TPTP Problem File: ITP076^1.p
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%------------------------------------------------------------------------------
% File : ITP076^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_850__5339526_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_850__5339526_1 [Des21]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 408 ( 258 unt; 52 typ; 0 def)
% Number of atoms : 781 ( 477 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 2218 ( 71 ~; 9 |; 60 &;1909 @)
% ( 0 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 179 ( 179 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 48 usr; 7 con; 0-3 aty)
% Number of variables : 871 ( 62 ^; 784 !; 25 ?; 871 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:21.238
%------------------------------------------------------------------------------
% 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__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (48)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
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__Nat__Onat,type,
zero_zero_nat: nat ).
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_Ohf_ORep__hf,type,
hF_Mirabelle_Rep_hf: hF_Mirabelle_hf > nat ).
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_Ohfunction,type,
hF_Mir199975595nction: hF_Mirabelle_hf > $o ).
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_Ohrelation,type,
hF_Mir434065167lation: hF_Mirabelle_hf > $o ).
thf(sy_c_HF__Mirabelle__glliljednj_Ohrestrict,type,
hF_Mir1653039215strict: 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_HF__Mirabelle__glliljednj_Ois__hpair,type,
hF_Mir137170979_hpair: hF_Mirabelle_hf > $o ).
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_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $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__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_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
sup_su1199008216e_hf_o: ( hF_Mirabelle_hf > $o ) > ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__HF____Mirabelle____glliljednj__Ohf,type,
sup_su638957495lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
sup_su1790843629lle_hf: set_HF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__HF____Mirabelle____glliljednj__Ohf_M_Eo_J,type,
top_to22270292e_hf_o: hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__HF____Mirabelle____glliljednj__Ohf_J,type,
top_to489427057lle_hf: set_HF_Mirabelle_hf ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
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_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oinsert_001t__HF____Mirabelle____glliljednj__Ohf,type,
insert9649339lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > set_HF_Mirabelle_hf ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Typedef_Otype__definition_001t__HF____Mirabelle____glliljednj__Ohf_001t__Nat__Onat,type,
type_d1794767497hf_nat: ( hF_Mirabelle_hf > nat ) > ( nat > hF_Mirabelle_hf ) > set_nat > $o ).
thf(sy_c_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_r,type,
r: hF_Mirabelle_hf ).
thf(sy_v_x,type,
x: hF_Mirabelle_hf ).
% Relevant facts (354)
thf(fact_0_hrelation__restr,axiom,
! [R: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( hF_Mir434065167lation @ ( hF_Mir1653039215strict @ R @ X ) ) ).
% hrelation_restr
thf(fact_1_hfunction__def,axiom,
( hF_Mir199975595nction
= ( ^ [R2: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y ) @ R2 )
=> ! [Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ Y2 ) @ R2 )
=> ( Y = Y2 ) ) ) ) ) ).
% hfunction_def
thf(fact_2_hrestrict__iff,axiom,
! [Z: hF_Mirabelle_hf,R: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Z @ ( hF_Mir1653039215strict @ R @ A ) )
= ( ( hF_Mirabelle_hmem @ Z @ R )
& ? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( Z
= ( hF_Mirabelle_hpair @ X2 @ Y ) )
& ( hF_Mirabelle_hmem @ X2 @ A ) ) ) ) ).
% hrestrict_iff
thf(fact_3_hpair__iff,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A2 @ B )
= ( hF_Mirabelle_hpair @ A3 @ B2 ) )
= ( ( A2 = A3 )
& ( B = B2 ) ) ) ).
% hpair_iff
thf(fact_4_Rep__hf__inject,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_Rep_hf @ X )
= ( hF_Mirabelle_Rep_hf @ Y3 ) )
= ( X = Y3 ) ) ).
% Rep_hf_inject
thf(fact_5_hf__equalityI,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A2 )
= ( hF_Mirabelle_hmem @ X3 @ B ) )
=> ( A2 = B ) ) ).
% hf_equalityI
thf(fact_6_hf__ext,axiom,
( ( ^ [Y4: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] : Y4 = Z2 )
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A4 )
= ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% hf_ext
thf(fact_7_hmem__ne,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ Y3 )
=> ( X != Y3 ) ) ).
% hmem_ne
thf(fact_8_replacement,axiom,
! [X: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ X )
=> ( ( R3 @ U @ V )
=> ( ( R3 @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ? [Z3: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z3 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X )
& ( R3 @ U2 @ V3 ) ) ) ) ) ).
% replacement
thf(fact_9_binary__union,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
? [Z3: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z3 )
= ( ( hF_Mirabelle_hmem @ U3 @ X )
| ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ).
% binary_union
thf(fact_10_hmem__not__sym,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
~ ( ( hF_Mirabelle_hmem @ X @ Y3 )
& ( hF_Mirabelle_hmem @ Y3 @ X ) ) ).
% hmem_not_sym
thf(fact_11_hpair__inject,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A3: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A2 @ B )
= ( hF_Mirabelle_hpair @ A3 @ B2 ) )
=> ~ ( ( A2 = A3 )
=> ( B != B2 ) ) ) ).
% hpair_inject
thf(fact_12_union__of__set,axiom,
! [X: hF_Mirabelle_hf] :
? [Z3: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z3 )
= ( ? [Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y @ X )
& ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ) ).
% union_of_set
thf(fact_13_comprehension,axiom,
! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
? [Z3: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z3 )
= ( ( hF_Mirabelle_hmem @ U3 @ X )
& ( P @ U3 ) ) ) ).
% comprehension
thf(fact_14_hmem__not__refl,axiom,
! [X: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X @ X ) ).
% hmem_not_refl
thf(fact_15_hpair__neq__fst,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A2 @ B )
!= A2 ) ).
% hpair_neq_fst
thf(fact_16_hpair__neq__snd,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A2 @ B )
!= B ) ).
% hpair_neq_snd
thf(fact_17_replacement__fun,axiom,
! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
? [Z3: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z3 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X )
& ( V3
= ( F @ U2 ) ) ) ) ) ).
% replacement_fun
thf(fact_18_hfst__conv,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A2 @ B ) )
= A2 ) ).
% hfst_conv
thf(fact_19_hsnd__conv,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A2 @ B ) )
= B ) ).
% hsnd_conv
thf(fact_20_hrelation__def,axiom,
( hF_Mir434065167lation
= ( ^ [R2: hF_Mirabelle_hf] :
! [Z4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Z4 @ R2 )
=> ( hF_Mir137170979_hpair @ Z4 ) ) ) ) ).
% hrelation_def
thf(fact_21_is__hpair__def,axiom,
( hF_Mir137170979_hpair
= ( ^ [Z4: hF_Mirabelle_hf] :
? [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( Z4
= ( hF_Mirabelle_hpair @ X2 @ Y ) ) ) ) ).
% is_hpair_def
thf(fact_22_PrimReplace__iff,axiom,
! [A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ A )
=> ( ( R3 @ U @ V )
=> ( ( R3 @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir1248913145eplace @ A @ R3 ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A )
& ( R3 @ U2 @ V4 ) ) ) ) ) ).
% PrimReplace_iff
thf(fact_23_HUnion__iff,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A ) )
= ( ? [Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y @ A )
& ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ).
% HUnion_iff
thf(fact_24_Replace__iff,axiom,
! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A @ R3 ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A )
& ( R3 @ U2 @ V4 )
& ! [Y: hF_Mirabelle_hf] :
( ( R3 @ U2 @ Y )
=> ( Y = V4 ) ) ) ) ) ).
% Replace_iff
thf(fact_25_HCollect__iff,axiom,
! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A ) )
= ( ( P @ X )
& ( hF_Mirabelle_hmem @ X @ A ) ) ) ).
% HCollect_iff
thf(fact_26_RepFun__iff,axiom,
! [V4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A @ F ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A )
& ( V4
= ( F @ U2 ) ) ) ) ) ).
% RepFun_iff
thf(fact_27_Rep__hf__inverse,axiom,
! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_Abs_hf @ ( hF_Mirabelle_Rep_hf @ X ) )
= X ) ).
% Rep_hf_inverse
thf(fact_28_RepFun__cong,axiom,
! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( A = B4 )
=> ( ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ B4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( hF_Mirabelle_RepFun @ A @ F )
= ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% RepFun_cong
thf(fact_29_Replace__cong,axiom,
! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( A = B4 )
=> ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ B4 )
=> ( ( P @ X3 @ Y5 )
= ( Q @ X3 @ Y5 ) ) )
=> ( ( hF_Mirabelle_Replace @ A @ P )
= ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% Replace_cong
thf(fact_30_hmem__Sup__ne,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ Y3 )
=> ( ( hF_Mirabelle_HUnion @ X )
!= Y3 ) ) ).
% hmem_Sup_ne
thf(fact_31_Abs__hf__inverse,axiom,
! [Y3: nat] :
( ( member_nat @ Y3 @ top_top_set_nat )
=> ( ( hF_Mirabelle_Rep_hf @ ( hF_Mirabelle_Abs_hf @ Y3 ) )
= Y3 ) ) ).
% Abs_hf_inverse
thf(fact_32_zero__notin__hpair,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ zero_z189798548lle_hf @ ( hF_Mirabelle_hpair @ X @ Y3 ) ) ).
% zero_notin_hpair
thf(fact_33_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% hmem_def
thf(fact_34_HCollect__hempty,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HCollect_hempty
thf(fact_35_HUnion__hempty,axiom,
( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HUnion_hempty
thf(fact_36_Replace__0,axiom,
! [R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ zero_z189798548lle_hf @ R3 )
= zero_z189798548lle_hf ) ).
% Replace_0
thf(fact_37_RepFun__0,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
= zero_z189798548lle_hf ) ).
% RepFun_0
thf(fact_38_RepFun__hinsert,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ ( hF_Mirabelle_hinsert @ A2 @ B ) @ F )
= ( hF_Mirabelle_hinsert @ ( F @ A2 ) @ ( hF_Mirabelle_RepFun @ B @ F ) ) ) ).
% RepFun_hinsert
thf(fact_39_hrelation__hunion,axiom,
! [F: hF_Mirabelle_hf,G: hF_Mirabelle_hf] :
( ( hF_Mir434065167lation @ ( sup_su638957495lle_hf @ F @ G ) )
= ( ( hF_Mir434065167lation @ F )
& ( hF_Mir434065167lation @ G ) ) ) ).
% hrelation_hunion
thf(fact_40_hmem__hinsert,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ A2 @ ( hF_Mirabelle_hinsert @ B @ C ) )
= ( ( A2 = B )
| ( hF_Mirabelle_hmem @ A2 @ C ) ) ) ).
% hmem_hinsert
thf(fact_41_singleton__eq__iff,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ A2 @ zero_z189798548lle_hf )
= ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
= ( A2 = B ) ) ).
% singleton_eq_iff
thf(fact_42_hunion__iff,axiom,
! [X: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A2 @ B ) )
= ( ( hF_Mirabelle_hmem @ X @ A2 )
| ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% hunion_iff
thf(fact_43_hunion__hempty__left,axiom,
! [A: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A )
= A ) ).
% hunion_hempty_left
thf(fact_44_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
( ( member1367349282lle_hf @ A2 @ ( collec2046588256lle_hf @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_HF_Mirabelle_hf] :
( ( collec2046588256lle_hf
@ ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Collect__cong,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ! [X3: hF_Mirabelle_hf] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec2046588256lle_hf @ P )
= ( collec2046588256lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_49_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_50_hunion__hempty__right,axiom,
! [A: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A @ zero_z189798548lle_hf )
= A ) ).
% hunion_hempty_right
thf(fact_51_RepFun__hunion,axiom,
! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A @ B4 ) @ F )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% RepFun_hunion
thf(fact_52_HUnion__hunion,axiom,
! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A @ B4 ) )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% HUnion_hunion
thf(fact_53_Replace__hunion,axiom,
! [A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R3: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A @ B4 ) @ R3 )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A @ R3 ) @ ( hF_Mirabelle_Replace @ B4 @ R3 ) ) ) ).
% Replace_hunion
thf(fact_54_HUnion__hinsert,axiom,
! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
= ( sup_su638957495lle_hf @ A2 @ ( hF_Mirabelle_HUnion @ A ) ) ) ).
% HUnion_hinsert
thf(fact_55_hf__induct__ax,axiom,
! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X3: hF_Mirabelle_hf] :
( ( P @ X3 )
=> ! [Y5: hF_Mirabelle_hf] :
( ( P @ Y5 )
=> ( P @ ( hF_Mirabelle_hinsert @ Y5 @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% hf_induct_ax
thf(fact_56_hinsert__eq__sup,axiom,
( hF_Mirabelle_hinsert
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) ) ) ).
% hinsert_eq_sup
thf(fact_57_hinsert__commute,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y3 @ Z ) )
= ( hF_Mirabelle_hinsert @ Y3 @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% hinsert_commute
thf(fact_58_HF__Mirabelle__glliljednj_Odoubleton__eq__iff,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ A2 @ ( hF_Mirabelle_hinsert @ B @ zero_z189798548lle_hf ) )
= ( hF_Mirabelle_hinsert @ C @ ( hF_Mirabelle_hinsert @ D @ zero_z189798548lle_hf ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% HF_Mirabelle_glliljednj.doubleton_eq_iff
thf(fact_59_hinsert__nonempty,axiom,
! [A2: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ A2 @ A )
!= zero_z189798548lle_hf ) ).
% hinsert_nonempty
thf(fact_60_hunion__hinsert__left,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
= ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A @ B4 ) ) ) ).
% hunion_hinsert_left
thf(fact_61_hunion__hinsert__right,axiom,
! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
= ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A ) ) ) ).
% hunion_hinsert_right
thf(fact_62_hf__cases,axiom,
! [Y3: hF_Mirabelle_hf] :
( ( Y3 != zero_z189798548lle_hf )
=> ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
( ( Y3
= ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
=> ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% hf_cases
thf(fact_63_hf__induct,axiom,
! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf] :
( ( P @ X3 )
=> ( ( P @ Y5 )
=> ( ~ ( hF_Mirabelle_hmem @ X3 @ Y5 )
=> ( P @ ( hF_Mirabelle_hinsert @ X3 @ Y5 ) ) ) ) )
=> ( P @ Z ) ) ) ).
% hf_induct
thf(fact_64_hpair__def,axiom,
( hF_Mirabelle_hpair
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% hpair_def
thf(fact_65_hpair__def_H,axiom,
( hF_Mirabelle_hpair
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ A4 @ zero_z189798548lle_hf ) ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A4 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% hpair_def'
thf(fact_66_hinsert__iff,axiom,
! [Z: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( Z
= ( hF_Mirabelle_hinsert @ Y3 @ X ) )
= ( ! [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ Z )
= ( ( hF_Mirabelle_hmem @ U2 @ X )
| ( U2 = Y3 ) ) ) ) ) ).
% hinsert_iff
thf(fact_67_hemptyE,axiom,
! [A2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% hemptyE
thf(fact_68_hempty__iff,axiom,
! [Z: hF_Mirabelle_hf] :
( ( Z = zero_z189798548lle_hf )
= ( ! [X2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X2 @ Z ) ) ) ).
% hempty_iff
thf(fact_69_hmem__hempty,axiom,
! [A2: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A2 @ zero_z189798548lle_hf ) ).
% hmem_hempty
thf(fact_70_hpair__nonzero,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ X @ Y3 )
!= zero_z189798548lle_hf ) ).
% hpair_nonzero
thf(fact_71_Rep__hf,axiom,
! [X: hF_Mirabelle_hf] : ( member_nat @ ( hF_Mirabelle_Rep_hf @ X ) @ top_top_set_nat ) ).
% Rep_hf
thf(fact_72_Rep__hf__cases,axiom,
! [Y3: nat] :
( ( member_nat @ Y3 @ top_top_set_nat )
=> ~ ! [X3: hF_Mirabelle_hf] :
( Y3
!= ( hF_Mirabelle_Rep_hf @ X3 ) ) ) ).
% Rep_hf_cases
thf(fact_73_Rep__hf__induct,axiom,
! [Y3: nat,P: nat > $o] :
( ( member_nat @ Y3 @ top_top_set_nat )
=> ( ! [X3: hF_Mirabelle_hf] : ( P @ ( hF_Mirabelle_Rep_hf @ X3 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_hf_induct
thf(fact_74_Abs__hf__cases,axiom,
! [X: hF_Mirabelle_hf] :
~ ! [Y5: nat] :
( ( X
= ( hF_Mirabelle_Abs_hf @ Y5 ) )
=> ~ ( member_nat @ Y5 @ top_top_set_nat ) ) ).
% Abs_hf_cases
thf(fact_75_Abs__hf__induct,axiom,
! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
( ! [Y5: nat] :
( ( member_nat @ Y5 @ top_top_set_nat )
=> ( P @ ( hF_Mirabelle_Abs_hf @ Y5 ) ) )
=> ( P @ X ) ) ).
% Abs_hf_induct
thf(fact_76_Abs__hf__inject,axiom,
! [X: nat,Y3: nat] :
( ( member_nat @ X @ top_top_set_nat )
=> ( ( member_nat @ Y3 @ top_top_set_nat )
=> ( ( ( hF_Mirabelle_Abs_hf @ X )
= ( hF_Mirabelle_Abs_hf @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% Abs_hf_inject
thf(fact_77_hrelation__0,axiom,
hF_Mir434065167lation @ zero_z189798548lle_hf ).
% hrelation_0
thf(fact_78_sup__top__left,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ top_to489427057lle_hf @ X )
= top_to489427057lle_hf ) ).
% sup_top_left
thf(fact_79_sup__top__left,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ top_top_nat_o @ X )
= top_top_nat_o ) ).
% sup_top_left
thf(fact_80_sup__top__left,axiom,
! [X: hF_Mirabelle_hf > $o] :
( ( sup_su1199008216e_hf_o @ top_to22270292e_hf_o @ X )
= top_to22270292e_hf_o ) ).
% sup_top_left
thf(fact_81_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_82_sup__top__right,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ top_to489427057lle_hf )
= top_to489427057lle_hf ) ).
% sup_top_right
thf(fact_83_sup__top__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ top_top_nat_o )
= top_top_nat_o ) ).
% sup_top_right
thf(fact_84_sup__top__right,axiom,
! [X: hF_Mirabelle_hf > $o] :
( ( sup_su1199008216e_hf_o @ X @ top_to22270292e_hf_o )
= top_to22270292e_hf_o ) ).
% sup_top_right
thf(fact_85_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_86_HInter__iff,axiom,
! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( A != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A ) )
= ( ! [Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y @ A )
=> ( hF_Mirabelle_hmem @ X @ Y ) ) ) ) ) ).
% HInter_iff
thf(fact_87_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_88_sup_Oidem,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_89_sup_Oidem,axiom,
! [A2: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_90_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_91_sup__idem,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ X )
= X ) ).
% sup_idem
thf(fact_92_sup__idem,axiom,
! [X: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ X )
= X ) ).
% sup_idem
thf(fact_93_sup_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_94_sup_Oleft__idem,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A2 @ B ) )
= ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_95_sup_Oleft__idem,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ A2 @ B ) )
= ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% sup.left_idem
thf(fact_96_sup__left__idem,axiom,
! [X: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
= ( sup_sup_set_nat @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_97_sup__left__idem,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
= ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_98_sup__left__idem,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
= ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% sup_left_idem
thf(fact_99_sup_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ B )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_100_sup_Oright__idem,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ B )
= ( sup_su1790843629lle_hf @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_101_sup_Oright__idem,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ B )
= ( sup_su638957495lle_hf @ A2 @ B ) ) ).
% sup.right_idem
thf(fact_102_HInter__hempty,axiom,
( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HInter_hempty
thf(fact_103_sup__left__commute,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_104_sup__left__commute,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
= ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_105_sup__left__commute,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
= ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_106_sup_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_107_sup_Oleft__commute,axiom,
! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ B @ ( sup_su1790843629lle_hf @ A2 @ C ) )
= ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_108_sup_Oleft__commute,axiom,
! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A2 @ C ) )
= ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_109_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_110_sup__commute,axiom,
( sup_su1790843629lle_hf
= ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_111_sup__commute,axiom,
( sup_su638957495lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_112_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_113_sup_Ocommute,axiom,
( sup_su1790843629lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_114_sup_Ocommute,axiom,
( sup_su638957495lle_hf
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A4 ) ) ) ).
% sup.commute
thf(fact_115_sup__assoc,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_116_sup__assoc,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
= ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_117_sup__assoc,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
= ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_118_sup_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_119_sup_Oassoc,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ A2 @ B ) @ C )
= ( sup_su1790843629lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B @ C ) ) ) ).
% sup.assoc
thf(fact_120_sup_Oassoc,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A2 @ B ) @ C )
= ( sup_su638957495lle_hf @ A2 @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% sup.assoc
thf(fact_121_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B4
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A2 @ B4 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_122_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
( ( B4
= ( sup_su1790843629lle_hf @ K @ B ) )
=> ( ( sup_su1790843629lle_hf @ A2 @ B4 )
= ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_123_boolean__algebra__cancel_Osup2,axiom,
! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( B4
= ( sup_su638957495lle_hf @ K @ B ) )
=> ( ( sup_su638957495lle_hf @ A2 @ B4 )
= ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_124_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_125_boolean__algebra__cancel_Osup1,axiom,
! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( A
= ( sup_su1790843629lle_hf @ K @ A2 ) )
=> ( ( sup_su1790843629lle_hf @ A @ B )
= ( sup_su1790843629lle_hf @ K @ ( sup_su1790843629lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_126_boolean__algebra__cancel_Osup1,axiom,
! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A
= ( sup_su638957495lle_hf @ K @ A2 ) )
=> ( ( sup_su638957495lle_hf @ A @ B )
= ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_127_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_128_inf__sup__aci_I5_J,axiom,
( sup_su1790843629lle_hf
= ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( sup_su1790843629lle_hf @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_129_inf__sup__aci_I5_J,axiom,
( sup_su638957495lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_130_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y3 ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_131_inf__sup__aci_I6_J,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ ( sup_su1790843629lle_hf @ X @ Y3 ) @ Z )
= ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_132_inf__sup__aci_I6_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ Z )
= ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_133_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_134_inf__sup__aci_I7_J,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ Y3 @ Z ) )
= ( sup_su1790843629lle_hf @ Y3 @ ( sup_su1790843629lle_hf @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_135_inf__sup__aci_I7_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
= ( sup_su638957495lle_hf @ Y3 @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_136_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
= ( sup_sup_set_nat @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_137_inf__sup__aci_I8_J,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
= ( sup_su1790843629lle_hf @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_138_inf__sup__aci_I8_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
= ( sup_su638957495lle_hf @ X @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_139_iso__tuple__UNIV__I,axiom,
! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% iso_tuple_UNIV_I
thf(fact_140_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_141_UNIV__I,axiom,
! [X: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X @ top_to489427057lle_hf ) ).
% UNIV_I
thf(fact_142_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_143_top__apply,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : top_top_o ) ) ).
% top_apply
thf(fact_144_top__apply,axiom,
( top_to22270292e_hf_o
= ( ^ [X2: hF_Mirabelle_hf] : top_top_o ) ) ).
% top_apply
thf(fact_145_HInter__hinsert,axiom,
! [A: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( A != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A2 @ A ) )
= ( inf_in956532509lle_hf @ A2 @ ( hF_Mirabelle_HInter @ A ) ) ) ) ).
% HInter_hinsert
thf(fact_146_type__definition__hf,axiom,
type_d1794767497hf_nat @ hF_Mirabelle_Rep_hf @ hF_Mirabelle_Abs_hf @ top_top_set_nat ).
% type_definition_hf
thf(fact_147_inf__right__idem,axiom,
! [X: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X @ Y3 ) ) ).
% inf_right_idem
thf(fact_148_inf__right__idem,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Y3 )
= ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% inf_right_idem
thf(fact_149_inf__right__idem,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Y3 )
= ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% inf_right_idem
thf(fact_150_inf_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_151_inf_Oright__idem,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ B )
= ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_152_inf_Oright__idem,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ B )
= ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_153_inf__left__idem,axiom,
! [X: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
= ( inf_inf_set_nat @ X @ Y3 ) ) ).
% inf_left_idem
thf(fact_154_inf__left__idem,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
= ( inf_in923488851lle_hf @ X @ Y3 ) ) ).
% inf_left_idem
thf(fact_155_inf__left__idem,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
= ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% inf_left_idem
thf(fact_156_inf_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_157_inf_Oleft__idem,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ A2 @ B ) )
= ( inf_in923488851lle_hf @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_158_inf_Oleft__idem,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ A2 @ B ) )
= ( inf_in956532509lle_hf @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_159_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_160_inf__idem,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X @ X )
= X ) ).
% inf_idem
thf(fact_161_inf__idem,axiom,
! [X: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ X )
= X ) ).
% inf_idem
thf(fact_162_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_163_inf_Oidem,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_164_inf_Oidem,axiom,
! [A2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_165_inf__top_Oright__neutral,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ A2 @ top_to489427057lle_hf )
= A2 ) ).
% inf_top.right_neutral
thf(fact_166_inf__top_Oright__neutral,axiom,
! [A2: nat > $o] :
( ( inf_inf_nat_o @ A2 @ top_top_nat_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_167_inf__top_Oright__neutral,axiom,
! [A2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ A2 @ top_to22270292e_hf_o )
= A2 ) ).
% inf_top.right_neutral
thf(fact_168_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_169_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( top_to489427057lle_hf
= ( inf_in923488851lle_hf @ A2 @ B ) )
= ( ( A2 = top_to489427057lle_hf )
& ( B = top_to489427057lle_hf ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_170_inf__top_Oneutr__eq__iff,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( top_top_nat_o
= ( inf_inf_nat_o @ A2 @ B ) )
= ( ( A2 = top_top_nat_o )
& ( B = top_top_nat_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_171_inf__top_Oneutr__eq__iff,axiom,
! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
( ( top_to22270292e_hf_o
= ( inf_in307783154e_hf_o @ A2 @ B ) )
= ( ( A2 = top_to22270292e_hf_o )
& ( B = top_to22270292e_hf_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_172_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B ) )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_173_inf__top_Oleft__neutral,axiom,
! [A2: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_174_inf__top_Oleft__neutral,axiom,
! [A2: nat > $o] :
( ( inf_inf_nat_o @ top_top_nat_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_175_inf__top_Oleft__neutral,axiom,
! [A2: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_176_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_177_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( ( inf_in923488851lle_hf @ A2 @ B )
= top_to489427057lle_hf )
= ( ( A2 = top_to489427057lle_hf )
& ( B = top_to489427057lle_hf ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_178_inf__top_Oeq__neutr__iff,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( ( inf_inf_nat_o @ A2 @ B )
= top_top_nat_o )
= ( ( A2 = top_top_nat_o )
& ( B = top_top_nat_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_179_inf__top_Oeq__neutr__iff,axiom,
! [A2: hF_Mirabelle_hf > $o,B: hF_Mirabelle_hf > $o] :
( ( ( inf_in307783154e_hf_o @ A2 @ B )
= top_to22270292e_hf_o )
= ( ( A2 = top_to22270292e_hf_o )
& ( B = top_to22270292e_hf_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_180_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_181_top__eq__inf__iff,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( top_to489427057lle_hf
= ( inf_in923488851lle_hf @ X @ Y3 ) )
= ( ( X = top_to489427057lle_hf )
& ( Y3 = top_to489427057lle_hf ) ) ) ).
% top_eq_inf_iff
thf(fact_182_top__eq__inf__iff,axiom,
! [X: nat > $o,Y3: nat > $o] :
( ( top_top_nat_o
= ( inf_inf_nat_o @ X @ Y3 ) )
= ( ( X = top_top_nat_o )
& ( Y3 = top_top_nat_o ) ) ) ).
% top_eq_inf_iff
thf(fact_183_top__eq__inf__iff,axiom,
! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
( ( top_to22270292e_hf_o
= ( inf_in307783154e_hf_o @ X @ Y3 ) )
= ( ( X = top_to22270292e_hf_o )
& ( Y3 = top_to22270292e_hf_o ) ) ) ).
% top_eq_inf_iff
thf(fact_184_top__eq__inf__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y3 ) )
= ( ( X = top_top_set_nat )
& ( Y3 = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_185_inf__eq__top__iff,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( ( inf_in923488851lle_hf @ X @ Y3 )
= top_to489427057lle_hf )
= ( ( X = top_to489427057lle_hf )
& ( Y3 = top_to489427057lle_hf ) ) ) ).
% inf_eq_top_iff
thf(fact_186_inf__eq__top__iff,axiom,
! [X: nat > $o,Y3: nat > $o] :
( ( ( inf_inf_nat_o @ X @ Y3 )
= top_top_nat_o )
= ( ( X = top_top_nat_o )
& ( Y3 = top_top_nat_o ) ) ) ).
% inf_eq_top_iff
thf(fact_187_inf__eq__top__iff,axiom,
! [X: hF_Mirabelle_hf > $o,Y3: hF_Mirabelle_hf > $o] :
( ( ( inf_in307783154e_hf_o @ X @ Y3 )
= top_to22270292e_hf_o )
= ( ( X = top_to22270292e_hf_o )
& ( Y3 = top_to22270292e_hf_o ) ) ) ).
% inf_eq_top_iff
thf(fact_188_inf__eq__top__iff,axiom,
! [X: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y3 )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y3 = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_189_inf__top__right,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X @ top_to489427057lle_hf )
= X ) ).
% inf_top_right
thf(fact_190_inf__top__right,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ X @ top_top_nat_o )
= X ) ).
% inf_top_right
thf(fact_191_inf__top__right,axiom,
! [X: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ X @ top_to22270292e_hf_o )
= X ) ).
% inf_top_right
thf(fact_192_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_193_inf__top__left,axiom,
! [X: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ top_to489427057lle_hf @ X )
= X ) ).
% inf_top_left
thf(fact_194_inf__top__left,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ top_top_nat_o @ X )
= X ) ).
% inf_top_left
thf(fact_195_inf__top__left,axiom,
! [X: hF_Mirabelle_hf > $o] :
( ( inf_in307783154e_hf_o @ top_to22270292e_hf_o @ X )
= X ) ).
% inf_top_left
thf(fact_196_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_197_inf__sup__absorb,axiom,
! [X: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_198_inf__sup__absorb,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X @ ( sup_su1790843629lle_hf @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_199_inf__sup__absorb,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y3 ) )
= X ) ).
% inf_sup_absorb
thf(fact_200_sup__inf__absorb,axiom,
! [X: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_201_sup__inf__absorb,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ X @ ( inf_in923488851lle_hf @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_202_sup__inf__absorb,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
= X ) ).
% sup_inf_absorb
thf(fact_203_hinter__iff,axiom,
! [U4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U4 @ ( inf_in956532509lle_hf @ X @ Y3 ) )
= ( ( hF_Mirabelle_hmem @ U4 @ X )
& ( hF_Mirabelle_hmem @ U4 @ Y3 ) ) ) ).
% hinter_iff
thf(fact_204_hinter__hempty__left,axiom,
! [A: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A )
= zero_z189798548lle_hf ) ).
% hinter_hempty_left
thf(fact_205_hinter__hempty__right,axiom,
! [A: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% hinter_hempty_right
thf(fact_206_Abs__hf__0,axiom,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= zero_z189798548lle_hf ) ).
% Abs_hf_0
thf(fact_207_inf__left__commute,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_208_inf__left__commute,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) )
= ( inf_in923488851lle_hf @ Y3 @ ( inf_in923488851lle_hf @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_209_inf__left__commute,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
= ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_210_inf_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_211_inf_Oleft__commute,axiom,
! [B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ B @ ( inf_in923488851lle_hf @ A2 @ C ) )
= ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_212_inf_Oleft__commute,axiom,
! [B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ B @ ( inf_in956532509lle_hf @ A2 @ C ) )
= ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_213_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_214_inf__commute,axiom,
( inf_in923488851lle_hf
= ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_215_inf__commute,axiom,
( inf_in956532509lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% inf_commute
thf(fact_216_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_217_inf_Ocommute,axiom,
( inf_in923488851lle_hf
= ( ^ [A4: set_HF_Mirabelle_hf,B3: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_218_inf_Ocommute,axiom,
( inf_in956532509lle_hf
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A4 ) ) ) ).
% inf.commute
thf(fact_219_inf__assoc,axiom,
! [X: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y3 ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_220_inf__assoc,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
= ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_221_inf__assoc,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
= ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_222_inf_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_223_inf_Oassoc,axiom,
! [A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,C: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ A2 @ B ) @ C )
= ( inf_in923488851lle_hf @ A2 @ ( inf_in923488851lle_hf @ B @ C ) ) ) ).
% inf.assoc
thf(fact_224_inf_Oassoc,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ A2 @ B ) @ C )
= ( inf_in956532509lle_hf @ A2 @ ( inf_in956532509lle_hf @ B @ C ) ) ) ).
% inf.assoc
thf(fact_225_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B4
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B4 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_226_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf] :
( ( B4
= ( inf_in923488851lle_hf @ K @ B ) )
=> ( ( inf_in923488851lle_hf @ A2 @ B4 )
= ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_227_boolean__algebra__cancel_Oinf2,axiom,
! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf] :
( ( B4
= ( inf_in956532509lle_hf @ K @ B ) )
=> ( ( inf_in956532509lle_hf @ A2 @ B4 )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_228_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_229_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_HF_Mirabelle_hf,K: set_HF_Mirabelle_hf,A2: set_HF_Mirabelle_hf,B: set_HF_Mirabelle_hf] :
( ( A
= ( inf_in923488851lle_hf @ K @ A2 ) )
=> ( ( inf_in923488851lle_hf @ A @ B )
= ( inf_in923488851lle_hf @ K @ ( inf_in923488851lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_230_boolean__algebra__cancel_Oinf1,axiom,
! [A: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A
= ( inf_in956532509lle_hf @ K @ A2 ) )
=> ( ( inf_in956532509lle_hf @ A @ B )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_231_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_232_inf__sup__aci_I1_J,axiom,
( inf_in923488851lle_hf
= ( ^ [X2: set_HF_Mirabelle_hf,Y: set_HF_Mirabelle_hf] : ( inf_in923488851lle_hf @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_233_inf__sup__aci_I1_J,axiom,
( inf_in956532509lle_hf
= ( ^ [X2: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_234_inf__sup__aci_I2_J,axiom,
! [X: set_HF_Mirabelle_hf,Y3: set_HF_Mirabelle_hf,Z: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( inf_in923488851lle_hf @ X @ Y3 ) @ Z )
= ( inf_in923488851lle_hf @ X @ ( inf_in923488851lle_hf @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_235_inf__sup__aci_I2_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ Z )
= ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_236_inf__sup__aci_I3_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
= ( inf_in956532509lle_hf @ Y3 @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_237_inf__sup__aci_I4_J,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y3 ) )
= ( inf_in956532509lle_hf @ X @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_238_sup__inf__distrib2,axiom,
! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ Z ) @ X )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_239_sup__inf__distrib1,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_240_inf__sup__distrib2,axiom,
! [Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y3 @ Z ) @ X )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y3 @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_241_inf__sup__distrib1,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_242_distrib__imp2,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X3 @ ( inf_in956532509lle_hf @ Y5 @ Z3 ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X3 @ Y5 ) @ ( sup_su638957495lle_hf @ X3 @ Z3 ) ) )
=> ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y3 @ Z ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y3 ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_243_distrib__imp1,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf,Y5: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X3 @ ( sup_su638957495lle_hf @ Y5 @ Z3 ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X3 @ Y5 ) @ ( inf_in956532509lle_hf @ X3 @ Z3 ) ) )
=> ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y3 @ Z ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y3 ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_244_foundation,axiom,
! [Z: hF_Mirabelle_hf] :
( ( Z != zero_z189798548lle_hf )
=> ? [W: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ W @ Z )
& ( ( inf_in956532509lle_hf @ W @ Z )
= zero_z189798548lle_hf ) ) ) ).
% foundation
thf(fact_245_hinter__hinsert__right,axiom,
! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
= ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A ) )
= ( inf_in956532509lle_hf @ B4 @ A ) ) ) ) ).
% hinter_hinsert_right
thf(fact_246_hinter__hinsert__left,axiom,
! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
= ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A @ B4 ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A ) @ B4 )
= ( inf_in956532509lle_hf @ A @ B4 ) ) ) ) ).
% hinter_hinsert_left
thf(fact_247_zero__reorient,axiom,
! [X: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X )
= ( X = zero_z189798548lle_hf ) ) ).
% zero_reorient
thf(fact_248_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_249_UNIV__eq__I,axiom,
! [A: set_HF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A )
=> ( top_to489427057lle_hf = A ) ) ).
% UNIV_eq_I
thf(fact_250_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_251_UNIV__witness,axiom,
? [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ top_to489427057lle_hf ) ).
% UNIV_witness
thf(fact_252_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_253_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_254_Un__UNIV__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_255_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_256_type__copy__ex__RepI,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,F2: nat > $o] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( ? [X4: nat] : ( F2 @ X4 ) )
= ( ? [B3: hF_Mirabelle_hf] : ( F2 @ ( Rep @ B3 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_257_type__copy__obj__one__point__absE,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,S: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ top_top_set_nat )
=> ~ ! [X3: nat] :
( S
!= ( Abs @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_258_HF__hfset,axiom,
! [A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
= A2 ) ).
% HF_hfset
thf(fact_259_type__definition_ORep,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( member_nat @ ( Rep @ X ) @ A ) ) ).
% type_definition.Rep
thf(fact_260_type__definition__def,axiom,
( type_d1794767497hf_nat
= ( ^ [Rep2: hF_Mirabelle_hf > nat,Abs2: nat > hF_Mirabelle_hf,A6: set_nat] :
( ! [X2: hF_Mirabelle_hf] : ( member_nat @ ( Rep2 @ X2 ) @ A6 )
& ! [X2: hF_Mirabelle_hf] :
( ( Abs2 @ ( Rep2 @ X2 ) )
= X2 )
& ! [Y: nat] :
( ( member_nat @ Y @ A6 )
=> ( ( Rep2 @ ( Abs2 @ Y ) )
= Y ) ) ) ) ) ).
% type_definition_def
thf(fact_261_Int__UNIV,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B4 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_262_UnCI,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_263_UnCI,axiom,
! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( ~ ( member1367349282lle_hf @ C @ B4 )
=> ( member1367349282lle_hf @ C @ A ) )
=> ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% UnCI
thf(fact_264_Un__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_265_Un__iff,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
= ( ( member1367349282lle_hf @ C @ A )
| ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_266_UnE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_267_UnE,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) )
=> ( ~ ( member1367349282lle_hf @ C @ A )
=> ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% UnE
thf(fact_268_UnI1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_269_UnI1,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ A )
=> ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% UnI1
thf(fact_270_UnI2,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_271_UnI2,axiom,
! [C: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ B4 )
=> ( member1367349282lle_hf @ C @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% UnI2
thf(fact_272_Int__UNIV__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_273_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_274_type__definition_ORep__inverse,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_275_type__definition_OAbs__inverse,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( Rep @ ( Abs @ Y3 ) )
= Y3 ) ) ) ).
% type_definition.Abs_inverse
thf(fact_276_type__definition_ORep__inject,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( ( Rep @ X )
= ( Rep @ Y3 ) )
= ( X = Y3 ) ) ) ).
% type_definition.Rep_inject
thf(fact_277_type__definition_ORep__induct,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat,P: nat > $o] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ! [X3: hF_Mirabelle_hf] : ( P @ ( Rep @ X3 ) )
=> ( P @ Y3 ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_278_type__definition_OAbs__inject,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: nat,Y3: nat] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( Abs @ X )
= ( Abs @ Y3 ) )
= ( X = Y3 ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_279_type__definition_OAbs__induct,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( P @ ( Abs @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_280_type__definition_ORep__cases,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,Y3: nat] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ( ( member_nat @ Y3 @ A )
=> ~ ! [X3: hF_Mirabelle_hf] :
( Y3
!= ( Rep @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_281_type__definition_OAbs__cases,axiom,
! [Rep: hF_Mirabelle_hf > nat,Abs: nat > hF_Mirabelle_hf,A: set_nat,X: hF_Mirabelle_hf] :
( ( type_d1794767497hf_nat @ Rep @ Abs @ A )
=> ~ ! [Y5: nat] :
( ( X
= ( Abs @ Y5 ) )
=> ~ ( member_nat @ Y5 @ A ) ) ) ).
% type_definition.Abs_cases
thf(fact_282_type__definition_Ointro,axiom,
! [Rep: hF_Mirabelle_hf > nat,A: set_nat,Abs: nat > hF_Mirabelle_hf] :
( ! [X3: hF_Mirabelle_hf] : ( member_nat @ ( Rep @ X3 ) @ A )
=> ( ! [X3: hF_Mirabelle_hf] :
( ( Abs @ ( Rep @ X3 ) )
= X3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( ( Rep @ ( Abs @ Y5 ) )
= Y5 ) )
=> ( type_d1794767497hf_nat @ Rep @ Abs @ A ) ) ) ) ).
% type_definition.intro
thf(fact_283_top__empty__eq,axiom,
( top_to22270292e_hf_o
= ( ^ [X2: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X2 @ top_to489427057lle_hf ) ) ) ).
% top_empty_eq
thf(fact_284_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_285_hcard__0,axiom,
( ( hF_Mirabelle_hcard @ zero_z189798548lle_hf )
= zero_zero_nat ) ).
% hcard_0
thf(fact_286_hinsert__def,axiom,
( hF_Mirabelle_hinsert
= ( ^ [A4: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_HF @ ( insert9649339lle_hf @ A4 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ) ).
% hinsert_def
thf(fact_287_insert__absorb2,axiom,
! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ X @ A ) )
= ( insert9649339lle_hf @ X @ A ) ) ).
% insert_absorb2
thf(fact_288_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_289_insert__iff,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
= ( ( A2 = B )
| ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_290_insertCI,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_291_insertCI,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ~ ( member1367349282lle_hf @ A2 @ B4 )
=> ( A2 = B ) )
=> ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% insertCI
thf(fact_292_IntI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_293_IntI,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ A )
=> ( ( member1367349282lle_hf @ C @ B4 )
=> ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_294_Int__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_295_Int__iff,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
= ( ( member1367349282lle_hf @ C @ A )
& ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_296_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_297_Int__insert__left__if0,axiom,
! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ A2 @ C2 )
=> ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
= ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_298_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_299_Int__insert__left__if1,axiom,
! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ C2 )
=> ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
= ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_300_insert__inter__insert,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ A ) @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_301_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_302_Int__insert__right__if0,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ A2 @ A )
=> ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( inf_in923488851lle_hf @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_303_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_304_Int__insert__right__if1,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A )
=> ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_305_Un__insert__left,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,C2: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
= ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_306_Un__insert__right,axiom,
! [A: set_HF_Mirabelle_hf,A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( sup_su1790843629lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( insert9649339lle_hf @ A2 @ ( sup_su1790843629lle_hf @ A @ B4 ) ) ) ).
% Un_insert_right
thf(fact_307_insert__UNIV,axiom,
! [X: hF_Mirabelle_hf] :
( ( insert9649339lle_hf @ X @ top_to489427057lle_hf )
= top_to489427057lle_hf ) ).
% insert_UNIV
thf(fact_308_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_309_IntE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_310_IntE,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
=> ~ ( ( member1367349282lle_hf @ C @ A )
=> ~ ( member1367349282lle_hf @ C @ B4 ) ) ) ).
% IntE
thf(fact_311_IntD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_312_IntD1,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
=> ( member1367349282lle_hf @ C @ A ) ) ).
% IntD1
thf(fact_313_IntD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_314_IntD2,axiom,
! [C: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ C @ ( inf_in923488851lle_hf @ A @ B4 ) )
=> ( member1367349282lle_hf @ C @ B4 ) ) ).
% IntD2
thf(fact_315_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_316_Int__insert__left,axiom,
! [A2: hF_Mirabelle_hf,C2: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( ( member1367349282lle_hf @ A2 @ C2 )
=> ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
= ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) )
& ( ~ ( member1367349282lle_hf @ A2 @ C2 )
=> ( ( inf_in923488851lle_hf @ ( insert9649339lle_hf @ A2 @ B4 ) @ C2 )
= ( inf_in923488851lle_hf @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_317_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_318_Int__insert__right,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ( ( member1367349282lle_hf @ A2 @ A )
=> ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( insert9649339lle_hf @ A2 @ ( inf_in923488851lle_hf @ A @ B4 ) ) ) )
& ( ~ ( member1367349282lle_hf @ A2 @ A )
=> ( ( inf_in923488851lle_hf @ A @ ( insert9649339lle_hf @ A2 @ B4 ) )
= ( inf_in923488851lle_hf @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_319_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat @ A2 @ B6 ) )
& ~ ( member_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_320_mk__disjoint__insert,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A )
=> ? [B6: set_HF_Mirabelle_hf] :
( ( A
= ( insert9649339lle_hf @ A2 @ B6 ) )
& ~ ( member1367349282lle_hf @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_321_insert__commute,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( insert9649339lle_hf @ X @ ( insert9649339lle_hf @ Y3 @ A ) )
= ( insert9649339lle_hf @ Y3 @ ( insert9649339lle_hf @ X @ A ) ) ) ).
% insert_commute
thf(fact_322_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B4 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B4
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_323_insert__eq__iff,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ A2 @ A )
=> ( ~ ( member1367349282lle_hf @ B @ B4 )
=> ( ( ( insert9649339lle_hf @ A2 @ A )
= ( insert9649339lle_hf @ B @ B4 ) )
= ( ( ( A2 = B )
=> ( A = B4 ) )
& ( ( A2 != B )
=> ? [C3: set_HF_Mirabelle_hf] :
( ( A
= ( insert9649339lle_hf @ B @ C3 ) )
& ~ ( member1367349282lle_hf @ B @ C3 )
& ( B4
= ( insert9649339lle_hf @ A2 @ C3 ) )
& ~ ( member1367349282lle_hf @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_324_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_325_insert__absorb,axiom,
! [A2: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ A )
=> ( ( insert9649339lle_hf @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_326_insert__ident,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_327_insert__ident,axiom,
! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] :
( ~ ( member1367349282lle_hf @ X @ A )
=> ( ~ ( member1367349282lle_hf @ X @ B4 )
=> ( ( ( insert9649339lle_hf @ X @ A )
= ( insert9649339lle_hf @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_328_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_329_Set_Oset__insert,axiom,
! [X: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ X @ A )
=> ~ ! [B6: set_HF_Mirabelle_hf] :
( ( A
= ( insert9649339lle_hf @ X @ B6 ) )
=> ( member1367349282lle_hf @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_330_insertI2,axiom,
! [A2: nat,B4: set_nat,B: nat] :
( ( member_nat @ A2 @ B4 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_331_insertI2,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ B4 )
=> ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ B4 ) ) ) ).
% insertI2
thf(fact_332_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_333_insertI1,axiom,
! [A2: hF_Mirabelle_hf,B4: set_HF_Mirabelle_hf] : ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ A2 @ B4 ) ) ).
% insertI1
thf(fact_334_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_335_insertE,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: set_HF_Mirabelle_hf] :
( ( member1367349282lle_hf @ A2 @ ( insert9649339lle_hf @ B @ A ) )
=> ( ( A2 != B )
=> ( member1367349282lle_hf @ A2 @ A ) ) ) ).
% insertE
thf(fact_336_hfset__hinsert,axiom,
! [A2: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_hinsert @ A2 @ B ) )
= ( insert9649339lle_hf @ A2 @ ( hF_Mirabelle_hfset @ B ) ) ) ).
% hfset_hinsert
thf(fact_337_hcard__union__inter,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( plus_plus_nat @ ( hF_Mirabelle_hcard @ ( sup_su638957495lle_hf @ X @ Y3 ) ) @ ( hF_Mirabelle_hcard @ ( inf_in956532509lle_hf @ X @ Y3 ) ) )
= ( plus_plus_nat @ ( hF_Mirabelle_hcard @ X ) @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ).
% hcard_union_inter
thf(fact_338_hcard__hinsert__if,axiom,
! [X: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ Y3 )
=> ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
= ( hF_Mirabelle_hcard @ Y3 ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ Y3 )
=> ( ( hF_Mirabelle_hcard @ ( hF_Mirabelle_hinsert @ X @ Y3 ) )
= ( suc @ ( hF_Mirabelle_hcard @ Y3 ) ) ) ) ) ).
% hcard_hinsert_if
thf(fact_339_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_340_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_341_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_342_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_343_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_344_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_345_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_346_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_347_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_348_add_Oleft__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add.left_neutral
thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_350_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_351_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_352_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A2: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_353_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
% Conjectures (2)
thf(conj_0,hypothesis,
hF_Mir199975595nction @ r ).
thf(conj_1,conjecture,
hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ).
%------------------------------------------------------------------------------