TPTP Problem File: SLH0872^1.p
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%------------------------------------------------------------------------------
% 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_Calc2/0020_Soundness/prob_00063_002253__13464098_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1687 ( 780 unt; 404 typ; 0 def)
% Number of atoms : 3800 (1968 equ; 0 cnn)
% Maximal formula atoms : 42 ( 2 avg)
% Number of connectives : 14342 ( 436 ~; 97 |; 417 &;12031 @)
% ( 0 <=>;1361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1657 (1657 >; 0 *; 0 +; 0 <<)
% Number of symbols : 369 ( 366 usr; 25 con; 0-12 aty)
% Number of variables : 4906 ( 683 ^;3741 !; 482 ?;4906 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:46:16.101
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
fset_P8989946509869081563ist_fm: $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__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
list_l1925138706763304843ist_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_J,type,
list_list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
list_list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_list_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_set_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J_J,type,
set_set_set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J,type,
product_prod_rule_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
fset_list_fm: $tType ).
thf(ty_n_t__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__List__Olist_It__Set__Oset_It__SeCaV__Otm_J_J,type,
list_set_tm: $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__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
set_set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
set_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $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__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Prover__Orule,type,
rule: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (366)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
comple8784269564784259782ist_fm: set_set_list_fm > set_list_fm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
comple2134933779557159616set_fm: set_set_fm > set_fm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Otm_J,type,
comple2138885804642794802set_tm: set_set_tm > set_tm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
comple4084446694820577554set_tm: set_set_set_tm > set_set_tm ).
thf(sy_c_FSet_Ofimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fimage4743371125182381497ist_fm: ( list_fm > produc6018962875968178549ist_fm ) > fset_list_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fmembe3754813877001230652ist_fm: produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_FSet_Ofset__of__list_001t__List__Olist_It__SeCaV__Ofm_J,type,
fset_of_list_list_fm: list_list_fm > fset_list_fm ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_I_Eo_J,type,
if_list_o: $o > list_o > list_o > list_o ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
if_list_list_fm: $o > list_list_fm > list_list_fm > list_list_fm ).
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__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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
if_list_set_nat: $o > list_set_nat > list_set_nat > list_set_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_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
append_list_list_fm: list_list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Ofm_J,type,
append_list_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Otm_J,type,
append_list_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
append_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
append_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__Nat__Onat_J,type,
append_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_list_fm_list_fm: list_list_fm > ( list_fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
bind_list_fm_fm: list_list_fm > ( list_fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
bind_list_fm_tm: list_list_fm > ( list_fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_fm_list_fm: list_fm > ( fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__Nat__Onat,type,
bind_fm_nat: list_fm > ( fm > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
bind_fm_fm: list_fm > ( fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
bind_fm_tm: list_fm > ( fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_tm_list_fm: list_tm > ( tm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__Nat__Onat,type,
bind_tm_nat: list_tm > ( tm > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
bind_tm_fm: list_tm > ( tm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Ofm,type,
bind_set_nat_fm: list_set_nat > ( set_nat > list_fm ) > list_fm ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
concat_list_list_fm: list_l1925138706763304843ist_fm > list_list_list_fm ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__SeCaV__Ofm_J,type,
concat_list_fm: list_list_list_fm > list_list_fm ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__SeCaV__Otm_J,type,
concat_list_tm: list_list_list_tm > list_list_tm ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
concat_fm: list_list_fm > list_fm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Oconcat_001t__Set__Oset_It__Nat__Onat_J,type,
concat_set_nat: list_list_set_nat > list_set_nat ).
thf(sy_c_List_Ofilter_001_Eo,type,
filter_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
filter_list_list_fm: ( list_list_fm > $o ) > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__SeCaV__Ofm_J,type,
filter_list_fm: ( list_fm > $o ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__SeCaV__Otm_J,type,
filter_list_tm: ( list_tm > $o ) > list_list_tm > list_list_tm ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__SeCaV__Ofm,type,
filter_fm: ( fm > $o ) > list_fm > list_fm ).
thf(sy_c_List_Ofilter_001t__SeCaV__Otm,type,
filter_tm: ( tm > $o ) > list_tm > list_tm ).
thf(sy_c_List_Ofilter_001t__Set__Oset_It__Nat__Onat_J,type,
filter_set_nat: ( set_nat > $o ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Oinsert_001_Eo,type,
insert_o: $o > list_o > list_o ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
insert_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
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_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_OCons_001_Eo,type,
cons_o: $o > list_o > list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
cons_list_list_fm: list_list_fm > list_list_list_fm > list_list_list_fm ).
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__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_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_ONil_001_Eo,type,
nil_o: list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
nil_list_list_fm: list_list_list_fm ).
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_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__List__Olist_It__SeCaV__Ofm_J,type,
case_list_o_list_fm: $o > ( list_fm > list_list_fm > $o ) > list_list_fm > $o ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__SeCaV__Ofm,type,
case_list_o_fm: $o > ( fm > list_fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__SeCaV__Otm,type,
case_list_o_tm: $o > ( tm > list_tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
case_l2744553060881268634ist_fm: list_list_list_fm > ( list_fm > list_list_fm > list_list_list_fm ) > list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
case_l7658988447939845536ist_fm: list_list_fm > ( list_fm > list_list_fm > list_list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__SeCaV__Ofm,type,
case_l1147175924207067418_fm_fm: list_list_fm > ( fm > list_fm > list_list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_001t__SeCaV__Otm,type,
case_l799553655970854810_tm_tm: list_list_tm > ( tm > list_tm > list_list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
case_list_list_fm_fm: list_fm > ( fm > list_fm > list_fm ) > list_fm > list_fm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
case_list_list_tm_tm: list_tm > ( tm > list_tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
map_o_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
map_li2158052760755490954ist_fm: ( list_list_fm > list_list_list_fm ) > list_list_list_fm > list_l1925138706763304843ist_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
map_li4351931137408529412ist_fm: ( list_list_fm > list_list_fm ) > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
map_li9121411909794442256ist_fm: ( list_fm > list_list_list_fm ) > list_list_fm > list_l1925138706763304843ist_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
map_li1351512418201717386ist_fm: ( list_fm > list_list_fm ) > list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_li1108997747876207612ist_tm: ( list_fm > list_list_tm ) > list_list_fm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_list_fm_list_fm: ( list_fm > list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
map_list_fm_list_tm: ( list_fm > list_tm ) > list_list_fm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
map_list_fm_fm: ( list_fm > fm ) > list_list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
map_list_fm_tm: ( list_fm > tm ) > list_list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_li6264597563971819530ist_tm: ( list_tm > list_list_tm ) > list_list_tm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_list_tm_list_fm: ( list_tm > list_fm ) > list_list_tm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
map_list_tm_list_tm: ( list_tm > list_tm ) > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
map_li5423145413338040381et_nat: ( list_tm > list_set_nat ) > list_list_tm > list_list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
map_fm_list_list_fm: ( fm > list_list_fm ) > list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_fm_list_list_tm: ( fm > list_list_tm ) > list_fm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_fm_list_fm: ( fm > list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_fm_list_tm: ( fm > list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
map_fm_fm: ( fm > fm ) > list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
map_fm_tm: ( fm > tm ) > list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Otm_J,type,
map_fm_set_tm: ( fm > set_tm ) > list_fm > list_set_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_tm_list_list_tm: ( tm > list_list_tm ) > list_tm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_tm_list_fm: ( tm > list_fm ) > list_tm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
map_tm_list_set_nat: ( tm > list_set_nat ) > list_tm > list_list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
map_tm_fm: ( tm > fm ) > list_tm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
map_tm_set_nat: ( tm > set_nat ) > list_tm > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Otm_J,type,
map_tm_set_tm: ( tm > set_tm ) > list_tm > list_set_tm ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_001t__SeCaV__Ofm,type,
rec_li6905007533772093688_tm_fm: list_list_tm > ( fm > list_fm > list_list_tm > list_list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_001t__SeCaV__Otm,type,
rec_li6905007533773012074_tm_tm: list_list_tm > ( tm > list_tm > list_list_tm > list_list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
rec_list_list_fm_tm: list_fm > ( tm > list_tm > list_fm > list_fm ) > list_tm > list_fm ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__SeCaV__Otm,type,
rec_li8667420360015564823nat_tm: list_set_nat > ( tm > list_tm > list_set_nat > list_set_nat ) > list_tm > list_set_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_list_fm2: list_list_list_fm > set_list_list_fm ).
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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
set_list_set_nat2: list_list_set_nat > set_list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
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_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__SeCaV__Otm_J,type,
set_set_tm2: list_set_tm > set_set_tm ).
thf(sy_c_List_Olist_Otl_001_Eo,type,
tl_o: list_o > list_o ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__SeCaV__Ofm_J,type,
tl_list_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__SeCaV__Otm_J,type,
tl_list_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__SeCaV__Ofm,type,
tl_fm: list_fm > list_fm ).
thf(sy_c_List_Olist_Otl_001t__SeCaV__Otm,type,
tl_tm: list_tm > list_tm ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_It__Nat__Onat_J,type,
tl_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_list_fm_list_fm: ( list_fm > list_list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
maps_list_fm_fm: ( list_fm > list_fm ) > list_list_fm > list_fm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
maps_list_fm_tm: ( list_fm > list_tm ) > list_list_fm > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_fm_list_fm: ( fm > list_list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
maps_fm_fm: ( fm > list_fm ) > list_fm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
maps_fm_tm: ( fm > list_tm ) > list_fm > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_tm_list_fm: ( tm > list_list_fm ) > list_tm > list_list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
maps_tm_fm: ( tm > list_fm ) > list_tm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
maps_tm_tm: ( tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc373462945560358120ist_fm: list_list_list_fm > list_list_list_fm ).
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_Oremdups_001_Eo,type,
remdups_o: list_o > list_o ).
thf(sy_c_List_Oremdups_001t__List__Olist_It__SeCaV__Ofm_J,type,
remdups_list_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oremdups_001t__List__Olist_It__SeCaV__Otm_J,type,
remdups_list_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
remdups_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups_001t__SeCaV__Ofm,type,
remdups_fm: list_fm > list_fm ).
thf(sy_c_List_Oremdups_001t__SeCaV__Otm,type,
remdups_tm: list_tm > list_tm ).
thf(sy_c_List_Oremdups_001t__Set__Oset_It__Nat__Onat_J,type,
remdups_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Oset__Cons_001_Eo,type,
set_Cons_o: set_o > set_list_o > set_list_o ).
thf(sy_c_List_Oset__Cons_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_Cons_list_fm: set_list_fm > set_list_list_fm > set_list_list_fm ).
thf(sy_c_List_Oset__Cons_001t__Nat__Onat,type,
set_Cons_nat: set_nat > set_list_nat > set_list_nat ).
thf(sy_c_List_Oset__Cons_001t__SeCaV__Ofm,type,
set_Cons_fm: set_fm > set_list_fm > set_list_fm ).
thf(sy_c_List_Oset__Cons_001t__SeCaV__Otm,type,
set_Cons_tm: set_tm > set_list_tm > set_list_tm ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__SeCaV__Ofm_J,type,
subseqs_list_fm: list_list_fm > list_list_list_fm ).
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_Osubseqs_001t__Set__Oset_It__Nat__Onat_J,type,
subseqs_set_nat: list_set_nat > list_list_set_nat ).
thf(sy_c_List_Otranspose_001t__List__Olist_It__SeCaV__Ofm_J,type,
transpose_list_fm: list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Otranspose_001t__SeCaV__Ofm,type,
transpose_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Otranspose_001t__SeCaV__Otm,type,
transpose_tm: list_list_tm > list_list_tm ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
bot_bo6461889142629771335ist_fm: fset_P8989946509869081563ist_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Otm_J,type,
bot_bot_set_tm: set_tm ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $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__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__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_001_Eo,type,
ord_less_eq_o: $o > $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_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le7838213414353715577ist_fm: set_list_fm > set_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
ord_le5844446314808584147set_fm: set_set_fm > set_set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
ord_le5601931644483074373set_tm: set_set_tm > set_set_tm > $o ).
thf(sy_c_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__Prover__Orule_001t__SeCaV__Ofm,type,
product_Pair_rule_fm: rule > fm > product_prod_rule_fm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Prover__Orule_001t__SeCaV__Ofm_001_Eo,type,
produc3561889649859641891e_fm_o: ( rule > fm > $o ) > product_prod_rule_fm > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Prover__Orule_001t__SeCaV__Ofm_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc1325496751214513674ist_fm: ( rule > fm > list_list_fm ) > product_prod_rule_fm > list_list_fm ).
thf(sy_c_ProverLemmas_Oaffects,type,
affects: rule > fm > $o ).
thf(sy_c_Prover_ObranchDone,type,
branchDone: list_fm > $o ).
thf(sy_c_Prover_ObranchDone__rel,type,
branchDone_rel: list_fm > list_fm > $o ).
thf(sy_c_Prover_Ochildren,type,
children: list_tm > rule > list_fm > list_list_fm ).
thf(sy_c_Prover_Oeff,type,
eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Prover_Oeffect,type,
effect: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_Prover_OgenerateNew,type,
generateNew: list_tm > nat ).
thf(sy_c_Prover_Olist__prod_001t__SeCaV__Ofm,type,
list_prod_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_Prover_Olist__prod_001t__SeCaV__Otm,type,
list_prod_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_Prover_Oparts,type,
parts: list_tm > rule > fm > list_list_fm ).
thf(sy_c_Prover_Orule_OAlphaImp,type,
alphaImp: rule ).
thf(sy_c_Prover_Orule_Ocase__rule_001_Eo,type,
case_rule_o: $o > $o > $o > $o > $o > $o > $o > $o > $o > $o > $o > rule > $o ).
thf(sy_c_Prover_Orule_Ocase__rule_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
case_r8401956329264079908ist_fm: list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > list_list_fm > rule > list_list_fm ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_Prover_Osubterms,type,
subterms: list_fm > list_tm ).
thf(sy_c_SeCaV_Oext_001t__List__Olist_It__SeCaV__Ofm_J,type,
ext_list_fm: list_list_fm > list_list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__Nat__Onat,type,
ext_nat: list_nat > list_nat > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Oext_001t__Set__Oset_It__Nat__Onat_J,type,
ext_set_nat: list_set_nat > list_set_nat > $o ).
thf(sy_c_SeCaV_Ofm_OCon,type,
con: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ODis,type,
dis: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_OExi,type,
exi: fm > fm ).
thf(sy_c_SeCaV_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ONeg,type,
neg: fm > fm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_SeCaV_Ofm_Ocase__fm_001_Eo,type,
case_fm_o: ( nat > list_tm > $o ) > ( fm > fm > $o ) > ( fm > fm > $o ) > ( fm > fm > $o ) > ( fm > $o ) > ( fm > $o ) > ( fm > $o ) > fm > $o ).
thf(sy_c_SeCaV_Ofm_Ocase__fm_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
case_fm_list_list_fm: ( nat > list_tm > list_list_fm ) > ( fm > fm > list_list_fm ) > ( fm > fm > list_list_fm ) > ( fm > fm > list_list_fm ) > ( fm > list_list_fm ) > ( fm > list_list_fm ) > ( fm > list_list_fm ) > fm > list_list_fm ).
thf(sy_c_SeCaV_Omember_001_Eo,type,
member_o: $o > list_o > $o ).
thf(sy_c_SeCaV_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > list_list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
member_nat: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__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_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > list_set_nat > $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_Oparams_H,type,
params2: fm > set_nat ).
thf(sy_c_SeCaV_Oparams_H_H,type,
params3: fm > set_nat ).
thf(sy_c_SeCaV_Oparams_H_H__rel,type,
params_rel: fm > fm > $o ).
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_Oparamst_H_H,type,
paramst3: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H_H__rel,type,
paramst_rel: tm > tm > $o ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osequent__calculus,type,
sequent_calculus: list_fm > $o ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_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_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
collect_list_list_fm: ( list_list_fm > $o ) > set_list_list_fm ).
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__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_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__SeCaV__Ofm,type,
image_o_fm: ( $o > fm ) > set_o > set_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_o_set_fm: ( $o > set_fm ) > set_o > set_set_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_o_set_tm: ( $o > set_tm ) > set_o > set_set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
image_3687226712311829663ist_fm: ( list_list_fm > set_list_fm ) > set_list_list_fm > set_set_list_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
image_list_fm_fm: ( list_fm > fm ) > set_list_fm > set_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
image_list_fm_tm: ( list_fm > tm ) > set_list_fm > set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_list_fm_set_fm: ( list_fm > set_fm ) > set_list_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Otm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_list_tm_set_tm: ( list_tm > set_tm ) > set_list_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_8726355809080528601et_nat: ( list_set_nat > set_set_nat ) > set_list_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Ofm,type,
image_nat_fm: ( nat > fm ) > set_nat > set_fm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Otm,type,
image_nat_tm: ( nat > tm ) > set_nat > set_tm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_nat_set_fm: ( nat > set_fm ) > set_nat > set_set_fm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_nat_set_tm: ( nat > set_tm ) > set_nat > set_set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001_Eo,type,
image_fm_o: ( fm > $o ) > set_fm > set_o ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_fm_list_fm: ( fm > list_fm ) > set_fm > set_list_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
image_fm_list_tm: ( fm > list_tm ) > set_fm > set_list_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Nat__Onat,type,
image_fm_nat: ( fm > nat ) > set_fm > set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
image_fm_fm: ( fm > fm ) > set_fm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
image_fm_tm: ( fm > tm ) > set_fm > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_I_Eo_J,type,
image_fm_set_o: ( fm > set_o ) > set_fm > set_set_o ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
image_fm_set_list_fm: ( fm > set_list_fm ) > set_fm > set_set_list_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
image_fm_set_nat: ( fm > set_nat ) > set_fm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_fm_set_fm: ( fm > set_fm ) > set_fm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_fm_set_tm: ( fm > set_tm ) > set_fm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_fm_set_set_nat: ( fm > set_set_nat ) > set_fm > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001_Eo,type,
image_tm_o: ( tm > $o ) > set_tm > set_o ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
image_tm_list_tm: ( tm > list_tm ) > set_tm > set_list_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Nat__Onat,type,
image_tm_nat: ( tm > nat ) > set_tm > set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
image_tm_fm: ( tm > fm ) > set_tm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
image_tm_tm: ( tm > tm ) > set_tm > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_I_Eo_J,type,
image_tm_set_o: ( tm > set_o ) > set_tm > set_set_o ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
image_tm_set_list_fm: ( tm > set_list_fm ) > set_tm > set_set_list_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
image_tm_set_nat: ( tm > set_nat ) > set_tm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_tm_set_fm: ( tm > set_fm ) > set_tm > set_set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_tm_set_tm: ( tm > set_tm ) > set_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_tm_set_set_nat: ( tm > set_set_nat ) > set_tm > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_001_Eo,type,
image_set_list_fm_o: ( set_list_fm > $o ) > set_set_list_fm > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__SeCaV__Ofm_J,type,
image_set_nat_set_fm: ( set_nat > set_fm ) > set_set_nat > set_set_fm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_set_nat_set_tm: ( set_nat > set_tm ) > set_set_nat > set_set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001_Eo,type,
image_set_fm_o: ( set_fm > $o ) > set_set_fm > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_fm_set_nat: ( set_fm > set_nat ) > set_set_fm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_set_fm_set_tm: ( set_fm > set_tm ) > set_set_fm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_1496149073759408202et_nat: ( set_fm > set_set_nat ) > set_set_fm > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Ofm_J_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
image_1809285061380348183set_tm: ( set_fm > set_set_tm ) > set_set_fm > set_set_set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001_Eo,type,
image_set_tm_o: ( set_tm > $o ) > set_set_tm > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_tm_set_nat: ( set_tm > set_nat ) > set_set_tm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_set_tm_set_tm: ( set_tm > set_tm ) > set_set_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_5490068892692554428et_nat: ( set_tm > set_set_nat ) > set_set_tm > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
image_9072780396932801317set_tm: ( set_tm > set_set_tm ) > set_set_tm > set_set_set_tm ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__SeCaV__Otm,type,
insert_tm2: tm > set_tm > set_tm ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__SeCaV__Ofm_J,type,
the_elem_list_fm: set_list_fm > list_fm ).
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_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
prefixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Ofm,type,
prefixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Otm,type,
prefixes_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__SeCaV__Ofm_J,type,
sublists_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Ofm,type,
sublists_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Otm,type,
sublists_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
suffixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Ofm,type,
suffixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Otm,type,
suffixes_tm: list_tm > list_list_tm ).
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_Oaccp_001t__SeCaV__Ofm,type,
accp_fm: ( fm > fm > $o ) > fm > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__SeCaV__Otm,type,
accp_tm: ( tm > tm > $o ) > tm > $o ).
thf(sy_c_member_001_Eo,type,
member_o2: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
member_list_list_fm: list_list_fm > set_list_list_fm > $o ).
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_fm2: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm2: tm > set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
member_set_list_fm: set_list_fm > set_set_list_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Ofm_J,type,
member_set_fm: set_fm > set_set_fm > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Otm_J,type,
member_set_tm: set_tm > set_set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_A,type,
a: list_tm ).
thf(sy_v_Aa____,type,
aa: list_tm ).
thf(sy_v_p____,type,
p: fm ).
thf(sy_v_pa____,type,
pa: fm ).
thf(sy_v_pre,type,
pre2: list_fm ).
thf(sy_v_prea____,type,
prea: list_fm ).
thf(sy_v_q____,type,
q: fm ).
thf(sy_v_r,type,
r: rule ).
thf(sy_v_z,type,
z: list_fm ).
thf(sy_v_za____,type,
za: list_fm ).
% Relevant facts (1269)
thf(fact_0_local_OAlphaImp_I1_J,axiom,
r = alphaImp ).
% local.AlphaImp(1)
thf(fact_1_Cons_Oprems_I1_J,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( children @ aa @ r @ ( cons_fm @ p @ za ) ) ) )
=> ( sequent_calculus @ ( append_fm @ prea @ X ) ) ) ).
% Cons.prems(1)
thf(fact_2_local_OAlphaImp_I2_J,axiom,
( p
= ( imp @ pa @ q ) ) ).
% local.AlphaImp(2)
thf(fact_3_Neg,axiom,
! [P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P ) ) @ Z ) ) ) ).
% Neg
thf(fact_4__092_060open_062_092_060forall_062z_H_092_060in_062set_A_Ichildren_A_Iremdups_A_IA_A_064_AsubtermFms_A_Iconcat_A_Iparts_AA_Ar_Ap_____J_J_J_J_Ar_Az_J_O_A_I_092_060tturnstile_062_A_Ipre_A_064_A_091Neg_Ap_M_Aq_093_J_A_064_Az_H_J_092_060close_062,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) @ r @ za ) ) )
=> ( sequent_calculus @ ( append_fm @ ( append_fm @ prea @ ( cons_fm @ ( neg @ pa ) @ ( cons_fm @ q @ nil_fm ) ) ) @ X ) ) ) ).
% \<open>\<forall>z'\<in>set (children (remdups (A @ subtermFms (concat (parts A r p__)))) r z). (\<tturnstile> (pre @ [Neg p, q]) @ z')\<close>
thf(fact_5_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_6_append_Oassoc,axiom,
! [A: list_fm,B: list_fm,C: list_fm] :
( ( append_fm @ ( append_fm @ A @ B ) @ C )
= ( append_fm @ A @ ( append_fm @ B @ C ) ) ) ).
% append.assoc
thf(fact_7_append_Oassoc,axiom,
! [A: list_tm,B: list_tm,C: list_tm] :
( ( append_tm @ ( append_tm @ A @ B ) @ C )
= ( append_tm @ A @ ( append_tm @ B @ C ) ) ) ).
% append.assoc
thf(fact_8_append__assoc,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( append_fm @ ( append_fm @ Xs @ Ys ) @ Zs )
= ( append_fm @ Xs @ ( append_fm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_9_append__assoc,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( append_tm @ ( append_tm @ Xs @ Ys ) @ Zs )
= ( append_tm @ Xs @ ( append_tm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_10_append__same__eq,axiom,
! [Ys: list_fm,Xs: list_fm,Zs: list_fm] :
( ( ( append_fm @ Ys @ Xs )
= ( append_fm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_11_append__same__eq,axiom,
! [Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( append_tm @ Ys @ Xs )
= ( append_tm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_12_same__append__eq,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_13_same__append__eq,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_14_list_Oinject,axiom,
! [X21: fm,X22: list_fm,Y21: fm,Y22: list_fm] :
( ( ( cons_fm @ X21 @ X22 )
= ( cons_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_15_list_Oinject,axiom,
! [X21: list_fm,X22: list_list_fm,Y21: list_fm,Y22: list_list_fm] :
( ( ( cons_list_fm @ X21 @ X22 )
= ( cons_list_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_16_list_Oinject,axiom,
! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X22 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_append__Cons,axiom,
! [X2: fm,Xs: list_fm,Ys: list_fm] :
( ( append_fm @ ( cons_fm @ X2 @ Xs ) @ Ys )
= ( cons_fm @ X2 @ ( append_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_18_append__Cons,axiom,
! [X2: list_fm,Xs: list_list_fm,Ys: list_list_fm] :
( ( append_list_fm @ ( cons_list_fm @ X2 @ Xs ) @ Ys )
= ( cons_list_fm @ X2 @ ( append_list_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_19_append__Cons,axiom,
! [X2: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X2 @ Xs ) @ Ys )
= ( cons_tm @ X2 @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_20_Cons__eq__appendI,axiom,
! [X2: fm,Xs1: list_fm,Ys: list_fm,Xs: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_fm @ Xs1 @ Zs ) )
=> ( ( cons_fm @ X2 @ Xs )
= ( append_fm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_21_Cons__eq__appendI,axiom,
! [X2: list_fm,Xs1: list_list_fm,Ys: list_list_fm,Xs: list_list_fm,Zs: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_fm @ Xs1 @ Zs ) )
=> ( ( cons_list_fm @ X2 @ Xs )
= ( append_list_fm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_22_Cons__eq__appendI,axiom,
! [X2: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs ) )
=> ( ( cons_tm @ X2 @ Xs )
= ( append_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_23__092_060open_062_092_060forall_062z_H_092_060in_062set_A_Ichildren_A_Iremdups_A_IA_A_064_AsubtermFms_A_Iconcat_A_Iparts_AA_Ar_Ap_____J_J_J_J_Ar_Az_J_O_A_I_092_060tturnstile_062_Apre_A_064_ANeg_Ap_A_D_Aq_A_D_Az_H_J_092_060close_062,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) @ r @ za ) ) )
=> ( sequent_calculus @ ( append_fm @ prea @ ( cons_fm @ ( neg @ pa ) @ ( cons_fm @ q @ X ) ) ) ) ) ).
% \<open>\<forall>z'\<in>set (children (remdups (A @ subtermFms (concat (parts A r p__)))) r z). (\<tturnstile> pre @ Neg p # q # z')\<close>
thf(fact_24_fm_Oinject_I2_J,axiom,
! [X21: fm,X22: fm,Y21: fm,Y22: fm] :
( ( ( imp @ X21 @ X22 )
= ( imp @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fm.inject(2)
thf(fact_25_append__is__Nil__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= nil_fm )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_26_append__is__Nil__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= nil_list_fm )
= ( ( Xs = nil_list_fm )
& ( Ys = nil_list_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_27_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_28_Nil__is__append__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( nil_fm
= ( append_fm @ Xs @ Ys ) )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_29_Nil__is__append__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( nil_list_fm
= ( append_list_fm @ Xs @ Ys ) )
= ( ( Xs = nil_list_fm )
& ( Ys = nil_list_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_30_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys ) )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_31_self__append__conv2,axiom,
! [Y: list_fm,Xs: list_fm] :
( ( Y
= ( append_fm @ Xs @ Y ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_32_self__append__conv2,axiom,
! [Y: list_list_fm,Xs: list_list_fm] :
( ( Y
= ( append_list_fm @ Xs @ Y ) )
= ( Xs = nil_list_fm ) ) ).
% self_append_conv2
thf(fact_33_self__append__conv2,axiom,
! [Y: list_tm,Xs: list_tm] :
( ( Y
= ( append_tm @ Xs @ Y ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_34_append__self__conv2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_35_append__self__conv2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_fm ) ) ).
% append_self_conv2
thf(fact_36_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_37_self__append__conv,axiom,
! [Y: list_fm,Ys: list_fm] :
( ( Y
= ( append_fm @ Y @ Ys ) )
= ( Ys = nil_fm ) ) ).
% self_append_conv
thf(fact_38_self__append__conv,axiom,
! [Y: list_list_fm,Ys: list_list_fm] :
( ( Y
= ( append_list_fm @ Y @ Ys ) )
= ( Ys = nil_list_fm ) ) ).
% self_append_conv
thf(fact_39_self__append__conv,axiom,
! [Y: list_tm,Ys: list_tm] :
( ( Y
= ( append_tm @ Y @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_40_append__self__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_fm ) ) ).
% append_self_conv
thf(fact_41_append__self__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_fm ) ) ).
% append_self_conv
thf(fact_42_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_43_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_44_append__Nil2,axiom,
! [Xs: list_list_fm] :
( ( append_list_fm @ Xs @ nil_list_fm )
= Xs ) ).
% append_Nil2
thf(fact_45_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_46_append_Oright__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ A @ nil_fm )
= A ) ).
% append.right_neutral
thf(fact_47_append_Oright__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ A @ nil_list_fm )
= A ) ).
% append.right_neutral
thf(fact_48_append_Oright__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ A @ nil_tm )
= A ) ).
% append.right_neutral
thf(fact_49_list_Omap__disc__iff,axiom,
! [F: fm > fm,A: list_fm] :
( ( ( map_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_50_list_Omap__disc__iff,axiom,
! [F: fm > tm,A: list_fm] :
( ( ( map_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_51_list_Omap__disc__iff,axiom,
! [F: tm > tm,A: list_tm] :
( ( ( map_tm_tm @ F @ A )
= nil_tm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_52_list_Omap__disc__iff,axiom,
! [F: tm > fm,A: list_tm] :
( ( ( map_tm_fm @ F @ A )
= nil_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_53_list_Omap__disc__iff,axiom,
! [F: list_fm > fm,A: list_list_fm] :
( ( ( map_list_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_54_list_Omap__disc__iff,axiom,
! [F: fm > list_fm,A: list_fm] :
( ( ( map_fm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_55_list_Omap__disc__iff,axiom,
! [F: tm > list_fm,A: list_tm] :
( ( ( map_tm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_56_list_Omap__disc__iff,axiom,
! [F: list_fm > tm,A: list_list_fm] :
( ( ( map_list_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_57_list_Omap__disc__iff,axiom,
! [F: fm > list_tm,A: list_fm] :
( ( ( map_fm_list_tm @ F @ A )
= nil_list_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_58_list_Omap__disc__iff,axiom,
! [F: tm > set_nat,A: list_tm] :
( ( ( map_tm_set_nat @ F @ A )
= nil_set_nat )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_59_Nil__is__map__conv,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( nil_fm
= ( map_fm_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_60_Nil__is__map__conv,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( nil_tm
= ( map_fm_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_61_Nil__is__map__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( nil_tm
= ( map_tm_tm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_62_Nil__is__map__conv,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( nil_fm
= ( map_tm_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_63_Nil__is__map__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( nil_fm
= ( map_list_fm_fm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_64_Nil__is__map__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( nil_list_fm
= ( map_fm_list_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_65_Nil__is__map__conv,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( nil_list_fm
= ( map_tm_list_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_66_Nil__is__map__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( nil_tm
= ( map_list_fm_tm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_67_Nil__is__map__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( nil_list_tm
= ( map_fm_list_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_68_Nil__is__map__conv,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( nil_set_nat
= ( map_tm_set_nat @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_69_map__is__Nil__conv,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_70_map__is__Nil__conv,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( ( map_fm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_71_map__is__Nil__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_72_map__is__Nil__conv,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( ( map_tm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_73_map__is__Nil__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( ( map_list_fm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_list_fm ) ) ).
% map_is_Nil_conv
thf(fact_74_map__is__Nil__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( ( map_fm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_75_map__is__Nil__conv,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( ( map_tm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_76_map__is__Nil__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( ( map_list_fm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_list_fm ) ) ).
% map_is_Nil_conv
thf(fact_77_map__is__Nil__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( ( map_fm_list_tm @ F @ Xs )
= nil_list_tm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_78_map__is__Nil__conv,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( ( map_tm_set_nat @ F @ Xs )
= nil_set_nat )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_79_map__eq__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,G: fm > list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Xs ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_80_map__eq__conv,axiom,
! [F: tm > set_nat,Xs: list_tm,G: tm > set_nat] :
( ( ( map_tm_set_nat @ F @ Xs )
= ( map_tm_set_nat @ G @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_81_map__eq__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,G: tm > list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_82_map__eq__conv,axiom,
! [F: tm > fm,Xs: list_tm,G: tm > fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( map_tm_fm @ G @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_83_map__append,axiom,
! [F: fm > fm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_fm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_fm @ ( map_fm_fm @ F @ Xs ) @ ( map_fm_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_84_map__append,axiom,
! [F: fm > tm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_tm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_tm @ ( map_fm_tm @ F @ Xs ) @ ( map_fm_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_85_map__append,axiom,
! [F: tm > tm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_tm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( map_tm_tm @ F @ Xs ) @ ( map_tm_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_86_map__append,axiom,
! [F: fm > list_tm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_list_tm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_list_tm @ ( map_fm_list_tm @ F @ Xs ) @ ( map_fm_list_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_87_map__append,axiom,
! [F: tm > set_nat,Xs: list_tm,Ys: list_tm] :
( ( map_tm_set_nat @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_set_nat @ ( map_tm_set_nat @ F @ Xs ) @ ( map_tm_set_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_88_map__append,axiom,
! [F: tm > list_tm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_list_tm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( map_tm_list_tm @ F @ Xs ) @ ( map_tm_list_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_89_map__append,axiom,
! [F: tm > fm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_fm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_fm @ ( map_tm_fm @ F @ Xs ) @ ( map_tm_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_90_concat__append,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( concat_tm @ ( append_list_tm @ Xs @ Ys ) )
= ( append_tm @ ( concat_tm @ Xs ) @ ( concat_tm @ Ys ) ) ) ).
% concat_append
thf(fact_91_concat__append,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( concat_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( append_fm @ ( concat_fm @ Xs ) @ ( concat_fm @ Ys ) ) ) ).
% concat_append
thf(fact_92_remdups__eq__nil__iff,axiom,
! [X2: list_fm] :
( ( ( remdups_fm @ X2 )
= nil_fm )
= ( X2 = nil_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_93_remdups__eq__nil__iff,axiom,
! [X2: list_list_fm] :
( ( ( remdups_list_fm @ X2 )
= nil_list_fm )
= ( X2 = nil_list_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_94_remdups__eq__nil__iff,axiom,
! [X2: list_tm] :
( ( ( remdups_tm @ X2 )
= nil_tm )
= ( X2 = nil_tm ) ) ).
% remdups_eq_nil_iff
thf(fact_95_remdups__eq__nil__right__iff,axiom,
! [X2: list_fm] :
( ( nil_fm
= ( remdups_fm @ X2 ) )
= ( X2 = nil_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_96_remdups__eq__nil__right__iff,axiom,
! [X2: list_list_fm] :
( ( nil_list_fm
= ( remdups_list_fm @ X2 ) )
= ( X2 = nil_list_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_97_remdups__eq__nil__right__iff,axiom,
! [X2: list_tm] :
( ( nil_tm
= ( remdups_tm @ X2 ) )
= ( X2 = nil_tm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_98_set__remdups,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( remdups_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_99_set__remdups,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( remdups_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_100_set__remdups,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( remdups_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_remdups
thf(fact_101_set__remdups,axiom,
! [Xs: list_set_nat] :
( ( set_set_nat2 @ ( remdups_set_nat @ Xs ) )
= ( set_set_nat2 @ Xs ) ) ).
% set_remdups
thf(fact_102_append1__eq__conv,axiom,
! [Xs: list_fm,X2: fm,Ys: list_fm,Y: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) )
= ( append_fm @ Ys @ ( cons_fm @ Y @ nil_fm ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_103_append1__eq__conv,axiom,
! [Xs: list_list_fm,X2: list_fm,Ys: list_list_fm,Y: list_fm] :
( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) )
= ( append_list_fm @ Ys @ ( cons_list_fm @ Y @ nil_list_fm ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_104_append1__eq__conv,axiom,
! [Xs: list_tm,X2: tm,Ys: list_tm,Y: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_105_Nil__eq__concat__conv,axiom,
! [Xss: list_list_list_fm] :
( ( nil_list_fm
= ( concat_list_fm @ Xss ) )
= ( ! [X3: list_list_fm] :
( ( member_list_list_fm @ X3 @ ( set_list_list_fm2 @ Xss ) )
=> ( X3 = nil_list_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_106_Nil__eq__concat__conv,axiom,
! [Xss: list_list_tm] :
( ( nil_tm
= ( concat_tm @ Xss ) )
= ( ! [X3: list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xss ) )
=> ( X3 = nil_tm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_107_Nil__eq__concat__conv,axiom,
! [Xss: list_list_fm] :
( ( nil_fm
= ( concat_fm @ Xss ) )
= ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xss ) )
=> ( X3 = nil_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_108_concat__eq__Nil__conv,axiom,
! [Xss: list_list_list_fm] :
( ( ( concat_list_fm @ Xss )
= nil_list_fm )
= ( ! [X3: list_list_fm] :
( ( member_list_list_fm @ X3 @ ( set_list_list_fm2 @ Xss ) )
=> ( X3 = nil_list_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_109_concat__eq__Nil__conv,axiom,
! [Xss: list_list_tm] :
( ( ( concat_tm @ Xss )
= nil_tm )
= ( ! [X3: list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xss ) )
=> ( X3 = nil_tm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_110_concat__eq__Nil__conv,axiom,
! [Xss: list_list_fm] :
( ( ( concat_fm @ Xss )
= nil_fm )
= ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xss ) )
=> ( X3 = nil_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_111__C_K_C,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( parts @ aa @ r @ p ) ) )
=> ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) @ r @ za ) ) )
=> ( sequent_calculus @ ( append_fm @ prea @ ( append_fm @ X @ Xa ) ) ) ) ) ).
% "*"
thf(fact_112__092_060open_062_092_060forall_062z_H_092_060in_062set_A_Ilist__prod_A_Iparts_AA_Ar_Ap_____J_A_Ichildren_A_Iremdups_A_IA_A_064_AsubtermFms_A_Iconcat_A_Iparts_AA_Ar_Ap_____J_J_J_J_Ar_Az_J_J_O_A_I_092_060tturnstile_062_Apre_A_064_Az_H_J_092_060close_062,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( list_prod_fm @ ( parts @ aa @ r @ p ) @ ( children @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) @ r @ za ) ) ) )
=> ( sequent_calculus @ ( append_fm @ prea @ X ) ) ) ).
% \<open>\<forall>z'\<in>set (list_prod (parts A r p__) (children (remdups (A @ subtermFms (concat (parts A r p__)))) r z)). (\<tturnstile> pre @ z')\<close>
thf(fact_113_assms_I1_J,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( children @ a @ r @ z ) ) )
=> ( sequent_calculus @ ( append_fm @ pre2 @ X ) ) ) ).
% assms(1)
thf(fact_114_list_Osimps_I8_J,axiom,
! [F: fm > fm] :
( ( map_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_115_list_Osimps_I8_J,axiom,
! [F: fm > tm] :
( ( map_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_116_list_Osimps_I8_J,axiom,
! [F: tm > tm] :
( ( map_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% list.simps(8)
thf(fact_117_list_Osimps_I8_J,axiom,
! [F: tm > fm] :
( ( map_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% list.simps(8)
thf(fact_118_list_Osimps_I8_J,axiom,
! [F: fm > list_fm] :
( ( map_fm_list_fm @ F @ nil_fm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_119_list_Osimps_I8_J,axiom,
! [F: list_fm > fm] :
( ( map_list_fm_fm @ F @ nil_list_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_120_list_Osimps_I8_J,axiom,
! [F: list_fm > tm] :
( ( map_list_fm_tm @ F @ nil_list_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_121_list_Osimps_I8_J,axiom,
! [F: tm > list_fm] :
( ( map_tm_list_fm @ F @ nil_tm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_122_list_Osimps_I8_J,axiom,
! [F: fm > list_tm] :
( ( map_fm_list_tm @ F @ nil_fm )
= nil_list_tm ) ).
% list.simps(8)
thf(fact_123_list_Osimps_I8_J,axiom,
! [F: tm > set_nat] :
( ( map_tm_set_nat @ F @ nil_tm )
= nil_set_nat ) ).
% list.simps(8)
thf(fact_124_concat_Osimps_I1_J,axiom,
( ( concat_list_fm @ nil_list_list_fm )
= nil_list_fm ) ).
% concat.simps(1)
thf(fact_125_concat_Osimps_I1_J,axiom,
( ( concat_tm @ nil_list_tm )
= nil_tm ) ).
% concat.simps(1)
thf(fact_126_concat_Osimps_I1_J,axiom,
( ( concat_fm @ nil_list_fm )
= nil_fm ) ).
% concat.simps(1)
thf(fact_127_remdups_Osimps_I1_J,axiom,
( ( remdups_fm @ nil_fm )
= nil_fm ) ).
% remdups.simps(1)
thf(fact_128_remdups_Osimps_I1_J,axiom,
( ( remdups_list_fm @ nil_list_fm )
= nil_list_fm ) ).
% remdups.simps(1)
thf(fact_129_remdups_Osimps_I1_J,axiom,
( ( remdups_tm @ nil_tm )
= nil_tm ) ).
% remdups.simps(1)
thf(fact_130_list_Omap__cong,axiom,
! [X2: list_fm,Ya: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ( X2 = Ya )
=> ( ! [Z2: fm] :
( ( member_fm2 @ Z2 @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_fm_list_tm @ F @ X2 )
= ( map_fm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_131_list_Omap__cong,axiom,
! [X2: list_tm,Ya: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ( X2 = Ya )
=> ( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_set_nat @ F @ X2 )
= ( map_tm_set_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_132_list_Omap__cong,axiom,
! [X2: list_tm,Ya: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ( X2 = Ya )
=> ( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_list_tm @ F @ X2 )
= ( map_tm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_133_list_Omap__cong,axiom,
! [X2: list_tm,Ya: list_tm,F: tm > fm,G: tm > fm] :
( ( X2 = Ya )
=> ( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_fm @ F @ X2 )
= ( map_tm_fm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_134_list_Omap__cong0,axiom,
! [X2: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ! [Z2: fm] :
( ( member_fm2 @ Z2 @ ( set_fm2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_fm_list_tm @ F @ X2 )
= ( map_fm_list_tm @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_135_list_Omap__cong0,axiom,
! [X2: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_set_nat @ F @ X2 )
= ( map_tm_set_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_136_list_Omap__cong0,axiom,
! [X2: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_list_tm @ F @ X2 )
= ( map_tm_list_tm @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_137_list_Omap__cong0,axiom,
! [X2: list_tm,F: tm > fm,G: tm > fm] :
( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_tm_fm @ F @ X2 )
= ( map_tm_fm @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_138_list_Oinj__map__strong,axiom,
! [X2: list_fm,Xa2: list_fm,F: fm > list_tm,Fa: fm > list_tm] :
( ! [Z2: fm,Za: fm] :
( ( member_fm2 @ Z2 @ ( set_fm2 @ X2 ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_fm_list_tm @ F @ X2 )
= ( map_fm_list_tm @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_139_list_Oinj__map__strong,axiom,
! [X2: list_tm,Xa2: list_tm,F: tm > set_nat,Fa: tm > set_nat] :
( ! [Z2: tm,Za: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_tm_set_nat @ F @ X2 )
= ( map_tm_set_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_140_list_Oinj__map__strong,axiom,
! [X2: list_tm,Xa2: list_tm,F: tm > list_tm,Fa: tm > list_tm] :
( ! [Z2: tm,Za: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_tm_list_tm @ F @ X2 )
= ( map_tm_list_tm @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_141_list_Oinj__map__strong,axiom,
! [X2: list_tm,Xa2: list_tm,F: tm > fm,Fa: tm > fm] :
( ! [Z2: tm,Za: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ X2 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_tm_fm @ F @ X2 )
= ( map_tm_fm @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_142_mem__Collect__eq,axiom,
! [A: fm,P2: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_143_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_144_mem__Collect__eq,axiom,
! [A: $o,P2: $o > $o] :
( ( member_o2 @ A @ ( collect_o @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A: tm,P2: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A: list_fm,P2: list_fm > $o] :
( ( member_list_fm2 @ A @ ( collect_list_fm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_147_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X3: tm] : ( member_tm2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A2: set_list_fm] :
( ( collect_list_fm
@ ^ [X3: list_fm] : ( member_list_fm2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_152_Collect__cong,axiom,
! [P2: list_fm > $o,Q: list_fm > $o] :
( ! [X4: list_fm] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_fm @ P2 )
= ( collect_list_fm @ Q ) ) ) ).
% Collect_cong
thf(fact_153_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat2 @ Z2 @ ( set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_154_list_Omap__ident__strong,axiom,
! [T: list_o,F: $o > $o] :
( ! [Z2: $o] :
( ( member_o2 @ Z2 @ ( set_o2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_o_o @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_155_list_Omap__ident__strong,axiom,
! [T: list_list_fm,F: list_fm > list_fm] :
( ! [Z2: list_fm] :
( ( member_list_fm2 @ Z2 @ ( set_list_fm2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_list_fm_list_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_156_list_Omap__ident__strong,axiom,
! [T: list_fm,F: fm > fm] :
( ! [Z2: fm] :
( ( member_fm2 @ Z2 @ ( set_fm2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_fm_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_157_list_Omap__ident__strong,axiom,
! [T: list_tm,F: tm > tm] :
( ! [Z2: tm] :
( ( member_tm2 @ Z2 @ ( set_tm2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_tm_tm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_158_list_Omap__ident__strong,axiom,
! [T: list_set_nat,F: set_nat > set_nat] :
( ! [Z2: set_nat] :
( ( member_set_nat2 @ Z2 @ ( set_set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_set_nat_set_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_159_map__ext,axiom,
! [Xs: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_160_map__ext,axiom,
! [Xs: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_set_nat @ F @ Xs )
= ( map_tm_set_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_161_map__ext,axiom,
! [Xs: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_162_map__ext,axiom,
! [Xs: list_tm,F: tm > fm,G: tm > fm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_fm @ F @ Xs )
= ( map_tm_fm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_163_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_164_map__idI,axiom,
! [Xs: list_o,F: $o > $o] :
( ! [X4: $o] :
( ( member_o2 @ X4 @ ( set_o2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_o_o @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_165_map__idI,axiom,
! [Xs: list_list_fm,F: list_fm > list_fm] :
( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_list_fm_list_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_166_map__idI,axiom,
! [Xs: list_fm,F: fm > fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_fm_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_167_map__idI,axiom,
! [Xs: list_tm,F: tm > tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_tm_tm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_168_map__idI,axiom,
! [Xs: list_set_nat,F: set_nat > set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_set_nat_set_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_169_map__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ( Xs = Ys )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_170_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_set_nat @ F @ Xs )
= ( map_tm_set_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_171_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_172_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > fm,G: tm > fm] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_fm @ F @ Xs )
= ( map_tm_fm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_173_map__concat,axiom,
! [F: tm > tm,Xs: list_list_tm] :
( ( map_tm_tm @ F @ ( concat_tm @ Xs ) )
= ( concat_tm @ ( map_list_tm_list_tm @ ( map_tm_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_174_map__concat,axiom,
! [F: fm > tm,Xs: list_list_fm] :
( ( map_fm_tm @ F @ ( concat_fm @ Xs ) )
= ( concat_tm @ ( map_list_fm_list_tm @ ( map_fm_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_175_map__concat,axiom,
! [F: fm > fm,Xs: list_list_fm] :
( ( map_fm_fm @ F @ ( concat_fm @ Xs ) )
= ( concat_fm @ ( map_list_fm_list_fm @ ( map_fm_fm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_176_map__concat,axiom,
! [F: fm > list_tm,Xs: list_list_fm] :
( ( map_fm_list_tm @ F @ ( concat_fm @ Xs ) )
= ( concat_list_tm @ ( map_li1108997747876207612ist_tm @ ( map_fm_list_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_177_map__concat,axiom,
! [F: tm > set_nat,Xs: list_list_tm] :
( ( map_tm_set_nat @ F @ ( concat_tm @ Xs ) )
= ( concat_set_nat @ ( map_li5423145413338040381et_nat @ ( map_tm_set_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_178_map__concat,axiom,
! [F: tm > list_tm,Xs: list_list_tm] :
( ( map_tm_list_tm @ F @ ( concat_tm @ Xs ) )
= ( concat_list_tm @ ( map_li6264597563971819530ist_tm @ ( map_tm_list_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_179_map__concat,axiom,
! [F: tm > fm,Xs: list_list_tm] :
( ( map_tm_fm @ F @ ( concat_tm @ Xs ) )
= ( concat_fm @ ( map_list_tm_list_fm @ ( map_tm_fm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_180_ex__map__conv,axiom,
! [Ys: list_list_tm,F: fm > list_tm] :
( ( ? [Xs2: list_fm] :
( Ys
= ( map_fm_list_tm @ F @ Xs2 ) ) )
= ( ! [X3: list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Ys ) )
=> ? [Y2: fm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_181_ex__map__conv,axiom,
! [Ys: list_list_tm,F: tm > list_tm] :
( ( ? [Xs2: list_tm] :
( Ys
= ( map_tm_list_tm @ F @ Xs2 ) ) )
= ( ! [X3: list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Ys ) )
=> ? [Y2: tm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_182_ex__map__conv,axiom,
! [Ys: list_fm,F: tm > fm] :
( ( ? [Xs2: list_tm] :
( Ys
= ( map_tm_fm @ F @ Xs2 ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Ys ) )
=> ? [Y2: tm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_183_ex__map__conv,axiom,
! [Ys: list_set_nat,F: tm > set_nat] :
( ( ? [Xs2: list_tm] :
( Ys
= ( map_tm_set_nat @ F @ Xs2 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ? [Y2: tm] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_184_remdups__remdups,axiom,
! [Xs: list_tm] :
( ( remdups_tm @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ Xs ) ) ).
% remdups_remdups
thf(fact_185_remdups__map__remdups,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( remdups_tm @ ( map_tm_tm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_tm @ ( map_tm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_186_remdups__map__remdups,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( remdups_list_tm @ ( map_fm_list_tm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_list_tm @ ( map_fm_list_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_187_remdups__map__remdups,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( remdups_set_nat @ ( map_tm_set_nat @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_set_nat @ ( map_tm_set_nat @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_188_remdups__map__remdups,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( remdups_list_tm @ ( map_tm_list_tm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_list_tm @ ( map_tm_list_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_189_remdups__map__remdups,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( remdups_fm @ ( map_tm_fm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_fm @ ( map_tm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_190_concat__eq__append__conv,axiom,
! [Xss: list_list_list_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ( ( concat_list_fm @ Xss )
= ( append_list_fm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_list_fm )
=> ( ( Ys = nil_list_fm )
& ( Zs = nil_list_fm ) ) )
& ( ( Xss != nil_list_list_fm )
=> ? [Xss1: list_list_list_fm,Xs2: list_list_fm,Xs3: list_list_fm,Xss2: list_list_list_fm] :
( ( Xss
= ( append_list_list_fm @ Xss1 @ ( cons_list_list_fm @ ( append_list_fm @ Xs2 @ Xs3 ) @ Xss2 ) ) )
& ( Ys
= ( append_list_fm @ ( concat_list_fm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_list_fm @ Xs3 @ ( concat_list_fm @ Xss2 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_191_concat__eq__append__conv,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs = nil_tm ) ) )
& ( ( Xss != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs2: list_tm,Xs3: list_tm,Xss2: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs2 @ Xs3 ) @ Xss2 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_tm @ Xs3 @ ( concat_tm @ Xss2 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_192_concat__eq__append__conv,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_fm )
=> ( ( Ys = nil_fm )
& ( Zs = nil_fm ) ) )
& ( ( Xss != nil_list_fm )
=> ? [Xss1: list_list_fm,Xs2: list_fm,Xs3: list_fm,Xss2: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss1 @ ( cons_list_fm @ ( append_fm @ Xs2 @ Xs3 ) @ Xss2 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_fm @ Xs3 @ ( concat_fm @ Xss2 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_193_transpose_Ocases,axiom,
! [X2: list_list_list_fm] :
( ( X2 != nil_list_list_fm )
=> ( ! [Xss3: list_list_list_fm] :
( X2
!= ( cons_list_list_fm @ nil_list_fm @ Xss3 ) )
=> ~ ! [X4: list_fm,Xs4: list_list_fm,Xss3: list_list_list_fm] :
( X2
!= ( cons_list_list_fm @ ( cons_list_fm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_194_transpose_Ocases,axiom,
! [X2: list_list_tm] :
( ( X2 != nil_list_tm )
=> ( ! [Xss3: list_list_tm] :
( X2
!= ( cons_list_tm @ nil_tm @ Xss3 ) )
=> ~ ! [X4: tm,Xs4: list_tm,Xss3: list_list_tm] :
( X2
!= ( cons_list_tm @ ( cons_tm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_195_transpose_Ocases,axiom,
! [X2: list_list_fm] :
( ( X2 != nil_list_fm )
=> ( ! [Xss3: list_list_fm] :
( X2
!= ( cons_list_fm @ nil_fm @ Xss3 ) )
=> ~ ! [X4: fm,Xs4: list_fm,Xss3: list_list_fm] :
( X2
!= ( cons_list_fm @ ( cons_fm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_196_concat_Osimps_I2_J,axiom,
! [X2: list_tm,Xs: list_list_tm] :
( ( concat_tm @ ( cons_list_tm @ X2 @ Xs ) )
= ( append_tm @ X2 @ ( concat_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_197_concat_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( concat_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_fm @ X2 @ ( concat_fm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_198_fm_Odistinct_I21_J,axiom,
! [X21: fm,X22: fm,X7: fm] :
( ( imp @ X21 @ X22 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_199_remdups_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remdups_nat @ ( cons_nat @ X2 @ Xs ) )
= ( remdups_nat @ Xs ) ) )
& ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( remdups_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( remdups_nat @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_200_remdups_Osimps_I2_J,axiom,
! [X2: $o,Xs: list_o] :
( ( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( ( remdups_o @ ( cons_o @ X2 @ Xs ) )
= ( remdups_o @ Xs ) ) )
& ( ~ ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( ( remdups_o @ ( cons_o @ X2 @ Xs ) )
= ( cons_o @ X2 @ ( remdups_o @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_201_remdups_Osimps_I2_J,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( remdups_set_nat @ ( cons_set_nat @ X2 @ Xs ) )
= ( remdups_set_nat @ Xs ) ) )
& ( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( remdups_set_nat @ ( cons_set_nat @ X2 @ Xs ) )
= ( cons_set_nat @ X2 @ ( remdups_set_nat @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_202_remdups_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X2 @ Xs ) )
= ( remdups_fm @ Xs ) ) )
& ( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X2 @ Xs ) )
= ( cons_fm @ X2 @ ( remdups_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_203_remdups_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( remdups_list_fm @ Xs ) ) )
& ( ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( cons_list_fm @ X2 @ ( remdups_list_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_204_remdups_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X2 @ Xs ) )
= ( remdups_tm @ Xs ) ) )
& ( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X2 @ Xs ) )
= ( cons_tm @ X2 @ ( remdups_tm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_205_map__eq__Cons__conv,axiom,
! [F: fm > fm,Xs: list_fm,Y: fm,Ys: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_206_map__eq__Cons__conv,axiom,
! [F: tm > fm,Xs: list_tm,Y: fm,Ys: list_fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_207_map__eq__Cons__conv,axiom,
! [F: fm > tm,Xs: list_fm,Y: tm,Ys: list_tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_208_map__eq__Cons__conv,axiom,
! [F: tm > tm,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_209_map__eq__Cons__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,Y: list_tm,Ys: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_list_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_210_map__eq__Cons__conv,axiom,
! [F: tm > set_nat,Xs: list_tm,Y: set_nat,Ys: list_set_nat] :
( ( ( map_tm_set_nat @ F @ Xs )
= ( cons_set_nat @ Y @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_set_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_211_map__eq__Cons__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,Y: list_tm,Ys: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_list_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_212_map__eq__Cons__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm,Y: fm,Ys: list_fm] :
( ( ( map_list_fm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
= ( ? [Z3: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_list_fm_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_213_map__eq__Cons__conv,axiom,
! [F: fm > list_fm,Xs: list_fm,Y: list_fm,Ys: list_list_fm] :
( ( ( map_fm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_list_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_214_map__eq__Cons__conv,axiom,
! [F: tm > list_fm,Xs: list_tm,Y: list_fm,Ys: list_list_fm] :
( ( ( map_tm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_list_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_215_Cons__eq__map__conv,axiom,
! [X2: fm,Xs: list_fm,F: fm > fm,Ys: list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_fm_fm @ F @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_216_Cons__eq__map__conv,axiom,
! [X2: fm,Xs: list_fm,F: tm > fm,Ys: list_tm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_tm_fm @ F @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_217_Cons__eq__map__conv,axiom,
! [X2: tm,Xs: list_tm,F: fm > tm,Ys: list_fm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_fm_tm @ F @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_218_Cons__eq__map__conv,axiom,
! [X2: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_219_Cons__eq__map__conv,axiom,
! [X2: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X2 @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_220_Cons__eq__map__conv,axiom,
! [X2: set_nat,Xs: list_set_nat,F: tm > set_nat,Ys: list_tm] :
( ( ( cons_set_nat @ X2 @ Xs )
= ( map_tm_set_nat @ F @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_set_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_221_Cons__eq__map__conv,axiom,
! [X2: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X2 @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_222_Cons__eq__map__conv,axiom,
! [X2: fm,Xs: list_fm,F: list_fm > fm,Ys: list_list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_list_fm_fm @ F @ Ys ) )
= ( ? [Z3: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_list_fm_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_223_Cons__eq__map__conv,axiom,
! [X2: list_fm,Xs: list_list_fm,F: fm > list_fm,Ys: list_fm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( map_fm_list_fm @ F @ Ys ) )
= ( ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_list_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_224_Cons__eq__map__conv,axiom,
! [X2: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
= ( ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_225_map__eq__Cons__D,axiom,
! [F: fm > fm,Xs: list_fm,Y: fm,Ys: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_fm_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_226_map__eq__Cons__D,axiom,
! [F: tm > fm,Xs: list_tm,Y: fm,Ys: list_fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_227_map__eq__Cons__D,axiom,
! [F: fm > tm,Xs: list_fm,Y: tm,Ys: list_tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_fm_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_228_map__eq__Cons__D,axiom,
! [F: tm > tm,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_229_map__eq__Cons__D,axiom,
! [F: fm > list_tm,Xs: list_fm,Y: list_tm,Ys: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_fm_list_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_230_map__eq__Cons__D,axiom,
! [F: tm > set_nat,Xs: list_tm,Y: set_nat,Ys: list_set_nat] :
( ( ( map_tm_set_nat @ F @ Xs )
= ( cons_set_nat @ Y @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_set_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_231_map__eq__Cons__D,axiom,
! [F: tm > list_tm,Xs: list_tm,Y: list_tm,Ys: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_list_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_232_map__eq__Cons__D,axiom,
! [F: list_fm > fm,Xs: list_list_fm,Y: fm,Ys: list_fm] :
( ( ( map_list_fm_fm @ F @ Xs )
= ( cons_fm @ Y @ Ys ) )
=> ? [Z2: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_list_fm_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_233_map__eq__Cons__D,axiom,
! [F: fm > list_fm,Xs: list_fm,Y: list_fm,Ys: list_list_fm] :
( ( ( map_fm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_fm_list_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_234_map__eq__Cons__D,axiom,
! [F: tm > list_fm,Xs: list_tm,Y: list_fm,Ys: list_list_fm] :
( ( ( map_tm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_tm_list_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_235_Cons__eq__map__D,axiom,
! [X2: fm,Xs: list_fm,F: fm > fm,Ys: list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_fm_fm @ F @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_fm_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_236_Cons__eq__map__D,axiom,
! [X2: fm,Xs: list_fm,F: tm > fm,Ys: list_tm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_tm_fm @ F @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_237_Cons__eq__map__D,axiom,
! [X2: tm,Xs: list_tm,F: fm > tm,Ys: list_fm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_fm_tm @ F @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_fm_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_238_Cons__eq__map__D,axiom,
! [X2: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_239_Cons__eq__map__D,axiom,
! [X2: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X2 @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_240_Cons__eq__map__D,axiom,
! [X2: set_nat,Xs: list_set_nat,F: tm > set_nat,Ys: list_tm] :
( ( ( cons_set_nat @ X2 @ Xs )
= ( map_tm_set_nat @ F @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_set_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_241_Cons__eq__map__D,axiom,
! [X2: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X2 @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_242_Cons__eq__map__D,axiom,
! [X2: fm,Xs: list_fm,F: list_fm > fm,Ys: list_list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( map_list_fm_fm @ F @ Ys ) )
=> ? [Z2: list_fm,Zs3: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_list_fm_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_243_Cons__eq__map__D,axiom,
! [X2: list_fm,Xs: list_list_fm,F: fm > list_fm,Ys: list_fm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( map_fm_list_fm @ F @ Ys ) )
=> ? [Z2: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_fm_list_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_244_Cons__eq__map__D,axiom,
! [X2: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
=> ? [Z2: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z2 @ Zs3 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_245_list_Osimps_I9_J,axiom,
! [F: fm > fm,X21: fm,X22: list_fm] :
( ( map_fm_fm @ F @ ( cons_fm @ X21 @ X22 ) )
= ( cons_fm @ ( F @ X21 ) @ ( map_fm_fm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_246_list_Osimps_I9_J,axiom,
! [F: fm > tm,X21: fm,X22: list_fm] :
( ( map_fm_tm @ F @ ( cons_fm @ X21 @ X22 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_fm_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_247_list_Osimps_I9_J,axiom,
! [F: tm > fm,X21: tm,X22: list_tm] :
( ( map_tm_fm @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_fm @ ( F @ X21 ) @ ( map_tm_fm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_248_list_Osimps_I9_J,axiom,
! [F: tm > tm,X21: tm,X22: list_tm] :
( ( map_tm_tm @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_tm_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_249_list_Osimps_I9_J,axiom,
! [F: fm > list_tm,X21: fm,X22: list_fm] :
( ( map_fm_list_tm @ F @ ( cons_fm @ X21 @ X22 ) )
= ( cons_list_tm @ ( F @ X21 ) @ ( map_fm_list_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_250_list_Osimps_I9_J,axiom,
! [F: fm > list_fm,X21: fm,X22: list_fm] :
( ( map_fm_list_fm @ F @ ( cons_fm @ X21 @ X22 ) )
= ( cons_list_fm @ ( F @ X21 ) @ ( map_fm_list_fm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_251_list_Osimps_I9_J,axiom,
! [F: list_fm > fm,X21: list_fm,X22: list_list_fm] :
( ( map_list_fm_fm @ F @ ( cons_list_fm @ X21 @ X22 ) )
= ( cons_fm @ ( F @ X21 ) @ ( map_list_fm_fm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_252_list_Osimps_I9_J,axiom,
! [F: list_fm > tm,X21: list_fm,X22: list_list_fm] :
( ( map_list_fm_tm @ F @ ( cons_list_fm @ X21 @ X22 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_list_fm_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_253_list_Osimps_I9_J,axiom,
! [F: tm > set_nat,X21: tm,X22: list_tm] :
( ( map_tm_set_nat @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_set_nat @ ( F @ X21 ) @ ( map_tm_set_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_254_list_Osimps_I9_J,axiom,
! [F: tm > list_tm,X21: tm,X22: list_tm] :
( ( map_tm_list_tm @ F @ ( cons_tm @ X21 @ X22 ) )
= ( cons_list_tm @ ( F @ X21 ) @ ( map_tm_list_tm @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_255_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_256_set__ConsD,axiom,
! [Y: $o,X2: $o,Xs: list_o] :
( ( member_o2 @ Y @ ( set_o2 @ ( cons_o @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_o2 @ Y @ ( set_o2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_257_set__ConsD,axiom,
! [Y: set_nat,X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_258_set__ConsD,axiom,
! [Y: fm,X2: fm,Xs: list_fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_fm2 @ Y @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_259_set__ConsD,axiom,
! [Y: list_fm,X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ Y @ ( set_list_fm2 @ ( cons_list_fm @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_fm2 @ Y @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_260_set__ConsD,axiom,
! [Y: tm,X2: tm,Xs: list_tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_tm2 @ Y @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_261_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_262_list_Oset__cases,axiom,
! [E: $o,A: list_o] :
( ( member_o2 @ E @ ( set_o2 @ A ) )
=> ( ! [Z22: list_o] :
( A
!= ( cons_o @ E @ Z22 ) )
=> ~ ! [Z1: $o,Z22: list_o] :
( ( A
= ( cons_o @ Z1 @ Z22 ) )
=> ~ ( member_o2 @ E @ ( set_o2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_263_list_Oset__cases,axiom,
! [E: set_nat,A: list_set_nat] :
( ( member_set_nat2 @ E @ ( set_set_nat2 @ A ) )
=> ( ! [Z22: list_set_nat] :
( A
!= ( cons_set_nat @ E @ Z22 ) )
=> ~ ! [Z1: set_nat,Z22: list_set_nat] :
( ( A
= ( cons_set_nat @ Z1 @ Z22 ) )
=> ~ ( member_set_nat2 @ E @ ( set_set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_264_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_265_list_Oset__cases,axiom,
! [E: list_fm,A: list_list_fm] :
( ( member_list_fm2 @ E @ ( set_list_fm2 @ A ) )
=> ( ! [Z22: list_list_fm] :
( A
!= ( cons_list_fm @ E @ Z22 ) )
=> ~ ! [Z1: list_fm,Z22: list_list_fm] :
( ( A
= ( cons_list_fm @ Z1 @ Z22 ) )
=> ~ ( member_list_fm2 @ E @ ( set_list_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_266_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_267_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_268_list_Oset__intros_I1_J,axiom,
! [X21: $o,X22: list_o] : ( member_o2 @ X21 @ ( set_o2 @ ( cons_o @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_269_list_Oset__intros_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] : ( member_set_nat2 @ X21 @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_270_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_271_list_Oset__intros_I1_J,axiom,
! [X21: list_fm,X22: list_list_fm] : ( member_list_fm2 @ X21 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_272_list_Oset__intros_I1_J,axiom,
! [X21: tm,X22: list_tm] : ( member_tm2 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_273_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_274_list_Oset__intros_I2_J,axiom,
! [Y: $o,X22: list_o,X21: $o] :
( ( member_o2 @ Y @ ( set_o2 @ X22 ) )
=> ( member_o2 @ Y @ ( set_o2 @ ( cons_o @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_275_list_Oset__intros_I2_J,axiom,
! [Y: set_nat,X22: list_set_nat,X21: set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ X22 ) )
=> ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_276_list_Oset__intros_I2_J,axiom,
! [Y: fm,X22: list_fm,X21: fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ X22 ) )
=> ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_277_list_Oset__intros_I2_J,axiom,
! [Y: list_fm,X22: list_list_fm,X21: list_fm] :
( ( member_list_fm2 @ Y @ ( set_list_fm2 @ X22 ) )
=> ( member_list_fm2 @ Y @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_278_list_Oset__intros_I2_J,axiom,
! [Y: tm,X22: list_tm,X21: tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ X22 ) )
=> ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_279_map__eq__append__conv,axiom,
! [F: fm > fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_280_map__eq__append__conv,axiom,
! [F: fm > tm,Xs: list_fm,Ys: list_tm,Zs: list_tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_281_map__eq__append__conv,axiom,
! [F: tm > tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_282_map__eq__append__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_list_tm @ F @ Us ) )
& ( Zs
= ( map_fm_list_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_283_map__eq__append__conv,axiom,
! [F: tm > set_nat,Xs: list_tm,Ys: list_set_nat,Zs: list_set_nat] :
( ( ( map_tm_set_nat @ F @ Xs )
= ( append_set_nat @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_set_nat @ F @ Us ) )
& ( Zs
= ( map_tm_set_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_284_map__eq__append__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_285_map__eq__append__conv,axiom,
! [F: tm > fm,Xs: list_tm,Ys: list_fm,Zs: list_fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_286_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs: list_fm,F: fm > fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs )
= ( map_fm_fm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_287_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs: list_tm,F: fm > tm,Xs: list_fm] :
( ( ( append_tm @ Ys @ Zs )
= ( map_fm_tm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_288_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs: list_tm,F: tm > tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( map_tm_tm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_289_append__eq__map__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,F: fm > list_tm,Xs: list_fm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( map_fm_list_tm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_list_tm @ F @ Us ) )
& ( Zs
= ( map_fm_list_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_290_append__eq__map__conv,axiom,
! [Ys: list_set_nat,Zs: list_set_nat,F: tm > set_nat,Xs: list_tm] :
( ( ( append_set_nat @ Ys @ Zs )
= ( map_tm_set_nat @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_set_nat @ F @ Us ) )
& ( Zs
= ( map_tm_set_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_291_append__eq__map__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,F: tm > list_tm,Xs: list_tm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( map_tm_list_tm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_292_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs: list_fm,F: tm > fm,Xs: list_tm] :
( ( ( append_fm @ Ys @ Zs )
= ( map_tm_fm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_293_list__nonempty__induct,axiom,
! [Xs: list_fm,P2: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X4: fm] : ( P2 @ ( cons_fm @ X4 @ nil_fm ) )
=> ( ! [X4: fm,Xs4: list_fm] :
( ( Xs4 != nil_fm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( cons_fm @ X4 @ Xs4 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_294_list__nonempty__induct,axiom,
! [Xs: list_list_fm,P2: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X4: list_fm] : ( P2 @ ( cons_list_fm @ X4 @ nil_list_fm ) )
=> ( ! [X4: list_fm,Xs4: list_list_fm] :
( ( Xs4 != nil_list_fm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_295_list__nonempty__induct,axiom,
! [Xs: list_tm,P2: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X4: tm] : ( P2 @ ( cons_tm @ X4 @ nil_tm ) )
=> ( ! [X4: tm,Xs4: list_tm] :
( ( Xs4 != nil_tm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( cons_tm @ X4 @ Xs4 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_296_list__induct2_H,axiom,
! [P2: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P2 @ nil_fm @ nil_fm )
=> ( ! [X4: fm,Xs4: list_fm] : ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ nil_fm )
=> ( ! [Y3: fm,Ys2: list_fm] : ( P2 @ nil_fm @ ( cons_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: fm,Xs4: list_fm,Y3: fm,Ys2: list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ ( cons_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_297_list__induct2_H,axiom,
! [P2: list_fm > list_list_fm > $o,Xs: list_fm,Ys: list_list_fm] :
( ( P2 @ nil_fm @ nil_list_fm )
=> ( ! [X4: fm,Xs4: list_fm] : ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys2: list_list_fm] : ( P2 @ nil_fm @ ( cons_list_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: fm,Xs4: list_fm,Y3: list_fm,Ys2: list_list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ ( cons_list_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_298_list__induct2_H,axiom,
! [P2: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P2 @ nil_fm @ nil_tm )
=> ( ! [X4: fm,Xs4: list_fm] : ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ nil_tm )
=> ( ! [Y3: tm,Ys2: list_tm] : ( P2 @ nil_fm @ ( cons_tm @ Y3 @ Ys2 ) )
=> ( ! [X4: fm,Xs4: list_fm,Y3: tm,Ys2: list_tm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_fm @ X4 @ Xs4 ) @ ( cons_tm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_299_list__induct2_H,axiom,
! [P2: list_list_fm > list_fm > $o,Xs: list_list_fm,Ys: list_fm] :
( ( P2 @ nil_list_fm @ nil_fm )
=> ( ! [X4: list_fm,Xs4: list_list_fm] : ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ nil_fm )
=> ( ! [Y3: fm,Ys2: list_fm] : ( P2 @ nil_list_fm @ ( cons_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: list_fm,Xs4: list_list_fm,Y3: fm,Ys2: list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ ( cons_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_300_list__induct2_H,axiom,
! [P2: list_list_fm > list_list_fm > $o,Xs: list_list_fm,Ys: list_list_fm] :
( ( P2 @ nil_list_fm @ nil_list_fm )
=> ( ! [X4: list_fm,Xs4: list_list_fm] : ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys2: list_list_fm] : ( P2 @ nil_list_fm @ ( cons_list_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: list_fm,Xs4: list_list_fm,Y3: list_fm,Ys2: list_list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ ( cons_list_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_301_list__induct2_H,axiom,
! [P2: list_list_fm > list_tm > $o,Xs: list_list_fm,Ys: list_tm] :
( ( P2 @ nil_list_fm @ nil_tm )
=> ( ! [X4: list_fm,Xs4: list_list_fm] : ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ nil_tm )
=> ( ! [Y3: tm,Ys2: list_tm] : ( P2 @ nil_list_fm @ ( cons_tm @ Y3 @ Ys2 ) )
=> ( ! [X4: list_fm,Xs4: list_list_fm,Y3: tm,Ys2: list_tm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_list_fm @ X4 @ Xs4 ) @ ( cons_tm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_302_list__induct2_H,axiom,
! [P2: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P2 @ nil_tm @ nil_fm )
=> ( ! [X4: tm,Xs4: list_tm] : ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ nil_fm )
=> ( ! [Y3: fm,Ys2: list_fm] : ( P2 @ nil_tm @ ( cons_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: tm,Xs4: list_tm,Y3: fm,Ys2: list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ ( cons_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_303_list__induct2_H,axiom,
! [P2: list_tm > list_list_fm > $o,Xs: list_tm,Ys: list_list_fm] :
( ( P2 @ nil_tm @ nil_list_fm )
=> ( ! [X4: tm,Xs4: list_tm] : ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys2: list_list_fm] : ( P2 @ nil_tm @ ( cons_list_fm @ Y3 @ Ys2 ) )
=> ( ! [X4: tm,Xs4: list_tm,Y3: list_fm,Ys2: list_list_fm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ ( cons_list_fm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_304_list__induct2_H,axiom,
! [P2: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P2 @ nil_tm @ nil_tm )
=> ( ! [X4: tm,Xs4: list_tm] : ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ nil_tm )
=> ( ! [Y3: tm,Ys2: list_tm] : ( P2 @ nil_tm @ ( cons_tm @ Y3 @ Ys2 ) )
=> ( ! [X4: tm,Xs4: list_tm,Y3: tm,Ys2: list_tm] :
( ( P2 @ Xs4 @ Ys2 )
=> ( P2 @ ( cons_tm @ X4 @ Xs4 ) @ ( cons_tm @ Y3 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_305_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y2: fm,Ys3: list_fm] :
( Xs
= ( cons_fm @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_306_neq__Nil__conv,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
= ( ? [Y2: list_fm,Ys3: list_list_fm] :
( Xs
= ( cons_list_fm @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_307_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y2: tm,Ys3: list_tm] :
( Xs
= ( cons_tm @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_308_remdups__adj_Ocases,axiom,
! [X2: list_fm] :
( ( X2 != nil_fm )
=> ( ! [X4: fm] :
( X2
!= ( cons_fm @ X4 @ nil_fm ) )
=> ~ ! [X4: fm,Y3: fm,Xs4: list_fm] :
( X2
!= ( cons_fm @ X4 @ ( cons_fm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_309_remdups__adj_Ocases,axiom,
! [X2: list_list_fm] :
( ( X2 != nil_list_fm )
=> ( ! [X4: list_fm] :
( X2
!= ( cons_list_fm @ X4 @ nil_list_fm ) )
=> ~ ! [X4: list_fm,Y3: list_fm,Xs4: list_list_fm] :
( X2
!= ( cons_list_fm @ X4 @ ( cons_list_fm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_310_remdups__adj_Ocases,axiom,
! [X2: list_tm] :
( ( X2 != nil_tm )
=> ( ! [X4: tm] :
( X2
!= ( cons_tm @ X4 @ nil_tm ) )
=> ~ ! [X4: tm,Y3: tm,Xs4: list_tm] :
( X2
!= ( cons_tm @ X4 @ ( cons_tm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_311_list_Oexhaust,axiom,
! [Y: list_fm] :
( ( Y != nil_fm )
=> ~ ! [X212: fm,X222: list_fm] :
( Y
!= ( cons_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_312_list_Oexhaust,axiom,
! [Y: list_list_fm] :
( ( Y != nil_list_fm )
=> ~ ! [X212: list_fm,X222: list_list_fm] :
( Y
!= ( cons_list_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_313_list_Oexhaust,axiom,
! [Y: list_tm] :
( ( Y != nil_tm )
=> ~ ! [X212: tm,X222: list_tm] :
( Y
!= ( cons_tm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_314_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X22: list_fm] :
( ( List
= ( cons_fm @ X21 @ X22 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_315_list_OdiscI,axiom,
! [List: list_list_fm,X21: list_fm,X22: list_list_fm] :
( ( List
= ( cons_list_fm @ X21 @ X22 ) )
=> ( List != nil_list_fm ) ) ).
% list.discI
thf(fact_316_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X22: list_tm] :
( ( List
= ( cons_tm @ X21 @ X22 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_317_list_Odistinct_I1_J,axiom,
! [X21: fm,X22: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_318_list_Odistinct_I1_J,axiom,
! [X21: list_fm,X22: list_list_fm] :
( nil_list_fm
!= ( cons_list_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_319_list_Odistinct_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_320_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_fm @ nil_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_321_eq__Nil__appendI,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_fm @ nil_list_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_322_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_323_append_Oleft__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ nil_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_324_append_Oleft__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_325_append_Oleft__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ nil_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_326_append__Nil,axiom,
! [Ys: list_fm] :
( ( append_fm @ nil_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_327_append__Nil,axiom,
! [Ys: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_328_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_329_remdups__append2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( remdups_fm @ ( append_fm @ Xs @ ( remdups_fm @ Ys ) ) )
= ( remdups_fm @ ( append_fm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_330_remdups__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( remdups_tm @ ( append_tm @ Xs @ ( remdups_tm @ Ys ) ) )
= ( remdups_tm @ ( append_tm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_331_SeCaV_OAlphaImp,axiom,
! [P: fm,Q2: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ Q2 @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P @ Q2 ) @ Z ) ) ) ).
% SeCaV.AlphaImp
thf(fact_332_BetaImp,axiom,
! [P: fm,Z: list_fm,Q2: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P @ Q2 ) ) @ Z ) ) ) ) ).
% BetaImp
thf(fact_333_not__Cons__self2,axiom,
! [X2: fm,Xs: list_fm] :
( ( cons_fm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_334_not__Cons__self2,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( cons_list_fm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_335_not__Cons__self2,axiom,
! [X2: tm,Xs: list_tm] :
( ( cons_tm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_336_append__eq__append__conv2,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ts: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs @ Ts ) )
= ( ? [Us: list_fm] :
( ( ( Xs
= ( append_fm @ Zs @ Us ) )
& ( ( append_fm @ Us @ Ys )
= Ts ) )
| ( ( ( append_fm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_fm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_337_append__eq__append__conv2,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,Ts: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Ts ) )
= ( ? [Us: list_tm] :
( ( ( Xs
= ( append_tm @ Zs @ Us ) )
& ( ( append_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_338_append__eq__appendI,axiom,
! [Xs: list_fm,Xs1: list_fm,Zs: list_fm,Ys: list_fm,Us2: list_fm] :
( ( ( append_fm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_fm @ Xs1 @ Us2 ) )
=> ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_339_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs: list_tm,Ys: list_tm,Us2: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_tm @ Xs1 @ Us2 ) )
=> ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_340_split__list__first__prop__iff,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_set_nat,X3: set_nat] :
( ? [Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: set_nat] :
( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_341_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: fm] :
( ( member_fm2 @ Y2 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_342_split__list__first__prop__iff,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ( ? [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: list_fm] :
( ( member_list_fm2 @ Y2 @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_343_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( set_tm2 @ Ys3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_344_split__list__last__prop__iff,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_set_nat,X3: set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: set_nat] :
( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Zs2 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_345_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_fm,X3: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: fm] :
( ( member_fm2 @ Y2 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_346_split__list__last__prop__iff,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ( ? [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_list_fm,X3: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: list_fm] :
( ( member_list_fm2 @ Y2 @ ( set_list_fm2 @ Zs2 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_347_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
& ( P2 @ X3 ) ) )
= ( ? [Ys3: list_tm,X3: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( set_tm2 @ Zs2 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_348_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_349_in__set__conv__decomp__first,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys3: list_o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) )
& ~ ( member_o2 @ X2 @ ( set_o2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_350_in__set__conv__decomp__first,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X2 @ Zs2 ) ) )
& ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_351_in__set__conv__decomp__first,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ~ ( member_fm2 @ X2 @ ( set_fm2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_352_in__set__conv__decomp__first,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X2 @ Zs2 ) ) )
& ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_353_in__set__conv__decomp__first,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X2 @ Zs2 ) ) )
& ~ ( member_tm2 @ X2 @ ( set_tm2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_354_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_355_in__set__conv__decomp__last,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys3: list_o,Zs2: list_o] :
( ( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) )
& ~ ( member_o2 @ X2 @ ( set_o2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_356_in__set__conv__decomp__last,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs2: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X2 @ Zs2 ) ) )
& ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_357_in__set__conv__decomp__last,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ~ ( member_fm2 @ X2 @ ( set_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_358_in__set__conv__decomp__last,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X2 @ Zs2 ) ) )
& ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_359_in__set__conv__decomp__last,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X2 @ Zs2 ) ) )
& ~ ( member_tm2 @ X2 @ ( set_tm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_360_split__list__first__propE,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_nat,X4: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_361_split__list__first__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_fm,X4: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_362_split__list__first__propE,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_fm,X4: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_363_split__list__first__propE,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_tm,X4: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_364_split__list__last__propE,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_nat,X4: set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_365_split__list__last__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_fm,X4: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_366_split__list__last__propE,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_fm,X4: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_367_split__list__last__propE,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_tm,X4: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_368_split__list__first__prop,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_nat,X4: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_369_split__list__first__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_fm,X4: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_370_split__list__first__prop,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_fm,X4: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_371_split__list__first__prop,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_tm,X4: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_372_split__list__last__prop,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_nat,X4: set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_373_split__list__last__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_fm,X4: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_374_split__list__last__prop,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_fm,X4: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_375_split__list__last__prop,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_tm,X4: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_376_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_377_in__set__conv__decomp,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
= ( ? [Ys3: list_o,Zs2: list_o] :
( Xs
= ( append_o @ Ys3 @ ( cons_o @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_378_in__set__conv__decomp,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys3: list_set_nat,Zs2: list_set_nat] :
( Xs
= ( append_set_nat @ Ys3 @ ( cons_set_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_379_in__set__conv__decomp,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
= ( ? [Ys3: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_380_in__set__conv__decomp,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_381_in__set__conv__decomp,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
= ( ? [Ys3: list_tm,Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_382_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys4: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_383_append__Cons__eq__iff,axiom,
! [X2: $o,Xs: list_o,Ys: list_o,Xs5: list_o,Ys4: list_o] :
( ~ ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( ~ ( member_o2 @ X2 @ ( set_o2 @ Ys ) )
=> ( ( ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) )
= ( append_o @ Xs5 @ ( cons_o @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_384_append__Cons__eq__iff,axiom,
! [X2: set_nat,Xs: list_set_nat,Ys: list_set_nat,Xs5: list_set_nat,Ys4: list_set_nat] :
( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ( ( ( append_set_nat @ Xs @ ( cons_set_nat @ X2 @ Ys ) )
= ( append_set_nat @ Xs5 @ ( cons_set_nat @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_385_append__Cons__eq__iff,axiom,
! [X2: fm,Xs: list_fm,Ys: list_fm,Xs5: list_fm,Ys4: list_fm] :
( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm2 @ X2 @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X2 @ Ys ) )
= ( append_fm @ Xs5 @ ( cons_fm @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_386_append__Cons__eq__iff,axiom,
! [X2: list_fm,Xs: list_list_fm,Ys: list_list_fm,Xs5: list_list_fm,Ys4: list_list_fm] :
( ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Ys ) )
=> ( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ Ys ) )
= ( append_list_fm @ Xs5 @ ( cons_list_fm @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_387_append__Cons__eq__iff,axiom,
! [X2: tm,Xs: list_tm,Ys: list_tm,Xs5: list_tm,Ys4: list_tm] :
( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm2 @ X2 @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X2 @ Ys ) )
= ( append_tm @ Xs5 @ ( cons_tm @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_388_split__list__propE,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_set_nat,X4: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_389_split__list__propE,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_fm,X4: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_390_split__list__propE,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_fm,X4: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_391_split__list__propE,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_tm,X4: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_392_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_393_split__list__first,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) )
& ~ ( member_o2 @ X2 @ ( set_o2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_394_split__list__first,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_395_split__list__first,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ~ ( member_fm2 @ X2 @ ( set_fm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_396_split__list__first,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_397_split__list__first,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ~ ( member_tm2 @ X2 @ ( set_tm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_398_split__list__prop,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ? [X: set_nat] :
( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_set_nat,X4: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs3 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_399_split__list__prop,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ? [X: fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_fm,X4: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_400_split__list__prop,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ? [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_fm,X4: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_401_split__list__prop,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ? [X: tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_tm,X4: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs3 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_402_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_403_split__list__last,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) )
& ~ ( member_o2 @ X2 @ ( set_o2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_404_split__list__last,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_405_split__list__last,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ~ ( member_fm2 @ X2 @ ( set_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_406_split__list__last,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_407_split__list__last,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ~ ( member_tm2 @ X2 @ ( set_tm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_408_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_409_split__list,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs3: list_o] :
( Xs
= ( append_o @ Ys2 @ ( cons_o @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_410_split__list,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_411_split__list,axiom,
! [X2: fm,Xs: list_fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_412_split__list,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_413_split__list,axiom,
! [X2: tm,Xs: list_tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_414_rev__nonempty__induct,axiom,
! [Xs: list_fm,P2: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X4: fm] : ( P2 @ ( cons_fm @ X4 @ nil_fm ) )
=> ( ! [X4: fm,Xs4: list_fm] :
( ( Xs4 != nil_fm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append_fm @ Xs4 @ ( cons_fm @ X4 @ nil_fm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_415_rev__nonempty__induct,axiom,
! [Xs: list_list_fm,P2: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X4: list_fm] : ( P2 @ ( cons_list_fm @ X4 @ nil_list_fm ) )
=> ( ! [X4: list_fm,Xs4: list_list_fm] :
( ( Xs4 != nil_list_fm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append_list_fm @ Xs4 @ ( cons_list_fm @ X4 @ nil_list_fm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_416_rev__nonempty__induct,axiom,
! [Xs: list_tm,P2: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X4: tm] : ( P2 @ ( cons_tm @ X4 @ nil_tm ) )
=> ( ! [X4: tm,Xs4: list_tm] :
( ( Xs4 != nil_tm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append_tm @ Xs4 @ ( cons_tm @ X4 @ nil_tm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_417_append__eq__Cons__conv,axiom,
! [Ys: list_fm,Zs: list_fm,X2: fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs )
= ( cons_fm @ X2 @ Xs ) )
= ( ( ( Ys = nil_fm )
& ( Zs
= ( cons_fm @ X2 @ Xs ) ) )
| ? [Ys5: list_fm] :
( ( Ys
= ( cons_fm @ X2 @ Ys5 ) )
& ( ( append_fm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_418_append__eq__Cons__conv,axiom,
! [Ys: list_list_fm,Zs: list_list_fm,X2: list_fm,Xs: list_list_fm] :
( ( ( append_list_fm @ Ys @ Zs )
= ( cons_list_fm @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_fm )
& ( Zs
= ( cons_list_fm @ X2 @ Xs ) ) )
| ? [Ys5: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X2 @ Ys5 ) )
& ( ( append_list_fm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_419_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs: list_tm,X2: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( cons_tm @ X2 @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs
= ( cons_tm @ X2 @ Xs ) ) )
| ? [Ys5: list_tm] :
( ( Ys
= ( cons_tm @ X2 @ Ys5 ) )
& ( ( append_tm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_420_Cons__eq__append__conv,axiom,
! [X2: fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Ys = nil_fm )
& ( ( cons_fm @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_fm] :
( ( ( cons_fm @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_fm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_421_Cons__eq__append__conv,axiom,
! [X2: list_fm,Xs: list_list_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( append_list_fm @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_fm )
& ( ( cons_list_fm @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_fm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_422_Cons__eq__append__conv,axiom,
! [X2: tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_tm] :
( ( ( cons_tm @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_tm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_423_rev__exhaust,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ~ ! [Ys2: list_fm,Y3: fm] :
( Xs
!= ( append_fm @ Ys2 @ ( cons_fm @ Y3 @ nil_fm ) ) ) ) ).
% rev_exhaust
thf(fact_424_rev__exhaust,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ~ ! [Ys2: list_list_fm,Y3: list_fm] :
( Xs
!= ( append_list_fm @ Ys2 @ ( cons_list_fm @ Y3 @ nil_list_fm ) ) ) ) ).
% rev_exhaust
thf(fact_425_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys2: list_tm,Y3: tm] :
( Xs
!= ( append_tm @ Ys2 @ ( cons_tm @ Y3 @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_426_rev__induct,axiom,
! [P2: list_fm > $o,Xs: list_fm] :
( ( P2 @ nil_fm )
=> ( ! [X4: fm,Xs4: list_fm] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append_fm @ Xs4 @ ( cons_fm @ X4 @ nil_fm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_427_rev__induct,axiom,
! [P2: list_list_fm > $o,Xs: list_list_fm] :
( ( P2 @ nil_list_fm )
=> ( ! [X4: list_fm,Xs4: list_list_fm] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append_list_fm @ Xs4 @ ( cons_list_fm @ X4 @ nil_list_fm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_428_rev__induct,axiom,
! [P2: list_tm > $o,Xs: list_tm] :
( ( P2 @ nil_tm )
=> ( ! [X4: tm,Xs4: list_tm] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append_tm @ Xs4 @ ( cons_tm @ X4 @ nil_tm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_429_subtermFm_Osimps_I2_J,axiom,
! [P: fm,Q2: fm] :
( ( subtermFm @ ( imp @ P @ Q2 ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q2 ) ) ) ).
% subtermFm.simps(2)
thf(fact_430__092_060open_062_092_060forall_062z_H_092_060in_062_123hs_A_064_Ats_A_124hs_Ats_O_Ahs_A_092_060in_062_Aset_A_Iparts_AA_Ar_Ap_____J_A_092_060and_062_Ats_A_092_060in_062_Aset_A_Ichildren_A_Iremdups_A_IA_A_064_AsubtermFms_A_Iconcat_A_Iparts_AA_Ar_Ap_____J_J_J_J_Ar_Az_J_125_O_A_I_092_060tturnstile_062_Apre_A_064_Az_H_J_092_060close_062,axiom,
! [X: list_fm] :
( ( member_list_fm2 @ X
@ ( collect_list_fm
@ ^ [Uu: list_fm] :
? [Hs: list_fm,Ts2: list_fm] :
( ( Uu
= ( append_fm @ Hs @ Ts2 ) )
& ( member_list_fm2 @ Hs @ ( set_list_fm2 @ ( parts @ aa @ r @ p ) ) )
& ( member_list_fm2 @ Ts2 @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) @ r @ za ) ) ) ) ) )
=> ( sequent_calculus @ ( append_fm @ prea @ X ) ) ) ).
% \<open>\<forall>z'\<in>{hs @ ts |hs ts. hs \<in> set (parts A r p__) \<and> ts \<in> set (children (remdups (A @ subtermFms (concat (parts A r p__)))) r z)}. (\<tturnstile> pre @ z')\<close>
thf(fact_431_subtermFm_Osimps_I7_J,axiom,
! [P: fm] :
( ( subtermFm @ ( neg @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(7)
thf(fact_432_bind__simps_I2_J,axiom,
! [X2: fm,Xs: list_fm,F: fm > list_fm] :
( ( bind_fm_fm @ ( cons_fm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_433_bind__simps_I2_J,axiom,
! [X2: fm,Xs: list_fm,F: fm > list_tm] :
( ( bind_fm_tm @ ( cons_fm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_434_bind__simps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm,F: list_fm > list_fm] :
( ( bind_list_fm_fm @ ( cons_list_fm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_list_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_435_bind__simps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm,F: list_fm > list_tm] :
( ( bind_list_fm_tm @ ( cons_list_fm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_list_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_436_bind__simps_I2_J,axiom,
! [X2: tm,Xs: list_tm,F: tm > list_fm] :
( ( bind_tm_fm @ ( cons_tm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_tm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_437_bind__simps_I2_J,axiom,
! [X2: tm,Xs: list_tm,F: tm > list_tm] :
( ( bind_tm_tm @ ( cons_tm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_tm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_438_Basic,axiom,
! [P: fm,Z: list_fm] :
( ( member_fm @ ( neg @ P ) @ Z )
=> ( sequent_calculus @ ( cons_fm @ P @ Z ) ) ) ).
% Basic
thf(fact_439_maps__def,axiom,
( maps_fm_tm
= ( ^ [F2: fm > list_tm,Xs2: list_fm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_440_maps__def,axiom,
( maps_tm_tm
= ( ^ [F2: tm > list_tm,Xs2: list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_441_concat__map__maps,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( concat_tm @ ( map_fm_list_tm @ F @ Xs ) )
= ( maps_fm_tm @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_442_concat__map__maps,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( concat_tm @ ( map_tm_list_tm @ F @ Xs ) )
= ( maps_tm_tm @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_443_the__elem__set,axiom,
! [X2: set_nat] :
( ( the_elem_set_nat @ ( set_set_nat2 @ ( cons_set_nat @ X2 @ nil_set_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_444_the__elem__set,axiom,
! [X2: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X2 @ nil_fm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_445_the__elem__set,axiom,
! [X2: list_fm] :
( ( the_elem_list_fm @ ( set_list_fm2 @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_446_the__elem__set,axiom,
! [X2: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X2 @ nil_tm ) ) )
= X2 ) ).
% the_elem_set
thf(fact_447_maps__simps_I1_J,axiom,
! [F: fm > list_fm,X2: fm,Xs: list_fm] :
( ( maps_fm_fm @ F @ ( cons_fm @ X2 @ Xs ) )
= ( append_fm @ ( F @ X2 ) @ ( maps_fm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_448_maps__simps_I1_J,axiom,
! [F: fm > list_tm,X2: fm,Xs: list_fm] :
( ( maps_fm_tm @ F @ ( cons_fm @ X2 @ Xs ) )
= ( append_tm @ ( F @ X2 ) @ ( maps_fm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_449_maps__simps_I1_J,axiom,
! [F: list_fm > list_fm,X2: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_fm @ F @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_fm @ ( F @ X2 ) @ ( maps_list_fm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_450_maps__simps_I1_J,axiom,
! [F: list_fm > list_tm,X2: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_tm @ F @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_tm @ ( F @ X2 ) @ ( maps_list_fm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_451_maps__simps_I1_J,axiom,
! [F: tm > list_fm,X2: tm,Xs: list_tm] :
( ( maps_tm_fm @ F @ ( cons_tm @ X2 @ Xs ) )
= ( append_fm @ ( F @ X2 ) @ ( maps_tm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_452_maps__simps_I1_J,axiom,
! [F: tm > list_tm,X2: tm,Xs: list_tm] :
( ( maps_tm_tm @ F @ ( cons_tm @ X2 @ Xs ) )
= ( append_tm @ ( F @ X2 ) @ ( maps_tm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_453_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_454_not__in__set__insert,axiom,
! [X2: $o,Xs: list_o] :
( ~ ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( ( insert_o @ X2 @ Xs )
= ( cons_o @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_455_not__in__set__insert,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X2 @ Xs )
= ( cons_set_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_456_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_457_not__in__set__insert,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ~ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X2 @ Xs )
= ( cons_list_fm @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_458_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_459_member,axiom,
( member_nat
= ( ^ [P3: nat,Z3: list_nat] : ( member_nat2 @ P3 @ ( set_nat2 @ Z3 ) ) ) ) ).
% member
thf(fact_460_member,axiom,
( member_o
= ( ^ [P3: $o,Z3: list_o] : ( member_o2 @ P3 @ ( set_o2 @ Z3 ) ) ) ) ).
% member
thf(fact_461_member,axiom,
( member_list_fm
= ( ^ [P3: list_fm,Z3: list_list_fm] : ( member_list_fm2 @ P3 @ ( set_list_fm2 @ Z3 ) ) ) ) ).
% member
thf(fact_462_member,axiom,
( member_fm
= ( ^ [P3: fm,Z3: list_fm] : ( member_fm2 @ P3 @ ( set_fm2 @ Z3 ) ) ) ) ).
% member
thf(fact_463_member,axiom,
( member_tm
= ( ^ [P3: tm,Z3: list_tm] : ( member_tm2 @ P3 @ ( set_tm2 @ Z3 ) ) ) ) ).
% member
thf(fact_464_member,axiom,
( member_set_nat
= ( ^ [P3: set_nat,Z3: list_set_nat] : ( member_set_nat2 @ P3 @ ( set_set_nat2 @ Z3 ) ) ) ) ).
% member
thf(fact_465_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_466_in__set__insert,axiom,
! [X2: $o,Xs: list_o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( ( insert_o @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_467_in__set__insert,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_468_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_469_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_470_in__set__insert,axiom,
! [X2: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_471_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_472_bind__simps_I1_J,axiom,
! [F: fm > list_list_fm] :
( ( bind_fm_list_fm @ nil_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_473_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_474_bind__simps_I1_J,axiom,
! [F: list_fm > list_fm] :
( ( bind_list_fm_fm @ nil_list_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_475_bind__simps_I1_J,axiom,
! [F: list_fm > list_list_fm] :
( ( bind_list_fm_list_fm @ nil_list_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_476_bind__simps_I1_J,axiom,
! [F: list_fm > list_tm] :
( ( bind_list_fm_tm @ nil_list_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_477_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_478_bind__simps_I1_J,axiom,
! [F: tm > list_list_fm] :
( ( bind_tm_list_fm @ nil_tm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_479_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_480_insert__Nil,axiom,
! [X2: fm] :
( ( insert_fm @ X2 @ nil_fm )
= ( cons_fm @ X2 @ nil_fm ) ) ).
% insert_Nil
thf(fact_481_insert__Nil,axiom,
! [X2: list_fm] :
( ( insert_list_fm @ X2 @ nil_list_fm )
= ( cons_list_fm @ X2 @ nil_list_fm ) ) ).
% insert_Nil
thf(fact_482_insert__Nil,axiom,
! [X2: tm] :
( ( insert_tm @ X2 @ nil_tm )
= ( cons_tm @ X2 @ nil_tm ) ) ).
% insert_Nil
thf(fact_483_concat__map__singleton,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( concat_list_tm
@ ( map_fm_list_list_tm
@ ^ [X3: fm] : ( cons_list_tm @ ( F @ X3 ) @ nil_list_tm )
@ Xs ) )
= ( map_fm_list_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_484_concat__map__singleton,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( concat_set_nat
@ ( map_tm_list_set_nat
@ ^ [X3: tm] : ( cons_set_nat @ ( F @ X3 ) @ nil_set_nat )
@ Xs ) )
= ( map_tm_set_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_485_concat__map__singleton,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( concat_list_tm
@ ( map_tm_list_list_tm
@ ^ [X3: tm] : ( cons_list_tm @ ( F @ X3 ) @ nil_list_tm )
@ Xs ) )
= ( map_tm_list_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_486_concat__map__singleton,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( concat_fm
@ ( map_tm_list_fm
@ ^ [X3: tm] : ( cons_fm @ ( F @ X3 ) @ nil_fm )
@ Xs ) )
= ( map_tm_fm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_487_concat__map__singleton,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( concat_tm
@ ( map_fm_list_tm
@ ^ [X3: fm] : ( cons_tm @ ( F @ X3 ) @ nil_tm )
@ Xs ) )
= ( map_fm_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_488_concat__map__singleton,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( concat_tm
@ ( map_tm_list_tm
@ ^ [X3: tm] : ( cons_tm @ ( F @ X3 ) @ nil_tm )
@ Xs ) )
= ( map_tm_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_489_list__prod_Osimps_I2_J,axiom,
! [Hs2: list_list_tm,T: list_tm,Ts: list_list_tm] :
( ( list_prod_tm @ Hs2 @ ( cons_list_tm @ T @ Ts ) )
= ( append_list_tm
@ ( map_list_tm_list_tm
@ ^ [H: list_tm] : ( append_tm @ H @ T )
@ Hs2 )
@ ( list_prod_tm @ Hs2 @ Ts ) ) ) ).
% list_prod.simps(2)
thf(fact_490_list__prod_Osimps_I2_J,axiom,
! [Hs2: list_list_fm,T: list_fm,Ts: list_list_fm] :
( ( list_prod_fm @ Hs2 @ ( cons_list_fm @ T @ Ts ) )
= ( append_list_fm
@ ( map_list_fm_list_fm
@ ^ [H: list_fm] : ( append_fm @ H @ T )
@ Hs2 )
@ ( list_prod_fm @ Hs2 @ Ts ) ) ) ).
% list_prod.simps(2)
thf(fact_491_list__prod_Osimps_I1_J,axiom,
! [Uu2: list_list_fm] :
( ( list_prod_fm @ Uu2 @ nil_list_fm )
= nil_list_fm ) ).
% list_prod.simps(1)
thf(fact_492_children_Osimps_I1_J,axiom,
! [Uu2: list_tm,Uv: rule] :
( ( children @ Uu2 @ Uv @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% children.simps(1)
thf(fact_493_insert__remdups,axiom,
! [X2: tm,Xs: list_tm] :
( ( insert_tm @ X2 @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ ( insert_tm @ X2 @ Xs ) ) ) ).
% insert_remdups
thf(fact_494_concat__eq__appendD,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_tm )
=> ? [Xss12: list_list_tm,Xs4: list_tm,Xs6: list_tm,Xss22: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss12 @ ( cons_list_tm @ ( append_tm @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append_tm @ Xs6 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_495_concat__eq__appendD,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_fm )
=> ? [Xss12: list_list_fm,Xs4: list_fm,Xs6: list_fm,Xss22: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss12 @ ( cons_list_fm @ ( append_fm @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append_fm @ Xs6 @ ( concat_fm @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_496_maps__simps_I2_J,axiom,
! [F: fm > list_fm] :
( ( maps_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_497_maps__simps_I2_J,axiom,
! [F: fm > list_list_fm] :
( ( maps_fm_list_fm @ F @ nil_fm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_498_maps__simps_I2_J,axiom,
! [F: fm > list_tm] :
( ( maps_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_499_maps__simps_I2_J,axiom,
! [F: list_fm > list_fm] :
( ( maps_list_fm_fm @ F @ nil_list_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_500_maps__simps_I2_J,axiom,
! [F: list_fm > list_list_fm] :
( ( maps_list_fm_list_fm @ F @ nil_list_fm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_501_maps__simps_I2_J,axiom,
! [F: list_fm > list_tm] :
( ( maps_list_fm_tm @ F @ nil_list_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_502_maps__simps_I2_J,axiom,
! [F: tm > list_fm] :
( ( maps_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_503_maps__simps_I2_J,axiom,
! [F: tm > list_list_fm] :
( ( maps_tm_list_fm @ F @ nil_tm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_504_maps__simps_I2_J,axiom,
! [F: tm > list_tm] :
( ( maps_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_505_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: fm,Q2: fm,Z: list_fm] :
( ( member_fm @ P @ ( cons_fm @ Q2 @ Z ) )
= ( ( P != Q2 )
=> ( member_fm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_506_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: list_fm,Q2: list_fm,Z: list_list_fm] :
( ( member_list_fm @ P @ ( cons_list_fm @ Q2 @ Z ) )
= ( ( P != Q2 )
=> ( member_list_fm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_507_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: tm,Q2: tm,Z: list_tm] :
( ( member_tm @ P @ ( cons_tm @ Q2 @ Z ) )
= ( ( P != Q2 )
=> ( member_tm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_508_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_509_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: list_fm] :
~ ( member_list_fm @ P @ nil_list_fm ) ).
% SeCaV.member.simps(1)
thf(fact_510_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_511_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_512_List_Oinsert__def,axiom,
( insert_o
= ( ^ [X3: $o,Xs2: list_o] : ( if_list_o @ ( member_o2 @ X3 @ ( set_o2 @ Xs2 ) ) @ Xs2 @ ( cons_o @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_513_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X3: set_nat,Xs2: list_set_nat] : ( if_list_set_nat @ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_set_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_514_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X3: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm2 @ X3 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_515_List_Oinsert__def,axiom,
( insert_list_fm
= ( ^ [X3: list_fm,Xs2: list_list_fm] : ( if_list_list_fm @ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_list_fm @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_516_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X3: tm,Xs2: list_tm] : ( if_list_tm @ ( member_tm2 @ X3 @ ( set_tm2 @ Xs2 ) ) @ Xs2 @ ( cons_tm @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_517_children_Osimps_I2_J,axiom,
! [A2: list_tm,R: rule,P: fm,Z: list_fm] :
( ( children @ A2 @ R @ ( cons_fm @ P @ Z ) )
= ( list_prod_fm @ ( parts @ A2 @ R @ P ) @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ A2 @ R @ P ) ) ) ) ) ) @ R @ Z ) ) ) ).
% children.simps(2)
thf(fact_518_List_Obind__def,axiom,
( bind_fm_tm
= ( ^ [Xs2: list_fm,F2: fm > list_tm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_519_List_Obind__def,axiom,
( bind_tm_tm
= ( ^ [Xs2: list_tm,F2: tm > list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs2 ) ) ) ) ).
% List.bind_def
thf(fact_520_set__children__Cons,axiom,
! [A2: list_tm,R: rule,P: fm,Z: list_fm] :
( ( set_list_fm2 @ ( children @ A2 @ R @ ( cons_fm @ P @ Z ) ) )
= ( collect_list_fm
@ ^ [Uu: list_fm] :
? [Hs: list_fm,Ts2: list_fm] :
( ( Uu
= ( append_fm @ Hs @ Ts2 ) )
& ( member_list_fm2 @ Hs @ ( set_list_fm2 @ ( parts @ A2 @ R @ P ) ) )
& ( member_list_fm2 @ Ts2 @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ A2 @ R @ P ) ) ) ) ) ) @ R @ Z ) ) ) ) ) ) ).
% set_children_Cons
thf(fact_521_list__prod__is__cartesian,axiom,
! [Hs2: list_list_tm,Ts: list_list_tm] :
( ( set_list_tm2 @ ( list_prod_tm @ Hs2 @ Ts ) )
= ( collect_list_tm
@ ^ [Uu: list_tm] :
? [H: list_tm,T2: list_tm] :
( ( Uu
= ( append_tm @ H @ T2 ) )
& ( member_list_tm @ H @ ( set_list_tm2 @ Hs2 ) )
& ( member_list_tm @ T2 @ ( set_list_tm2 @ Ts ) ) ) ) ) ).
% list_prod_is_cartesian
thf(fact_522_list__prod__is__cartesian,axiom,
! [Hs2: list_list_fm,Ts: list_list_fm] :
( ( set_list_fm2 @ ( list_prod_fm @ Hs2 @ Ts ) )
= ( collect_list_fm
@ ^ [Uu: list_fm] :
? [H: list_fm,T2: list_fm] :
( ( Uu
= ( append_fm @ H @ T2 ) )
& ( member_list_fm2 @ H @ ( set_list_fm2 @ Hs2 ) )
& ( member_list_fm2 @ T2 @ ( set_list_fm2 @ Ts ) ) ) ) ) ).
% list_prod_is_cartesian
thf(fact_523_prefixes__snoc,axiom,
! [Xs: list_fm,X2: fm] :
( ( prefixes_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) )
= ( append_list_fm @ ( prefixes_fm @ Xs ) @ ( cons_list_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) @ nil_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_524_prefixes__snoc,axiom,
! [Xs: list_list_fm,X2: list_fm] :
( ( prefixes_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( append_list_list_fm @ ( prefixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) @ nil_list_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_525_prefixes__snoc,axiom,
! [Xs: list_tm,X2: tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( cons_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) @ nil_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_526_suffixes__snoc,axiom,
! [Xs: list_fm,X2: fm] :
( ( suffixes_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) )
= ( cons_list_fm @ nil_fm
@ ( map_list_fm_list_fm
@ ^ [Ys3: list_fm] : ( append_fm @ Ys3 @ ( cons_fm @ X2 @ nil_fm ) )
@ ( suffixes_fm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_527_suffixes__snoc,axiom,
! [Xs: list_list_fm,X2: list_fm] :
( ( suffixes_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( cons_list_list_fm @ nil_list_fm
@ ( map_li4351931137408529412ist_fm
@ ^ [Ys3: list_list_fm] : ( append_list_fm @ Ys3 @ ( cons_list_fm @ X2 @ nil_list_fm ) )
@ ( suffixes_list_fm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_528_suffixes__snoc,axiom,
! [Xs: list_tm,X2: tm] :
( ( suffixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) )
= ( cons_list_tm @ nil_tm
@ ( map_list_tm_list_tm
@ ^ [Ys3: list_tm] : ( append_tm @ Ys3 @ ( cons_tm @ X2 @ nil_tm ) )
@ ( suffixes_tm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_529_prefixes__eq__snoc,axiom,
! [Ys: list_list_fm,Xs: list_list_list_fm,X2: list_list_fm] :
( ( ( prefixes_list_fm @ Ys )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X2 @ nil_list_list_fm ) ) )
= ( ( ( ( Ys = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z3: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( append_list_fm @ Zs2 @ ( cons_list_fm @ Z3 @ nil_list_fm ) ) )
& ( Xs
= ( prefixes_list_fm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_530_prefixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X2: list_tm] :
( ( ( prefixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X2 @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( append_tm @ Zs2 @ ( cons_tm @ Z3 @ nil_tm ) ) )
& ( Xs
= ( prefixes_tm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_531_prefixes__eq__snoc,axiom,
! [Ys: list_fm,Xs: list_list_fm,X2: list_fm] :
( ( ( prefixes_fm @ Ys )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( ( ( ( Ys = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( append_fm @ Zs2 @ ( cons_fm @ Z3 @ nil_fm ) ) )
& ( Xs
= ( prefixes_fm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_532_set__Cons__def,axiom,
( set_Cons_nat
= ( ^ [A3: set_nat,XS: set_list_nat] :
( collect_list_nat
@ ^ [Z3: list_nat] :
? [X3: nat,Xs2: list_nat] :
( ( Z3
= ( cons_nat @ X3 @ Xs2 ) )
& ( member_nat2 @ X3 @ A3 )
& ( member_list_nat @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_533_set__Cons__def,axiom,
( set_Cons_o
= ( ^ [A3: set_o,XS: set_list_o] :
( collect_list_o
@ ^ [Z3: list_o] :
? [X3: $o,Xs2: list_o] :
( ( Z3
= ( cons_o @ X3 @ Xs2 ) )
& ( member_o2 @ X3 @ A3 )
& ( member_list_o @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_534_set__Cons__def,axiom,
( set_Cons_fm
= ( ^ [A3: set_fm,XS: set_list_fm] :
( collect_list_fm
@ ^ [Z3: list_fm] :
? [X3: fm,Xs2: list_fm] :
( ( Z3
= ( cons_fm @ X3 @ Xs2 ) )
& ( member_fm2 @ X3 @ A3 )
& ( member_list_fm2 @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_535_set__Cons__def,axiom,
( set_Cons_list_fm
= ( ^ [A3: set_list_fm,XS: set_list_list_fm] :
( collect_list_list_fm
@ ^ [Z3: list_list_fm] :
? [X3: list_fm,Xs2: list_list_fm] :
( ( Z3
= ( cons_list_fm @ X3 @ Xs2 ) )
& ( member_list_fm2 @ X3 @ A3 )
& ( member_list_list_fm @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_536_set__Cons__def,axiom,
( set_Cons_tm
= ( ^ [A3: set_tm,XS: set_list_tm] :
( collect_list_tm
@ ^ [Z3: list_tm] :
? [X3: tm,Xs2: list_tm] :
( ( Z3
= ( cons_tm @ X3 @ Xs2 ) )
& ( member_tm2 @ X3 @ A3 )
& ( member_list_tm @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_537_ext_Osimps_I2_J,axiom,
! [Y: list_fm,P: fm,Z: list_fm] :
( ( ext_fm @ Y @ ( cons_fm @ P @ Z ) )
= ( ( ( member_fm @ P @ Y )
=> ( ext_fm @ Y @ Z ) )
& ( member_fm @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_538_ext_Osimps_I2_J,axiom,
! [Y: list_list_fm,P: list_fm,Z: list_list_fm] :
( ( ext_list_fm @ Y @ ( cons_list_fm @ P @ Z ) )
= ( ( ( member_list_fm @ P @ Y )
=> ( ext_list_fm @ Y @ Z ) )
& ( member_list_fm @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_539_ext_Osimps_I2_J,axiom,
! [Y: list_tm,P: tm,Z: list_tm] :
( ( ext_tm @ Y @ ( cons_tm @ P @ Z ) )
= ( ( ( member_tm @ P @ Y )
=> ( ext_tm @ Y @ Z ) )
& ( member_tm @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_540_product__lists_Osimps_I2_J,axiom,
! [Xs: list_list_fm,Xss: list_list_list_fm] :
( ( produc373462945560358120ist_fm @ ( cons_list_list_fm @ Xs @ Xss ) )
= ( concat_list_list_fm
@ ( map_li9121411909794442256ist_fm
@ ^ [X3: list_fm] : ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X3 ) @ ( produc373462945560358120ist_fm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_541_product__lists_Osimps_I2_J,axiom,
! [Xs: list_tm,Xss: list_list_tm] :
( ( product_lists_tm @ ( cons_list_tm @ Xs @ Xss ) )
= ( concat_list_tm
@ ( map_tm_list_list_tm
@ ^ [X3: tm] : ( map_list_tm_list_tm @ ( cons_tm @ X3 ) @ ( product_lists_tm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_542_product__lists_Osimps_I2_J,axiom,
! [Xs: list_fm,Xss: list_list_fm] :
( ( product_lists_fm @ ( cons_list_fm @ Xs @ Xss ) )
= ( concat_list_fm
@ ( map_fm_list_list_fm
@ ^ [X3: fm] : ( map_list_fm_list_fm @ ( cons_fm @ X3 ) @ ( product_lists_fm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_543_suffixes__eq__snoc,axiom,
! [Ys: list_list_fm,Xs: list_list_list_fm,X2: list_list_fm] :
( ( ( suffixes_list_fm @ Ys )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X2 @ nil_list_list_fm ) ) )
= ( ( ( ( Ys = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z3: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z3 @ Zs2 ) )
& ( Xs
= ( suffixes_list_fm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_544_suffixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X2: list_tm] :
( ( ( suffixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X2 @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z3: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs2 ) )
& ( Xs
= ( suffixes_tm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_545_suffixes__eq__snoc,axiom,
! [Ys: list_fm,Xs: list_list_fm,X2: list_fm] :
( ( ( suffixes_fm @ Ys )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( ( ( ( Ys = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z3: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs2 ) )
& ( Xs
= ( suffixes_fm @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_546_prefixes__not__Nil,axiom,
! [Xs: list_fm] :
( ( prefixes_fm @ Xs )
!= nil_list_fm ) ).
% prefixes_not_Nil
thf(fact_547_suffixes__not__Nil,axiom,
! [Xs: list_fm] :
( ( suffixes_fm @ Xs )
!= nil_list_fm ) ).
% suffixes_not_Nil
thf(fact_548_Ext,axiom,
! [Z: list_fm,Y: list_fm] :
( ( sequent_calculus @ Z )
=> ( ( ext_fm @ Y @ Z )
=> ( sequent_calculus @ Y ) ) ) ).
% Ext
thf(fact_549_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_550_ext_Osimps_I1_J,axiom,
! [Y: list_list_fm] : ( ext_list_fm @ Y @ nil_list_fm ) ).
% ext.simps(1)
thf(fact_551_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_552_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_553_prefixes_Osimps_I1_J,axiom,
( ( prefixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% prefixes.simps(1)
thf(fact_554_prefixes_Osimps_I1_J,axiom,
( ( prefixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_555_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_556_suffixes_Osimps_I1_J,axiom,
( ( suffixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% suffixes.simps(1)
thf(fact_557_suffixes_Osimps_I1_J,axiom,
( ( suffixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_558_prefixes_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( prefixes_fm @ ( cons_fm @ X2 @ Xs ) )
= ( cons_list_fm @ nil_fm @ ( map_list_fm_list_fm @ ( cons_fm @ X2 ) @ ( prefixes_fm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_559_prefixes_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( prefixes_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( cons_list_list_fm @ nil_list_fm @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X2 ) @ ( prefixes_list_fm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_560_prefixes_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( prefixes_tm @ ( cons_tm @ X2 @ Xs ) )
= ( cons_list_tm @ nil_tm @ ( map_list_tm_list_tm @ ( cons_tm @ X2 ) @ ( prefixes_tm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_561_suffixes_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( suffixes_fm @ ( cons_fm @ X2 @ Xs ) )
= ( append_list_fm @ ( suffixes_fm @ Xs ) @ ( cons_list_fm @ ( cons_fm @ X2 @ Xs ) @ nil_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_562_suffixes_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( suffixes_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_list_list_fm @ ( suffixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( cons_list_fm @ X2 @ Xs ) @ nil_list_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_563_suffixes_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( suffixes_tm @ ( cons_tm @ X2 @ Xs ) )
= ( append_list_tm @ ( suffixes_tm @ Xs ) @ ( cons_list_tm @ ( cons_tm @ X2 @ Xs ) @ nil_list_tm ) ) ) ).
% suffixes.simps(2)
thf(fact_564_product__lists_Osimps_I1_J,axiom,
( ( produc373462945560358120ist_fm @ nil_list_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_565_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_566_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_567_sublists_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( sublists_fm @ ( cons_fm @ X2 @ Xs ) )
= ( append_list_fm @ ( sublists_fm @ Xs ) @ ( map_list_fm_list_fm @ ( cons_fm @ X2 ) @ ( prefixes_fm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_568_sublists_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( sublists_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_list_list_fm @ ( sublists_list_fm @ Xs ) @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X2 ) @ ( prefixes_list_fm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_569_sublists_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( sublists_tm @ ( cons_tm @ X2 @ Xs ) )
= ( append_list_tm @ ( sublists_tm @ Xs ) @ ( map_list_tm_list_tm @ ( cons_tm @ X2 ) @ ( prefixes_tm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_570_parts__preserves__unaffected,axiom,
! [R: rule,P: fm,Z4: list_fm,A2: list_tm] :
( ~ ( affects @ R @ P )
=> ( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ ( parts @ A2 @ R @ P ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z4 ) ) ) ) ).
% parts_preserves_unaffected
thf(fact_571_children__preserves__unaffected,axiom,
! [P: fm,Z: list_fm,R: rule,Z4: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ~ ( affects @ R @ P )
=> ( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z4 ) ) ) ) ) ).
% children_preserves_unaffected
thf(fact_572_sublists_Osimps_I1_J,axiom,
( ( sublists_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% sublists.simps(1)
thf(fact_573_sublists_Osimps_I1_J,axiom,
( ( sublists_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% sublists.simps(1)
thf(fact_574_sublists_Osimps_I1_J,axiom,
( ( sublists_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% sublists.simps(1)
thf(fact_575_subseqs_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( subseqs_fm @ ( cons_fm @ X2 @ Xs ) )
= ( append_list_fm @ ( map_list_fm_list_fm @ ( cons_fm @ X2 ) @ ( subseqs_fm @ Xs ) ) @ ( subseqs_fm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_576_subseqs_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( subseqs_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_list_list_fm @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X2 ) @ ( subseqs_list_fm @ Xs ) ) @ ( subseqs_list_fm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_577_subseqs_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( subseqs_tm @ ( cons_tm @ X2 @ Xs ) )
= ( append_list_tm @ ( map_list_tm_list_tm @ ( cons_tm @ X2 ) @ ( subseqs_tm @ Xs ) ) @ ( subseqs_tm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_578_map__rec,axiom,
( map_fm_list_tm
= ( ^ [F2: fm > list_tm] :
( rec_li6905007533772093688_tm_fm @ nil_list_tm
@ ^ [X3: fm,Uu: list_fm] : ( cons_list_tm @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_579_map__rec,axiom,
( map_tm_set_nat
= ( ^ [F2: tm > set_nat] :
( rec_li8667420360015564823nat_tm @ nil_set_nat
@ ^ [X3: tm,Uu: list_tm] : ( cons_set_nat @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_580_map__rec,axiom,
( map_tm_list_tm
= ( ^ [F2: tm > list_tm] :
( rec_li6905007533773012074_tm_tm @ nil_list_tm
@ ^ [X3: tm,Uu: list_tm] : ( cons_list_tm @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_581_map__rec,axiom,
( map_tm_fm
= ( ^ [F2: tm > fm] :
( rec_list_list_fm_tm @ nil_fm
@ ^ [X3: tm,Uu: list_tm] : ( cons_fm @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_582_A,axiom,
ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ ( append_fm @ prea @ ( cons_fm @ p @ za ) ) ) ) ) @ ( paramsts @ ( remdups_tm @ ( append_tm @ aa @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ aa @ r @ p ) ) ) ) ) ) ) ).
% A
thf(fact_583_suffixes__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( suffixes_fm @ ( append_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( suffixes_fm @ Ys )
@ ( map_list_fm_list_fm
@ ^ [Xs3: list_fm] : ( append_fm @ Xs3 @ Ys )
@ ( tl_list_fm @ ( suffixes_fm @ Xs ) ) ) ) ) ).
% suffixes_append
thf(fact_584_suffixes__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( suffixes_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( suffixes_tm @ Ys )
@ ( map_list_tm_list_tm
@ ^ [Xs3: list_tm] : ( append_tm @ Xs3 @ Ys )
@ ( tl_list_tm @ ( suffixes_tm @ Xs ) ) ) ) ) ).
% suffixes_append
thf(fact_585_Cons_Oprems_I2_J,axiom,
ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ ( append_fm @ prea @ ( cons_fm @ p @ za ) ) ) ) ) @ ( paramsts @ aa ) ).
% Cons.prems(2)
thf(fact_586_assms_I2_J,axiom,
ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ ( append_fm @ pre2 @ z ) ) ) ) @ ( paramsts @ a ) ).
% assms(2)
thf(fact_587_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_588_list_Oset__map,axiom,
! [F: fm > fm,V: list_fm] :
( ( set_fm2 @ ( map_fm_fm @ F @ V ) )
= ( image_fm_fm @ F @ ( set_fm2 @ V ) ) ) ).
% list.set_map
thf(fact_589_list_Oset__map,axiom,
! [F: tm > fm,V: list_tm] :
( ( set_fm2 @ ( map_tm_fm @ F @ V ) )
= ( image_tm_fm @ F @ ( set_tm2 @ V ) ) ) ).
% list.set_map
thf(fact_590_list_Oset__map,axiom,
! [F: fm > tm,V: list_fm] :
( ( set_tm2 @ ( map_fm_tm @ F @ V ) )
= ( image_fm_tm @ F @ ( set_fm2 @ V ) ) ) ).
% list.set_map
thf(fact_591_list_Oset__map,axiom,
! [F: tm > tm,V: list_tm] :
( ( set_tm2 @ ( map_tm_tm @ F @ V ) )
= ( image_tm_tm @ F @ ( set_tm2 @ V ) ) ) ).
% list.set_map
thf(fact_592_list_Oset__map,axiom,
! [F: fm > set_tm,V: list_fm] :
( ( set_set_tm2 @ ( map_fm_set_tm @ F @ V ) )
= ( image_fm_set_tm @ F @ ( set_fm2 @ V ) ) ) ).
% list.set_map
thf(fact_593_list_Oset__map,axiom,
! [F: fm > list_tm,V: list_fm] :
( ( set_list_tm2 @ ( map_fm_list_tm @ F @ V ) )
= ( image_fm_list_tm @ F @ ( set_fm2 @ V ) ) ) ).
% list.set_map
thf(fact_594_list_Oset__map,axiom,
! [F: tm > set_tm,V: list_tm] :
( ( set_set_tm2 @ ( map_tm_set_tm @ F @ V ) )
= ( image_tm_set_tm @ F @ ( set_tm2 @ V ) ) ) ).
% list.set_map
thf(fact_595_list_Oset__map,axiom,
! [F: tm > list_tm,V: list_tm] :
( ( set_list_tm2 @ ( map_tm_list_tm @ F @ V ) )
= ( image_tm_list_tm @ F @ ( set_tm2 @ V ) ) ) ).
% list.set_map
thf(fact_596_list_Oset__map,axiom,
! [F: fm > list_fm,V: list_fm] :
( ( set_list_fm2 @ ( map_fm_list_fm @ F @ V ) )
= ( image_fm_list_fm @ F @ ( set_fm2 @ V ) ) ) ).
% list.set_map
thf(fact_597_tl__append2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs != nil_fm )
=> ( ( tl_fm @ ( append_fm @ Xs @ Ys ) )
= ( append_fm @ ( tl_fm @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_598_tl__append2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( tl_list_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( tl_list_fm @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_599_tl__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( tl_tm @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_600_SeCaV_Oext,axiom,
( ext_list_fm
= ( ^ [Y2: list_list_fm,Z3: list_list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Z3 ) @ ( set_list_fm2 @ Y2 ) ) ) ) ).
% SeCaV.ext
thf(fact_601_SeCaV_Oext,axiom,
( ext_set_nat
= ( ^ [Y2: list_set_nat,Z3: list_set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Z3 ) @ ( set_set_nat2 @ Y2 ) ) ) ) ).
% SeCaV.ext
thf(fact_602_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y2: list_nat,Z3: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z3 ) @ ( set_nat2 @ Y2 ) ) ) ) ).
% SeCaV.ext
thf(fact_603_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y2: list_tm,Z3: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z3 ) @ ( set_tm2 @ Y2 ) ) ) ) ).
% SeCaV.ext
thf(fact_604_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y2: list_fm,Z3: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z3 ) @ ( set_fm2 @ Y2 ) ) ) ) ).
% SeCaV.ext
thf(fact_605_Cons_Ohyps,axiom,
! [A2: list_tm,Pre: list_fm] :
( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ ( children @ A2 @ r @ za ) ) )
=> ( sequent_calculus @ ( append_fm @ Pre @ X4 ) ) )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ ( append_fm @ Pre @ za ) ) ) ) @ ( paramsts @ A2 ) )
=> ( sequent_calculus @ ( append_fm @ Pre @ za ) ) ) ) ).
% Cons.hyps
thf(fact_606_ih,axiom,
! [Pre: list_fm,A2: list_tm] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ ( append_fm @ Pre @ za ) ) ) ) @ ( paramsts @ A2 ) )
=> ( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ ( children @ A2 @ r @ za ) ) )
=> ( sequent_calculus @ ( append_fm @ Pre @ X4 ) ) )
=> ( sequent_calculus @ ( append_fm @ Pre @ za ) ) ) ) ).
% ih
thf(fact_607_tl__def,axiom,
( tl_fm
= ( case_list_list_fm_fm @ nil_fm
@ ^ [X213: fm,X223: list_fm] : X223 ) ) ).
% tl_def
thf(fact_608_tl__def,axiom,
( tl_list_fm
= ( case_l7658988447939845536ist_fm @ nil_list_fm
@ ^ [X213: list_fm,X223: list_list_fm] : X223 ) ) ).
% tl_def
thf(fact_609_tl__def,axiom,
( tl_tm
= ( case_list_list_tm_tm @ nil_tm
@ ^ [X213: tm,X223: list_tm] : X223 ) ) ).
% tl_def
thf(fact_610_tl__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( tl_fm @ ( append_fm @ Xs @ Ys ) )
= ( case_list_list_fm_fm @ ( tl_fm @ Ys )
@ ^ [Z3: fm,Zs2: list_fm] : ( append_fm @ Zs2 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_611_tl__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( case_list_list_tm_tm @ ( tl_tm @ Ys )
@ ^ [Z3: tm,Zs2: list_tm] : ( append_tm @ Zs2 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_612_list_Ocase__distrib,axiom,
! [H2: list_tm > list_tm,F1: list_tm,F22: tm > list_tm > list_tm,List: list_tm] :
( ( H2 @ ( case_list_list_tm_tm @ F1 @ F22 @ List ) )
= ( case_list_list_tm_tm @ ( H2 @ F1 )
@ ^ [X1: tm,X23: list_tm] : ( H2 @ ( F22 @ X1 @ X23 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_613_list_Osel_I3_J,axiom,
! [X21: fm,X22: list_fm] :
( ( tl_fm @ ( cons_fm @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_614_list_Osel_I3_J,axiom,
! [X21: list_fm,X22: list_list_fm] :
( ( tl_list_fm @ ( cons_list_fm @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_615_list_Osel_I3_J,axiom,
! [X21: tm,X22: list_tm] :
( ( tl_tm @ ( cons_tm @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_616_list_Osel_I2_J,axiom,
( ( tl_fm @ nil_fm )
= nil_fm ) ).
% list.sel(2)
thf(fact_617_list_Osel_I2_J,axiom,
( ( tl_list_fm @ nil_list_fm )
= nil_list_fm ) ).
% list.sel(2)
thf(fact_618_list_Osel_I2_J,axiom,
( ( tl_tm @ nil_tm )
= nil_tm ) ).
% list.sel(2)
thf(fact_619_map__tl,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( map_fm_list_tm @ F @ ( tl_fm @ Xs ) )
= ( tl_list_tm @ ( map_fm_list_tm @ F @ Xs ) ) ) ).
% map_tl
thf(fact_620_map__tl,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( map_tm_set_nat @ F @ ( tl_tm @ Xs ) )
= ( tl_set_nat @ ( map_tm_set_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_621_map__tl,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( map_tm_list_tm @ F @ ( tl_tm @ Xs ) )
= ( tl_list_tm @ ( map_tm_list_tm @ F @ Xs ) ) ) ).
% map_tl
thf(fact_622_map__tl,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( map_tm_fm @ F @ ( tl_tm @ Xs ) )
= ( tl_fm @ ( map_tm_fm @ F @ Xs ) ) ) ).
% map_tl
thf(fact_623_subset__code_I1_J,axiom,
! [Xs: list_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
= ( ! [X3: $o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( member_o2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_624_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B2: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B2 )
= ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_625_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_626_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_627_subset__code_I1_J,axiom,
! [Xs: list_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B2 )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_628_subset__code_I1_J,axiom,
! [Xs: list_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B2 )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_629_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_fm] :
( ( set_fm2 @ ( bind_fm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_fm_set_fm
@ ^ [X3: fm] : ( set_fm2 @ ( F @ X3 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_630_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_fm] :
( ( set_fm2 @ ( bind_tm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_tm_set_fm
@ ^ [X3: tm] : ( set_fm2 @ ( F @ X3 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_631_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_nat] :
( ( set_nat2 @ ( bind_fm_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( set_nat2 @ ( F @ X3 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_632_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_nat] :
( ( set_nat2 @ ( bind_tm_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( set_nat2 @ ( F @ X3 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_633_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_tm] :
( ( set_tm2 @ ( bind_fm_tm @ Xs @ F ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X3: fm] : ( set_tm2 @ ( F @ X3 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_634_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_tm] :
( ( set_tm2 @ ( bind_tm_tm @ Xs @ F ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X3: tm] : ( set_tm2 @ ( F @ X3 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_635_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_list_fm] :
( ( set_list_fm2 @ ( bind_fm_list_fm @ Xs @ F ) )
= ( comple8784269564784259782ist_fm
@ ( image_fm_set_list_fm
@ ^ [X3: fm] : ( set_list_fm2 @ ( F @ X3 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_636_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_list_fm] :
( ( set_list_fm2 @ ( bind_tm_list_fm @ Xs @ F ) )
= ( comple8784269564784259782ist_fm
@ ( image_tm_set_list_fm
@ ^ [X3: tm] : ( set_list_fm2 @ ( F @ X3 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_637_set__list__bind,axiom,
! [Xs: list_list_fm,F: list_fm > list_fm] :
( ( set_fm2 @ ( bind_list_fm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_list_fm_set_fm
@ ^ [X3: list_fm] : ( set_fm2 @ ( F @ X3 ) )
@ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_638_set__list__bind,axiom,
! [Xs: list_set_nat,F: set_nat > list_fm] :
( ( set_fm2 @ ( bind_set_nat_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_set_nat_set_fm
@ ^ [X3: set_nat] : ( set_fm2 @ ( F @ X3 ) )
@ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_639_params_Osimps_I7_J,axiom,
! [P: fm] :
( ( params @ ( neg @ P ) )
= ( params @ P ) ) ).
% params.simps(7)
thf(fact_640_list_Osimps_I5_J,axiom,
! [F1: list_tm,F22: tm > list_tm > list_tm,X21: tm,X22: list_tm] :
( ( case_list_list_tm_tm @ F1 @ F22 @ ( cons_tm @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_641_list_Osimps_I4_J,axiom,
! [F1: list_tm,F22: tm > list_tm > list_tm] :
( ( case_list_list_tm_tm @ F1 @ F22 @ nil_tm )
= F1 ) ).
% list.simps(4)
thf(fact_642_Nil__tl,axiom,
! [Xs: list_fm] :
( ( nil_fm
= ( tl_fm @ Xs ) )
= ( ( Xs = nil_fm )
| ? [X3: fm] :
( Xs
= ( cons_fm @ X3 @ nil_fm ) ) ) ) ).
% Nil_tl
thf(fact_643_Nil__tl,axiom,
! [Xs: list_list_fm] :
( ( nil_list_fm
= ( tl_list_fm @ Xs ) )
= ( ( Xs = nil_list_fm )
| ? [X3: list_fm] :
( Xs
= ( cons_list_fm @ X3 @ nil_list_fm ) ) ) ) ).
% Nil_tl
thf(fact_644_Nil__tl,axiom,
! [Xs: list_tm] :
( ( nil_tm
= ( tl_tm @ Xs ) )
= ( ( Xs = nil_tm )
| ? [X3: tm] :
( Xs
= ( cons_tm @ X3 @ nil_tm ) ) ) ) ).
% Nil_tl
thf(fact_645_tl__Nil,axiom,
! [Xs: list_fm] :
( ( ( tl_fm @ Xs )
= nil_fm )
= ( ( Xs = nil_fm )
| ? [X3: fm] :
( Xs
= ( cons_fm @ X3 @ nil_fm ) ) ) ) ).
% tl_Nil
thf(fact_646_tl__Nil,axiom,
! [Xs: list_list_fm] :
( ( ( tl_list_fm @ Xs )
= nil_list_fm )
= ( ( Xs = nil_list_fm )
| ? [X3: list_fm] :
( Xs
= ( cons_list_fm @ X3 @ nil_list_fm ) ) ) ) ).
% tl_Nil
thf(fact_647_tl__Nil,axiom,
! [Xs: list_tm] :
( ( ( tl_tm @ Xs )
= nil_tm )
= ( ( Xs = nil_tm )
| ? [X3: tm] :
( Xs
= ( cons_tm @ X3 @ nil_tm ) ) ) ) ).
% tl_Nil
thf(fact_648_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X2: nat] :
( ( A != nil_nat )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_649_list_Oset__sel_I2_J,axiom,
! [A: list_o,X2: $o] :
( ( A != nil_o )
=> ( ( member_o2 @ X2 @ ( set_o2 @ ( tl_o @ A ) ) )
=> ( member_o2 @ X2 @ ( set_o2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_650_list_Oset__sel_I2_J,axiom,
! [A: list_list_fm,X2: list_fm] :
( ( A != nil_list_fm )
=> ( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ ( tl_list_fm @ A ) ) )
=> ( member_list_fm2 @ X2 @ ( set_list_fm2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_651_list_Oset__sel_I2_J,axiom,
! [A: list_fm,X2: fm] :
( ( A != nil_fm )
=> ( ( member_fm2 @ X2 @ ( set_fm2 @ ( tl_fm @ A ) ) )
=> ( member_fm2 @ X2 @ ( set_fm2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_652_list_Oset__sel_I2_J,axiom,
! [A: list_tm,X2: tm] :
( ( A != nil_tm )
=> ( ( member_tm2 @ X2 @ ( set_tm2 @ ( tl_tm @ A ) ) )
=> ( member_tm2 @ X2 @ ( set_tm2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_653_list_Oset__sel_I2_J,axiom,
! [A: list_set_nat,X2: set_nat] :
( ( A != nil_set_nat )
=> ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ ( tl_set_nat @ A ) ) )
=> ( member_set_nat2 @ X2 @ ( set_set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_654_subseqs__refl,axiom,
! [Xs: list_fm] : ( member_list_fm2 @ Xs @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ).
% subseqs_refl
thf(fact_655_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_656_image__set,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( image_fm_fm @ F @ ( set_fm2 @ Xs ) )
= ( set_fm2 @ ( map_fm_fm @ F @ Xs ) ) ) ).
% image_set
thf(fact_657_image__set,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( image_fm_tm @ F @ ( set_fm2 @ Xs ) )
= ( set_tm2 @ ( map_fm_tm @ F @ Xs ) ) ) ).
% image_set
thf(fact_658_image__set,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( image_tm_fm @ F @ ( set_tm2 @ Xs ) )
= ( set_fm2 @ ( map_tm_fm @ F @ Xs ) ) ) ).
% image_set
thf(fact_659_image__set,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( image_tm_tm @ F @ ( set_tm2 @ Xs ) )
= ( set_tm2 @ ( map_tm_tm @ F @ Xs ) ) ) ).
% image_set
thf(fact_660_image__set,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( image_list_fm_fm @ F @ ( set_list_fm2 @ Xs ) )
= ( set_fm2 @ ( map_list_fm_fm @ F @ Xs ) ) ) ).
% image_set
thf(fact_661_image__set,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( image_list_fm_tm @ F @ ( set_list_fm2 @ Xs ) )
= ( set_tm2 @ ( map_list_fm_tm @ F @ Xs ) ) ) ).
% image_set
thf(fact_662_image__set,axiom,
! [F: fm > set_tm,Xs: list_fm] :
( ( image_fm_set_tm @ F @ ( set_fm2 @ Xs ) )
= ( set_set_tm2 @ ( map_fm_set_tm @ F @ Xs ) ) ) ).
% image_set
thf(fact_663_image__set,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( image_fm_list_tm @ F @ ( set_fm2 @ Xs ) )
= ( set_list_tm2 @ ( map_fm_list_tm @ F @ Xs ) ) ) ).
% image_set
thf(fact_664_image__set,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( image_fm_list_fm @ F @ ( set_fm2 @ Xs ) )
= ( set_list_fm2 @ ( map_fm_list_fm @ F @ Xs ) ) ) ).
% image_set
thf(fact_665_tl__append__if,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( Xs = nil_fm )
=> ( ( tl_fm @ ( append_fm @ Xs @ Ys ) )
= ( tl_fm @ Ys ) ) )
& ( ( Xs != nil_fm )
=> ( ( tl_fm @ ( append_fm @ Xs @ Ys ) )
= ( append_fm @ ( tl_fm @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_666_tl__append__if,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( Xs = nil_list_fm )
=> ( ( tl_list_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( tl_list_fm @ Ys ) ) )
& ( ( Xs != nil_list_fm )
=> ( ( tl_list_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( tl_list_fm @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_667_tl__append__if,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( Xs = nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( tl_tm @ Ys ) ) )
& ( ( Xs != nil_tm )
=> ( ( tl_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( tl_tm @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_668_list_Omap__sel_I2_J,axiom,
! [A: list_fm,F: fm > list_tm] :
( ( A != nil_fm )
=> ( ( tl_list_tm @ ( map_fm_list_tm @ F @ A ) )
= ( map_fm_list_tm @ F @ ( tl_fm @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_669_list_Omap__sel_I2_J,axiom,
! [A: list_tm,F: tm > set_nat] :
( ( A != nil_tm )
=> ( ( tl_set_nat @ ( map_tm_set_nat @ F @ A ) )
= ( map_tm_set_nat @ F @ ( tl_tm @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_670_list_Omap__sel_I2_J,axiom,
! [A: list_tm,F: tm > list_tm] :
( ( A != nil_tm )
=> ( ( tl_list_tm @ ( map_tm_list_tm @ F @ A ) )
= ( map_tm_list_tm @ F @ ( tl_tm @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_671_list_Omap__sel_I2_J,axiom,
! [A: list_tm,F: tm > fm] :
( ( A != nil_tm )
=> ( ( tl_fm @ ( map_tm_fm @ F @ A ) )
= ( map_tm_fm @ F @ ( tl_tm @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_672_set__subset__Cons,axiom,
! [Xs: list_set_nat,X2: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ ( cons_set_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_673_set__subset__Cons,axiom,
! [Xs: list_list_fm,X2: list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ ( cons_list_fm @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_674_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_675_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_676_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_677_Cons__in__subseqsD,axiom,
! [Y: fm,Ys: list_fm,Xs: list_fm] :
( ( member_list_fm2 @ ( cons_fm @ Y @ Ys ) @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
=> ( member_list_fm2 @ Ys @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_678_Cons__in__subseqsD,axiom,
! [Y: list_fm,Ys: list_list_fm,Xs: list_list_fm] :
( ( member_list_list_fm @ ( cons_list_fm @ Y @ Ys ) @ ( set_list_list_fm2 @ ( subseqs_list_fm @ Xs ) ) )
=> ( member_list_list_fm @ Ys @ ( set_list_list_fm2 @ ( subseqs_list_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_679_Cons__in__subseqsD,axiom,
! [Y: tm,Ys: list_tm,Xs: list_tm] :
( ( member_list_tm @ ( cons_tm @ Y @ Ys ) @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
=> ( member_list_tm @ Ys @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_680_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_681_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_682_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_683_prefixes__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( prefixes_fm @ ( append_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( prefixes_fm @ Xs ) @ ( map_list_fm_list_fm @ ( append_fm @ Xs ) @ ( tl_list_fm @ ( prefixes_fm @ Ys ) ) ) ) ) ).
% prefixes_append
thf(fact_684_prefixes__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( map_list_tm_list_tm @ ( append_tm @ Xs ) @ ( tl_list_tm @ ( prefixes_tm @ Ys ) ) ) ) ) ).
% prefixes_append
thf(fact_685_UN__I,axiom,
! [A: fm,A2: set_fm,B: fm,B2: fm > set_fm] :
( ( member_fm2 @ A @ A2 )
=> ( ( member_fm2 @ B @ ( B2 @ A ) )
=> ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_686_UN__I,axiom,
! [A: fm,A2: set_fm,B: $o,B2: fm > set_o] :
( ( member_fm2 @ A @ A2 )
=> ( ( member_o2 @ B @ ( B2 @ A ) )
=> ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_687_UN__I,axiom,
! [A: nat,A2: set_nat,B: fm,B2: nat > set_fm] :
( ( member_nat2 @ A @ A2 )
=> ( ( member_fm2 @ B @ ( B2 @ A ) )
=> ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_688_UN__I,axiom,
! [A: nat,A2: set_nat,B: $o,B2: nat > set_o] :
( ( member_nat2 @ A @ A2 )
=> ( ( member_o2 @ B @ ( B2 @ A ) )
=> ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_689_UN__I,axiom,
! [A: $o,A2: set_o,B: fm,B2: $o > set_fm] :
( ( member_o2 @ A @ A2 )
=> ( ( member_fm2 @ B @ ( B2 @ A ) )
=> ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_690_UN__I,axiom,
! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
( ( member_o2 @ A @ A2 )
=> ( ( member_o2 @ B @ ( B2 @ A ) )
=> ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_691_UN__I,axiom,
! [A: tm,A2: set_tm,B: fm,B2: tm > set_fm] :
( ( member_tm2 @ A @ A2 )
=> ( ( member_fm2 @ B @ ( B2 @ A ) )
=> ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_692_UN__I,axiom,
! [A: tm,A2: set_tm,B: $o,B2: tm > set_o] :
( ( member_tm2 @ A @ A2 )
=> ( ( member_o2 @ B @ ( B2 @ A ) )
=> ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_tm_set_o @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_693_UN__I,axiom,
! [A: fm,A2: set_fm,B: nat,B2: fm > set_nat] :
( ( member_fm2 @ A @ A2 )
=> ( ( member_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_694_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( member_nat2 @ B @ ( B2 @ A ) )
=> ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_695_UN__iff,axiom,
! [B: nat,B2: fm > set_nat,A2: set_fm] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ( member_nat2 @ B @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_696_UN__iff,axiom,
! [B: nat,B2: tm > set_nat,A2: set_tm] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ( member_nat2 @ B @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_697_UN__iff,axiom,
! [B: tm,B2: tm > set_tm,A2: set_tm] :
( ( member_tm2 @ B @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ( member_tm2 @ B @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_698_UN__iff,axiom,
! [B: tm,B2: fm > set_tm,A2: set_fm] :
( ( member_tm2 @ B @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ( member_tm2 @ B @ ( B2 @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_699_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_700_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_701_SUP__identity__eq,axiom,
! [A2: set_set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_set_tm_set_tm
@ ^ [X3: set_tm] : X3
@ A2 ) )
= ( comple2138885804642794802set_tm @ A2 ) ) ).
% SUP_identity_eq
thf(fact_702_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_703_ball__UN,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_704_ball__UN,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_705_ball__UN,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_706_ball__UN,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_707_bex__UN,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_708_bex__UN,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_709_bex__UN,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_710_bex__UN,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_711_UN__ball__bex__simps_I2_J,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_712_UN__ball__bex__simps_I2_J,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_713_UN__ball__bex__simps_I2_J,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_714_UN__ball__bex__simps_I2_J,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_715_UN__ball__bex__simps_I4_J,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_716_UN__ball__bex__simps_I4_J,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_717_UN__ball__bex__simps_I4_J,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_718_UN__ball__bex__simps_I4_J,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X3 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_719_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ X3 )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_720_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_tm,P2: tm > $o] :
( ( ? [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ A2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: set_tm] :
( ( member_set_tm @ X3 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ X3 )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_721_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ X3 )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_722_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_tm,P2: tm > $o] :
( ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( comple2138885804642794802set_tm @ A2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: set_tm] :
( ( member_set_tm @ X3 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ X3 )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_723_UnionI,axiom,
! [X5: set_list_fm,C2: set_set_list_fm,A2: list_fm] :
( ( member_set_list_fm @ X5 @ C2 )
=> ( ( member_list_fm2 @ A2 @ X5 )
=> ( member_list_fm2 @ A2 @ ( comple8784269564784259782ist_fm @ C2 ) ) ) ) ).
% UnionI
thf(fact_724_UnionI,axiom,
! [X5: set_fm,C2: set_set_fm,A2: fm] :
( ( member_set_fm @ X5 @ C2 )
=> ( ( member_fm2 @ A2 @ X5 )
=> ( member_fm2 @ A2 @ ( comple2134933779557159616set_fm @ C2 ) ) ) ) ).
% UnionI
thf(fact_725_UnionI,axiom,
! [X5: set_o,C2: set_set_o,A2: $o] :
( ( member_set_o @ X5 @ C2 )
=> ( ( member_o2 @ A2 @ X5 )
=> ( member_o2 @ A2 @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_726_UnionI,axiom,
! [X5: set_nat,C2: set_set_nat,A2: nat] :
( ( member_set_nat2 @ X5 @ C2 )
=> ( ( member_nat2 @ A2 @ X5 )
=> ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_727_UnionI,axiom,
! [X5: set_tm,C2: set_set_tm,A2: tm] :
( ( member_set_tm @ X5 @ C2 )
=> ( ( member_tm2 @ A2 @ X5 )
=> ( member_tm2 @ A2 @ ( comple2138885804642794802set_tm @ C2 ) ) ) ) ).
% UnionI
thf(fact_728_Union__iff,axiom,
! [A2: list_fm,C2: set_set_list_fm] :
( ( member_list_fm2 @ A2 @ ( comple8784269564784259782ist_fm @ C2 ) )
= ( ? [X3: set_list_fm] :
( ( member_set_list_fm @ X3 @ C2 )
& ( member_list_fm2 @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_729_Union__iff,axiom,
! [A2: fm,C2: set_set_fm] :
( ( member_fm2 @ A2 @ ( comple2134933779557159616set_fm @ C2 ) )
= ( ? [X3: set_fm] :
( ( member_set_fm @ X3 @ C2 )
& ( member_fm2 @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_730_Union__iff,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o2 @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X3: set_o] :
( ( member_set_o @ X3 @ C2 )
& ( member_o2 @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_731_Union__iff,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X3: set_nat] :
( ( member_set_nat2 @ X3 @ C2 )
& ( member_nat2 @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_732_Union__iff,axiom,
! [A2: tm,C2: set_set_tm] :
( ( member_tm2 @ A2 @ ( comple2138885804642794802set_tm @ C2 ) )
= ( ? [X3: set_tm] :
( ( member_set_tm @ X3 @ C2 )
& ( member_tm2 @ A2 @ X3 ) ) ) ) ).
% Union_iff
thf(fact_733_set__concat,axiom,
! [Xs: list_list_list_fm] :
( ( set_list_fm2 @ ( concat_list_fm @ Xs ) )
= ( comple8784269564784259782ist_fm @ ( image_3687226712311829663ist_fm @ set_list_fm2 @ ( set_list_list_fm2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_734_set__concat,axiom,
! [Xs: list_list_fm] :
( ( set_fm2 @ ( concat_fm @ Xs ) )
= ( comple2134933779557159616set_fm @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_735_set__concat,axiom,
! [Xs: list_list_set_nat] :
( ( set_set_nat2 @ ( concat_set_nat @ Xs ) )
= ( comple548664676211718543et_nat @ ( image_8726355809080528601et_nat @ set_set_nat2 @ ( set_list_set_nat2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_736_set__concat,axiom,
! [Xs: list_list_nat] :
( ( set_nat2 @ ( concat_nat @ Xs ) )
= ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_737_set__concat,axiom,
! [Xs: list_list_tm] :
( ( set_tm2 @ ( concat_tm @ Xs ) )
= ( comple2138885804642794802set_tm @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ Xs ) ) ) ) ).
% set_concat
thf(fact_738_Sup__set__def,axiom,
( comple2134933779557159616set_fm
= ( ^ [A3: set_set_fm] :
( collect_fm
@ ^ [X3: fm] : ( complete_Sup_Sup_o @ ( image_set_fm_o @ ( member_fm2 @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_739_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o2 @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_740_Sup__set__def,axiom,
( comple8784269564784259782ist_fm
= ( ^ [A3: set_set_list_fm] :
( collect_list_fm
@ ^ [X3: list_fm] : ( complete_Sup_Sup_o @ ( image_set_list_fm_o @ ( member_list_fm2 @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_741_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X3: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat2 @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_742_Sup__set__def,axiom,
( comple2138885804642794802set_tm
= ( ^ [A3: set_set_tm] :
( collect_tm
@ ^ [X3: tm] : ( complete_Sup_Sup_o @ ( image_set_tm_o @ ( member_tm2 @ X3 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_743_list_Odisc__eq__case_I1_J,axiom,
! [List: list_fm] :
( ( List = nil_fm )
= ( case_list_o_fm @ $true
@ ^ [Uu: fm,Uv2: list_fm] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_744_list_Odisc__eq__case_I1_J,axiom,
! [List: list_list_fm] :
( ( List = nil_list_fm )
= ( case_list_o_list_fm @ $true
@ ^ [Uu: list_fm,Uv2: list_list_fm] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_745_list_Odisc__eq__case_I1_J,axiom,
! [List: list_tm] :
( ( List = nil_tm )
= ( case_list_o_tm @ $true
@ ^ [Uu: tm,Uv2: list_tm] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_746_list_Odisc__eq__case_I2_J,axiom,
! [List: list_fm] :
( ( List != nil_fm )
= ( case_list_o_fm @ $false
@ ^ [Uu: fm,Uv2: list_fm] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_747_list_Odisc__eq__case_I2_J,axiom,
! [List: list_list_fm] :
( ( List != nil_list_fm )
= ( case_list_o_list_fm @ $false
@ ^ [Uu: list_fm,Uv2: list_list_fm] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_748_list_Odisc__eq__case_I2_J,axiom,
! [List: list_tm] :
( ( List != nil_tm )
= ( case_list_o_tm @ $false
@ ^ [Uu: tm,Uv2: list_tm] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_749_parts__in__children,axiom,
! [P: fm,Z: list_fm,Z4: list_fm,A2: list_tm,R: rule] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z ) ) )
=> ? [B3: list_tm,Xs4: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B3 ) )
& ( member_list_fm2 @ Xs4 @ ( set_list_fm2 @ ( parts @ B3 @ R @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs4 ) @ ( set_fm2 @ Z4 ) ) ) ) ) ).
% parts_in_children
thf(fact_750_paramsts__subset,axiom,
! [A2: list_tm,B2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B2 ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A2 ) @ ( paramsts @ B2 ) ) ) ).
% paramsts_subset
thf(fact_751_Inf_OINF__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > set_nat,D: fm > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_fm_set_nat @ C2 @ A2 ) )
= ( Inf @ ( image_fm_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_752_Inf_OINF__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > set_tm,D: fm > set_tm,Inf: set_set_tm > set_tm] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_fm_set_tm @ C2 @ A2 ) )
= ( Inf @ ( image_fm_set_tm @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_753_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_754_Inf_OINF__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > set_nat,D: tm > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_tm_set_nat @ C2 @ A2 ) )
= ( Inf @ ( image_tm_set_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_755_Inf_OINF__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > set_tm,D: tm > set_tm,Inf: set_set_tm > set_tm] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_tm_set_tm @ C2 @ A2 ) )
= ( Inf @ ( image_tm_set_tm @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_756_Sup_OSUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > set_nat,D: fm > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_fm_set_nat @ C2 @ A2 ) )
= ( Sup @ ( image_fm_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_757_Sup_OSUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > set_tm,D: fm > set_tm,Sup: set_set_tm > set_tm] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_fm_set_tm @ C2 @ A2 ) )
= ( Sup @ ( image_fm_set_tm @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_758_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_759_Sup_OSUP__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > set_nat,D: tm > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_tm_set_nat @ C2 @ A2 ) )
= ( Sup @ ( image_tm_set_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_760_Sup_OSUP__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > set_tm,D: tm > set_tm,Sup: set_set_tm > set_tm] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_tm_set_tm @ C2 @ A2 ) )
= ( Sup @ ( image_tm_set_tm @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_761_UnionE,axiom,
! [A2: list_fm,C2: set_set_list_fm] :
( ( member_list_fm2 @ A2 @ ( comple8784269564784259782ist_fm @ C2 ) )
=> ~ ! [X6: set_list_fm] :
( ( member_list_fm2 @ A2 @ X6 )
=> ~ ( member_set_list_fm @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_762_UnionE,axiom,
! [A2: fm,C2: set_set_fm] :
( ( member_fm2 @ A2 @ ( comple2134933779557159616set_fm @ C2 ) )
=> ~ ! [X6: set_fm] :
( ( member_fm2 @ A2 @ X6 )
=> ~ ( member_set_fm @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_763_UnionE,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o2 @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X6: set_o] :
( ( member_o2 @ A2 @ X6 )
=> ~ ( member_set_o @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_764_UnionE,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat2 @ A2 @ X6 )
=> ~ ( member_set_nat2 @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_765_UnionE,axiom,
! [A2: tm,C2: set_set_tm] :
( ( member_tm2 @ A2 @ ( comple2138885804642794802set_tm @ C2 ) )
=> ~ ! [X6: set_tm] :
( ( member_tm2 @ A2 @ X6 )
=> ~ ( member_set_tm @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_766_subset__subseqs,axiom,
! [X5: set_list_fm,Xs: list_list_fm] :
( ( ord_le7838213414353715577ist_fm @ X5 @ ( set_list_fm2 @ Xs ) )
=> ( member_set_list_fm @ X5 @ ( image_3687226712311829663ist_fm @ set_list_fm2 @ ( set_list_list_fm2 @ ( subseqs_list_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_767_subset__subseqs,axiom,
! [X5: set_set_nat,Xs: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_set_nat @ X5 @ ( image_8726355809080528601et_nat @ set_set_nat2 @ ( set_list_set_nat2 @ ( subseqs_set_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_768_subset__subseqs,axiom,
! [X5: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_769_subset__subseqs,axiom,
! [X5: set_tm,Xs: list_tm] :
( ( ord_less_eq_set_tm @ X5 @ ( set_tm2 @ Xs ) )
=> ( member_set_tm @ X5 @ ( image_list_tm_set_tm @ set_tm2 @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_770_subset__subseqs,axiom,
! [X5: set_fm,Xs: list_fm] :
( ( ord_less_eq_set_fm @ X5 @ ( set_fm2 @ Xs ) )
=> ( member_set_fm @ X5 @ ( image_list_fm_set_fm @ set_fm2 @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_771_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_772_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_773_subtermFm__subset__params,axiom,
! [P: fm,A2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P ) ) @ ( set_tm2 @ A2 ) )
=> ( ord_less_eq_set_nat @ ( params @ P ) @ ( paramsts @ A2 ) ) ) ).
% subtermFm_subset_params
thf(fact_774_Sup__eqI,axiom,
! [A2: set_set_fm,X2: set_fm] :
( ! [Y3: set_fm] :
( ( member_set_fm @ Y3 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ X2 ) )
=> ( ! [Y3: set_fm] :
( ! [Z5: set_fm] :
( ( member_set_fm @ Z5 @ A2 )
=> ( ord_less_eq_set_fm @ Z5 @ Y3 ) )
=> ( ord_less_eq_set_fm @ X2 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_775_Sup__eqI,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat2 @ Y3 @ A2 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ! [Y3: set_nat] :
( ! [Z5: set_nat] :
( ( member_set_nat2 @ Z5 @ A2 )
=> ( ord_less_eq_set_nat @ Z5 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_776_Sup__eqI,axiom,
! [A2: set_o,X2: $o] :
( ! [Y3: $o] :
( ( member_o2 @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X2 ) )
=> ( ! [Y3: $o] :
( ! [Z5: $o] :
( ( member_o2 @ Z5 @ A2 )
=> ( ord_less_eq_o @ Z5 @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_777_Sup__eqI,axiom,
! [A2: set_set_tm,X2: set_tm] :
( ! [Y3: set_tm] :
( ( member_set_tm @ Y3 @ A2 )
=> ( ord_less_eq_set_tm @ Y3 @ X2 ) )
=> ( ! [Y3: set_tm] :
( ! [Z5: set_tm] :
( ( member_set_tm @ Z5 @ A2 )
=> ( ord_less_eq_set_tm @ Z5 @ Y3 ) )
=> ( ord_less_eq_set_tm @ X2 @ Y3 ) )
=> ( ( comple2138885804642794802set_tm @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_778_Sup__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ! [A4: set_fm] :
( ( member_set_fm @ A4 @ A2 )
=> ? [X: set_fm] :
( ( member_set_fm @ X @ B2 )
& ( ord_less_eq_set_fm @ A4 @ X ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Sup_mono
thf(fact_779_Sup__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat2 @ A4 @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat2 @ X @ B2 )
& ( ord_less_eq_set_nat @ A4 @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_mono
thf(fact_780_Sup__mono,axiom,
! [A2: set_o,B2: set_o] :
( ! [A4: $o] :
( ( member_o2 @ A4 @ A2 )
=> ? [X: $o] :
( ( member_o2 @ X @ B2 )
& ( ord_less_eq_o @ A4 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_mono
thf(fact_781_Sup__mono,axiom,
! [A2: set_set_tm,B2: set_set_tm] :
( ! [A4: set_tm] :
( ( member_set_tm @ A4 @ A2 )
=> ? [X: set_tm] :
( ( member_set_tm @ X @ B2 )
& ( ord_less_eq_set_tm @ A4 @ X ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ ( comple2138885804642794802set_tm @ B2 ) ) ) ).
% Sup_mono
thf(fact_782_Sup__least,axiom,
! [A2: set_set_fm,Z: set_fm] :
( ! [X4: set_fm] :
( ( member_set_fm @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ X4 @ Z ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_783_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ X4 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_784_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_o @ X4 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_785_Sup__least,axiom,
! [A2: set_set_tm,Z: set_tm] :
( ! [X4: set_tm] :
( ( member_set_tm @ X4 @ A2 )
=> ( ord_less_eq_set_tm @ X4 @ Z ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_786_Sup__upper,axiom,
! [X2: set_fm,A2: set_set_fm] :
( ( member_set_fm @ X2 @ A2 )
=> ( ord_less_eq_set_fm @ X2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Sup_upper
thf(fact_787_Sup__upper,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat2 @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_788_Sup__upper,axiom,
! [X2: $o,A2: set_o] :
( ( member_o2 @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_789_Sup__upper,axiom,
! [X2: set_tm,A2: set_set_tm] :
( ( member_set_tm @ X2 @ A2 )
=> ( ord_less_eq_set_tm @ X2 @ ( comple2138885804642794802set_tm @ A2 ) ) ) ).
% Sup_upper
thf(fact_790_Sup__le__iff,axiom,
! [A2: set_set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ B )
= ( ! [X3: set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_791_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_792_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X3: $o] :
( ( member_o2 @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_793_Sup__le__iff,axiom,
! [A2: set_set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ B )
= ( ! [X3: set_tm] :
( ( member_set_tm @ X3 @ A2 )
=> ( ord_less_eq_set_tm @ X3 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_794_Sup__upper2,axiom,
! [U: set_fm,A2: set_set_fm,V: set_fm] :
( ( member_set_fm @ U @ A2 )
=> ( ( ord_less_eq_set_fm @ V @ U )
=> ( ord_less_eq_set_fm @ V @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_795_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat2 @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_796_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o2 @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_797_Sup__upper2,axiom,
! [U: set_tm,A2: set_set_tm,V: set_tm] :
( ( member_set_tm @ U @ A2 )
=> ( ( ord_less_eq_set_tm @ V @ U )
=> ( ord_less_eq_set_tm @ V @ ( comple2138885804642794802set_tm @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_798_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > $o,D: fm > $o] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ C2 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_799_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > $o,D: nat > $o] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_800_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_801_SUP__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > $o,D: tm > $o] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_tm_o @ C2 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_tm_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_802_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > nat,D: fm > nat] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_fm_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_fm_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_803_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D: nat > nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_804_SUP__cong,axiom,
! [A2: set_o,B2: set_o,C2: $o > nat,D: $o > nat] :
( ( A2 = B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_805_SUP__cong,axiom,
! [A2: set_tm,B2: set_tm,C2: tm > nat,D: tm > nat] :
( ( A2 = B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_tm_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_tm_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_806_SUP__cong,axiom,
! [A2: set_fm,B2: set_fm,C2: fm > set_nat,D: fm > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_807_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A2 = B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_808_Union__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Union_mono
thf(fact_809_Union__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_mono
thf(fact_810_Union__mono,axiom,
! [A2: set_set_tm,B2: set_set_tm] :
( ( ord_le5601931644483074373set_tm @ A2 @ B2 )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ ( comple2138885804642794802set_tm @ B2 ) ) ) ).
% Union_mono
thf(fact_811_Union__least,axiom,
! [A2: set_set_fm,C2: set_fm] :
( ! [X6: set_fm] :
( ( member_set_fm @ X6 @ A2 )
=> ( ord_less_eq_set_fm @ X6 @ C2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_812_Union__least,axiom,
! [A2: set_set_nat,C2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat2 @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ X6 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_813_Union__least,axiom,
! [A2: set_set_tm,C2: set_tm] :
( ! [X6: set_tm] :
( ( member_set_tm @ X6 @ A2 )
=> ( ord_less_eq_set_tm @ X6 @ C2 ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_814_Union__upper,axiom,
! [B2: set_fm,A2: set_set_fm] :
( ( member_set_fm @ B2 @ A2 )
=> ( ord_less_eq_set_fm @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Union_upper
thf(fact_815_Union__upper,axiom,
! [B2: set_nat,A2: set_set_nat] :
( ( member_set_nat2 @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_816_Union__upper,axiom,
! [B2: set_tm,A2: set_set_tm] :
( ( member_set_tm @ B2 @ A2 )
=> ( ord_less_eq_set_tm @ B2 @ ( comple2138885804642794802set_tm @ A2 ) ) ) ).
% Union_upper
thf(fact_817_Union__subsetI,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ! [X4: set_fm] :
( ( member_set_fm @ X4 @ A2 )
=> ? [Y4: set_fm] :
( ( member_set_fm @ Y4 @ B2 )
& ( ord_less_eq_set_fm @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Union_subsetI
thf(fact_818_Union__subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ A2 )
=> ? [Y4: set_nat] :
( ( member_set_nat2 @ Y4 @ B2 )
& ( ord_less_eq_set_nat @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_subsetI
thf(fact_819_Union__subsetI,axiom,
! [A2: set_set_tm,B2: set_set_tm] :
( ! [X4: set_tm] :
( ( member_set_tm @ X4 @ A2 )
=> ? [Y4: set_tm] :
( ( member_set_tm @ Y4 @ B2 )
& ( ord_less_eq_set_tm @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ ( comple2138885804642794802set_tm @ B2 ) ) ) ).
% Union_subsetI
thf(fact_820_SUP__commute,axiom,
! [F: fm > fm > set_nat,B2: set_fm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [I: fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [J: fm] :
( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [I: fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_821_SUP__commute,axiom,
! [F: fm > tm > set_nat,B2: set_tm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [I: fm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [J: tm] :
( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [I: fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_822_SUP__commute,axiom,
! [F: tm > fm > set_nat,B2: set_fm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [I: tm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [J: fm] :
( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [I: tm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_823_SUP__commute,axiom,
! [F: tm > tm > set_nat,B2: set_tm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [I: tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [J: tm] :
( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [I: tm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_824_SUP__commute,axiom,
! [F: tm > tm > set_tm,B2: set_tm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [I: tm] : ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [J: tm] :
( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [I: tm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_825_SUP__commute,axiom,
! [F: tm > fm > set_tm,B2: set_fm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [I: tm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [J: fm] :
( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [I: tm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_826_SUP__commute,axiom,
! [F: fm > tm > set_tm,B2: set_tm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [I: fm] : ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [J: tm] :
( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [I: fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_827_SUP__commute,axiom,
! [F: fm > fm > set_tm,B2: set_fm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [I: fm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [J: fm] :
( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [I: fm] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_828_image__Union,axiom,
! [F: fm > set_nat,S: set_set_fm] :
( ( image_fm_set_nat @ F @ ( comple2134933779557159616set_fm @ S ) )
= ( comple548664676211718543et_nat @ ( image_1496149073759408202et_nat @ ( image_fm_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_829_image__Union,axiom,
! [F: fm > set_tm,S: set_set_fm] :
( ( image_fm_set_tm @ F @ ( comple2134933779557159616set_fm @ S ) )
= ( comple4084446694820577554set_tm @ ( image_1809285061380348183set_tm @ ( image_fm_set_tm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_830_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_831_image__Union,axiom,
! [F: nat > tm,S: set_set_nat] :
( ( image_nat_tm @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple2138885804642794802set_tm @ ( image_set_nat_set_tm @ ( image_nat_tm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_832_image__Union,axiom,
! [F: tm > set_nat,S: set_set_tm] :
( ( image_tm_set_nat @ F @ ( comple2138885804642794802set_tm @ S ) )
= ( comple548664676211718543et_nat @ ( image_5490068892692554428et_nat @ ( image_tm_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_833_image__Union,axiom,
! [F: tm > set_tm,S: set_set_tm] :
( ( image_tm_set_tm @ F @ ( comple2138885804642794802set_tm @ S ) )
= ( comple4084446694820577554set_tm @ ( image_9072780396932801317set_tm @ ( image_tm_set_tm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_834_image__Union,axiom,
! [F: tm > nat,S: set_set_tm] :
( ( image_tm_nat @ F @ ( comple2138885804642794802set_tm @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_tm_set_nat @ ( image_tm_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_835_image__Union,axiom,
! [F: tm > tm,S: set_set_tm] :
( ( image_tm_tm @ F @ ( comple2138885804642794802set_tm @ S ) )
= ( comple2138885804642794802set_tm @ ( image_set_tm_set_tm @ ( image_tm_tm @ F ) @ S ) ) ) ).
% image_Union
thf(fact_836_UN__UN__flatten,axiom,
! [C2: fm > set_nat,B2: fm > set_fm,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_837_UN__UN__flatten,axiom,
! [C2: fm > set_nat,B2: tm > set_fm,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_838_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B2: fm > set_nat,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_839_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B2: tm > set_nat,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_840_UN__UN__flatten,axiom,
! [C2: tm > set_nat,B2: tm > set_tm,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_841_UN__UN__flatten,axiom,
! [C2: tm > set_nat,B2: fm > set_tm,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_842_UN__UN__flatten,axiom,
! [C2: fm > set_tm,B2: tm > set_fm,A2: set_tm] :
( ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [Y2: tm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_843_UN__UN__flatten,axiom,
! [C2: fm > set_tm,B2: fm > set_fm,A2: set_fm] :
( ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [Y2: fm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_844_UN__UN__flatten,axiom,
! [C2: nat > set_tm,B2: fm > set_nat,A2: set_fm] :
( ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [Y2: fm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_845_UN__UN__flatten,axiom,
! [C2: nat > set_tm,B2: tm > set_nat,A2: set_tm] :
( ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [Y2: tm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ Y2 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_846_UN__E,axiom,
! [B: fm,B2: fm > set_fm,A2: set_fm] :
( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ~ ( member_fm2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_847_UN__E,axiom,
! [B: fm,B2: nat > set_fm,A2: set_nat] :
( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ~ ( member_fm2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_848_UN__E,axiom,
! [B: fm,B2: $o > set_fm,A2: set_o] :
( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ~ ( member_fm2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_849_UN__E,axiom,
! [B: fm,B2: tm > set_fm,A2: set_tm] :
( ( member_fm2 @ B @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) )
=> ~ ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ~ ( member_fm2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_850_UN__E,axiom,
! [B: $o,B2: fm > set_o,A2: set_fm] :
( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_fm_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ~ ( member_o2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_851_UN__E,axiom,
! [B: $o,B2: nat > set_o,A2: set_nat] :
( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ~ ( member_o2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_852_UN__E,axiom,
! [B: $o,B2: $o > set_o,A2: set_o] :
( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ~ ( member_o2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_853_UN__E,axiom,
! [B: $o,B2: tm > set_o,A2: set_tm] :
( ( member_o2 @ B @ ( comple90263536869209701_set_o @ ( image_tm_set_o @ B2 @ A2 ) ) )
=> ~ ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ~ ( member_o2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_854_UN__E,axiom,
! [B: nat,B2: fm > set_nat,A2: set_fm] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ~ ( member_nat2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_855_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ~ ( member_nat2 @ B @ ( B2 @ X4 ) ) ) ) ).
% UN_E
thf(fact_856_UN__extend__simps_I8_J,axiom,
! [B2: fm > set_nat,A2: set_set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_set_fm_set_nat
@ ^ [Y2: set_fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_857_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y2: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_858_UN__extend__simps_I8_J,axiom,
! [B2: tm > set_nat,A2: set_set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_set_tm_set_nat
@ ^ [Y2: set_tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ Y2 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ ( comple2138885804642794802set_tm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_859_UN__extend__simps_I8_J,axiom,
! [B2: fm > set_tm,A2: set_set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_set_fm_set_tm
@ ^ [Y2: set_fm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_860_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_tm,A2: set_set_nat] :
( ( comple2138885804642794802set_tm
@ ( image_set_nat_set_tm
@ ^ [Y2: set_nat] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ B2 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_861_UN__extend__simps_I8_J,axiom,
! [B2: tm > set_tm,A2: set_set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_set_tm_set_tm
@ ^ [Y2: set_tm] : ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ Y2 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ ( comple2138885804642794802set_tm @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_862_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_nat,B2: fm > set_fm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_863_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_nat,B2: tm > set_fm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_864_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B2: fm > set_nat,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_865_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B2: tm > set_nat,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_866_UN__extend__simps_I9_J,axiom,
! [C2: tm > set_nat,B2: fm > set_tm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_867_UN__extend__simps_I9_J,axiom,
! [C2: tm > set_nat,B2: tm > set_tm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_868_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_tm,B2: tm > set_fm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X3: tm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_869_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_tm,B2: fm > set_fm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X3: fm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_870_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_tm,B2: tm > set_nat,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X3: tm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_871_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_tm,B2: fm > set_nat,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X3: fm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ X3 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_872_SUP__eq,axiom,
! [A2: set_fm,B2: set_fm,F: fm > $o,G: fm > $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm2 @ J2 @ B2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_873_SUP__eq,axiom,
! [A2: set_fm,B2: set_nat,F: fm > $o,G: nat > $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat2 @ J2 @ B2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_874_SUP__eq,axiom,
! [A2: set_fm,B2: set_o,F: fm > $o,G: $o > $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ? [X: $o] :
( ( member_o2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: $o] :
( ( member_o2 @ J2 @ B2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_875_SUP__eq,axiom,
! [A2: set_fm,B2: set_tm,F: fm > $o,G: tm > $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: tm] :
( ( member_tm2 @ J2 @ B2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_tm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_876_SUP__eq,axiom,
! [A2: set_nat,B2: set_fm,F: nat > $o,G: fm > $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm2 @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_877_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat2 @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_878_SUP__eq,axiom,
! [A2: set_nat,B2: set_o,F: nat > $o,G: $o > $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ? [X: $o] :
( ( member_o2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: $o] :
( ( member_o2 @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_879_SUP__eq,axiom,
! [A2: set_nat,B2: set_tm,F: nat > $o,G: tm > $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: tm] :
( ( member_tm2 @ J2 @ B2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_tm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_880_SUP__eq,axiom,
! [A2: set_o,B2: set_fm,F: $o > $o,G: fm > $o] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: fm] :
( ( member_fm2 @ J2 @ B2 )
=> ? [X: $o] :
( ( member_o2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_881_SUP__eq,axiom,
! [A2: set_o,B2: set_nat,F: $o > $o,G: nat > $o] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ? [X: nat] :
( ( member_nat2 @ X @ B2 )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat2 @ J2 @ B2 )
=> ? [X: $o] :
( ( member_o2 @ X @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_882_Sup__subset__mono,axiom,
! [A2: set_set_fm,B2: set_set_fm] :
( ( ord_le5844446314808584147set_fm @ A2 @ B2 )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ ( comple2134933779557159616set_fm @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_883_Sup__subset__mono,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_884_Sup__subset__mono,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_885_Sup__subset__mono,axiom,
! [A2: set_set_tm,B2: set_set_tm] :
( ( ord_le5601931644483074373set_tm @ A2 @ B2 )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ ( comple2138885804642794802set_tm @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_886_SUP__upper2,axiom,
! [I3: fm,A2: set_fm,U: $o,F: fm > $o] :
( ( member_fm2 @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_887_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: $o,F: nat > $o] :
( ( member_nat2 @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_888_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: $o,F: $o > $o] :
( ( member_o2 @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_889_SUP__upper2,axiom,
! [I3: tm,A2: set_tm,U: $o,F: tm > $o] :
( ( member_tm2 @ I3 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I3 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_890_SUP__upper2,axiom,
! [I3: fm,A2: set_fm,U: set_fm,F: fm > set_fm] :
( ( member_fm2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_891_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: set_fm,F: nat > set_fm] :
( ( member_nat2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_892_SUP__upper2,axiom,
! [I3: $o,A2: set_o,U: set_fm,F: $o > set_fm] :
( ( member_o2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_893_SUP__upper2,axiom,
! [I3: tm,A2: set_tm,U: set_fm,F: tm > set_fm] :
( ( member_tm2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_894_SUP__upper2,axiom,
! [I3: fm,A2: set_fm,U: set_nat,F: fm > set_nat] :
( ( member_fm2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_895_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I3 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I3 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_896_SUP__le__iff,axiom,
! [F: fm > set_nat,A2: set_fm,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ U )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_897_SUP__le__iff,axiom,
! [F: tm > set_nat,A2: set_tm,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ A2 ) ) @ U )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_898_SUP__le__iff,axiom,
! [F: tm > set_tm,A2: set_tm,U: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ F @ A2 ) ) @ U )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_899_SUP__le__iff,axiom,
! [F: fm > set_tm,A2: set_fm,U: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ F @ A2 ) ) @ U )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_900_SUP__upper,axiom,
! [I3: fm,A2: set_fm,F: fm > $o] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_901_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > $o] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_902_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > $o] :
( ( member_o2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_903_SUP__upper,axiom,
! [I3: tm,A2: set_tm,F: tm > $o] :
( ( member_tm2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_904_SUP__upper,axiom,
! [I3: fm,A2: set_fm,F: fm > set_fm] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_905_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > set_fm] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_906_SUP__upper,axiom,
! [I3: $o,A2: set_o,F: $o > set_fm] :
( ( member_o2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_907_SUP__upper,axiom,
! [I3: tm,A2: set_tm,F: tm > set_fm] :
( ( member_tm2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_908_SUP__upper,axiom,
! [I3: fm,A2: set_fm,F: fm > set_nat] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_909_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_910_SUP__mono_H,axiom,
! [F: fm > set_nat,G: fm > set_nat,A2: set_fm] :
( ! [X4: fm] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_911_SUP__mono_H,axiom,
! [F: tm > set_nat,G: tm > set_nat,A2: set_tm] :
( ! [X4: tm] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_912_SUP__mono_H,axiom,
! [F: tm > set_tm,G: tm > set_tm,A2: set_tm] :
( ! [X4: tm] : ( ord_less_eq_set_tm @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_913_SUP__mono_H,axiom,
! [F: fm > set_tm,G: fm > set_tm,A2: set_fm] :
( ! [X4: fm] : ( ord_less_eq_set_tm @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_914_SUP__least,axiom,
! [A2: set_fm,F: fm > $o,U: $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_915_SUP__least,axiom,
! [A2: set_nat,F: nat > $o,U: $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_916_SUP__least,axiom,
! [A2: set_o,F: $o > $o,U: $o] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_917_SUP__least,axiom,
! [A2: set_tm,F: tm > $o,U: $o] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_918_SUP__least,axiom,
! [A2: set_fm,F: fm > set_fm,U: set_fm] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_919_SUP__least,axiom,
! [A2: set_nat,F: nat > set_fm,U: set_fm] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_920_SUP__least,axiom,
! [A2: set_o,F: $o > set_fm,U: set_fm] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_921_SUP__least,axiom,
! [A2: set_tm,F: tm > set_fm,U: set_fm] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_922_SUP__least,axiom,
! [A2: set_fm,F: fm > set_nat,U: set_nat] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_923_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_924_SUP__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_nat,G: fm > set_nat] :
( ! [N: fm] :
( ( member_fm2 @ N @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_925_SUP__mono,axiom,
! [A2: set_fm,B2: set_tm,F: fm > set_nat,G: tm > set_nat] :
( ! [N: fm] :
( ( member_fm2 @ N @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_926_SUP__mono,axiom,
! [A2: set_nat,B2: set_fm,F: nat > set_nat,G: fm > set_nat] :
( ! [N: nat] :
( ( member_nat2 @ N @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_927_SUP__mono,axiom,
! [A2: set_nat,B2: set_tm,F: nat > set_nat,G: tm > set_nat] :
( ! [N: nat] :
( ( member_nat2 @ N @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_928_SUP__mono,axiom,
! [A2: set_o,B2: set_fm,F: $o > set_nat,G: fm > set_nat] :
( ! [N: $o] :
( ( member_o2 @ N @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_929_SUP__mono,axiom,
! [A2: set_o,B2: set_tm,F: $o > set_nat,G: tm > set_nat] :
( ! [N: $o] :
( ( member_o2 @ N @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_930_SUP__mono,axiom,
! [A2: set_tm,B2: set_fm,F: tm > set_nat,G: fm > set_nat] :
( ! [N: tm] :
( ( member_tm2 @ N @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_931_SUP__mono,axiom,
! [A2: set_tm,B2: set_tm,F: tm > set_nat,G: tm > set_nat] :
( ! [N: tm] :
( ( member_tm2 @ N @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_set_nat @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_932_SUP__mono,axiom,
! [A2: set_fm,B2: set_tm,F: fm > set_tm,G: tm > set_tm] :
( ! [N: fm] :
( ( member_fm2 @ N @ A2 )
=> ? [X: tm] :
( ( member_tm2 @ X @ B2 )
& ( ord_less_eq_set_tm @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_933_SUP__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_tm,G: fm > set_tm] :
( ! [N: fm] :
( ( member_fm2 @ N @ A2 )
=> ? [X: fm] :
( ( member_fm2 @ X @ B2 )
& ( ord_less_eq_set_tm @ ( F @ N ) @ ( G @ X ) ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_934_SUP__eqI,axiom,
! [A2: set_fm,F: fm > $o,X2: $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I4: fm] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_935_SUP__eqI,axiom,
! [A2: set_nat,F: nat > $o,X2: $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_936_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X2: $o] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I4: $o] :
( ( member_o2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_937_SUP__eqI,axiom,
! [A2: set_tm,F: tm > $o,X2: $o] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: $o] :
( ! [I4: tm] :
( ( member_tm2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_938_SUP__eqI,axiom,
! [A2: set_fm,F: fm > set_fm,X2: set_fm] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_fm] :
( ! [I4: fm] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X2 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_939_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_fm,X2: set_fm] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_fm] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X2 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_940_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_fm,X2: set_fm] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_fm] :
( ! [I4: $o] :
( ( member_o2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X2 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_941_SUP__eqI,axiom,
! [A2: set_tm,F: tm > set_fm,X2: set_fm] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_fm] :
( ! [I4: tm] :
( ( member_tm2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X2 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_942_SUP__eqI,axiom,
! [A2: set_fm,F: fm > set_nat,X2: set_nat] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_nat] :
( ! [I4: fm] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_943_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X2: set_nat] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X2 ) )
=> ( ! [Y3: set_nat] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_944_image__UN,axiom,
! [F: nat > nat,B2: fm > set_nat,A2: set_fm] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( image_nat_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_945_image__UN,axiom,
! [F: nat > nat,B2: tm > set_nat,A2: set_tm] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( image_nat_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_946_image__UN,axiom,
! [F: nat > tm,B2: fm > set_nat,A2: set_fm] :
( ( image_nat_tm @ F @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X3: fm] : ( image_nat_tm @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_947_image__UN,axiom,
! [F: nat > tm,B2: tm > set_nat,A2: set_tm] :
( ( image_nat_tm @ F @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X3: tm] : ( image_nat_tm @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_948_image__UN,axiom,
! [F: tm > nat,B2: tm > set_tm,A2: set_tm] :
( ( image_tm_nat @ F @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X3: tm] : ( image_tm_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_949_image__UN,axiom,
! [F: tm > nat,B2: fm > set_tm,A2: set_fm] :
( ( image_tm_nat @ F @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X3: fm] : ( image_tm_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_950_image__UN,axiom,
! [F: tm > tm,B2: tm > set_tm,A2: set_tm] :
( ( image_tm_tm @ F @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X3: tm] : ( image_tm_tm @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_951_image__UN,axiom,
! [F: tm > tm,B2: fm > set_tm,A2: set_fm] :
( ( image_tm_tm @ F @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X3: fm] : ( image_tm_tm @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_952_image__UN,axiom,
! [F: tm > set_nat,B2: tm > set_tm,A2: set_tm] :
( ( image_tm_set_nat @ F @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_tm_set_set_nat
@ ^ [X3: tm] : ( image_tm_set_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_953_image__UN,axiom,
! [F: tm > set_nat,B2: fm > set_tm,A2: set_fm] :
( ( image_tm_set_nat @ F @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_fm_set_set_nat
@ ^ [X3: fm] : ( image_tm_set_nat @ F @ ( B2 @ X3 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_954_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_955_UN__extend__simps_I10_J,axiom,
! [B2: fm > set_nat,F: fm > fm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [A5: fm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ ( image_fm_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_956_UN__extend__simps_I10_J,axiom,
! [B2: tm > set_nat,F: fm > tm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [A5: fm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ ( image_fm_tm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_957_UN__extend__simps_I10_J,axiom,
! [B2: fm > set_nat,F: tm > fm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [A5: tm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ ( image_tm_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_958_UN__extend__simps_I10_J,axiom,
! [B2: tm > set_nat,F: tm > tm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [A5: tm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ ( image_tm_tm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_959_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_tm,F: nat > nat,A2: set_nat] :
( ( comple2138885804642794802set_tm
@ ( image_nat_set_tm
@ ^ [A5: nat] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_960_UN__extend__simps_I10_J,axiom,
! [B2: tm > set_tm,F: tm > tm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [A5: tm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ ( image_tm_tm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_961_UN__extend__simps_I10_J,axiom,
! [B2: fm > set_tm,F: tm > fm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [A5: tm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ ( image_tm_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_962_UN__extend__simps_I10_J,axiom,
! [B2: tm > set_tm,F: fm > tm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [A5: fm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ ( image_fm_tm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_963_UN__extend__simps_I10_J,axiom,
! [B2: fm > set_tm,F: fm > fm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [A5: fm] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ ( image_fm_fm @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_964_UN__subset__iff,axiom,
! [A2: fm > set_nat,I5: set_fm,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ A2 @ I5 ) ) @ B2 )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_965_UN__subset__iff,axiom,
! [A2: tm > set_nat,I5: set_tm,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ A2 @ I5 ) ) @ B2 )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_966_UN__subset__iff,axiom,
! [A2: tm > set_tm,I5: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ A2 @ I5 ) ) @ B2 )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ I5 )
=> ( ord_less_eq_set_tm @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_967_UN__subset__iff,axiom,
! [A2: fm > set_tm,I5: set_fm,B2: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ A2 @ I5 ) ) @ B2 )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ I5 )
=> ( ord_less_eq_set_tm @ ( A2 @ X3 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_968_UN__upper,axiom,
! [A: fm,A2: set_fm,B2: fm > set_fm] :
( ( member_fm2 @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ A ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_969_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_fm] :
( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ A ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_970_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_fm] :
( ( member_o2 @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ A ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_971_UN__upper,axiom,
! [A: tm,A2: set_tm,B2: tm > set_fm] :
( ( member_tm2 @ A @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ A ) @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_972_UN__upper,axiom,
! [A: fm,A2: set_fm,B2: fm > set_nat] :
( ( member_fm2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_973_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_974_UN__upper,axiom,
! [A: $o,A2: set_o,B2: $o > set_nat] :
( ( member_o2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_975_UN__upper,axiom,
! [A: tm,A2: set_tm,B2: tm > set_nat] :
( ( member_tm2 @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_976_UN__upper,axiom,
! [A: fm,A2: set_fm,B2: fm > set_tm] :
( ( member_fm2 @ A @ A2 )
=> ( ord_less_eq_set_tm @ ( B2 @ A ) @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_977_UN__upper,axiom,
! [A: nat,A2: set_nat,B2: nat > set_tm] :
( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_set_tm @ ( B2 @ A ) @ ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_978_UN__least,axiom,
! [A2: set_fm,B2: fm > set_fm,C2: set_fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_979_UN__least,axiom,
! [A2: set_nat,B2: nat > set_fm,C2: set_fm] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_980_UN__least,axiom,
! [A2: set_o,B2: $o > set_fm,C2: set_fm] :
( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_981_UN__least,axiom,
! [A2: set_tm,B2: tm > set_fm,C2: set_fm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_982_UN__least,axiom,
! [A2: set_fm,B2: fm > set_nat,C2: set_nat] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_983_UN__least,axiom,
! [A2: set_nat,B2: nat > set_nat,C2: set_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_984_UN__least,axiom,
! [A2: set_o,B2: $o > set_nat,C2: set_nat] :
( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_985_UN__least,axiom,
! [A2: set_tm,B2: tm > set_nat,C2: set_nat] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_986_UN__least,axiom,
! [A2: set_fm,B2: fm > set_tm,C2: set_tm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_tm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_987_UN__least,axiom,
! [A2: set_nat,B2: nat > set_tm,C2: set_tm] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_tm @ ( B2 @ X4 ) @ C2 ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ B2 @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_988_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_989_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_fm,G: nat > set_fm] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_990_UN__mono,axiom,
! [A2: set_tm,B2: set_tm,F: tm > set_fm,G: tm > set_fm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_991_UN__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_fm,G: fm > set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_992_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_993_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_994_UN__mono,axiom,
! [A2: set_tm,B2: set_tm,F: tm > set_nat,G: tm > set_nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_995_UN__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_nat,G: fm > set_nat] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_996_UN__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_tm,G: $o > set_tm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_o_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_o_set_tm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_997_UN__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_tm,G: nat > set_tm] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ F @ A2 ) ) @ ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_998_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_999_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1000_SUP__subset__mono,axiom,
! [A2: set_tm,B2: set_tm,F: tm > $o,G: tm > $o] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_tm_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1001_SUP__subset__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > $o,G: fm > $o] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_fm_o @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1002_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_fm,G: $o > set_fm] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1003_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_fm,G: nat > set_fm] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1004_SUP__subset__mono,axiom,
! [A2: set_tm,B2: set_tm,F: tm > set_fm,G: tm > set_fm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1005_SUP__subset__mono,axiom,
! [A2: set_fm,B2: set_fm,F: fm > set_fm,G: fm > set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1006_SUP__subset__mono,axiom,
! [A2: set_o,B2: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1007_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1008_SUP__UNION,axiom,
! [F: nat > $o,G: fm > set_nat,A2: set_fm] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_fm_o
@ ^ [Y2: fm] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1009_SUP__UNION,axiom,
! [F: nat > $o,G: tm > set_nat,A2: set_tm] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_tm_o
@ ^ [Y2: tm] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1010_SUP__UNION,axiom,
! [F: tm > $o,G: tm > set_tm,A2: set_tm] :
( ( complete_Sup_Sup_o @ ( image_tm_o @ F @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_tm_o
@ ^ [Y2: tm] : ( complete_Sup_Sup_o @ ( image_tm_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1011_SUP__UNION,axiom,
! [F: tm > $o,G: fm > set_tm,A2: set_fm] :
( ( complete_Sup_Sup_o @ ( image_tm_o @ F @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_fm_o
@ ^ [Y2: fm] : ( complete_Sup_Sup_o @ ( image_tm_o @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1012_SUP__UNION,axiom,
! [F: fm > set_nat,G: fm > set_fm,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1013_SUP__UNION,axiom,
! [F: fm > set_nat,G: tm > set_fm,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1014_SUP__UNION,axiom,
! [F: nat > set_nat,G: fm > set_nat,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1015_SUP__UNION,axiom,
! [F: nat > set_nat,G: tm > set_nat,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1016_SUP__UNION,axiom,
! [F: tm > set_nat,G: tm > set_tm,A2: set_tm] :
( ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [Y2: tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1017_SUP__UNION,axiom,
! [F: tm > set_nat,G: fm > set_tm,A2: set_fm] :
( ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [Y2: fm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ F @ ( G @ Y2 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1018_transpose_Oelims,axiom,
! [X2: list_list_list_fm,Y: list_list_list_fm] :
( ( ( transpose_list_fm @ X2 )
= Y )
=> ( ( ( X2 = nil_list_list_fm )
=> ( Y != nil_list_list_fm ) )
=> ( ! [Xss3: list_list_list_fm] :
( ( X2
= ( cons_list_list_fm @ nil_list_fm @ Xss3 ) )
=> ( Y
!= ( transpose_list_fm @ Xss3 ) ) )
=> ~ ! [X4: list_fm,Xs4: list_list_fm,Xss3: list_list_list_fm] :
( ( X2
= ( cons_list_list_fm @ ( cons_list_fm @ X4 @ Xs4 ) @ Xss3 ) )
=> ( Y
!= ( cons_list_list_fm
@ ( cons_list_fm @ X4
@ ( concat_list_fm
@ ( map_li4351931137408529412ist_fm
@ ( case_l7658988447939845536ist_fm @ nil_list_fm
@ ^ [H: list_fm,T2: list_list_fm] : ( cons_list_fm @ H @ nil_list_fm ) )
@ Xss3 ) ) )
@ ( transpose_list_fm
@ ( cons_list_list_fm @ Xs4
@ ( concat_list_list_fm
@ ( map_li2158052760755490954ist_fm
@ ( case_l2744553060881268634ist_fm @ nil_list_list_fm
@ ^ [H: list_fm,T2: list_list_fm] : ( cons_list_list_fm @ T2 @ nil_list_list_fm ) )
@ Xss3 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1019_transpose_Oelims,axiom,
! [X2: list_list_fm,Y: list_list_fm] :
( ( ( transpose_fm @ X2 )
= Y )
=> ( ( ( X2 = nil_list_fm )
=> ( Y != nil_list_fm ) )
=> ( ! [Xss3: list_list_fm] :
( ( X2
= ( cons_list_fm @ nil_fm @ Xss3 ) )
=> ( Y
!= ( transpose_fm @ Xss3 ) ) )
=> ~ ! [X4: fm,Xs4: list_fm,Xss3: list_list_fm] :
( ( X2
= ( cons_list_fm @ ( cons_fm @ X4 @ Xs4 ) @ Xss3 ) )
=> ( Y
!= ( cons_list_fm
@ ( cons_fm @ X4
@ ( concat_fm
@ ( map_list_fm_list_fm
@ ( case_list_list_fm_fm @ nil_fm
@ ^ [H: fm,T2: list_fm] : ( cons_fm @ H @ nil_fm ) )
@ Xss3 ) ) )
@ ( transpose_fm
@ ( cons_list_fm @ Xs4
@ ( concat_list_fm
@ ( map_li1351512418201717386ist_fm
@ ( case_l1147175924207067418_fm_fm @ nil_list_fm
@ ^ [H: fm,T2: list_fm] : ( cons_list_fm @ T2 @ nil_list_fm ) )
@ Xss3 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1020_transpose_Oelims,axiom,
! [X2: list_list_tm,Y: list_list_tm] :
( ( ( transpose_tm @ X2 )
= Y )
=> ( ( ( X2 = nil_list_tm )
=> ( Y != nil_list_tm ) )
=> ( ! [Xss3: list_list_tm] :
( ( X2
= ( cons_list_tm @ nil_tm @ Xss3 ) )
=> ( Y
!= ( transpose_tm @ Xss3 ) ) )
=> ~ ! [X4: tm,Xs4: list_tm,Xss3: list_list_tm] :
( ( X2
= ( cons_list_tm @ ( cons_tm @ X4 @ Xs4 ) @ Xss3 ) )
=> ( Y
!= ( cons_list_tm
@ ( cons_tm @ X4
@ ( concat_tm
@ ( map_list_tm_list_tm
@ ( case_list_list_tm_tm @ nil_tm
@ ^ [H: tm,T2: list_tm] : ( cons_tm @ H @ nil_tm ) )
@ Xss3 ) ) )
@ ( transpose_tm
@ ( cons_list_tm @ Xs4
@ ( concat_list_tm
@ ( map_li6264597563971819530ist_tm
@ ( case_l799553655970854810_tm_tm @ nil_list_tm
@ ^ [H: tm,T2: list_tm] : ( cons_list_tm @ T2 @ nil_list_tm ) )
@ Xss3 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1021_transpose_Osimps_I3_J,axiom,
! [X2: list_fm,Xs: list_list_fm,Xss: list_list_list_fm] :
( ( transpose_list_fm @ ( cons_list_list_fm @ ( cons_list_fm @ X2 @ Xs ) @ Xss ) )
= ( cons_list_list_fm
@ ( cons_list_fm @ X2
@ ( concat_list_fm
@ ( map_li4351931137408529412ist_fm
@ ( case_l7658988447939845536ist_fm @ nil_list_fm
@ ^ [H: list_fm,T2: list_list_fm] : ( cons_list_fm @ H @ nil_list_fm ) )
@ Xss ) ) )
@ ( transpose_list_fm
@ ( cons_list_list_fm @ Xs
@ ( concat_list_list_fm
@ ( map_li2158052760755490954ist_fm
@ ( case_l2744553060881268634ist_fm @ nil_list_list_fm
@ ^ [H: list_fm,T2: list_list_fm] : ( cons_list_list_fm @ T2 @ nil_list_list_fm ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1022_transpose_Osimps_I3_J,axiom,
! [X2: fm,Xs: list_fm,Xss: list_list_fm] :
( ( transpose_fm @ ( cons_list_fm @ ( cons_fm @ X2 @ Xs ) @ Xss ) )
= ( cons_list_fm
@ ( cons_fm @ X2
@ ( concat_fm
@ ( map_list_fm_list_fm
@ ( case_list_list_fm_fm @ nil_fm
@ ^ [H: fm,T2: list_fm] : ( cons_fm @ H @ nil_fm ) )
@ Xss ) ) )
@ ( transpose_fm
@ ( cons_list_fm @ Xs
@ ( concat_list_fm
@ ( map_li1351512418201717386ist_fm
@ ( case_l1147175924207067418_fm_fm @ nil_list_fm
@ ^ [H: fm,T2: list_fm] : ( cons_list_fm @ T2 @ nil_list_fm ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1023_transpose_Osimps_I3_J,axiom,
! [X2: tm,Xs: list_tm,Xss: list_list_tm] :
( ( transpose_tm @ ( cons_list_tm @ ( cons_tm @ X2 @ Xs ) @ Xss ) )
= ( cons_list_tm
@ ( cons_tm @ X2
@ ( concat_tm
@ ( map_list_tm_list_tm
@ ( case_list_list_tm_tm @ nil_tm
@ ^ [H: tm,T2: list_tm] : ( cons_tm @ H @ nil_tm ) )
@ Xss ) ) )
@ ( transpose_tm
@ ( cons_list_tm @ Xs
@ ( concat_list_tm
@ ( map_li6264597563971819530ist_tm
@ ( case_l799553655970854810_tm_tm @ nil_list_tm
@ ^ [H: tm,T2: list_tm] : ( cons_list_tm @ T2 @ nil_list_tm ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1024_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_1025_news__paramss,axiom,
( news
= ( ^ [I: nat,Z3: list_fm] :
~ ( member_nat2 @ I @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ Z3 ) ) ) ) ) ) ).
% news_paramss
thf(fact_1026_transpose__aux__filter__tail,axiom,
! [Xss: list_list_list_fm] :
( ( concat_list_list_fm
@ ( map_li2158052760755490954ist_fm
@ ( case_l2744553060881268634ist_fm @ nil_list_list_fm
@ ^ [H: list_fm,T2: list_list_fm] : ( cons_list_list_fm @ T2 @ nil_list_list_fm ) )
@ Xss ) )
= ( map_li4351931137408529412ist_fm @ tl_list_fm
@ ( filter_list_list_fm
@ ^ [Ys3: list_list_fm] : ( Ys3 != nil_list_fm )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_1027_transpose__aux__filter__tail,axiom,
! [Xss: list_list_tm] :
( ( concat_list_tm
@ ( map_li6264597563971819530ist_tm
@ ( case_l799553655970854810_tm_tm @ nil_list_tm
@ ^ [H: tm,T2: list_tm] : ( cons_list_tm @ T2 @ nil_list_tm ) )
@ Xss ) )
= ( map_list_tm_list_tm @ tl_tm
@ ( filter_list_tm
@ ^ [Ys3: list_tm] : ( Ys3 != nil_tm )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_1028_transpose__aux__filter__tail,axiom,
! [Xss: list_list_fm] :
( ( concat_list_fm
@ ( map_li1351512418201717386ist_fm
@ ( case_l1147175924207067418_fm_fm @ nil_list_fm
@ ^ [H: fm,T2: list_fm] : ( cons_list_fm @ T2 @ nil_list_fm ) )
@ Xss ) )
= ( map_list_fm_list_fm @ tl_fm
@ ( filter_list_fm
@ ^ [Ys3: list_fm] : ( Ys3 != nil_fm )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_1029_image__eqI,axiom,
! [B: fm,F: fm > fm,X2: fm,A2: set_fm] :
( ( B
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A2 )
=> ( member_fm2 @ B @ ( image_fm_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1030_image__eqI,axiom,
! [B: nat,F: fm > nat,X2: fm,A2: set_fm] :
( ( B
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A2 )
=> ( member_nat2 @ B @ ( image_fm_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1031_image__eqI,axiom,
! [B: $o,F: fm > $o,X2: fm,A2: set_fm] :
( ( B
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A2 )
=> ( member_o2 @ B @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1032_image__eqI,axiom,
! [B: tm,F: fm > tm,X2: fm,A2: set_fm] :
( ( B
= ( F @ X2 ) )
=> ( ( member_fm2 @ X2 @ A2 )
=> ( member_tm2 @ B @ ( image_fm_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1033_image__eqI,axiom,
! [B: fm,F: nat > fm,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_fm2 @ B @ ( image_nat_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1034_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1035_image__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_o2 @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1036_image__eqI,axiom,
! [B: tm,F: nat > tm,X2: nat,A2: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_tm2 @ B @ ( image_nat_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1037_image__eqI,axiom,
! [B: fm,F: $o > fm,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o2 @ X2 @ A2 )
=> ( member_fm2 @ B @ ( image_o_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1038_image__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,A2: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o2 @ X2 @ A2 )
=> ( member_nat2 @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1039_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1040_subset__antisym,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1041_subset__antisym,axiom,
! [A2: set_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ A2 @ B2 )
=> ( ( ord_less_eq_set_fm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_1042_subsetI,axiom,
! [A2: set_list_fm,B2: set_list_fm] :
( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ A2 )
=> ( member_list_fm2 @ X4 @ B2 ) )
=> ( ord_le7838213414353715577ist_fm @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1043_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X4: $o] :
( ( member_o2 @ X4 @ A2 )
=> ( member_o2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1044_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( member_nat2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1045_subsetI,axiom,
! [A2: set_tm,B2: set_tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ A2 )
=> ( member_tm2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_tm @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1046_subsetI,axiom,
! [A2: set_fm,B2: set_fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ A2 )
=> ( member_fm2 @ X4 @ B2 ) )
=> ( ord_less_eq_set_fm @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1047_filter__True,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( filter_list_fm @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1048_filter__True,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( filter_fm @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1049_filter__True,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( filter_tm @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1050_filter__True,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( filter_set_nat @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_1051_filter__append,axiom,
! [P2: fm > $o,Xs: list_fm,Ys: list_fm] :
( ( filter_fm @ P2 @ ( append_fm @ Xs @ Ys ) )
= ( append_fm @ ( filter_fm @ P2 @ Xs ) @ ( filter_fm @ P2 @ Ys ) ) ) ).
% filter_append
thf(fact_1052_filter__append,axiom,
! [P2: tm > $o,Xs: list_tm,Ys: list_tm] :
( ( filter_tm @ P2 @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( filter_tm @ P2 @ Xs ) @ ( filter_tm @ P2 @ Ys ) ) ) ).
% filter_append
thf(fact_1053_set__filter,axiom,
! [P2: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P2 @ Xs ) )
= ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1054_set__filter,axiom,
! [P2: $o > $o,Xs: list_o] :
( ( set_o2 @ ( filter_o @ P2 @ Xs ) )
= ( collect_o
@ ^ [X3: $o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1055_set__filter,axiom,
! [P2: list_fm > $o,Xs: list_list_fm] :
( ( set_list_fm2 @ ( filter_list_fm @ P2 @ Xs ) )
= ( collect_list_fm
@ ^ [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1056_set__filter,axiom,
! [P2: fm > $o,Xs: list_fm] :
( ( set_fm2 @ ( filter_fm @ P2 @ Xs ) )
= ( collect_fm
@ ^ [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1057_set__filter,axiom,
! [P2: tm > $o,Xs: list_tm] :
( ( set_tm2 @ ( filter_tm @ P2 @ Xs ) )
= ( collect_tm
@ ^ [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1058_set__filter,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( set_set_nat2 @ ( filter_set_nat @ P2 @ Xs ) )
= ( collect_set_nat
@ ^ [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X3 ) ) ) ) ).
% set_filter
thf(fact_1059_filter__False,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
=> ~ ( P2 @ X4 ) )
=> ( ( filter_list_fm @ P2 @ Xs )
= nil_list_fm ) ) ).
% filter_False
thf(fact_1060_filter__False,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ~ ( P2 @ X4 ) )
=> ( ( filter_fm @ P2 @ Xs )
= nil_fm ) ) ).
% filter_False
thf(fact_1061_filter__False,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ~ ( P2 @ X4 ) )
=> ( ( filter_tm @ P2 @ Xs )
= nil_tm ) ) ).
% filter_False
thf(fact_1062_filter__False,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ ( set_set_nat2 @ Xs ) )
=> ~ ( P2 @ X4 ) )
=> ( ( filter_set_nat @ P2 @ Xs )
= nil_set_nat ) ) ).
% filter_False
thf(fact_1063_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o2 @ $true ) ) ).
% Sup_bool_def
thf(fact_1064_less__eq__set__def,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A3: set_list_fm,B4: set_list_fm] :
( ord_le6518561683347902116t_fm_o
@ ^ [X3: list_fm] : ( member_list_fm2 @ X3 @ A3 )
@ ^ [X3: list_fm] : ( member_list_fm2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1065_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B4: set_o] :
( ord_less_eq_o_o
@ ^ [X3: $o] : ( member_o2 @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1066_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1067_less__eq__set__def,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B4: set_tm] :
( ord_less_eq_tm_o
@ ^ [X3: tm] : ( member_tm2 @ X3 @ A3 )
@ ^ [X3: tm] : ( member_tm2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1068_less__eq__set__def,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B4: set_fm] :
( ord_less_eq_fm_o
@ ^ [X3: fm] : ( member_fm2 @ X3 @ A3 )
@ ^ [X3: fm] : ( member_fm2 @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1069_filter_Osimps_I2_J,axiom,
! [P2: fm > $o,X2: fm,Xs: list_fm] :
( ( ( P2 @ X2 )
=> ( ( filter_fm @ P2 @ ( cons_fm @ X2 @ Xs ) )
= ( cons_fm @ X2 @ ( filter_fm @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( filter_fm @ P2 @ ( cons_fm @ X2 @ Xs ) )
= ( filter_fm @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_1070_filter_Osimps_I2_J,axiom,
! [P2: list_fm > $o,X2: list_fm,Xs: list_list_fm] :
( ( ( P2 @ X2 )
=> ( ( filter_list_fm @ P2 @ ( cons_list_fm @ X2 @ Xs ) )
= ( cons_list_fm @ X2 @ ( filter_list_fm @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( filter_list_fm @ P2 @ ( cons_list_fm @ X2 @ Xs ) )
= ( filter_list_fm @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_1071_filter_Osimps_I2_J,axiom,
! [P2: tm > $o,X2: tm,Xs: list_tm] :
( ( ( P2 @ X2 )
=> ( ( filter_tm @ P2 @ ( cons_tm @ X2 @ Xs ) )
= ( cons_tm @ X2 @ ( filter_tm @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( filter_tm @ P2 @ ( cons_tm @ X2 @ Xs ) )
= ( filter_tm @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_1072_filter_Osimps_I1_J,axiom,
! [P2: fm > $o] :
( ( filter_fm @ P2 @ nil_fm )
= nil_fm ) ).
% filter.simps(1)
thf(fact_1073_filter_Osimps_I1_J,axiom,
! [P2: list_fm > $o] :
( ( filter_list_fm @ P2 @ nil_list_fm )
= nil_list_fm ) ).
% filter.simps(1)
thf(fact_1074_filter_Osimps_I1_J,axiom,
! [P2: tm > $o] :
( ( filter_tm @ P2 @ nil_tm )
= nil_tm ) ).
% filter.simps(1)
thf(fact_1075_filter__id__conv,axiom,
! [P2: list_fm > $o,Xs: list_list_fm] :
( ( ( filter_list_fm @ P2 @ Xs )
= Xs )
= ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_1076_filter__id__conv,axiom,
! [P2: fm > $o,Xs: list_fm] :
( ( ( filter_fm @ P2 @ Xs )
= Xs )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_1077_filter__id__conv,axiom,
! [P2: tm > $o,Xs: list_tm] :
( ( ( filter_tm @ P2 @ Xs )
= Xs )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_1078_filter__id__conv,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( ( filter_set_nat @ P2 @ Xs )
= Xs )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( P2 @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_1079_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P2: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_nat @ P2 @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1080_filter__cong,axiom,
! [Xs: list_o,Ys: list_o,P2: $o > $o,Q: $o > $o] :
( ( Xs = Ys )
=> ( ! [X4: $o] :
( ( member_o2 @ X4 @ ( set_o2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_o @ P2 @ Xs )
= ( filter_o @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1081_filter__cong,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,P2: list_fm > $o,Q: list_fm > $o] :
( ( Xs = Ys )
=> ( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_list_fm @ P2 @ Xs )
= ( filter_list_fm @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1082_filter__cong,axiom,
! [Xs: list_fm,Ys: list_fm,P2: fm > $o,Q: fm > $o] :
( ( Xs = Ys )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_fm @ P2 @ Xs )
= ( filter_fm @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1083_filter__cong,axiom,
! [Xs: list_tm,Ys: list_tm,P2: tm > $o,Q: tm > $o] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_tm @ P2 @ Xs )
= ( filter_tm @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1084_filter__cong,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( Xs = Ys )
=> ( ! [X4: set_nat] :
( ( member_set_nat2 @ X4 @ ( set_set_nat2 @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_set_nat @ P2 @ Xs )
= ( filter_set_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1085_remdups__filter,axiom,
! [P2: tm > $o,Xs: list_tm] :
( ( remdups_tm @ ( filter_tm @ P2 @ Xs ) )
= ( filter_tm @ P2 @ ( remdups_tm @ Xs ) ) ) ).
% remdups_filter
thf(fact_1086_transpose_Osimps_I1_J,axiom,
( ( transpose_fm @ nil_list_fm )
= nil_list_fm ) ).
% transpose.simps(1)
thf(fact_1087_filter__empty__conv,axiom,
! [P2: list_fm > $o,Xs: list_list_fm] :
( ( ( filter_list_fm @ P2 @ Xs )
= nil_list_fm )
= ( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_1088_filter__empty__conv,axiom,
! [P2: fm > $o,Xs: list_fm] :
( ( ( filter_fm @ P2 @ Xs )
= nil_fm )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_1089_filter__empty__conv,axiom,
! [P2: tm > $o,Xs: list_tm] :
( ( ( filter_tm @ P2 @ Xs )
= nil_tm )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_1090_filter__empty__conv,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( ( filter_set_nat @ P2 @ Xs )
= nil_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_1091_empty__filter__conv,axiom,
! [P2: fm > $o,Xs: list_fm] :
( ( nil_fm
= ( filter_fm @ P2 @ Xs ) )
= ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_1092_empty__filter__conv,axiom,
! [P2: tm > $o,Xs: list_tm] :
( ( nil_tm
= ( filter_tm @ P2 @ Xs ) )
= ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_1093_empty__filter__conv,axiom,
! [P2: set_nat > $o,Xs: list_set_nat] :
( ( nil_set_nat
= ( filter_set_nat @ P2 @ Xs ) )
= ( ! [X3: set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_1094_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_1095_p0,axiom,
( paramsts
= ( ^ [Ts2: list_tm] : ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst @ Ts2 ) ) ) ) ) ).
% p0
thf(fact_1096_s1_I2_J,axiom,
( new_list
= ( ^ [C3: nat,L: list_tm] :
~ ( member_nat2 @ C3 @ ( paramsts @ L ) ) ) ) ).
% s1(2)
thf(fact_1097_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_1098_s1_I1_J,axiom,
( new_term
= ( ^ [C3: nat,T2: tm] :
~ ( member_nat2 @ C3 @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_1099_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_1100_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L2: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L2 ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L2 ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_1101_params_Osimps_I2_J,axiom,
! [P: fm,Q2: fm] :
( ( params @ ( imp @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q2 ) ) ) ).
% params.simps(2)
thf(fact_1102_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_1103_paramst__sub__term_I2_J,axiom,
! [M: nat,S2: tm,L2: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S2 @ L2 ) ) @ ( sup_sup_set_nat @ ( paramst @ S2 ) @ ( paramsts @ L2 ) ) ) ).
% paramst_sub_term(2)
thf(fact_1104_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_1105_sub__list_Osimps_I2_J,axiom,
! [V: nat,S2: tm,T: tm,L2: list_tm] :
( ( sub_list @ V @ S2 @ ( cons_tm @ T @ L2 ) )
= ( cons_tm @ ( sub_term @ V @ S2 @ T ) @ ( sub_list @ V @ S2 @ L2 ) ) ) ).
% sub_list.simps(2)
thf(fact_1106_sub__list_Osimps_I1_J,axiom,
! [V: nat,S2: tm] :
( ( sub_list @ V @ S2 @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_1107_s5_I1_J,axiom,
( sub_term
= ( ^ [V2: nat,S3: tm,T2: tm] : ( substt @ T2 @ S3 @ V2 ) ) ) ).
% s5(1)
thf(fact_1108_params__sub,axiom,
! [M: nat,T: tm,P: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P ) ) ) ).
% params_sub
thf(fact_1109_sub_Osimps_I2_J,axiom,
! [V: nat,S2: tm,P: fm,Q2: fm] :
( ( sub @ V @ S2 @ ( imp @ P @ Q2 ) )
= ( imp @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q2 ) ) ) ).
% sub.simps(2)
thf(fact_1110_sub_Osimps_I7_J,axiom,
! [V: nat,S2: tm,P: fm] :
( ( sub @ V @ S2 @ ( neg @ P ) )
= ( neg @ ( sub @ V @ S2 @ P ) ) ) ).
% sub.simps(7)
thf(fact_1111_paramsts_Osimps_I1_J,axiom,
( ( paramsts @ nil_tm )
= bot_bot_set_nat ) ).
% paramsts.simps(1)
thf(fact_1112_paramst_H_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst2 @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst2 @ Ts ) ) ) ) ) ).
% paramst'.simps(2)
thf(fact_1113_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_1114_sub__term_Osimps_I2_J,axiom,
! [V: nat,S2: tm,I3: nat,L2: list_tm] :
( ( sub_term @ V @ S2 @ ( fun @ I3 @ L2 ) )
= ( fun @ I3 @ ( sub_list @ V @ S2 @ L2 ) ) ) ).
% sub_term.simps(2)
thf(fact_1115_new__term_Osimps_I2_J,axiom,
! [C: nat,I3: nat,L2: list_tm] :
( ( new_term @ C @ ( fun @ I3 @ L2 ) )
= ( ( I3 != C )
& ( ( I3 != C )
=> ( new_list @ C @ L2 ) ) ) ) ).
% new_term.simps(2)
thf(fact_1116_params__subtermFm,axiom,
! [P: fm,X: nat] :
( ( member_nat2 @ X @ ( params @ P ) )
=> ? [L3: list_tm] : ( member_tm2 @ ( fun @ X @ L3 ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% params_subtermFm
thf(fact_1117_paramst_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A @ bot_bot_set_nat ) @ ( paramsts @ Ts ) ) ) ).
% paramst.simps(2)
thf(fact_1118_fun__arguments__subterm,axiom,
! [N2: nat,Ts: list_tm,P: fm] :
( ( member_tm2 @ ( fun @ N2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% fun_arguments_subterm
thf(fact_1119_sub__const__transfer,axiom,
! [M: nat,A: nat,P: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P )
!= ( sub @ M @ T @ P ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P ) ) ) ) ) ).
% sub_const_transfer
thf(fact_1120_paramst_H_H_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst3 @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts ) ) ) ) ) ).
% paramst''.simps(2)
thf(fact_1121_sub__term__const__transfer_I2_J,axiom,
! [M: nat,A: nat,Ts: list_tm,S2: tm] :
( ( ( sub_list @ M @ ( fun @ A @ nil_tm ) @ Ts )
!= ( sub_list @ M @ S2 @ Ts ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm )
@ ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [T2: tm] : ( set_tm2 @ ( subtermTm @ T2 ) )
@ ( set_tm2 @ ( sub_list @ M @ ( fun @ A @ nil_tm ) @ Ts ) ) ) ) ) ) ).
% sub_term_const_transfer(2)
thf(fact_1122_subtermTm_Osimps_I1_J,axiom,
! [N2: nat,Ts: list_tm] :
( ( subtermTm @ ( fun @ N2 @ Ts ) )
= ( cons_tm @ ( fun @ N2 @ Ts ) @ ( remdups_tm @ ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ) ) ).
% subtermTm.simps(1)
thf(fact_1123_p1_H,axiom,
paramst3 = paramst ).
% p1'
thf(fact_1124_subtermTm__refl,axiom,
! [T: tm] : ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_1125_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_1126_paramst__subtermTm_I1_J,axiom,
! [T: tm,X: nat] :
( ( member_nat2 @ X @ ( paramst @ T ) )
=> ? [L3: list_tm] : ( member_tm2 @ ( fun @ X @ L3 ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_1127_subterm__Fun__refl,axiom,
! [Ts: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N2 @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_1128_paramst__subtermTm_I2_J,axiom,
! [Ts: list_tm,X: nat] :
( ( member_nat2 @ X @ ( paramsts @ Ts ) )
=> ? [L3: list_tm] :
( member_tm2 @ ( fun @ X @ L3 )
@ ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [T2: tm] : ( set_tm2 @ ( subtermTm @ T2 ) )
@ ( set_tm2 @ Ts ) ) ) ) ) ).
% paramst_subtermTm(2)
thf(fact_1129_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_1130_paramst_H_H_Oelims,axiom,
! [X2: tm,Y: set_nat] :
( ( ( paramst3 @ X2 )
= Y )
=> ( ( ? [N: nat] :
( X2
= ( var @ N ) )
=> ( Y != bot_bot_set_nat ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( ( X2
= ( fun @ A4 @ Ts3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) ) ) ) ) ) ).
% paramst''.elims
thf(fact_1131_paramst_H_H_Opelims,axiom,
! [X2: tm,Y: set_nat] :
( ( ( paramst3 @ X2 )
= Y )
=> ( ( accp_tm @ paramst_rel @ X2 )
=> ( ! [N: nat] :
( ( X2
= ( var @ N ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_tm @ paramst_rel @ ( var @ N ) ) ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( ( X2
= ( fun @ A4 @ Ts3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) ) )
=> ~ ( accp_tm @ paramst_rel @ ( fun @ A4 @ Ts3 ) ) ) ) ) ) ) ).
% paramst''.pelims
thf(fact_1132_tm_Oinject_I2_J,axiom,
! [X24: nat,Y23: nat] :
( ( ( var @ X24 )
= ( var @ Y23 ) )
= ( X24 = Y23 ) ) ).
% tm.inject(2)
thf(fact_1133_paramst_H_H_Ocases,axiom,
! [X2: tm] :
( ! [N: nat] :
( X2
!= ( var @ N ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( X2
!= ( fun @ A4 @ Ts3 ) ) ) ).
% paramst''.cases
thf(fact_1134_tm_Oexhaust,axiom,
! [Y: tm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X25: nat] :
( Y
!= ( var @ X25 ) ) ) ).
% tm.exhaust
thf(fact_1135_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X24: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X24 ) ) ).
% tm.distinct(1)
thf(fact_1136_new__term_Osimps_I1_J,axiom,
! [C: nat,N2: nat] : ( new_term @ C @ ( var @ N2 ) ) ).
% new_term.simps(1)
thf(fact_1137_paramst_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst.simps(1)
thf(fact_1138_paramst_H_H_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst3 @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst''.simps(1)
thf(fact_1139_paramst_H_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst2 @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst'.simps(1)
thf(fact_1140_subtermTm_Osimps_I2_J,axiom,
! [N2: nat] :
( ( subtermTm @ ( var @ N2 ) )
= ( cons_tm @ ( var @ N2 ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_1141_subterms__def,axiom,
( subterms
= ( ^ [Z3: list_fm] : ( case_list_list_tm_tm @ ( cons_tm @ ( fun @ zero_zero_nat @ nil_tm ) @ nil_tm ) @ cons_tm @ ( remdups_tm @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z3 ) ) ) ) ) ) ).
% subterms_def
thf(fact_1142_subtermFm_Osimps_I1_J,axiom,
! [Uu2: nat,Ts: list_tm] :
( ( subtermFm @ ( pre @ Uu2 @ Ts ) )
= ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ).
% subtermFm.simps(1)
thf(fact_1143_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_1144_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1145_fm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X21: fm,X22: fm] :
( ( pre @ X11 @ X12 )
!= ( imp @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_1146_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_1147_params_Osimps_I1_J,axiom,
! [B: nat,Ts: list_tm] :
( ( params @ ( pre @ B @ Ts ) )
= ( paramsts @ Ts ) ) ).
% params.simps(1)
thf(fact_1148_sub_Osimps_I1_J,axiom,
! [V: nat,S2: tm,I3: nat,L2: list_tm] :
( ( sub @ V @ S2 @ ( pre @ I3 @ L2 ) )
= ( pre @ I3 @ ( sub_list @ V @ S2 @ L2 ) ) ) ).
% sub.simps(1)
thf(fact_1149_subterm__Pre__refl,axiom,
! [Ts: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N2 @ Ts ) ) ) ) ).
% subterm_Pre_refl
thf(fact_1150_set__subterms,axiom,
! [Z: list_fm] :
( ( ( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P3: fm] : ( set_tm2 @ ( subtermFm @ P3 ) )
@ ( set_fm2 @ Z ) ) )
= bot_bot_set_tm )
=> ( ( set_tm2 @ ( subterms @ Z ) )
= ( insert_tm2 @ ( fun @ zero_zero_nat @ nil_tm ) @ bot_bot_set_tm ) ) )
& ( ( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P3: fm] : ( set_tm2 @ ( subtermFm @ P3 ) )
@ ( set_fm2 @ Z ) ) )
!= bot_bot_set_tm )
=> ( ( set_tm2 @ ( subterms @ Z ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P3: fm] : ( set_tm2 @ ( subtermFm @ P3 ) )
@ ( set_fm2 @ Z ) ) ) ) ) ) ).
% set_subterms
thf(fact_1151_params_H_Osimps_I1_J,axiom,
! [B: nat,Ts: list_tm] :
( ( params2 @ ( pre @ B @ Ts ) )
= ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst2 @ Ts ) ) ) ) ).
% params'.simps(1)
thf(fact_1152_p2,axiom,
params2 = params ).
% p2
thf(fact_1153_params_H_Osimps_I7_J,axiom,
! [P: fm] :
( ( params2 @ ( neg @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(7)
thf(fact_1154_params_H_Osimps_I2_J,axiom,
! [P: fm,Q2: fm] :
( ( params2 @ ( imp @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q2 ) ) ) ).
% params'.simps(2)
thf(fact_1155_DeltaExi,axiom,
! [I3: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I3 @ nil_tm ) @ P ) ) @ Z ) )
=> ( ( news @ I3 @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P ) ) @ Z ) ) ) ) ).
% DeltaExi
thf(fact_1156_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1157_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1158_fm_Oinject_I5_J,axiom,
! [X52: fm,Y52: fm] :
( ( ( exi @ X52 )
= ( exi @ Y52 ) )
= ( X52 = Y52 ) ) ).
% fm.inject(5)
thf(fact_1159_params_H_Osimps_I5_J,axiom,
! [P: fm] :
( ( params2 @ ( exi @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(5)
thf(fact_1160_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1161_le__trans,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1162_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1163_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_1164_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_1165_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1166_fm_Odistinct_I17_J,axiom,
! [X21: fm,X22: fm,X52: fm] :
( ( imp @ X21 @ X22 )
!= ( exi @ X52 ) ) ).
% fm.distinct(17)
thf(fact_1167_subtermFm_Osimps_I5_J,axiom,
! [P: fm] :
( ( subtermFm @ ( exi @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(5)
thf(fact_1168_fm_Odistinct_I39_J,axiom,
! [X52: fm,X7: fm] :
( ( exi @ X52 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_1169_params_Osimps_I5_J,axiom,
! [P: fm] :
( ( params @ ( exi @ P ) )
= ( params @ P ) ) ).
% params.simps(5)
thf(fact_1170_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X52: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X52 ) ) ).
% fm.distinct(7)
thf(fact_1171_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1172_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_1173_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_1174_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1175_GammaExi,axiom,
! [T: tm,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T @ P ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( exi @ P ) @ Z ) ) ) ).
% GammaExi
thf(fact_1176_DeltaUni,axiom,
! [I3: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I3 @ nil_tm ) @ P ) @ Z ) )
=> ( ( news @ I3 @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P ) @ Z ) ) ) ) ).
% DeltaUni
thf(fact_1177_params_H_H_Osimps_I1_J,axiom,
! [B: nat,Ts: list_tm] :
( ( params3 @ ( pre @ B @ Ts ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts ) ) ) ) ).
% params''.simps(1)
thf(fact_1178_tm_Osize__gen_I2_J,axiom,
! [X24: nat] :
( ( size_tm @ ( var @ X24 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_1179_fm_Oinject_I6_J,axiom,
! [X62: fm,Y6: fm] :
( ( ( uni @ X62 )
= ( uni @ Y6 ) )
= ( X62 = Y6 ) ) ).
% fm.inject(6)
thf(fact_1180_p2_H,axiom,
params3 = params ).
% p2'
thf(fact_1181_params_H_H_Osimps_I5_J,axiom,
! [P: fm] :
( ( params3 @ ( exi @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(5)
thf(fact_1182_fm_Odistinct_I37_J,axiom,
! [X52: fm,X62: fm] :
( ( exi @ X52 )
!= ( uni @ X62 ) ) ).
% fm.distinct(37)
thf(fact_1183_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X62: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X62 ) ) ).
% fm.distinct(9)
thf(fact_1184_params_Osimps_I6_J,axiom,
! [P: fm] :
( ( params @ ( uni @ P ) )
= ( params @ P ) ) ).
% params.simps(6)
thf(fact_1185_params_H_H_Osimps_I7_J,axiom,
! [P: fm] :
( ( params3 @ ( neg @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(7)
thf(fact_1186_params_H_H_Osimps_I6_J,axiom,
! [P: fm] :
( ( params3 @ ( uni @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(6)
thf(fact_1187_fm_Odistinct_I41_J,axiom,
! [X62: fm,X7: fm] :
( ( uni @ X62 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_1188_subtermFm_Osimps_I6_J,axiom,
! [P: fm] :
( ( subtermFm @ ( uni @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(6)
thf(fact_1189_fm_Odistinct_I19_J,axiom,
! [X21: fm,X22: fm,X62: fm] :
( ( imp @ X21 @ X22 )
!= ( uni @ X62 ) ) ).
% fm.distinct(19)
thf(fact_1190_params_H_Osimps_I6_J,axiom,
! [P: fm] :
( ( params2 @ ( uni @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(6)
thf(fact_1191_params_H_H_Osimps_I2_J,axiom,
! [P: fm,Q2: fm] :
( ( params3 @ ( imp @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q2 ) ) ) ).
% params''.simps(2)
thf(fact_1192_GammaUni,axiom,
! [T: tm,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T @ P ) ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( uni @ P ) ) @ Z ) ) ) ).
% GammaUni
thf(fact_1193_sequent__calculus_Ocases,axiom,
! [A: list_fm] :
( ( sequent_calculus @ A )
=> ( ! [P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ P4 @ Z2 ) )
=> ~ ( member_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ! [P4: fm,Q3: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( dis @ P4 @ Q3 ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ ( cons_fm @ Q3 @ Z2 ) ) ) )
=> ( ! [P4: fm,Q3: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( imp @ P4 @ Q3 ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ Q3 @ Z2 ) ) ) )
=> ( ! [P4: fm,Q3: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( con @ P4 @ Q3 ) ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ ( neg @ Q3 ) @ Z2 ) ) ) )
=> ( ! [P4: fm,Z2: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( con @ P4 @ Q3 ) @ Z2 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ Q3 @ Z2 ) ) ) )
=> ( ! [P4: fm,Z2: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( neg @ ( imp @ P4 @ Q3 ) ) @ Z2 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z2 ) ) ) )
=> ( ! [P4: fm,Z2: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( neg @ ( dis @ P4 @ Q3 ) ) @ Z2 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z2 ) ) ) )
=> ( ! [T3: tm,P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( exi @ P4 ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T3 @ P4 ) @ Z2 ) ) )
=> ( ! [T3: tm,P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( uni @ P4 ) ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T3 @ P4 ) ) @ Z2 ) ) )
=> ( ! [I2: nat,P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( uni @ P4 ) @ Z2 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P4 ) @ Z2 ) )
=> ~ ( news @ I2 @ ( cons_fm @ P4 @ Z2 ) ) ) )
=> ( ! [I2: nat,P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( exi @ P4 ) ) @ Z2 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P4 ) ) @ Z2 ) )
=> ~ ( news @ I2 @ ( cons_fm @ P4 @ Z2 ) ) ) )
=> ( ! [P4: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( neg @ P4 ) ) @ Z2 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ Z2 ) ) )
=> ~ ! [Z2: list_fm] :
( ( sequent_calculus @ Z2 )
=> ~ ( ext_fm @ A @ Z2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sequent_calculus.cases
thf(fact_1194_sequent__calculus_Osimps,axiom,
( sequent_calculus
= ( ^ [A5: list_fm] :
( ? [P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ P3 @ Z3 ) )
& ( member_fm @ ( neg @ P3 ) @ Z3 ) )
| ? [P3: fm,Q4: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( dis @ P3 @ Q4 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ P3 @ ( cons_fm @ Q4 @ Z3 ) ) ) )
| ? [P3: fm,Q4: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( imp @ P3 @ Q4 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ Q4 @ Z3 ) ) ) )
| ? [P3: fm,Q4: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( con @ P3 @ Q4 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ ( neg @ Q4 ) @ Z3 ) ) ) )
| ? [P3: fm,Z3: list_fm,Q4: fm] :
( ( A5
= ( cons_fm @ ( con @ P3 @ Q4 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ Q4 @ Z3 ) ) )
| ? [P3: fm,Z3: list_fm,Q4: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( imp @ P3 @ Q4 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q4 ) @ Z3 ) ) )
| ? [P3: fm,Z3: list_fm,Q4: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( dis @ P3 @ Q4 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P3 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q4 ) @ Z3 ) ) )
| ? [T2: tm,P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( exi @ P3 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T2 @ P3 ) @ Z3 ) ) )
| ? [T2: tm,P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( uni @ P3 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T2 @ P3 ) ) @ Z3 ) ) )
| ? [I: nat,P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( uni @ P3 ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P3 ) @ Z3 ) )
& ( news @ I @ ( cons_fm @ P3 @ Z3 ) ) )
| ? [I: nat,P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( exi @ P3 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P3 ) ) @ Z3 ) )
& ( news @ I @ ( cons_fm @ P3 @ Z3 ) ) )
| ? [P3: fm,Z3: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( neg @ P3 ) ) @ Z3 ) )
& ( sequent_calculus @ ( cons_fm @ P3 @ Z3 ) ) )
| ? [Z3: list_fm,Y2: list_fm] :
( ( A5 = Y2 )
& ( sequent_calculus @ Z3 )
& ( ext_fm @ Y2 @ Z3 ) ) ) ) ) ).
% sequent_calculus.simps
thf(fact_1195_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_1196_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_1197_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X62: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X62 ) ) ).
% fm.distinct(33)
thf(fact_1198_fm_Odistinct_I27_J,axiom,
! [X31: fm,X32: fm,X62: fm] :
( ( dis @ X31 @ X32 )
!= ( uni @ X62 ) ) ).
% fm.distinct(27)
thf(fact_1199_sub_Osimps_I4_J,axiom,
! [V: nat,S2: tm,P: fm,Q2: fm] :
( ( sub @ V @ S2 @ ( con @ P @ Q2 ) )
= ( con @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q2 ) ) ) ).
% sub.simps(4)
thf(fact_1200_sub_Osimps_I3_J,axiom,
! [V: nat,S2: tm,P: fm,Q2: fm] :
( ( sub @ V @ S2 @ ( dis @ P @ Q2 ) )
= ( dis @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q2 ) ) ) ).
% sub.simps(3)
thf(fact_1201_fm_Odistinct_I13_J,axiom,
! [X21: fm,X22: fm,X31: fm,X32: fm] :
( ( imp @ X21 @ X22 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(13)
thf(fact_1202_fm_Odistinct_I15_J,axiom,
! [X21: fm,X22: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X22 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_1203_fm_Odistinct_I23_J,axiom,
! [X31: fm,X32: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X32 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_1204_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_1205_fm_Odistinct_I29_J,axiom,
! [X31: fm,X32: fm,X7: fm] :
( ( dis @ X31 @ X32 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_1206_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_1207_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_1208_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X52: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X52 ) ) ).
% fm.distinct(31)
thf(fact_1209_fm_Odistinct_I25_J,axiom,
! [X31: fm,X32: fm,X52: fm] :
( ( dis @ X31 @ X32 )
!= ( exi @ X52 ) ) ).
% fm.distinct(25)
thf(fact_1210_AlphaDis,axiom,
! [P: fm,Q2: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ ( cons_fm @ Q2 @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P @ Q2 ) @ Z ) ) ) ).
% AlphaDis
thf(fact_1211_BetaCon,axiom,
! [P: fm,Z: list_fm,Q2: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q2 @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P @ Q2 ) @ Z ) ) ) ) ).
% BetaCon
thf(fact_1212_params_H_H_Osimps_I4_J,axiom,
! [P: fm,Q2: fm] :
( ( params3 @ ( con @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q2 ) ) ) ).
% params''.simps(4)
thf(fact_1213_params_H_H_Osimps_I3_J,axiom,
! [P: fm,Q2: fm] :
( ( params3 @ ( dis @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q2 ) ) ) ).
% params''.simps(3)
thf(fact_1214_params_Osimps_I3_J,axiom,
! [P: fm,Q2: fm] :
( ( params @ ( dis @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q2 ) ) ) ).
% params.simps(3)
thf(fact_1215_params_Osimps_I4_J,axiom,
! [P: fm,Q2: fm] :
( ( params @ ( con @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q2 ) ) ) ).
% params.simps(4)
thf(fact_1216_subtermFm_Osimps_I4_J,axiom,
! [P: fm,Q2: fm] :
( ( subtermFm @ ( con @ P @ Q2 ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q2 ) ) ) ).
% subtermFm.simps(4)
thf(fact_1217_subtermFm_Osimps_I3_J,axiom,
! [P: fm,Q2: fm] :
( ( subtermFm @ ( dis @ P @ Q2 ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q2 ) ) ) ).
% subtermFm.simps(3)
thf(fact_1218_params_H_Osimps_I4_J,axiom,
! [P: fm,Q2: fm] :
( ( params2 @ ( con @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q2 ) ) ) ).
% params'.simps(4)
thf(fact_1219_params_H_Osimps_I3_J,axiom,
! [P: fm,Q2: fm] :
( ( params2 @ ( dis @ P @ Q2 ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q2 ) ) ) ).
% params'.simps(3)
thf(fact_1220_params_H_H_Ocases,axiom,
! [X2: fm] :
( ! [B5: nat,Ts3: list_tm] :
( X2
!= ( pre @ B5 @ Ts3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( imp @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( dis @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( con @ P4 @ Q3 ) )
=> ( ! [P4: fm] :
( X2
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X2
!= ( uni @ P4 ) )
=> ~ ! [P4: fm] :
( X2
!= ( neg @ P4 ) ) ) ) ) ) ) ) ).
% params''.cases
thf(fact_1221_fm_Oexhaust,axiom,
! [Y: fm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( pre @ X112 @ X122 ) )
=> ( ! [X212: fm,X222: fm] :
( Y
!= ( imp @ X212 @ X222 ) )
=> ( ! [X312: fm,X322: fm] :
( Y
!= ( dis @ X312 @ X322 ) )
=> ( ! [X412: fm,X422: fm] :
( Y
!= ( con @ X412 @ X422 ) )
=> ( ! [X53: fm] :
( Y
!= ( exi @ X53 ) )
=> ( ! [X63: fm] :
( Y
!= ( uni @ X63 ) )
=> ~ ! [X72: fm] :
( Y
!= ( neg @ X72 ) ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_1222_Neg__exhaust,axiom,
! [X2: fm] :
( ! [I2: nat,Ts3: list_tm] :
( X2
!= ( pre @ I2 @ Ts3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( imp @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( dis @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( con @ P4 @ Q3 ) )
=> ( ! [P4: fm] :
( X2
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X2
!= ( uni @ P4 ) )
=> ( ! [I2: nat,Ts3: list_tm] :
( X2
!= ( neg @ ( pre @ I2 @ Ts3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( neg @ ( imp @ P4 @ Q3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( neg @ ( dis @ P4 @ Q3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X2
!= ( neg @ ( con @ P4 @ Q3 ) ) )
=> ( ! [P4: fm] :
( X2
!= ( neg @ ( exi @ P4 ) ) )
=> ( ! [P4: fm] :
( X2
!= ( neg @ ( uni @ P4 ) ) )
=> ~ ! [P4: fm] :
( X2
!= ( neg @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Neg_exhaust
thf(fact_1223_AlphaCon,axiom,
! [P: fm,Q2: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ ( neg @ Q2 ) @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P @ Q2 ) ) @ Z ) ) ) ).
% AlphaCon
thf(fact_1224_BetaDis,axiom,
! [P: fm,Z: list_fm,Q2: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P @ Q2 ) ) @ Z ) ) ) ) ).
% BetaDis
thf(fact_1225_branchDone_Ocases,axiom,
! [X2: list_fm] :
( ( X2 != nil_fm )
=> ( ! [P4: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ! [V3: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( X2
!= ( cons_fm @ ( uni @ V3 ) @ Z2 ) ) ) ) ) ) ) ) ) ).
% branchDone.cases
thf(fact_1226_params_H_H_Oelims,axiom,
! [X2: fm,Y: set_nat] :
( ( ( params3 @ X2 )
= Y )
=> ( ! [B5: nat,Ts3: list_tm] :
( ( X2
= ( pre @ B5 @ Ts3 ) )
=> ( Y
!= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( imp @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( dis @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( con @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm] :
( ( X2
= ( exi @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) )
=> ( ! [P4: fm] :
( ( X2
= ( uni @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) )
=> ~ ! [P4: fm] :
( ( X2
= ( neg @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) ) ) ) ) ) ) ) ) ).
% params''.elims
thf(fact_1227_params_H_H_Opelims,axiom,
! [X2: fm,Y: set_nat] :
( ( ( params3 @ X2 )
= Y )
=> ( ( accp_fm @ params_rel @ X2 )
=> ( ! [B5: nat,Ts3: list_tm] :
( ( X2
= ( pre @ B5 @ Ts3 ) )
=> ( ( Y
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) )
=> ~ ( accp_fm @ params_rel @ ( pre @ B5 @ Ts3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( imp @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( imp @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( dis @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( dis @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X2
= ( con @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( con @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm] :
( ( X2
= ( exi @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( exi @ P4 ) ) ) )
=> ( ! [P4: fm] :
( ( X2
= ( uni @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( uni @ P4 ) ) ) )
=> ~ ! [P4: fm] :
( ( X2
= ( neg @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% params''.pelims
thf(fact_1228_branchDone_Oelims_I1_J,axiom,
! [X2: list_fm,Y: $o] :
( ( ( branchDone @ X2 )
= Y )
=> ( ( ( X2 = nil_fm )
=> Y )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(1)
thf(fact_1229_branchDone_Oelims_I3_J,axiom,
! [X2: list_fm] :
( ~ ( branchDone @ X2 )
=> ( ( X2 != nil_fm )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(3)
thf(fact_1230_branchDone_Osimps_I1_J,axiom,
~ ( branchDone @ nil_fm ) ).
% branchDone.simps(1)
thf(fact_1231_branchDone__contradiction,axiom,
( branchDone
= ( ^ [Z3: list_fm] :
? [P3: fm] :
( ( member_fm2 @ P3 @ ( set_fm2 @ Z3 ) )
& ( member_fm2 @ ( neg @ P3 ) @ ( set_fm2 @ Z3 ) ) ) ) ) ).
% branchDone_contradiction
thf(fact_1232_branchDone_Osimps_I2_J,axiom,
! [P: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( neg @ P ) @ Z ) )
= ( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
| ( member_fm2 @ ( neg @ ( neg @ P ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(2)
thf(fact_1233_branchDone_Osimps_I8_J,axiom,
! [V: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( uni @ V ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(8)
thf(fact_1234_branchDone_Osimps_I4_J,axiom,
! [V: fm,Va2: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( imp @ V @ Va2 ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( imp @ V @ Va2 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(4)
thf(fact_1235_branchDone_Osimps_I7_J,axiom,
! [V: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( exi @ V ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(7)
thf(fact_1236_branchDone_Osimps_I6_J,axiom,
! [V: fm,Va2: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( con @ V @ Va2 ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( con @ V @ Va2 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(6)
thf(fact_1237_branchDone_Osimps_I5_J,axiom,
! [V: fm,Va2: fm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( dis @ V @ Va2 ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( dis @ V @ Va2 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(5)
thf(fact_1238_branchDone_Osimps_I3_J,axiom,
! [V: nat,Va2: list_tm,Z: list_fm] :
( ( branchDone @ ( cons_fm @ ( pre @ V @ Va2 ) @ Z ) )
= ( ( member_fm2 @ ( neg @ ( pre @ V @ Va2 ) ) @ ( set_fm2 @ Z ) )
| ( branchDone @ Z ) ) ) ).
% branchDone.simps(3)
thf(fact_1239_branchDone_Oelims_I2_J,axiom,
! [X2: list_fm] :
( ( branchDone @ X2 )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(2)
thf(fact_1240_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 ) )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(3)
thf(fact_1241_branchDone_Opelims_I1_J,axiom,
! [X2: list_fm,Y: $o] :
( ( ( branchDone @ X2 )
= Y )
=> ( ( accp_list_fm @ branchDone_rel @ X2 )
=> ( ( ( X2 = nil_fm )
=> ( ~ Y
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) ) )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z2 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z2 ) ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(1)
thf(fact_1242_branchDone_Opelims_I2_J,axiom,
! [X2: list_fm] :
( ( branchDone @ X2 )
=> ( ( accp_list_fm @ branchDone_rel @ X2 )
=> ( ! [P4: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z2 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ( ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) )
=> ~ ! [V3: fm,Z2: list_fm] :
( ( X2
= ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z2 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z2 ) )
| ( branchDone @ Z2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(2)
thf(fact_1243_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_1244_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1245_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_1246_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1247_Suc__le__D,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1248_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_1249_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1250_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_1251_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( P2 @ M4 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_1252_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P2 @ M )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1253_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X4 @ Z2 ) ) )
=> ( ! [N: nat] : ( R2 @ N @ ( suc @ N ) )
=> ( R2 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1254_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1255_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M5: nat] :
( ( P2 @ X2 )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M3: nat] :
( ( P2 @ M3 )
=> ~ ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1256_substts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm,S2: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts ) @ S2 @ K )
= ( cons_tm @ ( substt @ T @ S2 @ K ) @ ( substts @ Ts @ S2 @ K ) ) ) ).
% substts.simps(2)
thf(fact_1257_substt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm,S2: tm,K: nat] :
( ( substt @ ( fun @ A @ Ts ) @ S2 @ K )
= ( fun @ A @ ( substts @ Ts @ S2 @ K ) ) ) ).
% substt.simps(2)
thf(fact_1258_s5_I2_J,axiom,
( sub_list
= ( ^ [V2: nat,S3: tm,L: list_tm] : ( substts @ L @ S3 @ V2 ) ) ) ).
% s5(2)
thf(fact_1259_substts_Osimps_I1_J,axiom,
! [S2: tm,K: nat] :
( ( substts @ nil_tm @ S2 @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_1260_tm_Osize_I4_J,axiom,
! [X24: nat] :
( ( size_size_tm @ ( var @ X24 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_1261_parts__in__effect,axiom,
! [P: fm,Z: list_fm,B2: list_tm,Z4: list_fm,R: rule,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z4 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
=> ? [C4: list_tm,Xs4: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ C4 ) )
& ( member_list_fm2 @ Xs4 @ ( set_list_fm2 @ ( parts @ C4 @ R @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs4 ) @ ( set_fm2 @ Z4 ) ) ) ) ) ).
% parts_in_effect
thf(fact_1262_effect__preserves__unaffected,axiom,
! [P: fm,Z: list_fm,R: rule,B2: list_tm,Z4: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ~ ( affects @ R @ P )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z4 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z4 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_1263_ne__effect__not__branchDone,axiom,
! [B2: list_tm,Z4: list_fm,R: rule,A2: list_tm,Z: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z4 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
=> ~ ( branchDone @ Z ) ) ).
% ne_effect_not_branchDone
thf(fact_1264_eff__children,axiom,
! [Z: list_fm,R: rule,A2: list_tm,Ss: fset_P8989946509869081563ist_fm] :
( ~ ( branchDone @ Z )
=> ( ( eff @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) @ Ss )
=> ! [X: list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z ) ) ) ) @ R @ Z ) ) )
=> ? [B3: list_tm] : ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B3 @ X ) @ Ss ) ) ) ) ).
% eff_children
thf(fact_1265_eff__def,axiom,
( eff
= ( ^ [R3: rule,S3: produc6018962875968178549ist_fm] :
( ^ [Y8: fset_P8989946509869081563ist_fm,Z6: fset_P8989946509869081563ist_fm] : ( Y8 = Z6 )
@ ( effect @ R3 @ S3 ) ) ) ) ).
% eff_def
thf(fact_1266_effect_Osimps,axiom,
! [Z: list_fm,R: rule,A2: list_tm] :
( ( ( branchDone @ Z )
=> ( ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) )
= bot_bo6461889142629771335ist_fm ) )
& ( ~ ( branchDone @ Z )
=> ( ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) )
= ( fimage4743371125182381497ist_fm
@ ^ [Z7: list_fm] : ( produc1414352766439514085ist_fm @ ( remdups_tm @ ( append_tm @ A2 @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z7 ) ) ) ) @ Z7 )
@ ( fset_of_list_list_fm @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z ) ) ) ) @ R @ Z ) ) ) ) ) ) ).
% effect.simps
thf(fact_1267_parts__def,axiom,
( parts
= ( ^ [A3: list_tm,R3: rule,F2: fm] :
( produc1325496751214513674ist_fm
@ ^ [A5: rule,B6: fm] :
( case_r8401956329264079908ist_fm
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ P3 @ ( cons_fm @ Q4 @ nil_fm ) ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ Q4 @ nil_fm ) ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ ( neg @ P3 ) @ ( cons_fm @ ( neg @ Q4 ) @ nil_fm ) ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ P3 @ nil_fm ) @ ( cons_list_fm @ ( cons_fm @ Q4 @ nil_fm ) @ nil_list_fm ) )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ P3 @ nil_fm ) @ ( cons_list_fm @ ( cons_fm @ ( neg @ Q4 ) @ nil_fm ) @ nil_list_fm ) )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm,Q4: fm] : ( cons_list_fm @ ( cons_fm @ ( neg @ P3 ) @ nil_fm ) @ ( cons_list_fm @ ( cons_fm @ ( neg @ Q4 ) @ nil_fm ) @ nil_list_fm ) )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm] : ( cons_list_fm @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ ( generateNew @ A3 ) @ nil_tm ) @ P3 ) @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm] : ( cons_list_fm @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ ( generateNew @ A3 ) @ nil_tm ) @ P3 ) ) @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm] : ( cons_list_fm @ ( cons_fm @ P3 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm] :
( cons_list_fm
@ ( cons_fm @ ( exi @ P3 )
@ ( map_tm_fm
@ ^ [T2: tm] : ( sub @ zero_zero_nat @ T2 @ P3 )
@ A3 ) )
@ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ B6 )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ( case_fm_list_list_fm
@ ^ [Nat: nat,List2: list_tm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fm1: fm,Fm2: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm )
@ ^ [P3: fm] :
( cons_list_fm
@ ( cons_fm @ ( neg @ ( uni @ P3 ) )
@ ( map_tm_fm
@ ^ [T2: tm] : ( neg @ ( sub @ zero_zero_nat @ T2 @ P3 ) )
@ A3 ) )
@ nil_list_fm )
@ ^ [Fma: fm] : ( cons_list_fm @ ( cons_fm @ F2 @ nil_fm ) @ nil_list_fm ) )
@ B6 )
@ A5 )
@ ( product_Pair_rule_fm @ R3 @ F2 ) ) ) ) ).
% parts_def
thf(fact_1268_affects__def,axiom,
( affects
= ( ^ [R3: rule,P3: fm] :
( produc3561889649859641891e_fm_o
@ ^ [X3: rule,Xa3: fm] :
( case_rule_o
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false )
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false )
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm,Xb: fm] : $true
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false )
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Y2: fm] : $true
@ ^ [Fm: fm] : $false
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm] : $true
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false )
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Y2: fm] : $true )
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Y2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ Xa3 )
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm: fm] : $false
@ ^ [Fm: fm] : $false
@ ( case_fm_o
@ ^ [Nat: nat,List2: list_tm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fm1: fm,Fm2: fm] : $false
@ ^ [Fma: fm] : $false
@ ^ [Y2: fm] : $false
@ ^ [Fma: fm] : $false )
@ Xa3 )
@ X3 )
@ ( product_Pair_rule_fm @ R3 @ P3 ) ) ) ) ).
% affects_def
% Helper facts (13)
thf(help_If_2_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y: list_o] :
( ( if_list_o @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X2: list_o,Y: list_o] :
( ( if_list_o @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X2: list_fm,Y: list_fm] :
( ( if_list_fm @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X2: list_fm,Y: list_fm] :
( ( if_list_fm @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X2: list_tm,Y: list_tm] :
( ( if_list_tm @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X2: list_tm,Y: list_tm] :
( ( if_list_tm @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X2: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X2: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X2: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X2: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
sequent_calculus @ ( append_fm @ prea @ ( cons_fm @ ( neg @ pa ) @ ( cons_fm @ q @ za ) ) ) ).
%------------------------------------------------------------------------------