TPTP Problem File: SLH0465^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0017_Countermodel/prob_00211_007075__12968846_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1520 ( 586 unt; 240 typ; 0 def)
% Number of atoms : 3707 (1311 equ; 0 cnn)
% Maximal formula atoms : 42 ( 2 avg)
% Number of connectives : 12106 ( 486 ~; 130 |; 189 &;9435 @)
% ( 0 <=>;1866 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 723 ( 723 >; 0 *; 0 +; 0 <<)
% Number of symbols : 216 ( 213 usr; 19 con; 0-8 aty)
% Number of variables : 4009 ( 215 ^;3628 !; 166 ?;4009 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:03.656
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
set_set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
set_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (213)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
archim6058952711729229775r_real: real > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Hintikka_OHintikka,type,
hintikka: set_fm > $o ).
thf(sy_c_Hintikka_Oterms,type,
terms: set_fm > set_tm ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Real__Oreal_J,type,
if_list_real: $o > list_real > list_real > list_real ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Ofm_J,type,
if_list_fm: $o > list_fm > list_fm > list_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Otm_J,type,
if_list_tm: $o > list_tm > list_tm > list_tm ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
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_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
sup_sup_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Otm_J,type,
sup_sup_set_tm: set_tm > set_tm > set_tm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Real__Oreal,type,
can_select_real: ( real > $o ) > set_real > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
can_select_fm: ( fm > $o ) > set_fm > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
can_select_tm: ( tm > $o ) > set_tm > $o ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Real__Oreal,type,
coset_real: list_real > set_real ).
thf(sy_c_List_Ocoset_001t__SeCaV__Ofm,type,
coset_fm: list_fm > set_fm ).
thf(sy_c_List_Ocoset_001t__SeCaV__Otm,type,
coset_tm: list_tm > set_tm ).
thf(sy_c_List_Ogen__length_001t__SeCaV__Ofm,type,
gen_length_fm: nat > list_fm > nat ).
thf(sy_c_List_Ogen__length_001t__SeCaV__Otm,type,
gen_length_tm: nat > list_tm > nat ).
thf(sy_c_List_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Ofm_J,type,
cons_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
set_list_int2: list_list_int > set_list_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Otm_J,type,
set_list_tm2: list_list_tm > set_list_tm ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
set_tm2: list_tm > set_tm ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
list_ex1_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Real__Oreal,type,
nth_real: list_real > nat > real ).
thf(sy_c_List_Onth_001t__SeCaV__Ofm,type,
nth_fm: list_fm > nat > fm ).
thf(sy_c_List_Onth_001t__SeCaV__Otm,type,
nth_tm: list_tm > nat > tm ).
thf(sy_c_List_Onths_001t__Int__Oint,type,
nths_int: list_int > set_nat > list_int ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__Real__Oreal,type,
nths_real: list_real > set_nat > list_real ).
thf(sy_c_List_Onths_001t__SeCaV__Ofm,type,
nths_fm: list_fm > set_nat > list_fm ).
thf(sy_c_List_Onths_001t__SeCaV__Otm,type,
nths_tm: list_tm > set_nat > list_tm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Ofm,type,
product_lists_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
subseqs_int: list_int > list_list_int ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__SeCaV__Ofm,type,
union_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Ounion_001t__SeCaV__Otm,type,
union_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Otm_J,type,
size_size_list_tm: list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
ord_le5601931644483074373set_tm: set_set_tm > set_set_tm > $o ).
thf(sy_c_Prover_ObranchDone,type,
branchDone: list_fm > $o ).
thf(sy_c_Prover_ObranchDone__rel,type,
branchDone_rel: list_fm > list_fm > $o ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_SeCaV_Oext_001t__Int__Oint,type,
ext_int: list_int > list_int > $o ).
thf(sy_c_SeCaV_Oext_001t__Nat__Onat,type,
ext_nat: list_nat > list_nat > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Ofm_OCon,type,
con: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ODis,type,
dis: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_OExi,type,
exi: fm > fm ).
thf(sy_c_SeCaV_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ONeg,type,
neg: fm > fm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_SeCaV_Ofm_Osize__fm,type,
size_fm: fm > nat ).
thf(sy_c_SeCaV_Oinc__list,type,
inc_list: list_tm > list_tm ).
thf(sy_c_SeCaV_Oinc__term,type,
inc_term: tm > tm ).
thf(sy_c_SeCaV_Oliftt,type,
liftt: tm > tm ).
thf(sy_c_SeCaV_Oliftts,type,
liftts: list_tm > list_tm ).
thf(sy_c_SeCaV_Omember_001t__Int__Oint,type,
member_int: int > list_int > $o ).
thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
member_nat: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__Real__Oreal,type,
member_real: real > list_real > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
member_fm: fm > list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
member_tm: tm > list_tm > $o ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Onews,type,
news: nat > list_fm > $o ).
thf(sy_c_SeCaV_Oparams,type,
params: fm > set_nat ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H,type,
paramst2: tm > set_nat ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osequent__calculus,type,
sequent_calculus: list_fm > $o ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Osubst,type,
subst: fm > tm > nat > fm ).
thf(sy_c_SeCaV_Osubstt,type,
substt: tm > tm > nat > tm ).
thf(sy_c_SeCaV_Osubstts,type,
substts: list_tm > tm > nat > list_tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
collect_tm: ( tm > $o ) > set_tm ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__SeCaV__Ofm,type,
pow_fm: set_fm > set_set_fm ).
thf(sy_c_Set_OPow_001t__SeCaV__Otm,type,
pow_tm: set_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_list_fm_set_fm: ( list_fm > set_fm ) > set_list_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Otm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_list_tm_set_tm: ( list_tm > set_tm ) > set_list_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Ofm,type,
image_nat_fm: ( nat > fm ) > set_nat > set_fm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Otm,type,
image_nat_tm: ( nat > tm ) > set_nat > set_tm ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__SeCaV__Ofm,type,
image_real_fm: ( real > fm ) > set_real > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Real__Oreal,type,
image_fm_real: ( fm > real ) > set_fm > set_real ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
image_fm_fm: ( fm > fm ) > set_fm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
image_fm_tm: ( fm > tm ) > set_fm > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Nat__Onat,type,
image_tm_nat: ( tm > nat ) > set_tm > set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Real__Oreal,type,
image_tm_real: ( tm > real ) > set_tm > set_real ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
image_tm_fm: ( tm > fm ) > set_tm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
image_tm_tm: ( tm > tm ) > set_tm > set_tm ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Ofm,type,
the_elem_fm: set_fm > fm ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Otm,type,
the_elem_tm: set_tm > tm ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Ofm,type,
sublists_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Otm,type,
sublists_tm: list_tm > list_list_tm ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__SeCaV__Ofm_J,type,
accp_list_fm: ( list_fm > list_fm > $o ) > list_fm > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int2: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm2: tm > set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Ofm_J,type,
member_set_fm: set_fm > set_set_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Otm_J,type,
member_set_tm: set_tm > set_set_tm > $o ).
thf(sy_v_S,type,
s: set_fm ).
thf(sy_v_pa____,type,
pa: fm ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x____,type,
x: fm ).
% Relevant facts (1263)
thf(fact_0_assms,axiom,
hintikka @ s ).
% assms
thf(fact_1__092_060open_062Neg_Ax_A_092_060in_062_AS_092_060close_062,axiom,
member_fm2 @ ( neg @ x ) @ s ).
% \<open>Neg x \<in> S\<close>
thf(fact_2_Exi,axiom,
( x
= ( exi @ pa ) ) ).
% Exi
thf(fact_3_Hintikka_ODeltaExi,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( exi @ P ) ) @ H )
=> ? [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X @ P ) ) @ H ) ) ) ) ).
% Hintikka.DeltaExi
thf(fact_4_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_5_sub_Osimps_I7_J,axiom,
! [V: nat,S: tm,P: fm] :
( ( sub @ V @ S @ ( neg @ P ) )
= ( neg @ ( sub @ V @ S @ P ) ) ) ).
% sub.simps(7)
thf(fact_6_Hintikka_OGammaExi,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( exi @ P ) @ H )
=> ! [X2: tm] :
( ( member_tm2 @ X2 @ ( terms @ H ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X2 @ P ) @ H ) ) ) ) ).
% Hintikka.GammaExi
thf(fact_7_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_8_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_9_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_10_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_11_Hintikka_OGammaUni,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( uni @ P ) ) @ H )
=> ! [X2: tm] :
( ( member_tm2 @ X2 @ ( terms @ H ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X2 @ P ) ) @ H ) ) ) ) ).
% Hintikka.GammaUni
thf(fact_12_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_13_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_14_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_15_Hintikka_ONeg,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( neg @ P ) ) @ H )
=> ( member_fm2 @ P @ H ) ) ) ).
% Hintikka.Neg
thf(fact_16_fm_Oinject_I5_J,axiom,
! [X5: fm,Y5: fm] :
( ( ( exi @ X5 )
= ( exi @ Y5 ) )
= ( X5 = Y5 ) ) ).
% fm.inject(5)
thf(fact_17_fm_Oinject_I6_J,axiom,
! [X6: fm,Y6: fm] :
( ( ( uni @ X6 )
= ( uni @ Y6 ) )
= ( X6 = Y6 ) ) ).
% fm.inject(6)
thf(fact_18_fm_Odistinct_I37_J,axiom,
! [X5: fm,X6: fm] :
( ( exi @ X5 )
!= ( uni @ X6 ) ) ).
% fm.distinct(37)
thf(fact_19_fm_Odistinct_I41_J,axiom,
! [X6: fm,X7: fm] :
( ( uni @ X6 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_20_fm_Odistinct_I39_J,axiom,
! [X5: fm,X7: fm] :
( ( exi @ X5 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_21_Hintikka_ODeltaUni,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( uni @ P ) @ H )
=> ? [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X @ P ) @ H ) ) ) ) ).
% Hintikka.DeltaUni
thf(fact_22_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_23_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_24_s6,axiom,
( sub
= ( ^ [V2: nat,S2: tm,P2: fm] : ( subst @ P2 @ S2 @ V2 ) ) ) ).
% s6
thf(fact_25_Hintikka_OBasic,axiom,
! [H: set_fm,N: nat,Ts: list_tm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( pre @ N @ Ts ) @ H )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N @ Ts ) ) @ H ) ) ) ).
% Hintikka.Basic
thf(fact_26_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_27_Hintikka_OAlphaImp,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( imp @ P @ Q ) @ H )
=> ( ( member_fm2 @ ( neg @ P ) @ H )
& ( member_fm2 @ Q @ H ) ) ) ) ).
% Hintikka.AlphaImp
thf(fact_28_Hintikka_OAlphaCon,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( con @ P @ Q ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P ) @ H )
& ( member_fm2 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.AlphaCon
thf(fact_29_Hintikka_OBetaImp,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( imp @ P @ Q ) ) @ H )
=> ( ( member_fm2 @ P @ H )
| ( member_fm2 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.BetaImp
thf(fact_30_Hintikka_OBetaDis,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( dis @ P @ Q ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P ) @ H )
| ( member_fm2 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.BetaDis
thf(fact_31_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_32_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_33_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_34_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_35_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_36_fm_Oinject_I4_J,axiom,
! [X41: fm,X42: fm,Y41: fm,Y42: fm] :
( ( ( con @ X41 @ X42 )
= ( con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(4)
thf(fact_37_fm_Oinject_I3_J,axiom,
! [X31: fm,X322: fm,Y31: fm,Y32: fm] :
( ( ( dis @ X31 @ X322 )
= ( dis @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X322 = Y32 ) ) ) ).
% fm.inject(3)
thf(fact_38_fm_Oinject_I2_J,axiom,
! [X21: fm,X222: fm,Y21: fm,Y22: fm] :
( ( ( imp @ X21 @ X222 )
= ( imp @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% fm.inject(2)
thf(fact_39_fm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( pre @ X11 @ X12 )
= ( pre @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fm.inject(1)
thf(fact_40_char_Oinject,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y2: $o,Y3: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y2 @ Y3 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 )
& ( X32 = Y3 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_41_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_42_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_43_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_44_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_45_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_46_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_47_fm_Odistinct_I23_J,axiom,
! [X31: fm,X322: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X322 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_48_fm_Odistinct_I15_J,axiom,
! [X21: fm,X222: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X222 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_49_fm_Odistinct_I13_J,axiom,
! [X21: fm,X222: fm,X31: fm,X322: fm] :
( ( imp @ X21 @ X222 )
!= ( dis @ X31 @ X322 ) ) ).
% fm.distinct(13)
thf(fact_50_mem__Collect__eq,axiom,
! [A: tm,P3: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: fm,P3: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: real,P3: real > $o] :
( ( member_real2 @ A @ ( collect_real @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: nat,P3: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X9: tm] : ( member_tm2 @ X9 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X9: fm] : ( member_fm2 @ X9 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X9: real] : ( member_real2 @ X9 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X9: nat] : ( member_nat2 @ X9 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_fm_Odistinct_I5_J,axiom,
! [X11: nat,X12: list_tm,X41: fm,X42: fm] :
( ( pre @ X11 @ X12 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_59_fm_Odistinct_I3_J,axiom,
! [X11: nat,X12: list_tm,X31: fm,X322: fm] :
( ( pre @ X11 @ X12 )
!= ( dis @ X31 @ X322 ) ) ).
% fm.distinct(3)
thf(fact_60_fm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X21: fm,X222: fm] :
( ( pre @ X11 @ X12 )
!= ( imp @ X21 @ X222 ) ) ).
% fm.distinct(1)
thf(fact_61_subst_Osimps_I4_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( con @ P @ Q ) @ S @ K )
= ( con @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(4)
thf(fact_62_subst_Osimps_I3_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( dis @ P @ Q ) @ S @ K )
= ( dis @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(3)
thf(fact_63_subst_Osimps_I2_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( imp @ P @ Q ) @ S @ K )
= ( imp @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(2)
thf(fact_64_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X13: $o,X23: $o,X33: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X13 @ X23 @ X33 @ X43 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_65_params_H_H_Ocases,axiom,
! [X3: fm] :
( ! [B: nat,Ts2: list_tm] :
( X3
!= ( pre @ B @ Ts2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( imp @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( dis @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( con @ P4 @ Q2 ) )
=> ( ! [P4: fm] :
( X3
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X3
!= ( uni @ P4 ) )
=> ~ ! [P4: fm] :
( X3
!= ( neg @ P4 ) ) ) ) ) ) ) ) ).
% params''.cases
thf(fact_66_fm_Oexhaust,axiom,
! [Y: fm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( pre @ X112 @ X122 ) )
=> ( ! [X212: fm,X223: fm] :
( Y
!= ( imp @ X212 @ X223 ) )
=> ( ! [X312: fm,X323: fm] :
( Y
!= ( dis @ X312 @ X323 ) )
=> ( ! [X412: fm,X422: fm] :
( Y
!= ( con @ X412 @ X422 ) )
=> ( ! [X52: fm] :
( Y
!= ( exi @ X52 ) )
=> ( ! [X62: fm] :
( Y
!= ( uni @ X62 ) )
=> ~ ! [X72: fm] :
( Y
!= ( neg @ X72 ) ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_67_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_68_size__neq__size__imp__neq,axiom,
! [X3: char,Y: char] :
( ( ( size_size_char @ X3 )
!= ( size_size_char @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_69_size__neq__size__imp__neq,axiom,
! [X3: fm,Y: fm] :
( ( ( size_size_fm @ X3 )
!= ( size_size_fm @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_70_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_71_fm_Odistinct_I29_J,axiom,
! [X31: fm,X322: fm,X7: fm] :
( ( dis @ X31 @ X322 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_72_fm_Odistinct_I21_J,axiom,
! [X21: fm,X222: fm,X7: fm] :
( ( imp @ X21 @ X222 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_73_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_74_fm_Odistinct_I17_J,axiom,
! [X21: fm,X222: fm,X5: fm] :
( ( imp @ X21 @ X222 )
!= ( exi @ X5 ) ) ).
% fm.distinct(17)
thf(fact_75_fm_Odistinct_I25_J,axiom,
! [X31: fm,X322: fm,X5: fm] :
( ( dis @ X31 @ X322 )
!= ( exi @ X5 ) ) ).
% fm.distinct(25)
thf(fact_76_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X5: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X5 ) ) ).
% fm.distinct(31)
thf(fact_77_fm_Odistinct_I19_J,axiom,
! [X21: fm,X222: fm,X6: fm] :
( ( imp @ X21 @ X222 )
!= ( uni @ X6 ) ) ).
% fm.distinct(19)
thf(fact_78_fm_Odistinct_I27_J,axiom,
! [X31: fm,X322: fm,X6: fm] :
( ( dis @ X31 @ X322 )
!= ( uni @ X6 ) ) ).
% fm.distinct(27)
thf(fact_79_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X6: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X6 ) ) ).
% fm.distinct(33)
thf(fact_80_sub_Osimps_I4_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( con @ P @ Q ) )
= ( con @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(4)
thf(fact_81_sub_Osimps_I3_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( dis @ P @ Q ) )
= ( dis @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(3)
thf(fact_82_sub_Osimps_I2_J,axiom,
! [V: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V @ S @ ( imp @ P @ Q ) )
= ( imp @ ( sub @ V @ S @ P ) @ ( sub @ V @ S @ Q ) ) ) ).
% sub.simps(2)
thf(fact_83_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X5: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X5 ) ) ).
% fm.distinct(7)
thf(fact_84_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X6: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X6 ) ) ).
% fm.distinct(9)
thf(fact_85_Hintikka_OAlphaDis,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( dis @ P @ Q ) @ H )
=> ( ( member_fm2 @ P @ H )
& ( member_fm2 @ Q @ H ) ) ) ) ).
% Hintikka.AlphaDis
thf(fact_86_Hintikka_OBetaCon,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( con @ P @ Q ) @ H )
=> ( ( member_fm2 @ P @ H )
| ( member_fm2 @ Q @ H ) ) ) ) ).
% Hintikka.BetaCon
thf(fact_87_subst_Osimps_I7_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( neg @ P ) @ S @ K )
= ( neg @ ( subst @ P @ S @ K ) ) ) ).
% subst.simps(7)
thf(fact_88_Hintikka_Ointro,axiom,
! [H: set_fm] :
( ! [N2: nat,Ts2: list_tm] :
( ( member_fm2 @ ( pre @ N2 @ Ts2 ) @ H )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N2 @ Ts2 ) ) @ H ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( dis @ P4 @ Q2 ) @ H )
=> ( ( member_fm2 @ P4 @ H )
& ( member_fm2 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( imp @ P4 @ Q2 ) @ H )
=> ( ( member_fm2 @ ( neg @ P4 ) @ H )
& ( member_fm2 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( neg @ ( con @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P4 ) @ H )
& ( member_fm2 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( con @ P4 @ Q2 ) @ H )
=> ( ( member_fm2 @ P4 @ H )
| ( member_fm2 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( neg @ ( imp @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm2 @ P4 @ H )
| ( member_fm2 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm2 @ ( neg @ ( dis @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P4 ) @ H )
| ( member_fm2 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm2 @ ( exi @ P4 ) @ H )
=> ! [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X @ P4 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm2 @ ( neg @ ( uni @ P4 ) ) @ H )
=> ! [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X @ P4 ) ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm2 @ ( uni @ P4 ) @ H )
=> ? [X2: tm] :
( ( member_tm2 @ X2 @ ( terms @ H ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X2 @ P4 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm2 @ ( neg @ ( exi @ P4 ) ) @ H )
=> ? [X2: tm] :
( ( member_tm2 @ X2 @ ( terms @ H ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X2 @ P4 ) ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ H )
=> ( member_fm2 @ P4 @ H ) )
=> ( hintikka @ H ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Hintikka.intro
thf(fact_89_Hintikka__def,axiom,
( hintikka
= ( ^ [H2: set_fm] :
( ! [N3: nat,Ts3: list_tm] :
( ( member_fm2 @ ( pre @ N3 @ Ts3 ) @ H2 )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N3 @ Ts3 ) ) @ H2 ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( dis @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm2 @ P2 @ H2 )
& ( member_fm2 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( imp @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P2 ) @ H2 )
& ( member_fm2 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( neg @ ( con @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P2 ) @ H2 )
& ( member_fm2 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( con @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm2 @ P2 @ H2 )
| ( member_fm2 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( neg @ ( imp @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm2 @ P2 @ H2 )
| ( member_fm2 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm2 @ ( neg @ ( dis @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P2 ) @ H2 )
| ( member_fm2 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm2 @ ( exi @ P2 ) @ H2 )
=> ! [X9: tm] :
( ( member_tm2 @ X9 @ ( terms @ H2 ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X9 @ P2 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm2 @ ( neg @ ( uni @ P2 ) ) @ H2 )
=> ! [X9: tm] :
( ( member_tm2 @ X9 @ ( terms @ H2 ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X9 @ P2 ) ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm2 @ ( uni @ P2 ) @ H2 )
=> ? [X9: tm] :
( ( member_tm2 @ X9 @ ( terms @ H2 ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X9 @ P2 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm2 @ ( neg @ ( exi @ P2 ) ) @ H2 )
=> ? [X9: tm] :
( ( member_tm2 @ X9 @ ( terms @ H2 ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X9 @ P2 ) ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm2 @ ( neg @ ( neg @ P2 ) ) @ H2 )
=> ( member_fm2 @ P2 @ H2 ) ) ) ) ) ).
% Hintikka_def
thf(fact_90_Neg__exhaust,axiom,
! [X3: fm] :
( ! [I: nat,Ts2: list_tm] :
( X3
!= ( pre @ I @ Ts2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( imp @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( dis @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( con @ P4 @ Q2 ) )
=> ( ! [P4: fm] :
( X3
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X3
!= ( uni @ P4 ) )
=> ( ! [I: nat,Ts2: list_tm] :
( X3
!= ( neg @ ( pre @ I @ Ts2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( neg @ ( imp @ P4 @ Q2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( neg @ ( dis @ P4 @ Q2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X3
!= ( neg @ ( con @ P4 @ Q2 ) ) )
=> ( ! [P4: fm] :
( X3
!= ( neg @ ( exi @ P4 ) ) )
=> ( ! [P4: fm] :
( X3
!= ( neg @ ( uni @ P4 ) ) )
=> ~ ! [P4: fm] :
( X3
!= ( neg @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Neg_exhaust
thf(fact_91_subst_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm,S: tm,K: nat] :
( ( subst @ ( pre @ B2 @ Ts ) @ S @ K )
= ( pre @ B2 @ ( substts @ Ts @ S @ K ) ) ) ).
% subst.simps(1)
thf(fact_92_sub_Osimps_I1_J,axiom,
! [V: nat,S: tm,I2: nat,L: list_tm] :
( ( sub @ V @ S @ ( pre @ I2 @ L ) )
= ( pre @ I2 @ ( sub_list @ V @ S @ L ) ) ) ).
% sub.simps(1)
thf(fact_93_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_94_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_95_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_96_fm_Osize__gen_I1_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_97_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_98_branchDone_Ocases,axiom,
! [X3: list_fm] :
( ( X3 != nil_fm )
=> ( ! [P4: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( X3
!= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ! [V3: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ~ ! [V3: fm,Z: list_fm] :
( X3
!= ( cons_fm @ ( uni @ V3 ) @ Z ) ) ) ) ) ) ) ) ) ).
% branchDone.cases
thf(fact_99_fm_Osize_I8_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size(8)
thf(fact_100_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_101_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_102_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_103_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_104_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_105_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_106_size__sub,axiom,
! [I2: nat,T: tm,P: fm] :
( ( size_size_fm @ ( sub @ I2 @ T @ P ) )
= ( size_size_fm @ P ) ) ).
% size_sub
thf(fact_107_s5_I2_J,axiom,
( sub_list
= ( ^ [V2: nat,S2: tm,L2: list_tm] : ( substts @ L2 @ S2 @ V2 ) ) ) ).
% s5(2)
thf(fact_108_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_109_int__if,axiom,
! [P3: $o,A: nat,B2: nat] :
( ( P3
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P3 @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P3
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P3 @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_110_nat__int__comparison_I1_J,axiom,
( ( ^ [Y9: nat,Z2: nat] : ( Y9 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_111_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_112_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_113_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_114_verit__comp__simplify1_I2_J,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_115_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_116_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_117_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_118_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_119_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_120_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_121_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_122_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_123_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_124_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_125_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_126_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_127_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_128_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_129_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_130_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_131_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_132_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_133_list_Oinject,axiom,
! [X21: fm,X222: list_fm,Y21: fm,Y22: list_fm] :
( ( ( cons_fm @ X21 @ X222 )
= ( cons_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_134_list_Oinject,axiom,
! [X21: tm,X222: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X222 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_135_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_136_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_137_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_138_order__refl,axiom,
! [X3: set_tm] : ( ord_less_eq_set_tm @ X3 @ X3 ) ).
% order_refl
thf(fact_139_order__refl,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% order_refl
thf(fact_140_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_141_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_142_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_143_dual__order_Orefl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% dual_order.refl
thf(fact_144_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_145_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_146_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_147_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_148_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_149_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_150_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_fm,P3: list_fm > list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: fm,Zs2: list_fm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_151_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_tm,P3: list_fm > list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: fm,Zs2: list_fm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_152_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_int,P3: list_fm > list_fm > list_fm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_fm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: fm,Zs2: list_fm,W: int,Ws2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_153_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_fm,P3: list_fm > list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: tm,Zs2: list_tm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_154_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_tm,P3: list_fm > list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: tm,Zs2: list_tm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_155_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_int,P3: list_fm > list_fm > list_tm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_tm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: tm,Zs2: list_tm,W: int,Ws2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_156_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_int,Ws: list_fm,P3: list_fm > list_fm > list_int > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_int @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: int,Zs2: list_int,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_157_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_int,Ws: list_tm,P3: list_fm > list_fm > list_int > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_int @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: int,Zs2: list_int,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_158_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_int,Ws: list_int,P3: list_fm > list_fm > list_int > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_int @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: int,Zs2: list_int,W: int,Ws2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_159_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,Ws: list_fm,P3: list_fm > list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_fm @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm,Z: fm,Zs2: list_fm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_160_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,P3: list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_161_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,P3: list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_162_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_int,P3: list_fm > list_fm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_fm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm,Z: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_163_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,P3: list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm,Z: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_164_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,P3: list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm,Z: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_165_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_int,P3: list_fm > list_tm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_tm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm,Z: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( ( size_size_list_tm @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_166_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_fm,P3: list_fm > list_int > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_int @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: int,Ys2: list_int,Z: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_167_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_tm,P3: list_fm > list_int > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_int @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: int,Ys2: list_int,Z: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) @ ( cons_tm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_168_list__induct3,axiom,
! [Xs: list_fm,Ys: list_int,Zs: list_int,P3: list_fm > list_int > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_int @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: int,Ys2: list_int,Z: int,Zs2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_169_list__induct3,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,P3: list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm,Y10: fm,Ys2: list_fm,Z: fm,Zs2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( ( size_size_list_fm @ Ys2 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) @ ( cons_fm @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_170_list__induct2,axiom,
! [Xs: list_fm,Ys: list_fm,P3: list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_171_list__induct2,axiom,
! [Xs: list_fm,Ys: list_tm,P3: list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_172_list__induct2,axiom,
! [Xs: list_fm,Ys: list_int,P3: list_fm > list_int > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm,Y10: int,Ys2: list_int] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_173_list__induct2,axiom,
! [Xs: list_tm,Ys: list_fm,P3: list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm,Y10: fm,Ys2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_174_list__induct2,axiom,
! [Xs: list_tm,Ys: list_tm,P3: list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm,Y10: tm,Ys2: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_175_list__induct2,axiom,
! [Xs: list_tm,Ys: list_int,P3: list_tm > list_int > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_int )
=> ( ! [X: tm,Xs2: list_tm,Y10: int,Ys2: list_int] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_176_list__induct2,axiom,
! [Xs: list_int,Ys: list_fm,P3: list_int > list_fm > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_int @ nil_fm )
=> ( ! [X: int,Xs2: list_int,Y10: fm,Ys2: list_fm] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_fm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_177_list__induct2,axiom,
! [Xs: list_int,Ys: list_tm,P3: list_int > list_tm > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_int @ nil_tm )
=> ( ! [X: int,Xs2: list_int,Y10: tm,Ys2: list_tm] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_tm @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_178_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P3: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P3 @ nil_int @ nil_int )
=> ( ! [X: int,Xs2: list_int,Y10: int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_179_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_180_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B2: nat] :
( ( P3 @ K )
=> ( ! [Y10: nat] :
( ( P3 @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B2 ) )
=> ? [X: nat] :
( ( P3 @ X )
& ! [Y13: nat] :
( ( P3 @ Y13 )
=> ( ord_less_eq_nat @ Y13 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_181_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_182_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_183_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_184_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_185_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_186_sub__list_Osimps_I1_J,axiom,
! [V: nat,S: tm] :
( ( sub_list @ V @ S @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_187_substts_Osimps_I1_J,axiom,
! [S: tm,K: nat] :
( ( substts @ nil_tm @ S @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_188_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_189_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_190_order__antisym__conv,axiom,
! [Y: set_tm,X3: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X3 )
=> ( ( ord_less_eq_set_tm @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_191_order__antisym__conv,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_192_order__antisym__conv,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_193_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_194_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_195_linorder__le__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_196_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_197_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_198_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_199_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_200_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_201_ord__le__eq__subst,axiom,
! [A: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_203_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_204_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C2: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_210_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_211_ord__eq__le__subst,axiom,
! [A: real,F: int > real,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_212_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_213_ord__eq__le__subst,axiom,
! [A: int,F: real > int,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_214_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_215_ord__eq__le__subst,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_216_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_217_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_218_linorder__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
| ( ord_less_eq_real @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_219_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_220_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_221_order__eq__refl,axiom,
! [X3: set_tm,Y: set_tm] :
( ( X3 = Y )
=> ( ord_less_eq_set_tm @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_222_order__eq__refl,axiom,
! [X3: real,Y: real] :
( ( X3 = Y )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_223_order__eq__refl,axiom,
! [X3: set_nat,Y: set_nat] :
( ( X3 = Y )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_224_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_225_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_226_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_227_order__subst2,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_228_order__subst2,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_229_order__subst2,axiom,
! [A: int,B2: int,F: int > real,C2: real] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_230_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_231_order__subst2,axiom,
! [A: real,B2: real,F: real > int,C2: int] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_232_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_233_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C2: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_234_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_235_order__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_236_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_237_order__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_238_order__subst1,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_239_order__subst1,axiom,
! [A: int,F: real > int,B2: real,C2: real] :
( ( ord_less_eq_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_240_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_eq_nat @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_241_order__subst1,axiom,
! [A: real,F: int > real,B2: int,C2: int] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_eq_int @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C2: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_eq_real @ X @ Y10 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_243_order__subst1,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ! [X: set_tm,Y10: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y10 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_244_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y9: nat,Z2: nat] : ( Y9 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_245_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y9: int,Z2: int] : ( Y9 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_246_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y9: set_tm,Z2: set_tm] : ( Y9 = Z2 ) )
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B3 )
& ( ord_less_eq_set_tm @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_247_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y9: real,Z2: real] : ( Y9 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_248_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y9: set_nat,Z2: set_nat] : ( Y9 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_249_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_250_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_251_antisym,axiom,
! [A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_252_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_253_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_254_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_255_dual__order_Otrans,axiom,
! [B2: int,A: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C2 @ B2 )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_256_dual__order_Otrans,axiom,
! [B2: set_tm,A: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C2 @ B2 )
=> ( ord_less_eq_set_tm @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_257_dual__order_Otrans,axiom,
! [B2: real,A: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_258_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_259_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_260_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_261_dual__order_Oantisym,axiom,
! [B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_262_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_263_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_264_dual__order_Oeq__iff,axiom,
( ( ^ [Y9: nat,Z2: nat] : ( Y9 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_265_dual__order_Oeq__iff,axiom,
( ( ^ [Y9: int,Z2: int] : ( Y9 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_266_dual__order_Oeq__iff,axiom,
( ( ^ [Y9: set_tm,Z2: set_tm] : ( Y9 = Z2 ) )
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ B3 @ A3 )
& ( ord_less_eq_set_tm @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_267_dual__order_Oeq__iff,axiom,
( ( ^ [Y9: real,Z2: real] : ( Y9 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_268_dual__order_Oeq__iff,axiom,
( ( ^ [Y9: set_nat,Z2: set_nat] : ( Y9 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_269_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: nat,B: nat] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_270_linorder__wlog,axiom,
! [P3: int > int > $o,A: int,B2: int] :
( ! [A4: int,B: int] :
( ( ord_less_eq_int @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: int,B: int] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_271_linorder__wlog,axiom,
! [P3: real > real > $o,A: real,B2: real] :
( ! [A4: real,B: real] :
( ( ord_less_eq_real @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: real,B: real] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_272_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_273_order__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_274_order__trans,axiom,
! [X3: set_tm,Y: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ( ord_less_eq_set_tm @ Y @ Z3 )
=> ( ord_less_eq_set_tm @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_275_order__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_276_order__trans,axiom,
! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_277_order_Otrans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_278_order_Otrans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_279_order_Otrans,axiom,
! [A: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).
% order.trans
thf(fact_280_order_Otrans,axiom,
! [A: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_281_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_282_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_283_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_284_order__antisym,axiom,
! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ( ord_less_eq_set_tm @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_285_order__antisym,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_eq_real @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_286_order__antisym,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_287_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_288_ord__le__eq__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_289_ord__le__eq__trans,axiom,
! [A: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_290_ord__le__eq__trans,axiom,
! [A: real,B2: real,C2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_291_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_292_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_293_ord__eq__le__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_294_ord__eq__le__trans,axiom,
! [A: set_tm,B2: set_tm,C2: set_tm] :
( ( A = B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_295_ord__eq__le__trans,axiom,
! [A: real,B2: real,C2: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_296_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_297_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y9: nat,Z2: nat] : ( Y9 = Z2 ) )
= ( ^ [X9: nat,Y14: nat] :
( ( ord_less_eq_nat @ X9 @ Y14 )
& ( ord_less_eq_nat @ Y14 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_298_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y9: int,Z2: int] : ( Y9 = Z2 ) )
= ( ^ [X9: int,Y14: int] :
( ( ord_less_eq_int @ X9 @ Y14 )
& ( ord_less_eq_int @ Y14 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_299_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y9: set_tm,Z2: set_tm] : ( Y9 = Z2 ) )
= ( ^ [X9: set_tm,Y14: set_tm] :
( ( ord_less_eq_set_tm @ X9 @ Y14 )
& ( ord_less_eq_set_tm @ Y14 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_300_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y9: real,Z2: real] : ( Y9 = Z2 ) )
= ( ^ [X9: real,Y14: real] :
( ( ord_less_eq_real @ X9 @ Y14 )
& ( ord_less_eq_real @ Y14 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_301_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y9: set_nat,Z2: set_nat] : ( Y9 = Z2 ) )
= ( ^ [X9: set_nat,Y14: set_nat] :
( ( ord_less_eq_set_nat @ X9 @ Y14 )
& ( ord_less_eq_set_nat @ Y14 @ X9 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_302_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_303_le__cases3,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_304_le__cases3,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X3 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_305_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_306_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_307_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_308_not__Cons__self2,axiom,
! [X3: fm,Xs: list_fm] :
( ( cons_fm @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_309_not__Cons__self2,axiom,
! [X3: tm,Xs: list_tm] :
( ( cons_tm @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_310_not__Cons__self2,axiom,
! [X3: int,Xs: list_int] :
( ( cons_int @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_311_impossible__Cons,axiom,
! [Xs: list_fm,Ys: list_fm,X3: fm] :
( ( ord_less_eq_nat @ ( size_size_list_fm @ Xs ) @ ( size_size_list_fm @ Ys ) )
=> ( Xs
!= ( cons_fm @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_312_impossible__Cons,axiom,
! [Xs: list_tm,Ys: list_tm,X3: tm] :
( ( ord_less_eq_nat @ ( size_size_list_tm @ Xs ) @ ( size_size_list_tm @ Ys ) )
=> ( Xs
!= ( cons_tm @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_313_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X3: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_314_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_315_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_316_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_317_list__nonempty__induct,axiom,
! [Xs: list_fm,P3: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X: fm] : ( P3 @ ( cons_fm @ X @ nil_fm ) )
=> ( ! [X: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_318_list__nonempty__induct,axiom,
! [Xs: list_tm,P3: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X: tm] : ( P3 @ ( cons_tm @ X @ nil_tm ) )
=> ( ! [X: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_319_list__nonempty__induct,axiom,
! [Xs: list_int,P3: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P3 @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_320_list__induct2_H,axiom,
! [P3: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P3 @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y10: fm,Ys2: list_fm] : ( P3 @ nil_fm @ ( cons_fm @ Y10 @ Ys2 ) )
=> ( ! [X: fm,Xs2: list_fm,Y10: fm,Ys2: list_fm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_321_list__induct2_H,axiom,
! [P3: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P3 @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y10: tm,Ys2: list_tm] : ( P3 @ nil_fm @ ( cons_tm @ Y10 @ Ys2 ) )
=> ( ! [X: fm,Xs2: list_fm,Y10: tm,Ys2: list_tm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_322_list__induct2_H,axiom,
! [P3: list_fm > list_int > $o,Xs: list_fm,Ys: list_int] :
( ( P3 @ nil_fm @ nil_int )
=> ( ! [X: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X @ Xs2 ) @ nil_int )
=> ( ! [Y10: int,Ys2: list_int] : ( P3 @ nil_fm @ ( cons_int @ Y10 @ Ys2 ) )
=> ( ! [X: fm,Xs2: list_fm,Y10: int,Ys2: list_int] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_fm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_323_list__induct2_H,axiom,
! [P3: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P3 @ nil_tm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y10: fm,Ys2: list_fm] : ( P3 @ nil_tm @ ( cons_fm @ Y10 @ Ys2 ) )
=> ( ! [X: tm,Xs2: list_tm,Y10: fm,Ys2: list_fm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_324_list__induct2_H,axiom,
! [P3: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P3 @ nil_tm @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y10: tm,Ys2: list_tm] : ( P3 @ nil_tm @ ( cons_tm @ Y10 @ Ys2 ) )
=> ( ! [X: tm,Xs2: list_tm,Y10: tm,Ys2: list_tm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_325_list__induct2_H,axiom,
! [P3: list_tm > list_int > $o,Xs: list_tm,Ys: list_int] :
( ( P3 @ nil_tm @ nil_int )
=> ( ! [X: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X @ Xs2 ) @ nil_int )
=> ( ! [Y10: int,Ys2: list_int] : ( P3 @ nil_tm @ ( cons_int @ Y10 @ Ys2 ) )
=> ( ! [X: tm,Xs2: list_tm,Y10: int,Ys2: list_int] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_tm @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_326_list__induct2_H,axiom,
! [P3: list_int > list_fm > $o,Xs: list_int,Ys: list_fm] :
( ( P3 @ nil_int @ nil_fm )
=> ( ! [X: int,Xs2: list_int] : ( P3 @ ( cons_int @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y10: fm,Ys2: list_fm] : ( P3 @ nil_int @ ( cons_fm @ Y10 @ Ys2 ) )
=> ( ! [X: int,Xs2: list_int,Y10: fm,Ys2: list_fm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_fm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_327_list__induct2_H,axiom,
! [P3: list_int > list_tm > $o,Xs: list_int,Ys: list_tm] :
( ( P3 @ nil_int @ nil_tm )
=> ( ! [X: int,Xs2: list_int] : ( P3 @ ( cons_int @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y10: tm,Ys2: list_tm] : ( P3 @ nil_int @ ( cons_tm @ Y10 @ Ys2 ) )
=> ( ! [X: int,Xs2: list_int,Y10: tm,Ys2: list_tm] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_tm @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_328_list__induct2_H,axiom,
! [P3: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P3 @ nil_int @ nil_int )
=> ( ! [X: int,Xs2: list_int] : ( P3 @ ( cons_int @ X @ Xs2 ) @ nil_int )
=> ( ! [Y10: int,Ys2: list_int] : ( P3 @ nil_int @ ( cons_int @ Y10 @ Ys2 ) )
=> ( ! [X: int,Xs2: list_int,Y10: int,Ys2: list_int] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y10 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_329_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y14: fm,Ys3: list_fm] :
( Xs
= ( cons_fm @ Y14 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_330_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y14: tm,Ys3: list_tm] :
( Xs
= ( cons_tm @ Y14 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_331_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y14: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y14 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_332_remdups__adj_Ocases,axiom,
! [X3: list_fm] :
( ( X3 != nil_fm )
=> ( ! [X: fm] :
( X3
!= ( cons_fm @ X @ nil_fm ) )
=> ~ ! [X: fm,Y10: fm,Xs2: list_fm] :
( X3
!= ( cons_fm @ X @ ( cons_fm @ Y10 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_333_remdups__adj_Ocases,axiom,
! [X3: list_tm] :
( ( X3 != nil_tm )
=> ( ! [X: tm] :
( X3
!= ( cons_tm @ X @ nil_tm ) )
=> ~ ! [X: tm,Y10: tm,Xs2: list_tm] :
( X3
!= ( cons_tm @ X @ ( cons_tm @ Y10 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_334_remdups__adj_Ocases,axiom,
! [X3: list_int] :
( ( X3 != nil_int )
=> ( ! [X: int] :
( X3
!= ( cons_int @ X @ nil_int ) )
=> ~ ! [X: int,Y10: int,Xs2: list_int] :
( X3
!= ( cons_int @ X @ ( cons_int @ Y10 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_335_transpose_Ocases,axiom,
! [X3: list_list_fm] :
( ( X3 != nil_list_fm )
=> ( ! [Xss: list_list_fm] :
( X3
!= ( cons_list_fm @ nil_fm @ Xss ) )
=> ~ ! [X: fm,Xs2: list_fm,Xss: list_list_fm] :
( X3
!= ( cons_list_fm @ ( cons_fm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_336_transpose_Ocases,axiom,
! [X3: list_list_tm] :
( ( X3 != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X3
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X: tm,Xs2: list_tm,Xss: list_list_tm] :
( X3
!= ( cons_list_tm @ ( cons_tm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_337_transpose_Ocases,axiom,
! [X3: list_list_int] :
( ( X3 != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X3
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X: int,Xs2: list_int,Xss: list_list_int] :
( X3
!= ( cons_list_int @ ( cons_int @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_338_min__list_Ocases,axiom,
! [X3: list_int] :
( ! [X: int,Xs2: list_int] :
( X3
!= ( cons_int @ X @ Xs2 ) )
=> ( X3 = nil_int ) ) ).
% min_list.cases
thf(fact_339_list_Oexhaust,axiom,
! [Y: list_fm] :
( ( Y != nil_fm )
=> ~ ! [X212: fm,X223: list_fm] :
( Y
!= ( cons_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_340_list_Oexhaust,axiom,
! [Y: list_tm] :
( ( Y != nil_tm )
=> ~ ! [X212: tm,X223: list_tm] :
( Y
!= ( cons_tm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_341_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X223: list_int] :
( Y
!= ( cons_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_342_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X222: list_fm] :
( ( List
= ( cons_fm @ X21 @ X222 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_343_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X222: list_tm] :
( ( List
= ( cons_tm @ X21 @ X222 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_344_list_OdiscI,axiom,
! [List: list_int,X21: int,X222: list_int] :
( ( List
= ( cons_int @ X21 @ X222 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_345_list_Odistinct_I1_J,axiom,
! [X21: fm,X222: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_346_list_Odistinct_I1_J,axiom,
! [X21: tm,X222: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_347_list_Odistinct_I1_J,axiom,
! [X21: int,X222: list_int] :
( nil_int
!= ( cons_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_348_conj__le__cong,axiom,
! [X3: int,X10: int,P3: $o,P5: $o] :
( ( X3 = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P3 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P3 )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_349_imp__le__cong,axiom,
! [X3: int,X10: int,P3: $o,P5: $o] :
( ( X3 = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P3 = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P3 )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_350_real__arch__simple,axiom,
! [X3: real] :
? [N2: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_351_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_352_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_353_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_354_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= nil_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_355_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_356_GammaUni,axiom,
! [T: tm,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T @ P ) ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( uni @ P ) ) @ Z3 ) ) ) ).
% GammaUni
thf(fact_357_insert__Nil,axiom,
! [X3: fm] :
( ( insert_fm @ X3 @ nil_fm )
= ( cons_fm @ X3 @ nil_fm ) ) ).
% insert_Nil
thf(fact_358_insert__Nil,axiom,
! [X3: tm] :
( ( insert_tm @ X3 @ nil_tm )
= ( cons_tm @ X3 @ nil_tm ) ) ).
% insert_Nil
thf(fact_359_insert__Nil,axiom,
! [X3: int] :
( ( insert_int @ X3 @ nil_int )
= ( cons_int @ X3 @ nil_int ) ) ).
% insert_Nil
thf(fact_360_Neg,axiom,
! [P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P ) ) @ Z3 ) ) ) ).
% Neg
thf(fact_361_AlphaDis,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ ( cons_fm @ Q @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P @ Q ) @ Z3 ) ) ) ).
% AlphaDis
thf(fact_362_BetaCon,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P @ Q ) @ Z3 ) ) ) ) ).
% BetaCon
thf(fact_363_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_fm @ N @ nil_fm )
= N ) ).
% gen_length_code(1)
thf(fact_364_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_tm @ N @ nil_tm )
= N ) ).
% gen_length_code(1)
thf(fact_365_AlphaImp,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ Q @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P @ Q ) @ Z3 ) ) ) ).
% AlphaImp
thf(fact_366_AlphaCon,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ ( neg @ Q ) @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P @ Q ) ) @ Z3 ) ) ) ).
% AlphaCon
thf(fact_367_BetaImp,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P @ Q ) ) @ Z3 ) ) ) ) ).
% BetaImp
thf(fact_368_BetaDis,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P @ Q ) ) @ Z3 ) ) ) ) ).
% BetaDis
thf(fact_369_n__lists_Osimps_I1_J,axiom,
! [Xs: list_fm] :
( ( n_lists_fm @ zero_zero_nat @ Xs )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% n_lists.simps(1)
thf(fact_370_n__lists_Osimps_I1_J,axiom,
! [Xs: list_tm] :
( ( n_lists_tm @ zero_zero_nat @ Xs )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% n_lists.simps(1)
thf(fact_371_GammaExi,axiom,
! [T: tm,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T @ P ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( exi @ P ) @ Z3 ) ) ) ).
% GammaExi
thf(fact_372_sublists_Osimps_I1_J,axiom,
( ( sublists_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% sublists.simps(1)
thf(fact_373_sublists_Osimps_I1_J,axiom,
( ( sublists_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% sublists.simps(1)
thf(fact_374_Basic,axiom,
! [P: fm,Z3: list_fm] :
( ( member_fm @ ( neg @ P ) @ Z3 )
=> ( sequent_calculus @ ( cons_fm @ P @ Z3 ) ) ) ).
% Basic
thf(fact_375_product__lists_Osimps_I1_J,axiom,
( ( product_lists_fm @ nil_list_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_376_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_377_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_378_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_379_branchDone_Oelims_I1_J,axiom,
! [X3: list_fm,Y: $o] :
( ( ( branchDone @ X3 )
= Y )
=> ( ( ( X3 = nil_fm )
=> Y )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(1)
thf(fact_380_branchDone_Oelims_I3_J,axiom,
! [X3: list_fm] :
( ~ ( branchDone @ X3 )
=> ( ( X3 != nil_fm )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(3)
thf(fact_381_nths__singleton,axiom,
! [A2: set_nat,X3: fm] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_fm @ ( cons_fm @ X3 @ nil_fm ) @ A2 )
= ( cons_fm @ X3 @ nil_fm ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_fm @ ( cons_fm @ X3 @ nil_fm ) @ A2 )
= nil_fm ) ) ) ).
% nths_singleton
thf(fact_382_nths__singleton,axiom,
! [A2: set_nat,X3: tm] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_tm @ ( cons_tm @ X3 @ nil_tm ) @ A2 )
= ( cons_tm @ X3 @ nil_tm ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_tm @ ( cons_tm @ X3 @ nil_tm ) @ A2 )
= nil_tm ) ) ) ).
% nths_singleton
thf(fact_383_nths__singleton,axiom,
! [A2: set_nat,X3: int] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X3 @ nil_int ) @ A2 )
= ( cons_int @ X3 @ nil_int ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X3 @ nil_int ) @ A2 )
= nil_int ) ) ) ).
% nths_singleton
thf(fact_384_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_385_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_386_of__int__eq__iff,axiom,
! [W2: int,Z3: int] :
( ( ( ring_1_of_int_real @ W2 )
= ( ring_1_of_int_real @ Z3 ) )
= ( W2 = Z3 ) ) ).
% of_int_eq_iff
thf(fact_387_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_fm @ nil_fm @ A2 )
= nil_fm ) ).
% nths_nil
thf(fact_388_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_tm @ nil_tm @ A2 )
= nil_tm ) ).
% nths_nil
thf(fact_389_in__set__insert,axiom,
! [X3: real,Xs: list_real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_390_in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_391_in__set__insert,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_392_in__set__insert,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_393_member,axiom,
( member_real
= ( ^ [P2: real,Z4: list_real] : ( member_real2 @ P2 @ ( set_real2 @ Z4 ) ) ) ) ).
% member
thf(fact_394_member,axiom,
( member_nat
= ( ^ [P2: nat,Z4: list_nat] : ( member_nat2 @ P2 @ ( set_nat2 @ Z4 ) ) ) ) ).
% member
thf(fact_395_member,axiom,
( member_fm
= ( ^ [P2: fm,Z4: list_fm] : ( member_fm2 @ P2 @ ( set_fm2 @ Z4 ) ) ) ) ).
% member
thf(fact_396_member,axiom,
( member_tm
= ( ^ [P2: tm,Z4: list_tm] : ( member_tm2 @ P2 @ ( set_tm2 @ Z4 ) ) ) ) ).
% member
thf(fact_397_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_398_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_399_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_400_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_401_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= zero_zero_int )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_402_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_real @ Z3 )
= zero_zero_real )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_403_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_404_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_405_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_406_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) ).
% of_int_of_nat_eq
thf(fact_407_not__in__set__insert,axiom,
! [X3: real,Xs: list_real] :
( ~ ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X3 @ Xs )
= ( cons_real @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_408_not__in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= ( cons_nat @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_409_not__in__set__insert,axiom,
! [X3: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X3 @ Xs )
= ( cons_fm @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_410_not__in__set__insert,axiom,
! [X3: tm,Xs: list_tm] :
( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X3 @ Xs )
= ( cons_tm @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_411_not__in__set__insert,axiom,
! [X3: int,Xs: list_int] :
( ~ ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
=> ( ( insert_int @ X3 @ Xs )
= ( cons_int @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_412_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_413_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_414_subset__code_I1_J,axiom,
! [Xs: list_real,B4: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B4 )
= ( ! [X9: real] :
( ( member_real2 @ X9 @ ( set_real2 @ Xs ) )
=> ( member_real2 @ X9 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_415_subset__code_I1_J,axiom,
! [Xs: list_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B4 )
= ( ! [X9: fm] :
( ( member_fm2 @ X9 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X9 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_416_subset__code_I1_J,axiom,
! [Xs: list_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B4 )
= ( ! [X9: tm] :
( ( member_tm2 @ X9 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X9 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_417_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X9 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_418_set__nths__subset,axiom,
! [Xs: list_fm,I3: set_nat] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( nths_fm @ Xs @ I3 ) ) @ ( set_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_419_set__nths__subset,axiom,
! [Xs: list_tm,I3: set_nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( nths_tm @ Xs @ I3 ) ) @ ( set_tm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_420_set__nths__subset,axiom,
! [Xs: list_nat,I3: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I3 ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_421_branchDone__contradiction,axiom,
( branchDone
= ( ^ [Z4: list_fm] :
? [P2: fm] :
( ( member_fm2 @ P2 @ ( set_fm2 @ Z4 ) )
& ( member_fm2 @ ( neg @ P2 ) @ ( set_fm2 @ Z4 ) ) ) ) ) ).
% branchDone_contradiction
thf(fact_422_in__set__nthsD,axiom,
! [X3: real,Xs: list_real,I3: set_nat] :
( ( member_real2 @ X3 @ ( set_real2 @ ( nths_real @ Xs @ I3 ) ) )
=> ( member_real2 @ X3 @ ( set_real2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_423_in__set__nthsD,axiom,
! [X3: nat,Xs: list_nat,I3: set_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ ( nths_nat @ Xs @ I3 ) ) )
=> ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_424_in__set__nthsD,axiom,
! [X3: fm,Xs: list_fm,I3: set_nat] :
( ( member_fm2 @ X3 @ ( set_fm2 @ ( nths_fm @ Xs @ I3 ) ) )
=> ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_425_in__set__nthsD,axiom,
! [X3: tm,Xs: list_tm,I3: set_nat] :
( ( member_tm2 @ X3 @ ( set_tm2 @ ( nths_tm @ Xs @ I3 ) ) )
=> ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_426_notin__set__nthsI,axiom,
! [X3: real,Xs: list_real,I3: set_nat] :
( ~ ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ~ ( member_real2 @ X3 @ ( set_real2 @ ( nths_real @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_427_notin__set__nthsI,axiom,
! [X3: nat,Xs: list_nat,I3: set_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X3 @ ( set_nat2 @ ( nths_nat @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_428_notin__set__nthsI,axiom,
! [X3: fm,Xs: list_fm,I3: set_nat] :
( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm2 @ X3 @ ( set_fm2 @ ( nths_fm @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_429_notin__set__nthsI,axiom,
! [X3: tm,Xs: list_tm,I3: set_nat] :
( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ~ ( member_tm2 @ X3 @ ( set_tm2 @ ( nths_tm @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_430_branchDone_Osimps_I2_J,axiom,
! [P: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( neg @ P ) @ Z3 ) )
= ( ( member_fm2 @ P @ ( set_fm2 @ Z3 ) )
| ( member_fm2 @ ( neg @ ( neg @ P ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(2)
thf(fact_431_ex__le__of__int,axiom,
! [X3: real] :
? [Z: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z ) ) ).
% ex_le_of_int
thf(fact_432_set__subset__Cons,axiom,
! [Xs: list_fm,X3: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_433_set__subset__Cons,axiom,
! [Xs: list_int,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_434_set__subset__Cons,axiom,
! [Xs: list_tm,X3: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_435_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_436_set__ConsD,axiom,
! [Y: real,X3: real,Xs: list_real] :
( ( member_real2 @ Y @ ( set_real2 @ ( cons_real @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_real2 @ Y @ ( set_real2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_437_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_438_set__ConsD,axiom,
! [Y: fm,X3: fm,Xs: list_fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_fm2 @ Y @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_439_set__ConsD,axiom,
! [Y: tm,X3: tm,Xs: list_tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_tm2 @ Y @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_440_set__ConsD,axiom,
! [Y: int,X3: int,Xs: list_int] :
( ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_int2 @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_441_list_Oset__cases,axiom,
! [E: real,A: list_real] :
( ( member_real2 @ E @ ( set_real2 @ A ) )
=> ( ! [Z22: list_real] :
( A
!= ( cons_real @ E @ Z22 ) )
=> ~ ! [Z1: real,Z22: list_real] :
( ( A
= ( cons_real @ Z1 @ Z22 ) )
=> ~ ( member_real2 @ E @ ( set_real2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_442_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_443_list_Oset__cases,axiom,
! [E: fm,A: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A ) )
=> ( ! [Z22: list_fm] :
( A
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_444_list_Oset__cases,axiom,
! [E: tm,A: list_tm] :
( ( member_tm2 @ E @ ( set_tm2 @ A ) )
=> ( ! [Z22: list_tm] :
( A
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm2 @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_445_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int2 @ E @ ( set_int2 @ A ) )
=> ( ! [Z22: list_int] :
( A
!= ( cons_int @ E @ Z22 ) )
=> ~ ! [Z1: int,Z22: list_int] :
( ( A
= ( cons_int @ Z1 @ Z22 ) )
=> ~ ( member_int2 @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_446_list_Oset__intros_I1_J,axiom,
! [X21: real,X222: list_real] : ( member_real2 @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_447_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_448_list_Oset__intros_I1_J,axiom,
! [X21: fm,X222: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_449_list_Oset__intros_I1_J,axiom,
! [X21: tm,X222: list_tm] : ( member_tm2 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_450_list_Oset__intros_I1_J,axiom,
! [X21: int,X222: list_int] : ( member_int2 @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_451_list_Oset__intros_I2_J,axiom,
! [Y: real,X222: list_real,X21: real] :
( ( member_real2 @ Y @ ( set_real2 @ X222 ) )
=> ( member_real2 @ Y @ ( set_real2 @ ( cons_real @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_452_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_453_list_Oset__intros_I2_J,axiom,
! [Y: fm,X222: list_fm,X21: fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ X222 ) )
=> ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_454_list_Oset__intros_I2_J,axiom,
! [Y: tm,X222: list_tm,X21: tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ X222 ) )
=> ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_455_list_Oset__intros_I2_J,axiom,
! [Y: int,X222: list_int,X21: int] :
( ( member_int2 @ Y @ ( set_int2 @ X222 ) )
=> ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_456_branchDone_Osimps_I1_J,axiom,
~ ( branchDone @ nil_fm ) ).
% branchDone.simps(1)
thf(fact_457_branchDone_Osimps_I7_J,axiom,
! [V: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( exi @ V ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(7)
thf(fact_458_branchDone_Osimps_I8_J,axiom,
! [V: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( uni @ V ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(8)
thf(fact_459_branchDone_Osimps_I4_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( imp @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( imp @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(4)
thf(fact_460_branchDone_Osimps_I5_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( dis @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( dis @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(5)
thf(fact_461_branchDone_Osimps_I6_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( con @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( con @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(6)
thf(fact_462_branchDone_Osimps_I3_J,axiom,
! [V: nat,Va2: list_tm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( pre @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( pre @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(3)
thf(fact_463_List_Oinsert__def,axiom,
( insert_real
= ( ^ [X9: real,Xs3: list_real] : ( if_list_real @ ( member_real2 @ X9 @ ( set_real2 @ Xs3 ) ) @ Xs3 @ ( cons_real @ X9 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_464_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X9: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X9 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X9 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_465_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X9: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm2 @ X9 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X9 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_466_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X9: tm,Xs3: list_tm] : ( if_list_tm @ ( member_tm2 @ X9 @ ( set_tm2 @ Xs3 ) ) @ Xs3 @ ( cons_tm @ X9 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_467_List_Oinsert__def,axiom,
( insert_int
= ( ^ [X9: int,Xs3: list_int] : ( if_list_int @ ( member_int2 @ X9 @ ( set_int2 @ Xs3 ) ) @ Xs3 @ ( cons_int @ X9 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_468_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( member_fm @ P @ ( cons_fm @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_fm @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_469_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: tm,Q: tm,Z3: list_tm] :
( ( member_tm @ P @ ( cons_tm @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_tm @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_470_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: int,Q: int,Z3: list_int] :
( ( member_int @ P @ ( cons_int @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_int @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_471_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_472_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_473_of__int__nonneg,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_nonneg
thf(fact_474_of__int__nonneg,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_int_nonneg
thf(fact_475_branchDone_Oelims_I2_J,axiom,
! [X3: list_fm] :
( ( branchDone @ X3 )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(2)
thf(fact_476_ext_Osimps_I2_J,axiom,
! [Y: list_fm,P: fm,Z3: list_fm] :
( ( ext_fm @ Y @ ( cons_fm @ P @ Z3 ) )
= ( ( ( member_fm @ P @ Y )
=> ( ext_fm @ Y @ Z3 ) )
& ( member_fm @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_477_ext_Osimps_I2_J,axiom,
! [Y: list_tm,P: tm,Z3: list_tm] :
( ( ext_tm @ Y @ ( cons_tm @ P @ Z3 ) )
= ( ( ( member_tm @ P @ Y )
=> ( ext_tm @ Y @ Z3 ) )
& ( member_tm @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_478_ext_Osimps_I2_J,axiom,
! [Y: list_int,P: int,Z3: list_int] :
( ( ext_int @ Y @ ( cons_int @ P @ Z3 ) )
= ( ( ( member_int @ P @ Y )
=> ( ext_int @ Y @ Z3 ) )
& ( member_int @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_479_branchDone_Opelims_I3_J,axiom,
! [X3: list_fm] :
( ~ ( branchDone @ X3 )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ( ( X3 = nil_fm )
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(3)
thf(fact_480_branchDone_Opelims_I1_J,axiom,
! [X3: list_fm,Y: $o] :
( ( ( branchDone @ X3 )
= Y )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ( ( X3 = nil_fm )
=> ( ~ Y
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) ) )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z ) ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z ) ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(1)
thf(fact_481_sequent__calculus_Osimps,axiom,
( sequent_calculus
= ( ^ [A3: list_fm] :
( ? [P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ P2 @ Z4 ) )
& ( member_fm @ ( neg @ P2 ) @ Z4 ) )
| ? [P2: fm,Q3: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( dis @ P2 @ Q3 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ ( cons_fm @ Q3 @ Z4 ) ) ) )
| ? [P2: fm,Q3: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( imp @ P2 @ Q3 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ ( cons_fm @ Q3 @ Z4 ) ) ) )
| ? [P2: fm,Q3: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( neg @ ( con @ P2 @ Q3 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) ) )
| ? [P2: fm,Z4: list_fm,Q3: fm] :
( ( A3
= ( cons_fm @ ( con @ P2 @ Q3 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ Q3 @ Z4 ) ) )
| ? [P2: fm,Z4: list_fm,Q3: fm] :
( ( A3
= ( cons_fm @ ( neg @ ( imp @ P2 @ Q3 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) )
| ? [P2: fm,Z4: list_fm,Q3: fm] :
( ( A3
= ( cons_fm @ ( neg @ ( dis @ P2 @ Q3 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) )
| ? [T2: tm,P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( exi @ P2 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T2 @ P2 ) @ Z4 ) ) )
| ? [T2: tm,P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( neg @ ( uni @ P2 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T2 @ P2 ) ) @ Z4 ) ) )
| ? [I4: nat,P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( uni @ P2 ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P2 ) @ Z4 ) )
& ( news @ I4 @ ( cons_fm @ P2 @ Z4 ) ) )
| ? [I4: nat,P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( neg @ ( exi @ P2 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P2 ) ) @ Z4 ) )
& ( news @ I4 @ ( cons_fm @ P2 @ Z4 ) ) )
| ? [P2: fm,Z4: list_fm] :
( ( A3
= ( cons_fm @ ( neg @ ( neg @ P2 ) ) @ Z4 ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z4 ) ) )
| ? [Z4: list_fm,Y14: list_fm] :
( ( A3 = Y14 )
& ( sequent_calculus @ Z4 )
& ( ext_fm @ Y14 @ Z4 ) ) ) ) ) ).
% sequent_calculus.simps
thf(fact_482_sequent__calculus_Ocases,axiom,
! [A: list_fm] :
( ( sequent_calculus @ A )
=> ( ! [P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ P4 @ Z ) )
=> ~ ( member_fm @ ( neg @ P4 ) @ Z ) )
=> ( ! [P4: fm,Q2: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( dis @ P4 @ Q2 ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ ( cons_fm @ Q2 @ Z ) ) ) )
=> ( ! [P4: fm,Q2: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( imp @ P4 @ Q2 ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ Q2 @ Z ) ) ) )
=> ( ! [P4: fm,Q2: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( con @ P4 @ Q2 ) ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ ( neg @ Q2 ) @ Z ) ) ) )
=> ( ! [P4: fm,Z: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( con @ P4 @ Q2 ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ Q2 @ Z ) ) ) )
=> ( ! [P4: fm,Z: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( neg @ ( imp @ P4 @ Q2 ) ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z ) ) ) )
=> ( ! [P4: fm,Z: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( neg @ ( dis @ P4 @ Q2 ) ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z ) ) ) )
=> ( ! [T3: tm,P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( exi @ P4 ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T3 @ P4 ) @ Z ) ) )
=> ( ! [T3: tm,P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( uni @ P4 ) ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T3 @ P4 ) ) @ Z ) ) )
=> ( ! [I: nat,P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( uni @ P4 ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P4 ) @ Z ) )
=> ~ ( news @ I @ ( cons_fm @ P4 @ Z ) ) ) )
=> ( ! [I: nat,P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( exi @ P4 ) ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P4 ) ) @ Z ) )
=> ~ ( news @ I @ ( cons_fm @ P4 @ Z ) ) ) )
=> ( ! [P4: fm,Z: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( neg @ P4 ) ) @ Z ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ Z ) ) )
=> ~ ! [Z: list_fm] :
( ( sequent_calculus @ Z )
=> ~ ( ext_fm @ A @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sequent_calculus.cases
thf(fact_483_branchDone_Opelims_I2_J,axiom,
! [X3: list_fm] :
( ( branchDone @ X3 )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ! [P4: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Va: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ( ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) )
=> ~ ! [V3: fm,Z: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(2)
thf(fact_484_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y14: list_fm,Z4: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z4 ) @ ( set_fm2 @ Y14 ) ) ) ) ).
% SeCaV.ext
thf(fact_485_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y14: list_tm,Z4: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z4 ) @ ( set_tm2 @ Y14 ) ) ) ) ).
% SeCaV.ext
thf(fact_486_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y14: list_nat,Z4: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z4 ) @ ( set_nat2 @ Y14 ) ) ) ) ).
% SeCaV.ext
thf(fact_487_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_488_news_Osimps_I1_J,axiom,
! [C2: nat] : ( news @ C2 @ nil_fm ) ).
% news.simps(1)
thf(fact_489_Cons__in__subseqsD,axiom,
! [Y: fm,Ys: list_fm,Xs: list_fm] :
( ( member_list_fm @ ( cons_fm @ Y @ Ys ) @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
=> ( member_list_fm @ Ys @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_490_Cons__in__subseqsD,axiom,
! [Y: tm,Ys: list_tm,Xs: list_tm] :
( ( member_list_tm @ ( cons_tm @ Y @ Ys ) @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
=> ( member_list_tm @ Ys @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_491_Cons__in__subseqsD,axiom,
! [Y: int,Ys: list_int,Xs: list_int] :
( ( member_list_int @ ( cons_int @ Y @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
=> ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_492_Ext,axiom,
! [Z3: list_fm,Y: list_fm] :
( ( sequent_calculus @ Z3 )
=> ( ( ext_fm @ Y @ Z3 )
=> ( sequent_calculus @ Y ) ) ) ).
% Ext
thf(fact_493_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_494_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_495_DeltaUni,axiom,
! [I2: nat,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P ) @ Z3 ) ) ) ) ).
% DeltaUni
thf(fact_496_DeltaExi,axiom,
! [I2: nat,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P ) ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P ) ) @ Z3 ) ) ) ) ).
% DeltaExi
thf(fact_497_subset__antisym,axiom,
! [A2: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ( ord_less_eq_set_tm @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_498_subset__antisym,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_499_subsetI,axiom,
! [A2: set_fm,B4: set_fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A2 )
=> ( member_fm2 @ X @ B4 ) )
=> ( ord_less_eq_set_fm @ A2 @ B4 ) ) ).
% subsetI
thf(fact_500_subsetI,axiom,
! [A2: set_real,B4: set_real] :
( ! [X: real] :
( ( member_real2 @ X @ A2 )
=> ( member_real2 @ X @ B4 ) )
=> ( ord_less_eq_set_real @ A2 @ B4 ) ) ).
% subsetI
thf(fact_501_subsetI,axiom,
! [A2: set_tm,B4: set_tm] :
( ! [X: tm] :
( ( member_tm2 @ X @ A2 )
=> ( member_tm2 @ X @ B4 ) )
=> ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).
% subsetI
thf(fact_502_subsetI,axiom,
! [A2: set_nat,B4: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ X @ B4 ) )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% subsetI
thf(fact_503_substt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm,S: tm,K: nat] :
( ( substt @ ( fun @ A @ Ts ) @ S @ K )
= ( fun @ A @ ( substts @ Ts @ S @ K ) ) ) ).
% substt.simps(2)
thf(fact_504_terms__cases,axiom,
! [T: tm,S3: set_fm] :
( ( member_tm2 @ T @ ( terms @ S3 ) )
=> ( ( T
= ( fun @ zero_zero_nat @ nil_tm ) )
| ? [X: fm] :
( ( member_fm2 @ X @ S3 )
& ( member_tm2 @ T @ ( set_tm2 @ ( subtermFm @ X ) ) ) ) ) ) ).
% terms_cases
thf(fact_505_the__elem__set,axiom,
! [X3: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X3 @ nil_fm ) ) )
= X3 ) ).
% the_elem_set
thf(fact_506_the__elem__set,axiom,
! [X3: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X3 @ nil_tm ) ) )
= X3 ) ).
% the_elem_set
thf(fact_507_the__elem__set,axiom,
! [X3: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X3 @ nil_int ) ) )
= X3 ) ).
% the_elem_set
thf(fact_508_sub__term_Osimps_I2_J,axiom,
! [V: nat,S: tm,I2: nat,L: list_tm] :
( ( sub_term @ V @ S @ ( fun @ I2 @ L ) )
= ( fun @ I2 @ ( sub_list @ V @ S @ L ) ) ) ).
% sub_term.simps(2)
thf(fact_509_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_510_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_tm @ ( coset_tm @ nil_tm ) @ ( set_tm2 @ nil_tm ) ) ).
% subset_code(3)
thf(fact_511_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_512_s5_I1_J,axiom,
( sub_term
= ( ^ [V2: nat,S2: tm,T2: tm] : ( substt @ T2 @ S2 @ V2 ) ) ) ).
% s5(1)
thf(fact_513_subtermFm_Osimps_I7_J,axiom,
! [P: fm] :
( ( subtermFm @ ( neg @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(7)
thf(fact_514_fun__arguments__subterm,axiom,
! [N: nat,Ts: list_tm,P: fm] :
( ( member_tm2 @ ( fun @ N @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% fun_arguments_subterm
thf(fact_515_subtermFm_Osimps_I5_J,axiom,
! [P: fm] :
( ( subtermFm @ ( exi @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(5)
thf(fact_516_subtermFm_Osimps_I6_J,axiom,
! [P: fm] :
( ( subtermFm @ ( uni @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(6)
thf(fact_517_subterm__Pre__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N @ Ts ) ) ) ) ).
% subterm_Pre_refl
thf(fact_518_sub__list_Osimps_I2_J,axiom,
! [V: nat,S: tm,T: tm,L: list_tm] :
( ( sub_list @ V @ S @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( sub_term @ V @ S @ T ) @ ( sub_list @ V @ S @ L ) ) ) ).
% sub_list.simps(2)
thf(fact_519_in__mono,axiom,
! [A2: set_fm,B4: set_fm,X3: fm] :
( ( ord_less_eq_set_fm @ A2 @ B4 )
=> ( ( member_fm2 @ X3 @ A2 )
=> ( member_fm2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_520_in__mono,axiom,
! [A2: set_real,B4: set_real,X3: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real2 @ X3 @ A2 )
=> ( member_real2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_521_in__mono,axiom,
! [A2: set_tm,B4: set_tm,X3: tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_tm2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_522_in__mono,axiom,
! [A2: set_nat,B4: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat2 @ X3 @ A2 )
=> ( member_nat2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_523_subsetD,axiom,
! [A2: set_fm,B4: set_fm,C2: fm] :
( ( ord_less_eq_set_fm @ A2 @ B4 )
=> ( ( member_fm2 @ C2 @ A2 )
=> ( member_fm2 @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_524_subsetD,axiom,
! [A2: set_real,B4: set_real,C2: real] :
( ( ord_less_eq_set_real @ A2 @ B4 )
=> ( ( member_real2 @ C2 @ A2 )
=> ( member_real2 @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_525_subsetD,axiom,
! [A2: set_tm,B4: set_tm,C2: tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ( member_tm2 @ C2 @ A2 )
=> ( member_tm2 @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_526_subsetD,axiom,
! [A2: set_nat,B4: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat2 @ C2 @ A2 )
=> ( member_nat2 @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_527_equalityE,axiom,
! [A2: set_tm,B4: set_tm] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ~ ( ord_less_eq_set_tm @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_528_equalityE,axiom,
! [A2: set_nat,B4: set_nat] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_529_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A5: set_fm,B5: set_fm] :
! [X9: fm] :
( ( member_fm2 @ X9 @ A5 )
=> ( member_fm2 @ X9 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_530_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X9: real] :
( ( member_real2 @ X9 @ A5 )
=> ( member_real2 @ X9 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_531_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B5: set_tm] :
! [X9: tm] :
( ( member_tm2 @ X9 @ A5 )
=> ( member_tm2 @ X9 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_532_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X9: nat] :
( ( member_nat2 @ X9 @ A5 )
=> ( member_nat2 @ X9 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_533_equalityD1,axiom,
! [A2: set_tm,B4: set_tm] :
( ( A2 = B4 )
=> ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_534_equalityD1,axiom,
! [A2: set_nat,B4: set_nat] :
( ( A2 = B4 )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_535_equalityD2,axiom,
! [A2: set_tm,B4: set_tm] :
( ( A2 = B4 )
=> ( ord_less_eq_set_tm @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_536_equalityD2,axiom,
! [A2: set_nat,B4: set_nat] :
( ( A2 = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_537_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A5: set_fm,B5: set_fm] :
! [T2: fm] :
( ( member_fm2 @ T2 @ A5 )
=> ( member_fm2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_538_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real2 @ T2 @ A5 )
=> ( member_real2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_539_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B5: set_tm] :
! [T2: tm] :
( ( member_tm2 @ T2 @ A5 )
=> ( member_tm2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_540_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A5 )
=> ( member_nat2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_541_subset__refl,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% subset_refl
thf(fact_542_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_543_Collect__mono,axiom,
! [P3: tm > $o,Q4: tm > $o] :
( ! [X: tm] :
( ( P3 @ X )
=> ( Q4 @ X ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P3 ) @ ( collect_tm @ Q4 ) ) ) ).
% Collect_mono
thf(fact_544_Collect__mono,axiom,
! [P3: nat > $o,Q4: nat > $o] :
( ! [X: nat] :
( ( P3 @ X )
=> ( Q4 @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_545_subset__trans,axiom,
! [A2: set_tm,B4: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ( ord_less_eq_set_tm @ B4 @ C3 )
=> ( ord_less_eq_set_tm @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_546_subset__trans,axiom,
! [A2: set_nat,B4: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_547_set__eq__subset,axiom,
( ( ^ [Y9: set_tm,Z2: set_tm] : ( Y9 = Z2 ) )
= ( ^ [A5: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B5 )
& ( ord_less_eq_set_tm @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_548_set__eq__subset,axiom,
( ( ^ [Y9: set_nat,Z2: set_nat] : ( Y9 = Z2 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_549_Collect__mono__iff,axiom,
! [P3: tm > $o,Q4: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P3 ) @ ( collect_tm @ Q4 ) )
= ( ! [X9: tm] :
( ( P3 @ X9 )
=> ( Q4 @ X9 ) ) ) ) ).
% Collect_mono_iff
thf(fact_550_Collect__mono__iff,axiom,
! [P3: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q4 ) )
= ( ! [X9: nat] :
( ( P3 @ X9 )
=> ( Q4 @ X9 ) ) ) ) ).
% Collect_mono_iff
thf(fact_551_subset__code_I2_J,axiom,
! [A2: set_real,Ys: list_real] :
( ( ord_less_eq_set_real @ A2 @ ( coset_real @ Ys ) )
= ( ! [X9: real] :
( ( member_real2 @ X9 @ ( set_real2 @ Ys ) )
=> ~ ( member_real2 @ X9 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_552_subset__code_I2_J,axiom,
! [A2: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A2 @ ( coset_fm @ Ys ) )
= ( ! [X9: fm] :
( ( member_fm2 @ X9 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm2 @ X9 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_553_subset__code_I2_J,axiom,
! [A2: set_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ A2 @ ( coset_tm @ Ys ) )
= ( ! [X9: tm] :
( ( member_tm2 @ X9 @ ( set_tm2 @ Ys ) )
=> ~ ( member_tm2 @ X9 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_554_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X9 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_555_substts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm,S: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts ) @ S @ K )
= ( cons_tm @ ( substt @ T @ S @ K ) @ ( substts @ Ts @ S @ K ) ) ) ).
% substts.simps(2)
thf(fact_556_sub__const__transfer,axiom,
! [M: nat,A: nat,P: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P )
!= ( sub @ M @ T @ P ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P ) ) ) ) ) ).
% sub_const_transfer
thf(fact_557_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T: tm,S: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T )
!= ( sub_term @ M @ S @ T ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_558_nth__equal__first__eq,axiom,
! [X3: real,Xs: list_real,N: nat] :
( ~ ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
=> ( ( ( nth_real @ ( cons_real @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_559_nth__equal__first__eq,axiom,
! [X3: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_560_nth__equal__first__eq,axiom,
! [X3: fm,Xs: list_fm,N: nat] :
( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( ( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_561_nth__equal__first__eq,axiom,
! [X3: tm,Xs: list_tm,N: nat] :
( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_tm @ Xs ) )
=> ( ( ( nth_tm @ ( cons_tm @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_562_nth__equal__first__eq,axiom,
! [X3: int,Xs: list_int,N: nat] :
( ~ ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= X3 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_563_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_564_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_565_params__subtermFm,axiom,
! [P: fm,X2: nat] :
( ( member_nat2 @ X2 @ ( params @ P ) )
=> ? [L3: list_tm] : ( member_tm2 @ ( fun @ X2 @ L3 ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% params_subtermFm
thf(fact_566_ceiling__le__zero,axiom,
! [X3: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
= ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_567_list__ex1__simps_I1_J,axiom,
! [P3: fm > $o] :
~ ( list_ex1_fm @ P3 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_568_list__ex1__simps_I1_J,axiom,
! [P3: tm > $o] :
~ ( list_ex1_tm @ P3 @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_569_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_570_ceiling__of__int,axiom,
! [Z3: int] :
( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z3 ) )
= Z3 ) ).
% ceiling_of_int
thf(fact_571_nth__Cons__0,axiom,
! [X3: fm,Xs: list_fm] :
( ( nth_fm @ ( cons_fm @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_572_nth__Cons__0,axiom,
! [X3: tm,Xs: list_tm] :
( ( nth_tm @ ( cons_tm @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_573_nth__Cons__0,axiom,
! [X3: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_574_ceiling__zero,axiom,
( ( archim7802044766580827645g_real @ zero_zero_real )
= zero_zero_int ) ).
% ceiling_zero
thf(fact_575_ceiling__of__nat,axiom,
! [N: nat] :
( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% ceiling_of_nat
thf(fact_576_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_577_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_578_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_579_of__nat__ceiling,axiom,
! [R: real] : ( ord_less_eq_real @ R @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R ) ) ) ) ).
% of_nat_ceiling
thf(fact_580_params_Osimps_I7_J,axiom,
! [P: fm] :
( ( params @ ( neg @ P ) )
= ( params @ P ) ) ).
% params.simps(7)
thf(fact_581_params_Osimps_I5_J,axiom,
! [P: fm] :
( ( params @ ( exi @ P ) )
= ( params @ P ) ) ).
% params.simps(5)
thf(fact_582_params_Osimps_I6_J,axiom,
! [P: fm] :
( ( params @ ( uni @ P ) )
= ( params @ P ) ) ).
% params.simps(6)
thf(fact_583_ceiling__mono,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% ceiling_mono
thf(fact_584_le__of__int__ceiling,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% le_of_int_ceiling
thf(fact_585_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_586_nat__mono,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_587_ex__nat,axiom,
( ( ^ [P6: nat > $o] :
? [X14: nat] : ( P6 @ X14 ) )
= ( ^ [P7: nat > $o] :
? [X9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X9 )
& ( P7 @ ( nat2 @ X9 ) ) ) ) ) ).
% ex_nat
thf(fact_588_all__nat,axiom,
( ( ^ [P6: nat > $o] :
! [X14: nat] : ( P6 @ X14 ) )
= ( ^ [P7: nat > $o] :
! [X9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X9 )
=> ( P7 @ ( nat2 @ X9 ) ) ) ) ) ).
% all_nat
thf(fact_589_eq__nat__nat__iff,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z5 ) )
= ( Z3 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_590_subtermTm__refl,axiom,
! [T: tm] : ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_591_subtermTm__le,axiom,
! [T: tm,S: tm] :
( ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ S ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S ) ) ) ) ).
% subtermTm_le
thf(fact_592_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P7: real > $o,Xs3: list_real] :
? [X9: real] :
( ( member_real2 @ X9 @ ( set_real2 @ Xs3 ) )
& ( P7 @ X9 )
& ! [Y14: real] :
( ( ( member_real2 @ Y14 @ ( set_real2 @ Xs3 ) )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_593_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P7: nat > $o,Xs3: list_nat] :
? [X9: nat] :
( ( member_nat2 @ X9 @ ( set_nat2 @ Xs3 ) )
& ( P7 @ X9 )
& ! [Y14: nat] :
( ( ( member_nat2 @ Y14 @ ( set_nat2 @ Xs3 ) )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_594_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P7: fm > $o,Xs3: list_fm] :
? [X9: fm] :
( ( member_fm2 @ X9 @ ( set_fm2 @ Xs3 ) )
& ( P7 @ X9 )
& ! [Y14: fm] :
( ( ( member_fm2 @ Y14 @ ( set_fm2 @ Xs3 ) )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_595_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P7: tm > $o,Xs3: list_tm] :
? [X9: tm] :
( ( member_tm2 @ X9 @ ( set_tm2 @ Xs3 ) )
& ( P7 @ X9 )
& ! [Y14: tm] :
( ( ( member_tm2 @ Y14 @ ( set_tm2 @ Xs3 ) )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_596_ceiling__le__iff,axiom,
! [X3: real,Z3: int] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ Z3 )
= ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% ceiling_le_iff
thf(fact_597_nat__le__iff,axiom,
! [X3: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
= ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_598_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_599_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_600_subterm__Fun__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_601_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_602_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_603_terms__downwards__closed,axiom,
! [T: tm,S3: set_fm] :
( ( member_tm2 @ T @ ( terms @ S3 ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( terms @ S3 ) ) ) ).
% terms_downwards_closed
thf(fact_604_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_605_nat__ceiling__le__eq,axiom,
! [X3: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
= ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_606_ceiling__le,axiom,
! [X3: real,A: int] :
( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ A ) ) ).
% ceiling_le
thf(fact_607_can__select__set__list__ex1,axiom,
! [P3: fm > $o,A2: list_fm] :
( ( can_select_fm @ P3 @ ( set_fm2 @ A2 ) )
= ( list_ex1_fm @ P3 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_608_can__select__set__list__ex1,axiom,
! [P3: tm > $o,A2: list_tm] :
( ( can_select_tm @ P3 @ ( set_tm2 @ A2 ) )
= ( list_ex1_tm @ P3 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_609_subtermFm__subset__params,axiom,
! [P: fm,A2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P ) ) @ ( set_tm2 @ A2 ) )
=> ( ord_less_eq_set_nat @ ( params @ P ) @ ( paramsts @ A2 ) ) ) ).
% subtermFm_subset_params
thf(fact_610_paramst__subtermTm_I1_J,axiom,
! [T: tm,X2: nat] :
( ( member_nat2 @ X2 @ ( paramst @ T ) )
=> ? [L3: list_tm] : ( member_tm2 @ ( fun @ X2 @ L3 ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_611_subset__subseqs,axiom,
! [X15: set_fm,Xs: list_fm] :
( ( ord_less_eq_set_fm @ X15 @ ( set_fm2 @ Xs ) )
=> ( member_set_fm @ X15 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_612_subset__subseqs,axiom,
! [X15: set_tm,Xs: list_tm] :
( ( ord_less_eq_set_tm @ X15 @ ( set_tm2 @ Xs ) )
=> ( member_set_tm @ X15 @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_613_subset__subseqs,axiom,
! [X15: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X15 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X15 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_614_complete__real,axiom,
! [S3: set_real] :
( ? [X2: real] : ( member_real2 @ X2 @ S3 )
=> ( ? [Z6: real] :
! [X: real] :
( ( member_real2 @ X @ S3 )
=> ( ord_less_eq_real @ X @ Z6 ) )
=> ? [Y10: real] :
( ! [X2: real] :
( ( member_real2 @ X2 @ S3 )
=> ( ord_less_eq_real @ X2 @ Y10 ) )
& ! [Z6: real] :
( ! [X: real] :
( ( member_real2 @ X @ S3 )
=> ( ord_less_eq_real @ X @ Z6 ) )
=> ( ord_less_eq_real @ Y10 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_615_subset__image__iff,axiom,
! [B4: set_tm,F: tm > tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ ( image_tm_tm @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B4
= ( image_tm_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_616_subset__image__iff,axiom,
! [B4: set_tm,F: nat > tm,A2: set_nat] :
( ( ord_less_eq_set_tm @ B4 @ ( image_nat_tm @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_617_subset__image__iff,axiom,
! [B4: set_nat,F: tm > nat,A2: set_tm] :
( ( ord_less_eq_set_nat @ B4 @ ( image_tm_nat @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B4
= ( image_tm_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_618_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B4
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_619_subset__imageE,axiom,
! [B4: set_tm,F: tm > tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ ( image_tm_tm @ F @ A2 ) )
=> ~ ! [C4: set_tm] :
( ( ord_less_eq_set_tm @ C4 @ A2 )
=> ( B4
!= ( image_tm_tm @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_620_subset__imageE,axiom,
! [B4: set_tm,F: nat > tm,A2: set_nat] :
( ( ord_less_eq_set_tm @ B4 @ ( image_nat_tm @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B4
!= ( image_nat_tm @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_621_subset__imageE,axiom,
! [B4: set_nat,F: tm > nat,A2: set_tm] :
( ( ord_less_eq_set_nat @ B4 @ ( image_tm_nat @ F @ A2 ) )
=> ~ ! [C4: set_tm] :
( ( ord_less_eq_set_tm @ C4 @ A2 )
=> ( B4
!= ( image_tm_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_622_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B4
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_623_image__subsetI,axiom,
! [A2: set_tm,F: tm > fm,B4: set_fm] :
( ! [X: tm] :
( ( member_tm2 @ X @ A2 )
=> ( member_fm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_fm @ ( image_tm_fm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_624_image__subsetI,axiom,
! [A2: set_tm,F: tm > real,B4: set_real] :
( ! [X: tm] :
( ( member_tm2 @ X @ A2 )
=> ( member_real2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_tm_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_625_image__subsetI,axiom,
! [A2: set_fm,F: fm > fm,B4: set_fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A2 )
=> ( member_fm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_fm @ ( image_fm_fm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_626_image__subsetI,axiom,
! [A2: set_fm,F: fm > real,B4: set_real] :
( ! [X: fm] :
( ( member_fm2 @ X @ A2 )
=> ( member_real2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_fm_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_627_image__subsetI,axiom,
! [A2: set_real,F: real > fm,B4: set_fm] :
( ! [X: real] :
( ( member_real2 @ X @ A2 )
=> ( member_fm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_fm @ ( image_real_fm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_628_image__subsetI,axiom,
! [A2: set_real,F: real > real,B4: set_real] :
( ! [X: real] :
( ( member_real2 @ X @ A2 )
=> ( member_real2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_real_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_629_image__subsetI,axiom,
! [A2: set_nat,F: nat > fm,B4: set_fm] :
( ! [X: nat] :
( ( member_nat2 @ X @ A2 )
=> ( member_fm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_fm @ ( image_nat_fm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_630_image__subsetI,axiom,
! [A2: set_nat,F: nat > real,B4: set_real] :
( ! [X: nat] :
( ( member_nat2 @ X @ A2 )
=> ( member_real2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_real @ ( image_nat_real @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_631_image__subsetI,axiom,
! [A2: set_tm,F: tm > tm,B4: set_tm] :
( ! [X: tm] :
( ( member_tm2 @ X @ A2 )
=> ( member_tm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_tm @ ( image_tm_tm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_632_image__subsetI,axiom,
! [A2: set_fm,F: fm > tm,B4: set_tm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A2 )
=> ( member_tm2 @ ( F @ X ) @ B4 ) )
=> ( ord_less_eq_set_tm @ ( image_fm_tm @ F @ A2 ) @ B4 ) ) ).
% image_subsetI
thf(fact_633_image__mono,axiom,
! [A2: set_tm,B4: set_tm,F: tm > tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ord_less_eq_set_tm @ ( image_tm_tm @ F @ A2 ) @ ( image_tm_tm @ F @ B4 ) ) ) ).
% image_mono
thf(fact_634_image__mono,axiom,
! [A2: set_tm,B4: set_tm,F: tm > nat] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ord_less_eq_set_nat @ ( image_tm_nat @ F @ A2 ) @ ( image_tm_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_635_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > tm] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_tm @ ( image_nat_tm @ F @ A2 ) @ ( image_nat_tm @ F @ B4 ) ) ) ).
% image_mono
thf(fact_636_image__mono,axiom,
! [A2: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_637_can__select__def,axiom,
( can_select_tm
= ( ^ [P7: tm > $o,A5: set_tm] :
? [X9: tm] :
( ( member_tm2 @ X9 @ A5 )
& ( P7 @ X9 )
& ! [Y14: tm] :
( ( ( member_tm2 @ Y14 @ A5 )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% can_select_def
thf(fact_638_can__select__def,axiom,
( can_select_fm
= ( ^ [P7: fm > $o,A5: set_fm] :
? [X9: fm] :
( ( member_fm2 @ X9 @ A5 )
& ( P7 @ X9 )
& ! [Y14: fm] :
( ( ( member_fm2 @ Y14 @ A5 )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% can_select_def
thf(fact_639_can__select__def,axiom,
( can_select_real
= ( ^ [P7: real > $o,A5: set_real] :
? [X9: real] :
( ( member_real2 @ X9 @ A5 )
& ( P7 @ X9 )
& ! [Y14: real] :
( ( ( member_real2 @ Y14 @ A5 )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% can_select_def
thf(fact_640_can__select__def,axiom,
( can_select_nat
= ( ^ [P7: nat > $o,A5: set_nat] :
? [X9: nat] :
( ( member_nat2 @ X9 @ A5 )
& ( P7 @ X9 )
& ! [Y14: nat] :
( ( ( member_nat2 @ Y14 @ A5 )
& ( P7 @ Y14 ) )
=> ( Y14 = X9 ) ) ) ) ) ).
% can_select_def
thf(fact_641_real__nat__ceiling__ge,axiom,
! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_642_params_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm] :
( ( params @ ( pre @ B2 @ Ts ) )
= ( paramsts @ Ts ) ) ).
% params.simps(1)
thf(fact_643_paramsts__subset,axiom,
! [A2: list_tm,B4: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B4 ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A2 ) @ ( paramsts @ B4 ) ) ) ).
% paramsts_subset
thf(fact_644_s1_I1_J,axiom,
( new_term
= ( ^ [C: nat,T2: tm] :
~ ( member_nat2 @ C @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_645_all__subset__image,axiom,
! [F: tm > tm,A2: set_tm,P3: set_tm > $o] :
( ( ! [B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ ( image_tm_tm @ F @ A2 ) )
=> ( P3 @ B5 ) ) )
= ( ! [B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A2 )
=> ( P3 @ ( image_tm_tm @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_646_all__subset__image,axiom,
! [F: nat > tm,A2: set_nat,P3: set_tm > $o] :
( ( ! [B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ ( image_nat_tm @ F @ A2 ) )
=> ( P3 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P3 @ ( image_nat_tm @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_647_all__subset__image,axiom,
! [F: tm > nat,A2: set_tm,P3: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_tm_nat @ F @ A2 ) )
=> ( P3 @ B5 ) ) )
= ( ! [B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A2 )
=> ( P3 @ ( image_tm_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_648_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P3: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
=> ( P3 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P3 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_649_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_650_paramst__sub__term_I2_J,axiom,
! [M: nat,S: tm,L: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S @ L ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramsts @ L ) ) ) ).
% paramst_sub_term(2)
thf(fact_651_s1_I2_J,axiom,
( new_list
= ( ^ [C: nat,L2: list_tm] :
~ ( member_nat2 @ C @ ( paramsts @ L2 ) ) ) ) ).
% s1(2)
thf(fact_652_subseqs__powset,axiom,
! [Xs: list_fm] :
( ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
= ( pow_fm @ ( set_fm2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_653_subseqs__powset,axiom,
! [Xs: list_tm] :
( ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
= ( pow_tm @ ( set_tm2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_654_Un__subset__iff,axiom,
! [A2: set_tm,B4: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B4 ) @ C3 )
= ( ( ord_less_eq_set_tm @ A2 @ C3 )
& ( ord_less_eq_set_tm @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_655_Un__subset__iff,axiom,
! [A2: set_nat,B4: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B4 ) @ C3 )
= ( ( ord_less_eq_set_nat @ A2 @ C3 )
& ( ord_less_eq_set_nat @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_656_PowI,axiom,
! [A2: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( member_set_tm @ A2 @ ( pow_tm @ B4 ) ) ) ).
% PowI
thf(fact_657_PowI,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( member_set_nat @ A2 @ ( pow_nat @ B4 ) ) ) ).
% PowI
thf(fact_658_Pow__iff,axiom,
! [A2: set_tm,B4: set_tm] :
( ( member_set_tm @ A2 @ ( pow_tm @ B4 ) )
= ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).
% Pow_iff
thf(fact_659_Pow__iff,axiom,
! [A2: set_nat,B4: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B4 ) )
= ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% Pow_iff
thf(fact_660_Un__Pow__subset,axiom,
! [A2: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B4 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A2 @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_661_Un__mono,axiom,
! [A2: set_tm,C3: set_tm,B4: set_tm,D: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C3 )
=> ( ( ord_less_eq_set_tm @ B4 @ D )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B4 ) @ ( sup_sup_set_tm @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_662_Un__mono,axiom,
! [A2: set_nat,C3: set_nat,B4: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B4 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B4 ) @ ( sup_sup_set_nat @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_663_Un__least,axiom,
! [A2: set_tm,C3: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C3 )
=> ( ( ord_less_eq_set_tm @ B4 @ C3 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_664_Un__least,axiom,
! [A2: set_nat,C3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B4 @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_665_Un__upper1,axiom,
! [A2: set_tm,B4: set_tm] : ( ord_less_eq_set_tm @ A2 @ ( sup_sup_set_tm @ A2 @ B4 ) ) ).
% Un_upper1
thf(fact_666_Un__upper1,axiom,
! [A2: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B4 ) ) ).
% Un_upper1
thf(fact_667_Un__upper2,axiom,
! [B4: set_tm,A2: set_tm] : ( ord_less_eq_set_tm @ B4 @ ( sup_sup_set_tm @ A2 @ B4 ) ) ).
% Un_upper2
thf(fact_668_Un__upper2,axiom,
! [B4: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A2 @ B4 ) ) ).
% Un_upper2
thf(fact_669_Un__absorb1,axiom,
! [A2: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ( sup_sup_set_tm @ A2 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_670_Un__absorb1,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( sup_sup_set_nat @ A2 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_671_Un__absorb2,axiom,
! [B4: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ A2 )
=> ( ( sup_sup_set_tm @ A2 @ B4 )
= A2 ) ) ).
% Un_absorb2
thf(fact_672_Un__absorb2,axiom,
! [B4: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B4 )
= A2 ) ) ).
% Un_absorb2
thf(fact_673_subset__UnE,axiom,
! [C3: set_tm,A2: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ C3 @ ( sup_sup_set_tm @ A2 @ B4 ) )
=> ~ ! [A6: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ A2 )
=> ! [B6: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ B4 )
=> ( C3
!= ( sup_sup_set_tm @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_674_subset__UnE,axiom,
! [C3: set_nat,A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A2 @ B4 ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A2 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B4 )
=> ( C3
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_675_subset__Un__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B5: set_tm] :
( ( sup_sup_set_tm @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_676_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_677_PowD,axiom,
! [A2: set_tm,B4: set_tm] :
( ( member_set_tm @ A2 @ ( pow_tm @ B4 ) )
=> ( ord_less_eq_set_tm @ A2 @ B4 ) ) ).
% PowD
thf(fact_678_PowD,axiom,
! [A2: set_nat,B4: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B4 ) )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% PowD
thf(fact_679_Pow__mono,axiom,
! [A2: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B4 )
=> ( ord_le5601931644483074373set_tm @ ( pow_tm @ A2 ) @ ( pow_tm @ B4 ) ) ) ).
% Pow_mono
thf(fact_680_Pow__mono,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B4 ) ) ) ).
% Pow_mono
thf(fact_681_new__term_Osimps_I2_J,axiom,
! [C2: nat,I2: nat,L: list_tm] :
( ( new_term @ C2 @ ( fun @ I2 @ L ) )
= ( ( I2 != C2 )
& ( ( I2 != C2 )
=> ( new_list @ C2 @ L ) ) ) ) ).
% new_term.simps(2)
thf(fact_682_new__list_Osimps_I2_J,axiom,
! [C2: nat,T: tm,L: list_tm] :
( ( new_list @ C2 @ ( cons_tm @ T @ L ) )
= ( ( ( new_term @ C2 @ T )
=> ( new_list @ C2 @ L ) )
& ( new_term @ C2 @ T ) ) ) ).
% new_list.simps(2)
thf(fact_683_params_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( con @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(4)
thf(fact_684_params_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( dis @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(3)
thf(fact_685_params_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( imp @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(2)
thf(fact_686_new__list_Osimps_I1_J,axiom,
! [C2: nat] : ( new_list @ C2 @ nil_tm ) ).
% new_list.simps(1)
thf(fact_687_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_688_paramst__sub__term_I1_J,axiom,
! [M: nat,S: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramst @ T ) ) ) ).
% paramst_sub_term(1)
thf(fact_689_params__sub,axiom,
! [M: nat,T: tm,P: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P ) ) ) ).
% params_sub
thf(fact_690_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X3 @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_691_le__sup__iff,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_int @ X3 @ Z3 )
& ( ord_less_eq_int @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_692_le__sup__iff,axiom,
! [X3: set_tm,Y: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_set_tm @ X3 @ Z3 )
& ( ord_less_eq_set_tm @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_693_le__sup__iff,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_real @ X3 @ Z3 )
& ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_694_le__sup__iff,axiom,
! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X3 @ Z3 )
& ( ord_less_eq_set_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_695_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_696_sup_Obounded__iff,axiom,
! [B2: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_697_sup_Obounded__iff,axiom,
! [B2: set_tm,C2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_tm @ B2 @ A )
& ( ord_less_eq_set_tm @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_698_sup_Obounded__iff,axiom,
! [B2: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_real @ B2 @ A )
& ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_699_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_700_set__union,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( union_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_701_set__union,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( set_tm2 @ ( union_tm @ Xs @ Ys ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).
% set_union
thf(fact_702_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_703_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_704_inf__sup__ord_I4_J,axiom,
! [Y: int,X3: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_705_inf__sup__ord_I4_J,axiom,
! [Y: set_tm,X3: set_tm] : ( ord_less_eq_set_tm @ Y @ ( sup_sup_set_tm @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_706_inf__sup__ord_I4_J,axiom,
! [Y: real,X3: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_707_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_708_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_709_inf__sup__ord_I3_J,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ X3 @ ( sup_sup_int @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_710_inf__sup__ord_I3_J,axiom,
! [X3: set_tm,Y: set_tm] : ( ord_less_eq_set_tm @ X3 @ ( sup_sup_set_tm @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_711_inf__sup__ord_I3_J,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ X3 @ ( sup_sup_real @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_712_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_713_le__supE,axiom,
! [A: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_714_le__supE,axiom,
! [A: int,B2: int,X3: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_int @ A @ X3 )
=> ~ ( ord_less_eq_int @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_715_le__supE,axiom,
! [A: set_tm,B2: set_tm,X3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_tm @ A @ X3 )
=> ~ ( ord_less_eq_set_tm @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_716_le__supE,axiom,
! [A: real,B2: real,X3: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_real @ A @ X3 )
=> ~ ( ord_less_eq_real @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_717_le__supE,axiom,
! [A: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_718_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_719_sup_OcoboundedI2,axiom,
! [C2: int,B2: int,A: int] :
( ( ord_less_eq_int @ C2 @ B2 )
=> ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_720_sup_OcoboundedI2,axiom,
! [C2: set_tm,B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ C2 @ B2 )
=> ( ord_less_eq_set_tm @ C2 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_721_sup_OcoboundedI2,axiom,
! [C2: real,B2: real,A: real] :
( ( ord_less_eq_real @ C2 @ B2 )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_722_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_723_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_724_sup_OcoboundedI1,axiom,
! [C2: int,A: int,B2: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_725_sup_OcoboundedI1,axiom,
! [C2: set_tm,A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C2 @ A )
=> ( ord_less_eq_set_tm @ C2 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_726_sup_OcoboundedI1,axiom,
! [C2: real,A: real,B2: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ C2 @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_727_sup_OcoboundedI1,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_728_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_729_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( sup_sup_int @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_730_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
( ( sup_sup_set_tm @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_731_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( sup_sup_real @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_732_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_733_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_734_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( sup_sup_int @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_735_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_tm
= ( ^ [B3: set_tm,A3: set_tm] :
( ( sup_sup_set_tm @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_736_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( sup_sup_real @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_737_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_738_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_739_sup_Ocobounded2,axiom,
! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_740_sup_Ocobounded2,axiom,
! [B2: set_tm,A: set_tm] : ( ord_less_eq_set_tm @ B2 @ ( sup_sup_set_tm @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_741_sup_Ocobounded2,axiom,
! [B2: real,A: real] : ( ord_less_eq_real @ B2 @ ( sup_sup_real @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_742_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_743_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_744_sup_Ocobounded1,axiom,
! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_745_sup_Ocobounded1,axiom,
! [A: set_tm,B2: set_tm] : ( ord_less_eq_set_tm @ A @ ( sup_sup_set_tm @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_746_sup_Ocobounded1,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_747_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_748_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_749_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( A3
= ( sup_sup_int @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_750_sup_Oorder__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [B3: set_tm,A3: set_tm] :
( A3
= ( sup_sup_set_tm @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_751_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( A3
= ( sup_sup_real @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_752_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_753_sup_OboundedI,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_754_sup_OboundedI,axiom,
! [B2: int,A: int,C2: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C2 @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_755_sup_OboundedI,axiom,
! [B2: set_tm,A: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C2 @ A )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_756_sup_OboundedI,axiom,
! [B2: real,A: real,C2: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C2 @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_757_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_758_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_759_sup_OboundedE,axiom,
! [B2: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_int @ B2 @ A )
=> ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_760_sup_OboundedE,axiom,
! [B2: set_tm,C2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_tm @ B2 @ A )
=> ~ ( ord_less_eq_set_tm @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_761_sup_OboundedE,axiom,
! [B2: real,C2: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_real @ B2 @ A )
=> ~ ( ord_less_eq_real @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_762_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_763_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_764_sup__absorb2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( sup_sup_int @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_765_sup__absorb2,axiom,
! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ( sup_sup_set_tm @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_766_sup__absorb2,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
=> ( ( sup_sup_real @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_767_sup__absorb2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_768_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_769_sup__absorb1,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( sup_sup_int @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_770_sup__absorb1,axiom,
! [Y: set_tm,X3: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X3 )
=> ( ( sup_sup_set_tm @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_771_sup__absorb1,axiom,
! [Y: real,X3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( sup_sup_real @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_772_sup__absorb1,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_773_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_774_sup_Oabsorb2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( sup_sup_int @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_775_sup_Oabsorb2,axiom,
! [A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( sup_sup_set_tm @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_776_sup_Oabsorb2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( sup_sup_real @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_777_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_778_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_779_sup_Oabsorb1,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( sup_sup_int @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_780_sup_Oabsorb1,axiom,
! [B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( sup_sup_set_tm @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_781_sup_Oabsorb1,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( sup_sup_real @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_782_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_783_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X: nat,Y10: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y10 ) )
=> ( ! [X: nat,Y10: nat] : ( ord_less_eq_nat @ Y10 @ ( F @ X @ Y10 ) )
=> ( ! [X: nat,Y10: nat,Z: nat] :
( ( ord_less_eq_nat @ Y10 @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( F @ Y10 @ Z ) @ X ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_784_sup__unique,axiom,
! [F: int > int > int,X3: int,Y: int] :
( ! [X: int,Y10: int] : ( ord_less_eq_int @ X @ ( F @ X @ Y10 ) )
=> ( ! [X: int,Y10: int] : ( ord_less_eq_int @ Y10 @ ( F @ X @ Y10 ) )
=> ( ! [X: int,Y10: int,Z: int] :
( ( ord_less_eq_int @ Y10 @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( F @ Y10 @ Z ) @ X ) ) )
=> ( ( sup_sup_int @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_785_sup__unique,axiom,
! [F: set_tm > set_tm > set_tm,X3: set_tm,Y: set_tm] :
( ! [X: set_tm,Y10: set_tm] : ( ord_less_eq_set_tm @ X @ ( F @ X @ Y10 ) )
=> ( ! [X: set_tm,Y10: set_tm] : ( ord_less_eq_set_tm @ Y10 @ ( F @ X @ Y10 ) )
=> ( ! [X: set_tm,Y10: set_tm,Z: set_tm] :
( ( ord_less_eq_set_tm @ Y10 @ X )
=> ( ( ord_less_eq_set_tm @ Z @ X )
=> ( ord_less_eq_set_tm @ ( F @ Y10 @ Z ) @ X ) ) )
=> ( ( sup_sup_set_tm @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_786_sup__unique,axiom,
! [F: real > real > real,X3: real,Y: real] :
( ! [X: real,Y10: real] : ( ord_less_eq_real @ X @ ( F @ X @ Y10 ) )
=> ( ! [X: real,Y10: real] : ( ord_less_eq_real @ Y10 @ ( F @ X @ Y10 ) )
=> ( ! [X: real,Y10: real,Z: real] :
( ( ord_less_eq_real @ Y10 @ X )
=> ( ( ord_less_eq_real @ Z @ X )
=> ( ord_less_eq_real @ ( F @ Y10 @ Z ) @ X ) ) )
=> ( ( sup_sup_real @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_787_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X: set_nat,Y10: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y10 ) )
=> ( ! [X: set_nat,Y10: set_nat] : ( ord_less_eq_set_nat @ Y10 @ ( F @ X @ Y10 ) )
=> ( ! [X: set_nat,Y10: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y10 @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y10 @ Z ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_788_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_789_sup_OorderI,axiom,
! [A: int,B2: int] :
( ( A
= ( sup_sup_int @ A @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% sup.orderI
thf(fact_790_sup_OorderI,axiom,
! [A: set_tm,B2: set_tm] :
( ( A
= ( sup_sup_set_tm @ A @ B2 ) )
=> ( ord_less_eq_set_tm @ B2 @ A ) ) ).
% sup.orderI
thf(fact_791_sup_OorderI,axiom,
! [A: real,B2: real] :
( ( A
= ( sup_sup_real @ A @ B2 ) )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% sup.orderI
thf(fact_792_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_793_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_794_sup_OorderE,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( A
= ( sup_sup_int @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_795_sup_OorderE,axiom,
! [B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( A
= ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_796_sup_OorderE,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( A
= ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_797_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_798_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X9: nat,Y14: nat] :
( ( sup_sup_nat @ X9 @ Y14 )
= Y14 ) ) ) ).
% le_iff_sup
thf(fact_799_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X9: int,Y14: int] :
( ( sup_sup_int @ X9 @ Y14 )
= Y14 ) ) ) ).
% le_iff_sup
thf(fact_800_le__iff__sup,axiom,
( ord_less_eq_set_tm
= ( ^ [X9: set_tm,Y14: set_tm] :
( ( sup_sup_set_tm @ X9 @ Y14 )
= Y14 ) ) ) ).
% le_iff_sup
thf(fact_801_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X9: real,Y14: real] :
( ( sup_sup_real @ X9 @ Y14 )
= Y14 ) ) ) ).
% le_iff_sup
thf(fact_802_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X9: set_nat,Y14: set_nat] :
( ( sup_sup_set_nat @ X9 @ Y14 )
= Y14 ) ) ) ).
% le_iff_sup
thf(fact_803_sup__least,axiom,
! [Y: nat,X3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_804_sup__least,axiom,
! [Y: int,X3: int,Z3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ Z3 @ X3 )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_805_sup__least,axiom,
! [Y: set_tm,X3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X3 )
=> ( ( ord_less_eq_set_tm @ Z3 @ X3 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_806_sup__least,axiom,
! [Y: real,X3: real,Z3: real] :
( ( ord_less_eq_real @ Y @ X3 )
=> ( ( ord_less_eq_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_807_sup__least,axiom,
! [Y: set_nat,X3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_808_sup__mono,axiom,
! [A: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_809_sup__mono,axiom,
! [A: int,C2: int,B2: int,D2: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B2 @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ ( sup_sup_int @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_810_sup__mono,axiom,
! [A: set_tm,C2: set_tm,B2: set_tm,D2: set_tm] :
( ( ord_less_eq_set_tm @ A @ C2 )
=> ( ( ord_less_eq_set_tm @ B2 @ D2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ ( sup_sup_set_tm @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_811_sup__mono,axiom,
! [A: real,C2: real,B2: real,D2: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B2 @ D2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ ( sup_sup_real @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_812_sup__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_813_sup_Omono,axiom,
! [C2: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_814_sup_Omono,axiom,
! [C2: int,A: int,D2: int,B2: int] :
( ( ord_less_eq_int @ C2 @ A )
=> ( ( ord_less_eq_int @ D2 @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D2 ) @ ( sup_sup_int @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_815_sup_Omono,axiom,
! [C2: set_tm,A: set_tm,D2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C2 @ A )
=> ( ( ord_less_eq_set_tm @ D2 @ B2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ C2 @ D2 ) @ ( sup_sup_set_tm @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_816_sup_Omono,axiom,
! [C2: real,A: real,D2: real,B2: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ( ord_less_eq_real @ D2 @ B2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ C2 @ D2 ) @ ( sup_sup_real @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_817_sup_Omono,axiom,
! [C2: set_nat,A: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_818_le__supI2,axiom,
! [X3: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_819_le__supI2,axiom,
! [X3: int,B2: int,A: int] :
( ( ord_less_eq_int @ X3 @ B2 )
=> ( ord_less_eq_int @ X3 @ ( sup_sup_int @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_820_le__supI2,axiom,
! [X3: set_tm,B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ B2 )
=> ( ord_less_eq_set_tm @ X3 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_821_le__supI2,axiom,
! [X3: real,B2: real,A: real] :
( ( ord_less_eq_real @ X3 @ B2 )
=> ( ord_less_eq_real @ X3 @ ( sup_sup_real @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_822_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_823_le__supI1,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_824_le__supI1,axiom,
! [X3: int,A: int,B2: int] :
( ( ord_less_eq_int @ X3 @ A )
=> ( ord_less_eq_int @ X3 @ ( sup_sup_int @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_825_le__supI1,axiom,
! [X3: set_tm,A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ A )
=> ( ord_less_eq_set_tm @ X3 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_826_le__supI1,axiom,
! [X3: real,A: real,B2: real] :
( ( ord_less_eq_real @ X3 @ A )
=> ( ord_less_eq_real @ X3 @ ( sup_sup_real @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_827_le__supI1,axiom,
! [X3: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_828_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_829_sup__ge2,axiom,
! [Y: int,X3: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_830_sup__ge2,axiom,
! [Y: set_tm,X3: set_tm] : ( ord_less_eq_set_tm @ Y @ ( sup_sup_set_tm @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_831_sup__ge2,axiom,
! [Y: real,X3: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_832_sup__ge2,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_833_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_834_sup__ge1,axiom,
! [X3: int,Y: int] : ( ord_less_eq_int @ X3 @ ( sup_sup_int @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_835_sup__ge1,axiom,
! [X3: set_tm,Y: set_tm] : ( ord_less_eq_set_tm @ X3 @ ( sup_sup_set_tm @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_836_sup__ge1,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ X3 @ ( sup_sup_real @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_837_sup__ge1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_838_le__supI,axiom,
! [A: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_839_le__supI,axiom,
! [A: int,X3: int,B2: int] :
( ( ord_less_eq_int @ A @ X3 )
=> ( ( ord_less_eq_int @ B2 @ X3 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_840_le__supI,axiom,
! [A: set_tm,X3: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ X3 )
=> ( ( ord_less_eq_set_tm @ B2 @ X3 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_841_le__supI,axiom,
! [A: real,X3: real,B2: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ B2 @ X3 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_842_le__supI,axiom,
! [A: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_843_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_844_paramst__liftt_I1_J,axiom,
! [T: tm] :
( ( paramst @ ( liftt @ T ) )
= ( paramst @ T ) ) ).
% paramst_liftt(1)
thf(fact_845_zero__le__floor,axiom,
! [X3: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% zero_le_floor
thf(fact_846_split__nat,axiom,
! [P3: nat > $o,I2: int] :
( ( P3 @ ( nat2 @ I2 ) )
= ( ! [N3: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P3 @ N3 ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P3 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_847_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_848_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_849_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_850_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_851_floor__of__int,axiom,
! [Z3: int] :
( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z3 ) )
= Z3 ) ).
% floor_of_int
thf(fact_852_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_853_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_854_floor__zero,axiom,
( ( archim6058952711729229775r_real @ zero_zero_real )
= zero_zero_int ) ).
% floor_zero
thf(fact_855_floor__of__nat,axiom,
! [N: nat] :
( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% floor_of_nat
thf(fact_856_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_857_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_858_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_859_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_860_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_861_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_862_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_863_floor__less__zero,axiom,
! [X3: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
= ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% floor_less_zero
thf(fact_864_zero__less__ceiling,axiom,
! [X3: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
= ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% zero_less_ceiling
thf(fact_865_less__ceiling__iff,axiom,
! [Z3: int,X3: real] :
( ( ord_less_int @ Z3 @ ( archim7802044766580827645g_real @ X3 ) )
= ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X3 ) ) ).
% less_ceiling_iff
thf(fact_866_ceiling__less__cancel,axiom,
! [X3: real,Y: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y ) )
=> ( ord_less_real @ X3 @ Y ) ) ).
% ceiling_less_cancel
thf(fact_867_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_868_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_869_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_870_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_871_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_872_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_873_floor__less__iff,axiom,
! [X3: real,Z3: int] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ Z3 )
= ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ) ).
% floor_less_iff
thf(fact_874_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_875_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_876_order__less__imp__not__less,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_877_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_878_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_879_order__less__imp__not__eq2,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_880_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_881_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_882_order__less__imp__not__eq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_883_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_884_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_885_linorder__less__linear,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_real @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_886_order__less__imp__triv,axiom,
! [X3: int,Y: int,P3: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_887_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_888_order__less__imp__triv,axiom,
! [X3: real,Y: real,P3: $o] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ X3 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_889_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_890_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_891_order__less__not__sym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_892_order__less__subst2,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_893_order__less__subst2,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_894_order__less__subst2,axiom,
! [A: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_895_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_896_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_897_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_898_order__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_899_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_900_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_901_order__less__subst1,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_902_order__less__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_903_order__less__subst1,axiom,
! [A: int,F: real > int,B2: real,C2: real] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_904_order__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_905_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_906_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C2: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_907_order__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C2: int] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_908_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C2: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_909_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C2: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_910_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_911_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_912_order__less__irrefl,axiom,
! [X3: real] :
~ ( ord_less_real @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_913_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_914_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_915_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > real,C2: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_916_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_917_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_918_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C2: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_919_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C2: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_920_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C2: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_921_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_922_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_923_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_924_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B2: int,C2: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ! [X: int,Y10: int] :
( ( ord_less_int @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_925_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_926_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_927_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y10: nat] :
( ( ord_less_nat @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_928_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_929_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_930_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C2: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ! [X: real,Y10: real] :
( ( ord_less_real @ X @ Y10 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ Y10 ) ) )
=> ( ord_less_real @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_931_order__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_932_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_933_order__less__trans,axiom,
! [X3: real,Y: real,Z3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_934_order__less__asym_H,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_935_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_936_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_937_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_938_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_939_linorder__neq__iff,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
= ( ( ord_less_real @ X3 @ Y )
| ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_940_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_941_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_942_order__less__asym,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ~ ( ord_less_real @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_943_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_944_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_945_linorder__neqE,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_946_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_947_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_948_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_949_order_Ostrict__implies__not__eq,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_950_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_951_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_952_dual__order_Ostrict__trans,axiom,
! [B2: int,A: int,C2: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C2 @ B2 )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_953_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_954_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C2: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_955_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_956_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_957_not__less__iff__gr__or__eq,axiom,
! [X3: real,Y: real] :
( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( ( ord_less_real @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_958_order_Ostrict__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_959_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_960_order_Ostrict__trans,axiom,
! [A: real,B2: real,C2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_961_linorder__less__wlog,axiom,
! [P3: int > int > $o,A: int,B2: int] :
( ! [A4: int,B: int] :
( ( ord_less_int @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: int] : ( P3 @ A4 @ A4 )
=> ( ! [A4: int,B: int] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_962_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B: nat] :
( ( ord_less_nat @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: nat] : ( P3 @ A4 @ A4 )
=> ( ! [A4: nat,B: nat] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_963_linorder__less__wlog,axiom,
! [P3: real > real > $o,A: real,B2: real] :
( ! [A4: real,B: real] :
( ( ord_less_real @ A4 @ B )
=> ( P3 @ A4 @ B ) )
=> ( ! [A4: real] : ( P3 @ A4 @ A4 )
=> ( ! [A4: real,B: real] :
( ( P3 @ B @ A4 )
=> ( P3 @ A4 @ B ) )
=> ( P3 @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_964_exists__least__iff,axiom,
( ( ^ [P6: nat > $o] :
? [X14: nat] : ( P6 @ X14 ) )
= ( ^ [P7: nat > $o] :
? [N3: nat] :
( ( P7 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P7 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_965_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_966_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_967_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_968_dual__order_Oasym,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_969_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_970_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_971_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_972_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_973_linorder__cases,axiom,
! [X3: real,Y: real] :
( ~ ( ord_less_real @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_974_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_975_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_976_antisym__conv3,axiom,
! [Y: real,X3: real] :
( ~ ( ord_less_real @ Y @ X3 )
=> ( ( ~ ( ord_less_real @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_977_less__induct,axiom,
! [P3: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y13: nat] :
( ( ord_less_nat @ Y13 @ X )
=> ( P3 @ Y13 ) )
=> ( P3 @ X ) )
=> ( P3 @ A ) ) ).
% less_induct
thf(fact_978_ord__less__eq__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_979_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_980_ord__less__eq__trans,axiom,
! [A: real,B2: real,C2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_981_ord__eq__less__trans,axiom,
! [A: int,B2: int,C2: int] :
( ( A = B2 )
=> ( ( ord_less_int @ B2 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_982_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_983_ord__eq__less__trans,axiom,
! [A: real,B2: real,C2: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_984_order_Oasym,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order.asym
thf(fact_985_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_986_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_987_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_988_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_989_less__imp__neq,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_990_dense,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X3 @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_991_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_992_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_993_gt__ex,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% gt_ex
thf(fact_994_lt__ex,axiom,
! [X3: int] :
? [Y10: int] : ( ord_less_int @ Y10 @ X3 ) ).
% lt_ex
thf(fact_995_lt__ex,axiom,
! [X3: real] :
? [Y10: real] : ( ord_less_real @ Y10 @ X3 ) ).
% lt_ex
thf(fact_996_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_997_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_998_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_999_floor__less__cancel,axiom,
! [X3: real,Y: real] :
( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y ) )
=> ( ord_less_real @ X3 @ Y ) ) ).
% floor_less_cancel
thf(fact_1000_pinf_I1_J,axiom,
! [P3: int > $o,P5: int > $o,Q4: int > $o,Q5: int > $o] :
( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1001_pinf_I1_J,axiom,
! [P3: nat > $o,P5: nat > $o,Q4: nat > $o,Q5: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1002_pinf_I1_J,axiom,
! [P3: real > $o,P5: real > $o,Q4: real > $o,Q5: real > $o] :
( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1003_pinf_I2_J,axiom,
! [P3: int > $o,P5: int > $o,Q4: int > $o,Q5: int > $o] :
( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1004_pinf_I2_J,axiom,
! [P3: nat > $o,P5: nat > $o,Q4: nat > $o,Q5: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1005_pinf_I2_J,axiom,
! [P3: real > $o,P5: real > $o,Q4: real > $o,Q5: real > $o] :
( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ Z6 @ X )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ Z6 @ X )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1006_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1007_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1008_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_1009_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1010_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1011_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_1012_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1013_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1014_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_1015_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1016_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1017_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_1018_minf_I1_J,axiom,
! [P3: int > $o,P5: int > $o,Q4: int > $o,Q5: int > $o] :
( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1019_minf_I1_J,axiom,
! [P3: nat > $o,P5: nat > $o,Q4: nat > $o,Q5: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1020_minf_I1_J,axiom,
! [P3: real > $o,P5: real > $o,Q4: real > $o,Q5: real > $o] :
( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P3 @ X2 )
& ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
& ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1021_minf_I2_J,axiom,
! [P3: int > $o,P5: int > $o,Q4: int > $o,Q5: int > $o] :
( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: int] :
! [X: int] :
( ( ord_less_int @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1022_minf_I2_J,axiom,
! [P3: nat > $o,P5: nat > $o,Q4: nat > $o,Q5: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1023_minf_I2_J,axiom,
! [P3: real > $o,P5: real > $o,Q4: real > $o,Q5: real > $o] :
( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ X @ Z6 )
=> ( ( P3 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z6: real] :
! [X: real] :
( ( ord_less_real @ X @ Z6 )
=> ( ( Q4 @ X )
= ( Q5 @ X ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P3 @ X2 )
| ( Q4 @ X2 ) )
= ( ( P5 @ X2 )
| ( Q5 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1024_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1025_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1026_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_1027_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1028_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1029_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_1030_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_1031_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_1032_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_1033_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_1034_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_1035_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_1036_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_1037_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_1038_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_1039_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1040_liftt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A @ Ts ) )
= ( fun @ A @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_1041_liftts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T @ Ts ) )
= ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_1042_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_1043_nat__floor__neg,axiom,
! [X3: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1044_le__nat__floor,axiom,
! [X3: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
=> ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_1045_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1046_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_1047_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_1048_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1049_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1050_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1051_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1052_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1053_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1054_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1055_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P3 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_1056_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P3 @ M3 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_1057_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1058_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1059_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1060_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1061_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1062_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1063_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1064_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1065_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1066_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1067_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1068_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1069_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P3 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P3 @ M3 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_1070_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X9: real,Y14: real] :
( ( ord_less_real @ X9 @ Y14 )
| ( X9 = Y14 ) ) ) ) ).
% less_eq_real_def
thf(fact_1071_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1072_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1073_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1074_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1075_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1076_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1077_inc__term_Osimps_I2_J,axiom,
! [I2: nat,L: list_tm] :
( ( inc_term @ ( fun @ I2 @ L ) )
= ( fun @ I2 @ ( inc_list @ L ) ) ) ).
% inc_term.simps(2)
thf(fact_1078_inc__list_Osimps_I2_J,axiom,
! [T: tm,L: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L ) ) ) ).
% inc_list.simps(2)
thf(fact_1079_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P3 @ I5 ) )
& ( P3 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1080_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1081_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1082_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_1083_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1084_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1085_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1086_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1087_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1088_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1089_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1090_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1091_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1092_nat__descend__induct,axiom,
! [N: nat,P3: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P3 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P3 @ I5 ) )
=> ( P3 @ K2 ) ) )
=> ( P3 @ M ) ) ) ).
% nat_descend_induct
thf(fact_1093_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= zero_zero_real )
= ( X3 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1094_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_1095_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1096_ln__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1097_ln__inj__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X3 )
= ( ln_ln_real @ Y ) )
= ( X3 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1098_ln__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1099_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_1100_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1101_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_1102_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1103_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1104_ln__less__self,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_1105_ln__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_ge_zero
thf(fact_1106_ln__bound,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_1107_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1108_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1109_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1110_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_1111_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1112_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1113_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1114_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1115_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1116_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1117_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1118_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1119_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1120_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1121_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1122_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1123_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1124_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1125_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_1126_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_1127_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1128_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1129_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1130_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1131_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1132_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1133_int__cases2,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1134_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1135_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1136_int__cases,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1137_int__of__nat__induct,axiom,
! [P3: int > $o,Z3: int] :
( ! [N2: nat] : ( P3 @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P3 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P3 @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1138_zero__induct__lemma,axiom,
! [P3: nat > $o,K: nat,I2: nat] :
( ( P3 @ K )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1139_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1140_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1141_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_1142_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1143_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R2 @ X @ X )
=> ( ! [X: nat,Y10: nat,Z: nat] :
( ( R2 @ X @ Y10 )
=> ( ( R2 @ Y10 @ Z )
=> ( R2 @ X @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1144_nat__induct__at__least,axiom,
! [M: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P3 @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1145_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P3 @ M3 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_1146_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1147_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1148_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1149_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1150_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1151_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1152_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1153_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_1154_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1155_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1161_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1162_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1163_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1164_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1165_nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) )
=> ( P3 @ N ) ) ) ).
% nat_induct
thf(fact_1166_diff__induct,axiom,
! [P3: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P3 @ X @ zero_zero_nat )
=> ( ! [Y10: nat] : ( P3 @ zero_zero_nat @ ( suc @ Y10 ) )
=> ( ! [X: nat,Y10: nat] :
( ( P3 @ X @ Y10 )
=> ( P3 @ ( suc @ X ) @ ( suc @ Y10 ) ) )
=> ( P3 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1167_zero__induct,axiom,
! [P3: nat > $o,K: nat] :
( ( P3 @ K )
=> ( ! [N2: nat] :
( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1168_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1169_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1170_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1171_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1172_exists__least__lemma,axiom,
! [P3: nat > $o] :
( ~ ( P3 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P3 @ X_12 )
=> ? [N2: nat] :
( ~ ( P3 @ N2 )
& ( P3 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1173_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1174_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_1175_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1176_negD,axiom,
! [X3: int] :
( ( ord_less_int @ X3 @ zero_zero_int )
=> ? [N2: nat] :
( X3
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1177_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1178_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1179_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1180_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1181_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1182_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1183_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1184_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1185_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1186_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1187_Ex__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P3 @ I4 ) ) )
= ( ( P3 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P3 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1188_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1189_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1190_All__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P3 @ I4 ) ) )
= ( ( P3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P3 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1191_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1192_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1193_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1194_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1195_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1196_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1197_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1198_less__Suc__induct,axiom,
! [I2: nat,J: nat,P3: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] : ( P3 @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P3 @ I @ J2 )
=> ( ( P3 @ J2 @ K2 )
=> ( P3 @ I @ K2 ) ) ) ) )
=> ( P3 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1199_strict__inc__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P3 @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P3 @ ( suc @ I ) )
=> ( P3 @ I ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1200_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1201_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1202_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1203_Ex__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P3 @ I4 ) ) )
= ( ( P3 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P3 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1204_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1205_All__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P3 @ I4 ) ) )
= ( ( P3 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P3 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1206_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1207_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1208_diff__less__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1209_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1210_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1211_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1212_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1213_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1214_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1215_inc__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P3 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ ( suc @ N2 ) )
=> ( P3 @ N2 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1216_dec__induct,axiom,
! [I2: nat,J: nat,P3: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P3 @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) ) )
=> ( P3 @ J ) ) ) ) ).
% dec_induct
thf(fact_1217_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1218_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1219_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1220_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1221_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P3 @ I5 ) )
& ( P3 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1222_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1223_nat__induct__non__zero,axiom,
! [N: nat,P3: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P3 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P3 @ N2 )
=> ( P3 @ ( suc @ N2 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1224_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1225_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1226_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1227_subst_Osimps_I5_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( exi @ P ) @ S @ K )
= ( exi @ ( subst @ P @ ( liftt @ S ) @ ( suc @ K ) ) ) ) ).
% subst.simps(5)
thf(fact_1228_subst_Osimps_I6_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( uni @ P ) @ S @ K )
= ( uni @ ( subst @ P @ ( liftt @ S ) @ ( suc @ K ) ) ) ) ).
% subst.simps(6)
thf(fact_1229_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1230_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1231_floor__eq4,axiom,
! [N: nat,X3: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
=> ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
= N ) ) ) ).
% floor_eq4
thf(fact_1232_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1233_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M5: nat,N2: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1234_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1235_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1236_int__le__induct,axiom,
! [I2: int,K: int,P3: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P3 @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P3 @ I )
=> ( P3 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1237_int__less__induct,axiom,
! [I2: int,K: int,P3: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P3 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P3 @ I )
=> ( P3 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1238_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_1239_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1240_ln__eq__minus__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= ( minus_minus_real @ X3 @ one_one_real ) )
=> ( X3 = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_1241_int__ops_I6_J,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1242_nat__diff__distrib_H,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X3 @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1243_nat__diff__distrib,axiom,
! [Z5: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z3 )
=> ( ( nat2 @ ( minus_minus_int @ Z3 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1244_zdiff__int__split,axiom,
! [P3: int > $o,X3: nat,Y: nat] :
( ( P3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P3 @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1245_ln__le__minus__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_1246_ln__one__minus__pos__upper__bound,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_1247_arsinh__minus__real,axiom,
! [X3: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X3 ) )
= ( uminus_uminus_real @ ( arsinh_real @ X3 ) ) ) ).
% arsinh_minus_real
thf(fact_1248_Bolzano,axiom,
! [A: real,B2: real,P3: real > real > $o] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ! [A4: real,B: real,C5: real] :
( ( P3 @ A4 @ B )
=> ( ( P3 @ B @ C5 )
=> ( ( ord_less_eq_real @ A4 @ B )
=> ( ( ord_less_eq_real @ B @ C5 )
=> ( P3 @ A4 @ C5 ) ) ) ) )
=> ( ! [X: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [A4: real,B: real] :
( ( ( ord_less_eq_real @ A4 @ X )
& ( ord_less_eq_real @ X @ B )
& ( ord_less_real @ ( minus_minus_real @ B @ A4 ) @ D3 ) )
=> ( P3 @ A4 @ B ) ) ) ) )
=> ( P3 @ A @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1249_list__decode_Ocases,axiom,
! [X3: nat] :
( ( X3 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X3
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1250_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1251_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1252_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1253_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1254_nat__abs__int__diff,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1255_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_nat @ I @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1256_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1257_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1258_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1259_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1260_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1261_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1262_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
% Helper facts (15)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X3: list_fm,Y: list_fm] :
( ( if_list_fm @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X3: list_fm,Y: list_fm] :
( ( if_list_fm @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y: list_tm] :
( ( if_list_tm @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y: list_tm] :
( ( if_list_tm @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X3: list_real,Y: list_real] :
( ( if_list_real @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X3: list_real,Y: list_real] :
( ( if_list_real @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [T4: tm] :
( ( member_tm2 @ T4 @ ( terms @ s ) )
=> ( ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ T4 @ pa ) ) @ s )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------