TPTP Problem File: SLH0148^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0017_Countermodel/prob_00216_007393__12973448_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1603 ( 607 unt; 325 typ; 0 def)
% Number of atoms : 3959 (1424 equ; 0 cnn)
% Maximal formula atoms : 42 ( 3 avg)
% Number of connectives : 13573 ( 606 ~; 158 |; 309 &;10625 @)
% ( 0 <=>;1875 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 57 ( 56 usr)
% Number of type conns : 1221 (1221 >; 0 *; 0 +; 0 <<)
% Number of symbols : 272 ( 269 usr; 13 con; 0-5 aty)
% Number of variables : 4526 ( 470 ^;3875 !; 181 ?;4526 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:04.384
%------------------------------------------------------------------------------
% Could-be-implicit typings (56)
thf(ty_n_t__Product____Type__Oprod_I_062_It__SeCaV__Otm_M_062_It__SeCaV__Otm_M_Eo_J_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc2002131169352006116ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc7963324949210141170ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J_J,type,
set_Pr1507011332596240839ist_tm: $tType ).
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__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Otm_J_J_J,type,
set_Pr7443884325468708281ist_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_Pr1916137733696349511ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr6479235367683667248st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr2131058311039847614st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Otm_J_J_J,type,
set_Pr3149961925030284086ist_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_Pr6845587370112701124ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc254973753779126261st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc5776448205642668775ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc6018962875968178549ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc3002719820330532825ist_tm: $tType ).
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__SeCaV__Otm_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc3714773664571398778st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1369765774831474952st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc106838685478687616ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc349353355804197390ist_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__List__Olist_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
list_P8031219080602320621_fm_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J_J,type,
set_Pr2455929065695642951_tm_tm: $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__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Otm_J_J,type,
set_Pr4464301228316855097_fm_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
set_Pr4706815898642364871_fm_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__Nat__Onat_J_J,type,
set_Pr1365117562694539290tm_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Nat__Onat_J_J,type,
set_Pr6019225798553204136fm_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Otm_J_J,type,
set_Pr4584624442413714624nat_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Ofm_J_J,type,
set_Pr4827139112739224398nat_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J,type,
product_prod_tm_tm: $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__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Otm_J,type,
product_prod_fm_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
product_prod_fm_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Otm_Mt__Nat__Onat_J,type,
product_prod_tm_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Nat__Onat_J,type,
product_prod_fm_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Otm_J,type,
product_prod_nat_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Ofm_J,type,
product_prod_nat_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__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__List__Olist_It__Nat__Onat_J,type,
list_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__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 (269)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hintikka_OHintikka,type,
hintikka: set_fm > $o ).
thf(sy_c_Hintikka_Oterms,type,
terms: set_fm > set_tm ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
if_lis6935098856802376371_fm_fm: $o > list_P8031219080602320621_fm_fm > list_P8031219080602320621_fm_fm > list_P8031219080602320621_fm_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Ofm_J,type,
if_list_fm: $o > list_fm > list_fm > list_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Otm_J,type,
if_list_tm: $o > list_tm > list_tm > list_tm ).
thf(sy_c_If_001t__SeCaV__Otm,type,
if_tm: $o > tm > tm > tm ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_M_Eo_J,type,
sup_su6818665535280656642m_fm_o: ( product_prod_fm_fm > $o ) > ( product_prod_fm_fm > $o ) > product_prod_fm_fm > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J,type,
sup_sup_fm_fm_o: ( fm > fm > $o ) > ( fm > fm > $o ) > fm > fm > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__SeCaV__Ofm_M_Eo_J,type,
sup_sup_fm_o: ( fm > $o ) > ( fm > $o ) > fm > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__SeCaV__Otm_M_Eo_J,type,
sup_sup_tm_o: ( tm > $o ) > ( tm > $o ) > tm > $o ).
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__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__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
sup_su5810838807072965531_fm_fm: set_Pr4706815898642364871_fm_fm > set_Pr4706815898642364871_fm_fm > set_Pr4706815898642364871_fm_fm ).
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__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
can_se8056291635612950372_fm_fm: ( product_prod_fm_fm > $o ) > set_Pr4706815898642364871_fm_fm > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
can_select_fm: ( fm > $o ) > set_fm > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
can_select_tm: ( tm > $o ) > set_tm > $o ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
coset_5303328301088503642_fm_fm: list_P8031219080602320621_fm_fm > set_Pr4706815898642364871_fm_fm ).
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__Nat__Onat,type,
gen_length_nat: nat > list_nat > 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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
insert7339457527920503555_fm_fm: product_prod_fm_fm > list_P8031219080602320621_fm_fm > list_P8031219080602320621_fm_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001t__SeCaV__Ofm,type,
lenlex_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olenlex_001t__SeCaV__Otm,type,
lenlex_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__SeCaV__Ofm,type,
lex_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olex_001t__SeCaV__Otm,type,
lex_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_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__List__Olist_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
cons_P2476253307934258077_fm_fm: product_prod_fm_fm > list_P8031219080602320621_fm_fm > list_P8031219080602320621_fm_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
list_a808454179579425435_fm_fm: ( product_prod_fm_fm > $o ) > list_P8031219080602320621_fm_fm > $o ).
thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Ofm,type,
list_all_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Otm,type,
list_all_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Otm_J,type,
set_list_tm2: list_list_tm > set_list_tm ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
set_Pr5149718152543245948_fm_fm: list_P8031219080602320621_fm_fm > set_Pr4706815898642364871_fm_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
set_tm2: list_tm > set_tm ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
list_e1430114047001372208_fm_fm: ( product_prod_fm_fm > $o ) > list_P8031219080602320621_fm_fm > $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__Nat__Onat_001t__Nat__Onat,type,
listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__SeCaV__Ofm,type,
listrel_nat_fm: set_Pr4827139112739224398nat_fm > set_Pr6845587370112701124ist_fm ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__SeCaV__Otm,type,
listrel_nat_tm: set_Pr4584624442413714624nat_tm > set_Pr3149961925030284086ist_tm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__Nat__Onat,type,
listrel_fm_nat: set_Pr6019225798553204136fm_nat > set_Pr2131058311039847614st_nat ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
listrel_fm_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
listrel_fm_tm: set_Pr4464301228316855097_fm_tm > set_Pr7443884325468708281ist_tm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Otm_001t__Nat__Onat,type,
listrel_tm_nat: set_Pr1365117562694539290tm_nat > set_Pr6479235367683667248st_nat ).
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_Olistrel_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
listrel_tm_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Nat__Onat,type,
listrelp_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__SeCaV__Ofm,type,
listrelp_nat_fm: ( nat > fm > $o ) > list_nat > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__SeCaV__Otm,type,
listrelp_nat_tm: ( nat > tm > $o ) > list_nat > list_tm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__Nat__Onat,type,
listrelp_fm_nat: ( fm > nat > $o ) > list_fm > list_nat > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
listrelp_fm_fm: ( fm > fm > $o ) > list_fm > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
listrelp_fm_tm: ( fm > tm > $o ) > list_fm > list_tm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__Nat__Onat,type,
listrelp_tm_nat: ( tm > nat > $o ) > list_tm > list_nat > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
listrelp_tm_fm: ( tm > fm > $o ) > list_tm > list_fm > $o ).
thf(sy_c_List_Olistrelp_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
listrelp_tm_tm: ( tm > tm > $o ) > list_tm > list_tm > $o ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
nth_Pr5768189175911290222_fm_fm: list_P8031219080602320621_fm_fm > nat > product_prod_fm_fm ).
thf(sy_c_List_Onth_001t__SeCaV__Ofm,type,
nth_fm: list_fm > nat > fm ).
thf(sy_c_List_Onth_001t__SeCaV__Otm,type,
nth_tm: list_tm > nat > tm ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
nths_P3473792751519839979_fm_fm: list_P8031219080602320621_fm_fm > set_nat > list_P8031219080602320621_fm_fm ).
thf(sy_c_List_Onths_001t__SeCaV__Ofm,type,
nths_fm: list_fm > set_nat > list_fm ).
thf(sy_c_List_Onths_001t__SeCaV__Otm,type,
nths_tm: list_tm > set_nat > list_tm ).
thf(sy_c_List_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_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Ofm,type,
product_lists_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__SeCaV__Ofm,type,
union_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Ounion_001t__SeCaV__Otm,type,
union_tm: list_tm > list_tm > list_tm ).
thf(sy_c_Nat_OSuc,type,
suc: nat > 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__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
size_s3074140853920721241_fm_fm: list_P8031219080602320621_fm_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Otm_J,type,
size_size_list_tm: list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_M_Eo_J,type,
ord_le1062710942104439338m_fm_o: ( product_prod_fm_fm > $o ) > ( product_prod_fm_fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__SeCaV__Ofm_M_Eo_J,type,
ord_less_fm_o: ( fm > $o ) > ( fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__SeCaV__Otm_M_Eo_J,type,
ord_less_tm_o: ( tm > $o ) > ( tm > $o ) > $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__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__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
ord_le5731113155793438835_fm_fm: set_Pr4706815898642364871_fm_fm > set_Pr4706815898642364871_fm_fm > $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_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J,type,
ord_le3799113821011214030t_fm_o: ( list_fm > list_fm > $o ) > ( list_fm > list_fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
ord_le6518561683347902116t_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__SeCaV__Otm_J_M_Eo_J,type,
ord_le2468657205176945586t_tm_o: ( list_tm > $o ) > ( list_tm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_M_Eo_J,type,
ord_le2785158094463560246m_fm_o: ( product_prod_fm_fm > $o ) > ( product_prod_fm_fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J,type,
ord_less_eq_fm_fm_o: ( fm > fm > $o ) > ( fm > fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Ofm_M_Eo_J,type,
ord_less_eq_fm_o: ( fm > $o ) > ( fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Otm_M_Eo_J,type,
ord_less_eq_tm_o: ( tm > $o ) > ( tm > $o ) > $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__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__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
ord_le966076687461445991_fm_fm: set_Pr4706815898642364871_fm_fm > set_Pr4706815898642364871_fm_fm > $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_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc7687578365188660450ist_fm: ( fm > fm > $o ) > list_fm > produc7963324949210141170ist_fm ).
thf(sy_c_Product__Type_OPair_001_062_It__SeCaV__Otm_M_062_It__SeCaV__Otm_M_Eo_J_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc1972851280741670356ist_tm: ( tm > tm > $o ) > list_tm > produc2002131169352006116ist_tm ).
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__Nat__Onat_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc7721762080035590080ist_fm: list_nat > list_fm > produc349353355804197390ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc7725714105121225266ist_tm: list_nat > list_tm > produc106838685478687616ist_tm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__Nat__Onat_J,type,
produc8654734596316469506st_nat: list_fm > list_nat > produc1369765774831474952st_nat ).
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__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc7867948443067789129ist_tm: list_fm > list_tm > produc3002719820330532825ist_tm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__Nat__Onat_J,type,
produc1776370449201617524st_nat: list_tm > list_nat > produc3714773664571398778st_nat ).
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__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__SeCaV__Ofm,type,
product_Pair_nat_fm: nat > fm > product_prod_nat_fm ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__SeCaV__Otm,type,
product_Pair_nat_tm: nat > tm > product_prod_nat_tm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Ofm_001t__Nat__Onat,type,
product_Pair_fm_nat: fm > nat > product_prod_fm_nat ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
product_Pair_fm_fm: fm > fm > product_prod_fm_fm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
product_Pair_fm_tm: fm > tm > product_prod_fm_tm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Otm_001t__Nat__Onat,type,
product_Pair_tm_nat: tm > nat > product_prod_tm_nat ).
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_Product__Type_OPair_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
product_Pair_tm_tm: tm > tm > product_prod_tm_tm ).
thf(sy_c_Prover_ObranchDone,type,
branchDone: list_fm > $o ).
thf(sy_c_Prover_ObranchDone__rel,type,
branchDone_rel: list_fm > list_fm > $o ).
thf(sy_c_Prover_OlistFunTm,type,
listFunTm: tm > list_nat ).
thf(sy_c_Prover_OlistFunTms,type,
listFunTms: list_tm > list_nat ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_SeCaV_Oext_001t__Nat__Onat,type,
ext_nat: list_nat > list_nat > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Ofm_OCon,type,
con: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ODis,type,
dis: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_OExi,type,
exi: fm > fm ).
thf(sy_c_SeCaV_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ONeg,type,
neg: fm > fm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_SeCaV_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__Nat__Onat,type,
member_nat: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
member8099233463853703858_fm_fm: product_prod_fm_fm > list_P8031219080602320621_fm_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
member_fm: fm > list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
member_tm: tm > list_tm > $o ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Onews,type,
news: nat > list_fm > $o ).
thf(sy_c_SeCaV_Oparams,type,
params: fm > set_nat ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H,type,
paramst2: tm > set_nat ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osemantics_001t__SeCaV__Otm,type,
semantics_tm: ( nat > tm ) > ( nat > list_tm > tm ) > ( nat > list_tm > $o ) > fm > $o ).
thf(sy_c_SeCaV_Osemantics__list_001t__Nat__Onat,type,
semantics_list_nat: ( nat > nat ) > ( nat > list_nat > nat ) > list_tm > list_nat ).
thf(sy_c_SeCaV_Osemantics__list_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
semant4740343338788880149_fm_fm: ( nat > product_prod_fm_fm ) > ( nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm ) > list_tm > list_P8031219080602320621_fm_fm ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Ofm,type,
semantics_list_fm: ( nat > fm ) > ( nat > list_fm > fm ) > list_tm > list_fm ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Otm,type,
semantics_list_tm: ( nat > tm ) > ( nat > list_tm > tm ) > list_tm > list_tm ).
thf(sy_c_SeCaV_Osemantics__term_001t__Nat__Onat,type,
semantics_term_nat: ( nat > nat ) > ( nat > list_nat > nat ) > tm > nat ).
thf(sy_c_SeCaV_Osemantics__term_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
semant8857523558198744295_fm_fm: ( nat > product_prod_fm_fm ) > ( nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm ) > tm > product_prod_fm_fm ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Ofm,type,
semantics_term_fm: ( nat > fm ) > ( nat > list_fm > fm ) > tm > fm ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Otm,type,
semantics_term_tm: ( nat > tm ) > ( nat > list_tm > tm ) > tm > tm ).
thf(sy_c_SeCaV_Osequent__calculus,type,
sequent_calculus: list_fm > $o ).
thf(sy_c_SeCaV_Oshift_001t__Nat__Onat_001t__SeCaV__Otm,type,
shift_nat_tm: ( nat > tm ) > nat > tm > nat > tm ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Osubst,type,
subst: fm > tm > nat > fm ).
thf(sy_c_SeCaV_Osubstt,type,
substt: tm > tm > nat > tm ).
thf(sy_c_SeCaV_Osubstts,type,
substts: list_tm > tm > nat > list_tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_SeCaV_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_SeCaV_Otm_Osize__tm,type,
size_tm: tm > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
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__List__Olist_It__SeCaV__Otm_J,type,
collect_list_tm: ( list_tm > $o ) > set_list_tm ).
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__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
collec7637684051871000146_fm_fm: ( product_prod_fm_fm > $o ) > set_Pr4706815898642364871_fm_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
collect_tm: ( tm > $o ) > set_tm ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
remove1805396771031664588_fm_fm: product_prod_fm_fm > set_Pr4706815898642364871_fm_fm > set_Pr4706815898642364871_fm_fm ).
thf(sy_c_Set_Oremove_001t__SeCaV__Ofm,type,
remove_fm: fm > set_fm > set_fm ).
thf(sy_c_Set_Oremove_001t__SeCaV__Otm,type,
remove_tm: tm > set_tm > set_tm ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Ofm,type,
the_elem_fm: set_fm > fm ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Otm,type,
the_elem_tm: set_tm > tm ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Usemantics_Ois__env_001t__Nat__Onat,type,
is_env_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Usemantics_Ois__env_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
is_env1422318314337916935_fm_fm: set_Pr4706815898642364871_fm_fm > ( nat > product_prod_fm_fm ) > $o ).
thf(sy_c_Usemantics_Ois__env_001t__SeCaV__Ofm,type,
is_env_fm: set_fm > ( nat > fm ) > $o ).
thf(sy_c_Usemantics_Ois__env_001t__SeCaV__Otm,type,
is_env_tm: set_tm > ( nat > tm ) > $o ).
thf(sy_c_Usemantics_Ois__fdenot_001t__Nat__Onat,type,
is_fdenot_nat: set_nat > ( nat > list_nat > nat ) > $o ).
thf(sy_c_Usemantics_Ois__fdenot_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
is_fde8755990660405543756_fm_fm: set_Pr4706815898642364871_fm_fm > ( nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm ) > $o ).
thf(sy_c_Usemantics_Ois__fdenot_001t__SeCaV__Ofm,type,
is_fdenot_fm: set_fm > ( nat > list_fm > fm ) > $o ).
thf(sy_c_Usemantics_Ois__fdenot_001t__SeCaV__Otm,type,
is_fdenot_tm: set_tm > ( nat > list_tm > tm ) > $o ).
thf(sy_c_Usemantics_Ousemantics_001t__SeCaV__Otm,type,
usemantics_tm: set_tm > ( nat > tm ) > ( nat > list_tm > tm ) > ( nat > list_tm > $o ) > fm > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__SeCaV__Ofm_J,type,
accp_list_fm: ( list_fm > list_fm > $o ) > list_fm > $o ).
thf(sy_c_Wellfounded_Omeasure_001t__SeCaV__Ofm,type,
measure_fm: ( fm > nat ) > set_Pr4706815898642364871_fm_fm ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
fChoic1084495941463685435_fm_fm: ( product_prod_fm_fm > $o ) > product_prod_fm_fm ).
thf(sy_c_fChoice_001t__SeCaV__Ofm,type,
fChoice_fm: ( fm > $o ) > fm ).
thf(sy_c_fChoice_001t__SeCaV__Otm,type,
fChoice_tm: ( tm > $o ) > tm ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member848049699933468325ist_fm: produc349353355804197390ist_fm > set_Pr6845587370112701124ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member605535029607958551ist_tm: produc106838685478687616ist_tm > set_Pr3149961925030284086ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member1868462118960745887st_nat: produc1369765774831474952st_nat > set_Pr2131058311039847614st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member1926098302810316688ist_fm: produc3245234490656042599ist_fm > set_Pr1916137733696349511ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member1683583632484806914ist_tm: produc3002719820330532825ist_tm > set_Pr7443884325468708281ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member4213470008700669713st_nat: produc3714773664571398778st_nat > set_Pr6479235367683667248st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member4457312017796942864ist_tm: produc5776448205642668775ist_tm > set_Pr1507011332596240839ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Ofm_J,type,
member2965011204816613423nat_fm: product_prod_nat_fm > set_Pr4827139112739224398nat_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__SeCaV__Otm_J,type,
member2968963229902248609nat_tm: product_prod_nat_tm > set_Pr4584624442413714624nat_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Nat__Onat_J,type,
member1425945901753563017fm_nat: product_prod_fm_nat > set_Pr6019225798553204136fm_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
member7780952600467998736_fm_fm: product_prod_fm_fm > set_Pr4706815898642364871_fm_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Otm_J,type,
member7784904625553633922_fm_tm: product_prod_fm_tm > set_Pr4464301228316855097_fm_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__Nat__Onat_J,type,
member3090325158824063739tm_nat: product_prod_tm_nat > set_Pr1365117562694539290tm_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Ofm_J,type,
member3117664881408846110_tm_fm: product_prod_tm_fm > set_Pr2698443736021152725_tm_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J,type,
member3121616906494481296_tm_tm: product_prod_tm_tm > set_Pr2455929065695642951_tm_tm > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm2: tm > set_tm > $o ).
thf(sy_v_S,type,
s: set_fm ).
thf(sy_v_pa____,type,
pa: fm ).
thf(sy_v_t____,type,
t: tm ).
thf(sy_v_x____,type,
x: fm ).
% Relevant facts (1262)
thf(fact_0_terms__downwards__closed,axiom,
! [T: tm,S: set_fm] :
( ( member_tm2 @ T @ ( terms @ S ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( terms @ S ) ) ) ).
% terms_downwards_closed
thf(fact_1__092_060open_062t_A_092_060in_062_Aterms_AS_092_060close_062,axiom,
member_tm2 @ t @ ( terms @ s ) ).
% \<open>t \<in> terms S\<close>
thf(fact_2_assms,axiom,
hintikka @ s ).
% assms
thf(fact_3_usemantics__E_I1_J,axiom,
! [T: tm,S: set_fm] :
( ( member_tm2 @ T @ ( terms @ S ) )
=> ( ( semantics_term_tm
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ S ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ S ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) )
@ T )
= T ) ) ).
% usemantics_E(1)
thf(fact_4_some__equality,axiom,
! [P: tm > $o,A: tm] :
( ( P @ A )
=> ( ! [X: tm] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_tm @ P )
= A ) ) ) ).
% some_equality
thf(fact_5_some__eq__trivial,axiom,
! [X2: tm] :
( ( fChoice_tm
@ ^ [Y: tm] : ( Y = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_6_some__sym__eq__trivial,axiom,
! [X2: tm] :
( ( fChoice_tm
@ ( ^ [Y2: tm,Z: tm] : ( Y2 = Z )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_7_tm_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( var @ X22 )
= ( var @ Y22 ) )
= ( X22 = Y22 ) ) ).
% tm.inject(2)
thf(fact_8_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_9_is__env__E,axiom,
! [S: set_fm] :
( is_env_tm @ ( terms @ S )
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ S ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) ) ) ).
% is_env_E
thf(fact_10_semantics__term_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,N2: nat] :
( ( semantics_term_tm @ E @ F @ ( var @ N2 ) )
= ( E @ N2 ) ) ).
% semantics_term.simps(1)
thf(fact_11_is__fdenot__F,axiom,
! [S: set_fm] :
( is_fdenot_tm @ ( terms @ S )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ S ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) ) ) ).
% is_fdenot_F
thf(fact_12_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X22: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X22 ) ) ).
% tm.distinct(1)
thf(fact_13_tm_Oexhaust,axiom,
! [Y3: tm] :
( ! [X112: nat,X122: list_tm] :
( Y3
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X23: nat] :
( Y3
!= ( var @ X23 ) ) ) ).
% tm.exhaust
thf(fact_14_paramst_H_H_Ocases,axiom,
! [X2: tm] :
( ! [N3: nat] :
( X2
!= ( var @ N3 ) )
=> ~ ! [A2: nat,Ts: list_tm] :
( X2
!= ( fun @ A2 @ Ts ) ) ) ).
% paramst''.cases
thf(fact_15__092_060open_062Neg_Ax_A_092_060in_062_AS_092_060close_062,axiom,
member_fm2 @ ( neg @ x ) @ s ).
% \<open>Neg x \<in> S\<close>
thf(fact_16_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_17_some__eq__imp,axiom,
! [P: tm > $o,A: tm,B: tm] :
( ( ( fChoice_tm @ P )
= A )
=> ( ( P @ B )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_18_tfl__some,axiom,
! [P2: tm > $o,X3: tm] :
( ( P2 @ X3 )
=> ( P2 @ ( fChoice_tm @ P2 ) ) ) ).
% tfl_some
thf(fact_19_Eps__cong,axiom,
! [P: tm > $o,Q: tm > $o] :
( ! [X: tm] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_tm @ P )
= ( fChoice_tm @ Q ) ) ) ).
% Eps_cong
thf(fact_20_someI,axiom,
! [P: tm > $o,X2: tm] :
( ( P @ X2 )
=> ( P @ ( fChoice_tm @ P ) ) ) ).
% someI
thf(fact_21_some1__equality,axiom,
! [P: tm > $o,A: tm] :
( ? [X3: tm] :
( ( P @ X3 )
& ! [Y4: tm] :
( ( P @ Y4 )
=> ( Y4 = X3 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_tm @ P )
= A ) ) ) ).
% some1_equality
thf(fact_22_some__eq__ex,axiom,
! [P: tm > $o] :
( ( P @ ( fChoice_tm @ P ) )
= ( ? [X4: tm] : ( P @ X4 ) ) ) ).
% some_eq_ex
thf(fact_23_someI2__bex,axiom,
! [A3: set_fm,P: fm > $o,Q: fm > $o] :
( ? [X3: fm] :
( ( member_fm2 @ X3 @ A3 )
& ( P @ X3 ) )
=> ( ! [X: fm] :
( ( ( member_fm2 @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_24_someI2__bex,axiom,
! [A3: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( P @ X3 ) )
=> ( ! [X: nat] :
( ( ( member_nat2 @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_25_someI2__bex,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,P: product_prod_fm_fm > $o,Q: product_prod_fm_fm > $o] :
( ? [X3: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X3 @ A3 )
& ( P @ X3 ) )
=> ( ! [X: product_prod_fm_fm] :
( ( ( member7780952600467998736_fm_fm @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoic1084495941463685435_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_26_someI2__bex,axiom,
! [A3: set_tm,P: tm > $o,Q: tm > $o] :
( ? [X3: tm] :
( ( member_tm2 @ X3 @ A3 )
& ( P @ X3 ) )
=> ( ! [X: tm] :
( ( ( member_tm2 @ X @ A3 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_27_someI2__ex,axiom,
! [P: tm > $o,Q: tm > $o] :
( ? [X_1: tm] : ( P @ X_1 )
=> ( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_tm @ P ) ) ) ) ).
% someI2_ex
thf(fact_28_someI__ex,axiom,
! [P: tm > $o] :
( ? [X_1: tm] : ( P @ X_1 )
=> ( P @ ( fChoice_tm @ P ) ) ) ).
% someI_ex
thf(fact_29_someI2,axiom,
! [P: tm > $o,A: tm,Q: tm > $o] :
( ( P @ A )
=> ( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_tm @ P ) ) ) ) ).
% someI2
thf(fact_30_subterm__Fun__refl,axiom,
! [Ts2: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts2 ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N2 @ Ts2 ) ) ) ) ).
% subterm_Fun_refl
thf(fact_31_usemantics__term_I1_J,axiom,
! [U: set_fm,E: nat > fm,F: nat > list_fm > fm,T: tm] :
( ( is_env_fm @ U @ E )
=> ( ( is_fdenot_fm @ U @ F )
=> ( member_fm2 @ ( semantics_term_fm @ E @ F @ T ) @ U ) ) ) ).
% usemantics_term(1)
thf(fact_32_usemantics__term_I1_J,axiom,
! [U: set_nat,E: nat > nat,F: nat > list_nat > nat,T: tm] :
( ( is_env_nat @ U @ E )
=> ( ( is_fdenot_nat @ U @ F )
=> ( member_nat2 @ ( semantics_term_nat @ E @ F @ T ) @ U ) ) ) ).
% usemantics_term(1)
thf(fact_33_usemantics__term_I1_J,axiom,
! [U: set_Pr4706815898642364871_fm_fm,E: nat > product_prod_fm_fm,F: nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm,T: tm] :
( ( is_env1422318314337916935_fm_fm @ U @ E )
=> ( ( is_fde8755990660405543756_fm_fm @ U @ F )
=> ( member7780952600467998736_fm_fm @ ( semant8857523558198744295_fm_fm @ E @ F @ T ) @ U ) ) ) ).
% usemantics_term(1)
thf(fact_34_usemantics__term_I1_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,T: tm] :
( ( is_env_tm @ U @ E )
=> ( ( is_fdenot_tm @ U @ F )
=> ( member_tm2 @ ( semantics_term_tm @ E @ F @ T ) @ U ) ) ) ).
% usemantics_term(1)
thf(fact_35_subtermTm__le,axiom,
! [T: tm,S2: tm] :
( ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ S2 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S2 ) ) ) ) ).
% subtermTm_le
thf(fact_36_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y: list_fm,Z2: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z2 ) @ ( set_fm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_37_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y: list_tm,Z2: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z2 ) @ ( set_tm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_38_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y: list_nat,Z2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z2 ) @ ( set_nat2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_39_subtermTm__refl,axiom,
! [T: tm] : ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_40_subsetI,axiom,
! [A3: set_fm,B2: set_fm] :
( ! [X: fm] :
( ( member_fm2 @ X @ A3 )
=> ( member_fm2 @ X @ B2 ) )
=> ( ord_less_eq_set_fm @ A3 @ B2 ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm] :
( ! [X: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X @ A3 )
=> ( member7780952600467998736_fm_fm @ X @ B2 ) )
=> ( ord_le966076687461445991_fm_fm @ A3 @ B2 ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A3: set_tm,B2: set_tm] :
( ! [X: tm] :
( ( member_tm2 @ X @ A3 )
=> ( member_tm2 @ X @ B2 ) )
=> ( ord_less_eq_set_tm @ A3 @ B2 ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A3: set_nat,B2: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( member_nat2 @ X @ B2 ) )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% subsetI
thf(fact_44_subset__antisym,axiom,
! [A3: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_45_subset__antisym,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_46_Hintikka_ONeg,axiom,
! [H: set_fm,P3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( neg @ P3 ) ) @ H )
=> ( member_fm2 @ P3 @ H ) ) ) ).
% Hintikka.Neg
thf(fact_47_order__refl,axiom,
! [X2: set_tm] : ( ord_less_eq_set_tm @ X2 @ X2 ) ).
% order_refl
thf(fact_48_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_49_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_50_dual__order_Orefl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% dual_order.refl
thf(fact_51_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_52_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_53_Exi,axiom,
( x
= ( exi @ pa ) ) ).
% Exi
thf(fact_54_subset__code_I1_J,axiom,
! [Xs: list_P8031219080602320621_fm_fm,B2: set_Pr4706815898642364871_fm_fm] :
( ( ord_le966076687461445991_fm_fm @ ( set_Pr5149718152543245948_fm_fm @ Xs ) @ B2 )
= ( ! [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( member7780952600467998736_fm_fm @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_55_subset__code_I1_J,axiom,
! [Xs: list_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B2 )
= ( ! [X5: fm] :
( ( member_fm2 @ X5 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_56_subset__code_I1_J,axiom,
! [Xs: list_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B2 )
= ( ! [X5: tm] :
( ( member_tm2 @ X5 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_57_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_58_fm_Oinject_I5_J,axiom,
! [X52: fm,Y5: fm] :
( ( ( exi @ X52 )
= ( exi @ Y5 ) )
= ( X52 = Y5 ) ) ).
% fm.inject(5)
thf(fact_59_less__eq__set__def,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
( ord_less_eq_fm_o
@ ^ [X5: fm] : ( member_fm2 @ X5 @ A4 )
@ ^ [X5: fm] : ( member_fm2 @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_60_less__eq__set__def,axiom,
( ord_le966076687461445991_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
( ord_le2785158094463560246m_fm_o
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ A4 )
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_61_less__eq__set__def,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ord_less_eq_tm_o
@ ^ [X5: tm] : ( member_tm2 @ X5 @ A4 )
@ ^ [X5: tm] : ( member_tm2 @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_62_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat2 @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat2 @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_63_fm_Odistinct_I39_J,axiom,
! [X52: fm,X7: fm] :
( ( exi @ X52 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_64_order__antisym__conv,axiom,
! [Y3: set_tm,X2: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X2 )
=> ( ( ord_less_eq_set_tm @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_65_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_66_order__antisym__conv,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_67_mem__Collect__eq,axiom,
! [A: tm,P: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: fm,P: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: product_prod_fm_fm,P: product_prod_fm_fm > $o] :
( ( member7780952600467998736_fm_fm @ A @ ( collec7637684051871000146_fm_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A3: set_tm] :
( ( collect_tm
@ ^ [X5: tm] : ( member_tm2 @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A3: set_fm] :
( ( collect_fm
@ ^ [X5: fm] : ( member_fm2 @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat2 @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A3: set_Pr4706815898642364871_fm_fm] :
( ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_75_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_76_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_80_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_tm > set_nat,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_nat > set_tm,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_95_order__eq__refl,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_tm @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_96_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_97_order__eq__refl,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_98_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A: set_tm,F: set_nat > set_tm,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A: set_nat,F: set_tm > set_nat,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A5: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B4 )
& ( ord_less_eq_set_tm @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_antisym,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_120_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_121_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_122_dual__order_Otrans,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ C @ B )
=> ( ord_less_eq_set_tm @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_123_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_124_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_125_dual__order_Oantisym,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_126_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_127_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_128_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A5: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ A5 )
& ( ord_less_eq_set_tm @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_129_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_130_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B5: nat] :
( ( ord_less_eq_nat @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: nat,B5: nat] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_132_order__trans,axiom,
! [X2: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ Z3 )
=> ( ord_less_eq_set_tm @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_133_order__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_135_order_Otrans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% order.trans
thf(fact_136_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_137_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_138_order__antisym,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_139_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_140_order__antisym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_141_ord__le__eq__trans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_142_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_143_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_144_ord__eq__le__trans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( A = B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_145_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_146_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_147_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [X5: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X5 @ Y )
& ( ord_less_eq_set_tm @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_148_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X5: nat,Y: nat] :
( ( ord_less_eq_nat @ X5 @ Y )
& ( ord_less_eq_nat @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_149_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X5: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y )
& ( ord_less_eq_set_nat @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_150_le__cases3,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_151_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_152_Collect__mono__iff,axiom,
! [P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) )
= ( ! [X5: tm] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_153_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_154_set__eq__subset,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B3 )
& ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_155_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_156_subset__trans,axiom,
! [A3: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ord_less_eq_set_tm @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_157_subset__trans,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_158_Collect__mono,axiom,
! [P: tm > $o,Q: tm > $o] :
( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) ) ) ).
% Collect_mono
thf(fact_159_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_160_subset__refl,axiom,
! [A3: set_tm] : ( ord_less_eq_set_tm @ A3 @ A3 ) ).
% subset_refl
thf(fact_161_subset__refl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_162_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
! [T2: fm] :
( ( member_fm2 @ T2 @ A4 )
=> ( member_fm2 @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_163_subset__iff,axiom,
( ord_le966076687461445991_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
! [T2: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ T2 @ A4 )
=> ( member7780952600467998736_fm_fm @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_164_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
! [T2: tm] :
( ( member_tm2 @ T2 @ A4 )
=> ( member_tm2 @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_165_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A4 )
=> ( member_nat2 @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_166_equalityD2,axiom,
! [A3: set_tm,B2: set_tm] :
( ( A3 = B2 )
=> ( ord_less_eq_set_tm @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_167_equalityD2,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_168_equalityD1,axiom,
! [A3: set_tm,B2: set_tm] :
( ( A3 = B2 )
=> ( ord_less_eq_set_tm @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_169_equalityD1,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3 = B2 )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_170_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
! [X5: fm] :
( ( member_fm2 @ X5 @ A4 )
=> ( member_fm2 @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_171_subset__eq,axiom,
( ord_le966076687461445991_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
! [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A4 )
=> ( member7780952600467998736_fm_fm @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_172_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
! [X5: tm] :
( ( member_tm2 @ X5 @ A4 )
=> ( member_tm2 @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_173_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [X5: nat] :
( ( member_nat2 @ X5 @ A4 )
=> ( member_nat2 @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_174_equalityE,axiom,
! [A3: set_tm,B2: set_tm] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ~ ( ord_less_eq_set_tm @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_175_equalityE,axiom,
! [A3: set_nat,B2: set_nat] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_176_subsetD,axiom,
! [A3: set_fm,B2: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A3 @ B2 )
=> ( ( member_fm2 @ C @ A3 )
=> ( member_fm2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_177_subsetD,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm,C: product_prod_fm_fm] :
( ( ord_le966076687461445991_fm_fm @ A3 @ B2 )
=> ( ( member7780952600467998736_fm_fm @ C @ A3 )
=> ( member7780952600467998736_fm_fm @ C @ B2 ) ) ) ).
% subsetD
thf(fact_178_subsetD,axiom,
! [A3: set_tm,B2: set_tm,C: tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( member_tm2 @ C @ A3 )
=> ( member_tm2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_179_subsetD,axiom,
! [A3: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_nat2 @ C @ A3 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_180_in__mono,axiom,
! [A3: set_fm,B2: set_fm,X2: fm] :
( ( ord_less_eq_set_fm @ A3 @ B2 )
=> ( ( member_fm2 @ X2 @ A3 )
=> ( member_fm2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_181_in__mono,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm,X2: product_prod_fm_fm] :
( ( ord_le966076687461445991_fm_fm @ A3 @ B2 )
=> ( ( member7780952600467998736_fm_fm @ X2 @ A3 )
=> ( member7780952600467998736_fm_fm @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_182_in__mono,axiom,
! [A3: set_tm,B2: set_tm,X2: tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( member_tm2 @ X2 @ A3 )
=> ( member_tm2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_183_in__mono,axiom,
! [A3: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_nat2 @ X2 @ A3 )
=> ( member_nat2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_184_Collect__subset,axiom,
! [A3: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ A3 )
& ( P @ X5 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_185_Collect__subset,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,P: product_prod_fm_fm > $o] :
( ord_le966076687461445991_fm_fm
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A3 )
& ( P @ X5 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_186_Collect__subset,axiom,
! [A3: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ A3 )
& ( P @ X5 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_187_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
& ( P @ X5 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_188_is__env__def,axiom,
( is_env_fm
= ( ^ [U2: set_fm,E2: nat > fm] :
! [N: nat] : ( member_fm2 @ ( E2 @ N ) @ U2 ) ) ) ).
% is_env_def
thf(fact_189_is__env__def,axiom,
( is_env_nat
= ( ^ [U2: set_nat,E2: nat > nat] :
! [N: nat] : ( member_nat2 @ ( E2 @ N ) @ U2 ) ) ) ).
% is_env_def
thf(fact_190_is__env__def,axiom,
( is_env1422318314337916935_fm_fm
= ( ^ [U2: set_Pr4706815898642364871_fm_fm,E2: nat > product_prod_fm_fm] :
! [N: nat] : ( member7780952600467998736_fm_fm @ ( E2 @ N ) @ U2 ) ) ) ).
% is_env_def
thf(fact_191_is__env__def,axiom,
( is_env_tm
= ( ^ [U2: set_tm,E2: nat > tm] :
! [N: nat] : ( member_tm2 @ ( E2 @ N ) @ U2 ) ) ) ).
% is_env_def
thf(fact_192__092_060open_062Neg_A_Isub_A0_At_Ap_J_A_092_060in_062_AS_092_060close_062,axiom,
member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ t @ pa ) ) @ s ).
% \<open>Neg (sub 0 t p) \<in> S\<close>
thf(fact_193_paramst__subtermTm_I1_J,axiom,
! [T: tm,X3: nat] :
( ( member_nat2 @ X3 @ ( paramst @ T ) )
=> ? [L2: list_tm] : ( member_tm2 @ ( fun @ X3 @ L2 ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_194__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_Aterms_AS_059_ANeg_A_Isub_A0_At_Ap_J_A_092_060in_062_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T3: tm] :
( ( member_tm2 @ T3 @ ( terms @ s ) )
=> ~ ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ T3 @ pa ) ) @ s ) ) ).
% \<open>\<And>thesis. (\<And>t. \<lbrakk>t \<in> terms S; Neg (sub 0 t p) \<in> S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_195_verit__sko__ex_H,axiom,
! [P: tm > $o,A3: $o] :
( ( ( P @ ( fChoice_tm @ P ) )
= A3 )
=> ( ( ? [X4: tm] : ( P @ X4 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_196_verit__sko__forall,axiom,
( ( ^ [P4: tm > $o] :
! [X6: tm] : ( P4 @ X6 ) )
= ( ^ [P5: tm > $o] :
( P5
@ ( fChoice_tm
@ ^ [X5: tm] :
~ ( P5 @ X5 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_197_verit__sko__forall_H,axiom,
! [P: tm > $o,A3: $o] :
( ( ( P
@ ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
= A3 )
=> ( ( ! [X4: tm] : ( P @ X4 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_198_verit__sko__forall_H_H,axiom,
! [B2: tm,A3: tm,P: tm > $o] :
( ( B2 = A3 )
=> ( ( ( fChoice_tm @ P )
= A3 )
= ( ( fChoice_tm @ P )
= B2 ) ) ) ).
% verit_sko_forall''
thf(fact_199_verit__sko__ex__indirect,axiom,
! [X2: tm,P: tm > $o] :
( ( X2
= ( fChoice_tm @ P ) )
=> ( ( ? [X4: tm] : ( P @ X4 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_200_verit__sko__ex__indirect2,axiom,
! [X2: tm,P: tm > $o,P6: tm > $o] :
( ( X2
= ( fChoice_tm @ P ) )
=> ( ! [X: tm] :
( ( P @ X )
= ( P6 @ X ) )
=> ( ( ? [X4: tm] : ( P6 @ X4 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_201_verit__sko__forall__indirect,axiom,
! [X2: tm,P: tm > $o] :
( ( X2
= ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
=> ( ( ! [X4: tm] : ( P @ X4 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_202_verit__sko__forall__indirect2,axiom,
! [X2: tm,P: tm > $o,P6: tm > $o] :
( ( X2
= ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
=> ( ! [X: tm] :
( ( P @ X )
= ( P6 @ X ) )
=> ( ( ! [X4: tm] : ( P6 @ X4 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_203_sub_Osimps_I7_J,axiom,
! [V: nat,S2: tm,P3: fm] :
( ( sub @ V @ S2 @ ( neg @ P3 ) )
= ( neg @ ( sub @ V @ S2 @ P3 ) ) ) ).
% sub.simps(7)
thf(fact_204_Hintikka_OGammaExi,axiom,
! [H: set_fm,P3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( exi @ P3 ) @ H )
=> ! [X3: tm] :
( ( member_tm2 @ X3 @ ( terms @ H ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X3 @ P3 ) @ H ) ) ) ) ).
% Hintikka.GammaExi
thf(fact_205_Hintikka_ODeltaExi,axiom,
! [H: set_fm,P3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( exi @ P3 ) ) @ H )
=> ? [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X @ P3 ) ) @ H ) ) ) ) ).
% Hintikka.DeltaExi
thf(fact_206_verit__comp__simplify1_I2_J,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_207_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_208_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_209_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_210_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_211_calculation,axiom,
( usemantics_tm @ ( terms @ s )
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ s ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ s ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [N: nat,Ts3: list_tm] : ( member_fm2 @ ( neg @ ( pre @ N @ Ts3 ) ) @ s )
@ ( sub @ zero_zero_nat @ t @ pa ) ) ).
% calculation
thf(fact_212_pred__subset__eq,axiom,
! [R: set_fm,S: set_fm] :
( ( ord_less_eq_fm_o
@ ^ [X5: fm] : ( member_fm2 @ X5 @ R )
@ ^ [X5: fm] : ( member_fm2 @ X5 @ S ) )
= ( ord_less_eq_set_fm @ R @ S ) ) ).
% pred_subset_eq
thf(fact_213_pred__subset__eq,axiom,
! [R: set_Pr4706815898642364871_fm_fm,S: set_Pr4706815898642364871_fm_fm] :
( ( ord_le2785158094463560246m_fm_o
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ R )
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ S ) )
= ( ord_le966076687461445991_fm_fm @ R @ S ) ) ).
% pred_subset_eq
thf(fact_214_pred__subset__eq,axiom,
! [R: set_tm,S: set_tm] :
( ( ord_less_eq_tm_o
@ ^ [X5: tm] : ( member_tm2 @ X5 @ R )
@ ^ [X5: tm] : ( member_tm2 @ X5 @ S ) )
= ( ord_less_eq_set_tm @ R @ S ) ) ).
% pred_subset_eq
thf(fact_215_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat2 @ X5 @ R )
@ ^ [X5: nat] : ( member_nat2 @ X5 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_216_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_217_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_218_s1_I1_J,axiom,
( new_term
= ( ^ [C3: nat,T2: tm] :
~ ( member_nat2 @ C3 @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_219_tm_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( size_tm @ ( var @ X22 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_220_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_221_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_222_fm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( pre @ X11 @ X12 )
= ( pre @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fm.inject(1)
thf(fact_223_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_224_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_225_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_226_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_227_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_228_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_229_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_230_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X52: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X52 ) ) ).
% fm.distinct(7)
thf(fact_231_usemantics_Osimps_I7_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( neg @ P3 ) )
= ( ~ ( usemantics_tm @ U @ E @ F @ G @ P3 ) ) ) ).
% usemantics.simps(7)
thf(fact_232_Hintikka_OBasic,axiom,
! [H: set_fm,N2: nat,Ts2: list_tm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( pre @ N2 @ Ts2 ) @ H )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N2 @ Ts2 ) ) @ H ) ) ) ).
% Hintikka.Basic
thf(fact_233_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_234_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_235_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_236_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_237_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_238_new__term_Osimps_I1_J,axiom,
! [C: nat,N2: nat] : ( new_term @ C @ ( var @ N2 ) ) ).
% new_term.simps(1)
thf(fact_239_local_Owf,axiom,
! [Q2: fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Q2 @ x ) @ ( measure_fm @ size_size_fm ) )
=> ( ( ( member_fm2 @ Q2 @ s )
=> ~ ( usemantics_tm @ ( terms @ s )
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ s ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ s ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [N: nat,Ts3: list_tm] : ( member_fm2 @ ( neg @ ( pre @ N @ Ts3 ) ) @ s )
@ Q2 ) )
& ( ( member_fm2 @ ( neg @ Q2 ) @ s )
=> ( usemantics_tm @ ( terms @ s )
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ s ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ s ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [N: nat,Ts3: list_tm] : ( member_fm2 @ ( neg @ ( pre @ N @ Ts3 ) ) @ s )
@ Q2 ) ) ) ) ).
% local.wf
thf(fact_240_new__term_Osimps_I2_J,axiom,
! [C: nat,I2: nat,L3: list_tm] :
( ( new_term @ C @ ( fun @ I2 @ L3 ) )
= ( ( I2 != C )
& ( ( I2 != C )
=> ( new_list @ C @ L3 ) ) ) ) ).
% new_term.simps(2)
thf(fact_241_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_242_Hintikka_OGammaUni,axiom,
! [H: set_fm,P3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( uni @ P3 ) ) @ H )
=> ! [X3: tm] :
( ( member_tm2 @ X3 @ ( terms @ H ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X3 @ P3 ) ) @ H ) ) ) ) ).
% Hintikka.GammaUni
thf(fact_243_tm_Osize_I4_J,axiom,
! [X22: nat] :
( ( size_size_tm @ ( var @ X22 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_244_member,axiom,
( member8099233463853703858_fm_fm
= ( ^ [P7: product_prod_fm_fm,Z2: list_P8031219080602320621_fm_fm] : ( member7780952600467998736_fm_fm @ P7 @ ( set_Pr5149718152543245948_fm_fm @ Z2 ) ) ) ) ).
% member
thf(fact_245_member,axiom,
( member_tm
= ( ^ [P7: tm,Z2: list_tm] : ( member_tm2 @ P7 @ ( set_tm2 @ Z2 ) ) ) ) ).
% member
thf(fact_246_member,axiom,
( member_nat
= ( ^ [P7: nat,Z2: list_nat] : ( member_nat2 @ P7 @ ( set_nat2 @ Z2 ) ) ) ) ).
% member
thf(fact_247_member,axiom,
( member_fm
= ( ^ [P7: fm,Z2: list_fm] : ( member_fm2 @ P7 @ ( set_fm2 @ Z2 ) ) ) ) ).
% member
thf(fact_248_usemantics_Osimps_I5_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( exi @ P3 ) )
= ( ? [X5: tm] :
( ( member_tm2 @ X5 @ U )
& ( usemantics_tm @ U @ ( shift_nat_tm @ E @ zero_zero_nat @ X5 ) @ F @ G @ P3 ) ) ) ) ).
% usemantics.simps(5)
thf(fact_249_conj__subset__def,axiom,
! [A3: set_tm,P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ A3
@ ( collect_tm
@ ^ [X5: tm] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq_set_tm @ A3 @ ( collect_tm @ P ) )
& ( ord_less_eq_set_tm @ A3 @ ( collect_tm @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_250_conj__subset__def,axiom,
! [A3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3
@ ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_251_prop__restrict,axiom,
! [X2: fm,Z4: set_fm,X8: set_fm,P: fm > $o] :
( ( member_fm2 @ X2 @ Z4 )
=> ( ( ord_less_eq_set_fm @ Z4
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ X8 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_252_prop__restrict,axiom,
! [X2: product_prod_fm_fm,Z4: set_Pr4706815898642364871_fm_fm,X8: set_Pr4706815898642364871_fm_fm,P: product_prod_fm_fm > $o] :
( ( member7780952600467998736_fm_fm @ X2 @ Z4 )
=> ( ( ord_le966076687461445991_fm_fm @ Z4
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ X8 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_253_prop__restrict,axiom,
! [X2: tm,Z4: set_tm,X8: set_tm,P: tm > $o] :
( ( member_tm2 @ X2 @ Z4 )
=> ( ( ord_less_eq_set_tm @ Z4
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ X8 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_254_prop__restrict,axiom,
! [X2: nat,Z4: set_nat,X8: set_nat,P: nat > $o] :
( ( member_nat2 @ X2 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ X8 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_255_fm_Oinject_I6_J,axiom,
! [X62: fm,Y62: fm] :
( ( ( uni @ X62 )
= ( uni @ Y62 ) )
= ( X62 = Y62 ) ) ).
% fm.inject(6)
thf(fact_256_size__sub,axiom,
! [I2: nat,T: tm,P3: fm] :
( ( size_size_fm @ ( sub @ I2 @ T @ P3 ) )
= ( size_size_fm @ P3 ) ) ).
% size_sub
thf(fact_257_subrelI,axiom,
! [R2: set_Pr4706815898642364871_fm_fm,S2: set_Pr4706815898642364871_fm_fm] :
( ! [X: fm,Y4: fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y4 ) @ R2 )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y4 ) @ S2 ) )
=> ( ord_le966076687461445991_fm_fm @ R2 @ S2 ) ) ).
% subrelI
thf(fact_258_ssubst__Pair__rhs,axiom,
! [R2: fm,S2: fm,R: set_Pr4706815898642364871_fm_fm,S3: fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_259_pred__equals__eq2,axiom,
! [R: set_Pr4706815898642364871_fm_fm,S: set_Pr4706815898642364871_fm_fm] :
( ( ( ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ R ) )
= ( ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_260_pred__subset__eq2,axiom,
! [R: set_Pr4706815898642364871_fm_fm,S: set_Pr4706815898642364871_fm_fm] :
( ( ord_less_eq_fm_fm_o
@ ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ R )
@ ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ S ) )
= ( ord_le966076687461445991_fm_fm @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_261_size__neq__size__imp__neq,axiom,
! [X2: fm,Y3: fm] :
( ( ( size_size_fm @ X2 )
!= ( size_size_fm @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_262_size__neq__size__imp__neq,axiom,
! [X2: tm,Y3: tm] :
( ( ( size_size_tm @ X2 )
!= ( size_size_tm @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_263_size__neq__size__imp__neq,axiom,
! [X2: char,Y3: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_264_fm_Odistinct_I41_J,axiom,
! [X62: fm,X7: fm] :
( ( uni @ X62 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_265_usemantics_Osimps_I6_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( uni @ P3 ) )
= ( ! [X5: tm] :
( ( member_tm2 @ X5 @ U )
=> ( usemantics_tm @ U @ ( shift_nat_tm @ E @ zero_zero_nat @ X5 ) @ F @ G @ P3 ) ) ) ) ).
% usemantics.simps(6)
thf(fact_266_fm_Odistinct_I37_J,axiom,
! [X52: fm,X62: fm] :
( ( exi @ X52 )
!= ( uni @ X62 ) ) ).
% fm.distinct(37)
thf(fact_267_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X62: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X62 ) ) ).
% fm.distinct(9)
thf(fact_268_fm_Osize_I8_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size(8)
thf(fact_269_Hintikka_ODeltaUni,axiom,
! [H: set_fm,P3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( uni @ P3 ) @ H )
=> ? [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X @ P3 ) @ H ) ) ) ) ).
% Hintikka.DeltaUni
thf(fact_270_Collect__restrict,axiom,
! [X8: set_fm,P: fm > $o] :
( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ X8 )
& ( P @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_271_Collect__restrict,axiom,
! [X8: set_Pr4706815898642364871_fm_fm,P: product_prod_fm_fm > $o] :
( ord_le966076687461445991_fm_fm
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ X8 )
& ( P @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_272_Collect__restrict,axiom,
! [X8: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ X8 )
& ( P @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_273_Collect__restrict,axiom,
! [X8: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ X8 )
& ( P @ X5 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_274_subst__lemma_H_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,T: tm,U: tm,I2: nat] :
( ( semantics_term_tm @ E @ F @ ( substt @ T @ U @ I2 ) )
= ( semantics_term_tm @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ U ) ) @ F @ T ) ) ).
% subst_lemma'(1)
thf(fact_275_lift__lemma_I1_J,axiom,
! [E: nat > tm,X2: tm,F: nat > list_tm > tm,T: tm] :
( ( semantics_term_tm @ ( shift_nat_tm @ E @ zero_zero_nat @ X2 ) @ F @ ( liftt @ T ) )
= ( semantics_term_tm @ E @ F @ T ) ) ).
% lift_lemma(1)
thf(fact_276_prod_Oinject,axiom,
! [X1: fm,X22: fm,Y1: fm,Y22: fm] :
( ( ( product_Pair_fm_fm @ X1 @ X22 )
= ( product_Pair_fm_fm @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_277_old_Oprod_Oinject,axiom,
! [A: fm,B: fm,A6: fm,B6: fm] :
( ( ( product_Pair_fm_fm @ A @ B )
= ( product_Pair_fm_fm @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_278_usubst__lemma,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,A: fm,T: tm,I2: nat] :
( ( usemantics_tm @ U @ E @ F @ G @ ( subst @ A @ T @ I2 ) )
= ( usemantics_tm @ U @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ T ) ) @ F @ G @ A ) ) ).
% usubst_lemma
thf(fact_279_subset__Collect__iff,axiom,
! [B2: set_fm,A3: set_fm,P: fm > $o] :
( ( ord_less_eq_set_fm @ B2 @ A3 )
=> ( ( ord_less_eq_set_fm @ B2
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) )
= ( ! [X5: fm] :
( ( member_fm2 @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_280_subset__Collect__iff,axiom,
! [B2: set_Pr4706815898642364871_fm_fm,A3: set_Pr4706815898642364871_fm_fm,P: product_prod_fm_fm > $o] :
( ( ord_le966076687461445991_fm_fm @ B2 @ A3 )
=> ( ( ord_le966076687461445991_fm_fm @ B2
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A3 )
& ( P @ X5 ) ) ) )
= ( ! [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_281_subset__Collect__iff,axiom,
! [B2: set_tm,A3: set_tm,P: tm > $o] :
( ( ord_less_eq_set_tm @ B2 @ A3 )
=> ( ( ord_less_eq_set_tm @ B2
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) )
= ( ! [X5: tm] :
( ( member_tm2 @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_282_subset__Collect__iff,axiom,
! [B2: set_nat,A3: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
& ( P @ X5 ) ) ) )
= ( ! [X5: nat] :
( ( member_nat2 @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_283_subset__CollectI,axiom,
! [B2: set_fm,A3: set_fm,Q: fm > $o,P: fm > $o] :
( ( ord_less_eq_set_fm @ B2 @ A3 )
=> ( ! [X: fm] :
( ( member_fm2 @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_fm
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ B2 )
& ( Q @ X5 ) ) )
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_284_subset__CollectI,axiom,
! [B2: set_Pr4706815898642364871_fm_fm,A3: set_Pr4706815898642364871_fm_fm,Q: product_prod_fm_fm > $o,P: product_prod_fm_fm > $o] :
( ( ord_le966076687461445991_fm_fm @ B2 @ A3 )
=> ( ! [X: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le966076687461445991_fm_fm
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ B2 )
& ( Q @ X5 ) ) )
@ ( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_285_subset__CollectI,axiom,
! [B2: set_tm,A3: set_tm,Q: tm > $o,P: tm > $o] :
( ( ord_less_eq_set_tm @ B2 @ A3 )
=> ( ! [X: tm] :
( ( member_tm2 @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ B2 )
& ( Q @ X5 ) ) )
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_286_subset__CollectI,axiom,
! [B2: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ B2 )
& ( Q @ X5 ) ) )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_287_subterm__Pre__refl,axiom,
! [Ts2: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts2 ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N2 @ Ts2 ) ) ) ) ).
% subterm_Pre_refl
thf(fact_288_fun__arguments__subterm,axiom,
! [N2: nat,Ts2: list_tm,P3: fm] :
( ( member_tm2 @ ( fun @ N2 @ Ts2 ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts2 ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).
% fun_arguments_subterm
thf(fact_289_s6,axiom,
( sub
= ( ^ [V2: nat,S4: tm,P7: fm] : ( subst @ P7 @ S4 @ V2 ) ) ) ).
% s6
thf(fact_290_paramst__liftt_I1_J,axiom,
! [T: tm] :
( ( paramst @ ( liftt @ T ) )
= ( paramst @ T ) ) ).
% paramst_liftt(1)
thf(fact_291_subst_Osimps_I7_J,axiom,
! [P3: fm,S2: tm,K: nat] :
( ( subst @ ( neg @ P3 ) @ S2 @ K )
= ( neg @ ( subst @ P3 @ S2 @ K ) ) ) ).
% subst.simps(7)
thf(fact_292_Pair__inject,axiom,
! [A: fm,B: fm,A6: fm,B6: fm] :
( ( ( product_Pair_fm_fm @ A @ B )
= ( product_Pair_fm_fm @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_293_prod__cases,axiom,
! [P: product_prod_fm_fm > $o,P3: product_prod_fm_fm] :
( ! [A2: fm,B5: fm] : ( P @ ( product_Pair_fm_fm @ A2 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_294_surj__pair,axiom,
! [P3: product_prod_fm_fm] :
? [X: fm,Y4: fm] :
( P3
= ( product_Pair_fm_fm @ X @ Y4 ) ) ).
% surj_pair
thf(fact_295_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_fm_fm] :
~ ! [A2: fm,B5: fm] :
( Y3
!= ( product_Pair_fm_fm @ A2 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_296_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_297_subtermFm_Osimps_I6_J,axiom,
! [P3: fm] :
( ( subtermFm @ ( uni @ P3 ) )
= ( subtermFm @ P3 ) ) ).
% subtermFm.simps(6)
thf(fact_298_subtermFm_Osimps_I5_J,axiom,
! [P3: fm] :
( ( subtermFm @ ( exi @ P3 ) )
= ( subtermFm @ P3 ) ) ).
% subtermFm.simps(5)
thf(fact_299_subtermFm_Osimps_I7_J,axiom,
! [P3: fm] :
( ( subtermFm @ ( neg @ P3 ) )
= ( subtermFm @ P3 ) ) ).
% subtermFm.simps(7)
thf(fact_300_subst__lemma,axiom,
! [E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,A: fm,T: tm,I2: nat] :
( ( semantics_tm @ E @ F @ G @ ( subst @ A @ T @ I2 ) )
= ( semantics_tm @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ T ) ) @ F @ G @ A ) ) ).
% subst_lemma
thf(fact_301_terms__cases,axiom,
! [T: tm,S: set_fm] :
( ( member_tm2 @ T @ ( terms @ S ) )
=> ( ( T
= ( fun @ zero_zero_nat @ nil_tm ) )
| ? [X: fm] :
( ( member_fm2 @ X @ S )
& ( member_tm2 @ T @ ( set_tm2 @ ( subtermFm @ X ) ) ) ) ) ) ).
% terms_cases
thf(fact_302_substt_Osimps_I2_J,axiom,
! [A: nat,Ts2: list_tm,S2: tm,K: nat] :
( ( substt @ ( fun @ A @ Ts2 ) @ S2 @ K )
= ( fun @ A @ ( substts @ Ts2 @ S2 @ K ) ) ) ).
% substt.simps(2)
thf(fact_303_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_304_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_305_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_306_substts_Osimps_I1_J,axiom,
! [S2: tm,K: nat] :
( ( substts @ nil_tm @ S2 @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_307_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_308_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_309_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_310_SeCaV_Omember_Osimps_I1_J,axiom,
! [P3: tm] :
~ ( member_tm @ P3 @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_311_SeCaV_Omember_Osimps_I1_J,axiom,
! [P3: nat] :
~ ( member_nat @ P3 @ nil_nat ) ).
% SeCaV.member.simps(1)
thf(fact_312_SeCaV_Omember_Osimps_I1_J,axiom,
! [P3: fm] :
~ ( member_fm @ P3 @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_313_ext_Osimps_I1_J,axiom,
! [Y3: list_tm] : ( ext_tm @ Y3 @ nil_tm ) ).
% ext.simps(1)
thf(fact_314_ext_Osimps_I1_J,axiom,
! [Y3: list_nat] : ( ext_nat @ Y3 @ nil_nat ) ).
% ext.simps(1)
thf(fact_315_ext_Osimps_I1_J,axiom,
! [Y3: list_fm] : ( ext_fm @ Y3 @ nil_fm ) ).
% ext.simps(1)
thf(fact_316_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_317_sub__const__transfer,axiom,
! [M: nat,A: nat,P3: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P3 )
!= ( sub @ M @ T @ P3 ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P3 ) ) ) ) ) ).
% sub_const_transfer
thf(fact_318_subst_Osimps_I1_J,axiom,
! [B: nat,Ts2: list_tm,S2: tm,K: nat] :
( ( subst @ ( pre @ B @ Ts2 ) @ S2 @ K )
= ( pre @ B @ ( substts @ Ts2 @ S2 @ K ) ) ) ).
% subst.simps(1)
thf(fact_319_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_320_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T: tm,S2: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T )
!= ( sub_term @ M @ S2 @ T ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_321_subst__lemma_H_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,Ts2: list_tm,U: tm,I2: nat] :
( ( semantics_list_tm @ E @ F @ ( substts @ Ts2 @ U @ I2 ) )
= ( semantics_list_tm @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ U ) ) @ F @ Ts2 ) ) ).
% subst_lemma'(2)
thf(fact_322_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_323_liftt_Osimps_I2_J,axiom,
! [A: nat,Ts2: list_tm] :
( ( liftt @ ( fun @ A @ Ts2 ) )
= ( fun @ A @ ( liftts @ Ts2 ) ) ) ).
% liftt.simps(2)
thf(fact_324_s5_I1_J,axiom,
( sub_term
= ( ^ [V2: nat,S4: tm,T2: tm] : ( substt @ T2 @ S4 @ V2 ) ) ) ).
% s5(1)
thf(fact_325_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_fm @ ( coset_fm @ nil_fm ) @ ( set_fm2 @ nil_fm ) ) ).
% subset_code(3)
thf(fact_326_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_tm @ ( coset_tm @ nil_tm ) @ ( set_tm2 @ nil_tm ) ) ).
% subset_code(3)
thf(fact_327_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_328_lift__lemma_I2_J,axiom,
! [E: nat > tm,X2: tm,F: nat > list_tm > tm,Ts2: list_tm] :
( ( semantics_list_tm @ ( shift_nat_tm @ E @ zero_zero_nat @ X2 ) @ F @ ( liftts @ Ts2 ) )
= ( semantics_list_tm @ E @ F @ Ts2 ) ) ).
% lift_lemma(2)
thf(fact_329_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_330_subset__code_I2_J,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,Ys: list_P8031219080602320621_fm_fm] :
( ( ord_le966076687461445991_fm_fm @ A3 @ ( coset_5303328301088503642_fm_fm @ Ys ) )
= ( ! [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ ( set_Pr5149718152543245948_fm_fm @ Ys ) )
=> ~ ( member7780952600467998736_fm_fm @ X5 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_331_subset__code_I2_J,axiom,
! [A3: set_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ A3 @ ( coset_fm @ Ys ) )
= ( ! [X5: fm] :
( ( member_fm2 @ X5 @ ( set_fm2 @ Ys ) )
=> ~ ( member_fm2 @ X5 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_332_subset__code_I2_J,axiom,
! [A3: set_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ A3 @ ( coset_tm @ Ys ) )
= ( ! [X5: tm] :
( ( member_tm2 @ X5 @ ( set_tm2 @ Ys ) )
=> ~ ( member_tm2 @ X5 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_333_subset__code_I2_J,axiom,
! [A3: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( coset_nat @ Ys ) )
= ( ! [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X5 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_334_semantics__list_Osimps_I1_J,axiom,
! [E: nat > nat,F: nat > list_nat > nat] :
( ( semantics_list_nat @ E @ F @ nil_tm )
= nil_nat ) ).
% semantics_list.simps(1)
thf(fact_335_semantics__list_Osimps_I1_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm] :
( ( semantics_list_fm @ E @ F @ nil_tm )
= nil_fm ) ).
% semantics_list.simps(1)
thf(fact_336_semantics__list_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm] :
( ( semantics_list_tm @ E @ F @ nil_tm )
= nil_tm ) ).
% semantics_list.simps(1)
thf(fact_337_usemantics_Osimps_I1_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,I2: nat,L3: list_tm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( pre @ I2 @ L3 ) )
= ( G @ I2 @ ( semantics_list_tm @ E @ F @ L3 ) ) ) ).
% usemantics.simps(1)
thf(fact_338_semantics_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,I2: nat,L3: list_tm] :
( ( semantics_tm @ E @ F @ G @ ( pre @ I2 @ L3 ) )
= ( G @ I2 @ ( semantics_list_tm @ E @ F @ L3 ) ) ) ).
% semantics.simps(1)
thf(fact_339_semantics__term_Osimps_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,I2: nat,L3: list_tm] :
( ( semantics_term_tm @ E @ F @ ( fun @ I2 @ L3 ) )
= ( F @ I2 @ ( semantics_list_tm @ E @ F @ L3 ) ) ) ).
% semantics_term.simps(2)
thf(fact_340_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_341_inc__term_Osimps_I2_J,axiom,
! [I2: nat,L3: list_tm] :
( ( inc_term @ ( fun @ I2 @ L3 ) )
= ( fun @ I2 @ ( inc_list @ L3 ) ) ) ).
% inc_term.simps(2)
thf(fact_342_set__n__lists,axiom,
! [N2: nat,Xs: list_fm] :
( ( set_list_fm2 @ ( n_lists_fm @ N2 @ Xs ) )
= ( collect_list_fm
@ ^ [Ys2: list_fm] :
( ( ( size_size_list_fm @ Ys2 )
= N2 )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Ys2 ) @ ( set_fm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_343_set__n__lists,axiom,
! [N2: nat,Xs: list_tm] :
( ( set_list_tm2 @ ( n_lists_tm @ N2 @ Xs ) )
= ( collect_list_tm
@ ^ [Ys2: list_tm] :
( ( ( size_size_list_tm @ Ys2 )
= N2 )
& ( ord_less_eq_set_tm @ ( set_tm2 @ Ys2 ) @ ( set_tm2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_344_set__n__lists,axiom,
! [N2: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N2 @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N2 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_345_usemantics__E_I2_J,axiom,
! [S: set_fm,Ts2: list_tm] :
( ( list_all_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) )
@ Ts2 )
=> ( ( semantics_list_tm
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ S ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ S ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ S ) ) ) )
@ Ts2 )
= Ts2 ) ) ).
% usemantics_E(2)
thf(fact_346_usemantics__term_I2_J,axiom,
! [U: set_nat,E: nat > nat,F: nat > list_nat > nat,Ts2: list_tm] :
( ( is_env_nat @ U @ E )
=> ( ( is_fdenot_nat @ U @ F )
=> ( list_all_nat
@ ^ [X5: nat] : ( member_nat2 @ X5 @ U )
@ ( semantics_list_nat @ E @ F @ Ts2 ) ) ) ) ).
% usemantics_term(2)
thf(fact_347_usemantics__term_I2_J,axiom,
! [U: set_Pr4706815898642364871_fm_fm,E: nat > product_prod_fm_fm,F: nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm,Ts2: list_tm] :
( ( is_env1422318314337916935_fm_fm @ U @ E )
=> ( ( is_fde8755990660405543756_fm_fm @ U @ F )
=> ( list_a808454179579425435_fm_fm
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ U )
@ ( semant4740343338788880149_fm_fm @ E @ F @ Ts2 ) ) ) ) ).
% usemantics_term(2)
thf(fact_348_usemantics__term_I2_J,axiom,
! [U: set_fm,E: nat > fm,F: nat > list_fm > fm,Ts2: list_tm] :
( ( is_env_fm @ U @ E )
=> ( ( is_fdenot_fm @ U @ F )
=> ( list_all_fm
@ ^ [X5: fm] : ( member_fm2 @ X5 @ U )
@ ( semantics_list_fm @ E @ F @ Ts2 ) ) ) ) ).
% usemantics_term(2)
thf(fact_349_usemantics__term_I2_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,Ts2: list_tm] :
( ( is_env_tm @ U @ E )
=> ( ( is_fdenot_tm @ U @ F )
=> ( list_all_tm
@ ^ [X5: tm] : ( member_tm2 @ X5 @ U )
@ ( semantics_list_tm @ E @ F @ Ts2 ) ) ) ) ).
% usemantics_term(2)
thf(fact_350_sub__term_Osimps_I2_J,axiom,
! [V: nat,S2: tm,I2: nat,L3: list_tm] :
( ( sub_term @ V @ S2 @ ( fun @ I2 @ L3 ) )
= ( fun @ I2 @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub_term.simps(2)
thf(fact_351_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_352_list__all__simps_I2_J,axiom,
! [P: tm > $o] : ( list_all_tm @ P @ nil_tm ) ).
% list_all_simps(2)
thf(fact_353_list__all__simps_I2_J,axiom,
! [P: fm > $o] : ( list_all_fm @ P @ nil_fm ) ).
% list_all_simps(2)
thf(fact_354_s5_I2_J,axiom,
( sub_list
= ( ^ [V2: nat,S4: tm,L: list_tm] : ( substts @ L @ S4 @ V2 ) ) ) ).
% s5(2)
thf(fact_355_list_Opred__True,axiom,
( ( list_all_tm
@ ^ [Uu: tm] : $true )
= ( ^ [Uu: list_tm] : $true ) ) ).
% list.pred_True
thf(fact_356_list_Opred__True,axiom,
( ( list_all_fm
@ ^ [Uu: fm] : $true )
= ( ^ [Uu: list_fm] : $true ) ) ).
% list.pred_True
thf(fact_357_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_358_list_Opred__inject_I1_J,axiom,
! [P: tm > $o] : ( list_all_tm @ P @ nil_tm ) ).
% list.pred_inject(1)
thf(fact_359_list_Opred__inject_I1_J,axiom,
! [P: fm > $o] : ( list_all_fm @ P @ nil_fm ) ).
% list.pred_inject(1)
thf(fact_360_list_Opred__mono__strong,axiom,
! [P: product_prod_fm_fm > $o,X2: list_P8031219080602320621_fm_fm,Pa: product_prod_fm_fm > $o] :
( ( list_a808454179579425435_fm_fm @ P @ X2 )
=> ( ! [Z5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ Z5 @ ( set_Pr5149718152543245948_fm_fm @ X2 ) )
=> ( ( P @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( list_a808454179579425435_fm_fm @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_361_list_Opred__mono__strong,axiom,
! [P: nat > $o,X2: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P @ X2 )
=> ( ! [Z5: nat] :
( ( member_nat2 @ Z5 @ ( set_nat2 @ X2 ) )
=> ( ( P @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( list_all_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_362_list_Opred__mono__strong,axiom,
! [P: tm > $o,X2: list_tm,Pa: tm > $o] :
( ( list_all_tm @ P @ X2 )
=> ( ! [Z5: tm] :
( ( member_tm2 @ Z5 @ ( set_tm2 @ X2 ) )
=> ( ( P @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( list_all_tm @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_363_list_Opred__mono__strong,axiom,
! [P: fm > $o,X2: list_fm,Pa: fm > $o] :
( ( list_all_fm @ P @ X2 )
=> ( ! [Z5: fm] :
( ( member_fm2 @ Z5 @ ( set_fm2 @ X2 ) )
=> ( ( P @ Z5 )
=> ( Pa @ Z5 ) ) )
=> ( list_all_fm @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_364_list__all__cong,axiom,
! [X2: list_P8031219080602320621_fm_fm,Ya: list_P8031219080602320621_fm_fm,P: product_prod_fm_fm > $o,Pa: product_prod_fm_fm > $o] :
( ( X2 = Ya )
=> ( ! [Z5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ Z5 @ ( set_Pr5149718152543245948_fm_fm @ Ya ) )
=> ( ( P @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( list_a808454179579425435_fm_fm @ P @ X2 )
= ( list_a808454179579425435_fm_fm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_365_list__all__cong,axiom,
! [X2: list_nat,Ya: list_nat,P: nat > $o,Pa: nat > $o] :
( ( X2 = Ya )
=> ( ! [Z5: nat] :
( ( member_nat2 @ Z5 @ ( set_nat2 @ Ya ) )
=> ( ( P @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( list_all_nat @ P @ X2 )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_366_list__all__cong,axiom,
! [X2: list_tm,Ya: list_tm,P: tm > $o,Pa: tm > $o] :
( ( X2 = Ya )
=> ( ! [Z5: tm] :
( ( member_tm2 @ Z5 @ ( set_tm2 @ Ya ) )
=> ( ( P @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( list_all_tm @ P @ X2 )
= ( list_all_tm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_367_list__all__cong,axiom,
! [X2: list_fm,Ya: list_fm,P: fm > $o,Pa: fm > $o] :
( ( X2 = Ya )
=> ( ! [Z5: fm] :
( ( member_fm2 @ Z5 @ ( set_fm2 @ Ya ) )
=> ( ( P @ Z5 )
= ( Pa @ Z5 ) ) )
=> ( ( list_all_fm @ P @ X2 )
= ( list_all_fm @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_368_list_Opred__mono,axiom,
! [P: tm > $o,Pa: tm > $o] :
( ( ord_less_eq_tm_o @ P @ Pa )
=> ( ord_le2468657205176945586t_tm_o @ ( list_all_tm @ P ) @ ( list_all_tm @ Pa ) ) ) ).
% list.pred_mono
thf(fact_369_list_Opred__mono,axiom,
! [P: fm > $o,Pa: fm > $o] :
( ( ord_less_eq_fm_o @ P @ Pa )
=> ( ord_le6518561683347902116t_fm_o @ ( list_all_fm @ P ) @ ( list_all_fm @ Pa ) ) ) ).
% list.pred_mono
thf(fact_370_is__fdenot__def,axiom,
( is_fdenot_nat
= ( ^ [U2: set_nat,F2: nat > list_nat > nat] :
! [I: nat,L: list_nat] :
( ( list_all_nat
@ ^ [X5: nat] : ( member_nat2 @ X5 @ U2 )
@ L )
=> ( member_nat2 @ ( F2 @ I @ L ) @ U2 ) ) ) ) ).
% is_fdenot_def
thf(fact_371_is__fdenot__def,axiom,
( is_fde8755990660405543756_fm_fm
= ( ^ [U2: set_Pr4706815898642364871_fm_fm,F2: nat > list_P8031219080602320621_fm_fm > product_prod_fm_fm] :
! [I: nat,L: list_P8031219080602320621_fm_fm] :
( ( list_a808454179579425435_fm_fm
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ U2 )
@ L )
=> ( member7780952600467998736_fm_fm @ ( F2 @ I @ L ) @ U2 ) ) ) ) ).
% is_fdenot_def
thf(fact_372_is__fdenot__def,axiom,
( is_fdenot_fm
= ( ^ [U2: set_fm,F2: nat > list_fm > fm] :
! [I: nat,L: list_fm] :
( ( list_all_fm
@ ^ [X5: fm] : ( member_fm2 @ X5 @ U2 )
@ L )
=> ( member_fm2 @ ( F2 @ I @ L ) @ U2 ) ) ) ) ).
% is_fdenot_def
thf(fact_373_is__fdenot__def,axiom,
( is_fdenot_tm
= ( ^ [U2: set_tm,F2: nat > list_tm > tm] :
! [I: nat,L: list_tm] :
( ( list_all_tm
@ ^ [X5: tm] : ( member_tm2 @ X5 @ U2 )
@ L )
=> ( member_tm2 @ ( F2 @ I @ L ) @ U2 ) ) ) ) ).
% is_fdenot_def
thf(fact_374_sub__list_Osimps_I1_J,axiom,
! [V: nat,S2: tm] :
( ( sub_list @ V @ S2 @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_375_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_tm @ N2 @ nil_tm )
= N2 ) ).
% gen_length_code(1)
thf(fact_376_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_nat @ N2 @ nil_nat )
= N2 ) ).
% gen_length_code(1)
thf(fact_377_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_fm @ N2 @ nil_fm )
= N2 ) ).
% gen_length_code(1)
thf(fact_378_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_379_sub_Osimps_I1_J,axiom,
! [V: nat,S2: tm,I2: nat,L3: list_tm] :
( ( sub @ V @ S2 @ ( pre @ I2 @ L3 ) )
= ( pre @ I2 @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub.simps(1)
thf(fact_380_params__subtermFm,axiom,
! [P3: fm,X3: nat] :
( ( member_nat2 @ X3 @ ( params @ P3 ) )
=> ? [L2: list_tm] : ( member_tm2 @ ( fun @ X3 @ L2 ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).
% params_subtermFm
thf(fact_381_list__ex1__simps_I1_J,axiom,
! [P: tm > $o] :
~ ( list_ex1_tm @ P @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_382_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_383_list__ex1__simps_I1_J,axiom,
! [P: fm > $o] :
~ ( list_ex1_fm @ P @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_384_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_tm @ N2 @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_tm @ N2 @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_385_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_386_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_fm @ N2 @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_fm @ N2 @ nil_fm )
= nil_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_387_sound__usemantics,axiom,
! [Z3: list_fm,U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o] :
( ( sequent_calculus @ Z3 )
=> ( ( is_env_tm @ U @ E )
=> ( ( is_fdenot_tm @ U @ F )
=> ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Z3 ) )
& ( usemantics_tm @ U @ E @ F @ G @ X ) ) ) ) ) ).
% sound_usemantics
thf(fact_388_listFunTm__paramst_I1_J,axiom,
! [T: tm] :
( ( set_nat2 @ ( listFunTm @ T ) )
= ( paramst @ T ) ) ).
% listFunTm_paramst(1)
thf(fact_389_list_Oinject,axiom,
! [X21: fm,X222: list_fm,Y21: fm,Y222: list_fm] :
( ( ( cons_fm @ X21 @ X222 )
= ( cons_fm @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_390_list_Oinject,axiom,
! [X21: tm,X222: list_tm,Y21: tm,Y222: list_tm] :
( ( ( cons_tm @ X21 @ X222 )
= ( cons_tm @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_391_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_392_list__all__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_nat @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_393_list__all__simps_I1_J,axiom,
! [P: tm > $o,X2: tm,Xs: list_tm] :
( ( list_all_tm @ P @ ( cons_tm @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_tm @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_394_list__all__simps_I1_J,axiom,
! [P: fm > $o,X2: fm,Xs: list_fm] :
( ( list_all_fm @ P @ ( cons_fm @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_fm @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_395_list_Opred__inject_I2_J,axiom,
! [P: nat > $o,A: nat,Aa: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_nat @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_396_list_Opred__inject_I2_J,axiom,
! [P: tm > $o,A: tm,Aa: list_tm] :
( ( list_all_tm @ P @ ( cons_tm @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_tm @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_397_list_Opred__inject_I2_J,axiom,
! [P: fm > $o,A: fm,Aa: list_fm] :
( ( list_all_fm @ P @ ( cons_fm @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_fm @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_398_list__ex1__simps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex1_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_nat
@ ^ [Y: nat] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_nat @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_399_list__ex1__simps_I2_J,axiom,
! [P: tm > $o,X2: tm,Xs: list_tm] :
( ( list_ex1_tm @ P @ ( cons_tm @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_tm
@ ^ [Y: tm] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_tm @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_400_list__ex1__simps_I2_J,axiom,
! [P: fm > $o,X2: fm,Xs: list_fm] :
( ( list_ex1_fm @ P @ ( cons_fm @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_fm
@ ^ [Y: fm] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_fm @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_401_not__Cons__self2,axiom,
! [X2: fm,Xs: list_fm] :
( ( cons_fm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_402_not__Cons__self2,axiom,
! [X2: tm,Xs: list_tm] :
( ( cons_tm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_403_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_404_Ext,axiom,
! [Z3: list_fm,Y3: list_fm] :
( ( sequent_calculus @ Z3 )
=> ( ( ext_fm @ Y3 @ Z3 )
=> ( sequent_calculus @ Y3 ) ) ) ).
% Ext
thf(fact_405_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_406_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_407_product__lists_Osimps_I1_J,axiom,
( ( product_lists_fm @ nil_list_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_408_transpose_Ocases,axiom,
! [X2: list_list_fm] :
( ( X2 != nil_list_fm )
=> ( ! [Xss: list_list_fm] :
( X2
!= ( cons_list_fm @ nil_fm @ Xss ) )
=> ~ ! [X: fm,Xs2: list_fm,Xss: list_list_fm] :
( X2
!= ( cons_list_fm @ ( cons_fm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_409_transpose_Ocases,axiom,
! [X2: list_list_tm] :
( ( X2 != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X2
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X: tm,Xs2: list_tm,Xss: list_list_tm] :
( X2
!= ( cons_list_tm @ ( cons_tm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_410_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs2: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_411_list_Odistinct_I1_J,axiom,
! [X21: fm,X222: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_412_list_Odistinct_I1_J,axiom,
! [X21: tm,X222: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_413_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_414_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X222: list_fm] :
( ( List
= ( cons_fm @ X21 @ X222 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_415_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X222: list_tm] :
( ( List
= ( cons_tm @ X21 @ X222 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_416_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_417_list_Oexhaust,axiom,
! [Y3: list_fm] :
( ( Y3 != nil_fm )
=> ~ ! [X212: fm,X223: list_fm] :
( Y3
!= ( cons_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_418_list_Oexhaust,axiom,
! [Y3: list_tm] :
( ( Y3 != nil_tm )
=> ~ ! [X212: tm,X223: list_tm] :
( Y3
!= ( cons_tm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_419_list_Oexhaust,axiom,
! [Y3: list_nat] :
( ( Y3 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y3
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_420_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_421_remdups__adj_Ocases,axiom,
! [X2: list_fm] :
( ( X2 != nil_fm )
=> ( ! [X: fm] :
( X2
!= ( cons_fm @ X @ nil_fm ) )
=> ~ ! [X: fm,Y4: fm,Xs2: list_fm] :
( X2
!= ( cons_fm @ X @ ( cons_fm @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_422_remdups__adj_Ocases,axiom,
! [X2: list_tm] :
( ( X2 != nil_tm )
=> ( ! [X: tm] :
( X2
!= ( cons_tm @ X @ nil_tm ) )
=> ~ ! [X: tm,Y4: tm,Xs2: list_tm] :
( X2
!= ( cons_tm @ X @ ( cons_tm @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_423_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y4: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_424_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y: fm,Ys2: list_fm] :
( Xs
= ( cons_fm @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_425_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y: tm,Ys2: list_tm] :
( Xs
= ( cons_tm @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_426_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_427_list__induct2_H,axiom,
! [P: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y4: fm,Ys3: list_fm] : ( P @ nil_fm @ ( cons_fm @ Y4 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_428_list__induct2_H,axiom,
! [P: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y4: tm,Ys3: list_tm] : ( P @ nil_fm @ ( cons_tm @ Y4 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_429_list__induct2_H,axiom,
! [P: list_fm > list_nat > $o,Xs: list_fm,Ys: list_nat] :
( ( P @ nil_fm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_fm @ ( cons_nat @ Y4 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y4: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_430_list__induct2_H,axiom,
! [P: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P @ nil_tm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y4: fm,Ys3: list_fm] : ( P @ nil_tm @ ( cons_fm @ Y4 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y4: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_431_list__induct2_H,axiom,
! [P: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P @ nil_tm @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y4: tm,Ys3: list_tm] : ( P @ nil_tm @ ( cons_tm @ Y4 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y4: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_432_list__induct2_H,axiom,
! [P: list_tm > list_nat > $o,Xs: list_tm,Ys: list_nat] :
( ( P @ nil_tm @ nil_nat )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_tm @ ( cons_nat @ Y4 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y4: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_433_list__induct2_H,axiom,
! [P: list_nat > list_fm > $o,Xs: list_nat,Ys: list_fm] :
( ( P @ nil_nat @ nil_fm )
=> ( ! [X: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y4: fm,Ys3: list_fm] : ( P @ nil_nat @ ( cons_fm @ Y4 @ Ys3 ) )
=> ( ! [X: nat,Xs2: list_nat,Y4: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_434_list__induct2_H,axiom,
! [P: list_nat > list_tm > $o,Xs: list_nat,Ys: list_tm] :
( ( P @ nil_nat @ nil_tm )
=> ( ! [X: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y4: tm,Ys3: list_tm] : ( P @ nil_nat @ ( cons_tm @ Y4 @ Ys3 ) )
=> ( ! [X: nat,Xs2: list_nat,Y4: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_435_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys3 ) )
=> ( ! [X: nat,Xs2: list_nat,Y4: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_436_list__nonempty__induct,axiom,
! [Xs: list_fm,P: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X: fm] : ( P @ ( cons_fm @ X @ nil_fm ) )
=> ( ! [X: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_437_list__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X: tm] : ( P @ ( cons_tm @ X @ nil_tm ) )
=> ( ! [X: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_tm @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_438_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_439_splice_Ocases,axiom,
! [X2: produc3245234490656042599ist_fm] :
( ! [Ys3: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ nil_fm @ Ys3 ) )
=> ~ ! [X: fm,Xs2: list_fm,Ys3: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_440_splice_Ocases,axiom,
! [X2: produc5776448205642668775ist_tm] :
( ! [Ys3: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ nil_tm @ Ys3 ) )
=> ~ ! [X: tm,Xs2: list_tm,Ys3: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_441_splice_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ~ ! [X: nat,Xs2: list_nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_442_shuffles_Ocases,axiom,
! [X2: produc3245234490656042599ist_fm] :
( ! [Ys3: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ nil_fm @ Ys3 ) )
=> ( ! [Xs2: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ Xs2 @ nil_fm ) )
=> ~ ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_443_shuffles_Ocases,axiom,
! [X2: produc5776448205642668775ist_tm] :
( ! [Ys3: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ nil_tm @ Ys3 ) )
=> ( ! [Xs2: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ Xs2 @ nil_tm ) )
=> ~ ! [X: tm,Xs2: list_tm,Y4: tm,Ys3: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_444_shuffles_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
=> ( ! [Xs2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X: nat,Xs2: list_nat,Y4: nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_445_sorted__wrt_Ocases,axiom,
! [X2: produc7963324949210141170ist_fm] :
( ! [P8: fm > fm > $o] :
( X2
!= ( produc7687578365188660450ist_fm @ P8 @ nil_fm ) )
=> ~ ! [P8: fm > fm > $o,X: fm,Ys3: list_fm] :
( X2
!= ( produc7687578365188660450ist_fm @ P8 @ ( cons_fm @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_446_sorted__wrt_Ocases,axiom,
! [X2: produc2002131169352006116ist_tm] :
( ! [P8: tm > tm > $o] :
( X2
!= ( produc1972851280741670356ist_tm @ P8 @ nil_tm ) )
=> ~ ! [P8: tm > tm > $o,X: tm,Ys3: list_tm] :
( X2
!= ( produc1972851280741670356ist_tm @ P8 @ ( cons_tm @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_447_sorted__wrt_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P8: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P8 @ nil_nat ) )
=> ~ ! [P8: nat > nat > $o,X: nat,Ys3: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P8 @ ( cons_nat @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_448_successively_Ocases,axiom,
! [X2: produc7963324949210141170ist_fm] :
( ! [P8: fm > fm > $o] :
( X2
!= ( produc7687578365188660450ist_fm @ P8 @ nil_fm ) )
=> ( ! [P8: fm > fm > $o,X: fm] :
( X2
!= ( produc7687578365188660450ist_fm @ P8 @ ( cons_fm @ X @ nil_fm ) ) )
=> ~ ! [P8: fm > fm > $o,X: fm,Y4: fm,Xs2: list_fm] :
( X2
!= ( produc7687578365188660450ist_fm @ P8 @ ( cons_fm @ X @ ( cons_fm @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_449_successively_Ocases,axiom,
! [X2: produc2002131169352006116ist_tm] :
( ! [P8: tm > tm > $o] :
( X2
!= ( produc1972851280741670356ist_tm @ P8 @ nil_tm ) )
=> ( ! [P8: tm > tm > $o,X: tm] :
( X2
!= ( produc1972851280741670356ist_tm @ P8 @ ( cons_tm @ X @ nil_tm ) ) )
=> ~ ! [P8: tm > tm > $o,X: tm,Y4: tm,Xs2: list_tm] :
( X2
!= ( produc1972851280741670356ist_tm @ P8 @ ( cons_tm @ X @ ( cons_tm @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_450_successively_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P8: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P8 @ nil_nat ) )
=> ( ! [P8: nat > nat > $o,X: nat] :
( X2
!= ( produc4727192421694094319st_nat @ P8 @ ( cons_nat @ X @ nil_nat ) ) )
=> ~ ! [P8: nat > nat > $o,X: nat,Y4: nat,Xs2: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P8 @ ( cons_nat @ X @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_451_set__ConsD,axiom,
! [Y3: product_prod_fm_fm,X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm] :
( ( member7780952600467998736_fm_fm @ Y3 @ ( set_Pr5149718152543245948_fm_fm @ ( cons_P2476253307934258077_fm_fm @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member7780952600467998736_fm_fm @ Y3 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_452_set__ConsD,axiom,
! [Y3: fm,X2: fm,Xs: list_fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member_fm2 @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_453_set__ConsD,axiom,
! [Y3: tm,X2: tm,Xs: list_tm] :
( ( member_tm2 @ Y3 @ ( set_tm2 @ ( cons_tm @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member_tm2 @ Y3 @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_454_set__ConsD,axiom,
! [Y3: nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ Y3 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y3 = X2 )
| ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_455_list_Oset__cases,axiom,
! [E: product_prod_fm_fm,A: list_P8031219080602320621_fm_fm] :
( ( member7780952600467998736_fm_fm @ E @ ( set_Pr5149718152543245948_fm_fm @ A ) )
=> ( ! [Z22: list_P8031219080602320621_fm_fm] :
( A
!= ( cons_P2476253307934258077_fm_fm @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_fm_fm,Z22: list_P8031219080602320621_fm_fm] :
( ( A
= ( cons_P2476253307934258077_fm_fm @ Z1 @ Z22 ) )
=> ~ ( member7780952600467998736_fm_fm @ E @ ( set_Pr5149718152543245948_fm_fm @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_456_list_Oset__cases,axiom,
! [E: fm,A: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A ) )
=> ( ! [Z22: list_fm] :
( A
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_457_list_Oset__cases,axiom,
! [E: tm,A: list_tm] :
( ( member_tm2 @ E @ ( set_tm2 @ A ) )
=> ( ! [Z22: list_tm] :
( A
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm2 @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_458_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_459_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_fm_fm,X222: list_P8031219080602320621_fm_fm] : ( member7780952600467998736_fm_fm @ X21 @ ( set_Pr5149718152543245948_fm_fm @ ( cons_P2476253307934258077_fm_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_460_list_Oset__intros_I1_J,axiom,
! [X21: fm,X222: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_461_list_Oset__intros_I1_J,axiom,
! [X21: tm,X222: list_tm] : ( member_tm2 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_462_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_463_list_Oset__intros_I2_J,axiom,
! [Y3: product_prod_fm_fm,X222: list_P8031219080602320621_fm_fm,X21: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ Y3 @ ( set_Pr5149718152543245948_fm_fm @ X222 ) )
=> ( member7780952600467998736_fm_fm @ Y3 @ ( set_Pr5149718152543245948_fm_fm @ ( cons_P2476253307934258077_fm_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_464_list_Oset__intros_I2_J,axiom,
! [Y3: fm,X222: list_fm,X21: fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ X222 ) )
=> ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_465_list_Oset__intros_I2_J,axiom,
! [Y3: tm,X222: list_tm,X21: tm] :
( ( member_tm2 @ Y3 @ ( set_tm2 @ X222 ) )
=> ( member_tm2 @ Y3 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_466_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X222: list_nat,X21: nat] :
( ( member_nat2 @ Y3 @ ( set_nat2 @ X222 ) )
=> ( member_nat2 @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_467_semantics__list_Osimps_I2_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm,T: tm,L3: list_tm] :
( ( semantics_list_fm @ E @ F @ ( cons_tm @ T @ L3 ) )
= ( cons_fm @ ( semantics_term_fm @ E @ F @ T ) @ ( semantics_list_fm @ E @ F @ L3 ) ) ) ).
% semantics_list.simps(2)
thf(fact_468_semantics__list_Osimps_I2_J,axiom,
! [E: nat > nat,F: nat > list_nat > nat,T: tm,L3: list_tm] :
( ( semantics_list_nat @ E @ F @ ( cons_tm @ T @ L3 ) )
= ( cons_nat @ ( semantics_term_nat @ E @ F @ T ) @ ( semantics_list_nat @ E @ F @ L3 ) ) ) ).
% semantics_list.simps(2)
thf(fact_469_semantics__list_Osimps_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,T: tm,L3: list_tm] :
( ( semantics_list_tm @ E @ F @ ( cons_tm @ T @ L3 ) )
= ( cons_tm @ ( semantics_term_tm @ E @ F @ T ) @ ( semantics_list_tm @ E @ F @ L3 ) ) ) ).
% semantics_list.simps(2)
thf(fact_470_params_Osimps_I7_J,axiom,
! [P3: fm] :
( ( params @ ( neg @ P3 ) )
= ( params @ P3 ) ) ).
% params.simps(7)
thf(fact_471_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: fm,Q3: fm,Z3: list_fm] :
( ( member_fm @ P3 @ ( cons_fm @ Q3 @ Z3 ) )
= ( ( P3 != Q3 )
=> ( member_fm @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_472_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: tm,Q3: tm,Z3: list_tm] :
( ( member_tm @ P3 @ ( cons_tm @ Q3 @ Z3 ) )
= ( ( P3 != Q3 )
=> ( member_tm @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_473_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: nat,Q3: nat,Z3: list_nat] :
( ( member_nat @ P3 @ ( cons_nat @ Q3 @ Z3 ) )
= ( ( P3 != Q3 )
=> ( member_nat @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_474_params_Osimps_I5_J,axiom,
! [P3: fm] :
( ( params @ ( exi @ P3 ) )
= ( params @ P3 ) ) ).
% params.simps(5)
thf(fact_475_params_Osimps_I6_J,axiom,
! [P3: fm] :
( ( params @ ( uni @ P3 ) )
= ( params @ P3 ) ) ).
% params.simps(6)
thf(fact_476_set__subset__Cons,axiom,
! [Xs: list_fm,X2: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_477_set__subset__Cons,axiom,
! [Xs: list_tm,X2: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_478_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_479_list__induct2,axiom,
! [Xs: list_fm,Ys: list_fm,P: list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_480_list__induct2,axiom,
! [Xs: list_fm,Ys: list_tm,P: list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_481_list__induct2,axiom,
! [Xs: list_fm,Ys: list_nat,P: list_fm > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_fm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: nat,Ys3: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_482_list__induct2,axiom,
! [Xs: list_tm,Ys: list_fm,P: list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_tm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm,Y4: fm,Ys3: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_483_list__induct2,axiom,
! [Xs: list_tm,Ys: list_tm,P: list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_tm @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm,Y4: tm,Ys3: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_484_list__induct2,axiom,
! [Xs: list_tm,Ys: list_nat,P: list_tm > list_nat > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_tm @ nil_nat )
=> ( ! [X: tm,Xs2: list_tm,Y4: nat,Ys3: list_nat] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_485_list__induct2,axiom,
! [Xs: list_nat,Ys: list_fm,P: list_nat > list_fm > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_nat @ nil_fm )
=> ( ! [X: nat,Xs2: list_nat,Y4: fm,Ys3: list_fm] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_486_list__induct2,axiom,
! [Xs: list_nat,Ys: list_tm,P: list_nat > list_tm > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_nat @ nil_tm )
=> ( ! [X: nat,Xs2: list_nat,Y4: tm,Ys3: list_tm] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_487_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat,Y4: nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_488_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,P: list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_489_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,P: list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_490_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_nat,P: list_fm > list_fm > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: nat,Zs2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_491_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,P: list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm,Z5: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_492_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,P: list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm,Z5: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_493_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_nat,P: list_fm > list_tm > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm,Z5: nat,Zs2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_494_list__induct3,axiom,
! [Xs: list_fm,Ys: list_nat,Zs: list_fm,P: list_fm > list_nat > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_nat @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: nat,Ys3: list_nat,Z5: fm,Zs2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_495_list__induct3,axiom,
! [Xs: list_fm,Ys: list_nat,Zs: list_tm,P: list_fm > list_nat > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_fm @ nil_nat @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: nat,Ys3: list_nat,Z5: tm,Zs2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_496_list__induct3,axiom,
! [Xs: list_fm,Ys: list_nat,Zs: list_nat,P: list_fm > list_nat > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_fm @ nil_nat @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: nat,Ys3: list_nat,Z5: nat,Zs2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_497_list__induct3,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,P: list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm,Y4: fm,Ys3: list_fm,Z5: fm,Zs2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_498_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_fm,P: list_fm > list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: fm,Zs2: list_fm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_499_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_tm,P: list_fm > list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: fm,Zs2: list_fm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_500_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_nat,P: list_fm > list_fm > list_fm > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: fm,Zs2: list_fm,W: nat,Ws2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_501_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_fm,P: list_fm > list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: tm,Zs2: list_tm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_502_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_tm,P: list_fm > list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: tm,Zs2: list_tm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_503_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_nat,P: list_fm > list_fm > list_tm > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: tm,Zs2: list_tm,W: nat,Ws2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_tm @ Z5 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_504_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_nat,Ws: list_fm,P: list_fm > list_fm > list_nat > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_nat @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: nat,Zs2: list_nat,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_505_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_nat,Ws: list_tm,P: list_fm > list_fm > list_nat > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_nat @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: nat,Zs2: list_nat,W: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_506_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_nat,Ws: list_nat,P: list_fm > list_fm > list_nat > list_nat > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_nat @ nil_nat )
=> ( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm,Z5: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) @ ( cons_nat @ Z5 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_507_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,Ws: list_fm,P: list_fm > list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm,Y4: tm,Ys3: list_tm,Z5: fm,Zs2: list_fm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) @ ( cons_fm @ Z5 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_508_impossible__Cons,axiom,
! [Xs: list_fm,Ys: list_fm,X2: fm] :
( ( ord_less_eq_nat @ ( size_size_list_fm @ Xs ) @ ( size_size_list_fm @ Ys ) )
=> ( Xs
!= ( cons_fm @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_509_impossible__Cons,axiom,
! [Xs: list_tm,Ys: list_tm,X2: tm] :
( ( ord_less_eq_nat @ ( size_size_list_tm @ Xs ) @ ( size_size_list_tm @ Ys ) )
=> ( Xs
!= ( cons_tm @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_510_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_511_list__ex1__iff,axiom,
( list_e1430114047001372208_fm_fm
= ( ^ [P5: product_prod_fm_fm > $o,Xs3: list_P8031219080602320621_fm_fm] :
? [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ ( set_Pr5149718152543245948_fm_fm @ Xs3 ) )
& ( P5 @ X5 )
& ! [Y: product_prod_fm_fm] :
( ( ( member7780952600467998736_fm_fm @ Y @ ( set_Pr5149718152543245948_fm_fm @ Xs3 ) )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_512_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P5: tm > $o,Xs3: list_tm] :
? [X5: tm] :
( ( member_tm2 @ X5 @ ( set_tm2 @ Xs3 ) )
& ( P5 @ X5 )
& ! [Y: tm] :
( ( ( member_tm2 @ Y @ ( set_tm2 @ Xs3 ) )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_513_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P5: nat > $o,Xs3: list_nat] :
? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs3 ) )
& ( P5 @ X5 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ ( set_nat2 @ Xs3 ) )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_514_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P5: fm > $o,Xs3: list_fm] :
? [X5: fm] :
( ( member_fm2 @ X5 @ ( set_fm2 @ Xs3 ) )
& ( P5 @ X5 )
& ! [Y: fm] :
( ( ( member_fm2 @ Y @ ( set_fm2 @ Xs3 ) )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_515_ext_Osimps_I2_J,axiom,
! [Y3: list_fm,P3: fm,Z3: list_fm] :
( ( ext_fm @ Y3 @ ( cons_fm @ P3 @ Z3 ) )
= ( ( ( member_fm @ P3 @ Y3 )
=> ( ext_fm @ Y3 @ Z3 ) )
& ( member_fm @ P3 @ Y3 ) ) ) ).
% ext.simps(2)
thf(fact_516_ext_Osimps_I2_J,axiom,
! [Y3: list_tm,P3: tm,Z3: list_tm] :
( ( ext_tm @ Y3 @ ( cons_tm @ P3 @ Z3 ) )
= ( ( ( member_tm @ P3 @ Y3 )
=> ( ext_tm @ Y3 @ Z3 ) )
& ( member_tm @ P3 @ Y3 ) ) ) ).
% ext.simps(2)
thf(fact_517_ext_Osimps_I2_J,axiom,
! [Y3: list_nat,P3: nat,Z3: list_nat] :
( ( ext_nat @ Y3 @ ( cons_nat @ P3 @ Z3 ) )
= ( ( ( member_nat @ P3 @ Y3 )
=> ( ext_nat @ Y3 @ Z3 ) )
& ( member_nat @ P3 @ Y3 ) ) ) ).
% ext.simps(2)
thf(fact_518_n__lists_Osimps_I1_J,axiom,
! [Xs: list_tm] :
( ( n_lists_tm @ zero_zero_nat @ Xs )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% n_lists.simps(1)
thf(fact_519_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_520_n__lists_Osimps_I1_J,axiom,
! [Xs: list_fm] :
( ( n_lists_fm @ zero_zero_nat @ Xs )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% n_lists.simps(1)
thf(fact_521_listFunTm_Osimps_I2_J,axiom,
! [N2: nat] :
( ( listFunTm @ ( var @ N2 ) )
= nil_nat ) ).
% listFunTm.simps(2)
thf(fact_522_can__select__set__list__ex1,axiom,
! [P: tm > $o,A3: list_tm] :
( ( can_select_tm @ P @ ( set_tm2 @ A3 ) )
= ( list_ex1_tm @ P @ A3 ) ) ).
% can_select_set_list_ex1
thf(fact_523_can__select__set__list__ex1,axiom,
! [P: nat > $o,A3: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A3 ) )
= ( list_ex1_nat @ P @ A3 ) ) ).
% can_select_set_list_ex1
thf(fact_524_can__select__set__list__ex1,axiom,
! [P: fm > $o,A3: list_fm] :
( ( can_select_fm @ P @ ( set_fm2 @ A3 ) )
= ( list_ex1_fm @ P @ A3 ) ) ).
% can_select_set_list_ex1
thf(fact_525_subtermFm__subset__params,axiom,
! [P3: fm,A3: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P3 ) ) @ ( set_tm2 @ A3 ) )
=> ( ord_less_eq_set_nat @ ( params @ P3 ) @ ( paramsts @ A3 ) ) ) ).
% subtermFm_subset_params
thf(fact_526_longest__common__prefix_Ocases,axiom,
! [X2: produc3245234490656042599ist_fm] :
( ! [X: fm,Xs2: list_fm,Y4: fm,Ys3: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y4 @ Ys3 ) ) )
=> ( ! [Uv: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ nil_fm @ Uv ) )
=> ~ ! [Uu2: list_fm] :
( X2
!= ( produc7863996417982153943ist_fm @ Uu2 @ nil_fm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_527_longest__common__prefix_Ocases,axiom,
! [X2: produc5776448205642668775ist_tm] :
( ! [X: tm,Xs2: list_tm,Y4: tm,Ys3: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y4 @ Ys3 ) ) )
=> ( ! [Uv: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ nil_tm @ Uv ) )
=> ~ ! [Uu2: list_tm] :
( X2
!= ( produc1418304791525149271ist_tm @ Uu2 @ nil_tm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_528_longest__common__prefix_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X: nat,Xs2: list_nat,Y4: nat,Ys3: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
=> ( ! [Uv: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Uu2 @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_529_GammaUni,axiom,
! [T: tm,P3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T @ P3 ) ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( uni @ P3 ) ) @ Z3 ) ) ) ).
% GammaUni
thf(fact_530_the__elem__set,axiom,
! [X2: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X2 @ nil_fm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_531_the__elem__set,axiom,
! [X2: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X2 @ nil_tm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_532_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_533_Cons__in__lex,axiom,
! [X2: tm,Xs: list_tm,Y3: tm,Ys: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( lex_tm @ R2 ) )
= ( ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X2 @ Y3 ) @ R2 )
& ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_534_Cons__in__lex,axiom,
! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_535_Cons__in__lex,axiom,
! [X2: fm,Xs: list_fm,Y3: fm,Ys: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) @ ( lex_fm @ R2 ) )
= ( ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
& ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lex_fm @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_536_not__in__set__insert,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm] :
( ~ ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( ( insert7339457527920503555_fm_fm @ X2 @ Xs )
= ( cons_P2476253307934258077_fm_fm @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_537_not__in__set__insert,axiom,
! [X2: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X2 @ Xs )
= ( cons_fm @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_538_not__in__set__insert,axiom,
! [X2: tm,Xs: list_tm] :
( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X2 @ Xs )
= ( cons_tm @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_539_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_540_in__set__insert,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm] :
( ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( ( insert7339457527920503555_fm_fm @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_541_in__set__insert,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_542_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_543_in__set__insert,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_544_paramst__liftt_I2_J,axiom,
! [Ts2: list_tm] :
( ( paramsts @ ( liftts @ Ts2 ) )
= ( paramsts @ Ts2 ) ) ).
% paramst_liftt(2)
thf(fact_545_s1_I2_J,axiom,
( new_list
= ( ^ [C3: nat,L: list_tm] :
~ ( member_nat2 @ C3 @ ( paramsts @ L ) ) ) ) ).
% s1(2)
thf(fact_546_insert__Nil,axiom,
! [X2: fm] :
( ( insert_fm @ X2 @ nil_fm )
= ( cons_fm @ X2 @ nil_fm ) ) ).
% insert_Nil
thf(fact_547_insert__Nil,axiom,
! [X2: tm] :
( ( insert_tm @ X2 @ nil_tm )
= ( cons_tm @ X2 @ nil_tm ) ) ).
% insert_Nil
thf(fact_548_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_549_can__select__def,axiom,
( can_select_tm
= ( ^ [P5: tm > $o,A4: set_tm] :
? [X5: tm] :
( ( member_tm2 @ X5 @ A4 )
& ( P5 @ X5 )
& ! [Y: tm] :
( ( ( member_tm2 @ Y @ A4 )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_550_can__select__def,axiom,
( can_select_fm
= ( ^ [P5: fm > $o,A4: set_fm] :
? [X5: fm] :
( ( member_fm2 @ X5 @ A4 )
& ( P5 @ X5 )
& ! [Y: fm] :
( ( ( member_fm2 @ Y @ A4 )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_551_can__select__def,axiom,
( can_select_nat
= ( ^ [P5: nat > $o,A4: set_nat] :
? [X5: nat] :
( ( member_nat2 @ X5 @ A4 )
& ( P5 @ X5 )
& ! [Y: nat] :
( ( ( member_nat2 @ Y @ A4 )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_552_can__select__def,axiom,
( can_se8056291635612950372_fm_fm
= ( ^ [P5: product_prod_fm_fm > $o,A4: set_Pr4706815898642364871_fm_fm] :
? [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A4 )
& ( P5 @ X5 )
& ! [Y: product_prod_fm_fm] :
( ( ( member7780952600467998736_fm_fm @ Y @ A4 )
& ( P5 @ Y ) )
=> ( Y = X5 ) ) ) ) ) ).
% can_select_def
thf(fact_553_Nil2__notin__lex,axiom,
! [Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ nil_tm ) @ ( lex_tm @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_554_Nil2__notin__lex,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_555_Nil2__notin__lex,axiom,
! [Xs: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ nil_fm ) @ ( lex_fm @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_556_Nil__notin__lex,axiom,
! [Ys: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Ys ) @ ( lex_tm @ R2 ) ) ).
% Nil_notin_lex
thf(fact_557_Nil__notin__lex,axiom,
! [Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R2 ) ) ).
% Nil_notin_lex
thf(fact_558_Nil__notin__lex,axiom,
! [Ys: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Ys ) @ ( lex_fm @ R2 ) ) ).
% Nil_notin_lex
thf(fact_559_Neg,axiom,
! [P3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P3 ) ) @ Z3 ) ) ) ).
% Neg
thf(fact_560_Basic,axiom,
! [P3: fm,Z3: list_fm] :
( ( member_fm @ ( neg @ P3 ) @ Z3 )
=> ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) ) ) ).
% Basic
thf(fact_561_sub__list_Osimps_I2_J,axiom,
! [V: nat,S2: tm,T: tm,L3: list_tm] :
( ( sub_list @ V @ S2 @ ( cons_tm @ T @ L3 ) )
= ( cons_tm @ ( sub_term @ V @ S2 @ T ) @ ( sub_list @ V @ S2 @ L3 ) ) ) ).
% sub_list.simps(2)
thf(fact_562_liftts_Osimps_I2_J,axiom,
! [T: tm,Ts2: list_tm] :
( ( liftts @ ( cons_tm @ T @ Ts2 ) )
= ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts2 ) ) ) ).
% liftts.simps(2)
thf(fact_563_substts_Osimps_I2_J,axiom,
! [T: tm,Ts2: list_tm,S2: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts2 ) @ S2 @ K )
= ( cons_tm @ ( substt @ T @ S2 @ K ) @ ( substts @ Ts2 @ S2 @ K ) ) ) ).
% substts.simps(2)
thf(fact_564_inc__list_Osimps_I2_J,axiom,
! [T: tm,L3: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L3 ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L3 ) ) ) ).
% inc_list.simps(2)
thf(fact_565_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L3: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L3 ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L3 ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_566_List_Oinsert__def,axiom,
( insert7339457527920503555_fm_fm
= ( ^ [X5: product_prod_fm_fm,Xs3: list_P8031219080602320621_fm_fm] : ( if_lis6935098856802376371_fm_fm @ ( member7780952600467998736_fm_fm @ X5 @ ( set_Pr5149718152543245948_fm_fm @ Xs3 ) ) @ Xs3 @ ( cons_P2476253307934258077_fm_fm @ X5 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_567_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X5: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm2 @ X5 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X5 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_568_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X5: tm,Xs3: list_tm] : ( if_list_tm @ ( member_tm2 @ X5 @ ( set_tm2 @ Xs3 ) ) @ Xs3 @ ( cons_tm @ X5 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_569_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X5: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X5 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_570_params_Osimps_I1_J,axiom,
! [B: nat,Ts2: list_tm] :
( ( params @ ( pre @ B @ Ts2 ) )
= ( paramsts @ Ts2 ) ) ).
% params.simps(1)
thf(fact_571_subtermTm_Osimps_I2_J,axiom,
! [N2: nat] :
( ( subtermTm @ ( var @ N2 ) )
= ( cons_tm @ ( var @ N2 ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_572_paramsts__subset,axiom,
! [A3: list_tm,B2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A3 ) @ ( set_tm2 @ B2 ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A3 ) @ ( paramsts @ B2 ) ) ) ).
% paramsts_subset
thf(fact_573_GammaExi,axiom,
! [T: tm,P3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T @ P3 ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( exi @ P3 ) @ Z3 ) ) ) ).
% GammaExi
thf(fact_574_DeltaExi,axiom,
! [I2: nat,P3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P3 ) ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P3 @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P3 ) ) @ Z3 ) ) ) ) ).
% DeltaExi
thf(fact_575_DeltaUni,axiom,
! [I2: nat,P3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P3 ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P3 @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P3 ) @ Z3 ) ) ) ) ).
% DeltaUni
thf(fact_576_listFunTm_Osimps_I1_J,axiom,
! [N2: nat,Ts2: list_tm] :
( ( listFunTm @ ( fun @ N2 @ Ts2 ) )
= ( cons_nat @ N2 @ ( listFunTms @ Ts2 ) ) ) ).
% listFunTm.simps(1)
thf(fact_577_listFunTm__paramst_I2_J,axiom,
! [Ts2: list_tm] :
( ( set_nat2 @ ( listFunTms @ Ts2 ) )
= ( paramsts @ Ts2 ) ) ).
% listFunTm_paramst(2)
thf(fact_578_paramst__sub__term_I2_J,axiom,
! [M: nat,S2: tm,L3: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S2 @ L3 ) ) @ ( sup_sup_set_nat @ ( paramst @ S2 ) @ ( paramsts @ L3 ) ) ) ).
% paramst_sub_term(2)
thf(fact_579_UnCI,axiom,
! [C: tm,B2: set_tm,A3: set_tm] :
( ( ~ ( member_tm2 @ C @ B2 )
=> ( member_tm2 @ C @ A3 ) )
=> ( member_tm2 @ C @ ( sup_sup_set_tm @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_580_UnCI,axiom,
! [C: fm,B2: set_fm,A3: set_fm] :
( ( ~ ( member_fm2 @ C @ B2 )
=> ( member_fm2 @ C @ A3 ) )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_581_UnCI,axiom,
! [C: product_prod_fm_fm,B2: set_Pr4706815898642364871_fm_fm,A3: set_Pr4706815898642364871_fm_fm] :
( ( ~ ( member7780952600467998736_fm_fm @ C @ B2 )
=> ( member7780952600467998736_fm_fm @ C @ A3 ) )
=> ( member7780952600467998736_fm_fm @ C @ ( sup_su5810838807072965531_fm_fm @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_582_UnCI,axiom,
! [C: nat,B2: set_nat,A3: set_nat] :
( ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ A3 ) )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_583_Un__iff,axiom,
! [C: tm,A3: set_tm,B2: set_tm] :
( ( member_tm2 @ C @ ( sup_sup_set_tm @ A3 @ B2 ) )
= ( ( member_tm2 @ C @ A3 )
| ( member_tm2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_584_Un__iff,axiom,
! [C: fm,A3: set_fm,B2: set_fm] :
( ( member_fm2 @ C @ ( sup_sup_set_fm @ A3 @ B2 ) )
= ( ( member_fm2 @ C @ A3 )
| ( member_fm2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_585_Un__iff,axiom,
! [C: product_prod_fm_fm,A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm] :
( ( member7780952600467998736_fm_fm @ C @ ( sup_su5810838807072965531_fm_fm @ A3 @ B2 ) )
= ( ( member7780952600467998736_fm_fm @ C @ A3 )
| ( member7780952600467998736_fm_fm @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_586_Un__iff,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
= ( ( member_nat2 @ C @ A3 )
| ( member_nat2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_587_member__remove,axiom,
! [X2: tm,Y3: tm,A3: set_tm] :
( ( member_tm2 @ X2 @ ( remove_tm @ Y3 @ A3 ) )
= ( ( member_tm2 @ X2 @ A3 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_588_member__remove,axiom,
! [X2: fm,Y3: fm,A3: set_fm] :
( ( member_fm2 @ X2 @ ( remove_fm @ Y3 @ A3 ) )
= ( ( member_fm2 @ X2 @ A3 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_589_member__remove,axiom,
! [X2: nat,Y3: nat,A3: set_nat] :
( ( member_nat2 @ X2 @ ( remove_nat @ Y3 @ A3 ) )
= ( ( member_nat2 @ X2 @ A3 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_590_member__remove,axiom,
! [X2: product_prod_fm_fm,Y3: product_prod_fm_fm,A3: set_Pr4706815898642364871_fm_fm] :
( ( member7780952600467998736_fm_fm @ X2 @ ( remove1805396771031664588_fm_fm @ Y3 @ A3 ) )
= ( ( member7780952600467998736_fm_fm @ X2 @ A3 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_591_Un__subset__iff,axiom,
! [A3: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A3 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_tm @ A3 @ C2 )
& ( ord_less_eq_set_tm @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_592_Un__subset__iff,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A3 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_593_s3,axiom,
( news
= ( ^ [C3: nat] :
( list_all_fm
@ ^ [P7: fm] :
~ ( member_nat2 @ C3 @ ( params @ P7 ) ) ) ) ) ).
% s3
thf(fact_594_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_595_UnE,axiom,
! [C: tm,A3: set_tm,B2: set_tm] :
( ( member_tm2 @ C @ ( sup_sup_set_tm @ A3 @ B2 ) )
=> ( ~ ( member_tm2 @ C @ A3 )
=> ( member_tm2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_596_UnE,axiom,
! [C: fm,A3: set_fm,B2: set_fm] :
( ( member_fm2 @ C @ ( sup_sup_set_fm @ A3 @ B2 ) )
=> ( ~ ( member_fm2 @ C @ A3 )
=> ( member_fm2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_597_UnE,axiom,
! [C: product_prod_fm_fm,A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm] :
( ( member7780952600467998736_fm_fm @ C @ ( sup_su5810838807072965531_fm_fm @ A3 @ B2 ) )
=> ( ~ ( member7780952600467998736_fm_fm @ C @ A3 )
=> ( member7780952600467998736_fm_fm @ C @ B2 ) ) ) ).
% UnE
thf(fact_598_UnE,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
=> ( ~ ( member_nat2 @ C @ A3 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_599_UnI1,axiom,
! [C: tm,A3: set_tm,B2: set_tm] :
( ( member_tm2 @ C @ A3 )
=> ( member_tm2 @ C @ ( sup_sup_set_tm @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_600_UnI1,axiom,
! [C: fm,A3: set_fm,B2: set_fm] :
( ( member_fm2 @ C @ A3 )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_601_UnI1,axiom,
! [C: product_prod_fm_fm,A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm] :
( ( member7780952600467998736_fm_fm @ C @ A3 )
=> ( member7780952600467998736_fm_fm @ C @ ( sup_su5810838807072965531_fm_fm @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_602_UnI1,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A3 )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_603_UnI2,axiom,
! [C: tm,B2: set_tm,A3: set_tm] :
( ( member_tm2 @ C @ B2 )
=> ( member_tm2 @ C @ ( sup_sup_set_tm @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_604_UnI2,axiom,
! [C: fm,B2: set_fm,A3: set_fm] :
( ( member_fm2 @ C @ B2 )
=> ( member_fm2 @ C @ ( sup_sup_set_fm @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_605_UnI2,axiom,
! [C: product_prod_fm_fm,B2: set_Pr4706815898642364871_fm_fm,A3: set_Pr4706815898642364871_fm_fm] :
( ( member7780952600467998736_fm_fm @ C @ B2 )
=> ( member7780952600467998736_fm_fm @ C @ ( sup_su5810838807072965531_fm_fm @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_606_UnI2,axiom,
! [C: nat,B2: set_nat,A3: set_nat] :
( ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_607_bex__Un,axiom,
! [A3: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( sup_sup_set_nat @ A3 @ B2 ) )
& ( P @ X5 ) ) )
= ( ? [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
& ( P @ X5 ) )
| ? [X5: nat] :
( ( member_nat2 @ X5 @ B2 )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_608_ball__Un,axiom,
! [A3: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat2 @ X5 @ ( sup_sup_set_nat @ A3 @ B2 ) )
=> ( P @ X5 ) ) )
= ( ! [X5: nat] :
( ( member_nat2 @ X5 @ A3 )
=> ( P @ X5 ) )
& ! [X5: nat] :
( ( member_nat2 @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_609_Un__assoc,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_610_Un__absorb,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_611_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_612_Un__left__absorb,axiom,
! [A3: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) )
= ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_613_Un__left__commute,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_614_Un__def,axiom,
( sup_sup_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( collect_tm
@ ^ [X5: tm] :
( ( member_tm2 @ X5 @ A4 )
| ( member_tm2 @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_615_Un__def,axiom,
( sup_sup_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
( collect_fm
@ ^ [X5: fm] :
( ( member_fm2 @ X5 @ A4 )
| ( member_fm2 @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_616_Un__def,axiom,
( sup_su5810838807072965531_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
( collec7637684051871000146_fm_fm
@ ^ [X5: product_prod_fm_fm] :
( ( member7780952600467998736_fm_fm @ X5 @ A4 )
| ( member7780952600467998736_fm_fm @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_617_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X5: nat] :
( ( member_nat2 @ X5 @ A4 )
| ( member_nat2 @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_618_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
| ( Q @ X5 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_619_subset__Un__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( sup_sup_set_tm @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_620_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_621_subset__UnE,axiom,
! [C2: set_tm,A3: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ C2 @ ( sup_sup_set_tm @ A3 @ B2 ) )
=> ~ ! [A7: set_tm] :
( ( ord_less_eq_set_tm @ A7 @ A3 )
=> ! [B7: set_tm] :
( ( ord_less_eq_set_tm @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_tm @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_622_subset__UnE,axiom,
! [C2: set_nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A3 )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_623_Un__absorb2,axiom,
! [B2: set_tm,A3: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A3 )
=> ( ( sup_sup_set_tm @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_624_Un__absorb2,axiom,
! [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_625_Un__absorb1,axiom,
! [A3: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( sup_sup_set_tm @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_626_Un__absorb1,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( sup_sup_set_nat @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_627_Un__upper2,axiom,
! [B2: set_tm,A3: set_tm] : ( ord_less_eq_set_tm @ B2 @ ( sup_sup_set_tm @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_628_Un__upper2,axiom,
! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_629_Un__upper1,axiom,
! [A3: set_tm,B2: set_tm] : ( ord_less_eq_set_tm @ A3 @ ( sup_sup_set_tm @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_630_Un__upper1,axiom,
! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_631_Un__least,axiom,
! [A3: set_tm,C2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ C2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A3 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_632_Un__least,axiom,
! [A3: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_633_Un__mono,axiom,
! [A3: set_tm,C2: set_tm,B2: set_tm,D: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ C2 )
=> ( ( ord_less_eq_set_tm @ B2 @ D )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A3 @ B2 ) @ ( sup_sup_set_tm @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_634_Un__mono,axiom,
! [A3: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_635_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts2: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts2 ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts2 ) ) ) ).
% paramsts.simps(2)
thf(fact_636_paramst__sub__term_I1_J,axiom,
! [M: nat,S2: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S2 @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S2 ) @ ( paramst @ T ) ) ) ).
% paramst_sub_term(1)
thf(fact_637_params__sub,axiom,
! [M: nat,T: tm,P3: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P3 ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P3 ) ) ) ).
% params_sub
thf(fact_638_le__sup__iff,axiom,
! [X2: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ X2 @ Y3 ) @ Z3 )
= ( ( ord_less_eq_set_tm @ X2 @ Z3 )
& ( ord_less_eq_set_tm @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_639_le__sup__iff,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z3 )
= ( ( ord_less_eq_nat @ X2 @ Z3 )
& ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_640_le__sup__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z3 )
= ( ( ord_less_eq_set_nat @ X2 @ Z3 )
& ( ord_less_eq_set_nat @ Y3 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_641_sup_Obounded__iff,axiom,
! [B: set_tm,C: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B @ C ) @ A )
= ( ( ord_less_eq_set_tm @ B @ A )
& ( ord_less_eq_set_tm @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_642_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_643_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_644_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_645_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_646_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_647_inf__sup__ord_I4_J,axiom,
! [Y3: set_tm,X2: set_tm] : ( ord_less_eq_set_tm @ Y3 @ ( sup_sup_set_tm @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_648_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_649_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_650_inf__sup__ord_I3_J,axiom,
! [X2: set_tm,Y3: set_tm] : ( ord_less_eq_set_tm @ X2 @ ( sup_sup_set_tm @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_651_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_652_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_653_le__supE,axiom,
! [A: set_tm,B: set_tm,X2: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_tm @ A @ X2 )
=> ~ ( ord_less_eq_set_tm @ B @ X2 ) ) ) ).
% le_supE
thf(fact_654_le__supE,axiom,
! [A: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_655_le__supE,axiom,
! [A: set_nat,B: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_656_sup__Un__eq2,axiom,
! [R: set_Pr4706815898642364871_fm_fm,S: set_Pr4706815898642364871_fm_fm] :
( ( sup_sup_fm_fm_o
@ ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ R )
@ ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ S ) )
= ( ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ ( sup_su5810838807072965531_fm_fm @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_657_sup__Un__eq,axiom,
! [R: set_tm,S: set_tm] :
( ( sup_sup_tm_o
@ ^ [X5: tm] : ( member_tm2 @ X5 @ R )
@ ^ [X5: tm] : ( member_tm2 @ X5 @ S ) )
= ( ^ [X5: tm] : ( member_tm2 @ X5 @ ( sup_sup_set_tm @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_658_sup__Un__eq,axiom,
! [R: set_fm,S: set_fm] :
( ( sup_sup_fm_o
@ ^ [X5: fm] : ( member_fm2 @ X5 @ R )
@ ^ [X5: fm] : ( member_fm2 @ X5 @ S ) )
= ( ^ [X5: fm] : ( member_fm2 @ X5 @ ( sup_sup_set_fm @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_659_sup__Un__eq,axiom,
! [R: set_Pr4706815898642364871_fm_fm,S: set_Pr4706815898642364871_fm_fm] :
( ( sup_su6818665535280656642m_fm_o
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ R )
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ S ) )
= ( ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ ( sup_su5810838807072965531_fm_fm @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_660_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X5: nat] : ( member_nat2 @ X5 @ R )
@ ^ [X5: nat] : ( member_nat2 @ X5 @ S ) )
= ( ^ [X5: nat] : ( member_nat2 @ X5 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_661_sup__set__def,axiom,
( sup_sup_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( collect_tm
@ ( sup_sup_tm_o
@ ^ [X5: tm] : ( member_tm2 @ X5 @ A4 )
@ ^ [X5: tm] : ( member_tm2 @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_662_sup__set__def,axiom,
( sup_sup_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
( collect_fm
@ ( sup_sup_fm_o
@ ^ [X5: fm] : ( member_fm2 @ X5 @ A4 )
@ ^ [X5: fm] : ( member_fm2 @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_663_sup__set__def,axiom,
( sup_su5810838807072965531_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
( collec7637684051871000146_fm_fm
@ ( sup_su6818665535280656642m_fm_o
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ A4 )
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_664_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X5: nat] : ( member_nat2 @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat2 @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_665_sup_OcoboundedI2,axiom,
! [C: set_tm,B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ C @ B )
=> ( ord_less_eq_set_tm @ C @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_666_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_667_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_668_sup_OcoboundedI1,axiom,
! [C: set_tm,A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ C @ A )
=> ( ord_less_eq_set_tm @ C @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_669_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_670_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_671_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B4: set_tm] :
( ( sup_sup_set_tm @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_672_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_673_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_674_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_tm
= ( ^ [B4: set_tm,A5: set_tm] :
( ( sup_sup_set_tm @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_675_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_676_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_677_sup_Ocobounded2,axiom,
! [B: set_tm,A: set_tm] : ( ord_less_eq_set_tm @ B @ ( sup_sup_set_tm @ A @ B ) ) ).
% sup.cobounded2
thf(fact_678_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_679_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_680_sup_Ocobounded1,axiom,
! [A: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ A @ ( sup_sup_set_tm @ A @ B ) ) ).
% sup.cobounded1
thf(fact_681_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_682_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_683_sup_Oorder__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [B4: set_tm,A5: set_tm] :
( A5
= ( sup_sup_set_tm @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_684_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_685_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_686_sup_OboundedI,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ C @ A )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_687_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_688_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_689_sup_OboundedE,axiom,
! [B: set_tm,C: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_tm @ B @ A )
=> ~ ( ord_less_eq_set_tm @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_690_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_691_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_692_sup__absorb2,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( sup_sup_set_tm @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_693_sup__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_694_sup__absorb2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_695_sup__absorb1,axiom,
! [Y3: set_tm,X2: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X2 )
=> ( ( sup_sup_set_tm @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_696_sup__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_697_sup__absorb1,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_698_sup_Oabsorb2,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( sup_sup_set_tm @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_699_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_700_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_701_sup_Oabsorb1,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( sup_sup_set_tm @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_702_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_703_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_704_sup__unique,axiom,
! [F: set_tm > set_tm > set_tm,X2: set_tm,Y3: set_tm] :
( ! [X: set_tm,Y4: set_tm] : ( ord_less_eq_set_tm @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: set_tm,Y4: set_tm] : ( ord_less_eq_set_tm @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: set_tm,Y4: set_tm,Z5: set_tm] :
( ( ord_less_eq_set_tm @ Y4 @ X )
=> ( ( ord_less_eq_set_tm @ Z5 @ X )
=> ( ord_less_eq_set_tm @ ( F @ Y4 @ Z5 ) @ X ) ) )
=> ( ( sup_sup_set_tm @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_705_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: nat,Y4: nat,Z5: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_less_eq_nat @ Z5 @ X )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z5 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_706_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
( ! [X: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y4 ) )
=> ( ! [X: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X @ Y4 ) )
=> ( ! [X: set_nat,Y4: set_nat,Z5: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X )
=> ( ( ord_less_eq_set_nat @ Z5 @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z5 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_707_sup_OorderI,axiom,
! [A: set_tm,B: set_tm] :
( ( A
= ( sup_sup_set_tm @ A @ B ) )
=> ( ord_less_eq_set_tm @ B @ A ) ) ).
% sup.orderI
thf(fact_708_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_709_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_710_sup_OorderE,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( A
= ( sup_sup_set_tm @ A @ B ) ) ) ).
% sup.orderE
thf(fact_711_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_712_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_713_le__iff__sup,axiom,
( ord_less_eq_set_tm
= ( ^ [X5: set_tm,Y: set_tm] :
( ( sup_sup_set_tm @ X5 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_714_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y: nat] :
( ( sup_sup_nat @ X5 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_715_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X5: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X5 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_716_sup__least,axiom,
! [Y3: set_tm,X2: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X2 )
=> ( ( ord_less_eq_set_tm @ Z3 @ X2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_717_sup__least,axiom,
! [Y3: nat,X2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_718_sup__least,axiom,
! [Y3: set_nat,X2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z3 ) @ X2 ) ) ) ).
% sup_least
thf(fact_719_sup__mono,axiom,
! [A: set_tm,C: set_tm,B: set_tm,D2: set_tm] :
( ( ord_less_eq_set_tm @ A @ C )
=> ( ( ord_less_eq_set_tm @ B @ D2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ ( sup_sup_set_tm @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_720_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_721_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_722_sup_Omono,axiom,
! [C: set_tm,A: set_tm,D2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ C @ A )
=> ( ( ord_less_eq_set_tm @ D2 @ B )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ C @ D2 ) @ ( sup_sup_set_tm @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_723_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_724_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_725_le__supI2,axiom,
! [X2: set_tm,B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ B )
=> ( ord_less_eq_set_tm @ X2 @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% le_supI2
thf(fact_726_le__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_727_le__supI2,axiom,
! [X2: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_728_le__supI1,axiom,
! [X2: set_tm,A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ A )
=> ( ord_less_eq_set_tm @ X2 @ ( sup_sup_set_tm @ A @ B ) ) ) ).
% le_supI1
thf(fact_729_le__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_730_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_731_sup__ge2,axiom,
! [Y3: set_tm,X2: set_tm] : ( ord_less_eq_set_tm @ Y3 @ ( sup_sup_set_tm @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_732_sup__ge2,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_733_sup__ge2,axiom,
! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_734_sup__ge1,axiom,
! [X2: set_tm,Y3: set_tm] : ( ord_less_eq_set_tm @ X2 @ ( sup_sup_set_tm @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_735_sup__ge1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_736_sup__ge1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_737_le__supI,axiom,
! [A: set_tm,X2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ X2 )
=> ( ( ord_less_eq_set_tm @ B @ X2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_738_le__supI,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_739_le__supI,axiom,
! [A: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_740_partition__set,axiom,
! [P: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ( sup_sup_set_tm @ ( set_tm2 @ Yes ) @ ( set_tm2 @ No ) )
= ( set_tm2 @ Xs ) ) ) ).
% partition_set
thf(fact_741_partition__set,axiom,
! [P: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_fm @ ( set_fm2 @ Yes ) @ ( set_fm2 @ No ) )
= ( set_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_742_partition__set,axiom,
! [P: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
( ( ( partition_nat @ P @ Xs )
= ( produc2694037385005941721st_nat @ Yes @ No ) )
=> ( ( sup_sup_set_nat @ ( set_nat2 @ Yes ) @ ( set_nat2 @ No ) )
= ( set_nat2 @ Xs ) ) ) ).
% partition_set
thf(fact_743_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_744_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_745_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_746_nth__equal__first__eq,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm,N2: nat] :
( ~ ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_s3074140853920721241_fm_fm @ Xs ) )
=> ( ( ( nth_Pr5768189175911290222_fm_fm @ ( cons_P2476253307934258077_fm_fm @ X2 @ Xs ) @ N2 )
= X2 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_747_nth__equal__first__eq,axiom,
! [X2: fm,Xs: list_fm,N2: nat] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ( ( nth_fm @ ( cons_fm @ X2 @ Xs ) @ N2 )
= X2 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_748_nth__equal__first__eq,axiom,
! [X2: tm,Xs: list_tm,N2: nat] :
( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( ( ( nth_tm @ ( cons_tm @ X2 @ Xs ) @ N2 )
= X2 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_749_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N2: nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
= X2 )
= ( N2 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_750_sequent__calculus_Ocases,axiom,
! [A: list_fm] :
( ( sequent_calculus @ A )
=> ( ! [P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ P9 @ Z5 ) )
=> ~ ( member_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ! [P9: fm,Q4: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( dis @ P9 @ Q4 ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P9 @ ( cons_fm @ Q4 @ Z5 ) ) ) )
=> ( ! [P9: fm,Q4: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( imp @ P9 @ Q4 ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P9 ) @ ( cons_fm @ Q4 @ Z5 ) ) ) )
=> ( ! [P9: fm,Q4: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( con @ P9 @ Q4 ) ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P9 ) @ ( cons_fm @ ( neg @ Q4 ) @ Z5 ) ) ) )
=> ( ! [P9: fm,Z5: list_fm,Q4: fm] :
( ( A
= ( cons_fm @ ( con @ P9 @ Q4 ) @ Z5 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P9 @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ Q4 @ Z5 ) ) ) )
=> ( ! [P9: fm,Z5: list_fm,Q4: fm] :
( ( A
= ( cons_fm @ ( neg @ ( imp @ P9 @ Q4 ) ) @ Z5 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P9 @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q4 ) @ Z5 ) ) ) )
=> ( ! [P9: fm,Z5: list_fm,Q4: fm] :
( ( A
= ( cons_fm @ ( neg @ ( dis @ P9 @ Q4 ) ) @ Z5 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q4 ) @ Z5 ) ) ) )
=> ( ! [T3: tm,P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( exi @ P9 ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T3 @ P9 ) @ Z5 ) ) )
=> ( ! [T3: tm,P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( uni @ P9 ) ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T3 @ P9 ) ) @ Z5 ) ) )
=> ( ! [I3: nat,P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( uni @ P9 ) @ Z5 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I3 @ nil_tm ) @ P9 ) @ Z5 ) )
=> ~ ( news @ I3 @ ( cons_fm @ P9 @ Z5 ) ) ) )
=> ( ! [I3: nat,P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( exi @ P9 ) ) @ Z5 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I3 @ nil_tm ) @ P9 ) ) @ Z5 ) )
=> ~ ( news @ I3 @ ( cons_fm @ P9 @ Z5 ) ) ) )
=> ( ! [P9: fm,Z5: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( neg @ P9 ) ) @ Z5 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P9 @ Z5 ) ) )
=> ~ ! [Z5: list_fm] :
( ( sequent_calculus @ Z5 )
=> ~ ( ext_fm @ A @ Z5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sequent_calculus.cases
thf(fact_751_sequent__calculus_Osimps,axiom,
( sequent_calculus
= ( ^ [A5: list_fm] :
( ? [P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ P7 @ Z2 ) )
& ( member_fm @ ( neg @ P7 ) @ Z2 ) )
| ? [P7: fm,Q5: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( dis @ P7 @ Q5 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ P7 @ ( cons_fm @ Q5 @ Z2 ) ) ) )
| ? [P7: fm,Q5: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( imp @ P7 @ Q5 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P7 ) @ ( cons_fm @ Q5 @ Z2 ) ) ) )
| ? [P7: fm,Q5: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( con @ P7 @ Q5 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P7 ) @ ( cons_fm @ ( neg @ Q5 ) @ Z2 ) ) ) )
| ? [P7: fm,Z2: list_fm,Q5: fm] :
( ( A5
= ( cons_fm @ ( con @ P7 @ Q5 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ P7 @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ Q5 @ Z2 ) ) )
| ? [P7: fm,Z2: list_fm,Q5: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( imp @ P7 @ Q5 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ P7 @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q5 ) @ Z2 ) ) )
| ? [P7: fm,Z2: list_fm,Q5: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( dis @ P7 @ Q5 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P7 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q5 ) @ Z2 ) ) )
| ? [T2: tm,P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( exi @ P7 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T2 @ P7 ) @ Z2 ) ) )
| ? [T2: tm,P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( uni @ P7 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T2 @ P7 ) ) @ Z2 ) ) )
| ? [I: nat,P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( uni @ P7 ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P7 ) @ Z2 ) )
& ( news @ I @ ( cons_fm @ P7 @ Z2 ) ) )
| ? [I: nat,P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( exi @ P7 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P7 ) ) @ Z2 ) )
& ( news @ I @ ( cons_fm @ P7 @ Z2 ) ) )
| ? [P7: fm,Z2: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( neg @ P7 ) ) @ Z2 ) )
& ( sequent_calculus @ ( cons_fm @ P7 @ Z2 ) ) )
| ? [Z2: list_fm,Y: list_fm] :
( ( A5 = Y )
& ( sequent_calculus @ Z2 )
& ( ext_fm @ Y @ Z2 ) ) ) ) ) ).
% sequent_calculus.simps
thf(fact_752_fm_Oinject_I4_J,axiom,
! [X41: fm,X42: fm,Y41: fm,Y42: fm] :
( ( ( con @ X41 @ X42 )
= ( con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(4)
thf(fact_753_fm_Oinject_I3_J,axiom,
! [X31: fm,X32: fm,Y31: fm,Y32: fm] :
( ( ( dis @ X31 @ X32 )
= ( dis @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(3)
thf(fact_754_fm_Oinject_I2_J,axiom,
! [X21: fm,X222: fm,Y21: fm,Y222: fm] :
( ( ( imp @ X21 @ X222 )
= ( imp @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% fm.inject(2)
thf(fact_755_nth__Cons__0,axiom,
! [X2: fm,Xs: list_fm] :
( ( nth_fm @ ( cons_fm @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_756_nth__Cons__0,axiom,
! [X2: tm,Xs: list_tm] :
( ( nth_tm @ ( cons_tm @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_757_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_758_fm_Odistinct_I23_J,axiom,
! [X31: fm,X32: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X32 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_759_fm_Odistinct_I15_J,axiom,
! [X21: fm,X222: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X222 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_760_fm_Odistinct_I13_J,axiom,
! [X21: fm,X222: fm,X31: fm,X32: fm] :
( ( imp @ X21 @ X222 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(13)
thf(fact_761_fm_Odistinct_I21_J,axiom,
! [X21: fm,X222: fm,X7: fm] :
( ( imp @ X21 @ X222 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_762_fm_Odistinct_I29_J,axiom,
! [X31: fm,X32: fm,X7: fm] :
( ( dis @ X31 @ X32 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_763_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_764_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X52: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X52 ) ) ).
% fm.distinct(31)
thf(fact_765_fm_Odistinct_I25_J,axiom,
! [X31: fm,X32: fm,X52: fm] :
( ( dis @ X31 @ X32 )
!= ( exi @ X52 ) ) ).
% fm.distinct(25)
thf(fact_766_fm_Odistinct_I17_J,axiom,
! [X21: fm,X222: fm,X52: fm] :
( ( imp @ X21 @ X222 )
!= ( exi @ X52 ) ) ).
% fm.distinct(17)
thf(fact_767_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X62: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X62 ) ) ).
% fm.distinct(33)
thf(fact_768_fm_Odistinct_I27_J,axiom,
! [X31: fm,X32: fm,X62: fm] :
( ( dis @ X31 @ X32 )
!= ( uni @ X62 ) ) ).
% fm.distinct(27)
thf(fact_769_fm_Odistinct_I19_J,axiom,
! [X21: fm,X222: fm,X62: fm] :
( ( imp @ X21 @ X222 )
!= ( uni @ X62 ) ) ).
% fm.distinct(19)
thf(fact_770_fm_Odistinct_I5_J,axiom,
! [X11: nat,X12: list_tm,X41: fm,X42: fm] :
( ( pre @ X11 @ X12 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_771_fm_Odistinct_I3_J,axiom,
! [X11: nat,X12: list_tm,X31: fm,X32: fm] :
( ( pre @ X11 @ X12 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_772_fm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X21: fm,X222: fm] :
( ( pre @ X11 @ X12 )
!= ( imp @ X21 @ X222 ) ) ).
% fm.distinct(1)
thf(fact_773_sub_Osimps_I4_J,axiom,
! [V: nat,S2: tm,P3: fm,Q3: fm] :
( ( sub @ V @ S2 @ ( con @ P3 @ Q3 ) )
= ( con @ ( sub @ V @ S2 @ P3 ) @ ( sub @ V @ S2 @ Q3 ) ) ) ).
% sub.simps(4)
thf(fact_774_sub_Osimps_I3_J,axiom,
! [V: nat,S2: tm,P3: fm,Q3: fm] :
( ( sub @ V @ S2 @ ( dis @ P3 @ Q3 ) )
= ( dis @ ( sub @ V @ S2 @ P3 ) @ ( sub @ V @ S2 @ Q3 ) ) ) ).
% sub.simps(3)
thf(fact_775_sub_Osimps_I2_J,axiom,
! [V: nat,S2: tm,P3: fm,Q3: fm] :
( ( sub @ V @ S2 @ ( imp @ P3 @ Q3 ) )
= ( imp @ ( sub @ V @ S2 @ P3 ) @ ( sub @ V @ S2 @ Q3 ) ) ) ).
% sub.simps(2)
thf(fact_776_Neg__exhaust,axiom,
! [X2: fm] :
( ! [I3: nat,Ts: list_tm] :
( X2
!= ( pre @ I3 @ Ts ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( imp @ P9 @ Q4 ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( dis @ P9 @ Q4 ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( con @ P9 @ Q4 ) )
=> ( ! [P9: fm] :
( X2
!= ( exi @ P9 ) )
=> ( ! [P9: fm] :
( X2
!= ( uni @ P9 ) )
=> ( ! [I3: nat,Ts: list_tm] :
( X2
!= ( neg @ ( pre @ I3 @ Ts ) ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( neg @ ( imp @ P9 @ Q4 ) ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( neg @ ( dis @ P9 @ Q4 ) ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( neg @ ( con @ P9 @ Q4 ) ) )
=> ( ! [P9: fm] :
( X2
!= ( neg @ ( exi @ P9 ) ) )
=> ( ! [P9: fm] :
( X2
!= ( neg @ ( uni @ P9 ) ) )
=> ~ ! [P9: fm] :
( X2
!= ( neg @ ( neg @ P9 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Neg_exhaust
thf(fact_777_params_H_H_Ocases,axiom,
! [X2: fm] :
( ! [B5: nat,Ts: list_tm] :
( X2
!= ( pre @ B5 @ Ts ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( imp @ P9 @ Q4 ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( dis @ P9 @ Q4 ) )
=> ( ! [P9: fm,Q4: fm] :
( X2
!= ( con @ P9 @ Q4 ) )
=> ( ! [P9: fm] :
( X2
!= ( exi @ P9 ) )
=> ( ! [P9: fm] :
( X2
!= ( uni @ P9 ) )
=> ~ ! [P9: fm] :
( X2
!= ( neg @ P9 ) ) ) ) ) ) ) ) ).
% params''.cases
thf(fact_778_fm_Oexhaust,axiom,
! [Y3: fm] :
( ! [X112: nat,X122: list_tm] :
( Y3
!= ( pre @ X112 @ X122 ) )
=> ( ! [X212: fm,X223: fm] :
( Y3
!= ( imp @ X212 @ X223 ) )
=> ( ! [X312: fm,X322: fm] :
( Y3
!= ( dis @ X312 @ X322 ) )
=> ( ! [X412: fm,X422: fm] :
( Y3
!= ( con @ X412 @ X422 ) )
=> ( ! [X53: fm] :
( Y3
!= ( exi @ X53 ) )
=> ( ! [X63: fm] :
( Y3
!= ( uni @ X63 ) )
=> ~ ! [X72: fm] :
( Y3
!= ( neg @ X72 ) ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_779_usemantics_Osimps_I4_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm,Q3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( con @ P3 @ Q3 ) )
= ( ( usemantics_tm @ U @ E @ F @ G @ P3 )
& ( usemantics_tm @ U @ E @ F @ G @ Q3 ) ) ) ).
% usemantics.simps(4)
thf(fact_780_usemantics_Osimps_I3_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm,Q3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( dis @ P3 @ Q3 ) )
= ( ( usemantics_tm @ U @ E @ F @ G @ P3 )
| ( usemantics_tm @ U @ E @ F @ G @ Q3 ) ) ) ).
% usemantics.simps(3)
thf(fact_781_usemantics_Osimps_I2_J,axiom,
! [U: set_tm,E: nat > tm,F: nat > list_tm > tm,G: nat > list_tm > $o,P3: fm,Q3: fm] :
( ( usemantics_tm @ U @ E @ F @ G @ ( imp @ P3 @ Q3 ) )
= ( ( usemantics_tm @ U @ E @ F @ G @ P3 )
=> ( usemantics_tm @ U @ E @ F @ G @ Q3 ) ) ) ).
% usemantics.simps(2)
thf(fact_782_subst_Osimps_I4_J,axiom,
! [P3: fm,Q3: fm,S2: tm,K: nat] :
( ( subst @ ( con @ P3 @ Q3 ) @ S2 @ K )
= ( con @ ( subst @ P3 @ S2 @ K ) @ ( subst @ Q3 @ S2 @ K ) ) ) ).
% subst.simps(4)
thf(fact_783_subst_Osimps_I3_J,axiom,
! [P3: fm,Q3: fm,S2: tm,K: nat] :
( ( subst @ ( dis @ P3 @ Q3 ) @ S2 @ K )
= ( dis @ ( subst @ P3 @ S2 @ K ) @ ( subst @ Q3 @ S2 @ K ) ) ) ).
% subst.simps(3)
thf(fact_784_subst_Osimps_I2_J,axiom,
! [P3: fm,Q3: fm,S2: tm,K: nat] :
( ( subst @ ( imp @ P3 @ Q3 ) @ S2 @ K )
= ( imp @ ( subst @ P3 @ S2 @ K ) @ ( subst @ Q3 @ S2 @ K ) ) ) ).
% subst.simps(2)
thf(fact_785_Hintikka_OAlphaDis,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( dis @ P3 @ Q3 ) @ H )
=> ( ( member_fm2 @ P3 @ H )
& ( member_fm2 @ Q3 @ H ) ) ) ) ).
% Hintikka.AlphaDis
thf(fact_786_Hintikka_OBetaCon,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( con @ P3 @ Q3 ) @ H )
=> ( ( member_fm2 @ P3 @ H )
| ( member_fm2 @ Q3 @ H ) ) ) ) ).
% Hintikka.BetaCon
thf(fact_787_branchDone_Ocases,axiom,
! [X2: list_fm] :
( ( X2 != nil_fm )
=> ( ! [P9: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ! [V3: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( X2
!= ( cons_fm @ ( uni @ V3 ) @ Z5 ) ) ) ) ) ) ) ) ) ).
% branchDone.cases
thf(fact_788_AlphaDis,axiom,
! [P3: fm,Q3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P3 @ ( cons_fm @ Q3 @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P3 @ Q3 ) @ Z3 ) ) ) ).
% AlphaDis
thf(fact_789_BetaCon,axiom,
! [P3: fm,Z3: list_fm,Q3: fm] :
( ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q3 @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P3 @ Q3 ) @ Z3 ) ) ) ) ).
% BetaCon
thf(fact_790_Hintikka_OAlphaImp,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( imp @ P3 @ Q3 ) @ H )
=> ( ( member_fm2 @ ( neg @ P3 ) @ H )
& ( member_fm2 @ Q3 @ H ) ) ) ) ).
% Hintikka.AlphaImp
thf(fact_791_Hintikka_OAlphaCon,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( con @ P3 @ Q3 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P3 ) @ H )
& ( member_fm2 @ ( neg @ Q3 ) @ H ) ) ) ) ).
% Hintikka.AlphaCon
thf(fact_792_Hintikka_OBetaImp,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( imp @ P3 @ Q3 ) ) @ H )
=> ( ( member_fm2 @ P3 @ H )
| ( member_fm2 @ ( neg @ Q3 ) @ H ) ) ) ) ).
% Hintikka.BetaImp
thf(fact_793_Hintikka_OBetaDis,axiom,
! [H: set_fm,P3: fm,Q3: fm] :
( ( hintikka @ H )
=> ( ( member_fm2 @ ( neg @ ( dis @ P3 @ Q3 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P3 ) @ H )
| ( member_fm2 @ ( neg @ Q3 ) @ H ) ) ) ) ).
% Hintikka.BetaDis
thf(fact_794_params_Osimps_I2_J,axiom,
! [P3: fm,Q3: fm] :
( ( params @ ( imp @ P3 @ Q3 ) )
= ( sup_sup_set_nat @ ( params @ P3 ) @ ( params @ Q3 ) ) ) ).
% params.simps(2)
thf(fact_795_params_Osimps_I3_J,axiom,
! [P3: fm,Q3: fm] :
( ( params @ ( dis @ P3 @ Q3 ) )
= ( sup_sup_set_nat @ ( params @ P3 ) @ ( params @ Q3 ) ) ) ).
% params.simps(3)
thf(fact_796_params_Osimps_I4_J,axiom,
! [P3: fm,Q3: fm] :
( ( params @ ( con @ P3 @ Q3 ) )
= ( sup_sup_set_nat @ ( params @ P3 ) @ ( params @ Q3 ) ) ) ).
% params.simps(4)
thf(fact_797_Hintikka_Ointro,axiom,
! [H: set_fm] :
( ! [N3: nat,Ts: list_tm] :
( ( member_fm2 @ ( pre @ N3 @ Ts ) @ H )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N3 @ Ts ) ) @ H ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( dis @ P9 @ Q4 ) @ H )
=> ( ( member_fm2 @ P9 @ H )
& ( member_fm2 @ Q4 @ H ) ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( imp @ P9 @ Q4 ) @ H )
=> ( ( member_fm2 @ ( neg @ P9 ) @ H )
& ( member_fm2 @ Q4 @ H ) ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( neg @ ( con @ P9 @ Q4 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P9 ) @ H )
& ( member_fm2 @ ( neg @ Q4 ) @ H ) ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( con @ P9 @ Q4 ) @ H )
=> ( ( member_fm2 @ P9 @ H )
| ( member_fm2 @ Q4 @ H ) ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( neg @ ( imp @ P9 @ Q4 ) ) @ H )
=> ( ( member_fm2 @ P9 @ H )
| ( member_fm2 @ ( neg @ Q4 ) @ H ) ) )
=> ( ! [P9: fm,Q4: fm] :
( ( member_fm2 @ ( neg @ ( dis @ P9 @ Q4 ) ) @ H )
=> ( ( member_fm2 @ ( neg @ P9 ) @ H )
| ( member_fm2 @ ( neg @ Q4 ) @ H ) ) )
=> ( ! [P9: fm] :
( ( member_fm2 @ ( exi @ P9 ) @ H )
=> ! [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X @ P9 ) @ H ) ) )
=> ( ! [P9: fm] :
( ( member_fm2 @ ( neg @ ( uni @ P9 ) ) @ H )
=> ! [X: tm] :
( ( member_tm2 @ X @ ( terms @ H ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X @ P9 ) ) @ H ) ) )
=> ( ! [P9: fm] :
( ( member_fm2 @ ( uni @ P9 ) @ H )
=> ? [X3: tm] :
( ( member_tm2 @ X3 @ ( terms @ H ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X3 @ P9 ) @ H ) ) )
=> ( ! [P9: fm] :
( ( member_fm2 @ ( neg @ ( exi @ P9 ) ) @ H )
=> ? [X3: tm] :
( ( member_tm2 @ X3 @ ( terms @ H ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X3 @ P9 ) ) @ H ) ) )
=> ( ! [P9: fm] :
( ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ H )
=> ( member_fm2 @ P9 @ H ) )
=> ( hintikka @ H ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Hintikka.intro
thf(fact_798_Hintikka__def,axiom,
( hintikka
= ( ^ [H2: set_fm] :
( ! [N: nat,Ts3: list_tm] :
( ( member_fm2 @ ( pre @ N @ Ts3 ) @ H2 )
=> ~ ( member_fm2 @ ( neg @ ( pre @ N @ Ts3 ) ) @ H2 ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( dis @ P7 @ Q5 ) @ H2 )
=> ( ( member_fm2 @ P7 @ H2 )
& ( member_fm2 @ Q5 @ H2 ) ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( imp @ P7 @ Q5 ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P7 ) @ H2 )
& ( member_fm2 @ Q5 @ H2 ) ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( neg @ ( con @ P7 @ Q5 ) ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P7 ) @ H2 )
& ( member_fm2 @ ( neg @ Q5 ) @ H2 ) ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( con @ P7 @ Q5 ) @ H2 )
=> ( ( member_fm2 @ P7 @ H2 )
| ( member_fm2 @ Q5 @ H2 ) ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( neg @ ( imp @ P7 @ Q5 ) ) @ H2 )
=> ( ( member_fm2 @ P7 @ H2 )
| ( member_fm2 @ ( neg @ Q5 ) @ H2 ) ) )
& ! [P7: fm,Q5: fm] :
( ( member_fm2 @ ( neg @ ( dis @ P7 @ Q5 ) ) @ H2 )
=> ( ( member_fm2 @ ( neg @ P7 ) @ H2 )
| ( member_fm2 @ ( neg @ Q5 ) @ H2 ) ) )
& ! [P7: fm] :
( ( member_fm2 @ ( exi @ P7 ) @ H2 )
=> ! [X5: tm] :
( ( member_tm2 @ X5 @ ( terms @ H2 ) )
=> ( member_fm2 @ ( sub @ zero_zero_nat @ X5 @ P7 ) @ H2 ) ) )
& ! [P7: fm] :
( ( member_fm2 @ ( neg @ ( uni @ P7 ) ) @ H2 )
=> ! [X5: tm] :
( ( member_tm2 @ X5 @ ( terms @ H2 ) )
=> ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X5 @ P7 ) ) @ H2 ) ) )
& ! [P7: fm] :
( ( member_fm2 @ ( uni @ P7 ) @ H2 )
=> ? [X5: tm] :
( ( member_tm2 @ X5 @ ( terms @ H2 ) )
& ( member_fm2 @ ( sub @ zero_zero_nat @ X5 @ P7 ) @ H2 ) ) )
& ! [P7: fm] :
( ( member_fm2 @ ( neg @ ( exi @ P7 ) ) @ H2 )
=> ? [X5: tm] :
( ( member_tm2 @ X5 @ ( terms @ H2 ) )
& ( member_fm2 @ ( neg @ ( sub @ zero_zero_nat @ X5 @ P7 ) ) @ H2 ) ) )
& ! [P7: fm] :
( ( member_fm2 @ ( neg @ ( neg @ P7 ) ) @ H2 )
=> ( member_fm2 @ P7 @ H2 ) ) ) ) ) ).
% Hintikka_def
thf(fact_799_BetaDis,axiom,
! [P3: fm,Z3: list_fm,Q3: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P3 @ Q3 ) ) @ Z3 ) ) ) ) ).
% BetaDis
thf(fact_800_BetaImp,axiom,
! [P3: fm,Z3: list_fm,Q3: fm] :
( ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P3 @ Q3 ) ) @ Z3 ) ) ) ) ).
% BetaImp
thf(fact_801_AlphaCon,axiom,
! [P3: fm,Q3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ ( neg @ Q3 ) @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P3 @ Q3 ) ) @ Z3 ) ) ) ).
% AlphaCon
thf(fact_802_AlphaImp,axiom,
! [P3: fm,Q3: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ Q3 @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P3 @ Q3 ) @ Z3 ) ) ) ).
% AlphaImp
thf(fact_803_Cons__in__subseqsD,axiom,
! [Y3: fm,Ys: list_fm,Xs: list_fm] :
( ( member_list_fm @ ( cons_fm @ Y3 @ Ys ) @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
=> ( member_list_fm @ Ys @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_804_Cons__in__subseqsD,axiom,
! [Y3: tm,Ys: list_tm,Xs: list_tm] :
( ( member_list_tm @ ( cons_tm @ Y3 @ Ys ) @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
=> ( member_list_tm @ Ys @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_805_Cons__in__subseqsD,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y3 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_806_partition_Osimps_I1_J,axiom,
! [P: tm > $o] :
( ( partition_tm @ P @ nil_tm )
= ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) ) ).
% partition.simps(1)
thf(fact_807_partition_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( partition_nat @ P @ nil_nat )
= ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).
% partition.simps(1)
thf(fact_808_partition_Osimps_I1_J,axiom,
! [P: fm > $o] :
( ( partition_fm @ P @ nil_fm )
= ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) ) ).
% partition.simps(1)
thf(fact_809_partition__P,axiom,
! [P: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Yes ) )
=> ( P @ X3 ) )
& ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ No ) )
=> ~ ( P @ X3 ) ) ) ) ).
% partition_P
thf(fact_810_partition__P,axiom,
! [P: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
( ( ( partition_nat @ P @ Xs )
= ( produc2694037385005941721st_nat @ Yes @ No ) )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Yes ) )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ No ) )
=> ~ ( P @ X3 ) ) ) ) ).
% partition_P
thf(fact_811_partition__P,axiom,
! [P: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Yes ) )
=> ( P @ X3 ) )
& ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ No ) )
=> ~ ( P @ X3 ) ) ) ) ).
% partition_P
thf(fact_812_branchDone_Oelims_I1_J,axiom,
! [X2: list_fm,Y3: $o] :
( ( ( branchDone @ X2 )
= Y3 )
=> ( ( ( X2 = nil_fm )
=> Y3 )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( Y3
= ( ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(1)
thf(fact_813_branchDone_Oelims_I3_J,axiom,
! [X2: list_fm] :
( ~ ( branchDone @ X2 )
=> ( ( X2 != nil_fm )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(3)
thf(fact_814_branchDone_Oelims_I2_J,axiom,
! [X2: list_fm] :
( ( branchDone @ X2 )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(2)
thf(fact_815_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_tm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ A1 @ A22 ) @ ( listrel_fm_tm @ R2 ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_tm ) )
=> ~ ! [X: fm,Y4: tm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X @ Y4 ) @ R2 )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs2 @ Ys3 ) @ ( listrel_fm_tm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_816_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_nat,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ A1 @ A22 ) @ ( listrel_fm_nat @ R2 ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_nat ) )
=> ~ ! [X: fm,Y4: nat,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member1425945901753563017fm_nat @ ( product_Pair_fm_nat @ X @ Y4 ) @ R2 )
=> ~ ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs2 @ Ys3 ) @ ( listrel_fm_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_817_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ A1 @ A22 ) @ ( listrel_tm_fm @ R2 ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_fm ) )
=> ~ ! [X: tm,Y4: fm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X @ Y4 ) @ R2 )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs2 @ Ys3 ) @ ( listrel_tm_fm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_818_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A1 @ A22 ) @ ( listrel_tm_tm @ R2 ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_tm ) )
=> ~ ! [X: tm,Y4: tm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y4 ) @ R2 )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs2 @ Ys3 ) @ ( listrel_tm_tm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_819_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_nat,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ A1 @ A22 ) @ ( listrel_tm_nat @ R2 ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_nat ) )
=> ~ ! [X: tm,Y4: nat,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member3090325158824063739tm_nat @ ( product_Pair_tm_nat @ X @ Y4 ) @ R2 )
=> ~ ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs2 @ Ys3 ) @ ( listrel_tm_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_820_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_fm,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ A1 @ A22 ) @ ( listrel_nat_fm @ R2 ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_fm ) )
=> ~ ! [X: nat,Y4: fm,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member2965011204816613423nat_fm @ ( product_Pair_nat_fm @ X @ Y4 ) @ R2 )
=> ~ ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs2 @ Ys3 ) @ ( listrel_nat_fm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_821_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_tm,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ A1 @ A22 ) @ ( listrel_nat_tm @ R2 ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_tm ) )
=> ~ ! [X: nat,Y4: tm,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member2968963229902248609nat_tm @ ( product_Pair_nat_tm @ X @ Y4 ) @ R2 )
=> ~ ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs2 @ Ys3 ) @ ( listrel_nat_tm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_822_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X: nat,Y4: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_823_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A1 @ A22 ) @ ( listrel_fm_fm @ R2 ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_fm ) )
=> ~ ! [X: fm,Y4: fm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y4 ) @ R2 )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs2 @ Ys3 ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_824_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_tm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ A1 @ A22 ) @ ( listrel_fm_tm @ R2 ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_tm ) )
| ? [X5: fm,Y: tm,Xs3: list_fm,Ys2: list_tm] :
( ( A1
= ( cons_fm @ X5 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y @ Ys2 ) )
& ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X5 @ Y ) @ R2 )
& ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs3 @ Ys2 ) @ ( listrel_fm_tm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_825_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_nat,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ A1 @ A22 ) @ ( listrel_fm_nat @ R2 ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_nat ) )
| ? [X5: fm,Y: nat,Xs3: list_fm,Ys2: list_nat] :
( ( A1
= ( cons_fm @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y @ Ys2 ) )
& ( member1425945901753563017fm_nat @ ( product_Pair_fm_nat @ X5 @ Y ) @ R2 )
& ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs3 @ Ys2 ) @ ( listrel_fm_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_826_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ A1 @ A22 ) @ ( listrel_tm_fm @ R2 ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_fm ) )
| ? [X5: tm,Y: fm,Xs3: list_tm,Ys2: list_fm] :
( ( A1
= ( cons_tm @ X5 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y @ Ys2 ) )
& ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X5 @ Y ) @ R2 )
& ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs3 @ Ys2 ) @ ( listrel_tm_fm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_827_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A1 @ A22 ) @ ( listrel_tm_tm @ R2 ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_tm ) )
| ? [X5: tm,Y: tm,Xs3: list_tm,Ys2: list_tm] :
( ( A1
= ( cons_tm @ X5 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y @ Ys2 ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X5 @ Y ) @ R2 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs3 @ Ys2 ) @ ( listrel_tm_tm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_828_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_nat,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ A1 @ A22 ) @ ( listrel_tm_nat @ R2 ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_nat ) )
| ? [X5: tm,Y: nat,Xs3: list_tm,Ys2: list_nat] :
( ( A1
= ( cons_tm @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y @ Ys2 ) )
& ( member3090325158824063739tm_nat @ ( product_Pair_tm_nat @ X5 @ Y ) @ R2 )
& ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs3 @ Ys2 ) @ ( listrel_tm_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_829_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_fm,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ A1 @ A22 ) @ ( listrel_nat_fm @ R2 ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_fm ) )
| ? [X5: nat,Y: fm,Xs3: list_nat,Ys2: list_fm] :
( ( A1
= ( cons_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y @ Ys2 ) )
& ( member2965011204816613423nat_fm @ ( product_Pair_nat_fm @ X5 @ Y ) @ R2 )
& ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs3 @ Ys2 ) @ ( listrel_nat_fm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_830_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_tm,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ A1 @ A22 ) @ ( listrel_nat_tm @ R2 ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_tm ) )
| ? [X5: nat,Y: tm,Xs3: list_nat,Ys2: list_tm] :
( ( A1
= ( cons_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y @ Ys2 ) )
& ( member2968963229902248609nat_tm @ ( product_Pair_nat_tm @ X5 @ Y ) @ R2 )
& ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs3 @ Ys2 ) @ ( listrel_nat_tm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_831_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X5: nat,Y: nat,Xs3: list_nat,Ys2: list_nat] :
( ( A1
= ( cons_nat @ X5 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y @ Ys2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_832_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A1 @ A22 ) @ ( listrel_fm_fm @ R2 ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_fm ) )
| ? [X5: fm,Y: fm,Xs3: list_fm,Ys2: list_fm] :
( ( A1
= ( cons_fm @ X5 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y @ Ys2 ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ R2 )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs3 @ Ys2 ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_833_branchDone__contradiction,axiom,
( branchDone
= ( ^ [Z2: list_fm] :
? [P7: fm] :
( ( member_fm2 @ P7 @ ( set_fm2 @ Z2 ) )
& ( member_fm2 @ ( neg @ P7 ) @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% branchDone_contradiction
thf(fact_834_listrel__Nil2,axiom,
! [Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ nil_tm ) @ ( listrel_tm_tm @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_835_listrel__Nil2,axiom,
! [Xs: list_nat,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs @ nil_tm ) @ ( listrel_nat_tm @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_836_listrel__Nil2,axiom,
! [Xs: list_fm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ nil_tm ) @ ( listrel_fm_tm @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_837_listrel__Nil2,axiom,
! [Xs: list_tm,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs @ nil_nat ) @ ( listrel_tm_nat @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_838_listrel__Nil2,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_839_listrel__Nil2,axiom,
! [Xs: list_fm,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs @ nil_nat ) @ ( listrel_fm_nat @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_840_listrel__Nil2,axiom,
! [Xs: list_tm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ nil_fm ) @ ( listrel_tm_fm @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_841_listrel__Nil2,axiom,
! [Xs: list_nat,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs @ nil_fm ) @ ( listrel_nat_fm @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_842_listrel__Nil2,axiom,
! [Xs: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ nil_fm ) @ ( listrel_fm_fm @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_843_listrel__Nil1,axiom,
! [Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Xs ) @ ( listrel_tm_tm @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_844_listrel__Nil1,axiom,
! [Xs: list_nat,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ nil_tm @ Xs ) @ ( listrel_tm_nat @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_845_listrel__Nil1,axiom,
! [Xs: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ nil_tm @ Xs ) @ ( listrel_tm_fm @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_846_listrel__Nil1,axiom,
! [Xs: list_tm,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ nil_nat @ Xs ) @ ( listrel_nat_tm @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_847_listrel__Nil1,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_848_listrel__Nil1,axiom,
! [Xs: list_fm,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ nil_nat @ Xs ) @ ( listrel_nat_fm @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_849_listrel__Nil1,axiom,
! [Xs: list_tm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ nil_fm @ Xs ) @ ( listrel_fm_tm @ R2 ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_850_listrel__Nil1,axiom,
! [Xs: list_nat,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ nil_fm @ Xs ) @ ( listrel_fm_nat @ R2 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_851_listrel__Nil1,axiom,
! [Xs: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Xs ) @ ( listrel_fm_fm @ R2 ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_852_listrel_ONil,axiom,
! [R2: set_Pr2455929065695642951_tm_tm] : ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) @ ( listrel_tm_tm @ R2 ) ) ).
% listrel.Nil
thf(fact_853_listrel_ONil,axiom,
! [R2: set_Pr1365117562694539290tm_nat] : ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ nil_tm @ nil_nat ) @ ( listrel_tm_nat @ R2 ) ) ).
% listrel.Nil
thf(fact_854_listrel_ONil,axiom,
! [R2: set_Pr2698443736021152725_tm_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ nil_tm @ nil_fm ) @ ( listrel_tm_fm @ R2 ) ) ).
% listrel.Nil
thf(fact_855_listrel_ONil,axiom,
! [R2: set_Pr4584624442413714624nat_tm] : ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ nil_nat @ nil_tm ) @ ( listrel_nat_tm @ R2 ) ) ).
% listrel.Nil
thf(fact_856_listrel_ONil,axiom,
! [R2: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R2 ) ) ).
% listrel.Nil
thf(fact_857_listrel_ONil,axiom,
! [R2: set_Pr4827139112739224398nat_fm] : ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ nil_nat @ nil_fm ) @ ( listrel_nat_fm @ R2 ) ) ).
% listrel.Nil
thf(fact_858_listrel_ONil,axiom,
! [R2: set_Pr4464301228316855097_fm_tm] : ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ nil_fm @ nil_tm ) @ ( listrel_fm_tm @ R2 ) ) ).
% listrel.Nil
thf(fact_859_listrel_ONil,axiom,
! [R2: set_Pr6019225798553204136fm_nat] : ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ nil_fm @ nil_nat ) @ ( listrel_fm_nat @ R2 ) ) ).
% listrel.Nil
thf(fact_860_listrel_ONil,axiom,
! [R2: set_Pr4706815898642364871_fm_fm] : ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) @ ( listrel_fm_fm @ R2 ) ) ).
% listrel.Nil
thf(fact_861_branchDone_Osimps_I2_J,axiom,
! [P3: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( neg @ P3 ) @ Z3 ) )
= ( ( member_fm2 @ P3 @ ( set_fm2 @ Z3 ) )
| ( member_fm2 @ ( neg @ ( neg @ P3 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(2)
thf(fact_862_listrel_OCons,axiom,
! [X2: fm,Y3: tm,R2: set_Pr4464301228316855097_fm_tm,Xs: list_fm,Ys: list_tm] :
( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X2 @ Y3 ) @ R2 )
=> ( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ Ys ) @ ( listrel_fm_tm @ R2 ) )
=> ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ ( cons_fm @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_fm_tm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_863_listrel_OCons,axiom,
! [X2: fm,Y3: nat,R2: set_Pr6019225798553204136fm_nat,Xs: list_fm,Ys: list_nat] :
( ( member1425945901753563017fm_nat @ ( product_Pair_fm_nat @ X2 @ Y3 ) @ R2 )
=> ( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs @ Ys ) @ ( listrel_fm_nat @ R2 ) )
=> ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ ( cons_fm @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_fm_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_864_listrel_OCons,axiom,
! [X2: tm,Y3: fm,R2: set_Pr2698443736021152725_tm_fm,Xs: list_tm,Ys: list_fm] :
( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X2 @ Y3 ) @ R2 )
=> ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ Ys ) @ ( listrel_tm_fm @ R2 ) )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( cons_tm @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_tm_fm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_865_listrel_OCons,axiom,
! [X2: tm,Y3: tm,R2: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X2 @ Y3 ) @ R2 )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel_tm_tm @ R2 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_tm_tm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_866_listrel_OCons,axiom,
! [X2: tm,Y3: nat,R2: set_Pr1365117562694539290tm_nat,Xs: list_tm,Ys: list_nat] :
( ( member3090325158824063739tm_nat @ ( product_Pair_tm_nat @ X2 @ Y3 ) @ R2 )
=> ( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs @ Ys ) @ ( listrel_tm_nat @ R2 ) )
=> ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ ( cons_tm @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_tm_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_867_listrel_OCons,axiom,
! [X2: nat,Y3: fm,R2: set_Pr4827139112739224398nat_fm,Xs: list_nat,Ys: list_fm] :
( ( member2965011204816613423nat_fm @ ( product_Pair_nat_fm @ X2 @ Y3 ) @ R2 )
=> ( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs @ Ys ) @ ( listrel_nat_fm @ R2 ) )
=> ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ ( cons_nat @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_nat_fm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_868_listrel_OCons,axiom,
! [X2: nat,Y3: tm,R2: set_Pr4584624442413714624nat_tm,Xs: list_nat,Ys: list_tm] :
( ( member2968963229902248609nat_tm @ ( product_Pair_nat_tm @ X2 @ Y3 ) @ R2 )
=> ( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs @ Ys ) @ ( listrel_nat_tm @ R2 ) )
=> ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ ( cons_nat @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_nat_tm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_869_listrel_OCons,axiom,
! [X2: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_870_listrel_OCons,axiom,
! [X2: fm,Y3: fm,R2: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Ys: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ R2 )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel_fm_fm @ R2 ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_fm_fm @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_871_listrel__Cons1,axiom,
! [Y3: fm,Ys: list_fm,Xs: list_tm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ ( cons_fm @ Y3 @ Ys ) @ Xs ) @ ( listrel_fm_tm @ R2 ) )
=> ~ ! [Y4: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Ys @ Ys3 ) @ ( listrel_fm_tm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_872_listrel__Cons1,axiom,
! [Y3: fm,Ys: list_fm,Xs: list_nat,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ ( cons_fm @ Y3 @ Ys ) @ Xs ) @ ( listrel_fm_nat @ R2 ) )
=> ~ ! [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member1425945901753563017fm_nat @ ( product_Pair_fm_nat @ Y3 @ Y4 ) @ R2 )
=> ~ ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Ys @ Ys3 ) @ ( listrel_fm_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_873_listrel__Cons1,axiom,
! [Y3: tm,Ys: list_tm,Xs: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( cons_tm @ Y3 @ Ys ) @ Xs ) @ ( listrel_tm_fm @ R2 ) )
=> ~ ! [Y4: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Ys @ Ys3 ) @ ( listrel_tm_fm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_874_listrel__Cons1,axiom,
! [Y3: tm,Ys: list_tm,Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ Y3 @ Ys ) @ Xs ) @ ( listrel_tm_tm @ R2 ) )
=> ~ ! [Y4: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Ys3 ) @ ( listrel_tm_tm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_875_listrel__Cons1,axiom,
! [Y3: tm,Ys: list_tm,Xs: list_nat,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ ( cons_tm @ Y3 @ Ys ) @ Xs ) @ ( listrel_tm_nat @ R2 ) )
=> ~ ! [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member3090325158824063739tm_nat @ ( product_Pair_tm_nat @ Y3 @ Y4 ) @ R2 )
=> ~ ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Ys @ Ys3 ) @ ( listrel_tm_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_876_listrel__Cons1,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_fm,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_fm @ R2 ) )
=> ~ ! [Y4: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member2965011204816613423nat_fm @ ( product_Pair_nat_fm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Ys @ Ys3 ) @ ( listrel_nat_fm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_877_listrel__Cons1,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_tm,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_tm @ R2 ) )
=> ~ ! [Y4: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( member2968963229902248609nat_tm @ ( product_Pair_nat_tm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Ys @ Ys3 ) @ ( listrel_nat_tm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_878_listrel__Cons1,axiom,
! [Y3: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y4 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_879_listrel__Cons1,axiom,
! [Y3: fm,Ys: list_fm,Xs: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ Y3 @ Ys ) @ Xs ) @ ( listrel_fm_fm @ R2 ) )
=> ~ ! [Y4: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y3 @ Y4 ) @ R2 )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Ys3 ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_880_listrel__Cons2,axiom,
! [Xs: list_tm,Y3: fm,Ys: list_fm,R2: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_tm_fm @ R2 ) )
=> ~ ! [X: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X @ Xs2 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X @ Y3 ) @ R2 )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs2 @ Ys ) @ ( listrel_tm_fm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_881_listrel__Cons2,axiom,
! [Xs: list_nat,Y3: fm,Ys: list_fm,R2: set_Pr4827139112739224398nat_fm] :
( ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_nat_fm @ R2 ) )
=> ~ ! [X: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs2 ) )
=> ( ( member2965011204816613423nat_fm @ ( product_Pair_nat_fm @ X @ Y3 ) @ R2 )
=> ~ ( member848049699933468325ist_fm @ ( produc7721762080035590080ist_fm @ Xs2 @ Ys ) @ ( listrel_nat_fm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_882_listrel__Cons2,axiom,
! [Xs: list_fm,Y3: tm,Ys: list_tm,R2: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_fm_tm @ R2 ) )
=> ~ ! [X: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X @ Xs2 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X @ Y3 ) @ R2 )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs2 @ Ys ) @ ( listrel_fm_tm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_883_listrel__Cons2,axiom,
! [Xs: list_tm,Y3: tm,Ys: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_tm_tm @ R2 ) )
=> ~ ! [X: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X @ Xs2 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R2 )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs2 @ Ys ) @ ( listrel_tm_tm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_884_listrel__Cons2,axiom,
! [Xs: list_nat,Y3: tm,Ys: list_tm,R2: set_Pr4584624442413714624nat_tm] :
( ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel_nat_tm @ R2 ) )
=> ~ ! [X: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs2 ) )
=> ( ( member2968963229902248609nat_tm @ ( product_Pair_nat_tm @ X @ Y3 ) @ R2 )
=> ~ ( member605535029607958551ist_tm @ ( produc7725714105121225266ist_tm @ Xs2 @ Ys ) @ ( listrel_nat_tm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_885_listrel__Cons2,axiom,
! [Xs: list_fm,Y3: nat,Ys: list_nat,R2: set_Pr6019225798553204136fm_nat] :
( ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_fm_nat @ R2 ) )
=> ~ ! [X: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X @ Xs2 ) )
=> ( ( member1425945901753563017fm_nat @ ( product_Pair_fm_nat @ X @ Y3 ) @ R2 )
=> ~ ( member1868462118960745887st_nat @ ( produc8654734596316469506st_nat @ Xs2 @ Ys ) @ ( listrel_fm_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_886_listrel__Cons2,axiom,
! [Xs: list_tm,Y3: nat,Ys: list_nat,R2: set_Pr1365117562694539290tm_nat] :
( ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_tm_nat @ R2 ) )
=> ~ ! [X: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X @ Xs2 ) )
=> ( ( member3090325158824063739tm_nat @ ( product_Pair_tm_nat @ X @ Y3 ) @ R2 )
=> ~ ( member4213470008700669713st_nat @ ( produc1776370449201617524st_nat @ Xs2 @ Ys ) @ ( listrel_tm_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_887_listrel__Cons2,axiom,
! [Xs: list_nat,Y3: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [X: nat,Xs2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_888_listrel__Cons2,axiom,
! [Xs: list_fm,Y3: fm,Ys: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ ( cons_fm @ Y3 @ Ys ) ) @ ( listrel_fm_fm @ R2 ) )
=> ~ ! [X: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X @ Xs2 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R2 )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs2 @ Ys ) @ ( listrel_fm_fm @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_889_branchDone_Osimps_I7_J,axiom,
! [V: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( exi @ V ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(7)
thf(fact_890_branchDone_Osimps_I8_J,axiom,
! [V: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( uni @ V ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(8)
thf(fact_891_branchDone_Osimps_I4_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( imp @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( imp @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(4)
thf(fact_892_branchDone_Osimps_I5_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( dis @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( dis @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(5)
thf(fact_893_branchDone_Osimps_I6_J,axiom,
! [V: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( con @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( con @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(6)
thf(fact_894_branchDone_Osimps_I3_J,axiom,
! [V: nat,Va2: list_tm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( pre @ V @ Va2 ) @ Z3 ) )
= ( ( member_fm2 @ ( neg @ ( pre @ V @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(3)
thf(fact_895_branchDone_Opelims_I1_J,axiom,
! [X2: list_fm,Y3: $o] :
( ( ( branchDone @ X2 )
= Y3 )
=> ( ( accp_list_fm @ branchDone_rel @ X2 )
=> ( ( ( X2 = nil_fm )
=> ( ~ Y3
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) ) )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P9 ) @ Z5 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z5 ) ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( ( Y3
= ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z5 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(1)
thf(fact_896_branchDone_Opelims_I3_J,axiom,
! [X2: list_fm] :
( ~ ( branchDone @ X2 )
=> ( ( accp_list_fm @ branchDone_rel @ X2 )
=> ( ( ( X2 = nil_fm )
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(3)
thf(fact_897_listrelp__listrel__eq,axiom,
! [R2: set_Pr4706815898642364871_fm_fm] :
( ( listrelp_fm_fm
@ ^ [X5: fm,Y: fm] : ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X5 @ Y ) @ R2 ) )
= ( ^ [X5: list_fm,Y: list_fm] : ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X5 @ Y ) @ ( listrel_fm_fm @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_898_branchDone_Opelims_I2_J,axiom,
! [X2: list_fm] :
( ( branchDone @ X2 )
=> ( ( accp_list_fm @ branchDone_rel @ X2 )
=> ( ! [P9: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P9 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ P9 @ ( set_fm2 @ Z5 ) )
| ( member_fm2 @ ( neg @ ( neg @ P9 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ( ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) )
=> ~ ! [V3: fm,Z5: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z5 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z5 ) )
| ( branchDone @ Z5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(2)
thf(fact_899_listrelp_OCons,axiom,
! [R2: fm > fm > $o,X2: fm,Y3: fm,Xs: list_fm,Ys: list_fm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_fm_fm @ R2 @ Xs @ Ys )
=> ( listrelp_fm_fm @ R2 @ ( cons_fm @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_900_listrelp_OCons,axiom,
! [R2: fm > tm > $o,X2: fm,Y3: tm,Xs: list_fm,Ys: list_tm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_fm_tm @ R2 @ Xs @ Ys )
=> ( listrelp_fm_tm @ R2 @ ( cons_fm @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_901_listrelp_OCons,axiom,
! [R2: fm > nat > $o,X2: fm,Y3: nat,Xs: list_fm,Ys: list_nat] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_fm_nat @ R2 @ Xs @ Ys )
=> ( listrelp_fm_nat @ R2 @ ( cons_fm @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_902_listrelp_OCons,axiom,
! [R2: tm > fm > $o,X2: tm,Y3: fm,Xs: list_tm,Ys: list_fm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_tm_fm @ R2 @ Xs @ Ys )
=> ( listrelp_tm_fm @ R2 @ ( cons_tm @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_903_listrelp_OCons,axiom,
! [R2: tm > tm > $o,X2: tm,Y3: tm,Xs: list_tm,Ys: list_tm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_tm_tm @ R2 @ Xs @ Ys )
=> ( listrelp_tm_tm @ R2 @ ( cons_tm @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_904_listrelp_OCons,axiom,
! [R2: tm > nat > $o,X2: tm,Y3: nat,Xs: list_tm,Ys: list_nat] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_tm_nat @ R2 @ Xs @ Ys )
=> ( listrelp_tm_nat @ R2 @ ( cons_tm @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_905_listrelp_OCons,axiom,
! [R2: nat > fm > $o,X2: nat,Y3: fm,Xs: list_nat,Ys: list_fm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_nat_fm @ R2 @ Xs @ Ys )
=> ( listrelp_nat_fm @ R2 @ ( cons_nat @ X2 @ Xs ) @ ( cons_fm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_906_listrelp_OCons,axiom,
! [R2: nat > tm > $o,X2: nat,Y3: tm,Xs: list_nat,Ys: list_tm] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_nat_tm @ R2 @ Xs @ Ys )
=> ( listrelp_nat_tm @ R2 @ ( cons_nat @ X2 @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_907_listrelp_OCons,axiom,
! [R2: nat > nat > $o,X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
( ( R2 @ X2 @ Y3 )
=> ( ( listrelp_nat_nat @ R2 @ Xs @ Ys )
=> ( listrelp_nat_nat @ R2 @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_908_listrelp_ONil,axiom,
! [R2: tm > tm > $o] : ( listrelp_tm_tm @ R2 @ nil_tm @ nil_tm ) ).
% listrelp.Nil
thf(fact_909_listrelp_ONil,axiom,
! [R2: tm > nat > $o] : ( listrelp_tm_nat @ R2 @ nil_tm @ nil_nat ) ).
% listrelp.Nil
thf(fact_910_listrelp_ONil,axiom,
! [R2: tm > fm > $o] : ( listrelp_tm_fm @ R2 @ nil_tm @ nil_fm ) ).
% listrelp.Nil
thf(fact_911_listrelp_ONil,axiom,
! [R2: nat > tm > $o] : ( listrelp_nat_tm @ R2 @ nil_nat @ nil_tm ) ).
% listrelp.Nil
thf(fact_912_listrelp_ONil,axiom,
! [R2: nat > nat > $o] : ( listrelp_nat_nat @ R2 @ nil_nat @ nil_nat ) ).
% listrelp.Nil
thf(fact_913_listrelp_ONil,axiom,
! [R2: nat > fm > $o] : ( listrelp_nat_fm @ R2 @ nil_nat @ nil_fm ) ).
% listrelp.Nil
thf(fact_914_listrelp_ONil,axiom,
! [R2: fm > tm > $o] : ( listrelp_fm_tm @ R2 @ nil_fm @ nil_tm ) ).
% listrelp.Nil
thf(fact_915_listrelp_ONil,axiom,
! [R2: fm > nat > $o] : ( listrelp_fm_nat @ R2 @ nil_fm @ nil_nat ) ).
% listrelp.Nil
thf(fact_916_listrelp_ONil,axiom,
! [R2: fm > fm > $o] : ( listrelp_fm_fm @ R2 @ nil_fm @ nil_fm ) ).
% listrelp.Nil
thf(fact_917_listrelp_Ocases,axiom,
! [R2: fm > fm > $o,A1: list_fm,A22: list_fm] :
( ( listrelp_fm_fm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_fm ) )
=> ~ ! [X: fm,Y4: fm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_fm_fm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_918_listrelp_Ocases,axiom,
! [R2: fm > tm > $o,A1: list_fm,A22: list_tm] :
( ( listrelp_fm_tm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_tm ) )
=> ~ ! [X: fm,Y4: tm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_fm_tm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_919_listrelp_Ocases,axiom,
! [R2: fm > nat > $o,A1: list_fm,A22: list_nat] :
( ( listrelp_fm_nat @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_nat ) )
=> ~ ! [X: fm,Y4: nat,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_fm_nat @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_920_listrelp_Ocases,axiom,
! [R2: tm > fm > $o,A1: list_tm,A22: list_fm] :
( ( listrelp_tm_fm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_fm ) )
=> ~ ! [X: tm,Y4: fm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_tm_fm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_921_listrelp_Ocases,axiom,
! [R2: tm > tm > $o,A1: list_tm,A22: list_tm] :
( ( listrelp_tm_tm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_tm ) )
=> ~ ! [X: tm,Y4: tm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_tm_tm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_922_listrelp_Ocases,axiom,
! [R2: tm > nat > $o,A1: list_tm,A22: list_nat] :
( ( listrelp_tm_nat @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_nat ) )
=> ~ ! [X: tm,Y4: nat,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_tm_nat @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_923_listrelp_Ocases,axiom,
! [R2: nat > fm > $o,A1: list_nat,A22: list_fm] :
( ( listrelp_nat_fm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_fm ) )
=> ~ ! [X: nat,Y4: fm,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_nat_fm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_924_listrelp_Ocases,axiom,
! [R2: nat > tm > $o,A1: list_nat,A22: list_tm] :
( ( listrelp_nat_tm @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_tm ) )
=> ~ ! [X: nat,Y4: tm,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_nat_tm @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_925_listrelp_Ocases,axiom,
! [R2: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( listrelp_nat_nat @ R2 @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X: nat,Y4: nat,Xs2: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( R2 @ X @ Y4 )
=> ~ ( listrelp_nat_nat @ R2 @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_926_listrelp_Osimps,axiom,
( listrelp_fm_fm
= ( ^ [R3: fm > fm > $o,A12: list_fm,A23: list_fm] :
( ( ( A12 = nil_fm )
& ( A23 = nil_fm ) )
| ? [X5: fm,Y: fm,Xs3: list_fm,Ys2: list_fm] :
( ( A12
= ( cons_fm @ X5 @ Xs3 ) )
& ( A23
= ( cons_fm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_fm_fm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_927_listrelp_Osimps,axiom,
( listrelp_fm_tm
= ( ^ [R3: fm > tm > $o,A12: list_fm,A23: list_tm] :
( ( ( A12 = nil_fm )
& ( A23 = nil_tm ) )
| ? [X5: fm,Y: tm,Xs3: list_fm,Ys2: list_tm] :
( ( A12
= ( cons_fm @ X5 @ Xs3 ) )
& ( A23
= ( cons_tm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_fm_tm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_928_listrelp_Osimps,axiom,
( listrelp_fm_nat
= ( ^ [R3: fm > nat > $o,A12: list_fm,A23: list_nat] :
( ( ( A12 = nil_fm )
& ( A23 = nil_nat ) )
| ? [X5: fm,Y: nat,Xs3: list_fm,Ys2: list_nat] :
( ( A12
= ( cons_fm @ X5 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_fm_nat @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_929_listrelp_Osimps,axiom,
( listrelp_tm_fm
= ( ^ [R3: tm > fm > $o,A12: list_tm,A23: list_fm] :
( ( ( A12 = nil_tm )
& ( A23 = nil_fm ) )
| ? [X5: tm,Y: fm,Xs3: list_tm,Ys2: list_fm] :
( ( A12
= ( cons_tm @ X5 @ Xs3 ) )
& ( A23
= ( cons_fm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_tm_fm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_930_listrelp_Osimps,axiom,
( listrelp_tm_tm
= ( ^ [R3: tm > tm > $o,A12: list_tm,A23: list_tm] :
( ( ( A12 = nil_tm )
& ( A23 = nil_tm ) )
| ? [X5: tm,Y: tm,Xs3: list_tm,Ys2: list_tm] :
( ( A12
= ( cons_tm @ X5 @ Xs3 ) )
& ( A23
= ( cons_tm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_tm_tm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_931_listrelp_Osimps,axiom,
( listrelp_tm_nat
= ( ^ [R3: tm > nat > $o,A12: list_tm,A23: list_nat] :
( ( ( A12 = nil_tm )
& ( A23 = nil_nat ) )
| ? [X5: tm,Y: nat,Xs3: list_tm,Ys2: list_nat] :
( ( A12
= ( cons_tm @ X5 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_tm_nat @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_932_listrelp_Osimps,axiom,
( listrelp_nat_fm
= ( ^ [R3: nat > fm > $o,A12: list_nat,A23: list_fm] :
( ( ( A12 = nil_nat )
& ( A23 = nil_fm ) )
| ? [X5: nat,Y: fm,Xs3: list_nat,Ys2: list_fm] :
( ( A12
= ( cons_nat @ X5 @ Xs3 ) )
& ( A23
= ( cons_fm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_nat_fm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_933_listrelp_Osimps,axiom,
( listrelp_nat_tm
= ( ^ [R3: nat > tm > $o,A12: list_nat,A23: list_tm] :
( ( ( A12 = nil_nat )
& ( A23 = nil_tm ) )
| ? [X5: nat,Y: tm,Xs3: list_nat,Ys2: list_tm] :
( ( A12
= ( cons_nat @ X5 @ Xs3 ) )
& ( A23
= ( cons_tm @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_nat_tm @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_934_listrelp_Osimps,axiom,
( listrelp_nat_nat
= ( ^ [R3: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( ( A12 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X5: nat,Y: nat,Xs3: list_nat,Ys2: list_nat] :
( ( A12
= ( cons_nat @ X5 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y @ Ys2 ) )
& ( R3 @ X5 @ Y )
& ( listrelp_nat_nat @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_935_accp__subset,axiom,
! [R1: list_fm > list_fm > $o,R22: list_fm > list_fm > $o] :
( ( ord_le3799113821011214030t_fm_o @ R1 @ R22 )
=> ( ord_le6518561683347902116t_fm_o @ ( accp_list_fm @ R22 ) @ ( accp_list_fm @ R1 ) ) ) ).
% accp_subset
thf(fact_936_accp__subset__induct,axiom,
! [D: list_fm > $o,R: list_fm > list_fm > $o,X2: list_fm,P: list_fm > $o] :
( ( ord_le6518561683347902116t_fm_o @ D @ ( accp_list_fm @ R ) )
=> ( ! [X: list_fm,Z5: list_fm] :
( ( D @ X )
=> ( ( R @ Z5 @ X )
=> ( D @ Z5 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: list_fm] :
( ( D @ X )
=> ( ! [Z6: list_fm] :
( ( R @ Z6 @ X )
=> ( P @ Z6 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_937_listrel__iff__nth,axiom,
! [Xs: list_fm,Ys: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel_fm_fm @ R2 ) )
= ( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
& ! [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs ) )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ ( nth_fm @ Xs @ N ) @ ( nth_fm @ Ys @ N ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_938_nths__singleton,axiom,
! [A3: set_nat,X2: fm] :
( ( ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_fm @ ( cons_fm @ X2 @ nil_fm ) @ A3 )
= ( cons_fm @ X2 @ nil_fm ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_fm @ ( cons_fm @ X2 @ nil_fm ) @ A3 )
= nil_fm ) ) ) ).
% nths_singleton
thf(fact_939_nths__singleton,axiom,
! [A3: set_nat,X2: tm] :
( ( ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_tm @ ( cons_tm @ X2 @ nil_tm ) @ A3 )
= ( cons_tm @ X2 @ nil_tm ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_tm @ ( cons_tm @ X2 @ nil_tm ) @ A3 )
= nil_tm ) ) ) ).
% nths_singleton
thf(fact_940_nths__singleton,axiom,
! [A3: set_nat,X2: nat] :
( ( ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A3 )
= ( cons_nat @ X2 @ nil_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A3 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_941_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_942_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_943_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_944_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_945_nths__nil,axiom,
! [A3: set_nat] :
( ( nths_tm @ nil_tm @ A3 )
= nil_tm ) ).
% nths_nil
thf(fact_946_nths__nil,axiom,
! [A3: set_nat] :
( ( nths_nat @ nil_nat @ A3 )
= nil_nat ) ).
% nths_nil
thf(fact_947_nths__nil,axiom,
! [A3: set_nat] :
( ( nths_fm @ nil_fm @ A3 )
= nil_fm ) ).
% nths_nil
thf(fact_948_in__measure,axiom,
! [X2: fm,Y3: fm,F: fm > nat] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y3 ) @ ( measure_fm @ F ) )
= ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% in_measure
thf(fact_949_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_950_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_951_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_952_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_953_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_954_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_955_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_956_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_957_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_958_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_959_leD,axiom,
! [Y3: set_tm,X2: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X2 )
=> ~ ( ord_less_set_tm @ X2 @ Y3 ) ) ).
% leD
thf(fact_960_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_961_leD,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_962_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_963_nless__le,axiom,
! [A: set_tm,B: set_tm] :
( ( ~ ( ord_less_set_tm @ A @ B ) )
= ( ~ ( ord_less_eq_set_tm @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_964_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_965_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_966_antisym__conv1,axiom,
! [X2: set_tm,Y3: set_tm] :
( ~ ( ord_less_set_tm @ X2 @ Y3 )
=> ( ( ord_less_eq_set_tm @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_967_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_968_antisym__conv1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_969_antisym__conv2,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_tm @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_970_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_971_antisym__conv2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_972_less__le__not__le,axiom,
( ord_less_set_tm
= ( ^ [X5: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X5 @ Y )
& ~ ( ord_less_eq_set_tm @ Y @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_973_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y: nat] :
( ( ord_less_eq_nat @ X5 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_974_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_975_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_976_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B4: set_tm] :
( ( ord_less_set_tm @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_977_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_978_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_979_order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [A5: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_980_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_981_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_982_order_Ostrict__trans1,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_tm @ B @ C )
=> ( ord_less_set_tm @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_983_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_984_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_985_order_Ostrict__trans2,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_set_tm @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_986_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_987_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_988_order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [A5: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B4 )
& ~ ( ord_less_eq_set_tm @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_989_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_990_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_991_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [B4: set_tm,A5: set_tm] :
( ( ord_less_set_tm @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_992_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_993_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_994_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [B4: set_tm,A5: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_995_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_996_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_997_dual__order_Ostrict__trans1,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_set_tm @ C @ B )
=> ( ord_less_set_tm @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_998_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_999_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1000_dual__order_Ostrict__trans2,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ C @ B )
=> ( ord_less_set_tm @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1001_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1002_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1003_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [B4: set_tm,A5: set_tm] :
( ( ord_less_eq_set_tm @ B4 @ A5 )
& ~ ( ord_less_eq_set_tm @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1004_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1005_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1006_order_Ostrict__implies__order,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1007_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1008_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1009_dual__order_Ostrict__implies__order,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_set_tm @ B @ A )
=> ( ord_less_eq_set_tm @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1010_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1011_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1012_order__le__less,axiom,
( ord_less_eq_set_tm
= ( ^ [X5: set_tm,Y: set_tm] :
( ( ord_less_set_tm @ X5 @ Y )
| ( X5 = Y ) ) ) ) ).
% order_le_less
thf(fact_1013_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y: nat] :
( ( ord_less_nat @ X5 @ Y )
| ( X5 = Y ) ) ) ) ).
% order_le_less
thf(fact_1014_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X5: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X5 @ Y )
| ( X5 = Y ) ) ) ) ).
% order_le_less
thf(fact_1015_order__less__le,axiom,
( ord_less_set_tm
= ( ^ [X5: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X5 @ Y )
& ( X5 != Y ) ) ) ) ).
% order_less_le
thf(fact_1016_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y: nat] :
( ( ord_less_eq_nat @ X5 @ Y )
& ( X5 != Y ) ) ) ) ).
% order_less_le
thf(fact_1017_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X5: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y )
& ( X5 != Y ) ) ) ) ).
% order_less_le
thf(fact_1018_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_1019_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_1020_order__less__imp__le,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( ord_less_set_tm @ X2 @ Y3 )
=> ( ord_less_eq_set_tm @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1021_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1022_order__less__imp__le,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1023_order__le__neq__trans,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_tm @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1024_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1025_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1026_order__neq__le__trans,axiom,
! [A: set_tm,B: set_tm] :
( ( A != B )
=> ( ( ord_less_eq_set_tm @ A @ B )
=> ( ord_less_set_tm @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1027_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1028_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1029_order__le__less__trans,axiom,
! [X2: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( ord_less_set_tm @ Y3 @ Z3 )
=> ( ord_less_set_tm @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1030_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1031_order__le__less__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z3 )
=> ( ord_less_set_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1032_order__less__le__trans,axiom,
! [X2: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_set_tm @ X2 @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ Z3 )
=> ( ord_less_set_tm @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1033_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1034_order__less__le__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z3 )
=> ( ord_less_set_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1035_order__le__less__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1036_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1037_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1038_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1039_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1040_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1041_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1042_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1043_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1044_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1045_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1046_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1047_order__less__le__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1048_order__less__le__subst1,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1049_order__less__le__subst1,axiom,
! [A: set_nat,F: set_tm > set_nat,B: set_tm,C: set_tm] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1050_order__less__le__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1051_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1052_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1053_order__less__le__subst1,axiom,
! [A: set_tm,F: set_nat > set_tm,B: set_nat,C: set_nat] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1054_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1055_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1056_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_tm @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1057_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1058_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1059_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1060_order__le__imp__less__or__eq,axiom,
! [X2: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y3 )
=> ( ( ord_less_set_tm @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1061_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1062_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1063_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_1064_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1065_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1066_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_1067_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1068_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1069_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1070_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1071_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1072_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1073_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1074_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1075_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_1076_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_1077_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_1078_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_1079_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_1080_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1081_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_1082_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1083_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1084_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_1085_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1086_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1087_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1088_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1089_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_1090_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_1091_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_1092_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1093_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1094_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1095_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1096_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1097_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B5: nat] :
( ( ord_less_nat @ A2 @ B5 )
=> ( P @ A2 @ B5 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B5: nat] :
( ( P @ B5 @ A2 )
=> ( P @ A2 @ B5 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1098_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N: nat] :
( ( P5 @ N )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1099_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1100_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1101_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_1102_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_1103_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1104_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1105_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1106_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1107_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_1108_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_1109_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1110_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1111_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1112_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1113_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1114_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1115_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1116_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1117_in__set__nthsD,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm,I4: set_nat] :
( ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ ( nths_P3473792751519839979_fm_fm @ Xs @ I4 ) ) )
=> ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1118_in__set__nthsD,axiom,
! [X2: tm,Xs: list_tm,I4: set_nat] :
( ( member_tm2 @ X2 @ ( set_tm2 @ ( nths_tm @ Xs @ I4 ) ) )
=> ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1119_in__set__nthsD,axiom,
! [X2: nat,Xs: list_nat,I4: set_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1120_in__set__nthsD,axiom,
! [X2: fm,Xs: list_fm,I4: set_nat] :
( ( member_fm2 @ X2 @ ( set_fm2 @ ( nths_fm @ Xs @ I4 ) ) )
=> ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1121_notin__set__nthsI,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm,I4: set_nat] :
( ~ ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ~ ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ ( nths_P3473792751519839979_fm_fm @ Xs @ I4 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1122_notin__set__nthsI,axiom,
! [X2: tm,Xs: list_tm,I4: set_nat] :
( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ~ ( member_tm2 @ X2 @ ( set_tm2 @ ( nths_tm @ Xs @ I4 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1123_notin__set__nthsI,axiom,
! [X2: nat,Xs: list_nat,I4: set_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1124_notin__set__nthsI,axiom,
! [X2: fm,Xs: list_fm,I4: set_nat] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ~ ( member_fm2 @ X2 @ ( set_fm2 @ ( nths_fm @ Xs @ I4 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1125_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1126_set__nths__subset,axiom,
! [Xs: list_fm,I4: set_nat] : ( ord_less_eq_set_fm @ ( set_fm2 @ ( nths_fm @ Xs @ I4 ) ) @ ( set_fm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1127_set__nths__subset,axiom,
! [Xs: list_tm,I4: set_nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ ( nths_tm @ Xs @ I4 ) ) @ ( set_tm2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1128_set__nths__subset,axiom,
! [Xs: list_nat,I4: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) @ ( set_nat2 @ Xs ) ) ).
% set_nths_subset
thf(fact_1129_length__pos__if__in__set,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm] :
( ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3074140853920721241_fm_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1130_length__pos__if__in__set,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1131_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1132_length__pos__if__in__set,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1133_nth__mem,axiom,
! [N2: nat,Xs: list_P8031219080602320621_fm_fm] :
( ( ord_less_nat @ N2 @ ( size_s3074140853920721241_fm_fm @ Xs ) )
=> ( member7780952600467998736_fm_fm @ ( nth_Pr5768189175911290222_fm_fm @ Xs @ N2 ) @ ( set_Pr5149718152543245948_fm_fm @ Xs ) ) ) ).
% nth_mem
thf(fact_1134_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_1135_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_1136_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_1137_list__ball__nth,axiom,
! [N2: nat,Xs: list_tm,P: tm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( ! [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_tm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1138_list__ball__nth,axiom,
! [N2: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1139_list__ball__nth,axiom,
! [N2: nat,Xs: list_fm,P: fm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ! [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1140_in__set__conv__nth,axiom,
! [X2: product_prod_fm_fm,Xs: list_P8031219080602320621_fm_fm] :
( ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s3074140853920721241_fm_fm @ Xs ) )
& ( ( nth_Pr5768189175911290222_fm_fm @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1141_in__set__conv__nth,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
& ( ( nth_tm @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1142_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1143_in__set__conv__nth,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1144_all__nth__imp__all__set,axiom,
! [Xs: list_P8031219080602320621_fm_fm,P: product_prod_fm_fm > $o,X2: product_prod_fm_fm] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3074140853920721241_fm_fm @ Xs ) )
=> ( P @ ( nth_Pr5768189175911290222_fm_fm @ Xs @ I3 ) ) )
=> ( ( member7780952600467998736_fm_fm @ X2 @ ( set_Pr5149718152543245948_fm_fm @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1145_all__nth__imp__all__set,axiom,
! [Xs: list_tm,P: tm > $o,X2: tm] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_tm @ Xs ) )
=> ( P @ ( nth_tm @ Xs @ I3 ) ) )
=> ( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1146_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1147_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P: fm > $o,X2: fm] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I3 ) ) )
=> ( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1148_all__set__conv__all__nth,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ! [X5: tm] :
( ( member_tm2 @ X5 @ ( set_tm2 @ Xs ) )
=> ( P @ X5 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( P @ ( nth_tm @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1149_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
=> ( P @ X5 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1150_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ! [X5: fm] :
( ( member_fm2 @ X5 @ ( set_fm2 @ Xs ) )
=> ( P @ X5 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
=> ( P @ ( nth_fm @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1151_list__all__length,axiom,
( list_all_tm
= ( ^ [P5: tm > $o,Xs3: list_tm] :
! [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_tm @ Xs3 ) )
=> ( P5 @ ( nth_tm @ Xs3 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_1152_list__all__length,axiom,
( list_all_fm
= ( ^ [P5: fm > $o,Xs3: list_fm] :
! [N: nat] :
( ( ord_less_nat @ N @ ( size_size_list_fm @ Xs3 ) )
=> ( P5 @ ( nth_fm @ Xs3 @ N ) ) ) ) ) ).
% list_all_length
thf(fact_1153_Cons__lenlex__iff,axiom,
! [M: tm,Ms: list_tm,N2: tm,Ns: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ M @ Ms ) @ ( cons_tm @ N2 @ Ns ) ) @ ( lenlex_tm @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_tm @ Ms ) @ ( size_size_list_tm @ Ns ) )
| ( ( ( size_size_list_tm @ Ms )
= ( size_size_list_tm @ Ns ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ M @ N2 ) @ R2 ) )
| ( ( M = N2 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ms @ Ns ) @ ( lenlex_tm @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1154_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N2: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N2 @ Ns ) ) @ ( lenlex_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ R2 ) )
| ( ( M = N2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1155_Cons__lenlex__iff,axiom,
! [M: fm,Ms: list_fm,N2: fm,Ns: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ M @ Ms ) @ ( cons_fm @ N2 @ Ns ) ) @ ( lenlex_fm @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_fm @ Ms ) @ ( size_size_list_fm @ Ns ) )
| ( ( ( size_size_list_fm @ Ms )
= ( size_size_list_fm @ Ns ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ M @ N2 ) @ R2 ) )
| ( ( M = N2 )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ms @ Ns ) @ ( lenlex_fm @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1156_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1157_psubsetI,axiom,
! [A3: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_tm @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1158_psubsetI,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_nat @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1159_Nil__lenlex__iff1,axiom,
! [Ns: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Ns ) @ ( lenlex_tm @ R2 ) )
= ( Ns != nil_tm ) ) ).
% Nil_lenlex_iff1
thf(fact_1160_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R2 ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_1161_Nil__lenlex__iff1,axiom,
! [Ns: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Ns ) @ ( lenlex_fm @ R2 ) )
= ( Ns != nil_fm ) ) ).
% Nil_lenlex_iff1
thf(fact_1162_psubsetE,axiom,
! [A3: set_tm,B2: set_tm] :
( ( ord_less_set_tm @ A3 @ B2 )
=> ~ ( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ord_less_eq_set_tm @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_1163_psubsetE,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_1164_psubset__eq,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1165_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1166_psubset__imp__subset,axiom,
! [A3: set_tm,B2: set_tm] :
( ( ord_less_set_tm @ A3 @ B2 )
=> ( ord_less_eq_set_tm @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1167_psubset__imp__subset,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1168_psubset__subset__trans,axiom,
! [A3: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_set_tm @ A3 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C2 )
=> ( ord_less_set_tm @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1169_psubset__subset__trans,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1170_subset__not__subset__eq,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B3 )
& ~ ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1171_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1172_subset__psubset__trans,axiom,
! [A3: set_tm,B2: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B2 )
=> ( ( ord_less_set_tm @ B2 @ C2 )
=> ( ord_less_set_tm @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1173_subset__psubset__trans,axiom,
! [A3: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1174_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_set_tm @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1175_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1176_lenlex__irreflexive,axiom,
! [R2: set_Pr4706815898642364871_fm_fm,Xs: list_fm] :
( ! [X: fm] :
~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ X ) @ R2 )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Xs ) @ ( lenlex_fm @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_1177_Nil__lenlex__iff2,axiom,
! [Ns: list_tm,R2: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ns @ nil_tm ) @ ( lenlex_tm @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1178_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1179_Nil__lenlex__iff2,axiom,
! [Ns: list_fm,R2: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ns @ nil_fm ) @ ( lenlex_fm @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1180_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_1181_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_1182_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_1183_less__set__def,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ord_less_tm_o
@ ^ [X5: tm] : ( member_tm2 @ X5 @ A4 )
@ ^ [X5: tm] : ( member_tm2 @ X5 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_1184_less__set__def,axiom,
( ord_less_set_fm
= ( ^ [A4: set_fm,B3: set_fm] :
( ord_less_fm_o
@ ^ [X5: fm] : ( member_fm2 @ X5 @ A4 )
@ ^ [X5: fm] : ( member_fm2 @ X5 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_1185_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_nat_o
@ ^ [X5: nat] : ( member_nat2 @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat2 @ X5 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_1186_less__set__def,axiom,
( ord_le5731113155793438835_fm_fm
= ( ^ [A4: set_Pr4706815898642364871_fm_fm,B3: set_Pr4706815898642364871_fm_fm] :
( ord_le1062710942104439338m_fm_o
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ A4 )
@ ^ [X5: product_prod_fm_fm] : ( member7780952600467998736_fm_fm @ X5 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_1187_psubsetD,axiom,
! [A3: set_fm,B2: set_fm,C: fm] :
( ( ord_less_set_fm @ A3 @ B2 )
=> ( ( member_fm2 @ C @ A3 )
=> ( member_fm2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1188_psubsetD,axiom,
! [A3: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ( ( member_nat2 @ C @ A3 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1189_psubsetD,axiom,
! [A3: set_Pr4706815898642364871_fm_fm,B2: set_Pr4706815898642364871_fm_fm,C: product_prod_fm_fm] :
( ( ord_le5731113155793438835_fm_fm @ A3 @ B2 )
=> ( ( member7780952600467998736_fm_fm @ C @ A3 )
=> ( member7780952600467998736_fm_fm @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1190_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1191_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1192_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_1193_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1194_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1195_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_1196_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1197_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1198_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_1199_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1200_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1201_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1202_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1203_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1204_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1205_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1206_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X: nat,Y4: nat] :
( ( P @ X @ Y4 )
=> ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1207_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1208_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1209_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1210_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1211_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
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_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_1214_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1215_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_1216_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_1217_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1218_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_1219_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1220_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1221_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y4: nat,Z5: nat] :
( ( R @ X @ Y4 )
=> ( ( R @ Y4 @ Z5 )
=> ( R @ X @ Z5 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1222_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1223_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1224_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1225_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_1226_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1227_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1228_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1229_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_1230_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1231_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1232_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_1233_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1234_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_1235_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1236_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1237_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1238_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_1239_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1240_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_1241_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1242_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1243_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_1244_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_1245_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_1246_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1247_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_1248_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1249_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1250_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1251_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1252_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_1253_liftt_Osimps_I1_J,axiom,
! [I2: nat] :
( ( liftt @ ( var @ I2 ) )
= ( var @ ( suc @ I2 ) ) ) ).
% liftt.simps(1)
thf(fact_1254_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1255_subst_Osimps_I5_J,axiom,
! [P3: fm,S2: tm,K: nat] :
( ( subst @ ( exi @ P3 ) @ S2 @ K )
= ( exi @ ( subst @ P3 @ ( liftt @ S2 ) @ ( suc @ K ) ) ) ) ).
% subst.simps(5)
thf(fact_1256_subst_Osimps_I6_J,axiom,
! [P3: fm,S2: tm,K: nat] :
( ( subst @ ( uni @ P3 ) @ S2 @ K )
= ( uni @ ( subst @ P3 @ ( liftt @ S2 ) @ ( suc @ K ) ) ) ) ).
% subst.simps(6)
thf(fact_1257_min__0R,axiom,
! [N2: nat] :
( ( ord_min_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1258_min__0L,axiom,
! [N2: nat] :
( ( ord_min_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% min_0L
thf(fact_1259_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_1260_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1261_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
% Helper facts (15)
thf(help_If_2_1_If_001t__SeCaV__Otm_T,axiom,
! [X2: tm,Y3: tm] :
( ( if_tm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__SeCaV__Otm_T,axiom,
! [X2: tm,Y3: tm] :
( ( if_tm @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X4: nat] : ( P @ X4 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__SeCaV__Ofm_T,axiom,
! [P: fm > $o] :
( ( P @ ( fChoice_fm @ P ) )
= ( ? [X4: fm] : ( P @ X4 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__SeCaV__Otm_T,axiom,
! [P: tm > $o] :
( ( P @ ( fChoice_tm @ P ) )
= ( ? [X4: tm] : ( P @ X4 ) ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X2: list_fm,Y3: list_fm] :
( ( if_list_fm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X2: list_fm,Y3: list_fm] :
( ( if_list_fm @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X2: list_tm,Y3: list_tm] :
( ( if_list_tm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X2: list_tm,Y3: list_tm] :
( ( if_list_tm @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_T,axiom,
! [P: product_prod_fm_fm > $o] :
( ( P @ ( fChoic1084495941463685435_fm_fm @ P ) )
= ( ? [X4: product_prod_fm_fm] : ( P @ X4 ) ) ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J_T,axiom,
! [X2: list_P8031219080602320621_fm_fm,Y3: list_P8031219080602320621_fm_fm] :
( ( if_lis6935098856802376371_fm_fm @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J_T,axiom,
! [X2: list_P8031219080602320621_fm_fm,Y3: list_P8031219080602320621_fm_fm] :
( ( if_lis6935098856802376371_fm_fm @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( semantics_term_tm
@ ^ [N: nat] :
( if_tm @ ( member_tm2 @ ( var @ N ) @ ( terms @ s ) ) @ ( var @ N )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm2 @ ( fun @ I @ L ) @ ( terms @ s ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm2 @ T2 @ ( terms @ s ) ) ) )
@ t )
= t ) ).
%------------------------------------------------------------------------------