TPTP Problem File: SLH0864^1.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_Calc2/0018_EPathHintikka/prob_00691_029074__13316916_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1613 ( 588 unt; 340 typ; 0 def)
% Number of atoms : 3310 (1092 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9786 ( 346 ~; 57 |; 175 &;7824 @)
% ( 0 <=>;1384 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 58 ( 57 usr)
% Number of type conns : 661 ( 661 >; 0 *; 0 +; 0 <<)
% Number of symbols : 286 ( 283 usr; 18 con; 0-3 aty)
% Number of variables : 3389 ( 285 ^;3040 !; 64 ?;3389 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:55.668
%------------------------------------------------------------------------------
% Could-be-implicit typings (57)
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
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thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
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thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
fset_P5628638355292684902m_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
set_Pr1822751329126368876m_rule: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
set_Pr7013894833640182997ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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produc4386893285136772327ist_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_Pr5202636777678657877ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc3245234490656042599ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
stream6017534608192929797ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Ofm_J_J,type,
set_Pr2698443736021152725_tm_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
stream_list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_list_list_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__SeCaV__Otm_J_J,type,
stream_stream_tm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__SeCaV__Ofm_J_J,type,
stream_stream_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Ofm_J,type,
product_prod_tm_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Stream__Ostream_It__Nat__Onat_J_J,type,
stream_stream_nat: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__SeCaV__Otm_J_J,type,
stream_list_tm: $tType ).
thf(ty_n_t__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J,type,
stream_list_fm: $tType ).
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__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
fset_list_fm: $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__Stream__Ostream_It__Prover__Orule_J,type,
stream_rule: $tType ).
thf(ty_n_t__Stream__Ostream_It__SeCaV__Otm_J,type,
stream_tm: $tType ).
thf(ty_n_t__Stream__Ostream_It__SeCaV__Ofm_J,type,
stream_fm: $tType ).
thf(ty_n_t__Stream__Ostream_It__Nat__Onat_J,type,
stream_nat: $tType ).
thf(ty_n_t__List__Olist_It__Prover__Orule_J,type,
list_rule: $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__FSet__Ofset_It__SeCaV__Otm_J,type,
fset_tm: $tType ).
thf(ty_n_t__FSet__Ofset_It__SeCaV__Ofm_J,type,
fset_fm: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
fset_nat: $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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Prover__Orule,type,
rule: $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 ).
% Explicit typings (283)
thf(sy_c_Abstract__Completeness_ORuleSystem_Opos_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
abstra4499547390127564210m_rule: stream2709947120125613254m_rule > produc340336539035504054m_rule > nat ).
thf(sy_c_EPathHintikka_Opseq,type,
pseq: produc340336539035504054m_rule > list_fm ).
thf(sy_c_EPathHintikka_Optms,type,
ptms: produc340336539035504054m_rule > list_tm ).
thf(sy_c_EPathHintikka_Otree__fms,type,
tree_fms: stream2709947120125613254m_rule > set_fm ).
thf(sy_c_FSet_Ofmember_001t__List__Olist_It__SeCaV__Ofm_J,type,
fmember_list_fm: list_fm > fset_list_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
fmember_nat: nat > fset_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fmembe3754813877001230652ist_fm: produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
fmembe1089128867255857199m_rule: produc340336539035504054m_rule > fset_P5628638355292684902m_rule > $o ).
thf(sy_c_FSet_Ofmember_001t__SeCaV__Ofm,type,
fmember_fm: fm > fset_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__SeCaV__Otm,type,
fmember_tm: tm > fset_tm > $o ).
thf(sy_c_FSet_Ofset__of__list_001t__List__Olist_It__SeCaV__Ofm_J,type,
fset_of_list_list_fm: list_list_fm > fset_list_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__Nat__Onat,type,
fset_of_list_nat: list_nat > fset_nat ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fset_o3222440871817023035ist_fm: list_P2887561121880082555ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
fset_o6059354669909737328m_rule: list_P2774625669004309958m_rule > fset_P5628638355292684902m_rule ).
thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Ofm,type,
fset_of_list_fm: list_fm > fset_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Otm,type,
fset_of_list_tm: list_tm > fset_tm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
minus_7547787838945083330ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
minus_639611354763871680ist_fm: set_list_fm > set_list_fm > set_list_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
minus_5980356457887379781m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__SeCaV__Ofm_J,type,
minus_minus_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__SeCaV__Otm_J,type,
minus_minus_set_tm: set_tm > set_tm > set_tm ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__List__Olist_It__SeCaV__Ofm_J_001t__Nat__Onat,type,
groups8933402702422637614fm_nat: ( list_fm > nat ) > nat > list_list_fm > nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__SeCaV__Ofm_001t__Nat__Onat,type,
groups2912550417867917864fm_nat: ( fm > nat ) > nat > list_fm > nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__SeCaV__Otm_001t__Nat__Onat,type,
groups4576929674938418586tm_nat: ( tm > nat ) > nat > list_tm > nat ).
thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
if_fse7999432387889793441ist_fm: $o > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
if_Pro4760001780252510779ist_fm: $o > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
sup_su5005723363340324783ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm ).
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__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
sup_sup_set_list_fm: set_list_fm > set_list_fm > set_list_fm ).
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__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sup_su6946459741510085528m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule ).
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_List_Ocan__select_001t__List__Olist_It__SeCaV__Ofm_J,type,
can_select_list_fm: ( list_fm > $o ) > set_list_fm > $o ).
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__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
can_se4506553828710032889m_rule: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule > $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__List__Olist_It__SeCaV__Ofm_J,type,
coset_list_fm: list_list_fm > set_list_fm ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
coset_4722139400431689347m_rule: list_P2774625669004309958m_rule > set_Pr1822751329126368876m_rule ).
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__List__Olist_It__SeCaV__Ofm_J,type,
gen_length_list_fm: nat > list_list_fm > nat ).
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_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__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
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__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
nil_Pr2808214839852828720m_rule: list_P2774625669004309958m_rule ).
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_Ohd_001t__List__Olist_It__SeCaV__Ofm_J,type,
hd_list_fm: list_list_fm > list_fm ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
hd_Pro7241777042969981963m_rule: list_P2774625669004309958m_rule > produc340336539035504054m_rule ).
thf(sy_c_List_Olist_Ohd_001t__SeCaV__Ofm,type,
hd_fm: list_fm > fm ).
thf(sy_c_List_Olist_Ohd_001t__SeCaV__Otm,type,
hd_tm: list_tm > tm ).
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__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
set_Pr5051287786238754058ist_fm: list_P2887561121880082555ist_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
set_Pr4534715572506550497m_rule: list_P2774625669004309958m_rule > set_Pr1822751329126368876m_rule ).
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__List__Olist_It__SeCaV__Ofm_J,type,
list_ex1_list_fm: ( list_fm > $o ) > list_list_fm > $o ).
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__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
list_e4959145360368724013m_rule: ( produc340336539035504054m_rule > $o ) > list_P2774625669004309958m_rule > $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_Olistrel_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
listre5805154493122130495ist_fm: set_Pr5202636777678657877ist_fm > set_Pr7013894833640182997ist_fm ).
thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
listre749368180268468182m_rule: set_Pr1822751329126368876m_rule > set_Pr4462384566710528898t_rule ).
thf(sy_c_List_Olistrel_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
listrel_tm_fm: set_Pr2698443736021152725_tm_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_Onth_001t__List__Olist_It__SeCaV__Ofm_J,type,
nth_list_fm: list_list_fm > nat > list_fm ).
thf(sy_c_List_Onth_001t__List__Olist_It__SeCaV__Otm_J,type,
nth_list_tm: list_list_tm > nat > list_tm ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
nth_Pr580027083122244092ist_fm: list_P2887561121880082555ist_fm > nat > produc6018962875968178549ist_fm ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
nth_Pr3936752564452695919m_rule: list_P2774625669004309958m_rule > nat > produc340336539035504054m_rule ).
thf(sy_c_List_Onth_001t__Prover__Orule,type,
nth_rule: list_rule > nat > rule ).
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_Opartition_001t__List__Olist_It__SeCaV__Ofm_J,type,
partition_list_fm: ( list_fm > $o ) > list_list_fm > produc4386893285136772327ist_fm ).
thf(sy_c_List_Opartition_001t__Nat__Onat,type,
partition_nat: ( nat > $o ) > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_List_Opartition_001t__SeCaV__Ofm,type,
partition_fm: ( fm > $o ) > list_fm > produc3245234490656042599ist_fm ).
thf(sy_c_List_Opartition_001t__SeCaV__Otm,type,
partition_tm: ( tm > $o ) > list_tm > produc5776448205642668775ist_tm ).
thf(sy_c_List_Orotate_001t__List__Olist_It__SeCaV__Ofm_J,type,
rotate_list_fm: nat > list_list_fm > list_list_fm ).
thf(sy_c_List_Orotate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
rotate8779165051853931260m_rule: nat > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Orotate_001t__SeCaV__Ofm,type,
rotate_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Orotate_001t__SeCaV__Otm,type,
rotate_tm: nat > list_tm > list_tm ).
thf(sy_c_List_Otake_001t__List__Olist_It__SeCaV__Ofm_J,type,
take_list_fm: nat > list_list_fm > list_list_fm ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
take_P3067526267515409992m_rule: nat > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Otake_001t__SeCaV__Ofm,type,
take_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Otake_001t__SeCaV__Otm,type,
take_tm: nat > list_tm > list_tm ).
thf(sy_c_List_Ounion_001t__List__Olist_It__SeCaV__Ofm_J,type,
union_list_fm: list_list_fm > list_list_fm > list_list_fm ).
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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
size_s115229985653309035ist_fm: list_list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
size_s9096087352182575069ist_tm: list_list_tm > 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__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
size_s3138477486474831591ist_fm: list_P2887561121880082555ist_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
size_s1575636608424004698m_rule: list_P2774625669004309958m_rule > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Prover__Orule_J,type,
size_size_list_rule: list_rule > 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__Prover__Orule,type,
size_size_rule: rule > 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__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le7716396445699002503ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_less_set_list_fm: set_list_fm > set_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
ord_le4093866177137961152m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le3182162295460067583ist_fm: fset_list_fm > fset_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Nat__Onat_J,type,
ord_less_eq_fset_nat: fset_nat > fset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le3986950534092794747ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__SeCaV__Ofm_J,type,
ord_less_eq_fset_fm: fset_fm > fset_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__SeCaV__Otm_J,type,
ord_less_eq_fset_tm: fset_tm > fset_tm > $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__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le7838213414353715577ist_fm: set_list_fm > set_list_fm > $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__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
ord_le6390412330253371084m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > $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_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc8321651870839017815ist_fm: list_list_fm > list_list_fm > produc4386893285136772327ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc8328755919010440805ist_fm: list_list_tm > list_list_fm > produc4393997333308195317ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_001t__List__Olist_It__Prover__Orule_J,type,
produc3723918873312807110t_rule: list_P2887561121880082555ist_fm > list_rule > produc1097132844357787852t_rule ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc7863996417982153943ist_fm: list_fm > list_fm > produc3245234490656042599ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc1414352766439514085ist_fm: list_tm > list_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc1418304791525149271ist_tm: list_tm > list_tm > produc5776448205642668775ist_tm ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
produc1733806532565653680m_rule: produc6018962875968178549ist_fm > rule > produc340336539035504054m_rule ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
produc6261311607089640965m_rule: produc340336539035504054m_rule > produc340336539035504054m_rule > produc8828831911945107917m_rule ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
product_Pair_tm_fm: tm > fm > product_prod_tm_fm ).
thf(sy_c_ProverLemmas_Oaffects,type,
affects: rule > fm > $o ).
thf(sy_c_Prover_Ochildren,type,
children: list_tm > rule > list_fm > list_list_fm ).
thf(sy_c_Prover_Oeffect,type,
effect: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_Prover_OgenerateNew,type,
generateNew: list_tm > nat ).
thf(sy_c_Prover_Oparts,type,
parts: list_tm > rule > fm > list_list_fm ).
thf(sy_c_Prover_Orule_ODeltaUni,type,
deltaUni: rule ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_SeCaV_Oext_001t__List__Olist_It__SeCaV__Ofm_J,type,
ext_list_fm: list_list_fm > list_list_fm > $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_OUni,type,
uni: fm > fm ).
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__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > list_list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
member_nat: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
member4200870186495857963m_rule: produc340336539035504054m_rule > list_P2774625669004309958m_rule > $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_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_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_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__List__Olist_It__SeCaV__Ofm_J,type,
collect_list_fm: ( list_fm > $o ) > set_list_fm ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
collec1009276759030335883m_rule: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule ).
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_Stream_Ocycle_001t__List__Olist_It__SeCaV__Ofm_J,type,
cycle_list_fm: list_list_fm > stream_list_fm ).
thf(sy_c_Stream_Ocycle_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
cycle_5335117900641983313m_rule: list_P2774625669004309958m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Ocycle_001t__SeCaV__Ofm,type,
cycle_fm: list_fm > stream_fm ).
thf(sy_c_Stream_Ocycle_001t__SeCaV__Otm,type,
cycle_tm: list_tm > stream_tm ).
thf(sy_c_Stream_Oflat_001t__List__Olist_It__SeCaV__Ofm_J,type,
flat_list_fm: stream_list_list_fm > stream_list_fm ).
thf(sy_c_Stream_Oflat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
flat_P7721466590633226428m_rule: stream2471014364565126742m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oflat_001t__SeCaV__Ofm,type,
flat_fm: stream_list_fm > stream_fm ).
thf(sy_c_Stream_Oflat_001t__SeCaV__Otm,type,
flat_tm: stream_list_tm > stream_tm ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_It__SeCaV__Ofm_J,type,
sdrop_list_fm: nat > stream_list_fm > stream_list_fm ).
thf(sy_c_Stream_Osdrop_001t__Nat__Onat,type,
sdrop_nat: nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
sdrop_9176333610110415838ist_fm: nat > stream4408948924543953275ist_fm > stream4408948924543953275ist_fm ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sdrop_8169176516188972301m_rule: nat > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sdrop_7192298464603511222m_rule: nat > stream6210534828274662995m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Stream_Osdrop_001t__Prover__Orule,type,
sdrop_rule: nat > stream_rule > stream_rule ).
thf(sy_c_Stream_Osdrop_001t__SeCaV__Ofm,type,
sdrop_fm: nat > stream_fm > stream_fm ).
thf(sy_c_Stream_Osdrop_001t__SeCaV__Otm,type,
sdrop_tm: nat > stream_tm > stream_tm ).
thf(sy_c_Stream_Oshift_001t__List__Olist_It__SeCaV__Ofm_J,type,
shift_list_fm: list_list_fm > stream_list_fm > stream_list_fm ).
thf(sy_c_Stream_Oshift_001t__Nat__Onat,type,
shift_nat: list_nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Oshift_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
shift_2334844276748245581m_rule: list_P2774625669004309958m_rule > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oshift_001t__SeCaV__Ofm,type,
shift_fm: list_fm > stream_fm > stream_fm ).
thf(sy_c_Stream_Oshift_001t__SeCaV__Otm,type,
shift_tm: list_tm > stream_tm > stream_tm ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_It__SeCaV__Ofm_J,type,
smerge_list_fm: stream6017534608192929797ist_fm > stream_list_fm ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
smerge193809993764105000m_rule: stream3752074346242807894m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osmerge_001t__SeCaV__Ofm,type,
smerge_fm: stream_stream_fm > stream_fm ).
thf(sy_c_Stream_Osmerge_001t__SeCaV__Otm,type,
smerge_tm: stream_stream_tm > stream_tm ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__SeCaV__Ofm_J,type,
snth_list_fm: stream_list_fm > nat > list_fm ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__SeCaV__Otm_J,type,
snth_list_tm: stream_list_tm > nat > list_tm ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
snth_P7093566783922538521ist_fm: stream4408948924543953275ist_fm > nat > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
snth_P6679518042731451922m_rule: stream2709947120125613254m_rule > nat > produc340336539035504054m_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_P8853764340393315953m_rule: stream6210534828274662995m_rule > nat > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Osnth_001t__Prover__Orule,type,
snth_rule: stream_rule > nat > rule ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Ofm,type,
snth_fm: stream_fm > nat > fm ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Otm,type,
snth_tm: stream_tm > nat > tm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J,type,
snth_stream_list_fm: stream6017534608192929797ist_fm > nat > stream_list_fm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_s6182113952396108578m_rule: stream3752074346242807894m_rule > nat > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Ofm_J,type,
snth_stream_fm: stream_stream_fm > nat > stream_fm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Otm_J,type,
snth_stream_tm: stream_stream_tm > nat > stream_tm ).
thf(sy_c_Stream_Ostake_001t__List__Olist_It__SeCaV__Ofm_J,type,
stake_list_fm: nat > stream_list_fm > list_list_fm ).
thf(sy_c_Stream_Ostake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
stake_5421812949518764133m_rule: nat > stream2709947120125613254m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_Stream_Ostake_001t__SeCaV__Ofm,type,
stake_fm: nat > stream_fm > list_fm ).
thf(sy_c_Stream_Ostake_001t__SeCaV__Otm,type,
stake_tm: nat > stream_tm > list_tm ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
shd_list_list_fm: stream_list_list_fm > list_list_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
shd_li4676821617271663642m_rule: stream2471014364565126742m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__SeCaV__Ofm_J,type,
shd_list_fm: stream_list_fm > list_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__SeCaV__Otm_J,type,
shd_list_tm: stream_list_tm > list_tm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Nat__Onat,type,
shd_nat: stream_nat > nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
shd_Pr3211216682057661985ist_fm: stream4408948924543953275ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
shd_Pr4562317740776619530m_rule: stream2709947120125613254m_rule > produc340336539035504054m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
shd_Pr4461660664618831993m_rule: stream6210534828274662995m_rule > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Prover__Orule,type,
shd_rule: stream_rule > rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Ofm,type,
shd_fm: stream_fm > fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Otm,type,
shd_tm: stream_tm > tm ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
sset_list_list_fm: stream_list_list_fm > set_list_list_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sset_l1945517436738345138m_rule: stream2471014364565126742m_rule > set_li5047378834958542076m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__SeCaV__Ofm_J,type,
sset_list_fm: stream_list_fm > set_list_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__SeCaV__Otm_J,type,
sset_list_tm: stream_list_tm > set_list_tm ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sset_P4484857331586881186m_rule: stream2709947120125613254m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Ofm,type,
sset_fm: stream_fm > set_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Otm,type,
sset_tm: stream_tm > set_tm ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
stl_list_list_fm: stream_list_list_fm > stream_list_list_fm ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
stl_li6523153919213261078m_rule: stream2471014364565126742m_rule > stream2471014364565126742m_rule ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__SeCaV__Ofm_J,type,
stl_list_fm: stream_list_fm > stream_list_fm ).
thf(sy_c_Stream_Ostream_Ostl_001t__List__Olist_It__SeCaV__Otm_J,type,
stl_list_tm: stream_list_tm > stream_list_tm ).
thf(sy_c_Stream_Ostream_Ostl_001t__Nat__Onat,type,
stl_nat: stream_nat > stream_nat ).
thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_Stream_Ostream_Ostl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
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stl_Pr2679468901532635773m_rule: stream6210534828274662995m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Stream_Ostream_Ostl_001t__Prover__Orule,type,
stl_rule: stream_rule > stream_rule ).
thf(sy_c_Stream_Ostream_Ostl_001t__SeCaV__Ofm,type,
stl_fm: stream_fm > stream_fm ).
thf(sy_c_Stream_Ostream_Ostl_001t__SeCaV__Otm,type,
stl_tm: stream_tm > stream_tm ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
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thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
szip_P2924820683901490861m_rule: stream4408948924543953275ist_fm > stream_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
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thf(sy_c_String_Ochar_Osize__char,type,
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thf(sy_c_member_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
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thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm2: 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,
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thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
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thf(sy_c_member_001t__SeCaV__Ofm,type,
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thf(sy_c_member_001t__SeCaV__Otm,type,
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thf(sy_v_B____,type,
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thf(sy_v_C____,type,
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thf(sy_v_n____,type,
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thf(sy_v_p____,type,
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thf(sy_v_pre____,type,
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thf(sy_v_r_H____,type,
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thf(sy_v_steps,type,
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suf: stream2709947120125613254m_rule ).
thf(sy_v_z_H____,type,
z: list_fm ).
% Relevant facts (1265)
thf(fact_0_pseq__in__tree__fms,axiom,
! [X: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P: fm] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Steps ) )
=> ( ( member_fm2 @ P @ ( set_fm2 @ ( pseq @ X ) ) )
=> ( member_fm2 @ P @ ( tree_fms @ Steps ) ) ) ) ).
% pseq_in_tree_fms
thf(fact_1_z_H_I2_J,axiom,
( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ suf ) )
= ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ b @ z ) @ r ) ) ).
% z'(2)
thf(fact_2_ori,axiom,
( ( sdrop_8169176516188972301m_rule @ n @ steps )
= ( shift_2334844276748245581m_rule @ pre @ suf ) ) ).
% ori
thf(fact_3_C_I1_J,axiom,
ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) @ ( set_tm2 @ c ) ).
% C(1)
thf(fact_4_C_I2_J,axiom,
member_fm2 @ ( sub @ zero_zero_nat @ ( fun @ ( generateNew @ c ) @ nil_tm ) @ p ) @ ( set_fm2 @ z ) ).
% C(2)
thf(fact_5__092_060open_062Uni_Ap_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
member_fm2 @ ( uni @ p ) @ ( tree_fms @ steps ) ).
% \<open>Uni p \<in> tree_fms steps\<close>
thf(fact_6_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_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062C_O_A_092_060lbrakk_062set_A_Iptms_A_Ishd_Asuf_J_J_A_092_060subseteq_062_Aset_AC_059_Asub_A0_A_IFun_A_IgenerateNew_AC_J_A_091_093_J_Ap_A_092_060in_062_Aset_Az_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [C: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) @ ( set_tm2 @ C ) )
=> ~ ( member_fm2 @ ( sub @ zero_zero_nat @ ( fun @ ( generateNew @ C ) @ nil_tm ) @ p ) @ ( set_fm2 @ z ) ) ) ).
% \<open>\<And>thesis. (\<And>C. \<lbrakk>set (ptms (shd suf)) \<subseteq> set C; sub 0 (Fun (generateNew C) []) p \<in> set z'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_n,axiom,
member_fm2 @ ( uni @ p ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).
% n
thf(fact_9_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_10__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_AUni_Ap_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: nat] :
~ ( member_fm2 @ ( uni @ p ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ steps ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n. Uni p \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_11_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_12_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_13_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_14_horner__sum__simps_I1_J,axiom,
! [F: tm > nat,A: nat] :
( ( groups4576929674938418586tm_nat @ F @ A @ nil_tm )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_15_horner__sum__simps_I1_J,axiom,
! [F: fm > nat,A: nat] :
( ( groups2912550417867917864fm_nat @ F @ A @ nil_fm )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_16_horner__sum__simps_I1_J,axiom,
! [F: list_fm > nat,A: nat] :
( ( groups8933402702422637614fm_nat @ F @ A @ nil_list_fm )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_17_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_18_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_19_fm_Oinject_I6_J,axiom,
! [X6: fm,Y6: fm] :
( ( ( uni @ X6 )
= ( uni @ Y6 ) )
= ( X6 = Y6 ) ) ).
% fm.inject(6)
thf(fact_20__092_060open_062Uni_Ap_A_092_060in_062_Aset_A_Ipseq_A_Ishd_Asuf_J_J_092_060close_062,axiom,
member_fm2 @ ( uni @ p ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ).
% \<open>Uni p \<in> set (pseq (shd suf))\<close>
thf(fact_21_subset__code_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,B: set_Pr1822751329126368876m_rule] :
( ( ord_le6390412330253371084m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ B )
= ( ! [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_22_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_23_subset__code_I1_J,axiom,
! [Xs: list_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_24_subset__code_I1_J,axiom,
! [Xs: list_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_25_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_26_sset__sdrop,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] : ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% sset_sdrop
thf(fact_27_sset__sdrop,axiom,
! [N2: nat,S: stream_tm] : ( ord_less_eq_set_tm @ ( sset_tm @ ( sdrop_tm @ N2 @ S ) ) @ ( sset_tm @ S ) ) ).
% sset_sdrop
thf(fact_28_sset__sdrop,axiom,
! [N2: nat,S: stream_fm] : ( ord_less_eq_set_fm @ ( sset_fm @ ( sdrop_fm @ N2 @ S ) ) @ ( sset_fm @ S ) ) ).
% sset_sdrop
thf(fact_29_sset__sdrop,axiom,
! [N2: nat,S: stream_nat] : ( ord_less_eq_set_nat @ ( sset_nat @ ( sdrop_nat @ N2 @ S ) ) @ ( sset_nat @ S ) ) ).
% sset_sdrop
thf(fact_30_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_31_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_32_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_33_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_34_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_35_sset__induct,axiom,
! [Y: fm,S: stream_fm,P2: fm > stream_fm > $o] :
( ( member_fm2 @ Y @ ( sset_fm @ S ) )
=> ( ! [S2: stream_fm] : ( P2 @ ( shd_fm @ S2 ) @ S2 )
=> ( ! [S2: stream_fm,Y2: fm] :
( ( member_fm2 @ Y2 @ ( sset_fm @ ( stl_fm @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl_fm @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_36_sset__induct,axiom,
! [Y: list_fm,S: stream_list_fm,P2: list_fm > stream_list_fm > $o] :
( ( member_list_fm2 @ Y @ ( sset_list_fm @ S ) )
=> ( ! [S2: stream_list_fm] : ( P2 @ ( shd_list_fm @ S2 ) @ S2 )
=> ( ! [S2: stream_list_fm,Y2: list_fm] :
( ( member_list_fm2 @ Y2 @ ( sset_list_fm @ ( stl_list_fm @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl_list_fm @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_37_sset__induct,axiom,
! [Y: tm,S: stream_tm,P2: tm > stream_tm > $o] :
( ( member_tm2 @ Y @ ( sset_tm @ S ) )
=> ( ! [S2: stream_tm] : ( P2 @ ( shd_tm @ S2 ) @ S2 )
=> ( ! [S2: stream_tm,Y2: tm] :
( ( member_tm2 @ Y2 @ ( sset_tm @ ( stl_tm @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl_tm @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_38_sset__induct,axiom,
! [Y: nat,S: stream_nat,P2: nat > stream_nat > $o] :
( ( member_nat2 @ Y @ ( sset_nat @ S ) )
=> ( ! [S2: stream_nat] : ( P2 @ ( shd_nat @ S2 ) @ S2 )
=> ( ! [S2: stream_nat,Y2: nat] :
( ( member_nat2 @ Y2 @ ( sset_nat @ ( stl_nat @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl_nat @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_39_sset__induct,axiom,
! [Y: produc340336539035504054m_rule,S: stream2709947120125613254m_rule,P2: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o] :
( ( member7231649785386036813m_rule @ Y @ ( sset_P4484857331586881186m_rule @ S ) )
=> ( ! [S2: stream2709947120125613254m_rule] : ( P2 @ ( shd_Pr4562317740776619530m_rule @ S2 ) @ S2 )
=> ( ! [S2: stream2709947120125613254m_rule,Y2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y2 @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl_Pr12655793849929990m_rule @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_40_z_H_I1_J,axiom,
fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ b @ z ) @ ( effect @ deltaUni @ ( produc1414352766439514085ist_fm @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ suf ) ) @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ) ).
% z'(1)
thf(fact_41_SeCaV_Oext,axiom,
( ext_list_fm
= ( ^ [Y3: list_list_fm,Z: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Z ) @ ( set_list_fm2 @ Y3 ) ) ) ) ).
% SeCaV.ext
thf(fact_42_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y3: list_tm,Z: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z ) @ ( set_tm2 @ Y3 ) ) ) ) ).
% SeCaV.ext
thf(fact_43_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y3: list_fm,Z: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z ) @ ( set_fm2 @ Y3 ) ) ) ) ).
% SeCaV.ext
thf(fact_44_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y3: list_nat,Z: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z ) @ ( set_nat2 @ Y3 ) ) ) ) ).
% SeCaV.ext
thf(fact_45_shift__left__inj,axiom,
! [Xs: list_P2774625669004309958m_rule,S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
( ( ( shift_2334844276748245581m_rule @ Xs @ S1 )
= ( shift_2334844276748245581m_rule @ Xs @ S22 ) )
= ( S1 = S22 ) ) ).
% shift_left_inj
thf(fact_46_subsetI,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ! [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ A2 )
=> ( member7231649785386036813m_rule @ X3 @ B ) )
=> ( ord_le6390412330253371084m_rule @ A2 @ B ) ) ).
% subsetI
thf(fact_47_subsetI,axiom,
! [A2: set_list_fm,B: set_list_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A2 )
=> ( member_list_fm2 @ X3 @ B ) )
=> ( ord_le7838213414353715577ist_fm @ A2 @ B ) ) ).
% subsetI
thf(fact_48_subsetI,axiom,
! [A2: set_tm,B: set_tm] :
( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ( member_tm2 @ X3 @ B ) )
=> ( ord_less_eq_set_tm @ A2 @ B ) ) ).
% subsetI
thf(fact_49_subsetI,axiom,
! [A2: set_fm,B: set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ( member_fm2 @ X3 @ B ) )
=> ( ord_less_eq_set_fm @ A2 @ B ) ) ).
% subsetI
thf(fact_50_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
=> ( member_nat2 @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_51_subset__antisym,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_52_subset__antisym,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( ord_less_eq_set_fm @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_53_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_54_prod_Oinject,axiom,
! [X1: produc6018962875968178549ist_fm,X22: rule,Y1: produc6018962875968178549ist_fm,Y22: rule] :
( ( ( produc1733806532565653680m_rule @ X1 @ X22 )
= ( produc1733806532565653680m_rule @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_55_prod_Oinject,axiom,
! [X1: list_tm,X22: list_fm,Y1: list_tm,Y22: list_fm] :
( ( ( produc1414352766439514085ist_fm @ X1 @ X22 )
= ( produc1414352766439514085ist_fm @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
! [A: produc6018962875968178549ist_fm,B2: rule,A3: produc6018962875968178549ist_fm,B3: rule] :
( ( ( produc1733806532565653680m_rule @ A @ B2 )
= ( produc1733806532565653680m_rule @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_57_old_Oprod_Oinject,axiom,
! [A: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_58_order__refl,axiom,
! [X: set_tm] : ( ord_less_eq_set_tm @ X @ X ) ).
% order_refl
thf(fact_59_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_60_order__refl,axiom,
! [X: set_fm] : ( ord_less_eq_set_fm @ X @ X ) ).
% order_refl
thf(fact_61_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_62_dual__order_Orefl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% dual_order.refl
thf(fact_63_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_64_dual__order_Orefl,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).
% dual_order.refl
thf(fact_65_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_66_sdrop__stl,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ N2 @ ( stl_Pr12655793849929990m_rule @ S ) )
= ( stl_Pr12655793849929990m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) ) ) ).
% sdrop_stl
thf(fact_67_shift_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ nil_Pr2808214839852828720m_rule @ S )
= S ) ).
% shift.simps(1)
thf(fact_68_shift_Osimps_I1_J,axiom,
! [S: stream_tm] :
( ( shift_tm @ nil_tm @ S )
= S ) ).
% shift.simps(1)
thf(fact_69_shift_Osimps_I1_J,axiom,
! [S: stream_fm] :
( ( shift_fm @ nil_fm @ S )
= S ) ).
% shift.simps(1)
thf(fact_70_shift_Osimps_I1_J,axiom,
! [S: stream_list_fm] :
( ( shift_list_fm @ nil_list_fm @ S )
= S ) ).
% shift.simps(1)
thf(fact_71__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062B_Az_H_Ar_H_O_A_092_060lbrakk_062_IB_M_Az_H_J_A_124_092_060in_062_124_Aeffect_ADeltaUni_A_Iptms_A_Ishd_Asuf_J_M_Apseq_A_Ishd_Asuf_J_J_059_Ashd_A_Istl_Asuf_J_A_061_A_I_IB_M_Az_H_J_M_Ar_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [B4: list_tm,Z2: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z2 ) @ ( effect @ deltaUni @ ( produc1414352766439514085ist_fm @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ suf ) ) @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ suf ) ) ) ) )
=> ! [R: rule] :
( ( shd_Pr4562317740776619530m_rule @ ( stl_Pr12655793849929990m_rule @ suf ) )
!= ( produc1733806532565653680m_rule @ ( produc1414352766439514085ist_fm @ B4 @ Z2 ) @ R ) ) ) ).
% \<open>\<And>thesis. (\<And>B z' r'. \<lbrakk>(B, z') |\<in>| effect DeltaUni (ptms (shd suf), pseq (shd suf)); shd (stl suf) = ((B, z'), r')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_72__092_060open_062_092_060And_062r_O_Aaffects_Ar_A_IUni_Ap_J_A_061_A_Ir_A_061_ADeltaUni_J_092_060close_062,axiom,
! [R2: rule] :
( ( affects @ R2 @ ( uni @ p ) )
= ( R2 = deltaUni ) ) ).
% \<open>\<And>r. affects r (Uni p) = (r = DeltaUni)\<close>
thf(fact_73_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_74_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_75_mem__Collect__eq,axiom,
! [A: fm,P2: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: produc340336539035504054m_rule,P2: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ A @ ( collec1009276759030335883m_rule @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: list_fm,P2: list_fm > $o] :
( ( member_list_fm2 @ A @ ( collect_list_fm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: tm,P2: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( collec1009276759030335883m_rule
@ ^ [X2: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_list_fm] :
( ( collect_list_fm
@ ^ [X2: list_fm] : ( member_list_fm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_86_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_87_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_88_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_89_order__antisym__conv,axiom,
! [Y: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X )
=> ( ( ord_less_eq_set_tm @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_90_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_91_order__antisym__conv,axiom,
! [Y: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X )
=> ( ( ord_less_eq_set_fm @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_92_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_93_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_94_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > nat,C3: nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C3: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_fm,C3: set_fm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C3: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A: set_fm,B2: set_fm,F: set_fm > nat,C3: nat] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_fm,C3: set_fm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_nat,C3: set_nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C3: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C3: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_ord__eq__le__subst,axiom,
! [A: set_fm,F: nat > set_fm,B2: nat,C3: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C3: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: nat,F: set_fm > nat,B2: set_fm,C3: set_fm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C3: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_tm > set_tm,B2: set_tm,C3: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: set_fm,F: set_tm > set_fm,B2: set_tm,C3: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_tm > set_nat,B2: set_tm,C3: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_115_order__eq__refl,axiom,
! [X: set_tm,Y: set_tm] :
( ( X = Y )
=> ( ord_less_eq_set_tm @ X @ Y ) ) ).
% order_eq_refl
thf(fact_116_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_117_order__eq__refl,axiom,
! [X: set_fm,Y: set_fm] :
( ( X = Y )
=> ( ord_less_eq_set_fm @ X @ Y ) ) ).
% order_eq_refl
thf(fact_118_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_119_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > nat,C3: nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C3: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_fm,C3: set_fm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C3: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A: set_fm,B2: set_fm,F: set_fm > nat,C3: nat] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_fm,C3: set_fm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_128_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_nat,C3: set_nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_129_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_130_order__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C3: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_131_order__subst1,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C3: set_tm] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A: nat,F: set_fm > nat,B2: set_fm,C3: set_fm] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_134_order__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B2: nat,C3: nat] :
( ( ord_less_eq_set_fm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_135_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C3: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_136_order__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_137_order__subst1,axiom,
! [A: set_tm,F: set_fm > set_tm,B2: set_fm,C3: set_fm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_138_order__subst1,axiom,
! [A: set_tm,F: set_nat > set_tm,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_139_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_tm,Z3: set_tm] : ( Y5 = Z3 ) )
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ( ord_less_eq_set_tm @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_140_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_141_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_fm,Z3: set_fm] : ( Y5 = Z3 ) )
= ( ^ [A4: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B5 )
& ( ord_less_eq_set_fm @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_142_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_143_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_144_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_145_antisym,axiom,
! [A: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_146_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_147_dual__order_Otrans,axiom,
! [B2: set_tm,A: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C3 @ B2 )
=> ( ord_less_eq_set_tm @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_148_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C3 @ B2 )
=> ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_149_dual__order_Otrans,axiom,
! [B2: set_fm,A: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( ( ord_less_eq_set_fm @ C3 @ B2 )
=> ( ord_less_eq_set_fm @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_150_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C3 @ B2 )
=> ( ord_less_eq_set_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_151_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_152_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_153_dual__order_Oantisym,axiom,
! [B2: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( ( ord_less_eq_set_fm @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_154_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_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_tm,Z3: set_tm] : ( Y5 = Z3 ) )
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ( ord_less_eq_set_tm @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_fm,Z3: set_fm] : ( Y5 = Z3 ) )
= ( ^ [A4: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A4 )
& ( ord_less_eq_set_fm @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_159_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_160_order__trans,axiom,
! [X: set_tm,Y: set_tm,Z4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( ord_less_eq_set_tm @ Y @ Z4 )
=> ( ord_less_eq_set_tm @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_161_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_162_order__trans,axiom,
! [X: set_fm,Y: set_fm,Z4: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ Z4 )
=> ( ord_less_eq_set_fm @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z4 )
=> ( ord_less_eq_set_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_164_order_Otrans,axiom,
! [A: set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ord_less_eq_set_tm @ A @ C3 ) ) ) ).
% order.trans
thf(fact_165_order_Otrans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_166_order_Otrans,axiom,
! [A: set_fm,B2: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ord_less_eq_set_fm @ A @ C3 ) ) ) ).
% order.trans
thf(fact_167_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_168_order__antisym,axiom,
! [X: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( ord_less_eq_set_tm @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_169_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_170_order__antisym,axiom,
! [X: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_171_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_172_ord__le__eq__trans,axiom,
! [A: set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq_set_tm @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_173_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_174_ord__le__eq__trans,axiom,
! [A: set_fm,B2: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq_set_fm @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_175_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_176_ord__eq__le__trans,axiom,
! [A: set_tm,B2: set_tm,C3: set_tm] :
( ( A = B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ord_less_eq_set_tm @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_177_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_178_ord__eq__le__trans,axiom,
! [A: set_fm,B2: set_fm,C3: set_fm] :
( ( A = B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ord_less_eq_set_fm @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_179_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_180_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_tm,Z3: set_tm] : ( Y5 = Z3 ) )
= ( ^ [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
& ( ord_less_eq_set_tm @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_181_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_182_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_fm,Z3: set_fm] : ( Y5 = Z3 ) )
= ( ^ [X2: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y3 )
& ( ord_less_eq_set_fm @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_183_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_184_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_185_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_186_Pair__inject,axiom,
! [A: produc6018962875968178549ist_fm,B2: rule,A3: produc6018962875968178549ist_fm,B3: rule] :
( ( ( produc1733806532565653680m_rule @ A @ B2 )
= ( produc1733806532565653680m_rule @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_187_Pair__inject,axiom,
! [A: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_188_prod__cases,axiom,
! [P2: produc340336539035504054m_rule > $o,P: produc340336539035504054m_rule] :
( ! [A5: produc6018962875968178549ist_fm,B6: rule] : ( P2 @ ( produc1733806532565653680m_rule @ A5 @ B6 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_189_prod__cases,axiom,
! [P2: produc6018962875968178549ist_fm > $o,P: produc6018962875968178549ist_fm] :
( ! [A5: list_tm,B6: list_fm] : ( P2 @ ( produc1414352766439514085ist_fm @ A5 @ B6 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_190_surj__pair,axiom,
! [P: produc340336539035504054m_rule] :
? [X3: produc6018962875968178549ist_fm,Y2: rule] :
( P
= ( produc1733806532565653680m_rule @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_191_surj__pair,axiom,
! [P: produc6018962875968178549ist_fm] :
? [X3: list_tm,Y2: list_fm] :
( P
= ( produc1414352766439514085ist_fm @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_192_old_Oprod_Oexhaust,axiom,
! [Y: produc340336539035504054m_rule] :
~ ! [A5: produc6018962875968178549ist_fm,B6: rule] :
( Y
!= ( produc1733806532565653680m_rule @ A5 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_193_old_Oprod_Oexhaust,axiom,
! [Y: produc6018962875968178549ist_fm] :
~ ! [A5: list_tm,B6: list_fm] :
( Y
!= ( produc1414352766439514085ist_fm @ A5 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_194_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_195_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_196_ext_Osimps_I1_J,axiom,
! [Y: list_list_fm] : ( ext_list_fm @ Y @ nil_list_fm ) ).
% ext.simps(1)
thf(fact_197_Collect__mono__iff,axiom,
! [P2: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P2 ) @ ( collect_tm @ Q ) )
= ( ! [X2: tm] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_198_Collect__mono__iff,axiom,
! [P2: fm > $o,Q: fm > $o] :
( ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q ) )
= ( ! [X2: fm] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_199_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_200_set__eq__subset,axiom,
( ( ^ [Y5: set_tm,Z3: set_tm] : ( Y5 = Z3 ) )
= ( ^ [A6: set_tm,B7: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B7 )
& ( ord_less_eq_set_tm @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_201_set__eq__subset,axiom,
( ( ^ [Y5: set_fm,Z3: set_fm] : ( Y5 = Z3 ) )
= ( ^ [A6: set_fm,B7: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B7 )
& ( ord_less_eq_set_fm @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_202_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A6: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_203_subset__trans,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ C4 )
=> ( ord_less_eq_set_tm @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_204_subset__trans,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( ord_less_eq_set_fm @ B @ C4 )
=> ( ord_less_eq_set_fm @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_205_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_206_Collect__mono,axiom,
! [P2: tm > $o,Q: tm > $o] :
( ! [X3: tm] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P2 ) @ ( collect_tm @ Q ) ) ) ).
% Collect_mono
thf(fact_207_Collect__mono,axiom,
! [P2: fm > $o,Q: fm > $o] :
( ! [X3: fm] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_fm @ ( collect_fm @ P2 ) @ ( collect_fm @ Q ) ) ) ).
% Collect_mono
thf(fact_208_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_209_subset__refl,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% subset_refl
thf(fact_210_subset__refl,axiom,
! [A2: set_fm] : ( ord_less_eq_set_fm @ A2 @ A2 ) ).
% subset_refl
thf(fact_211_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_212_subset__iff,axiom,
( ord_le6390412330253371084m_rule
= ( ^ [A6: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
! [T: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ T @ A6 )
=> ( member7231649785386036813m_rule @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_213_subset__iff,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A6: set_list_fm,B7: set_list_fm] :
! [T: list_fm] :
( ( member_list_fm2 @ T @ A6 )
=> ( member_list_fm2 @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_214_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
! [T: tm] :
( ( member_tm2 @ T @ A6 )
=> ( member_tm2 @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_215_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
! [T: fm] :
( ( member_fm2 @ T @ A6 )
=> ( member_fm2 @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_216_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [T: nat] :
( ( member_nat2 @ T @ A6 )
=> ( member_nat2 @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_217_equalityD2,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ( ord_less_eq_set_tm @ B @ A2 ) ) ).
% equalityD2
thf(fact_218_equalityD2,axiom,
! [A2: set_fm,B: set_fm] :
( ( A2 = B )
=> ( ord_less_eq_set_fm @ B @ A2 ) ) ).
% equalityD2
thf(fact_219_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_220_equalityD1,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ( ord_less_eq_set_tm @ A2 @ B ) ) ).
% equalityD1
thf(fact_221_equalityD1,axiom,
! [A2: set_fm,B: set_fm] :
( ( A2 = B )
=> ( ord_less_eq_set_fm @ A2 @ B ) ) ).
% equalityD1
thf(fact_222_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_223_subset__eq,axiom,
( ord_le6390412330253371084m_rule
= ( ^ [A6: set_Pr1822751329126368876m_rule,B7: set_Pr1822751329126368876m_rule] :
! [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ A6 )
=> ( member7231649785386036813m_rule @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_224_subset__eq,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A6: set_list_fm,B7: set_list_fm] :
! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A6 )
=> ( member_list_fm2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_225_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
! [X2: tm] :
( ( member_tm2 @ X2 @ A6 )
=> ( member_tm2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_226_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
! [X2: fm] :
( ( member_fm2 @ X2 @ A6 )
=> ( member_fm2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_227_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A6 )
=> ( member_nat2 @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_228_equalityE,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_tm @ A2 @ B )
=> ~ ( ord_less_eq_set_tm @ B @ A2 ) ) ) ).
% equalityE
thf(fact_229_equalityE,axiom,
! [A2: set_fm,B: set_fm] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_fm @ A2 @ B )
=> ~ ( ord_less_eq_set_fm @ B @ A2 ) ) ) ).
% equalityE
thf(fact_230_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_231_subsetD,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule,C3: produc340336539035504054m_rule] :
( ( ord_le6390412330253371084m_rule @ A2 @ B )
=> ( ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% subsetD
thf(fact_232_subsetD,axiom,
! [A2: set_list_fm,B: set_list_fm,C3: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A2 @ B )
=> ( ( member_list_fm2 @ C3 @ A2 )
=> ( member_list_fm2 @ C3 @ B ) ) ) ).
% subsetD
thf(fact_233_subsetD,axiom,
! [A2: set_tm,B: set_tm,C3: tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( member_tm2 @ C3 @ A2 )
=> ( member_tm2 @ C3 @ B ) ) ) ).
% subsetD
thf(fact_234_subsetD,axiom,
! [A2: set_fm,B: set_fm,C3: fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( member_fm2 @ C3 @ A2 )
=> ( member_fm2 @ C3 @ B ) ) ) ).
% subsetD
thf(fact_235_subsetD,axiom,
! [A2: set_nat,B: set_nat,C3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ B ) ) ) ).
% subsetD
thf(fact_236_in__mono,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule,X: produc340336539035504054m_rule] :
( ( ord_le6390412330253371084m_rule @ A2 @ B )
=> ( ( member7231649785386036813m_rule @ X @ A2 )
=> ( member7231649785386036813m_rule @ X @ B ) ) ) ).
% in_mono
thf(fact_237_in__mono,axiom,
! [A2: set_list_fm,B: set_list_fm,X: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A2 @ B )
=> ( ( member_list_fm2 @ X @ A2 )
=> ( member_list_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_238_in__mono,axiom,
! [A2: set_tm,B: set_tm,X: tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( member_tm2 @ X @ A2 )
=> ( member_tm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_239_in__mono,axiom,
! [A2: set_fm,B: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( member_fm2 @ X @ A2 )
=> ( member_fm2 @ X @ B ) ) ) ).
% in_mono
thf(fact_240_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ X @ B ) ) ) ).
% in_mono
thf(fact_241_sdrop_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_242_stream_Ocoinduct__strong,axiom,
! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
( ( R3 @ Stream @ Stream2 )
=> ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
( ( R3 @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
= ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
& ( ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) )
| ( ( stl_Pr12655793849929990m_rule @ Stream3 )
= ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_243_stream_Ocoinduct,axiom,
! [R3: stream2709947120125613254m_rule > stream2709947120125613254m_rule > $o,Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
( ( R3 @ Stream @ Stream2 )
=> ( ! [Stream3: stream2709947120125613254m_rule,Stream4: stream2709947120125613254m_rule] :
( ( R3 @ Stream3 @ Stream4 )
=> ( ( ( shd_Pr4562317740776619530m_rule @ Stream3 )
= ( shd_Pr4562317740776619530m_rule @ Stream4 ) )
& ( R3 @ ( stl_Pr12655793849929990m_rule @ Stream3 ) @ ( stl_Pr12655793849929990m_rule @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_244_stream_Oexpand,axiom,
! [Stream: stream2709947120125613254m_rule,Stream2: stream2709947120125613254m_rule] :
( ( ( ( shd_Pr4562317740776619530m_rule @ Stream )
= ( shd_Pr4562317740776619530m_rule @ Stream2 ) )
& ( ( stl_Pr12655793849929990m_rule @ Stream )
= ( stl_Pr12655793849929990m_rule @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_245_shd__sset,axiom,
! [A: stream_fm] : ( member_fm2 @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).
% shd_sset
thf(fact_246_shd__sset,axiom,
! [A: stream_list_fm] : ( member_list_fm2 @ ( shd_list_fm @ A ) @ ( sset_list_fm @ A ) ) ).
% shd_sset
thf(fact_247_shd__sset,axiom,
! [A: stream_tm] : ( member_tm2 @ ( shd_tm @ A ) @ ( sset_tm @ A ) ) ).
% shd_sset
thf(fact_248_shd__sset,axiom,
! [A: stream_nat] : ( member_nat2 @ ( shd_nat @ A ) @ ( sset_nat @ A ) ) ).
% shd_sset
thf(fact_249_shd__sset,axiom,
! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).
% shd_sset
thf(fact_250_stl__sset,axiom,
! [X: fm,A: stream_fm] :
( ( member_fm2 @ X @ ( sset_fm @ ( stl_fm @ A ) ) )
=> ( member_fm2 @ X @ ( sset_fm @ A ) ) ) ).
% stl_sset
thf(fact_251_stl__sset,axiom,
! [X: list_fm,A: stream_list_fm] :
( ( member_list_fm2 @ X @ ( sset_list_fm @ ( stl_list_fm @ A ) ) )
=> ( member_list_fm2 @ X @ ( sset_list_fm @ A ) ) ) ).
% stl_sset
thf(fact_252_stl__sset,axiom,
! [X: tm,A: stream_tm] :
( ( member_tm2 @ X @ ( sset_tm @ ( stl_tm @ A ) ) )
=> ( member_tm2 @ X @ ( sset_tm @ A ) ) ) ).
% stl_sset
thf(fact_253_stl__sset,axiom,
! [X: nat,A: stream_nat] :
( ( member_nat2 @ X @ ( sset_nat @ ( stl_nat @ A ) ) )
=> ( member_nat2 @ X @ ( sset_nat @ A ) ) ) ).
% stl_sset
thf(fact_254_stl__sset,axiom,
! [X: produc340336539035504054m_rule,A: stream2709947120125613254m_rule] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ ( stl_Pr12655793849929990m_rule @ A ) ) )
=> ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ A ) ) ) ).
% stl_sset
thf(fact_255_parts__in__effect,axiom,
! [P: fm,Z4: list_fm,B: list_tm,Z5: list_fm,R2: rule,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z4 ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z5 ) @ ( effect @ R2 @ ( produc1414352766439514085ist_fm @ A2 @ Z4 ) ) )
=> ? [C: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ C ) )
& ( member_list_fm2 @ Xs2 @ ( set_list_fm2 @ ( parts @ C @ R2 @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% parts_in_effect
thf(fact_256_fsubsetI,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ X3 @ B ) )
=> ( ord_le3986950534092794747ist_fm @ A2 @ B ) ) ).
% fsubsetI
thf(fact_257_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_258_effect__preserves__unaffected,axiom,
! [P: fm,Z4: list_fm,R2: rule,B: list_tm,Z5: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z4 ) )
=> ( ~ ( affects @ R2 @ P )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z5 ) @ ( effect @ R2 @ ( produc1414352766439514085ist_fm @ A2 @ Z4 ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_259_sdrop__szip,axiom,
! [N2: nat,S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
( ( sdrop_7192298464603511222m_rule @ N2 @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
= ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S22 ) ) ) ).
% sdrop_szip
thf(fact_260_sdrop__szip,axiom,
! [N2: nat,S1: stream4408948924543953275ist_fm,S22: stream_rule] :
( ( sdrop_8169176516188972301m_rule @ N2 @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
= ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S22 ) ) ) ).
% sdrop_szip
thf(fact_261_sset__cycle,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( ( sset_fm @ ( cycle_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_262_sset__cycle,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( sset_tm @ ( cycle_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_263_sset__cycle,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( sset_list_fm @ ( cycle_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_264_sset__cycle,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( sset_P4484857331586881186m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% sset_cycle
thf(fact_265_szip_Osimps_I1_J,axiom,
! [S1: stream_list_tm,S22: stream_list_fm] :
( ( shd_Pr3211216682057661985ist_fm @ ( szip_list_tm_list_fm @ S1 @ S22 ) )
= ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S1 ) @ ( shd_list_fm @ S22 ) ) ) ).
% szip.simps(1)
thf(fact_266_szip_Osimps_I1_J,axiom,
! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
= ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S22 ) ) ) ).
% szip.simps(1)
thf(fact_267_szip_Osimps_I1_J,axiom,
! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
= ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S22 ) ) ) ).
% szip.simps(1)
thf(fact_268_subset__code_I3_J,axiom,
~ ( ord_le7838213414353715577ist_fm @ ( coset_list_fm @ nil_list_fm ) @ ( set_list_fm2 @ nil_list_fm ) ) ).
% subset_code(3)
thf(fact_269_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_tm @ ( coset_tm @ nil_tm ) @ ( set_tm2 @ nil_tm ) ) ).
% subset_code(3)
thf(fact_270_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_271_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_272_member,axiom,
( member4200870186495857963m_rule
= ( ^ [P3: produc340336539035504054m_rule,Z: list_P2774625669004309958m_rule] : ( member7231649785386036813m_rule @ P3 @ ( set_Pr4534715572506550497m_rule @ Z ) ) ) ) ).
% member
thf(fact_273_member,axiom,
( member_nat
= ( ^ [P3: nat,Z: list_nat] : ( member_nat2 @ P3 @ ( set_nat2 @ Z ) ) ) ) ).
% member
thf(fact_274_member,axiom,
( member_fm
= ( ^ [P3: fm,Z: list_fm] : ( member_fm2 @ P3 @ ( set_fm2 @ Z ) ) ) ) ).
% member
thf(fact_275_member,axiom,
( member_tm
= ( ^ [P3: tm,Z: list_tm] : ( member_tm2 @ P3 @ ( set_tm2 @ Z ) ) ) ) ).
% member
thf(fact_276_member,axiom,
( member_list_fm
= ( ^ [P3: list_fm,Z: list_list_fm] : ( member_list_fm2 @ P3 @ ( set_list_fm2 @ Z ) ) ) ) ).
% member
thf(fact_277_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_278_parts__preserves__unaffected,axiom,
! [R2: rule,P: fm,Z5: list_fm,A2: list_tm] :
( ~ ( affects @ R2 @ P )
=> ( ( member_list_fm2 @ Z5 @ ( set_list_fm2 @ ( parts @ A2 @ R2 @ P ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ).
% parts_preserves_unaffected
thf(fact_279_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_280_size__neq__size__imp__neq,axiom,
! [X: fm,Y: fm] :
( ( ( size_size_fm @ X )
!= ( size_size_fm @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_281_size__neq__size__imp__neq,axiom,
! [X: rule,Y: rule] :
( ( ( size_size_rule @ X )
!= ( size_size_rule @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_282_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_283_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_284_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: list_fm] :
~ ( member_list_fm @ P @ nil_list_fm ) ).
% SeCaV.member.simps(1)
thf(fact_285_cycle__decomp,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( cycle_5335117900641983313m_rule @ U )
= ( shift_2334844276748245581m_rule @ U @ ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% cycle_decomp
thf(fact_286_cycle__decomp,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( cycle_tm @ U )
= ( shift_tm @ U @ ( cycle_tm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_287_cycle__decomp,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( cycle_fm @ U )
= ( shift_fm @ U @ ( cycle_fm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_288_cycle__decomp,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( cycle_list_fm @ U )
= ( shift_list_fm @ U @ ( cycle_list_fm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_289_subset__code_I2_J,axiom,
! [A2: set_Pr1822751329126368876m_rule,Ys: list_P2774625669004309958m_rule] :
( ( ord_le6390412330253371084m_rule @ A2 @ ( coset_4722139400431689347m_rule @ Ys ) )
= ( ! [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ ( set_Pr4534715572506550497m_rule @ Ys ) )
=> ~ ( member7231649785386036813m_rule @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_290_subset__code_I2_J,axiom,
! [A2: set_list_fm,Ys: list_list_fm] :
( ( ord_le7838213414353715577ist_fm @ A2 @ ( coset_list_fm @ Ys ) )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Ys ) )
=> ~ ( member_list_fm2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_291_subset__code_I2_J,axiom,
! [A2: set_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ A2 @ ( coset_tm @ Ys ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Ys ) )
=> ~ ( member_tm2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_292_subset__code_I2_J,axiom,
! [A2: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A2 @ ( coset_fm @ Ys ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_293_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_294_fset__eqI,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X3 @ A2 )
= ( fmembe3754813877001230652ist_fm @ X3 @ B ) )
=> ( A2 = B ) ) ).
% fset_eqI
thf(fact_295_fequalityCE,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm,C3: produc6018962875968178549ist_fm] :
( ( A2 = B )
=> ( ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ~ ( fmembe3754813877001230652ist_fm @ C3 @ B ) )
=> ~ ( ~ ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ) ).
% fequalityCE
thf(fact_296_eq__fmem__trans,axiom,
! [A: produc6018962875968178549ist_fm,B2: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm] :
( ( A = B2 )
=> ( ( fmembe3754813877001230652ist_fm @ B2 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ A @ A2 ) ) ) ).
% eq_fmem_trans
thf(fact_297_eqfset__imp__iff,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm,X: produc6018962875968178549ist_fm] :
( ( A2 = B )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A2 )
= ( fmembe3754813877001230652ist_fm @ X @ B ) ) ) ).
% eqfset_imp_iff
thf(fact_298_if__split__fmem1,axiom,
! [Q: $o,X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,B2: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( if_Pro4760001780252510779ist_fm @ Q @ X @ Y ) @ B2 )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ X @ B2 ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_299_if__split__fmem2,axiom,
! [A: produc6018962875968178549ist_fm,Q: $o,X: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ A @ ( if_fse7999432387889793441ist_fm @ Q @ X @ Y ) )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ A @ X ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ A @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_300_eqfelem__imp__iff,axiom,
! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm] :
( ( X = Y )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A2 )
= ( fmembe3754813877001230652ist_fm @ Y @ A2 ) ) ) ).
% eqfelem_imp_iff
thf(fact_301_fsubsetD,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm,C3: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A2 @ B )
=> ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% fsubsetD
thf(fact_302_fin__mono,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm,X: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A2 @ B )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A2 )
=> ( fmembe3754813877001230652ist_fm @ X @ B ) ) ) ).
% fin_mono
thf(fact_303_szip_Osimps_I2_J,axiom,
! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule] :
( ( stl_Pr2679468901532635773m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) )
= ( szip_P811719526838699976m_rule @ ( stl_Pr12655793849929990m_rule @ S1 ) @ ( stl_Pr12655793849929990m_rule @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_304_szip_Osimps_I2_J,axiom,
! [S1: stream4408948924543953275ist_fm,S22: stream_rule] :
( ( stl_Pr12655793849929990m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) )
= ( szip_P2924820683901490861m_rule @ ( stl_Pr1506262294867171877ist_fm @ S1 ) @ ( stl_rule @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_305_parts__in__children,axiom,
! [P: fm,Z4: list_fm,Z5: list_fm,A2: list_tm,R2: rule] :
( ( member_fm2 @ P @ ( set_fm2 @ Z4 ) )
=> ( ( member_list_fm2 @ Z5 @ ( set_list_fm2 @ ( children @ A2 @ R2 @ Z4 ) ) )
=> ? [B4: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B4 ) )
& ( member_list_fm2 @ Xs2 @ ( set_list_fm2 @ ( parts @ B4 @ R2 @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% parts_in_children
thf(fact_306_children__preserves__unaffected,axiom,
! [P: fm,Z4: list_fm,R2: rule,Z5: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z4 ) )
=> ( ~ ( affects @ R2 @ P )
=> ( ( member_list_fm2 @ Z5 @ ( set_list_fm2 @ ( children @ A2 @ R2 @ Z4 ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% children_preserves_unaffected
thf(fact_307_subterm__Fun__refl,axiom,
! [Ts: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N2 @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_308_sub__const__transfer,axiom,
! [M: nat,A: nat,P: fm,T2: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P )
!= ( sub @ M @ T2 @ P ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P ) ) ) ) ) ).
% sub_const_transfer
thf(fact_309_fset__of__list__subset,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) )
=> ( ord_le3182162295460067583ist_fm @ ( fset_of_list_list_fm @ Xs ) @ ( fset_of_list_list_fm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_310_fset__of__list__subset,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) )
=> ( ord_less_eq_fset_tm @ ( fset_of_list_tm @ Xs ) @ ( fset_of_list_tm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_311_fset__of__list__subset,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
=> ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_312_fset__of__list__subset,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs ) @ ( fset_of_list_nat @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_313_fun__arguments__subterm,axiom,
! [N2: nat,Ts: list_tm,P: fm] :
( ( member_tm2 @ ( fun @ N2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% fun_arguments_subterm
thf(fact_314_tree__fms__in__pseq,axiom,
! [P: fm,Steps: stream2709947120125613254m_rule] :
( ( member_fm2 @ P @ ( tree_fms @ Steps ) )
=> ? [N: nat] : ( member_fm2 @ P @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N ) ) ) ) ) ).
% tree_fms_in_pseq
thf(fact_315_list__ex1__simps_I1_J,axiom,
! [P2: tm > $o] :
~ ( list_ex1_tm @ P2 @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_316_list__ex1__simps_I1_J,axiom,
! [P2: fm > $o] :
~ ( list_ex1_fm @ P2 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_317_list__ex1__simps_I1_J,axiom,
! [P2: list_fm > $o] :
~ ( list_ex1_list_fm @ P2 @ nil_list_fm ) ).
% list_ex1_simps(1)
thf(fact_318_size__sub,axiom,
! [I: nat,T2: tm,P: fm] :
( ( size_size_fm @ ( sub @ I @ T2 @ P ) )
= ( size_size_fm @ P ) ) ).
% size_sub
thf(fact_319_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_320_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_321_length__0__conv,axiom,
! [Xs: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_fm ) ) ).
% length_0_conv
thf(fact_322_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
= ( snth_P6679518042731451922m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_323_sdrop__cycle__eq,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ ( size_s1575636608424004698m_rule @ U ) @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_324_sdrop__cycle__eq,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( sdrop_tm @ ( size_size_list_tm @ U ) @ ( cycle_tm @ U ) )
= ( cycle_tm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_325_sdrop__cycle__eq,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( sdrop_fm @ ( size_size_list_fm @ U ) @ ( cycle_fm @ U ) )
= ( cycle_fm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_326_sdrop__cycle__eq,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( sdrop_list_fm @ ( size_s115229985653309035ist_fm @ U ) @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_327_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S22: stream2709947120125613254m_rule,N2: nat] :
( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S22 ) @ N2 )
= ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N2 ) @ ( snth_P6679518042731451922m_rule @ S22 @ N2 ) ) ) ).
% snth_szip
thf(fact_328_snth__szip,axiom,
! [S1: stream4408948924543953275ist_fm,S22: stream_rule,N2: nat] :
( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S22 ) @ N2 )
= ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N2 ) @ ( snth_rule @ S22 @ N2 ) ) ) ).
% snth_szip
thf(fact_329_snth__szip,axiom,
! [S1: stream_list_tm,S22: stream_list_fm,N2: nat] :
( ( snth_P7093566783922538521ist_fm @ ( szip_list_tm_list_fm @ S1 @ S22 ) @ N2 )
= ( produc1414352766439514085ist_fm @ ( snth_list_tm @ S1 @ N2 ) @ ( snth_list_fm @ S22 @ N2 ) ) ) ).
% snth_szip
thf(fact_330_snth__sset,axiom,
! [S: stream_fm,N2: nat] : ( member_fm2 @ ( snth_fm @ S @ N2 ) @ ( sset_fm @ S ) ) ).
% snth_sset
thf(fact_331_snth__sset,axiom,
! [S: stream_list_fm,N2: nat] : ( member_list_fm2 @ ( snth_list_fm @ S @ N2 ) @ ( sset_list_fm @ S ) ) ).
% snth_sset
thf(fact_332_snth__sset,axiom,
! [S: stream_tm,N2: nat] : ( member_tm2 @ ( snth_tm @ S @ N2 ) @ ( sset_tm @ S ) ) ).
% snth_sset
thf(fact_333_snth__sset,axiom,
! [S: stream_nat,N2: nat] : ( member_nat2 @ ( snth_nat @ S @ N2 ) @ ( sset_nat @ S ) ) ).
% snth_sset
thf(fact_334_snth__sset,axiom,
! [S: stream2709947120125613254m_rule,N2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N2 ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% snth_sset
thf(fact_335_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_336_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_337_list_Osize_I3_J,axiom,
( ( size_s115229985653309035ist_fm @ nil_list_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_338_subtermTm__refl,axiom,
! [T2: tm] : ( member_tm2 @ T2 @ ( set_tm2 @ ( subtermTm @ T2 ) ) ) ).
% subtermTm_refl
thf(fact_339_snth_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( snth_P6679518042731451922m_rule @ S @ zero_zero_nat )
= ( shd_Pr4562317740776619530m_rule @ S ) ) ).
% snth.simps(1)
thf(fact_340_list__ex1__iff,axiom,
( list_e4959145360368724013m_rule
= ( ^ [P4: produc340336539035504054m_rule > $o,Xs3: list_P2774625669004309958m_rule] :
? [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ ( set_Pr4534715572506550497m_rule @ Xs3 ) )
& ( P4 @ X2 )
& ! [Y3: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y3 @ ( set_Pr4534715572506550497m_rule @ Xs3 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_341_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P4: nat > $o,Xs3: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs3 ) )
& ( P4 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs3 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_342_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P4: fm > $o,Xs3: list_fm] :
? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs3 ) )
& ( P4 @ X2 )
& ! [Y3: fm] :
( ( ( member_fm2 @ Y3 @ ( set_fm2 @ Xs3 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_343_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P4: tm > $o,Xs3: list_tm] :
? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs3 ) )
& ( P4 @ X2 )
& ! [Y3: tm] :
( ( ( member_tm2 @ Y3 @ ( set_tm2 @ Xs3 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_344_list__ex1__iff,axiom,
( list_ex1_list_fm
= ( ^ [P4: list_fm > $o,Xs3: list_list_fm] :
? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs3 ) )
& ( P4 @ X2 )
& ! [Y3: list_fm] :
( ( ( member_list_fm2 @ Y3 @ ( set_list_fm2 @ Xs3 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_345_fset__of__list__elem,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( fmembe1089128867255857199m_rule @ X @ ( fset_o6059354669909737328m_rule @ Xs ) )
= ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_346_fset__of__list__elem,axiom,
! [X: nat,Xs: list_nat] :
( ( fmember_nat @ X @ ( fset_of_list_nat @ Xs ) )
= ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_347_fset__of__list__elem,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X @ ( fset_o3222440871817023035ist_fm @ Xs ) )
= ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_348_fset__of__list__elem,axiom,
! [X: fm,Xs: list_fm] :
( ( fmember_fm @ X @ ( fset_of_list_fm @ Xs ) )
= ( member_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_349_fset__of__list__elem,axiom,
! [X: tm,Xs: list_tm] :
( ( fmember_tm @ X @ ( fset_of_list_tm @ Xs ) )
= ( member_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_350_fset__of__list__elem,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( fmember_list_fm @ X @ ( fset_of_list_list_fm @ Xs ) )
= ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_351_subtermTm__le,axiom,
! [T2: tm,S: tm] :
( ( member_tm2 @ T2 @ ( set_tm2 @ ( subtermTm @ S ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T2 ) ) @ ( set_tm2 @ ( subtermTm @ S ) ) ) ) ).
% subtermTm_le
thf(fact_352_can__select__set__list__ex1,axiom,
! [P2: fm > $o,A2: list_fm] :
( ( can_select_fm @ P2 @ ( set_fm2 @ A2 ) )
= ( list_ex1_fm @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_353_can__select__set__list__ex1,axiom,
! [P2: tm > $o,A2: list_tm] :
( ( can_select_tm @ P2 @ ( set_tm2 @ A2 ) )
= ( list_ex1_tm @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_354_can__select__set__list__ex1,axiom,
! [P2: list_fm > $o,A2: list_list_fm] :
( ( can_select_list_fm @ P2 @ ( set_list_fm2 @ A2 ) )
= ( list_ex1_list_fm @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_355_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T2: tm,S: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T2 )
!= ( sub_term @ M @ S @ T2 ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T2 ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_356_subtermFm_Osimps_I6_J,axiom,
! [P: fm] :
( ( subtermFm @ ( uni @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(6)
thf(fact_357_rule_Osize_I18_J,axiom,
( ( size_size_rule @ deltaUni )
= zero_zero_nat ) ).
% rule.size(18)
thf(fact_358_snth__sset__smerge,axiom,
! [Ss: stream_stream_fm,N2: nat,M: nat] : ( member_fm2 @ ( snth_fm @ ( snth_stream_fm @ Ss @ N2 ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_359_snth__sset__smerge,axiom,
! [Ss: stream6017534608192929797ist_fm,N2: nat,M: nat] : ( member_list_fm2 @ ( snth_list_fm @ ( snth_stream_list_fm @ Ss @ N2 ) @ M ) @ ( sset_list_fm @ ( smerge_list_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_360_snth__sset__smerge,axiom,
! [Ss: stream_stream_tm,N2: nat,M: nat] : ( member_tm2 @ ( snth_tm @ ( snth_stream_tm @ Ss @ N2 ) @ M ) @ ( sset_tm @ ( smerge_tm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_361_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N2: nat,M: nat] : ( member_nat2 @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_362_snth__sset__smerge,axiom,
! [Ss: stream3752074346242807894m_rule,N2: nat,M: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N2 ) @ M ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_363_paramst__subtermTm_I1_J,axiom,
! [T2: tm,X4: nat] :
( ( member_nat2 @ X4 @ ( paramst @ T2 ) )
=> ? [L: list_tm] : ( member_tm2 @ ( fun @ X4 @ L ) @ ( set_tm2 @ ( subtermTm @ T2 ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_364_sdrop__cycle__eq__mod__0,axiom,
! [U: list_P2774625669004309958m_rule,N2: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ U ) )
= zero_zero_nat )
=> ( ( sdrop_8169176516188972301m_rule @ N2 @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_365_sdrop__cycle__eq__mod__0,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_tm @ N2 @ ( cycle_tm @ U ) )
= ( cycle_tm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_366_sdrop__cycle__eq__mod__0,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_fm @ N2 @ ( cycle_fm @ U ) )
= ( cycle_fm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_367_sdrop__cycle__eq__mod__0,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_368_params__subtermFm,axiom,
! [P: fm,X4: nat] :
( ( member_nat2 @ X4 @ ( params @ P ) )
=> ? [L: list_tm] : ( member_tm2 @ ( fun @ X4 @ L ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% params_subtermFm
thf(fact_369_can__select__def,axiom,
( can_select_fm
= ( ^ [P4: fm > $o,A6: set_fm] :
? [X2: fm] :
( ( member_fm2 @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y3: fm] :
( ( ( member_fm2 @ Y3 @ A6 )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_370_can__select__def,axiom,
( can_se4506553828710032889m_rule
= ( ^ [P4: produc340336539035504054m_rule > $o,A6: set_Pr1822751329126368876m_rule] :
? [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y3: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y3 @ A6 )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_371_can__select__def,axiom,
( can_select_list_fm
= ( ^ [P4: list_fm > $o,A6: set_list_fm] :
? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y3: list_fm] :
( ( ( member_list_fm2 @ Y3 @ A6 )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_372_can__select__def,axiom,
( can_select_tm
= ( ^ [P4: tm > $o,A6: set_tm] :
? [X2: tm] :
( ( member_tm2 @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y3: tm] :
( ( ( member_tm2 @ Y3 @ A6 )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_373_can__select__def,axiom,
( can_select_nat
= ( ^ [P4: nat > $o,A6: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A6 )
& ( P4 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_374_params_Osimps_I6_J,axiom,
! [P: fm] :
( ( params @ ( uni @ P ) )
= ( params @ P ) ) ).
% params.simps(6)
thf(fact_375_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_376_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_377_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_378_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_379_mod__mod__trivial,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mod_trivial
thf(fact_380_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_381_s1_I1_J,axiom,
( new_term
= ( ^ [C2: nat,T: tm] :
~ ( member_nat2 @ C2 @ ( paramst @ T ) ) ) ) ).
% s1(1)
thf(fact_382_mod__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% mod_less_eq_dividend
thf(fact_383_paramsts__subset,axiom,
! [A2: list_tm,B: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A2 ) @ ( paramsts @ B ) ) ) ).
% paramsts_subset
thf(fact_384_s5_I1_J,axiom,
( sub_term
= ( ^ [V: nat,S3: tm,T: tm] : ( substt @ T @ S3 @ V ) ) ) ).
% s5(1)
thf(fact_385_sdrop__cycle,axiom,
! [U: list_P2774625669004309958m_rule,N2: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ N2 @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ ( rotate8779165051853931260m_rule @ ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_386_sdrop__cycle,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( sdrop_tm @ N2 @ ( cycle_tm @ U ) )
= ( cycle_tm @ ( rotate_tm @ ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_387_sdrop__cycle,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( sdrop_fm @ N2 @ ( cycle_fm @ U ) )
= ( cycle_fm @ ( rotate_fm @ ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_388_sdrop__cycle,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( sdrop_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ ( rotate_list_fm @ ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_389_new__term_Osimps_I2_J,axiom,
! [C3: nat,I: nat,L2: list_tm] :
( ( new_term @ C3 @ ( fun @ I @ L2 ) )
= ( ( I != C3 )
& ( ( I != C3 )
=> ( new_list @ C3 @ L2 ) ) ) ) ).
% new_term.simps(2)
thf(fact_390_pos__least,axiom,
! [N2: nat,Rs: stream2709947120125613254m_rule,R2: produc340336539035504054m_rule] :
( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ Rs ) )
= R2 )
=> ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R2 ) @ N2 ) ) ).
% pos_least
thf(fact_391_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_392_shift__snth__ge,axiom,
! [Xs: list_P2774625669004309958m_rule,P: nat,S: stream2709947120125613254m_rule] :
( ( ord_less_eq_nat @ ( size_s1575636608424004698m_rule @ Xs ) @ P )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ P )
= ( snth_P6679518042731451922m_rule @ S @ ( minus_minus_nat @ P @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_394_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_395_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_396_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_397_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_398_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_399_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_tm] :
( ( ( rotate_tm @ N2 @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% rotate_is_Nil_conv
thf(fact_400_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_fm] :
( ( ( rotate_fm @ N2 @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% rotate_is_Nil_conv
thf(fact_401_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ( rotate_list_fm @ N2 @ Xs )
= nil_list_fm )
= ( Xs = nil_list_fm ) ) ).
% rotate_is_Nil_conv
thf(fact_402_set__rotate,axiom,
! [N2: nat,Xs: list_fm] :
( ( set_fm2 @ ( rotate_fm @ N2 @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_403_set__rotate,axiom,
! [N2: nat,Xs: list_tm] :
( ( set_tm2 @ ( rotate_tm @ N2 @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate
thf(fact_404_set__rotate,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( set_list_fm2 @ ( rotate_list_fm @ N2 @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_405_s1_I2_J,axiom,
( new_list
= ( ^ [C2: nat,L3: list_tm] :
~ ( member_nat2 @ C2 @ ( paramsts @ L3 ) ) ) ) ).
% s1(2)
thf(fact_406_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_407_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_408_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_409_diff__right__commute,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C3 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C3 ) ) ).
% diff_right_commute
thf(fact_410_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_411_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_412_diff__le__mono2,axiom,
! [M: nat,N2: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_413_le__diff__iff_H,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C3 )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C3 @ A ) @ ( minus_minus_nat @ C3 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_414_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_415_diff__le__mono,axiom,
! [M: nat,N2: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_416_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_417_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_418_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_419_le__mod__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( modulo_modulo_nat @ M @ N2 )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_420_new__list_Osimps_I1_J,axiom,
! [C3: nat] : ( new_list @ C3 @ nil_tm ) ).
% new_list.simps(1)
thf(fact_421_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_422_cycle__nth,axiom,
! [Xs: list_P2774625669004309958m_rule,N2: nat] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( snth_P6679518042731451922m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) @ N2 )
= ( nth_Pr3936752564452695919m_rule @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_423_cycle__nth,axiom,
! [Xs: list_tm,N2: nat] :
( ( Xs != nil_tm )
=> ( ( snth_tm @ ( cycle_tm @ Xs ) @ N2 )
= ( nth_tm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_424_cycle__nth,axiom,
! [Xs: list_fm,N2: nat] :
( ( Xs != nil_fm )
=> ( ( snth_fm @ ( cycle_fm @ Xs ) @ N2 )
= ( nth_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_425_cycle__nth,axiom,
! [Xs: list_list_fm,N2: nat] :
( ( Xs != nil_list_fm )
=> ( ( snth_list_fm @ ( cycle_list_fm @ Xs ) @ N2 )
= ( nth_list_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_426_params__sub,axiom,
! [M: nat,T2: tm,P: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T2 @ P ) ) @ ( sup_sup_set_nat @ ( paramst @ T2 ) @ ( params @ P ) ) ) ).
% params_sub
thf(fact_427_sub__term_Osimps_I2_J,axiom,
! [V2: nat,S: tm,I: nat,L2: list_tm] :
( ( sub_term @ V2 @ S @ ( fun @ I @ L2 ) )
= ( fun @ I @ ( sub_list @ V2 @ S @ L2 ) ) ) ).
% sub_term.simps(2)
thf(fact_428_stake__cycle__eq,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( stake_tm @ ( size_size_list_tm @ U ) @ ( cycle_tm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_429_stake__cycle__eq,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( stake_fm @ ( size_size_list_fm @ U ) @ ( cycle_fm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_430_stake__cycle__eq,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( stake_list_fm @ ( size_s115229985653309035ist_fm @ U ) @ ( cycle_list_fm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_431_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_432_length__n__lists__elem,axiom,
! [Ys: list_fm,N2: nat,Xs: list_fm] :
( ( member_list_fm2 @ Ys @ ( set_list_fm2 @ ( n_lists_fm @ N2 @ Xs ) ) )
=> ( ( size_size_list_fm @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_433_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_434_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_435_UnCI,axiom,
! [C3: fm,B: set_fm,A2: set_fm] :
( ( ~ ( member_fm2 @ C3 @ B )
=> ( member_fm2 @ C3 @ A2 ) )
=> ( member_fm2 @ C3 @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_436_UnCI,axiom,
! [C3: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( ~ ( member7231649785386036813m_rule @ C3 @ B )
=> ( member7231649785386036813m_rule @ C3 @ A2 ) )
=> ( member7231649785386036813m_rule @ C3 @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnCI
thf(fact_437_UnCI,axiom,
! [C3: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( ~ ( member_list_fm2 @ C3 @ B )
=> ( member_list_fm2 @ C3 @ A2 ) )
=> ( member_list_fm2 @ C3 @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_438_UnCI,axiom,
! [C3: tm,B: set_tm,A2: set_tm] :
( ( ~ ( member_tm2 @ C3 @ B )
=> ( member_tm2 @ C3 @ A2 ) )
=> ( member_tm2 @ C3 @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_439_UnCI,axiom,
! [C3: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat2 @ C3 @ B )
=> ( member_nat2 @ C3 @ A2 ) )
=> ( member_nat2 @ C3 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_440_Un__iff,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( sup_sup_set_fm @ A2 @ B ) )
= ( ( member_fm2 @ C3 @ A2 )
| ( member_fm2 @ C3 @ B ) ) ) ).
% Un_iff
thf(fact_441_Un__iff,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( sup_su6946459741510085528m_rule @ A2 @ B ) )
= ( ( member7231649785386036813m_rule @ C3 @ A2 )
| ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% Un_iff
thf(fact_442_Un__iff,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( sup_sup_set_list_fm @ A2 @ B ) )
= ( ( member_list_fm2 @ C3 @ A2 )
| ( member_list_fm2 @ C3 @ B ) ) ) ).
% Un_iff
thf(fact_443_Un__iff,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( sup_sup_set_tm @ A2 @ B ) )
= ( ( member_tm2 @ C3 @ A2 )
| ( member_tm2 @ C3 @ B ) ) ) ).
% Un_iff
thf(fact_444_Un__iff,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat2 @ C3 @ A2 )
| ( member_nat2 @ C3 @ B ) ) ) ).
% Un_iff
thf(fact_445_fminus__iff,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( minus_7547787838945083330ist_fm @ A2 @ B ) )
= ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
& ~ ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% fminus_iff
thf(fact_446_fminusI,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( ~ ( fmembe3754813877001230652ist_fm @ C3 @ B )
=> ( fmembe3754813877001230652ist_fm @ C3 @ ( minus_7547787838945083330ist_fm @ A2 @ B ) ) ) ) ).
% fminusI
thf(fact_447_Un__subset__iff,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ C4 )
= ( ( ord_less_eq_set_tm @ A2 @ C4 )
& ( ord_less_eq_set_tm @ B @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_448_Un__subset__iff,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B ) @ C4 )
= ( ( ord_less_eq_set_fm @ A2 @ C4 )
& ( ord_less_eq_set_fm @ B @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_449_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C4 )
= ( ( ord_less_eq_set_nat @ A2 @ C4 )
& ( ord_less_eq_set_nat @ B @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_450_s5_I2_J,axiom,
( sub_list
= ( ^ [V: nat,S3: tm,L3: list_tm] : ( substts @ L3 @ S3 @ V ) ) ) ).
% s5(2)
thf(fact_451_stake__invert__Nil,axiom,
! [N2: nat,S: stream_tm] :
( ( ( stake_tm @ N2 @ S )
= nil_tm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_452_stake__invert__Nil,axiom,
! [N2: nat,S: stream_fm] :
( ( ( stake_fm @ N2 @ S )
= nil_fm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_453_stake__invert__Nil,axiom,
! [N2: nat,S: stream_list_fm] :
( ( ( stake_list_fm @ N2 @ S )
= nil_list_fm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_454_sset__shift,axiom,
! [Xs: list_fm,S: stream_fm] :
( ( sset_fm @ ( shift_fm @ Xs @ S ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( sset_fm @ S ) ) ) ).
% sset_shift
thf(fact_455_sset__shift,axiom,
! [Xs: list_tm,S: stream_tm] :
( ( sset_tm @ ( shift_tm @ Xs @ S ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( sset_tm @ S ) ) ) ).
% sset_shift
thf(fact_456_sset__shift,axiom,
! [Xs: list_list_fm,S: stream_list_fm] :
( ( sset_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( sset_list_fm @ S ) ) ) ).
% sset_shift
thf(fact_457_sset__shift,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( sset_P4484857331586881186m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( sup_su6946459741510085528m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ ( sset_P4484857331586881186m_rule @ S ) ) ) ).
% sset_shift
thf(fact_458_sset__shift,axiom,
! [Xs: list_nat,S: stream_nat] :
( ( sset_nat @ ( shift_nat @ Xs @ S ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( sset_nat @ S ) ) ) ).
% sset_shift
thf(fact_459_Diff__subset__conv,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ B ) @ C4 )
= ( ord_less_eq_set_tm @ A2 @ ( sup_sup_set_tm @ B @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_460_Diff__subset__conv,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ B ) @ C4 )
= ( ord_less_eq_set_fm @ A2 @ ( sup_sup_set_fm @ B @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_461_Diff__subset__conv,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C4 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_462_Diff__partition,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( sup_sup_set_tm @ A2 @ ( minus_minus_set_tm @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_463_Diff__partition,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( sup_sup_set_fm @ A2 @ ( minus_minus_set_fm @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_464_Diff__partition,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_465_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C4 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C4 ) @ ( minus_minus_set_nat @ B @ C4 ) ) ) ).
% Un_Diff
thf(fact_466_UnE,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( sup_sup_set_fm @ A2 @ B ) )
=> ( ~ ( member_fm2 @ C3 @ A2 )
=> ( member_fm2 @ C3 @ B ) ) ) ).
% UnE
thf(fact_467_UnE,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( sup_su6946459741510085528m_rule @ A2 @ B ) )
=> ( ~ ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% UnE
thf(fact_468_UnE,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( sup_sup_set_list_fm @ A2 @ B ) )
=> ( ~ ( member_list_fm2 @ C3 @ A2 )
=> ( member_list_fm2 @ C3 @ B ) ) ) ).
% UnE
thf(fact_469_UnE,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( sup_sup_set_tm @ A2 @ B ) )
=> ( ~ ( member_tm2 @ C3 @ A2 )
=> ( member_tm2 @ C3 @ B ) ) ) ).
% UnE
thf(fact_470_UnE,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ B ) ) ) ).
% UnE
thf(fact_471_UnI1,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ A2 )
=> ( member_fm2 @ C3 @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_472_UnI1,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( member7231649785386036813m_rule @ C3 @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnI1
thf(fact_473_UnI1,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ A2 )
=> ( member_list_fm2 @ C3 @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_474_UnI1,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ A2 )
=> ( member_tm2 @ C3 @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_475_UnI1,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_476_UnI2,axiom,
! [C3: fm,B: set_fm,A2: set_fm] :
( ( member_fm2 @ C3 @ B )
=> ( member_fm2 @ C3 @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_477_UnI2,axiom,
! [C3: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ B )
=> ( member7231649785386036813m_rule @ C3 @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnI2
thf(fact_478_UnI2,axiom,
! [C3: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( member_list_fm2 @ C3 @ B )
=> ( member_list_fm2 @ C3 @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_479_UnI2,axiom,
! [C3: tm,B: set_tm,A2: set_tm] :
( ( member_tm2 @ C3 @ B )
=> ( member_tm2 @ C3 @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_480_UnI2,axiom,
! [C3: nat,B: set_nat,A2: set_nat] :
( ( member_nat2 @ C3 @ B )
=> ( member_nat2 @ C3 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_481_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( P2 @ X2 ) )
| ? [X2: nat] :
( ( member_nat2 @ X2 @ B )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_482_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( P2 @ X2 ) )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ B )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_483_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C4 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C4 ) ) ) ).
% Un_assoc
thf(fact_484_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_485_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A6 ) ) ) ).
% Un_commute
thf(fact_486_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_487_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C4 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_488_subset__Un__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
( ( sup_sup_set_tm @ A6 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_489_subset__Un__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
( ( sup_sup_set_fm @ A6 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_490_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( ( sup_sup_set_nat @ A6 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_491_subset__UnE,axiom,
! [C4: set_tm,A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ C4 @ ( sup_sup_set_tm @ A2 @ B ) )
=> ~ ! [A7: set_tm] :
( ( ord_less_eq_set_tm @ A7 @ A2 )
=> ! [B8: set_tm] :
( ( ord_less_eq_set_tm @ B8 @ B )
=> ( C4
!= ( sup_sup_set_tm @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_492_subset__UnE,axiom,
! [C4: set_fm,A2: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ C4 @ ( sup_sup_set_fm @ A2 @ B ) )
=> ~ ! [A7: set_fm] :
( ( ord_less_eq_set_fm @ A7 @ A2 )
=> ! [B8: set_fm] :
( ( ord_less_eq_set_fm @ B8 @ B )
=> ( C4
!= ( sup_sup_set_fm @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_493_subset__UnE,axiom,
! [C4: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A2 )
=> ! [B8: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ B )
=> ( C4
!= ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_494_Un__absorb2,axiom,
! [B: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B @ A2 )
=> ( ( sup_sup_set_tm @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_495_Un__absorb2,axiom,
! [B: set_fm,A2: set_fm] :
( ( ord_less_eq_set_fm @ B @ A2 )
=> ( ( sup_sup_set_fm @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_496_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_497_Un__absorb1,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( sup_sup_set_tm @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_498_Un__absorb1,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( sup_sup_set_fm @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_499_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_500_Un__upper2,axiom,
! [B: set_tm,A2: set_tm] : ( ord_less_eq_set_tm @ B @ ( sup_sup_set_tm @ A2 @ B ) ) ).
% Un_upper2
thf(fact_501_Un__upper2,axiom,
! [B: set_fm,A2: set_fm] : ( ord_less_eq_set_fm @ B @ ( sup_sup_set_fm @ A2 @ B ) ) ).
% Un_upper2
thf(fact_502_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_503_Un__upper1,axiom,
! [A2: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ A2 @ ( sup_sup_set_tm @ A2 @ B ) ) ).
% Un_upper1
thf(fact_504_Un__upper1,axiom,
! [A2: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ A2 @ ( sup_sup_set_fm @ A2 @ B ) ) ).
% Un_upper1
thf(fact_505_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_506_Un__least,axiom,
! [A2: set_tm,C4: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C4 )
=> ( ( ord_less_eq_set_tm @ B @ C4 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ C4 ) ) ) ).
% Un_least
thf(fact_507_Un__least,axiom,
! [A2: set_fm,C4: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ C4 )
=> ( ( ord_less_eq_set_fm @ B @ C4 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B ) @ C4 ) ) ) ).
% Un_least
thf(fact_508_Un__least,axiom,
! [A2: set_nat,C4: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C4 ) ) ) ).
% Un_least
thf(fact_509_Un__mono,axiom,
! [A2: set_tm,C4: set_tm,B: set_tm,D: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C4 )
=> ( ( ord_less_eq_set_tm @ B @ D )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ ( sup_sup_set_tm @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_510_Un__mono,axiom,
! [A2: set_fm,C4: set_fm,B: set_fm,D: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ C4 )
=> ( ( ord_less_eq_set_fm @ B @ D )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A2 @ B ) @ ( sup_sup_set_fm @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_511_Un__mono,axiom,
! [A2: set_nat,C4: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C4 @ D ) ) ) ) ).
% Un_mono
thf(fact_512_double__diff,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ C4 )
=> ( ( minus_minus_set_tm @ B @ ( minus_minus_set_tm @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_513_double__diff,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( ord_less_eq_set_fm @ B @ C4 )
=> ( ( minus_minus_set_fm @ B @ ( minus_minus_set_fm @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_514_double__diff,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_515_Diff__subset,axiom,
! [A2: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_516_Diff__subset,axiom,
! [A2: set_fm,B: set_fm] : ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_517_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_518_Diff__mono,axiom,
! [A2: set_tm,C4: set_tm,D: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C4 )
=> ( ( ord_less_eq_set_tm @ D @ B )
=> ( ord_less_eq_set_tm @ ( minus_minus_set_tm @ A2 @ B ) @ ( minus_minus_set_tm @ C4 @ D ) ) ) ) ).
% Diff_mono
thf(fact_519_Diff__mono,axiom,
! [A2: set_fm,C4: set_fm,D: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ C4 )
=> ( ( ord_less_eq_set_fm @ D @ B )
=> ( ord_less_eq_set_fm @ ( minus_minus_set_fm @ A2 @ B ) @ ( minus_minus_set_fm @ C4 @ D ) ) ) ) ).
% Diff_mono
thf(fact_520_Diff__mono,axiom,
! [A2: set_nat,C4: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C4 @ D ) ) ) ) ).
% Diff_mono
thf(fact_521_fminusD2,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( minus_7547787838945083330ist_fm @ A2 @ B ) )
=> ~ ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ).
% fminusD2
thf(fact_522_fminusD1,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( minus_7547787838945083330ist_fm @ A2 @ B ) )
=> ( fmembe3754813877001230652ist_fm @ C3 @ A2 ) ) ).
% fminusD1
thf(fact_523_fminusE,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( minus_7547787838945083330ist_fm @ A2 @ B ) )
=> ~ ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% fminusE
thf(fact_524_stake_Osimps_I1_J,axiom,
! [S: stream_tm] :
( ( stake_tm @ zero_zero_nat @ S )
= nil_tm ) ).
% stake.simps(1)
thf(fact_525_stake_Osimps_I1_J,axiom,
! [S: stream_fm] :
( ( stake_fm @ zero_zero_nat @ S )
= nil_fm ) ).
% stake.simps(1)
thf(fact_526_stake_Osimps_I1_J,axiom,
! [S: stream_list_fm] :
( ( stake_list_fm @ zero_zero_nat @ S )
= nil_list_fm ) ).
% stake.simps(1)
thf(fact_527_sub__list_Osimps_I1_J,axiom,
! [V2: nat,S: tm] :
( ( sub_list @ V2 @ S @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_528_paramst__sub__term_I2_J,axiom,
! [M: nat,S: tm,L2: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S @ L2 ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramsts @ L2 ) ) ) ).
% paramst_sub_term(2)
thf(fact_529_stake__sdrop,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_530_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_531_substts_Osimps_I1_J,axiom,
! [S: tm,K: nat] :
( ( substts @ nil_tm @ S @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_532_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_tm @ N2 @ nil_tm )
= N2 ) ).
% gen_length_code(1)
thf(fact_533_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_fm @ N2 @ nil_fm )
= N2 ) ).
% gen_length_code(1)
thf(fact_534_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_list_fm @ N2 @ nil_list_fm )
= N2 ) ).
% gen_length_code(1)
thf(fact_535_paramst__sub__term_I1_J,axiom,
! [M: nat,S: tm,T2: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S @ T2 ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramst @ T2 ) ) ) ).
% paramst_sub_term(1)
thf(fact_536_sup_Obounded__iff,axiom,
! [B2: set_tm,C3: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C3 ) @ A )
= ( ( ord_less_eq_set_tm @ B2 @ A )
& ( ord_less_eq_set_tm @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_537_sup_Obounded__iff,axiom,
! [B2: nat,C3: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C3 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_538_sup_Obounded__iff,axiom,
! [B2: set_fm,C3: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C3 ) @ A )
= ( ( ord_less_eq_set_fm @ B2 @ A )
& ( ord_less_eq_set_fm @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_539_sup_Obounded__iff,axiom,
! [B2: set_nat,C3: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C3 ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_540_le__sup__iff,axiom,
! [X: set_tm,Y: set_tm,Z4: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_set_tm @ X @ Z4 )
& ( ord_less_eq_set_tm @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_541_le__sup__iff,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_nat @ X @ Z4 )
& ( ord_less_eq_nat @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_542_le__sup__iff,axiom,
! [X: set_fm,Y: set_fm,Z4: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_set_fm @ X @ Z4 )
& ( ord_less_eq_set_fm @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_543_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_set_nat @ X @ Z4 )
& ( ord_less_eq_set_nat @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_544_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_545_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_546_set__union,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( set_list_fm2 @ ( union_list_fm @ Xs @ Ys ) )
= ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_547_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_548_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_549_sup_OcoboundedI2,axiom,
! [C3: set_tm,B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ C3 @ B2 )
=> ( ord_less_eq_set_tm @ C3 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_550_sup_OcoboundedI2,axiom,
! [C3: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C3 @ B2 )
=> ( ord_less_eq_nat @ C3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_551_sup_OcoboundedI2,axiom,
! [C3: set_fm,B2: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ C3 @ B2 )
=> ( ord_less_eq_set_fm @ C3 @ ( sup_sup_set_fm @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_552_sup_OcoboundedI2,axiom,
! [C3: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ B2 )
=> ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_553_sup_OcoboundedI1,axiom,
! [C3: set_tm,A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C3 @ A )
=> ( ord_less_eq_set_tm @ C3 @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_554_sup_OcoboundedI1,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_eq_nat @ C3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_555_sup_OcoboundedI1,axiom,
! [C3: set_fm,A: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ C3 @ A )
=> ( ord_less_eq_set_fm @ C3 @ ( sup_sup_set_fm @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_556_sup_OcoboundedI1,axiom,
! [C3: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_557_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( sup_sup_set_tm @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_558_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_559_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
( ( sup_sup_set_fm @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_560_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_561_Diff__iff,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( minus_minus_set_fm @ A2 @ B ) )
= ( ( member_fm2 @ C3 @ A2 )
& ~ ( member_fm2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_562_Diff__iff,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( minus_5980356457887379781m_rule @ A2 @ B ) )
= ( ( member7231649785386036813m_rule @ C3 @ A2 )
& ~ ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_563_Diff__iff,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( minus_639611354763871680ist_fm @ A2 @ B ) )
= ( ( member_list_fm2 @ C3 @ A2 )
& ~ ( member_list_fm2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_564_Diff__iff,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( minus_minus_set_tm @ A2 @ B ) )
= ( ( member_tm2 @ C3 @ A2 )
& ~ ( member_tm2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_565_Diff__iff,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat2 @ C3 @ A2 )
& ~ ( member_nat2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_566_DiffI,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ A2 )
=> ( ~ ( member_fm2 @ C3 @ B )
=> ( member_fm2 @ C3 @ ( minus_minus_set_fm @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_567_DiffI,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( ~ ( member7231649785386036813m_rule @ C3 @ B )
=> ( member7231649785386036813m_rule @ C3 @ ( minus_5980356457887379781m_rule @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_568_DiffI,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ A2 )
=> ( ~ ( member_list_fm2 @ C3 @ B )
=> ( member_list_fm2 @ C3 @ ( minus_639611354763871680ist_fm @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_569_DiffI,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ A2 )
=> ( ~ ( member_tm2 @ C3 @ B )
=> ( member_tm2 @ C3 @ ( minus_minus_set_tm @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_570_DiffI,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ A2 )
=> ( ~ ( member_nat2 @ C3 @ B )
=> ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_571_funionCI,axiom,
! [C3: produc6018962875968178549ist_fm,B: fset_P8989946509869081563ist_fm,A2: fset_P8989946509869081563ist_fm] :
( ( ~ ( fmembe3754813877001230652ist_fm @ C3 @ B )
=> ( fmembe3754813877001230652ist_fm @ C3 @ A2 ) )
=> ( fmembe3754813877001230652ist_fm @ C3 @ ( sup_su5005723363340324783ist_fm @ A2 @ B ) ) ) ).
% funionCI
thf(fact_572_funion__iff,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( sup_su5005723363340324783ist_fm @ A2 @ B ) )
= ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
| ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% funion_iff
thf(fact_573_DiffD2,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( minus_minus_set_fm @ A2 @ B ) )
=> ~ ( member_fm2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_574_DiffD2,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( minus_5980356457887379781m_rule @ A2 @ B ) )
=> ~ ( member7231649785386036813m_rule @ C3 @ B ) ) ).
% DiffD2
thf(fact_575_DiffD2,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( minus_639611354763871680ist_fm @ A2 @ B ) )
=> ~ ( member_list_fm2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_576_DiffD2,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( minus_minus_set_tm @ A2 @ B ) )
=> ~ ( member_tm2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_577_DiffD2,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_578_DiffD1,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( minus_minus_set_fm @ A2 @ B ) )
=> ( member_fm2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_579_DiffD1,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( minus_5980356457887379781m_rule @ A2 @ B ) )
=> ( member7231649785386036813m_rule @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_580_DiffD1,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( minus_639611354763871680ist_fm @ A2 @ B ) )
=> ( member_list_fm2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_581_DiffD1,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( minus_minus_set_tm @ A2 @ B ) )
=> ( member_tm2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_582_DiffD1,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_583_DiffE,axiom,
! [C3: fm,A2: set_fm,B: set_fm] :
( ( member_fm2 @ C3 @ ( minus_minus_set_fm @ A2 @ B ) )
=> ~ ( ( member_fm2 @ C3 @ A2 )
=> ( member_fm2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_584_DiffE,axiom,
! [C3: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C3 @ ( minus_5980356457887379781m_rule @ A2 @ B ) )
=> ~ ( ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% DiffE
thf(fact_585_DiffE,axiom,
! [C3: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm2 @ C3 @ ( minus_639611354763871680ist_fm @ A2 @ B ) )
=> ~ ( ( member_list_fm2 @ C3 @ A2 )
=> ( member_list_fm2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_586_DiffE,axiom,
! [C3: tm,A2: set_tm,B: set_tm] :
( ( member_tm2 @ C3 @ ( minus_minus_set_tm @ A2 @ B ) )
=> ~ ( ( member_tm2 @ C3 @ A2 )
=> ( member_tm2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_587_DiffE,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_588_funionE,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ ( sup_su5005723363340324783ist_fm @ A2 @ B ) )
=> ( ~ ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% funionE
thf(fact_589_funionI1,axiom,
! [C3: produc6018962875968178549ist_fm,A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ ( sup_su5005723363340324783ist_fm @ A2 @ B ) ) ) ).
% funionI1
thf(fact_590_funionI2,axiom,
! [C3: produc6018962875968178549ist_fm,B: fset_P8989946509869081563ist_fm,A2: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ C3 @ B )
=> ( fmembe3754813877001230652ist_fm @ C3 @ ( sup_su5005723363340324783ist_fm @ A2 @ B ) ) ) ).
% funionI2
thf(fact_591_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_592_inf__sup__ord_I4_J,axiom,
! [Y: set_tm,X: set_tm] : ( ord_less_eq_set_tm @ Y @ ( sup_sup_set_tm @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_593_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_594_inf__sup__ord_I4_J,axiom,
! [Y: set_fm,X: set_fm] : ( ord_less_eq_set_fm @ Y @ ( sup_sup_set_fm @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_595_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_596_inf__sup__ord_I3_J,axiom,
! [X: set_tm,Y: set_tm] : ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_597_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_598_inf__sup__ord_I3_J,axiom,
! [X: set_fm,Y: set_fm] : ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_599_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_600_le__supE,axiom,
! [A: set_tm,B2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_tm @ A @ X )
=> ~ ( ord_less_eq_set_tm @ B2 @ X ) ) ) ).
% le_supE
thf(fact_601_le__supE,axiom,
! [A: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_602_le__supE,axiom,
! [A: set_fm,B2: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_fm @ A @ X )
=> ~ ( ord_less_eq_set_fm @ B2 @ X ) ) ) ).
% le_supE
thf(fact_603_le__supE,axiom,
! [A: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_604_le__supI,axiom,
! [A: set_tm,X: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ X )
=> ( ( ord_less_eq_set_tm @ B2 @ X )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_605_le__supI,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_606_le__supI,axiom,
! [A: set_fm,X: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A @ X )
=> ( ( ord_less_eq_set_fm @ B2 @ X )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_607_le__supI,axiom,
! [A: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_608_sup__ge1,axiom,
! [X: set_tm,Y: set_tm] : ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ X @ Y ) ) ).
% sup_ge1
thf(fact_609_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_610_sup__ge1,axiom,
! [X: set_fm,Y: set_fm] : ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ X @ Y ) ) ).
% sup_ge1
thf(fact_611_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_612_sup__ge2,axiom,
! [Y: set_tm,X: set_tm] : ( ord_less_eq_set_tm @ Y @ ( sup_sup_set_tm @ X @ Y ) ) ).
% sup_ge2
thf(fact_613_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_614_sup__ge2,axiom,
! [Y: set_fm,X: set_fm] : ( ord_less_eq_set_fm @ Y @ ( sup_sup_set_fm @ X @ Y ) ) ).
% sup_ge2
thf(fact_615_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_616_le__supI1,axiom,
! [X: set_tm,A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ X @ A )
=> ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_617_le__supI1,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_618_le__supI1,axiom,
! [X: set_fm,A: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ X @ A )
=> ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_619_le__supI1,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_620_le__supI2,axiom,
! [X: set_tm,B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ X @ B2 )
=> ( ord_less_eq_set_tm @ X @ ( sup_sup_set_tm @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_621_le__supI2,axiom,
! [X: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_622_le__supI2,axiom,
! [X: set_fm,B2: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ X @ B2 )
=> ( ord_less_eq_set_fm @ X @ ( sup_sup_set_fm @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_623_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_624_sup_Omono,axiom,
! [C3: set_tm,A: set_tm,D2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C3 @ A )
=> ( ( ord_less_eq_set_tm @ D2 @ B2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ C3 @ D2 ) @ ( sup_sup_set_tm @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_625_sup_Omono,axiom,
! [C3: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C3 @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C3 @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_626_sup_Omono,axiom,
! [C3: set_fm,A: set_fm,D2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ C3 @ A )
=> ( ( ord_less_eq_set_fm @ D2 @ B2 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ C3 @ D2 ) @ ( sup_sup_set_fm @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_627_sup_Omono,axiom,
! [C3: set_nat,A: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C3 @ D2 ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_628_sup__mono,axiom,
! [A: set_tm,C3: set_tm,B2: set_tm,D2: set_tm] :
( ( ord_less_eq_set_tm @ A @ C3 )
=> ( ( ord_less_eq_set_tm @ B2 @ D2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B2 ) @ ( sup_sup_set_tm @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_629_sup__mono,axiom,
! [A: nat,C3: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C3 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_630_sup__mono,axiom,
! [A: set_fm,C3: set_fm,B2: set_fm,D2: set_fm] :
( ( ord_less_eq_set_fm @ A @ C3 )
=> ( ( ord_less_eq_set_fm @ B2 @ D2 )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ A @ B2 ) @ ( sup_sup_set_fm @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_631_sup__mono,axiom,
! [A: set_nat,C3: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_632_sup__least,axiom,
! [Y: set_tm,X: set_tm,Z4: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X )
=> ( ( ord_less_eq_set_tm @ Z4 @ X )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_633_sup__least,axiom,
! [Y: nat,X: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z4 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_634_sup__least,axiom,
! [Y: set_fm,X: set_fm,Z4: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X )
=> ( ( ord_less_eq_set_fm @ Z4 @ X )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_635_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z4 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_636_le__iff__sup,axiom,
( ord_less_eq_set_tm
= ( ^ [X2: set_tm,Y3: set_tm] :
( ( sup_sup_set_tm @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_637_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_638_le__iff__sup,axiom,
( ord_less_eq_set_fm
= ( ^ [X2: set_fm,Y3: set_fm] :
( ( sup_sup_set_fm @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_639_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_640_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_641_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_642_sup_OorderE,axiom,
! [B2: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( A
= ( sup_sup_set_fm @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_643_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_644_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_645_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_646_sup_OorderI,axiom,
! [A: set_fm,B2: set_fm] :
( ( A
= ( sup_sup_set_fm @ A @ B2 ) )
=> ( ord_less_eq_set_fm @ B2 @ A ) ) ).
% sup.orderI
thf(fact_647_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_648_sup__unique,axiom,
! [F: set_tm > set_tm > set_tm,X: set_tm,Y: set_tm] :
( ! [X3: set_tm,Y2: set_tm] : ( ord_less_eq_set_tm @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_tm,Y2: set_tm] : ( ord_less_eq_set_tm @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_tm,Y2: set_tm,Z6: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X3 )
=> ( ( ord_less_eq_set_tm @ Z6 @ X3 )
=> ( ord_less_eq_set_tm @ ( F @ Y2 @ Z6 ) @ X3 ) ) )
=> ( ( sup_sup_set_tm @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_649_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z6: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z6 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z6 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_650_sup__unique,axiom,
! [F: set_fm > set_fm > set_fm,X: set_fm,Y: set_fm] :
( ! [X3: set_fm,Y2: set_fm] : ( ord_less_eq_set_fm @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_fm,Y2: set_fm] : ( ord_less_eq_set_fm @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_fm,Y2: set_fm,Z6: set_fm] :
( ( ord_less_eq_set_fm @ Y2 @ X3 )
=> ( ( ord_less_eq_set_fm @ Z6 @ X3 )
=> ( ord_less_eq_set_fm @ ( F @ Y2 @ Z6 ) @ X3 ) ) )
=> ( ( sup_sup_set_fm @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_651_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_nat,Y2: set_nat,Z6: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z6 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z6 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_652_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_653_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_654_sup_Oabsorb1,axiom,
! [B2: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( ( sup_sup_set_fm @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_655_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_656_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_657_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_658_sup_Oabsorb2,axiom,
! [A: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( sup_sup_set_fm @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_659_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_660_sup__absorb1,axiom,
! [Y: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X )
=> ( ( sup_sup_set_tm @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_661_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_662_sup__absorb1,axiom,
! [Y: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X )
=> ( ( sup_sup_set_fm @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_663_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_664_sup__absorb2,axiom,
! [X: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( sup_sup_set_tm @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_665_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_666_sup__absorb2,axiom,
! [X: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( sup_sup_set_fm @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_667_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_668_sup_OboundedE,axiom,
! [B2: set_tm,C3: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C3 ) @ A )
=> ~ ( ( ord_less_eq_set_tm @ B2 @ A )
=> ~ ( ord_less_eq_set_tm @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_669_sup_OboundedE,axiom,
! [B2: nat,C3: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C3 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_670_sup_OboundedE,axiom,
! [B2: set_fm,C3: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C3 ) @ A )
=> ~ ( ( ord_less_eq_set_fm @ B2 @ A )
=> ~ ( ord_less_eq_set_fm @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_671_sup_OboundedE,axiom,
! [B2: set_nat,C3: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C3 ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_672_sup_OboundedI,axiom,
! [B2: set_tm,A: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C3 @ A )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B2 @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_673_sup_OboundedI,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_674_sup_OboundedI,axiom,
! [B2: set_fm,A: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( ( ord_less_eq_set_fm @ C3 @ A )
=> ( ord_less_eq_set_fm @ ( sup_sup_set_fm @ B2 @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_675_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C3 @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_676_sup_Oorder__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( A4
= ( sup_sup_set_tm @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_677_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_678_sup_Oorder__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A4: set_fm] :
( A4
= ( sup_sup_set_fm @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_679_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_680_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_681_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_682_sup_Ocobounded1,axiom,
! [A: set_fm,B2: set_fm] : ( ord_less_eq_set_fm @ A @ ( sup_sup_set_fm @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_683_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_684_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_685_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_686_sup_Ocobounded2,axiom,
! [B2: set_fm,A: set_fm] : ( ord_less_eq_set_fm @ B2 @ ( sup_sup_set_fm @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_687_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_688_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( sup_sup_set_tm @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_689_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_690_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A4: set_fm] :
( ( sup_sup_set_fm @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_691_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_692_inc__term_Osimps_I2_J,axiom,
! [I: nat,L2: list_tm] :
( ( inc_term @ ( fun @ I @ L2 ) )
= ( fun @ I @ ( inc_list @ L2 ) ) ) ).
% inc_term.simps(2)
thf(fact_693_shift__snth,axiom,
! [N2: nat,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ N2 )
= ( nth_Pr3936752564452695919m_rule @ Xs @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ N2 )
= ( snth_P6679518042731451922m_rule @ S @ ( minus_minus_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ) ).
% shift_snth
thf(fact_694_shift__snth__less,axiom,
! [P: nat,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ P @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ P )
= ( nth_Pr3936752564452695919m_rule @ Xs @ P ) ) ) ).
% shift_snth_less
thf(fact_695_liftt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A @ Ts ) )
= ( fun @ A @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_696_hd__rotate__conv__nth,axiom,
! [Xs: list_tm,N2: nat] :
( ( Xs != nil_tm )
=> ( ( hd_tm @ ( rotate_tm @ N2 @ Xs ) )
= ( nth_tm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_697_hd__rotate__conv__nth,axiom,
! [Xs: list_fm,N2: nat] :
( ( Xs != nil_fm )
=> ( ( hd_fm @ ( rotate_fm @ N2 @ Xs ) )
= ( nth_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_698_hd__rotate__conv__nth,axiom,
! [Xs: list_list_fm,N2: nat] :
( ( Xs != nil_list_fm )
=> ( ( hd_list_fm @ ( rotate_list_fm @ N2 @ Xs ) )
= ( nth_list_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_699_partition__set,axiom,
! [P2: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P2 @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_fm @ ( set_fm2 @ Yes ) @ ( set_fm2 @ No ) )
= ( set_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_700_partition__set,axiom,
! [P2: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P2 @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ( sup_sup_set_tm @ ( set_tm2 @ Yes ) @ ( set_tm2 @ No ) )
= ( set_tm2 @ Xs ) ) ) ).
% partition_set
thf(fact_701_partition__set,axiom,
! [P2: list_fm > $o,Xs: list_list_fm,Yes: list_list_fm,No: list_list_fm] :
( ( ( partition_list_fm @ P2 @ Xs )
= ( produc8321651870839017815ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_list_fm @ ( set_list_fm2 @ Yes ) @ ( set_list_fm2 @ No ) )
= ( set_list_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_702_partition__set,axiom,
! [P2: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
( ( ( partition_nat @ P2 @ Xs )
= ( produc2694037385005941721st_nat @ Yes @ No ) )
=> ( ( sup_sup_set_nat @ ( set_nat2 @ Yes ) @ ( set_nat2 @ No ) )
= ( set_nat2 @ Xs ) ) ) ).
% partition_set
thf(fact_703_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_704_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_705_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_706_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_707_mod__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( modulo_modulo_nat @ M @ N2 )
= M ) ) ).
% mod_less
thf(fact_708_paramst__liftt_I1_J,axiom,
! [T2: tm] :
( ( paramst @ ( liftt @ T2 ) )
= ( paramst @ T2 ) ) ).
% paramst_liftt(1)
thf(fact_709_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_710_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_711_length__greater__0__conv,axiom,
! [Xs: list_tm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) )
= ( Xs != nil_tm ) ) ).
% length_greater_0_conv
thf(fact_712_length__greater__0__conv,axiom,
! [Xs: list_fm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) )
= ( Xs != nil_fm ) ) ).
% length_greater_0_conv
thf(fact_713_length__greater__0__conv,axiom,
! [Xs: list_list_fm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) )
= ( Xs != nil_list_fm ) ) ).
% length_greater_0_conv
thf(fact_714_shift__simps_I1_J,axiom,
! [Xs: list_tm,S: stream_tm] :
( ( ( Xs = nil_tm )
=> ( ( shd_tm @ ( shift_tm @ Xs @ S ) )
= ( shd_tm @ S ) ) )
& ( ( Xs != nil_tm )
=> ( ( shd_tm @ ( shift_tm @ Xs @ S ) )
= ( hd_tm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_715_shift__simps_I1_J,axiom,
! [Xs: list_fm,S: stream_fm] :
( ( ( Xs = nil_fm )
=> ( ( shd_fm @ ( shift_fm @ Xs @ S ) )
= ( shd_fm @ S ) ) )
& ( ( Xs != nil_fm )
=> ( ( shd_fm @ ( shift_fm @ Xs @ S ) )
= ( hd_fm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_716_shift__simps_I1_J,axiom,
! [Xs: list_list_fm,S: stream_list_fm] :
( ( ( Xs = nil_list_fm )
=> ( ( shd_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( shd_list_fm @ S ) ) )
& ( ( Xs != nil_list_fm )
=> ( ( shd_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( hd_list_fm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_717_shift__simps_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( Xs = nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( shd_Pr4562317740776619530m_rule @ S ) ) )
& ( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( hd_Pro7241777042969981963m_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_718_stake__nth,axiom,
! [M: nat,N2: nat,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ M @ N2 )
=> ( ( nth_Pr3936752564452695919m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ M )
= ( snth_P6679518042731451922m_rule @ S @ M ) ) ) ).
% stake_nth
thf(fact_719_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_720_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_721_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_722_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_723_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_724_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_725_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_726_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_727_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_728_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_729_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_730_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_731_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_732_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_733_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_734_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_735_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_736_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_737_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_738_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P2 @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P2 @ B6 @ A5 )
=> ( P2 @ A5 @ B6 ) )
=> ( P2 @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_739_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans
thf(fact_740_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_741_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C3 @ B2 )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_742_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_743_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_744_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_745_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_746_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_747_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_748_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_749_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_750_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_751_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_752_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_753_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_754_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_755_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_756_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_757_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_758_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_759_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_760_order__le__imp__less__or__eq,axiom,
! [X: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( ord_less_set_tm @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_761_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_762_order__le__imp__less__or__eq,axiom,
! [X: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( ord_less_set_fm @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_763_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_764_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_765_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C3: set_tm] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_766_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_767_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_fm,C3: set_fm] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_768_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C3: set_nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_769_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_770_order__less__le__subst1,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C3: set_tm] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_771_order__less__le__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C3: nat] :
( ( ord_less_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_772_order__less__le__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B2: nat,C3: nat] :
( ( ord_less_set_fm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_773_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C3: nat] :
( ( ord_less_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_774_order__less__le__subst1,axiom,
! [A: nat,F: set_fm > nat,B2: set_fm,C3: set_fm] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_775_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C3: set_nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_776_order__less__le__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_777_order__less__le__subst1,axiom,
! [A: set_fm,F: set_tm > set_fm,B2: set_tm,C3: set_tm] :
( ( ord_less_set_fm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_778_order__less__le__subst1,axiom,
! [A: set_nat,F: set_tm > set_nat,B2: set_tm,C3: set_tm] :
( ( ord_less_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_779_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_780_order__le__less__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > nat,C3: nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_781_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C3: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_set_tm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_782_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_fm,C3: set_fm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_set_fm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_783_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C3: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_784_order__le__less__subst2,axiom,
! [A: set_fm,B2: set_fm,F: set_fm > nat,C3: nat] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_fm,Y2: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_785_order__le__less__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_786_order__le__less__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_set_tm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_787_order__le__less__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_fm,C3: set_fm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_set_fm @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_788_order__le__less__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_nat,C3: set_nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C3 )
=> ( ! [X3: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_789_order__le__less__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C3: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_790_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_791_order__le__less__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B2: nat,C3: nat] :
( ( ord_less_eq_set_fm @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_792_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C3: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_793_order__less__le__trans,axiom,
! [X: set_tm,Y: set_tm,Z4: set_tm] :
( ( ord_less_set_tm @ X @ Y )
=> ( ( ord_less_eq_set_tm @ Y @ Z4 )
=> ( ord_less_set_tm @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_794_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_795_order__less__le__trans,axiom,
! [X: set_fm,Y: set_fm,Z4: set_fm] :
( ( ord_less_set_fm @ X @ Y )
=> ( ( ord_less_eq_set_fm @ Y @ Z4 )
=> ( ord_less_set_fm @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_796_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z4 )
=> ( ord_less_set_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_797_order__le__less__trans,axiom,
! [X: set_tm,Y: set_tm,Z4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( ord_less_set_tm @ Y @ Z4 )
=> ( ord_less_set_tm @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_798_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_799_order__le__less__trans,axiom,
! [X: set_fm,Y: set_fm,Z4: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( ord_less_set_fm @ Y @ Z4 )
=> ( ord_less_set_fm @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_800_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z4 )
=> ( ord_less_set_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_801_order__neq__le__trans,axiom,
! [A: set_tm,B2: set_tm] :
( ( A != B2 )
=> ( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ord_less_set_tm @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_802_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_803_order__neq__le__trans,axiom,
! [A: set_fm,B2: set_fm] :
( ( A != B2 )
=> ( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ord_less_set_fm @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_804_order__neq__le__trans,axiom,
! [A: set_nat,B2: set_nat] :
( ( A != B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_805_order__le__neq__trans,axiom,
! [A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_tm @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_806_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_807_order__le__neq__trans,axiom,
! [A: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_fm @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_808_order__le__neq__trans,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_809_order__less__imp__le,axiom,
! [X: set_tm,Y: set_tm] :
( ( ord_less_set_tm @ X @ Y )
=> ( ord_less_eq_set_tm @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_810_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_811_order__less__imp__le,axiom,
! [X: set_fm,Y: set_fm] :
( ( ord_less_set_fm @ X @ Y )
=> ( ord_less_eq_set_fm @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_812_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_813_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_814_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_815_order__less__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_816_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_817_order__less__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_818_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_819_order__le__less,axiom,
( ord_less_eq_set_tm
= ( ^ [X2: set_tm,Y3: set_tm] :
( ( ord_less_set_tm @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_820_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_821_order__le__less,axiom,
( ord_less_eq_set_fm
= ( ^ [X2: set_fm,Y3: set_fm] :
( ( ord_less_set_fm @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_822_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_823_dual__order_Ostrict__implies__order,axiom,
! [B2: set_tm,A: set_tm] :
( ( ord_less_set_tm @ B2 @ A )
=> ( ord_less_eq_set_tm @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_824_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_825_dual__order_Ostrict__implies__order,axiom,
! [B2: set_fm,A: set_fm] :
( ( ord_less_set_fm @ B2 @ A )
=> ( ord_less_eq_set_fm @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_826_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_827_order_Ostrict__implies__order,axiom,
! [A: set_tm,B2: set_tm] :
( ( ord_less_set_tm @ A @ B2 )
=> ( ord_less_eq_set_tm @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_828_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_829_order_Ostrict__implies__order,axiom,
! [A: set_fm,B2: set_fm] :
( ( ord_less_set_fm @ A @ B2 )
=> ( ord_less_eq_set_fm @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_830_order_Ostrict__implies__order,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_831_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ~ ( ord_less_eq_set_tm @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_832_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_833_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [B5: set_fm,A4: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A4 )
& ~ ( ord_less_eq_set_fm @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_834_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_835_dual__order_Ostrict__trans2,axiom,
! [B2: set_tm,A: set_tm,C3: set_tm] :
( ( ord_less_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C3 @ B2 )
=> ( ord_less_set_tm @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_836_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C3 @ B2 )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_837_dual__order_Ostrict__trans2,axiom,
! [B2: set_fm,A: set_fm,C3: set_fm] :
( ( ord_less_set_fm @ B2 @ A )
=> ( ( ord_less_eq_set_fm @ C3 @ B2 )
=> ( ord_less_set_fm @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_838_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C3 @ B2 )
=> ( ord_less_set_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_839_dual__order_Ostrict__trans1,axiom,
! [B2: set_tm,A: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_set_tm @ C3 @ B2 )
=> ( ord_less_set_tm @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_840_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C3 @ B2 )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_841_dual__order_Ostrict__trans1,axiom,
! [B2: set_fm,A: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ B2 @ A )
=> ( ( ord_less_set_fm @ C3 @ B2 )
=> ( ord_less_set_fm @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_842_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_set_nat @ C3 @ B2 )
=> ( ord_less_set_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_843_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_844_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_845_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [B5: set_fm,A4: set_fm] :
( ( ord_less_eq_set_fm @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_846_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_847_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_set_tm @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_848_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_849_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [B5: set_fm,A4: set_fm] :
( ( ord_less_set_fm @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_850_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_851_order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ~ ( ord_less_eq_set_tm @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_852_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_853_order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B5 )
& ~ ( ord_less_eq_set_fm @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_854_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_855_order_Ostrict__trans2,axiom,
! [A: set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C3 )
=> ( ord_less_set_tm @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_856_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_857_order_Ostrict__trans2,axiom,
! [A: set_fm,B2: set_fm,C3: set_fm] :
( ( ord_less_set_fm @ A @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ C3 )
=> ( ord_less_set_fm @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_858_order_Ostrict__trans2,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_859_order_Ostrict__trans1,axiom,
! [A: set_tm,B2: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_set_tm @ B2 @ C3 )
=> ( ord_less_set_tm @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_860_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_861_order_Ostrict__trans1,axiom,
! [A: set_fm,B2: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A @ B2 )
=> ( ( ord_less_set_fm @ B2 @ C3 )
=> ( ord_less_set_fm @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_862_order_Ostrict__trans1,axiom,
! [A: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C3 )
=> ( ord_less_set_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_863_order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_864_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_865_order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_866_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_867_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_set_tm @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_868_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_869_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B5: set_fm] :
( ( ord_less_set_fm @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_870_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_871_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_872_less__le__not__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
& ~ ( ord_less_eq_set_tm @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_873_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_874_less__le__not__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y3 )
& ~ ( ord_less_eq_set_fm @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_875_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_876_antisym__conv2,axiom,
! [X: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y )
=> ( ( ~ ( ord_less_set_tm @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_877_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_878_antisym__conv2,axiom,
! [X: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y )
=> ( ( ~ ( ord_less_set_fm @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_879_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_880_antisym__conv1,axiom,
! [X: set_tm,Y: set_tm] :
( ~ ( ord_less_set_tm @ X @ Y )
=> ( ( ord_less_eq_set_tm @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_881_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_882_antisym__conv1,axiom,
! [X: set_fm,Y: set_fm] :
( ~ ( ord_less_set_fm @ X @ Y )
=> ( ( ord_less_eq_set_fm @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_883_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_884_nless__le,axiom,
! [A: set_tm,B2: set_tm] :
( ( ~ ( ord_less_set_tm @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_tm @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_885_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_886_nless__le,axiom,
! [A: set_fm,B2: set_fm] :
( ( ~ ( ord_less_set_fm @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_fm @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_887_nless__le,axiom,
! [A: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_888_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_889_leD,axiom,
! [Y: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y @ X )
=> ~ ( ord_less_set_tm @ X @ Y ) ) ).
% leD
thf(fact_890_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_891_leD,axiom,
! [Y: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y @ X )
=> ~ ( ord_less_set_fm @ X @ Y ) ) ).
% leD
thf(fact_892_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_893_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_894_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_895_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_896_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_897_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_898_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_899_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_900_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_901_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_902_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_903_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_904_infinite__descent0,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P2 @ N )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_905_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_906_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_907_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_908_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_909_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_910_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_911_diff__less__mono2,axiom,
! [M: nat,N2: nat,L2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_912_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_913_list_Oset__sel_I1_J,axiom,
! [A: list_P2774625669004309958m_rule] :
( ( A != nil_Pr2808214839852828720m_rule )
=> ( member7231649785386036813m_rule @ ( hd_Pro7241777042969981963m_rule @ A ) @ ( set_Pr4534715572506550497m_rule @ A ) ) ) ).
% list.set_sel(1)
thf(fact_914_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_915_list_Oset__sel_I1_J,axiom,
! [A: list_fm] :
( ( A != nil_fm )
=> ( member_fm2 @ ( hd_fm @ A ) @ ( set_fm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_916_list_Oset__sel_I1_J,axiom,
! [A: list_tm] :
( ( A != nil_tm )
=> ( member_tm2 @ ( hd_tm @ A ) @ ( set_tm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_917_list_Oset__sel_I1_J,axiom,
! [A: list_list_fm] :
( ( A != nil_list_fm )
=> ( member_list_fm2 @ ( hd_list_fm @ A ) @ ( set_list_fm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_918_hd__in__set,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( member7231649785386036813m_rule @ ( hd_Pro7241777042969981963m_rule @ Xs ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% hd_in_set
thf(fact_919_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_920_hd__in__set,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( member_fm2 @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_921_hd__in__set,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( member_tm2 @ ( hd_tm @ Xs ) @ ( set_tm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_922_hd__in__set,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( member_list_fm2 @ ( hd_list_fm @ Xs ) @ ( set_list_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_923_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_924_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_925_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_926_diff__less__mono,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C3 ) @ ( minus_minus_nat @ B2 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_927_mod__less__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_928_mod__suff,axiom,
! [M: nat,P2: nat > $o,K: nat] :
( ! [N: nat] :
( ( ord_less_nat @ M @ N )
=> ( P2 @ ( modulo_modulo_nat @ N @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P2 @ I3 ) ) ) ) ).
% mod_suff
thf(fact_929_mod__hit,axiom,
! [K: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( ( modulo_modulo_nat @ N @ K )
= I3 ) ) ) ) ).
% mod_hit
thf(fact_930_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M: nat,N2: nat] :
( ! [M4: nat] : ( P2 @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ N @ ( modulo_modulo_nat @ M4 @ N ) )
=> ( P2 @ M4 @ N ) ) )
=> ( P2 @ M @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_931_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_932_cycle_Osimps_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( hd_Pro7241777042969981963m_rule @ Xs ) ) ).
% cycle.simps(1)
thf(fact_933_hd__conv__nth,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( hd_tm @ Xs )
= ( nth_tm @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_934_hd__conv__nth,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( ( hd_fm @ Xs )
= ( nth_fm @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_935_hd__conv__nth,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( hd_list_fm @ Xs )
= ( nth_list_fm @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_936_length__pos__if__in__set,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_937_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_938_length__pos__if__in__set,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_939_length__pos__if__in__set,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_940_length__pos__if__in__set,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_941_mod__le__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_942_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
=> ( P2 @ ( nth_fm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_943_all__set__conv__all__nth,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_tm @ Xs ) )
=> ( P2 @ ( nth_tm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_944_all__set__conv__all__nth,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P2 @ ( nth_list_fm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_945_all__nth__imp__all__set,axiom,
! [Xs: list_P2774625669004309958m_rule,P2: produc340336539035504054m_rule > $o,X: produc340336539035504054m_rule] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( P2 @ ( nth_Pr3936752564452695919m_rule @ Xs @ I2 ) ) )
=> ( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_946_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_947_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P2: fm > $o,X: fm] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( P2 @ ( nth_fm @ Xs @ I2 ) ) )
=> ( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_948_all__nth__imp__all__set,axiom,
! [Xs: list_tm,P2: tm > $o,X: tm] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( P2 @ ( nth_tm @ Xs @ I2 ) ) )
=> ( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_949_all__nth__imp__all__set,axiom,
! [Xs: list_list_fm,P2: list_fm > $o,X: list_fm] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P2 @ ( nth_list_fm @ Xs @ I2 ) ) )
=> ( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_950_in__set__conv__nth,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1575636608424004698m_rule @ Xs ) )
& ( ( nth_Pr3936752564452695919m_rule @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_951_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_952_in__set__conv__nth,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_953_in__set__conv__nth,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_tm @ Xs ) )
& ( ( nth_tm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_954_in__set__conv__nth,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s115229985653309035ist_fm @ Xs ) )
& ( ( nth_list_fm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_955_list__ball__nth,axiom,
! [N2: nat,Xs: list_fm,P2: fm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_956_list__ball__nth,axiom,
! [N2: nat,Xs: list_tm,P2: tm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_tm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_957_list__ball__nth,axiom,
! [N2: nat,Xs: list_list_fm,P2: list_fm > $o] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_list_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_958_nth__mem,axiom,
! [N2: nat,Xs: list_P2774625669004309958m_rule] :
( ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ ( nth_Pr3936752564452695919m_rule @ Xs @ N2 ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% nth_mem
thf(fact_959_nth__mem,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_960_nth__mem,axiom,
! [N2: nat,Xs: list_fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( member_fm2 @ ( nth_fm @ Xs @ N2 ) @ ( set_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_961_nth__mem,axiom,
! [N2: nat,Xs: list_tm] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( member_tm2 @ ( nth_tm @ Xs @ N2 ) @ ( set_tm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_962_nth__mem,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( member_list_fm2 @ ( nth_list_fm @ Xs @ N2 ) @ ( set_list_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_963_partition_Osimps_I1_J,axiom,
! [P2: tm > $o] :
( ( partition_tm @ P2 @ nil_tm )
= ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) ) ).
% partition.simps(1)
thf(fact_964_partition_Osimps_I1_J,axiom,
! [P2: fm > $o] :
( ( partition_fm @ P2 @ nil_fm )
= ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) ) ).
% partition.simps(1)
thf(fact_965_partition_Osimps_I1_J,axiom,
! [P2: list_fm > $o] :
( ( partition_list_fm @ P2 @ nil_list_fm )
= ( produc8321651870839017815ist_fm @ nil_list_fm @ nil_list_fm ) ) ).
% partition.simps(1)
thf(fact_966_partition__P,axiom,
! [P2: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P2 @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Yes ) )
=> ( P2 @ X4 ) )
& ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ No ) )
=> ~ ( P2 @ X4 ) ) ) ) ).
% partition_P
thf(fact_967_partition__P,axiom,
! [P2: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P2 @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Yes ) )
=> ( P2 @ X4 ) )
& ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ No ) )
=> ~ ( P2 @ X4 ) ) ) ) ).
% partition_P
thf(fact_968_partition__P,axiom,
! [P2: list_fm > $o,Xs: list_list_fm,Yes: list_list_fm,No: list_list_fm] :
( ( ( partition_list_fm @ P2 @ Xs )
= ( produc8321651870839017815ist_fm @ Yes @ No ) )
=> ( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Yes ) )
=> ( P2 @ X4 ) )
& ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ No ) )
=> ~ ( P2 @ X4 ) ) ) ) ).
% partition_P
thf(fact_969_flat__snth,axiom,
! [S: stream2471014364565126742m_rule,N2: nat] :
( ! [X3: list_P2774625669004309958m_rule] :
( ( member522667613741702877m_rule @ X3 @ ( sset_l1945517436738345138m_rule @ S ) )
=> ( X3 != nil_Pr2808214839852828720m_rule ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) )
=> ( ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ S ) @ N2 )
= ( nth_Pr3936752564452695919m_rule @ ( shd_li4676821617271663642m_rule @ S ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) )
=> ( ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ S ) @ N2 )
= ( snth_P6679518042731451922m_rule @ ( flat_P7721466590633226428m_rule @ ( stl_li6523153919213261078m_rule @ S ) ) @ ( minus_minus_nat @ N2 @ ( size_s1575636608424004698m_rule @ ( shd_li4676821617271663642m_rule @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_970_flat__snth,axiom,
! [S: stream_list_tm,N2: nat] :
( ! [X3: list_tm] :
( ( member_list_tm @ X3 @ ( sset_list_tm @ S ) )
=> ( X3 != nil_tm ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_size_list_tm @ ( shd_list_tm @ S ) ) )
=> ( ( snth_tm @ ( flat_tm @ S ) @ N2 )
= ( nth_tm @ ( shd_list_tm @ S ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_tm @ ( shd_list_tm @ S ) ) )
=> ( ( snth_tm @ ( flat_tm @ S ) @ N2 )
= ( snth_tm @ ( flat_tm @ ( stl_list_tm @ S ) ) @ ( minus_minus_nat @ N2 @ ( size_size_list_tm @ ( shd_list_tm @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_971_flat__snth,axiom,
! [S: stream_list_fm,N2: nat] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( sset_list_fm @ S ) )
=> ( X3 != nil_fm ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_size_list_fm @ ( shd_list_fm @ S ) ) )
=> ( ( snth_fm @ ( flat_fm @ S ) @ N2 )
= ( nth_fm @ ( shd_list_fm @ S ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_size_list_fm @ ( shd_list_fm @ S ) ) )
=> ( ( snth_fm @ ( flat_fm @ S ) @ N2 )
= ( snth_fm @ ( flat_fm @ ( stl_list_fm @ S ) ) @ ( minus_minus_nat @ N2 @ ( size_size_list_fm @ ( shd_list_fm @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_972_flat__snth,axiom,
! [S: stream_list_list_fm,N2: nat] :
( ! [X3: list_list_fm] :
( ( member_list_list_fm @ X3 @ ( sset_list_list_fm @ S ) )
=> ( X3 != nil_list_fm ) )
=> ( ( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ ( shd_list_list_fm @ S ) ) )
=> ( ( snth_list_fm @ ( flat_list_fm @ S ) @ N2 )
= ( nth_list_fm @ ( shd_list_list_fm @ S ) @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ ( shd_list_list_fm @ S ) ) )
=> ( ( snth_list_fm @ ( flat_list_fm @ S ) @ N2 )
= ( snth_list_fm @ ( flat_list_fm @ ( stl_list_list_fm @ S ) ) @ ( minus_minus_nat @ N2 @ ( size_s115229985653309035ist_fm @ ( shd_list_list_fm @ S ) ) ) ) ) ) ) ) ).
% flat_snth
thf(fact_973_nat__descend__induct,axiom,
! [N2: nat,P2: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P2 @ I3 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_974_complete__interval,axiom,
! [A: nat,B2: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B2 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B2 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C5 ) )
=> ( P2 @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_975_psubsetI,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_tm @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_976_psubsetI,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_fm @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_977_psubsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_978_psubsetE,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A2 @ B )
=> ~ ( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_less_eq_set_tm @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_979_psubsetE,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A2 @ B )
=> ~ ( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ord_less_eq_set_fm @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_980_psubsetE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_981_psubset__eq,axiom,
( ord_less_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_982_psubset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_983_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_984_psubset__imp__subset,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A2 @ B )
=> ( ord_less_eq_set_tm @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_985_psubset__imp__subset,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A2 @ B )
=> ( ord_less_eq_set_fm @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_986_psubset__imp__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_987_psubset__subset__trans,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ C4 )
=> ( ord_less_set_tm @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_988_psubset__subset__trans,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_set_fm @ A2 @ B )
=> ( ( ord_less_eq_set_fm @ B @ C4 )
=> ( ord_less_set_fm @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_989_psubset__subset__trans,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_990_subset__not__subset__eq,axiom,
( ord_less_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B7 )
& ~ ( ord_less_eq_set_tm @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_991_subset__not__subset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B7 )
& ~ ( ord_less_eq_set_fm @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_992_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B7 )
& ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_993_subset__psubset__trans,axiom,
! [A2: set_tm,B: set_tm,C4: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_set_tm @ B @ C4 )
=> ( ord_less_set_tm @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_994_subset__psubset__trans,axiom,
! [A2: set_fm,B: set_fm,C4: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ( ord_less_set_fm @ B @ C4 )
=> ( ord_less_set_fm @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_995_subset__psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C4 )
=> ( ord_less_set_nat @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_996_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B7: set_tm] :
( ( ord_less_set_tm @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_997_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B7: set_fm] :
( ( ord_less_set_fm @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_998_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B7: set_nat] :
( ( ord_less_set_nat @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_999_psubset__imp__ex__mem,axiom,
! [A2: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A2 @ B )
=> ? [B6: fm] : ( member_fm2 @ B6 @ ( minus_minus_set_fm @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1000_psubset__imp__ex__mem,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( ord_le4093866177137961152m_rule @ A2 @ B )
=> ? [B6: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ B6 @ ( minus_5980356457887379781m_rule @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1001_psubset__imp__ex__mem,axiom,
! [A2: set_list_fm,B: set_list_fm] :
( ( ord_less_set_list_fm @ A2 @ B )
=> ? [B6: list_fm] : ( member_list_fm2 @ B6 @ ( minus_639611354763871680ist_fm @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1002_psubset__imp__ex__mem,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A2 @ B )
=> ? [B6: tm] : ( member_tm2 @ B6 @ ( minus_minus_set_tm @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1003_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B6: nat] : ( member_nat2 @ B6 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1004_pfsubsetD,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm,C3: produc6018962875968178549ist_fm] :
( ( ord_le7716396445699002503ist_fm @ A2 @ B )
=> ( ( fmembe3754813877001230652ist_fm @ C3 @ A2 )
=> ( fmembe3754813877001230652ist_fm @ C3 @ B ) ) ) ).
% pfsubsetD
thf(fact_1005_pfsubset__imp__ex__fmem,axiom,
! [A2: fset_P8989946509869081563ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( ord_le7716396445699002503ist_fm @ A2 @ B )
=> ? [B6: produc6018962875968178549ist_fm] : ( fmembe3754813877001230652ist_fm @ B6 @ ( minus_7547787838945083330ist_fm @ B @ A2 ) ) ) ).
% pfsubset_imp_ex_fmem
thf(fact_1006_flat__unfold,axiom,
! [Ws: stream2471014364565126742m_rule] :
( ( ( shd_li4676821617271663642m_rule @ Ws )
!= nil_Pr2808214839852828720m_rule )
=> ( ( flat_P7721466590633226428m_rule @ Ws )
= ( shift_2334844276748245581m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) @ ( flat_P7721466590633226428m_rule @ ( stl_li6523153919213261078m_rule @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_1007_flat__unfold,axiom,
! [Ws: stream_list_tm] :
( ( ( shd_list_tm @ Ws )
!= nil_tm )
=> ( ( flat_tm @ Ws )
= ( shift_tm @ ( shd_list_tm @ Ws ) @ ( flat_tm @ ( stl_list_tm @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_1008_flat__unfold,axiom,
! [Ws: stream_list_fm] :
( ( ( shd_list_fm @ Ws )
!= nil_fm )
=> ( ( flat_fm @ Ws )
= ( shift_fm @ ( shd_list_fm @ Ws ) @ ( flat_fm @ ( stl_list_fm @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_1009_flat__unfold,axiom,
! [Ws: stream_list_list_fm] :
( ( ( shd_list_list_fm @ Ws )
!= nil_list_fm )
=> ( ( flat_list_fm @ Ws )
= ( shift_list_fm @ ( shd_list_list_fm @ Ws ) @ ( flat_list_fm @ ( stl_list_list_fm @ Ws ) ) ) ) ) ).
% flat_unfold
thf(fact_1010_flat_Osimps_I1_J,axiom,
! [Ws: stream2471014364565126742m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( flat_P7721466590633226428m_rule @ Ws ) )
= ( hd_Pro7241777042969981963m_rule @ ( shd_li4676821617271663642m_rule @ Ws ) ) ) ).
% flat.simps(1)
thf(fact_1011_minf_I8_J,axiom,
! [T2: nat] :
? [Z6: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z6 )
=> ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% minf(8)
thf(fact_1012_minf_I6_J,axiom,
! [T2: nat] :
? [Z6: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z6 )
=> ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% minf(6)
thf(fact_1013_pinf_I8_J,axiom,
! [T2: nat] :
? [Z6: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z6 @ X4 )
=> ( ord_less_eq_nat @ T2 @ X4 ) ) ).
% pinf(8)
thf(fact_1014_pinf_I6_J,axiom,
! [T2: nat] :
? [Z6: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z6 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).
% pinf(6)
thf(fact_1015_psubsetD,axiom,
! [A2: set_fm,B: set_fm,C3: fm] :
( ( ord_less_set_fm @ A2 @ B )
=> ( ( member_fm2 @ C3 @ A2 )
=> ( member_fm2 @ C3 @ B ) ) ) ).
% psubsetD
thf(fact_1016_psubsetD,axiom,
! [A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule,C3: produc340336539035504054m_rule] :
( ( ord_le4093866177137961152m_rule @ A2 @ B )
=> ( ( member7231649785386036813m_rule @ C3 @ A2 )
=> ( member7231649785386036813m_rule @ C3 @ B ) ) ) ).
% psubsetD
thf(fact_1017_psubsetD,axiom,
! [A2: set_list_fm,B: set_list_fm,C3: list_fm] :
( ( ord_less_set_list_fm @ A2 @ B )
=> ( ( member_list_fm2 @ C3 @ A2 )
=> ( member_list_fm2 @ C3 @ B ) ) ) ).
% psubsetD
thf(fact_1018_psubsetD,axiom,
! [A2: set_tm,B: set_tm,C3: tm] :
( ( ord_less_set_tm @ A2 @ B )
=> ( ( member_tm2 @ C3 @ A2 )
=> ( member_tm2 @ C3 @ B ) ) ) ).
% psubsetD
thf(fact_1019_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C3: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ B ) ) ) ).
% psubsetD
thf(fact_1020_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1021_stake__cycle__le,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_tm @ U ) )
=> ( ( stake_tm @ N2 @ ( cycle_tm @ U ) )
= ( take_tm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_1022_stake__cycle__le,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_fm @ U ) )
=> ( ( stake_fm @ N2 @ ( cycle_fm @ U ) )
= ( take_fm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_1023_stake__cycle__le,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) )
=> ( ( stake_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( take_list_fm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_1024_listrel__iff__nth,axiom,
! [Xs: list_P2887561121880082555ist_fm,Ys: list_rule,R2: set_Pr1822751329126368876m_rule] :
( ( member2627171955076190819t_rule @ ( produc3723918873312807110t_rule @ Xs @ Ys ) @ ( listre749368180268468182m_rule @ R2 ) )
= ( ( ( size_s3138477486474831591ist_fm @ Xs )
= ( size_size_list_rule @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ ( nth_Pr580027083122244092ist_fm @ Xs @ N3 ) @ ( nth_rule @ Ys @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_1025_listrel__iff__nth,axiom,
! [Xs: list_list_tm,Ys: list_list_fm,R2: set_Pr5202636777678657877ist_fm] :
( ( member5109946355746398750ist_fm @ ( produc8328755919010440805ist_fm @ Xs @ Ys ) @ ( listre5805154493122130495ist_fm @ R2 ) )
= ( ( ( size_s9096087352182575069ist_tm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s9096087352182575069ist_tm @ Xs ) )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( nth_list_tm @ Xs @ N3 ) @ ( nth_list_fm @ Ys @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_1026_listrel__iff__nth,axiom,
! [Xs: list_tm,Ys: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ Ys ) @ ( listrel_tm_fm @ R2 ) )
= ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_tm @ Xs ) )
=> ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ ( nth_tm @ Xs @ N3 ) @ ( nth_fm @ Ys @ N3 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_1027_add__right__cancel,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C3 @ A ) )
= ( B2 = C3 ) ) ).
% add_right_cancel
thf(fact_1028_add__left__cancel,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C3 ) )
= ( B2 = C3 ) ) ).
% add_left_cancel
thf(fact_1029_add__le__cancel__right,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1030_add__le__cancel__left,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1031_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1032_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1033_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1034_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1035_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1036_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1037_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1038_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1039_add__less__cancel__right,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1040_add__less__cancel__left,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1041_add__diff__cancel__left,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1042_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1043_add__diff__cancel__right,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1044_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_1045_mod__add__self2,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self2
thf(fact_1046_mod__add__self1,axiom,
! [B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self1
thf(fact_1047_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1048_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1049_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1050_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1051_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1052_sdrop__add,axiom,
! [N2: nat,M: nat,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ N2 @ ( sdrop_8169176516188972301m_rule @ M @ S ) )
= ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ M @ N2 ) @ S ) ) ).
% sdrop_add
thf(fact_1053_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1054_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1055_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1056_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1057_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1058_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1059_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1060_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1061_le__add__diff__inverse,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1062_le__add__diff__inverse2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1063_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1064_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1065_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1066_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1067_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1068_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_tm] :
( ( nil_tm
= ( take_tm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_tm ) ) ) ).
% take_eq_Nil2
thf(fact_1069_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_fm] :
( ( nil_fm
= ( take_fm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_fm ) ) ) ).
% take_eq_Nil2
thf(fact_1070_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( nil_list_fm
= ( take_list_fm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_list_fm ) ) ) ).
% take_eq_Nil2
thf(fact_1071_take__eq__Nil,axiom,
! [N2: nat,Xs: list_tm] :
( ( ( take_tm @ N2 @ Xs )
= nil_tm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_tm ) ) ) ).
% take_eq_Nil
thf(fact_1072_take__eq__Nil,axiom,
! [N2: nat,Xs: list_fm] :
( ( ( take_fm @ N2 @ Xs )
= nil_fm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_fm ) ) ) ).
% take_eq_Nil
thf(fact_1073_take__eq__Nil,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ( take_list_fm @ N2 @ Xs )
= nil_list_fm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_list_fm ) ) ) ).
% take_eq_Nil
thf(fact_1074_take0,axiom,
( ( take_tm @ zero_zero_nat )
= ( ^ [Xs3: list_tm] : nil_tm ) ) ).
% take0
thf(fact_1075_take0,axiom,
( ( take_fm @ zero_zero_nat )
= ( ^ [Xs3: list_fm] : nil_fm ) ) ).
% take0
thf(fact_1076_take0,axiom,
( ( take_list_fm @ zero_zero_nat )
= ( ^ [Xs3: list_list_fm] : nil_list_fm ) ) ).
% take0
thf(fact_1077_add__less__imp__less__right,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1078_add__less__imp__less__left,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1079_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% add_strict_right_mono
thf(fact_1080_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1081_add__strict__mono,axiom,
! [A: nat,B2: nat,C3: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C3 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1082_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1083_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1084_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1085_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1086_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1087_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1088_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1089_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1090_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1091_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1092_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1093_take__Nil,axiom,
! [N2: nat] :
( ( take_tm @ N2 @ nil_tm )
= nil_tm ) ).
% take_Nil
thf(fact_1094_take__Nil,axiom,
! [N2: nat] :
( ( take_fm @ N2 @ nil_fm )
= nil_fm ) ).
% take_Nil
thf(fact_1095_take__Nil,axiom,
! [N2: nat] :
( ( take_list_fm @ N2 @ nil_list_fm )
= nil_list_fm ) ).
% take_Nil
thf(fact_1096_in__set__takeD,axiom,
! [X: produc340336539035504054m_rule,N2: nat,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ ( take_P3067526267515409992m_rule @ N2 @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1097_in__set__takeD,axiom,
! [X: nat,N2: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N2 @ Xs ) ) )
=> ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1098_in__set__takeD,axiom,
! [X: fm,N2: nat,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) )
=> ( member_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1099_in__set__takeD,axiom,
! [X: tm,N2: nat,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ ( take_tm @ N2 @ Xs ) ) )
=> ( member_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1100_in__set__takeD,axiom,
! [X: list_fm,N2: nat,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( take_list_fm @ N2 @ Xs ) ) )
=> ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1101_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C3 @ A ) )
=> ( B2 = C3 ) ) ).
% add_right_imp_eq
thf(fact_1102_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C3 ) )
=> ( B2 = C3 ) ) ).
% add_left_imp_eq
thf(fact_1103_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C3: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C3 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% add.left_commute
thf(fact_1104_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_1105_add_Oassoc,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% add.assoc
thf(fact_1106_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1107_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1108_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1109_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1110_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1111_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1112_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1113_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1114_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1115_mod__add__eq,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C3 ) @ ( modulo_modulo_nat @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 ) ) ).
% mod_add_eq
thf(fact_1116_mod__add__cong,axiom,
! [A: nat,C3: nat,A3: nat,B2: nat,B3: nat] :
( ( ( modulo_modulo_nat @ A @ C3 )
= ( modulo_modulo_nat @ A3 @ C3 ) )
=> ( ( ( modulo_modulo_nat @ B2 @ C3 )
= ( modulo_modulo_nat @ B3 @ C3 ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C3 ) ) ) ) ).
% mod_add_cong
thf(fact_1117_mod__add__left__eq,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C3 ) @ B2 ) @ C3 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 ) ) ).
% mod_add_left_eq
thf(fact_1118_mod__add__right__eq,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C3 ) ) ).
% mod_add_right_eq
thf(fact_1119_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1120_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1121_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1122_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_1123_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1124_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N: nat] :
( L2
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_1125_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1126_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1127_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1128_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1129_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1130_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1131_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1132_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( P2 @ A5 @ B6 )
= ( P2 @ B6 @ A5 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B6: nat] :
( ( P2 @ A5 @ B6 )
=> ( P2 @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
=> ( P2 @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1133_add__implies__diff,axiom,
! [C3: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C3 @ B2 )
= A )
=> ( C3
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1134_diff__diff__eq,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C3 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% diff_diff_eq
thf(fact_1135_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1136_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1137_add__le__imp__le__right,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1138_add__le__imp__le__left,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1139_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
? [C2: nat] :
( B5
= ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_1140_add__right__mono,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ C3 ) ) ) ).
% add_right_mono
thf(fact_1141_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_1142_add__left__mono,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B2 ) ) ) ).
% add_left_mono
thf(fact_1143_add__mono,axiom,
! [A: nat,B2: nat,C3: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C3 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1144_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1145_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1146_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1147_take__0,axiom,
! [Xs: list_tm] :
( ( take_tm @ zero_zero_nat @ Xs )
= nil_tm ) ).
% take_0
thf(fact_1148_take__0,axiom,
! [Xs: list_fm] :
( ( take_fm @ zero_zero_nat @ Xs )
= nil_fm ) ).
% take_0
thf(fact_1149_take__0,axiom,
! [Xs: list_list_fm] :
( ( take_list_fm @ zero_zero_nat @ Xs )
= nil_list_fm ) ).
% take_0
thf(fact_1150_set__take__subset,axiom,
! [N2: nat,Xs: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ ( take_list_fm @ N2 @ Xs ) ) @ ( set_list_fm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1151_set__take__subset,axiom,
! [N2: nat,Xs: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( take_tm @ N2 @ Xs ) ) @ ( set_tm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1152_set__take__subset,axiom,
! [N2: nat,Xs: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) @ ( set_fm2 @ Xs ) ) ).
% set_take_subset
thf(fact_1153_set__take__subset,axiom,
! [N2: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N2 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1154_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1155_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1156_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1157_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1158_add__increasing2,axiom,
! [C3: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C3 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_increasing2
thf(fact_1159_add__decreasing2,axiom,
! [C3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1160_add__increasing,axiom,
! [A: nat,B2: nat,C3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C3 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_increasing
thf(fact_1161_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1162_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1163_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1164_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1165_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1166_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1167_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1168_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1169_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1170_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1171_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B2: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1172_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B2: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1173_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1174_min__0L,axiom,
! [N2: nat] :
( ( ord_min_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% min_0L
thf(fact_1175_min__0R,axiom,
! [N2: nat] :
( ( ord_min_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1176_children_Osimps_I1_J,axiom,
! [Uu: list_tm,Uv: rule] :
( ( children @ Uu @ Uv @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% children.simps(1)
thf(fact_1177_min__diff,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).
% min_diff
thf(fact_1178_sub__list_Osimps_I2_J,axiom,
! [V2: nat,S: tm,T2: tm,L2: list_tm] :
( ( sub_list @ V2 @ S @ ( cons_tm @ T2 @ L2 ) )
= ( cons_tm @ ( sub_term @ V2 @ S @ T2 ) @ ( sub_list @ V2 @ S @ L2 ) ) ) ).
% sub_list.simps(2)
thf(fact_1179_liftts_Osimps_I2_J,axiom,
! [T2: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T2 @ Ts ) )
= ( cons_tm @ ( liftt @ T2 ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_1180_substts_Osimps_I2_J,axiom,
! [T2: tm,Ts: list_tm,S: tm,K: nat] :
( ( substts @ ( cons_tm @ T2 @ Ts ) @ S @ K )
= ( cons_tm @ ( substt @ T2 @ S @ K ) @ ( substts @ Ts @ S @ K ) ) ) ).
% substts.simps(2)
thf(fact_1181_inc__list_Osimps_I2_J,axiom,
! [T2: tm,L2: list_tm] :
( ( inc_list @ ( cons_tm @ T2 @ L2 ) )
= ( cons_tm @ ( inc_term @ T2 ) @ ( inc_list @ L2 ) ) ) ).
% inc_list.simps(2)
thf(fact_1182_new__list_Osimps_I2_J,axiom,
! [C3: nat,T2: tm,L2: list_tm] :
( ( new_list @ C3 @ ( cons_tm @ T2 @ L2 ) )
= ( ( ( new_term @ C3 @ T2 )
=> ( new_list @ C3 @ L2 ) )
& ( new_term @ C3 @ T2 ) ) ) ).
% new_list.simps(2)
thf(fact_1183_paramsts_Osimps_I2_J,axiom,
! [T2: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T2 @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T2 ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_1184_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1185_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1186_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1187_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1188_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1189_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_1190_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1191_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1192_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1193_min__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_1194_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1195_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1196_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1197_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1198_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1199_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1200_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1201_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_1202_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1203_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z6: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z6 )
=> ( R3 @ X3 @ Z6 ) ) )
=> ( ! [N: nat] : ( R3 @ N @ ( suc @ N ) )
=> ( R3 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1204_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P2 @ M )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1205_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_1206_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1207_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1208_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1209_Suc__le__D,axiom,
! [N2: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
=> ? [M4: nat] :
( M5
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1210_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1211_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1212_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1213_Ex__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1214_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1215_All__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1216_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1217_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1218_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1219_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1220_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_1221_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_1222_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1223_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1224_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1225_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1226_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1227_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1228_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1229_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1230_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1231_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1232_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1233_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1234_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1235_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1236_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1237_mod__Suc,axiom,
! [M: nat,N2: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
!= N2 )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% mod_Suc
thf(fact_1238_mod__induct,axiom,
! [P2: nat > $o,N2: nat,P: nat,M: nat] :
( ( P2 @ N2 )
=> ( ( ord_less_nat @ N2 @ P )
=> ( ( ord_less_nat @ M @ P )
=> ( ! [N: nat] :
( ( ord_less_nat @ N @ P )
=> ( ( P2 @ N )
=> ( P2 @ ( modulo_modulo_nat @ ( suc @ N ) @ P ) ) ) )
=> ( P2 @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1239_mod__Suc__le__divisor,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_1240_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1241_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1242_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I2 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1243_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1244_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1245_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1246_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1247_All__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1248_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1249_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1250_Ex__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1251_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1252_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1253_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1254_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1255_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1256_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1257_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1258_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1259_mod__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% mod_Suc_eq
thf(fact_1260_mod__Suc__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% mod_Suc_Suc_eq
thf(fact_1261_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1262_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1263_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1264_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: fset_P8989946509869081563ist_fm,Y: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_fm2 @ ( sub @ zero_zero_nat @ ( fun @ ( generateNew @ c ) @ nil_tm ) @ p ) @ ( tree_fms @ steps ) ).
%------------------------------------------------------------------------------