf_in923488851lle_hf @ A @ B ) ) ).
% inf.right_idem
thf(fact_71_inf_Oright__idem,axiom,
! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ B )
= ( inf_in307783154e_hf_o @ A @ B ) ) ).
% inf.right_idem
thf(fact_72_inf_Oright__idem,axiom,
! [A: int,B: int] :
( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ B )
= ( inf_inf_int @ A @ B ) ) ).
% inf.right_idem
thf(fact_73_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_74_inf_Oright__idem,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B )
= ( inf_in956532509lle_hf @ A @ B ) ) ).
% inf.right_idem
thf(fact_75_inf__left__idem,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
= ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_76_inf__left__idem,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
= ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_77_inf__left__idem,axiom,
! [X2: int,Y2: int] :
( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
= ( inf_inf_int @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_78_inf__left__idem,axiom,
! [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
= ( inf_inf_nat @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_79_inf__left__idem,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
= ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_80_minus__apply,axiom,
( minus_1668263787e_hf_o
= ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% minus_apply
thf(fact_81_inf__apply,axiom,
( inf_in307783154e_hf_o
= ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_apply
thf(fact_82_inf_Oidem,axiom,
! [A: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A @ A )
= A ) ).
% inf.idem
thf(fact_83_inf_Oidem,axiom,
! [A: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ A @ A )
= A ) ).
% inf.idem
thf(fact_84_inf_Oidem,axiom,
! [A: int] :
( ( inf_inf_int @ A @ A )
= A ) ).
% inf.idem
thf(fact_85_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_86_inf_Oidem,axiom,
! [A: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A @ A )
= A ) ).
% inf.idem
thf(fact_87_inf__idem,axiom,
! [X2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_88_inf__idem,axiom,
! [X2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_89_inf__idem,axiom,
! [X2: int] :
( ( inf_inf_int @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_90_inf__idem,axiom,
! [X2: nat] :
( ( inf_inf_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_91_inf__idem,axiom,
! [X2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_92_inf_Oleft__idem,axiom,
! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ A @ B ) )
= ( inf_in923488851lle_hf @ A @ B ) ) ).
% inf.left_idem
thf(fact_93_inf_Oleft__idem,axiom,
! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ A @ B ) )
= ( inf_in307783154e_hf_o @ A @ B ) ) ).
% inf.left_idem
thf(fact_94_inf_Oleft__idem,axiom,
! [A: int,B: int] :
( ( inf_inf_int @ A @ ( inf_inf_int @ A @ B ) )
= ( inf_inf_int @ A @ B ) ) ).
% inf.left_idem
thf(fact_95_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_96_inf_Oleft__idem,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ A @ B ) )
= ( inf_in956532509lle_hf @ A @ B ) ) ).
% inf.left_idem
thf(fact_97_hfset__hdiff,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hfset @ ( minus_1232880740lle_hf @ X2 @ Y2 ) )
= ( minus_1450406810lle_hf @ ( hF_Mirabelle_hfset @ X2 ) @ ( hF_Mirabelle_hfset @ Y2 ) ) ) ).
% hfset_hdiff
thf(fact_98_zero__reorient,axiom,
! [X2: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X2 )
= ( X2 = zero_z189798548lle_hf ) ) ).
% zero_reorient
thf(fact_99_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_100_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_101_fun__diff__def,axiom,
( minus_1668263787e_hf_o
= ( ^ [A4: hF_Mirabelle_hf > $o,B4: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( minus_minus_o @ ( A4 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).
% fun_diff_def
thf(fact_102_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_103_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_104_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_105_inf__sup__aci_I4_J,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ X2 @ Y2 ) )
= ( inf_in923488851lle_hf @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_106_inf__sup__aci_I4_J,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) )
= ( inf_in307783154e_hf_o @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_107_inf__sup__aci_I4_J,axiom,
! [X2: int,Y2: int] :
( ( inf_inf_int @ X2 @ ( inf_inf_int @ X2 @ Y2 ) )
= ( inf_inf_int @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_108_inf__sup__aci_I4_J,axiom,
! [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y2 ) )
= ( inf_inf_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_109_inf__sup__aci_I4_J,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ X2 @ Y2 ) )
= ( inf_in956532509lle_hf @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_110_inf__sup__aci_I3_J,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
= ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_111_inf__sup__aci_I3_J,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
= ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_112_inf__sup__aci_I3_J,axiom,
! [X2: int,Y2: int,Z4: int] :
( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
= ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_113_inf__sup__aci_I3_J,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
= ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_114_inf__sup__aci_I3_J,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
= ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% inf_sup_aci(3)
thf(fact_115_inf__sup__aci_I2_J,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
= ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_116_inf__sup__aci_I2_J,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
= ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_117_inf__sup__aci_I2_J,axiom,
! [X2: int,Y2: int,Z4: int] :
( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
= ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_118_inf__sup__aci_I2_J,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
= ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_119_inf__sup__aci_I2_J,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
= ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% inf_sup_aci(2)
thf(fact_120_inf__sup__aci_I1_J,axiom,
( inf_in923488851lle_hf
= ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_121_inf__sup__aci_I1_J,axiom,
( inf_in307783154e_hf_o
= ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_122_inf__sup__aci_I1_J,axiom,
( inf_inf_int
= ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_123_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_124_inf__sup__aci_I1_J,axiom,
( inf_in956532509lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_125_inf__fun__def,axiom,
( inf_in307783154e_hf_o
= ( ^ [F2: hF_Mirabelle_hf > $o,G2: hF_Mirabelle_hf > $o,X3: hF_Mirabelle_hf] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_fun_def
thf(fact_126_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( A2
= ( inf_in923488851lle_hf @ K @ A ) )
=> ( ( inf_in923488851lle_hf @ A2 @ B )
= ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_127_boolean__algebra__cancel_Oinf1,axiom,
! [A2: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
( ( A2
= ( inf_in307783154e_hf_o @ K @ A ) )
=> ( ( inf_in307783154e_hf_o @ A2 @ B )
= ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_128_boolean__algebra__cancel_Oinf1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( inf_inf_int @ K @ A ) )
=> ( ( inf_inf_int @ A2 @ B )
= ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_129_boolean__algebra__cancel_Oinf1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A2 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_130_boolean__algebra__cancel_Oinf1,axiom,
! [A2: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A2
= ( inf_in956532509lle_hf @ K @ A ) )
=> ( ( inf_in956532509lle_hf @ A2 @ B )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_131_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( B3
= ( inf_in923488851lle_hf @ K @ B ) )
=> ( ( inf_in923488851lle_hf @ A @ B3 )
= ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_132_boolean__algebra__cancel_Oinf2,axiom,
! [B3: hF_Mirabelle_hf > $o,K: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o] :
( ( B3
= ( inf_in307783154e_hf_o @ K @ B ) )
=> ( ( inf_in307783154e_hf_o @ A @ B3 )
= ( inf_in307783154e_hf_o @ K @ ( inf_in307783154e_hf_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_133_boolean__algebra__cancel_Oinf2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( inf_inf_int @ K @ B ) )
=> ( ( inf_inf_int @ A @ B3 )
= ( inf_inf_int @ K @ ( inf_inf_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_134_boolean__algebra__cancel_Oinf2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B3 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_135_boolean__algebra__cancel_Oinf2,axiom,
! [B3: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( B3
= ( inf_in956532509lle_hf @ K @ B ) )
=> ( ( inf_in956532509lle_hf @ A @ B3 )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_136_inf_Oassoc,axiom,
! [A: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A @ B ) @ C )
= ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% inf.assoc
thf(fact_137_inf_Oassoc,axiom,
! [A: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ A @ B ) @ C )
= ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% inf.assoc
thf(fact_138_inf_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( inf_inf_int @ ( inf_inf_int @ A @ B ) @ C )
= ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% inf.assoc
thf(fact_139_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_140_inf_Oassoc,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C )
= ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% inf.assoc
thf(fact_141_inf__assoc,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X2 @ Y2 ) @ Z4 )
= ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) ) ) ).
% inf_assoc
thf(fact_142_inf__assoc,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ ( inf_in307783154e_hf_o @ X2 @ Y2 ) @ Z4 )
= ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) ) ) ).
% inf_assoc
thf(fact_143_inf__assoc,axiom,
! [X2: int,Y2: int,Z4: int] :
( ( inf_inf_int @ ( inf_inf_int @ X2 @ Y2 ) @ Z4 )
= ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) ) ) ).
% inf_assoc
thf(fact_144_inf__assoc,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Z4 )
= ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) ) ) ).
% inf_assoc
thf(fact_145_inf__assoc,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ Z4 )
= ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) ) ) ).
% inf_assoc
thf(fact_146_inf_Ocommute,axiom,
( inf_in923488851lle_hf
= ( ^ [A3: set_HF_Mirabelle_hf,B2: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_147_inf_Ocommute,axiom,
( inf_in307783154e_hf_o
= ( ^ [A3: hF_Mirabelle_hf > $o,B2: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_148_inf_Ocommute,axiom,
( inf_inf_int
= ( ^ [A3: int,B2: int] : ( inf_inf_int @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_149_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A3: nat,B2: nat] : ( inf_inf_nat @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_150_inf_Ocommute,axiom,
( inf_in956532509lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_151_inf__commute,axiom,
( inf_in923488851lle_hf
= ( ^ [X3: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_152_inf__commute,axiom,
( inf_in307783154e_hf_o
= ( ^ [X3: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] : ( inf_in307783154e_hf_o @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_153_inf__commute,axiom,
( inf_inf_int
= ( ^ [X3: int,Y3: int] : ( inf_inf_int @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_154_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_155_inf__commute,axiom,
( inf_in956532509lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_156_inf_Oleft__commute,axiom,
! [B: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A @ C ) )
= ( inf_in923488851lle_hf @ A @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_157_inf_Oleft__commute,axiom,
! [B: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf > $o,C: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ B @ ( inf_in307783154e_hf_o @ A @ C ) )
= ( inf_in307783154e_hf_o @ A @ ( inf_in307783154e_hf_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_158_inf_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( inf_inf_int @ B @ ( inf_inf_int @ A @ C ) )
= ( inf_inf_int @ A @ ( inf_inf_int @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_159_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_160_inf_Oleft__commute,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A @ C ) )
= ( inf_in956532509lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_161_inf__left__commute,axiom,
! [X2: set_HF_Mirabelle_hf,Y2: set_HF_Mirabelle_hf,Z4: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ ( inf_in923488851lle_hf @ Y2 @ Z4 ) )
= ( inf_in923488851lle_hf @ Y2 @ ( inf_in923488851lle_hf @ X2 @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_162_inf__left__commute,axiom,
! [X2: hF_Mirabelle_hf > $o,Y2: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ ( inf_in307783154e_hf_o @ Y2 @ Z4 ) )
= ( inf_in307783154e_hf_o @ Y2 @ ( inf_in307783154e_hf_o @ X2 @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_163_inf__left__commute,axiom,
! [X2: int,Y2: int,Z4: int] :
( ( inf_inf_int @ X2 @ ( inf_inf_int @ Y2 @ Z4 ) )
= ( inf_inf_int @ Y2 @ ( inf_inf_int @ X2 @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_164_inf__left__commute,axiom,
! [X2: nat,Y2: nat,Z4: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z4 ) )
= ( inf_inf_nat @ Y2 @ ( inf_inf_nat @ X2 @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_165_inf__left__commute,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z4 ) )
= ( inf_in956532509lle_hf @ Y2 @ ( inf_in956532509lle_hf @ X2 @ Z4 ) ) ) ).
% inf_left_commute
thf(fact_166_eq__iff__diff__eq__0,axiom,
( ( ^ [Y: int,Z: int] : Y = Z )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_167_HInter__hinsert,axiom,
! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( A2 != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A2 ) )
= ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A2 ) ) ) ) ).
% HInter_hinsert
thf(fact_168_HCollect__def,axiom,
( hF_Mir818139703ollect
= ( ^ [P2: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
( the_HF_Mirabelle_hf
@ ^ [Z3: hF_Mirabelle_hf] :
! [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ Z3 )
= ( ( P2 @ U2 )
& ( hF_Mirabelle_hmem @ U2 @ A4 ) ) ) ) ) ) ).
% HCollect_def
thf(fact_169_HF__hfset,axiom,
! [A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
= A ) ).
% HF_hfset
thf(fact_170_zero__notin__hpair,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) ) ).
% zero_notin_hpair
thf(fact_171_Abs__hf__0,axiom,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= zero_z189798548lle_hf ) ).
% Abs_hf_0
thf(fact_172_hfset__0,axiom,
( ( hF_Mirabelle_hfset @ zero_z189798548lle_hf )
= bot_bo53200981lle_hf ) ).
% hfset_0
thf(fact_173_hpair__iff,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A @ B )
= ( hF_Mirabelle_hpair @ A6 @ B5 ) )
= ( ( A = A6 )
& ( B = B5 ) ) ) ).
% hpair_iff
thf(fact_174_inf__bot__right,axiom,
! [X2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X2 @ bot_bo1263054448e_hf_o )
= bot_bo1263054448e_hf_o ) ).
% inf_bot_right
thf(fact_175_inf__bot__right,axiom,
! [X2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X2 @ bot_bo53200981lle_hf )
= bot_bo53200981lle_hf ) ).
% inf_bot_right
thf(fact_176_inf__bot__right,axiom,
! [X2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ bot_bo1001194783lle_hf )
= bot_bo1001194783lle_hf ) ).
% inf_bot_right
thf(fact_177_inf__bot__left,axiom,
! [X2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ bot_bo1263054448e_hf_o @ X2 )
= bot_bo1263054448e_hf_o ) ).
% inf_bot_left
thf(fact_178_inf__bot__left,axiom,
! [X2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ X2 )
= bot_bo53200981lle_hf ) ).
% inf_bot_left
thf(fact_179_inf__bot__left,axiom,
! [X2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ bot_bo1001194783lle_hf @ X2 )
= bot_bo1001194783lle_hf ) ).
% inf_bot_left
thf(fact_180_hmem__hinsert,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ A @ ( hF_Mirabelle_hinsert @ B @ C ) )
= ( ( A = B )
| ( hF_Mirabelle_hmem @ A @ C ) ) ) ).
% hmem_hinsert
thf(fact_181_singleton__eq__iff,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf )
= ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
= ( A = B ) ) ).
% singleton_eq_iff
thf(fact_182_RepFun__hinsert,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A @ B ) @ F )
= ( hF_Mirabelle_hinsert @ ( F @ A ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% RepFun_hinsert
thf(fact_183_hpair__def,axiom,
( hF_Mirabelle_hpair
= ( ^ [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 ) ) ) ) ).
% hpair_def
thf(fact_184_hpair__def_H,axiom,
( hF_Mirabelle_hpair
= ( ^ [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 ) ) ) ) ).
% hpair_def'
thf(fact_185_Zero__hf__def,axiom,
( zero_z189798548lle_hf
= ( hF_Mirabelle_HF @ bot_bo53200981lle_hf ) ) ).
% Zero_hf_def
thf(fact_186_hpair__neq__snd,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A @ B )
!= B ) ).
% hpair_neq_snd
thf(fact_187_hpair__neq__fst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A @ B )
!= A ) ).
% hpair_neq_fst
thf(fact_188_hpair__inject,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A6: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A @ B )
= ( hF_Mirabelle_hpair @ A6 @ B5 ) )
=> ~ ( ( A = A6 )
=> ( B != B5 ) ) ) ).
% hpair_inject
thf(fact_189_hinsert__commute,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ X2 @ ( hF_Mirabelle_hinsert @ Y2 @ Z4 ) )
= ( hF_Mirabelle_hinsert @ Y2 @ ( hF_Mirabelle_hinsert @ X2 @ Z4 ) ) ) ).
% hinsert_commute
thf(fact_190_hinsert__iff,axiom,
! [Z4: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( Z4
= ( hF_Mirabelle_hinsert @ Y2 @ X2 ) )
= ( ! [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ Z4 )
= ( ( hF_Mirabelle_hmem @ U2 @ X2 )
| ( U2 = Y2 ) ) ) ) ) ).
% hinsert_iff
thf(fact_191_hinsert__nonempty,axiom,
! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ A @ A2 )
!= zero_z189798548lle_hf ) ).
% hinsert_nonempty
thf(fact_192_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ A @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
= ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% HF_Mirabelle_glliljednj.doubleton_eq_iff
thf(fact_193_hf__induct__ax,axiom,
! [P: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( P @ ( hF_Mirabelle_hinsert @ Y4 @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% hf_induct_ax
thf(fact_194_hpair__nonzero,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ X2 @ Y2 )
!= zero_z189798548lle_hf ) ).
% hpair_nonzero
thf(fact_195_hf__cases,axiom,
! [Y2: hF_Mirabelle_hf] :
( ( Y2 != zero_z189798548lle_hf )
=> ~ ! [A7: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
( ( Y2
= ( hF_Mirabelle_hinsert @ A7 @ B6 ) )
=> ( hF_Mirabelle_hmem @ A7 @ B6 ) ) ) ).
% hf_cases
thf(fact_196_hf__induct,axiom,
! [P: hF_Mirabelle_hf > $o,Z4: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( P @ X )
=> ( ( P @ Y4 )
=> ( ~ ( hF_Mirabelle_hmem @ X @ Y4 )
=> ( P @ ( hF_Mirabelle_hinsert @ X @ Y4 ) ) ) ) )
=> ( P @ Z4 ) ) ) ).
% hf_induct
thf(fact_197_hinter__hinsert__right,axiom,
! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
= ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( inf_in956532509lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) )
= ( inf_in956532509lle_hf @ B3 @ A2 ) ) ) ) ).
% hinter_hinsert_right
thf(fact_198_hinter__hinsert__left,axiom,
! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
= ( hF_Mirabelle_hinsert @ X2 @ ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
= ( inf_in956532509lle_hf @ A2 @ B3 ) ) ) ) ).
% hinter_hinsert_left
thf(fact_199_hinsert__hdiff__if,axiom,
! [X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
= ( minus_1232880740lle_hf @ A2 @ B3 ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X2 @ B3 )
=> ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X2 @ A2 ) @ B3 )
= ( hF_Mirabelle_hinsert @ X2 @ ( minus_1232880740lle_hf @ A2 @ B3 ) ) ) ) ) ).
% hinsert_hdiff_if
thf(fact_200_hdiff__insert,axiom,
! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ A2 @ ( hF_Mirabelle_hinsert @ A @ B3 ) )
= ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A2 @ B3 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% hdiff_insert
thf(fact_201_Diff__disjoint,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ A2 ) )
= bot_bo53200981lle_hf ) ).
% Diff_disjoint
thf(fact_202_Diff__cancel,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A2 @ A2 )
= bot_bo53200981lle_hf ) ).
% Diff_cancel
thf(fact_203_empty__Diff,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ bot_bo53200981lle_hf @ A2 )
= bot_bo53200981lle_hf ) ).
% empty_Diff
thf(fact_204_Diff__empty,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A2 @ bot_bo53200981lle_hf )
= A2 ) ).
% Diff_empty
thf(fact_205_empty__iff,axiom,
! [C: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ C @ bot_bo53200981lle_hf ) ).
% empty_iff
thf(fact_206_all__not__in__conv,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( ! [X3: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ X3 @ A2 ) )
= ( A2 = bot_bo53200981lle_hf ) ) ).
% all_not_in_conv
thf(fact_207_Collect__empty__eq,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( ( collec2046588256lle_hf @ P )
= bot_bo53200981lle_hf )
= ( ! [X3: hF_Mirabelle_hf] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_208_empty__Collect__eq,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( bot_bo53200981lle_hf
= ( collec2046588256lle_hf @ P ) )
= ( ! [X3: hF_Mirabelle_hf] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_209_Int__iff,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
= ( ( member1367349282lle_hf @ C @ A2 )
& ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_210_IntI,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ A2 )
=> ( ( member1367349282lle_hf @ C @ B3 )
=> ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_211_DiffI,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ A2 )
=> ( ~ ( member1367349282lle_hf @ C @ B3 )
=> ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_212_Diff__iff,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
= ( ( member1367349282lle_hf @ C @ A2 )
& ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_213_Diff__idemp,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ B3 )
= ( minus_1450406810lle_hf @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_214_Int__left__commute,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
= ( inf_in923488851lle_hf @ B3 @ ( inf_in923488851lle_hf @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_215_Int__left__absorb,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
= ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_216_Collect__conj__eq,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in923488851lle_hf @ ( collec2046588256lle_hf @ P ) @ ( collec2046588256lle_hf @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_217_inf__set__def,axiom,
( inf_in923488851lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( collec2046588256lle_hf
@ ( inf_in307783154e_hf_o
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_218_Int__commute,axiom,
( inf_in923488851lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_219_Int__Collect,axiom,
! [X2: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( member1367349282lle_hf @ X2 @ ( inf_in923488851lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) ) )
= ( ( member1367349282lle_hf @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_220_Int__absorb,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_221_Int__assoc,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
= ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_222_Int__def,axiom,
( inf_in923488851lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A4 )
& ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_223_IntD2,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
=> ( member1367349282lle_hf @ C @ B3 ) ) ).
% IntD2
thf(fact_224_IntD1,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
=> ( member1367349282lle_hf @ C @ A2 ) ) ).
% IntD1
thf(fact_225_IntE,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A2 @ B3 ) )
=> ~ ( ( member1367349282lle_hf @ C @ A2 )
=> ~ ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% IntE
thf(fact_226_bot__set__def,axiom,
( bot_bo53200981lle_hf
= ( collec2046588256lle_hf @ bot_bo1263054448e_hf_o ) ) ).
% bot_set_def
thf(fact_227_bot__hf__def,axiom,
bot_bo1001194783lle_hf = zero_z189798548lle_hf ).
% bot_hf_def
thf(fact_228_emptyE,axiom,
! [A: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ A @ bot_bo53200981lle_hf ) ).
% emptyE
thf(fact_229_equals0D,axiom,
! [A2: set_HF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( A2 = bot_bo53200981lle_hf )
=> ~ ( member1367349282lle_hf @ A @ A2 ) ) ).
% equals0D
thf(fact_230_equals0I,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ! [Y4: hF_Mirabelle_hf] :
~ ( member1367349282lle_hf @ Y4 @ A2 )
=> ( A2 = bot_bo53200981lle_hf ) ) ).
% equals0I
thf(fact_231_ex__in__conv,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( ? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A2 ) )
= ( A2 != bot_bo53200981lle_hf ) ) ).
% ex_in_conv
thf(fact_232_Int__emptyI,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ! [X: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X @ A2 )
=> ~ ( member1367349282lle_hf @ X @ B3 ) )
=> ( ( inf_in923488851lle_hf @ A2 @ B3 )
= bot_bo53200981lle_hf ) ) ).
% Int_emptyI
thf(fact_233_disjoint__iff,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( ( inf_in923488851lle_hf @ A2 @ B3 )
= bot_bo53200981lle_hf )
= ( ! [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A2 )
=> ~ ( member1367349282lle_hf @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_234_Int__empty__left,axiom,
! [B3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ bot_bo53200981lle_hf @ B3 )
= bot_bo53200981lle_hf ) ).
% Int_empty_left
thf(fact_235_Int__empty__right,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ bot_bo53200981lle_hf )
= bot_bo53200981lle_hf ) ).
% Int_empty_right
thf(fact_236_disjoint__iff__not__equal,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( ( inf_in923488851lle_hf @ A2 @ B3 )
= bot_bo53200981lle_hf )
= ( ! [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A2 )
=> ! [Y3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ Y3 @ B3 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_237_DiffE,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
=> ~ ( ( member1367349282lle_hf @ C @ A2 )
=> ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% DiffE
thf(fact_238_DiffD1,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
=> ( member1367349282lle_hf @ C @ A2 ) ) ).
% DiffD1
thf(fact_239_DiffD2,axiom,
! [C: hF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
=> ~ ( member1367349282lle_hf @ C @ B3 ) ) ).
% DiffD2
thf(fact_240_Int__Diff,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ B3 ) @ C2 )
= ( inf_in923488851lle_hf @ A2 @ ( minus_1450406810lle_hf @ B3 @ C2 ) ) ) ).
% Int_Diff
thf(fact_241_Diff__Int2,axiom,
! [A2: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) )
= ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_242_Diff__Diff__Int,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( minus_1450406810lle_hf @ A2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
= ( inf_in923488851lle_hf @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_243_Diff__Int__distrib,axiom,
! [C2: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ C2 @ ( minus_1450406810lle_hf @ A2 @ B3 ) )
= ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ C2 @ A2 ) @ ( inf_in923488851lle_hf @ C2 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_244_Diff__Int__distrib2,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( minus_1450406810lle_hf @ A2 @ B3 ) @ C2 )
= ( minus_1450406810lle_hf @ ( inf_in923488851lle_hf @ A2 @ C2 ) @ ( inf_in923488851lle_hf @ B3 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_245_empty__def,axiom,
( bot_bo53200981lle_hf
= ( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] : $false ) ) ).
% empty_def
thf(fact_246_set__diff__eq,axiom,
( minus_1450406810lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X3 @ A4 )
& ~ ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_247_minus__set__def,axiom,
( minus_1450406810lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( collec2046588256lle_hf
@ ( minus_1668263787e_hf_o
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_248_Diff__triv,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( ( inf_in923488851lle_hf @ A2 @ B3 )
= bot_bo53200981lle_hf )
=> ( ( minus_1450406810lle_hf @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_249_hfst__conv,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
= A ) ).
% hfst_conv
thf(fact_250_hsnd__conv,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
= B ) ).
% hsnd_conv
thf(fact_251_the__equality,axiom,
! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
( ( P @ A )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( X = A ) )
=> ( ( the_HF_Mirabelle_hf @ P )
= A ) ) ) ).
% the_equality
thf(fact_252_the__eq__trivial,axiom,
! [A: hF_Mirabelle_hf] :
( ( the_HF_Mirabelle_hf
@ ^ [X3: hF_Mirabelle_hf] : X3 = A )
= A ) ).
% the_eq_trivial
thf(fact_253_inf1I,axiom,
! [A2: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf,B3: hF_Mirabelle_hf > $o] :
( ( A2 @ X2 )
=> ( ( B3 @ X2 )
=> ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 ) ) ) ).
% inf1I
thf(fact_254_the__sym__eq__trivial,axiom,
! [X2: hF_Mirabelle_hf] :
( ( the_HF_Mirabelle_hf
@ ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_255_inf1D2,axiom,
! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
=> ( B3 @ X2 ) ) ).
% inf1D2
thf(fact_256_inf1D1,axiom,
! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
=> ( A2 @ X2 ) ) ).
% inf1D1
thf(fact_257_inf1E,axiom,
! [A2: hF_Mirabelle_hf > $o,B3: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( inf_in307783154e_hf_o @ A2 @ B3 @ X2 )
=> ~ ( ( A2 @ X2 )
=> ~ ( B3 @ X2 ) ) ) ).
% inf1E
thf(fact_258_the1__equality,axiom,
! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
( ? [X4: hF_Mirabelle_hf] :
( ( P @ X4 )
& ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ( P @ A )
=> ( ( the_HF_Mirabelle_hf @ P )
= A ) ) ) ).
% the1_equality
thf(fact_259_the1I2,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ? [X4: hF_Mirabelle_hf] :
( ( P @ X4 )
& ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% the1I2
thf(fact_260_If__def,axiom,
( if_HF_Mirabelle_hf
= ( ^ [P2: $o,X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( the_HF_Mirabelle_hf
@ ^ [Z3: hF_Mirabelle_hf] :
( ( P2
=> ( Z3 = X3 ) )
& ( ~ P2
=> ( Z3 = Y3 ) ) ) ) ) ) ).
% If_def
thf(fact_261_theI2,axiom,
! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf,Q: hF_Mirabelle_hf > $o] :
( ( P @ A )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( X = A ) )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ) ).
% theI2
thf(fact_262_theI_H,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ? [X4: hF_Mirabelle_hf] :
( ( P @ X4 )
& ! [Y4: hF_Mirabelle_hf] :
( ( P @ Y4 )
=> ( Y4 = X4 ) ) )
=> ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ).
% theI'
thf(fact_263_theI,axiom,
! [P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
( ( P @ A )
=> ( ! [X: hF_Mirabelle_hf] :
( ( P @ X )
=> ( X = A ) )
=> ( P @ ( the_HF_Mirabelle_hf @ P ) ) ) ) ).
% theI
thf(fact_264_bot__apply,axiom,
( bot_bo1263054448e_hf_o
= ( ^ [X3: hF_Mirabelle_hf] : bot_bot_o ) ) ).
% bot_apply
thf(fact_265_inf__Int__eq,axiom,
! [R: set_HF_Mirabelle_hf,S: set_HF_Mirabelle_hf] :
( ( inf_in307783154e_hf_o
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ R )
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ S ) )
= ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ ( inf_in923488851lle_hf @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_266_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_267_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_268_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_269_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_270_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_271_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_272_bot__empty__eq,axiom,
( bot_bo1263054448e_hf_o
= ( ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ bot_bo53200981lle_hf ) ) ) ).
% bot_empty_eq
thf(fact_273_Collect__empty__eq__bot,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( ( collec2046588256lle_hf @ P )
= bot_bo53200981lle_hf )
= ( P = bot_bo1263054448e_hf_o ) ) ).
% Collect_empty_eq_bot
thf(fact_274_ord_OLeast__def,axiom,
( least_917110455lle_hf
= ( ^ [Less_eq: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,P2: hF_Mirabelle_hf > $o] :
( the_HF_Mirabelle_hf
@ ^ [X3: hF_Mirabelle_hf] :
( ( P2 @ X3 )
& ! [Y3: hF_Mirabelle_hf] :
( ( P2 @ Y3 )
=> ( Less_eq @ X3 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_275_PrimReplace__def,axiom,
( hF_Mir1248913145eplace
= ( ^ [A4: hF_Mirabelle_hf,R2: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( the_HF_Mirabelle_hf
@ ^ [Z3: hF_Mirabelle_hf] :
! [V5: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V5 @ Z3 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( R2 @ U2 @ V5 ) ) ) ) ) ) ) ).
% PrimReplace_def
thf(fact_276_hcard__0,axiom,
( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
= zero_zero_nat ) ).
% hcard_0
thf(fact_277_hfst__def,axiom,
( hF_Mirabelle_hfst
= ( ^ [P4: hF_Mirabelle_hf] :
( the_HF_Mirabelle_hf
@ ^ [X3: hF_Mirabelle_hf] :
? [Y3: hF_Mirabelle_hf] :
( P4
= ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% hfst_def
thf(fact_278_hsnd__def,axiom,
( hF_Mirabelle_hsnd
= ( ^ [P4: hF_Mirabelle_hf] :
( the_HF_Mirabelle_hf
@ ^ [Y3: hF_Mirabelle_hf] :
? [X3: hF_Mirabelle_hf] :
( P4
= ( hF_Mirabelle_hpair @ X3 @ Y3 ) ) ) ) ) ).
% hsnd_def
thf(fact_279_HUnion__def,axiom,
( hF_Mirabelle_HUnion
= ( ^ [A4: hF_Mirabelle_hf] :
( the_HF_Mirabelle_hf
@ ^ [Z3: hF_Mirabelle_hf] :
! [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ Z3 )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A4 )
& ( hF_Mirabelle_hmem @ U2 @ Y3 ) ) ) ) ) ) ) ).
% HUnion_def
thf(fact_280_hcard__hdiff1__less,axiom,
! [X2: hF_Mirabelle_hf,Z4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ Z4 )
=> ( ord_less_nat @ ( hF_Mirabelle_hcard @ ( minus_1232880740lle_hf @ Z4 @ ( hF_Mirabelle_hinsert @ X2 @ zero_z189798548lle_hf ) ) ) @ ( hF_Mirabelle_hcard @ Z4 ) ) ) ).
% hcard_hdiff1_less
thf(fact_281_Set_Ois__empty__def,axiom,
( is_emp566801209lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf] : A4 = bot_bo53200981lle_hf ) ) ).
% Set.is_empty_def
thf(fact_282_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_283_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_284_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_285_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_286_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_287_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_288_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_289_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_290_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_291_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_292_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_293_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_294_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_295_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_296_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_297_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_298_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_299_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_300_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_301_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_302_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_303_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_304_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_305_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_306_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_307_less__infI1,axiom,
! [A: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ X2 )
=> ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_308_less__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_nat @ A @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_309_less__infI1,axiom,
! [A: int,X2: int,B: int] :
( ( ord_less_int @ A @ X2 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_310_less__infI2,axiom,
! [B: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ X2 )
=> ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_311_less__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_nat @ B @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_312_less__infI2,axiom,
! [B: int,X2: int,A: int] :
( ( ord_less_int @ B @ X2 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_313_inf_Ostrict__boundedE,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
=> ~ ( ( ord_le1310584031lle_hf @ A @ B )
=> ~ ( ord_le1310584031lle_hf @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_314_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_315_inf_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_316_inf_Ostrict__order__iff,axiom,
( ord_le1310584031lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( A3
= ( inf_in956532509lle_hf @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_317_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_318_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( A3
= ( inf_inf_int @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_319_inf_Ostrict__coboundedI1,axiom,
! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ A @ C )
=> ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_320_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_321_inf_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_322_inf_Ostrict__coboundedI2,axiom,
! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le1310584031lle_hf @ B @ C )
=> ( ord_le1310584031lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_323_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_324_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_325_bot_Oextremum__strict,axiom,
! [A: set_HF_Mirabelle_hf] :
~ ( ord_le1344122901lle_hf @ A @ bot_bo53200981lle_hf ) ).
% bot.extremum_strict
thf(fact_326_bot_Oextremum__strict,axiom,
! [A: hF_Mirabelle_hf] :
~ ( ord_le1310584031lle_hf @ A @ bot_bo1001194783lle_hf ) ).
% bot.extremum_strict
thf(fact_327_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_328_bot_Onot__eq__extremum,axiom,
! [A: set_HF_Mirabelle_hf] :
( ( A != bot_bo53200981lle_hf )
= ( ord_le1344122901lle_hf @ bot_bo53200981lle_hf @ A ) ) ).
% bot.not_eq_extremum
thf(fact_329_bot_Onot__eq__extremum,axiom,
! [A: hF_Mirabelle_hf] :
( ( A != bot_bo1001194783lle_hf )
= ( ord_le1310584031lle_hf @ bot_bo1001194783lle_hf @ A ) ) ).
% bot.not_eq_extremum
thf(fact_330_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_331_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_332_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_333_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_334_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_335_of__nat__0,axiom,
( ( semiri1382578993at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_336_of__nat__0,axiom,
( ( semiri2019852685at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_337_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1382578993at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_338_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri2019852685at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_339_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1382578993at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_340_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri2019852685at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_341_psubset__imp__ex__mem,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] :
( ( ord_le1344122901lle_hf @ A2 @ B3 )
=> ? [B6: hF_Mirabelle_hf] : ( member1367349282lle_hf @ B6 @ ( minus_1450406810lle_hf @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_342_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_343_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_344_not__psubset__empty,axiom,
! [A2: set_HF_Mirabelle_hf] :
~ ( ord_le1344122901lle_hf @ A2 @ bot_bo53200981lle_hf ) ).
% not_psubset_empty
thf(fact_345_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri2019852685at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_346_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri2019852685at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_347_psubsetD,axiom,
! [A2: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le1344122901lle_hf @ A2 @ B3 )
=> ( ( member1367349282lle_hf @ C @ A2 )
=> ( member1367349282lle_hf @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_348_less__set__def,axiom,
( ord_le1344122901lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ord_le616625840e_hf_o
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 )
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_349_int__diff__cases,axiom,
! [Z4: int] :
~ ! [M3: nat,N2: nat] :
( Z4
!= ( minus_minus_int @ ( semiri2019852685at_int @ M3 ) @ ( semiri2019852685at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_350_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
% Helper facts (3)
thf(help_If_3_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( if_HF_Mirabelle_hf @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__HF____Mirabelle____glliljednj__Ohf_T,axiom,
! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( if_HF_Mirabelle_hf @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( hF_Mirabelle_HBex @ a
@ ^ [X3: hF_Mirabelle_hf] : p )
= ( ? [X3: hF_Mirabelle_hf] : ( hF_Mirabelle_hmem @ X3 @ a )
& p ) ) ).
%------------------------------------------------------------------------------