TPTP Problem File: ITP074^1.p
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%------------------------------------------------------------------------------
% File : ITP074^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_612__5334652_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_612__5334652_1 [Des21]
% Status : Theorem
% Rating : 0.38 v9.0.0, 0.50 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 376 ( 220 unt; 32 typ; 0 def)
% Number of atoms : 708 ( 335 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 2286 ( 42 ~; 7 |; 34 &;1996 @)
% ( 0 <=>; 207 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 86 ( 86 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 29 usr; 4 con; 0-2 aty)
% Number of variables : 876 ( 79 ^; 782 !; 15 ?; 876 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:44.587
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
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 ).
% Explicit typings (29)
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__Nat__Onat,type,
minus_minus_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_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_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__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
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__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1382578993at_nat: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__HF____Mirabelle____glliljednj__Ohf,type,
ord_le976219883lle_hf: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $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_member_001t__HF____Mirabelle____glliljednj__Ohf,type,
member1367349282lle_hf: hF_Mirabelle_hf > set_HF_Mirabelle_hf > $o ).
thf(sy_v_x,type,
x: hF_Mirabelle_hf ).
thf(sy_v_y,type,
y: hF_Mirabelle_hf ).
% Relevant facts (343)
thf(fact_0_hpair__iff,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A @ B )
= ( hF_Mirabelle_hpair @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% hpair_iff
thf(fact_1_hpair__inject,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ A @ B )
= ( hF_Mirabelle_hpair @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% hpair_inject
thf(fact_2_hpair__neq__fst,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A @ B )
!= A ) ).
% hpair_neq_fst
thf(fact_3_hpair__neq__snd,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hpair @ A @ B )
!= B ) ).
% hpair_neq_snd
thf(fact_4_hfst__conv,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hfst @ ( hF_Mirabelle_hpair @ A @ B ) )
= A ) ).
% hfst_conv
thf(fact_5_hsnd__conv,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hsnd @ ( hF_Mirabelle_hpair @ A @ B ) )
= B ) ).
% hsnd_conv
thf(fact_6_HInter__hempty,axiom,
( ( hF_Mirabelle_HInter @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HInter_hempty
thf(fact_7_HCollect__hempty,axiom,
! [P: hF_Mirabelle_hf > $o] :
( ( hF_Mir818139703ollect @ P @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HCollect_hempty
thf(fact_8_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_9_HUnion__hempty,axiom,
( ( hF_Mirabelle_HUnion @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% HUnion_hempty
thf(fact_10_zero__reorient,axiom,
! [X: hF_Mirabelle_hf] :
( ( zero_z189798548lle_hf = X )
= ( X = zero_z189798548lle_hf ) ) ).
% zero_reorient
thf(fact_11_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_12_RepFun__0,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ zero_z189798548lle_hf @ F )
= zero_z189798548lle_hf ) ).
% RepFun_0
thf(fact_13_hpair__def,axiom,
( hF_Mirabelle_hpair
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) @ ( hF_Mirabelle_hinsert @ ( hF_Mirabelle_hinsert @ A3 @ ( hF_Mirabelle_hinsert @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% hpair_def
thf(fact_14_hpair__def_H,axiom,
( hF_Mirabelle_hpair
= ( ^ [A3: hF_Mirabelle_hf,B3: 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 @ B3 @ zero_z189798548lle_hf ) ) @ zero_z189798548lle_hf ) ) ) ) ).
% hpair_def'
thf(fact_15_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_16_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_17_hinsert__commute,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ X @ ( hF_Mirabelle_hinsert @ Y @ Z ) )
= ( hF_Mirabelle_hinsert @ Y @ ( hF_Mirabelle_hinsert @ X @ Z ) ) ) ).
% hinsert_commute
thf(fact_18_hinsert__nonempty,axiom,
! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hinsert @ A @ A4 )
!= zero_z189798548lle_hf ) ).
% hinsert_nonempty
thf(fact_19_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_20_hf__induct__ax,axiom,
! [P: hF_Mirabelle_hf > $o,X: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X2: hF_Mirabelle_hf] :
( ( P @ X2 )
=> ! [Y2: hF_Mirabelle_hf] :
( ( P @ Y2 )
=> ( P @ ( hF_Mirabelle_hinsert @ Y2 @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% hf_induct_ax
thf(fact_21_HInter__hinsert,axiom,
! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( A4 != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_HInter @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
= ( inf_in956532509lle_hf @ A @ ( hF_Mirabelle_HInter @ A4 ) ) ) ) ).
% HInter_hinsert
thf(fact_22_Abs__hf__0,axiom,
( ( hF_Mirabelle_Abs_hf @ zero_zero_nat )
= zero_z189798548lle_hf ) ).
% Abs_hf_0
thf(fact_23_HInter__iff,axiom,
! [A4: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( A4 != zero_z189798548lle_hf )
=> ( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HInter @ A4 ) )
= ( ! [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A4 )
=> ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ) ).
% HInter_iff
thf(fact_24_HUnion__hinsert,axiom,
! [A: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HUnion @ ( hF_Mirabelle_hinsert @ A @ A4 ) )
= ( sup_su638957495lle_hf @ A @ ( hF_Mirabelle_HUnion @ A4 ) ) ) ).
% HUnion_hinsert
thf(fact_25_hdiff__insert,axiom,
! [A4: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ A4 @ ( hF_Mirabelle_hinsert @ A @ B4 ) )
= ( minus_1232880740lle_hf @ ( minus_1232880740lle_hf @ A4 @ B4 ) @ ( hF_Mirabelle_hinsert @ A @ zero_z189798548lle_hf ) ) ) ).
% hdiff_insert
thf(fact_26_hinsert__eq__sup,axiom,
( hF_Mirabelle_hinsert
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ ( hF_Mirabelle_hinsert @ A3 @ zero_z189798548lle_hf ) ) ) ) ).
% hinsert_eq_sup
thf(fact_27_hf__induct,axiom,
! [P: hF_Mirabelle_hf > $o,Z: hF_Mirabelle_hf] :
( ( P @ zero_z189798548lle_hf )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( P @ X2 )
=> ( ( P @ Y2 )
=> ( ~ ( hF_Mirabelle_hmem @ X2 @ Y2 )
=> ( P @ ( hF_Mirabelle_hinsert @ X2 @ Y2 ) ) ) ) )
=> ( P @ Z ) ) ) ).
% hf_induct
thf(fact_28_hf__cases,axiom,
! [Y: hF_Mirabelle_hf] :
( ( Y != zero_z189798548lle_hf )
=> ~ ! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf] :
( ( Y
= ( hF_Mirabelle_hinsert @ A5 @ B5 ) )
=> ( hF_Mirabelle_hmem @ A5 @ B5 ) ) ) ).
% hf_cases
thf(fact_29_Replace__hunion,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ R )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_Replace @ A4 @ R ) @ ( hF_Mirabelle_Replace @ B4 @ R ) ) ) ).
% Replace_hunion
thf(fact_30_HCollect__iff,axiom,
! [X: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( hF_Mir818139703ollect @ P @ A4 ) )
= ( ( P @ X )
& ( hF_Mirabelle_hmem @ X @ A4 ) ) ) ).
% HCollect_iff
thf(fact_31_hf__equalityI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A )
= ( hF_Mirabelle_hmem @ X2 @ B ) )
=> ( A = B ) ) ).
% hf_equalityI
thf(fact_32_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_33_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_34_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_35_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_36_hunion__iff,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( sup_su638957495lle_hf @ A @ B ) )
= ( ( hF_Mirabelle_hmem @ X @ A )
| ( hF_Mirabelle_hmem @ X @ B ) ) ) ).
% hunion_iff
thf(fact_37_hinter__iff,axiom,
! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ ( inf_in956532509lle_hf @ X @ Y ) )
= ( ( hF_Mirabelle_hmem @ U @ X )
& ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% hinter_iff
thf(fact_38_RepFun__iff,axiom,
! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_RepFun @ A4 @ F ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( V
= ( F @ U2 ) ) ) ) ) ).
% RepFun_iff
thf(fact_39_hunion__hempty__left,axiom,
! [A4: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ zero_z189798548lle_hf @ A4 )
= A4 ) ).
% hunion_hempty_left
thf(fact_40_hunion__hempty__right,axiom,
! [A4: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A4 @ zero_z189798548lle_hf )
= A4 ) ).
% hunion_hempty_right
thf(fact_41_HUnion__iff,axiom,
! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ ( hF_Mirabelle_HUnion @ A4 ) )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A4 )
& ( hF_Mirabelle_hmem @ X @ Y3 ) ) ) ) ).
% HUnion_iff
thf(fact_42_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_43_Collect__mem__eq,axiom,
! [A4: set_HF_Mirabelle_hf] :
( ( collec2046588256lle_hf
@ ^ [X3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [P: hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > $o] :
( ! [X2: hF_Mirabelle_hf] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec2046588256lle_hf @ P )
= ( collec2046588256lle_hf @ Q ) ) ) ).
% Collect_cong
thf(fact_45_hdiff__iff,axiom,
! [U: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ ( minus_1232880740lle_hf @ X @ Y ) )
= ( ( hF_Mirabelle_hmem @ U @ X )
& ~ ( hF_Mirabelle_hmem @ U @ Y ) ) ) ).
% hdiff_iff
thf(fact_46_hinter__hempty__left,axiom,
! [A4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ zero_z189798548lle_hf @ A4 )
= zero_z189798548lle_hf ) ).
% hinter_hempty_left
thf(fact_47_hinter__hempty__right,axiom,
! [A4: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A4 @ zero_z189798548lle_hf )
= zero_z189798548lle_hf ) ).
% hinter_hempty_right
thf(fact_48_Replace__iff,axiom,
! [V: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ V @ ( hF_Mirabelle_Replace @ A4 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( R @ U2 @ V )
& ! [Y3: hF_Mirabelle_hf] :
( ( R @ U2 @ Y3 )
=> ( Y3 = V ) ) ) ) ) ).
% Replace_iff
thf(fact_49_hdiff__zero,axiom,
! [X: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ X @ zero_z189798548lle_hf )
= X ) ).
% hdiff_zero
thf(fact_50_zero__hdiff,axiom,
! [X: hF_Mirabelle_hf] :
( ( minus_1232880740lle_hf @ zero_z189798548lle_hf @ X )
= zero_z189798548lle_hf ) ).
% zero_hdiff
thf(fact_51_RepFun__hunion,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ ( sup_su638957495lle_hf @ A4 @ B4 ) @ F )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_RepFun @ A4 @ F ) @ ( hF_Mirabelle_RepFun @ B4 @ F ) ) ) ).
% RepFun_hunion
thf(fact_52_HUnion__hunion,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HUnion @ ( sup_su638957495lle_hf @ A4 @ B4 ) )
= ( sup_su638957495lle_hf @ ( hF_Mirabelle_HUnion @ A4 ) @ ( hF_Mirabelle_HUnion @ B4 ) ) ) ).
% HUnion_hunion
thf(fact_53_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_54_replacement__fun,axiom,
! [X: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V2 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X )
& ( V2
= ( F @ U2 ) ) ) ) ) ).
% replacement_fun
thf(fact_55_hmem__not__refl,axiom,
! [X: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X @ X ) ).
% hmem_not_refl
thf(fact_56_comprehension,axiom,
! [X: 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 @ X )
& ( P @ U3 ) ) ) ).
% comprehension
thf(fact_57_union__of__set,axiom,
! [X: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ X )
& ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% union_of_set
thf(fact_58_hmem__not__sym,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
~ ( ( hF_Mirabelle_hmem @ X @ Y )
& ( hF_Mirabelle_hmem @ Y @ X ) ) ).
% hmem_not_sym
thf(fact_59_binary__union,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X )
| ( hF_Mirabelle_hmem @ U3 @ Y ) ) ) ).
% binary_union
thf(fact_60_replacement,axiom,
! [X: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U4 @ X )
=> ( ( R @ U4 @ V3 )
=> ( ( R @ U4 @ V4 )
=> ( V4 = V3 ) ) ) )
=> ? [Z2: hF_Mirabelle_hf] :
! [V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V2 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X )
& ( R @ U2 @ V2 ) ) ) ) ) ).
% replacement
thf(fact_61_hmem__ne,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ Y )
=> ( X != Y ) ) ).
% hmem_ne
thf(fact_62_hf__ext,axiom,
( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A3 )
= ( hF_Mirabelle_hmem @ X3 @ B3 ) ) ) ) ).
% hf_ext
thf(fact_63_hinter__hinsert__right,axiom,
! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
= ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
= ( inf_in956532509lle_hf @ B4 @ A4 ) ) ) ) ).
% hinter_hinsert_right
thf(fact_64_hinter__hinsert__left,axiom,
! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
= ( hF_Mirabelle_hinsert @ X @ ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( inf_in956532509lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
= ( inf_in956532509lle_hf @ A4 @ B4 ) ) ) ) ).
% hinter_hinsert_left
thf(fact_65_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_66_hinsert__hdiff__if,axiom,
! [X: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
= ( minus_1232880740lle_hf @ A4 @ B4 ) ) )
& ( ~ ( hF_Mirabelle_hmem @ X @ B4 )
=> ( ( minus_1232880740lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
= ( hF_Mirabelle_hinsert @ X @ ( minus_1232880740lle_hf @ A4 @ B4 ) ) ) ) ) ).
% hinsert_hdiff_if
thf(fact_67_hinsert__iff,axiom,
! [Z: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( Z
= ( hF_Mirabelle_hinsert @ Y @ X ) )
= ( ! [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ Z )
= ( ( hF_Mirabelle_hmem @ U2 @ X )
| ( U2 = Y ) ) ) ) ) ).
% hinsert_iff
thf(fact_68_hemptyE,axiom,
! [A: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% hemptyE
thf(fact_69_hempty__iff,axiom,
! [Z: hF_Mirabelle_hf] :
( ( Z = zero_z189798548lle_hf )
= ( ! [X3: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ X3 @ Z ) ) ) ).
% hempty_iff
thf(fact_70_hmem__hempty,axiom,
! [A: hF_Mirabelle_hf] :
~ ( hF_Mirabelle_hmem @ A @ zero_z189798548lle_hf ) ).
% hmem_hempty
thf(fact_71_hunion__hinsert__right,axiom,
! [B4: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ B4 @ ( hF_Mirabelle_hinsert @ X @ A4 ) )
= ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ B4 @ A4 ) ) ) ).
% hunion_hinsert_right
thf(fact_72_hunion__hinsert__left,axiom,
! [X: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( hF_Mirabelle_hinsert @ X @ A4 ) @ B4 )
= ( hF_Mirabelle_hinsert @ X @ ( sup_su638957495lle_hf @ A4 @ B4 ) ) ) ).
% hunion_hinsert_left
thf(fact_73_RepFun__cong,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( A4 = B4 )
=> ( ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ B4 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( hF_Mirabelle_RepFun @ A4 @ F )
= ( hF_Mirabelle_RepFun @ B4 @ G ) ) ) ) ).
% RepFun_cong
thf(fact_74_hmem__Sup__ne,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ Y )
=> ( ( hF_Mirabelle_HUnion @ X )
!= Y ) ) ).
% hmem_Sup_ne
thf(fact_75_Replace__cong,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( A4 = B4 )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ B4 )
=> ( ( P @ X2 @ Y2 )
= ( Q @ X2 @ Y2 ) ) )
=> ( ( hF_Mirabelle_Replace @ A4 @ P )
= ( hF_Mirabelle_Replace @ B4 @ Q ) ) ) ) ).
% Replace_cong
thf(fact_76_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_77_sup__inf__absorb,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_78_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_79_inf__sup__absorb,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_80_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_81_sup_Oright__idem,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ B )
= ( sup_su638957495lle_hf @ A @ B ) ) ).
% sup.right_idem
thf(fact_82_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_83_sup__left__idem,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
= ( sup_su638957495lle_hf @ X @ Y ) ) ).
% sup_left_idem
thf(fact_84_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_85_sup_Oleft__idem,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) )
= ( sup_su638957495lle_hf @ A @ B ) ) ).
% sup.left_idem
thf(fact_86_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_87_sup__idem,axiom,
! [X: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ X )
= X ) ).
% sup_idem
thf(fact_88_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_89_sup_Oidem,axiom,
! [A: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A @ A )
= A ) ).
% sup.idem
thf(fact_90_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_91_inf__right__idem,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y )
= ( inf_in956532509lle_hf @ X @ Y ) ) ).
% inf_right_idem
thf(fact_92_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_93_inf_Oidem,axiom,
! [A: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A @ A )
= A ) ).
% inf.idem
thf(fact_94_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_95_inf__idem,axiom,
! [X: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ X )
= X ) ).
% inf_idem
thf(fact_96_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_97_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_98_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_99_inf__left__idem,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
= ( inf_in956532509lle_hf @ X @ Y ) ) ).
% inf_left_idem
thf(fact_100_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_101_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_102_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_103_inf__sup__aci_I4_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ X @ Y ) )
= ( inf_in956532509lle_hf @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_104_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_105_inf__sup__aci_I3_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
= ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_106_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_107_inf__sup__aci_I2_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
= ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_108_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_109_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_110_boolean__algebra__cancel_Oinf1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A4 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_111_boolean__algebra__cancel_Oinf1,axiom,
! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A4
= ( inf_in956532509lle_hf @ K @ A ) )
=> ( ( inf_in956532509lle_hf @ A4 @ B )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_112_boolean__algebra__cancel_Oinf2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B4 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_113_boolean__algebra__cancel_Oinf2,axiom,
! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( B4
= ( inf_in956532509lle_hf @ K @ B ) )
=> ( ( inf_in956532509lle_hf @ A @ B4 )
= ( inf_in956532509lle_hf @ K @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_114_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_115_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_116_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_117_inf__assoc,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Z )
= ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_118_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A3: nat,B3: nat] : ( inf_inf_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_119_inf_Ocommute,axiom,
( inf_in956532509lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_120_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_121_inf__commute,axiom,
( inf_in956532509lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( inf_in956532509lle_hf @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_122_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_123_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_124_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_125_inf__left__commute,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
= ( inf_in956532509lle_hf @ Y @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_126_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_127_inf__sup__aci_I8_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) )
= ( sup_su638957495lle_hf @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_128_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_129_inf__sup__aci_I7_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
= ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_130_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_131_inf__sup__aci_I6_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
= ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_132_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_133_inf__sup__aci_I5_J,axiom,
( sup_su638957495lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_134_boolean__algebra__cancel_Osup1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A4 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_135_boolean__algebra__cancel_Osup1,axiom,
! [A4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A4
= ( sup_su638957495lle_hf @ K @ A ) )
=> ( ( sup_su638957495lle_hf @ A4 @ B )
= ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_136_boolean__algebra__cancel_Osup2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A @ B4 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_137_boolean__algebra__cancel_Osup2,axiom,
! [B4: hF_Mirabelle_hf,K: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( B4
= ( sup_su638957495lle_hf @ K @ B ) )
=> ( ( sup_su638957495lle_hf @ A @ B4 )
= ( sup_su638957495lle_hf @ K @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_138_sup_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_139_sup_Oassoc,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ C )
= ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% sup.assoc
thf(fact_140_sup__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_141_sup__assoc,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
= ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_142_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_143_sup_Ocommute,axiom,
( sup_su638957495lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_144_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_145_sup__commute,axiom,
( sup_su638957495lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] : ( sup_su638957495lle_hf @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_146_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_147_sup_Oleft__commute,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ B @ ( sup_su638957495lle_hf @ A @ C ) )
= ( sup_su638957495lle_hf @ A @ ( sup_su638957495lle_hf @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_148_sup__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_149_sup__left__commute,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
= ( sup_su638957495lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_150_distrib__imp1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y2: nat,Z2: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_151_distrib__imp1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X2 @ ( sup_su638957495lle_hf @ Y2 @ Z2 ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X2 @ Y2 ) @ ( inf_in956532509lle_hf @ X2 @ Z2 ) ) )
=> ( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_152_distrib__imp2,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y2: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_153_distrib__imp2,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X2 @ ( inf_in956532509lle_hf @ Y2 @ Z2 ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X2 @ Y2 ) @ ( sup_su638957495lle_hf @ X2 @ Z2 ) ) )
=> ( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_154_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_155_inf__sup__distrib1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_156_inf__sup__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_157_inf__sup__distrib2,axiom,
! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X )
= ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ X ) @ ( inf_in956532509lle_hf @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_158_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_159_sup__inf__distrib1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_160_sup__inf__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_161_sup__inf__distrib2,axiom,
! [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ Y @ Z ) @ X )
= ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ Y @ X ) @ ( sup_su638957495lle_hf @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_162_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_163_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_164_PrimReplace__iff,axiom,
! [A4: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V: hF_Mirabelle_hf] :
( ! [U4: hF_Mirabelle_hf,V3: hF_Mirabelle_hf,V4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U4 @ A4 )
=> ( ( R @ U4 @ V3 )
=> ( ( R @ U4 @ V4 )
=> ( V4 = V3 ) ) ) )
=> ( ( hF_Mirabelle_hmem @ V @ ( hF_Mir1248913145eplace @ A4 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A4 )
& ( R @ U2 @ V ) ) ) ) ) ).
% PrimReplace_iff
thf(fact_165_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_166_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_167_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_168_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_169_less__eq__insert1__iff,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( hF_Mirabelle_hinsert @ X @ Y ) @ Z )
= ( ( hF_Mirabelle_hmem @ X @ Z )
& ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% less_eq_insert1_iff
thf(fact_170_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member1367349282lle_hf @ A3 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% hmem_def
thf(fact_171_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_172_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_173_inf_Obounded__iff,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
= ( ( ord_le976219883lle_hf @ A @ B )
& ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_174_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_175_le__inf__iff,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) )
= ( ( ord_le976219883lle_hf @ X @ Y )
& ( ord_le976219883lle_hf @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_176_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_177_sup_Obounded__iff,axiom,
! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
= ( ( ord_le976219883lle_hf @ B @ A )
& ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_178_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_179_le__sup__iff,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ Z )
= ( ( ord_le976219883lle_hf @ X @ Z )
& ( ord_le976219883lle_hf @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_180_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_181_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_182_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_183_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_184_hsubsetI,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A4 )
=> ( hF_Mirabelle_hmem @ X2 @ B4 ) )
=> ( ord_le976219883lle_hf @ A4 @ B4 ) ) ).
% hsubsetI
thf(fact_185_less__eq__hempty,axiom,
! [U: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ U @ zero_z189798548lle_hf )
= ( U = zero_z189798548lle_hf ) ) ).
% less_eq_hempty
thf(fact_186_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_187_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_188_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_189_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_190_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_191_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_192_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_193_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_194_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_195_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_196_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_197_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_198_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_199_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_200_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_201_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_202_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_203_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_204_canonically__ordered__monoid__add__class_Ozero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% canonically_ordered_monoid_add_class.zero_le
thf(fact_205_inf_OcoboundedI2,axiom,
! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ C )
=> ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_206_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_207_inf_OcoboundedI1,axiom,
! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ C )
=> ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_208_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_209_inf_Oabsorb__iff2,axiom,
( ord_le976219883lle_hf
= ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_210_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_211_inf_Oabsorb__iff1,axiom,
( ord_le976219883lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_212_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_213_inf_Ocobounded2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_214_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_215_inf_Ocobounded1,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_216_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_217_inf_Oorder__iff,axiom,
( ord_le976219883lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( A3
= ( inf_in956532509lle_hf @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_218_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_219_inf__greatest,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y )
=> ( ( ord_le976219883lle_hf @ X @ Z )
=> ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_220_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_221_inf_OboundedI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( ord_le976219883lle_hf @ A @ C )
=> ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_222_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_223_inf_OboundedE,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ ( inf_in956532509lle_hf @ B @ C ) )
=> ~ ( ( ord_le976219883lle_hf @ A @ B )
=> ~ ( ord_le976219883lle_hf @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_224_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_225_inf__absorb2,axiom,
! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y @ X )
=> ( ( inf_in956532509lle_hf @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_226_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_227_inf__absorb1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y )
=> ( ( inf_in956532509lle_hf @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_228_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_229_inf_Oabsorb2,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( inf_in956532509lle_hf @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_230_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_231_inf_Oabsorb1,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( inf_in956532509lle_hf @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_232_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_233_le__iff__inf,axiom,
( ord_le976219883lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( inf_in956532509lle_hf @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_234_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_235_inf__unique,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X2 @ Y2 )
=> ( ( ord_le976219883lle_hf @ X2 @ Z2 )
=> ( ord_le976219883lle_hf @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_in956532509lle_hf @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_236_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_237_inf_OorderI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A
= ( inf_in956532509lle_hf @ A @ B ) )
=> ( ord_le976219883lle_hf @ A @ B ) ) ).
% inf.orderI
thf(fact_238_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_239_inf_OorderE,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( A
= ( inf_in956532509lle_hf @ A @ B ) ) ) ).
% inf.orderE
thf(fact_240_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_241_le__infI2,axiom,
! [B: hF_Mirabelle_hf,X: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ X )
=> ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_242_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_243_le__infI1,axiom,
! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ X )
=> ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_244_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_245_inf__mono,axiom,
! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ C )
=> ( ( ord_le976219883lle_hf @ B @ D )
=> ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ A @ B ) @ ( inf_in956532509lle_hf @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_246_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_247_le__infI,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ A )
=> ( ( ord_le976219883lle_hf @ X @ B )
=> ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) ) ) ) ).
% le_infI
thf(fact_248_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_249_le__infE,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ ( inf_in956532509lle_hf @ A @ B ) )
=> ~ ( ( ord_le976219883lle_hf @ X @ A )
=> ~ ( ord_le976219883lle_hf @ X @ B ) ) ) ).
% le_infE
thf(fact_250_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_251_inf__le2,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_252_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_253_inf__le1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_254_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_255_inf__sup__ord_I1_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_256_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_257_inf__sup__ord_I2_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_258_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_259_sup_OcoboundedI2,axiom,
! [C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ C @ B )
=> ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_260_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_261_sup_OcoboundedI1,axiom,
! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ C @ A )
=> ( ord_le976219883lle_hf @ C @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_262_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_263_sup_Oabsorb__iff2,axiom,
( ord_le976219883lle_hf
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_264_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_265_sup_Oabsorb__iff1,axiom,
( ord_le976219883lle_hf
= ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_266_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_267_sup_Ocobounded2,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ B @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% sup.cobounded2
thf(fact_268_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_269_sup_Ocobounded1,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ A @ ( sup_su638957495lle_hf @ A @ B ) ) ).
% sup.cobounded1
thf(fact_270_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_271_sup_Oorder__iff,axiom,
( ord_le976219883lle_hf
= ( ^ [B3: hF_Mirabelle_hf,A3: hF_Mirabelle_hf] :
( A3
= ( sup_su638957495lle_hf @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_272_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_273_sup_OboundedI,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( ord_le976219883lle_hf @ C @ A )
=> ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_274_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_275_sup_OboundedE,axiom,
! [B: hF_Mirabelle_hf,C: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ B @ C ) @ A )
=> ~ ( ( ord_le976219883lle_hf @ B @ A )
=> ~ ( ord_le976219883lle_hf @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_276_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_277_sup__absorb2,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ Y )
=> ( ( sup_su638957495lle_hf @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_278_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_279_sup__absorb1,axiom,
! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y @ X )
=> ( ( sup_su638957495lle_hf @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_280_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_281_sup_Oabsorb2,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ B )
=> ( ( sup_su638957495lle_hf @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_282_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_283_sup_Oabsorb1,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( sup_su638957495lle_hf @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_284_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_285_sup__unique,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf,Z2: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y2 @ X2 )
=> ( ( ord_le976219883lle_hf @ Z2 @ X2 )
=> ( ord_le976219883lle_hf @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_su638957495lle_hf @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_286_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_287_sup_OorderI,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( A
= ( sup_su638957495lle_hf @ A @ B ) )
=> ( ord_le976219883lle_hf @ B @ A ) ) ).
% sup.orderI
thf(fact_288_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_289_sup_OorderE,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( A
= ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% sup.orderE
thf(fact_290_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_291_le__iff__sup,axiom,
( ord_le976219883lle_hf
= ( ^ [X3: hF_Mirabelle_hf,Y3: hF_Mirabelle_hf] :
( ( sup_su638957495lle_hf @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_292_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_293_sup__least,axiom,
! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Y @ X )
=> ( ( ord_le976219883lle_hf @ Z @ X )
=> ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_294_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_295_sup__mono,axiom,
! [A: hF_Mirabelle_hf,C: hF_Mirabelle_hf,B: hF_Mirabelle_hf,D: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ C )
=> ( ( ord_le976219883lle_hf @ B @ D )
=> ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ ( sup_su638957495lle_hf @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_296_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_297_sup_Omono,axiom,
! [C: hF_Mirabelle_hf,A: hF_Mirabelle_hf,D: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ C @ A )
=> ( ( ord_le976219883lle_hf @ D @ B )
=> ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ C @ D ) @ ( sup_su638957495lle_hf @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_298_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_299_le__supI2,axiom,
! [X: hF_Mirabelle_hf,B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ B )
=> ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% le_supI2
thf(fact_300_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_301_le__supI1,axiom,
! [X: hF_Mirabelle_hf,A: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ X @ A )
=> ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ A @ B ) ) ) ).
% le_supI1
thf(fact_302_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_303_sup__ge2,axiom,
! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% sup_ge2
thf(fact_304_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_305_sup__ge1,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% sup_ge1
thf(fact_306_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_307_le__supI,axiom,
! [A: hF_Mirabelle_hf,X: hF_Mirabelle_hf,B: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A @ X )
=> ( ( ord_le976219883lle_hf @ B @ X )
=> ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_308_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_309_le__supE,axiom,
! [A: hF_Mirabelle_hf,B: hF_Mirabelle_hf,X: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ A @ B ) @ X )
=> ~ ( ( ord_le976219883lle_hf @ A @ X )
=> ~ ( ord_le976219883lle_hf @ B @ X ) ) ) ).
% le_supE
thf(fact_310_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_311_inf__sup__ord_I3_J,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_312_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_313_inf__sup__ord_I4_J,axiom,
! [Y: hF_Mirabelle_hf,X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ Y @ ( sup_su638957495lle_hf @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_314_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_315_less__eq__hf__def,axiom,
( ord_le976219883lle_hf
= ( ^ [A6: hF_Mirabelle_hf,B6: hF_Mirabelle_hf] :
! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ A6 )
=> ( hF_Mirabelle_hmem @ X3 @ B6 ) ) ) ) ).
% less_eq_hf_def
thf(fact_316_rev__hsubsetD,axiom,
! [C: hF_Mirabelle_hf,A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ C @ A4 )
=> ( ( ord_le976219883lle_hf @ A4 @ B4 )
=> ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% rev_hsubsetD
thf(fact_317_hsubsetCE,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A4 @ B4 )
=> ( ( hF_Mirabelle_hmem @ C @ A4 )
=> ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% hsubsetCE
thf(fact_318_hsubsetD,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf,C: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ A4 @ B4 )
=> ( ( hF_Mirabelle_hmem @ C @ A4 )
=> ( hF_Mirabelle_hmem @ C @ B4 ) ) ) ).
% hsubsetD
thf(fact_319_HF__Mirabelle__glliljednj_Ozero__le,axiom,
! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ zero_z189798548lle_hf @ X ) ).
% HF_Mirabelle_glliljednj.zero_le
thf(fact_320_hf__equalityE,axiom,
! [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( A4 = B4 )
=> ~ ( ( ord_le976219883lle_hf @ A4 @ B4 )
=> ~ ( ord_le976219883lle_hf @ B4 @ A4 ) ) ) ).
% hf_equalityE
thf(fact_321_distrib__inf__le,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ ( inf_in956532509lle_hf @ X @ Y ) @ ( inf_in956532509lle_hf @ X @ Z ) ) @ ( inf_in956532509lle_hf @ X @ ( sup_su638957495lle_hf @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_322_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_323_distrib__sup__le,axiom,
! [X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ ( sup_su638957495lle_hf @ X @ ( inf_in956532509lle_hf @ Y @ Z ) ) @ ( inf_in956532509lle_hf @ ( sup_su638957495lle_hf @ X @ Y ) @ ( sup_su638957495lle_hf @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_324_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_325_less__eq__insert2__iff,axiom,
! [Z: hF_Mirabelle_hf,X: hF_Mirabelle_hf,Y: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ Z @ ( hF_Mirabelle_hinsert @ X @ Y ) )
= ( ( ord_le976219883lle_hf @ Z @ Y )
| ? [U2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hinsert @ X @ U2 )
= Z )
& ~ ( hF_Mirabelle_hmem @ X @ U2 )
& ( ord_le976219883lle_hf @ U2 @ Y ) ) ) ) ).
% less_eq_insert2_iff
thf(fact_326_order__refl,axiom,
! [X: hF_Mirabelle_hf] : ( ord_le976219883lle_hf @ X @ X ) ).
% order_refl
thf(fact_327_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_328_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_329_HF__hfset,axiom,
! [A: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A ) )
= A ) ).
% HF_hfset
thf(fact_330_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_331_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1382578993at_nat @ M )
= ( semiri1382578993at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_332_of__nat__0,axiom,
( ( semiri1382578993at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_333_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_334_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_335_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_336_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_337_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_338_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1382578993at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_339_dual__order_Oantisym,axiom,
! [B: hF_Mirabelle_hf,A: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B @ A )
=> ( ( ord_le976219883lle_hf @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_340_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_341_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: hF_Mirabelle_hf,Z3: hF_Mirabelle_hf] : ( Y4 = Z3 ) )
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
( ( ord_le976219883lle_hf @ B3 @ A3 )
& ( ord_le976219883lle_hf @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_342_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
% Conjectures (1)
thf(conj_0,conjecture,
( ( hF_Mirabelle_hpair @ x @ y )
!= zero_z189798548lle_hf ) ).
%------------------------------------------------------------------------------