TPTP Problem File: SLH0874^1.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_00125_004592__13560218_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1659 ( 782 unt; 374 typ; 0 def)
% Number of atoms : 3713 (2050 equ; 0 cnn)
% Maximal formula atoms : 42 ( 2 avg)
% Number of connectives : 13537 ( 419 ~; 92 |; 461 &;11194 @)
% ( 0 <=>;1371 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1524 (1524 >; 0 *; 0 +; 0 <<)
% Number of symbols : 339 ( 336 usr; 25 con; 0-3 aty)
% Number of variables : 4648 ( 223 ^;3873 !; 552 ?;4648 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:46:25.585
%------------------------------------------------------------------------------
% 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__Otm_J_J_J,type,
set_list_list_tm: $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__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 (336)
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__List__Olist_It__SeCaV__Otm_J_J,type,
append_list_list_tm: list_list_list_tm > list_list_list_tm > list_list_list_tm ).
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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
append_list_set_nat: list_list_set_nat > list_list_set_nat > list_list_set_nat ).
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__List__Olist_It__SeCaV__Otm_J,type,
bind_fm_list_tm: list_fm > ( fm > list_list_tm ) > list_list_tm ).
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__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
bind_fm_set_nat: list_fm > ( fm > list_set_nat ) > list_set_nat ).
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_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__List__Olist_It__SeCaV__Otm_J_J,type,
cons_list_list_tm: list_list_tm > list_list_list_tm > list_list_list_tm ).
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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
cons_list_set_nat: list_set_nat > list_list_set_nat > list_list_set_nat ).
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_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__List__Olist_It__SeCaV__Otm_J_J,type,
nil_list_list_tm: list_list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
nil_list_set_nat: list_list_set_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_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__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__List__Olist_It__SeCaV__Otm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_li4116520515254442628ist_tm: ( list_list_tm > list_list_tm ) > list_list_list_tm > list_list_list_tm ).
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__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__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Ofm,type,
map_list_tm_fm: ( list_tm > fm ) > list_list_tm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
map_list_tm_tm: ( list_tm > tm ) > list_list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
map_li9190756048810191986et_nat: ( list_set_nat > list_set_nat ) > list_list_set_nat > 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__Nat__Onat_J,type,
map_fm_set_nat: ( fm > set_nat ) > list_fm > list_set_nat ).
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__SeCaV__Ofm,type,
map_set_nat_fm: ( set_nat > fm ) > list_set_nat > list_fm ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Otm,type,
map_set_nat_tm: ( set_nat > tm ) > list_set_nat > list_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_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__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_list_tm2: list_list_list_tm > set_list_list_tm ).
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_Omap__tailrec_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_ta7425747110069464646ist_tm: ( fm > list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
map_tailrec_fm_fm: ( fm > fm ) > list_fm > list_fm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
map_tailrec_fm_tm: ( fm > tm ) > list_fm > list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_ta7538967730773405780ist_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
map_tailrec_tm_fm: ( tm > fm ) > list_tm > list_fm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_tailrec_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
map_ta6199207329629434205et_nat: ( tm > set_nat ) > list_tm > 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__List__Olist_It__SeCaV__Otm_J,type,
maps_fm_list_tm: ( fm > list_list_tm ) > list_fm > list_list_tm ).
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__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
maps_fm_set_nat: ( fm > list_set_nat ) > list_fm > list_set_nat ).
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_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_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_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_Olist__prod_001t__List__Olist_It__SeCaV__Ofm_J,type,
list_prod_list_fm: list_list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_Prover_Olist__prod_001t__List__Olist_It__SeCaV__Otm_J,type,
list_prod_list_tm: list_list_list_tm > list_list_list_tm > list_list_list_tm ).
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_Olist__prod_001t__Set__Oset_It__Nat__Onat_J,type,
list_prod_set_nat: list_list_set_nat > list_list_set_nat > list_list_set_nat ).
thf(sy_c_Prover_Oparts,type,
parts: list_tm > rule > fm > list_list_fm ).
thf(sy_c_Prover_Orule_OBetaDis,type,
betaDis: rule ).
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_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__List__Olist_It__SeCaV__Otm_J_J,type,
collect_list_list_tm: ( list_list_tm > $o ) > set_list_list_tm ).
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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_list_set_nat: ( list_set_nat > $o ) > set_list_set_nat ).
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_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__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__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__Ofm_J,type,
image_tm_list_fm: ( tm > list_fm ) > set_tm > set_list_fm ).
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_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__List__Olist_It__SeCaV__Otm_J_J,type,
member_list_list_tm: list_list_tm > set_list_list_tm > $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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $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 (1271)
thf(fact_0_local_OBetaDis_I1_J,axiom,
r = betaDis ).
% local.BetaDis(1)
thf(fact_1_local_OBetaDis_I2_J,axiom,
( p
= ( neg @ ( dis @ pa @ q ) ) ) ).
% local.BetaDis(2)
thf(fact_2_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_3__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_4__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_5_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_6_set__remdups,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( remdups_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_7_set__remdups,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( remdups_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_8_set__remdups,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( remdups_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_remdups
thf(fact_9_set__remdups,axiom,
! [Xs: list_set_nat] :
( ( set_set_nat2 @ ( remdups_set_nat @ Xs ) )
= ( set_set_nat2 @ Xs ) ) ).
% set_remdups
thf(fact_10_concat__append,axiom,
! [Xs: list_list_list_fm,Ys: list_list_list_fm] :
( ( concat_list_fm @ ( append_list_list_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( concat_list_fm @ Xs ) @ ( concat_list_fm @ Ys ) ) ) ).
% concat_append
thf(fact_11_concat__append,axiom,
! [Xs: list_list_list_tm,Ys: list_list_list_tm] :
( ( concat_list_tm @ ( append_list_list_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( concat_list_tm @ Xs ) @ ( concat_list_tm @ Ys ) ) ) ).
% concat_append
thf(fact_12_concat__append,axiom,
! [Xs: list_list_set_nat,Ys: list_list_set_nat] :
( ( concat_set_nat @ ( append_list_set_nat @ Xs @ Ys ) )
= ( append_set_nat @ ( concat_set_nat @ Xs ) @ ( concat_set_nat @ Ys ) ) ) ).
% concat_append
thf(fact_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_map__append,axiom,
! [F: tm > list_fm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_list_fm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_list_fm @ ( map_tm_list_fm @ F @ Xs ) @ ( map_tm_list_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_23_map__append,axiom,
! [F: fm > list_fm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_list_fm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_list_fm @ ( map_fm_list_fm @ F @ Xs ) @ ( map_fm_list_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_24_map__append,axiom,
! [F: fm > set_nat,Xs: list_fm,Ys: list_fm] :
( ( map_fm_set_nat @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_set_nat @ ( map_fm_set_nat @ F @ Xs ) @ ( map_fm_set_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_25_map__eq__conv,axiom,
! [F: fm > fm,Xs: list_fm,G: fm > fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( map_fm_fm @ G @ Xs ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_26_map__eq__conv,axiom,
! [F: fm > tm,Xs: list_fm,G: fm > tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( map_fm_tm @ G @ Xs ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_27_map__eq__conv,axiom,
! [F: tm > fm,Xs: list_tm,G: tm > fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( map_tm_fm @ G @ Xs ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_28_map__eq__conv,axiom,
! [F: tm > tm,Xs: list_tm,G: tm > tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_29_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 ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_30_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 ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_31_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 ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_32_Neg,axiom,
! [P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P ) ) @ Z ) ) ) ).
% Neg
thf(fact_33_remdups_Osimps_I2_J,axiom,
! [X3: nat,Xs: list_nat] :
( ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( remdups_nat @ ( cons_nat @ X3 @ Xs ) )
= ( remdups_nat @ Xs ) ) )
& ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( remdups_nat @ ( cons_nat @ X3 @ Xs ) )
= ( cons_nat @ X3 @ ( remdups_nat @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_34_remdups_Osimps_I2_J,axiom,
! [X3: $o,Xs: list_o] :
( ( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( ( remdups_o @ ( cons_o @ X3 @ Xs ) )
= ( remdups_o @ Xs ) ) )
& ( ~ ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( ( remdups_o @ ( cons_o @ X3 @ Xs ) )
= ( cons_o @ X3 @ ( remdups_o @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_35_remdups_Osimps_I2_J,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( remdups_set_nat @ ( cons_set_nat @ X3 @ Xs ) )
= ( remdups_set_nat @ Xs ) ) )
& ( ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( remdups_set_nat @ ( cons_set_nat @ X3 @ Xs ) )
= ( cons_set_nat @ X3 @ ( remdups_set_nat @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_36_remdups_Osimps_I2_J,axiom,
! [X3: fm,Xs: list_fm] :
( ( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X3 @ Xs ) )
= ( remdups_fm @ Xs ) ) )
& ( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X3 @ Xs ) )
= ( cons_fm @ X3 @ ( remdups_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_37_remdups_Osimps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( remdups_list_fm @ Xs ) ) )
& ( ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( cons_list_fm @ X3 @ ( remdups_list_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_38_remdups_Osimps_I2_J,axiom,
! [X3: tm,Xs: list_tm] :
( ( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X3 @ Xs ) )
= ( remdups_tm @ Xs ) ) )
& ( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X3 @ Xs ) )
= ( cons_tm @ X3 @ ( remdups_tm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_39_split__list,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_40_split__list,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs: list_o] :
( Xs
= ( append_o @ Ys2 @ ( cons_o @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_41_split__list,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
=> ? [Ys2: list_list_tm,Zs: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_42_split__list,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_43_split__list,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_44_split__list,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_45_split__list,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs ) ) ) ) ).
% split_list
thf(fact_46_split__list__last,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_47_split__list__last,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X3 @ Zs ) ) )
& ~ ( member_o2 @ X3 @ ( set_o2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_48_split__list__last,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
=> ? [Ys2: list_list_tm,Zs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X3 @ Zs ) ) )
& ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_49_split__list__last,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs ) ) )
& ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_50_split__list__last,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs ) ) )
& ~ ( member_fm2 @ X3 @ ( set_fm2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_51_split__list__last,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs ) ) )
& ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_52_split__list__last,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs ) ) )
& ~ ( member_tm2 @ X3 @ ( set_tm2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_53_split__list__prop,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_tm,X4: list_tm] :
( ? [Zs: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_54_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] :
( ? [Zs: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_55_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] :
( ? [Zs: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_56_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] :
( ? [Zs: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_57_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] :
( ? [Zs: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_58_split__list__first,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_59_split__list__first,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ? [Ys2: list_o,Zs: list_o] :
( ( Xs
= ( append_o @ Ys2 @ ( cons_o @ X3 @ Zs ) ) )
& ~ ( member_o2 @ X3 @ ( set_o2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_60_split__list__first,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
=> ? [Ys2: list_list_tm,Zs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X3 @ Zs ) ) )
& ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_61_split__list__first,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ? [Ys2: list_set_nat,Zs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X3 @ Zs ) ) )
& ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_62_split__list__first,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ? [Ys2: list_fm,Zs: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X3 @ Zs ) ) )
& ~ ( member_fm2 @ X3 @ ( set_fm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_63_split__list__first,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ? [Ys2: list_list_fm,Zs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X3 @ Zs ) ) )
& ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_64_split__list__first,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ? [Ys2: list_tm,Zs: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X3 @ Zs ) ) )
& ~ ( member_tm2 @ X3 @ ( set_tm2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_65_split__list__propE,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_tm,X4: list_tm] :
( ? [Zs: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_66_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] :
( ? [Zs: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_67_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] :
( ? [Zs: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_68_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] :
( ? [Zs: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_69_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] :
( ? [Zs: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_70_append__Cons__eq__iff,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat,Xs2: list_nat,Ys3: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) )
= ( append_nat @ Xs2 @ ( cons_nat @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_71_append__Cons__eq__iff,axiom,
! [X3: $o,Xs: list_o,Ys: list_o,Xs2: list_o,Ys3: list_o] :
( ~ ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( ~ ( member_o2 @ X3 @ ( set_o2 @ Ys ) )
=> ( ( ( append_o @ Xs @ ( cons_o @ X3 @ Ys ) )
= ( append_o @ Xs2 @ ( cons_o @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_72_append__Cons__eq__iff,axiom,
! [X3: list_tm,Xs: list_list_tm,Ys: list_list_tm,Xs2: list_list_tm,Ys3: list_list_tm] :
( ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
=> ( ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Ys ) )
=> ( ( ( append_list_tm @ Xs @ ( cons_list_tm @ X3 @ Ys ) )
= ( append_list_tm @ Xs2 @ ( cons_list_tm @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_73_append__Cons__eq__iff,axiom,
! [X3: set_nat,Xs: list_set_nat,Ys: list_set_nat,Xs2: list_set_nat,Ys3: list_set_nat] :
( ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( ( append_set_nat @ Xs @ ( cons_set_nat @ X3 @ Ys ) )
= ( append_set_nat @ Xs2 @ ( cons_set_nat @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_74_append__Cons__eq__iff,axiom,
! [X3: fm,Xs: list_fm,Ys: list_fm,Xs2: list_fm,Ys3: list_fm] :
( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X3 @ Ys ) )
= ( append_fm @ Xs2 @ ( cons_fm @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_75_append__Cons__eq__iff,axiom,
! [X3: list_fm,Xs: list_list_fm,Ys: list_list_fm,Xs2: list_list_fm,Ys3: list_list_fm] :
( ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Ys ) )
=> ( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X3 @ Ys ) )
= ( append_list_fm @ Xs2 @ ( cons_list_fm @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_76_append__Cons__eq__iff,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm,Xs2: list_tm,Ys3: list_tm] :
( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X3 @ Ys ) )
= ( append_tm @ Xs2 @ ( cons_tm @ X3 @ Ys3 ) ) )
= ( ( Xs = Xs2 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_77_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_78_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_79_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_80_same__append__eq,axiom,
! [Xs: list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_81_same__append__eq,axiom,
! [Xs: list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_82_same__append__eq,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,Zs2: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= ( append_list_fm @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_83_same__append__eq,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs2: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_84_same__append__eq,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,Zs2: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= ( append_set_nat @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_85_append__same__eq,axiom,
! [Ys: list_tm,Xs: list_tm,Zs2: list_tm] :
( ( ( append_tm @ Ys @ Xs )
= ( append_tm @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_86_append__same__eq,axiom,
! [Ys: list_fm,Xs: list_fm,Zs2: list_fm] :
( ( ( append_fm @ Ys @ Xs )
= ( append_fm @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_87_append__same__eq,axiom,
! [Ys: list_list_fm,Xs: list_list_fm,Zs2: list_list_fm] :
( ( ( append_list_fm @ Ys @ Xs )
= ( append_list_fm @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_88_append__same__eq,axiom,
! [Ys: list_list_tm,Xs: list_list_tm,Zs2: list_list_tm] :
( ( ( append_list_tm @ Ys @ Xs )
= ( append_list_tm @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_89_append__same__eq,axiom,
! [Ys: list_set_nat,Xs: list_set_nat,Zs2: list_set_nat] :
( ( ( append_set_nat @ Ys @ Xs )
= ( append_set_nat @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_90_append__assoc,axiom,
! [Xs: list_tm,Ys: list_tm,Zs2: list_tm] :
( ( append_tm @ ( append_tm @ Xs @ Ys ) @ Zs2 )
= ( append_tm @ Xs @ ( append_tm @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_91_append__assoc,axiom,
! [Xs: list_fm,Ys: list_fm,Zs2: list_fm] :
( ( append_fm @ ( append_fm @ Xs @ Ys ) @ Zs2 )
= ( append_fm @ Xs @ ( append_fm @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_92_append__assoc,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,Zs2: list_list_fm] :
( ( append_list_fm @ ( append_list_fm @ Xs @ Ys ) @ Zs2 )
= ( append_list_fm @ Xs @ ( append_list_fm @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_93_append__assoc,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs2: list_list_tm] :
( ( append_list_tm @ ( append_list_tm @ Xs @ Ys ) @ Zs2 )
= ( append_list_tm @ Xs @ ( append_list_tm @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_94_append__assoc,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,Zs2: list_set_nat] :
( ( append_set_nat @ ( append_set_nat @ Xs @ Ys ) @ Zs2 )
= ( append_set_nat @ Xs @ ( append_set_nat @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_95_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_96_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_97_append_Oassoc,axiom,
! [A: list_list_fm,B: list_list_fm,C: list_list_fm] :
( ( append_list_fm @ ( append_list_fm @ A @ B ) @ C )
= ( append_list_fm @ A @ ( append_list_fm @ B @ C ) ) ) ).
% append.assoc
thf(fact_98_append_Oassoc,axiom,
! [A: list_list_tm,B: list_list_tm,C: list_list_tm] :
( ( append_list_tm @ ( append_list_tm @ A @ B ) @ C )
= ( append_list_tm @ A @ ( append_list_tm @ B @ C ) ) ) ).
% append.assoc
thf(fact_99_append_Oassoc,axiom,
! [A: list_set_nat,B: list_set_nat,C: list_set_nat] :
( ( append_set_nat @ ( append_set_nat @ A @ B ) @ C )
= ( append_set_nat @ A @ ( append_set_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_100_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_101_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_102_concat_Osimps_I2_J,axiom,
! [X3: list_list_fm,Xs: list_list_list_fm] :
( ( concat_list_fm @ ( cons_list_list_fm @ X3 @ Xs ) )
= ( append_list_fm @ X3 @ ( concat_list_fm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_103_concat_Osimps_I2_J,axiom,
! [X3: list_list_tm,Xs: list_list_list_tm] :
( ( concat_list_tm @ ( cons_list_list_tm @ X3 @ Xs ) )
= ( append_list_tm @ X3 @ ( concat_list_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_104_concat_Osimps_I2_J,axiom,
! [X3: list_set_nat,Xs: list_list_set_nat] :
( ( concat_set_nat @ ( cons_list_set_nat @ X3 @ Xs ) )
= ( append_set_nat @ X3 @ ( concat_set_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_105_concat_Osimps_I2_J,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( concat_tm @ ( cons_list_tm @ X3 @ Xs ) )
= ( append_tm @ X3 @ ( concat_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_106_concat_Osimps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( concat_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( append_fm @ X3 @ ( concat_fm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_107_fm_Odistinct_I29_J,axiom,
! [X31: fm,X32: fm,X7: fm] :
( ( dis @ X31 @ X32 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_108_not__Cons__self2,axiom,
! [X3: fm,Xs: list_fm] :
( ( cons_fm @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_109_not__Cons__self2,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( cons_list_fm @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_110_not__Cons__self2,axiom,
! [X3: tm,Xs: list_tm] :
( ( cons_tm @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_111_append__eq__append__conv2,axiom,
! [Xs: list_tm,Ys: list_tm,Zs2: list_tm,Ts: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs2 @ Ts ) )
= ( ? [Us: list_tm] :
( ( ( Xs
= ( append_tm @ Zs2 @ Us ) )
& ( ( append_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_112_append__eq__append__conv2,axiom,
! [Xs: list_fm,Ys: list_fm,Zs2: list_fm,Ts: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs2 @ Ts ) )
= ( ? [Us: list_fm] :
( ( ( Xs
= ( append_fm @ Zs2 @ Us ) )
& ( ( append_fm @ Us @ Ys )
= Ts ) )
| ( ( ( append_fm @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_fm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_113_append__eq__append__conv2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,Zs2: list_list_fm,Ts: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= ( append_list_fm @ Zs2 @ Ts ) )
= ( ? [Us: list_list_fm] :
( ( ( Xs
= ( append_list_fm @ Zs2 @ Us ) )
& ( ( append_list_fm @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_fm @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_list_fm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_114_append__eq__append__conv2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs2: list_list_tm,Ts: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Zs2 @ Ts ) )
= ( ? [Us: list_list_tm] :
( ( ( Xs
= ( append_list_tm @ Zs2 @ Us ) )
& ( ( append_list_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_tm @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_list_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_115_append__eq__append__conv2,axiom,
! [Xs: list_set_nat,Ys: list_set_nat,Zs2: list_set_nat,Ts: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= ( append_set_nat @ Zs2 @ Ts ) )
= ( ? [Us: list_set_nat] :
( ( ( Xs
= ( append_set_nat @ Zs2 @ Us ) )
& ( ( append_set_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_set_nat @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append_set_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_116_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs2: list_tm,Ys: list_tm,Us2: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_tm @ Xs1 @ Us2 ) )
=> ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_117_append__eq__appendI,axiom,
! [Xs: list_fm,Xs1: list_fm,Zs2: list_fm,Ys: list_fm,Us2: list_fm] :
( ( ( append_fm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_fm @ Xs1 @ Us2 ) )
=> ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_118_append__eq__appendI,axiom,
! [Xs: list_list_fm,Xs1: list_list_fm,Zs2: list_list_fm,Ys: list_list_fm,Us2: list_list_fm] :
( ( ( append_list_fm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_list_fm @ Xs1 @ Us2 ) )
=> ( ( append_list_fm @ Xs @ Ys )
= ( append_list_fm @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_119_append__eq__appendI,axiom,
! [Xs: list_list_tm,Xs1: list_list_tm,Zs2: list_list_tm,Ys: list_list_tm,Us2: list_list_tm] :
( ( ( append_list_tm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_list_tm @ Xs1 @ Us2 ) )
=> ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_120_append__eq__appendI,axiom,
! [Xs: list_set_nat,Xs1: list_set_nat,Zs2: list_set_nat,Ys: list_set_nat,Us2: list_set_nat] :
( ( ( append_set_nat @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_set_nat @ Xs1 @ Us2 ) )
=> ( ( append_set_nat @ Xs @ Ys )
= ( append_set_nat @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_121_AlphaDis,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ ( cons_fm @ Q @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P @ Q ) @ Z ) ) ) ).
% AlphaDis
thf(fact_122_remdups__remdups,axiom,
! [Xs: list_tm] :
( ( remdups_tm @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ Xs ) ) ).
% remdups_remdups
thf(fact_123_remdups__remdups,axiom,
! [Xs: list_set_nat] :
( ( remdups_set_nat @ ( remdups_set_nat @ Xs ) )
= ( remdups_set_nat @ Xs ) ) ).
% remdups_remdups
thf(fact_124_remdups__remdups,axiom,
! [Xs: list_fm] :
( ( remdups_fm @ ( remdups_fm @ Xs ) )
= ( remdups_fm @ Xs ) ) ).
% remdups_remdups
thf(fact_125_remdups__remdups,axiom,
! [Xs: list_list_fm] :
( ( remdups_list_fm @ ( remdups_list_fm @ Xs ) )
= ( remdups_list_fm @ Xs ) ) ).
% remdups_remdups
thf(fact_126_SeCaV_OBetaDis,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P @ Q ) ) @ Z ) ) ) ) ).
% SeCaV.BetaDis
thf(fact_127_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_128_set__ConsD,axiom,
! [Y: $o,X3: $o,Xs: list_o] :
( ( member_o2 @ Y @ ( set_o2 @ ( cons_o @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_o2 @ Y @ ( set_o2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_129_set__ConsD,axiom,
! [Y: set_nat,X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ Y @ ( set_set_nat2 @ ( cons_set_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_set_nat2 @ Y @ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_130_set__ConsD,axiom,
! [Y: fm,X3: fm,Xs: list_fm] :
( ( member_fm2 @ Y @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_fm2 @ Y @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_131_set__ConsD,axiom,
! [Y: list_fm,X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ Y @ ( set_list_fm2 @ ( cons_list_fm @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_list_fm2 @ Y @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_132_set__ConsD,axiom,
! [Y: tm,X3: tm,Xs: list_tm] :
( ( member_tm2 @ Y @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_tm2 @ Y @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_133_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_134_list_Oset__cases,axiom,
! [E: $o,A: list_o] :
( ( member_o2 @ E @ ( set_o2 @ A ) )
=> ( ! [Z2: list_o] :
( A
!= ( cons_o @ E @ Z2 ) )
=> ~ ! [Z1: $o,Z2: list_o] :
( ( A
= ( cons_o @ Z1 @ Z2 ) )
=> ~ ( member_o2 @ E @ ( set_o2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_135_list_Oset__cases,axiom,
! [E: set_nat,A: list_set_nat] :
( ( member_set_nat2 @ E @ ( set_set_nat2 @ A ) )
=> ( ! [Z2: list_set_nat] :
( A
!= ( cons_set_nat @ E @ Z2 ) )
=> ~ ! [Z1: set_nat,Z2: list_set_nat] :
( ( A
= ( cons_set_nat @ Z1 @ Z2 ) )
=> ~ ( member_set_nat2 @ E @ ( set_set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_136_list_Oset__cases,axiom,
! [E: fm,A: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A ) )
=> ( ! [Z2: list_fm] :
( A
!= ( cons_fm @ E @ Z2 ) )
=> ~ ! [Z1: fm,Z2: list_fm] :
( ( A
= ( cons_fm @ Z1 @ Z2 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_137_list_Oset__cases,axiom,
! [E: list_fm,A: list_list_fm] :
( ( member_list_fm2 @ E @ ( set_list_fm2 @ A ) )
=> ( ! [Z2: list_list_fm] :
( A
!= ( cons_list_fm @ E @ Z2 ) )
=> ~ ! [Z1: list_fm,Z2: list_list_fm] :
( ( A
= ( cons_list_fm @ Z1 @ Z2 ) )
=> ~ ( member_list_fm2 @ E @ ( set_list_fm2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_138_list_Oset__cases,axiom,
! [E: tm,A: list_tm] :
( ( member_tm2 @ E @ ( set_tm2 @ A ) )
=> ( ! [Z2: list_tm] :
( A
!= ( cons_tm @ E @ Z2 ) )
=> ~ ! [Z1: tm,Z2: list_tm] :
( ( A
= ( cons_tm @ Z1 @ Z2 ) )
=> ~ ( member_tm2 @ E @ ( set_tm2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_Cons__eq__appendI,axiom,
! [X3: list_tm,Xs1: list_list_tm,Ys: list_list_tm,Xs: list_list_tm,Zs2: list_list_tm] :
( ( ( cons_list_tm @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_tm @ Xs1 @ Zs2 ) )
=> ( ( cons_list_tm @ X3 @ Xs )
= ( append_list_tm @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_152_Cons__eq__appendI,axiom,
! [X3: set_nat,Xs1: list_set_nat,Ys: list_set_nat,Xs: list_set_nat,Zs2: list_set_nat] :
( ( ( cons_set_nat @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_set_nat @ Xs1 @ Zs2 ) )
=> ( ( cons_set_nat @ X3 @ Xs )
= ( append_set_nat @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_153_Cons__eq__appendI,axiom,
! [X3: fm,Xs1: list_fm,Ys: list_fm,Xs: list_fm,Zs2: list_fm] :
( ( ( cons_fm @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_fm @ Xs1 @ Zs2 ) )
=> ( ( cons_fm @ X3 @ Xs )
= ( append_fm @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_154_Cons__eq__appendI,axiom,
! [X3: list_fm,Xs1: list_list_fm,Ys: list_list_fm,Xs: list_list_fm,Zs2: list_list_fm] :
( ( ( cons_list_fm @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_fm @ Xs1 @ Zs2 ) )
=> ( ( cons_list_fm @ X3 @ Xs )
= ( append_list_fm @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_155_Cons__eq__appendI,axiom,
! [X3: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs2: list_tm] :
( ( ( cons_tm @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs2 ) )
=> ( ( cons_tm @ X3 @ Xs )
= ( append_tm @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_156_append__Cons,axiom,
! [X3: list_tm,Xs: list_list_tm,Ys: list_list_tm] :
( ( append_list_tm @ ( cons_list_tm @ X3 @ Xs ) @ Ys )
= ( cons_list_tm @ X3 @ ( append_list_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_157_append__Cons,axiom,
! [X3: set_nat,Xs: list_set_nat,Ys: list_set_nat] :
( ( append_set_nat @ ( cons_set_nat @ X3 @ Xs ) @ Ys )
= ( cons_set_nat @ X3 @ ( append_set_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_158_append__Cons,axiom,
! [X3: fm,Xs: list_fm,Ys: list_fm] :
( ( append_fm @ ( cons_fm @ X3 @ Xs ) @ Ys )
= ( cons_fm @ X3 @ ( append_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_159_append__Cons,axiom,
! [X3: list_fm,Xs: list_list_fm,Ys: list_list_fm] :
( ( append_list_fm @ ( cons_list_fm @ X3 @ Xs ) @ Ys )
= ( cons_list_fm @ X3 @ ( append_list_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_160_append__Cons,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X3 @ Xs ) @ Ys )
= ( cons_tm @ X3 @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_161_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,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_fm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_162_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,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_fm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_163_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,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_tm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_164_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,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_tm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_165_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,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_list_tm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_166_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,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_set_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_167_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,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_list_tm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_168_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,Zs3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_list_fm_fm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_169_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,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_fm_list_fm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_170_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,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_tm_list_fm @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_171_Cons__eq__map__conv,axiom,
! [X3: fm,Xs: list_fm,F: fm > fm,Ys: list_fm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_fm_fm @ F @ Ys ) )
= ( ? [Z3: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_fm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_172_Cons__eq__map__conv,axiom,
! [X3: fm,Xs: list_fm,F: tm > fm,Ys: list_tm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_tm_fm @ F @ Ys ) )
= ( ? [Z3: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_fm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_173_Cons__eq__map__conv,axiom,
! [X3: tm,Xs: list_tm,F: fm > tm,Ys: list_fm] :
( ( ( cons_tm @ X3 @ Xs )
= ( map_fm_tm @ F @ Ys ) )
= ( ? [Z3: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_tm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_174_Cons__eq__map__conv,axiom,
! [X3: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X3 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
= ( ? [Z3: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_175_Cons__eq__map__conv,axiom,
! [X3: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X3 @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
= ( ? [Z3: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_176_Cons__eq__map__conv,axiom,
! [X3: set_nat,Xs: list_set_nat,F: tm > set_nat,Ys: list_tm] :
( ( ( cons_set_nat @ X3 @ Xs )
= ( map_tm_set_nat @ F @ Ys ) )
= ( ? [Z3: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_set_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_177_Cons__eq__map__conv,axiom,
! [X3: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X3 @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
= ( ? [Z3: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_178_Cons__eq__map__conv,axiom,
! [X3: fm,Xs: list_fm,F: list_fm > fm,Ys: list_list_fm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_list_fm_fm @ F @ Ys ) )
= ( ? [Z3: list_fm,Zs3: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_list_fm_fm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_179_Cons__eq__map__conv,axiom,
! [X3: list_fm,Xs: list_list_fm,F: fm > list_fm,Ys: list_fm] :
( ( ( cons_list_fm @ X3 @ Xs )
= ( map_fm_list_fm @ F @ Ys ) )
= ( ? [Z3: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_fm_list_fm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_180_Cons__eq__map__conv,axiom,
! [X3: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X3 @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
= ( ? [Z3: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z3 @ Zs3 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_181_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 ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Xs
= ( cons_fm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_fm_fm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_182_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 ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_tm_fm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_183_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 ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Xs
= ( cons_fm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_fm_tm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_184_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 ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_tm_tm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_185_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 ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Xs
= ( cons_fm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_fm_list_tm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_186_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 ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_tm_set_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_187_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 ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_tm_list_tm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_188_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 ) )
=> ? [Z4: list_fm,Zs: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_list_fm_fm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_189_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 ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Xs
= ( cons_fm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_fm_list_fm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_190_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 ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Xs
= ( cons_tm @ Z4 @ Zs ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_tm_list_fm @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_191_Cons__eq__map__D,axiom,
! [X3: fm,Xs: list_fm,F: fm > fm,Ys: list_fm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_fm_fm @ F @ Ys ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Ys
= ( cons_fm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_fm_fm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_192_Cons__eq__map__D,axiom,
! [X3: fm,Xs: list_fm,F: tm > fm,Ys: list_tm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_tm_fm @ F @ Ys ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_tm_fm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_193_Cons__eq__map__D,axiom,
! [X3: tm,Xs: list_tm,F: fm > tm,Ys: list_fm] :
( ( ( cons_tm @ X3 @ Xs )
= ( map_fm_tm @ F @ Ys ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Ys
= ( cons_fm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_fm_tm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_194_Cons__eq__map__D,axiom,
! [X3: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X3 @ Xs )
= ( map_tm_tm @ F @ Ys ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_195_Cons__eq__map__D,axiom,
! [X3: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X3 @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Ys
= ( cons_fm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_196_Cons__eq__map__D,axiom,
! [X3: set_nat,Xs: list_set_nat,F: tm > set_nat,Ys: list_tm] :
( ( ( cons_set_nat @ X3 @ Xs )
= ( map_tm_set_nat @ F @ Ys ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_tm_set_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_197_Cons__eq__map__D,axiom,
! [X3: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X3 @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_198_Cons__eq__map__D,axiom,
! [X3: fm,Xs: list_fm,F: list_fm > fm,Ys: list_list_fm] :
( ( ( cons_fm @ X3 @ Xs )
= ( map_list_fm_fm @ F @ Ys ) )
=> ? [Z4: list_fm,Zs: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_list_fm_fm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_199_Cons__eq__map__D,axiom,
! [X3: list_fm,Xs: list_list_fm,F: fm > list_fm,Ys: list_fm] :
( ( ( cons_list_fm @ X3 @ Xs )
= ( map_fm_list_fm @ F @ Ys ) )
=> ? [Z4: fm,Zs: list_fm] :
( ( Ys
= ( cons_fm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_fm_list_fm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_200_Cons__eq__map__D,axiom,
! [X3: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X3 @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
=> ? [Z4: tm,Zs: list_tm] :
( ( Ys
= ( cons_tm @ Z4 @ Zs ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_ex__map__conv,axiom,
! [Ys: list_list_tm,F: fm > list_tm] :
( ( ? [Xs3: list_fm] :
( Ys
= ( map_fm_list_tm @ F @ Xs3 ) ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Ys ) )
=> ? [Y2: fm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_212_ex__map__conv,axiom,
! [Ys: list_list_tm,F: tm > list_tm] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_list_tm @ F @ Xs3 ) ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Ys ) )
=> ? [Y2: tm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_213_ex__map__conv,axiom,
! [Ys: list_fm,F: fm > fm] :
( ( ? [Xs3: list_fm] :
( Ys
= ( map_fm_fm @ F @ Xs3 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Ys ) )
=> ? [Y2: fm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_214_ex__map__conv,axiom,
! [Ys: list_fm,F: tm > fm] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_fm @ F @ Xs3 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Ys ) )
=> ? [Y2: tm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_215_ex__map__conv,axiom,
! [Ys: list_tm,F: fm > tm] :
( ( ? [Xs3: list_fm] :
( Ys
= ( map_fm_tm @ F @ Xs3 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Ys ) )
=> ? [Y2: fm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_216_ex__map__conv,axiom,
! [Ys: list_tm,F: tm > tm] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_tm @ F @ Xs3 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Ys ) )
=> ? [Y2: tm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_217_ex__map__conv,axiom,
! [Ys: list_set_nat,F: tm > set_nat] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_set_nat @ F @ Xs3 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Ys ) )
=> ? [Y2: tm] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_218_mem__Collect__eq,axiom,
! [A: fm,P2: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
! [A: $o,P2: $o > $o] :
( ( member_o2 @ A @ ( collect_o @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
! [A: tm,P2: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_222_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_223_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
! [A2: set_list_fm] :
( ( collect_list_fm
@ ^ [X2: list_fm] : ( member_list_fm2 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_228_Collect__cong,axiom,
! [P2: list_fm > $o,Q2: list_fm > $o] :
( ! [X4: list_fm] :
( ( P2 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collect_list_fm @ P2 )
= ( collect_list_fm @ Q2 ) ) ) ).
% Collect_cong
thf(fact_229_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_230_map__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > fm,G: fm > fm] :
( ( Xs = Ys )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_fm @ F @ Xs )
= ( map_fm_fm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_231_map__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > tm,G: fm > tm] :
( ( Xs = Ys )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_tm @ F @ Xs )
= ( map_fm_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_232_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_233_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_234_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_235_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > tm,G: tm > tm] :
( ( Xs = Ys )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_map__ext,axiom,
! [Xs: list_fm,F: fm > fm,G: fm > fm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_fm @ F @ Xs )
= ( map_fm_fm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_244_map__ext,axiom,
! [Xs: list_fm,F: fm > tm,G: fm > tm] :
( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_fm_tm @ F @ Xs )
= ( map_fm_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_245_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_246_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_247_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_248_map__ext,axiom,
! [Xs: list_tm,F: tm > tm,G: tm > tm] :
( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_249_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z4: nat] :
( ( member_nat2 @ Z4 @ ( set_nat2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_250_list_Omap__ident__strong,axiom,
! [T: list_o,F: $o > $o] :
( ! [Z4: $o] :
( ( member_o2 @ Z4 @ ( set_o2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_o_o @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_251_list_Omap__ident__strong,axiom,
! [T: list_list_fm,F: list_fm > list_fm] :
( ! [Z4: list_fm] :
( ( member_list_fm2 @ Z4 @ ( set_list_fm2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_list_fm_list_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_252_list_Omap__ident__strong,axiom,
! [T: list_fm,F: fm > fm] :
( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_fm_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_253_list_Omap__ident__strong,axiom,
! [T: list_tm,F: tm > tm] :
( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_tm_tm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_254_list_Omap__ident__strong,axiom,
! [T: list_set_nat,F: set_nat > set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat2 @ Z4 @ ( set_set_nat2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_set_nat_set_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_255_list_Oinj__map__strong,axiom,
! [X3: list_fm,Xa2: list_fm,F: fm > list_tm,Fa: fm > list_tm] :
( ! [Z4: fm,Za: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_fm_list_tm @ F @ X3 )
= ( map_fm_list_tm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_256_list_Oinj__map__strong,axiom,
! [X3: list_fm,Xa2: list_fm,F: fm > fm,Fa: fm > fm] :
( ! [Z4: fm,Za: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_fm_fm @ F @ X3 )
= ( map_fm_fm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_257_list_Oinj__map__strong,axiom,
! [X3: list_fm,Xa2: list_fm,F: fm > tm,Fa: fm > tm] :
( ! [Z4: fm,Za: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_fm_tm @ F @ X3 )
= ( map_fm_tm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_258_list_Oinj__map__strong,axiom,
! [X3: list_tm,Xa2: list_tm,F: tm > set_nat,Fa: tm > set_nat] :
( ! [Z4: tm,Za: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_tm_set_nat @ F @ X3 )
= ( map_tm_set_nat @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_259_list_Oinj__map__strong,axiom,
! [X3: list_tm,Xa2: list_tm,F: tm > list_tm,Fa: tm > list_tm] :
( ! [Z4: tm,Za: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_tm_list_tm @ F @ X3 )
= ( map_tm_list_tm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_260_list_Oinj__map__strong,axiom,
! [X3: list_tm,Xa2: list_tm,F: tm > fm,Fa: tm > fm] :
( ! [Z4: tm,Za: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_tm_fm @ F @ X3 )
= ( map_tm_fm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_261_list_Oinj__map__strong,axiom,
! [X3: list_tm,Xa2: list_tm,F: tm > tm,Fa: tm > tm] :
( ! [Z4: tm,Za: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_tm_tm @ F @ X3 )
= ( map_tm_tm @ Fa @ Xa2 ) )
=> ( X3 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_262_list_Omap__cong0,axiom,
! [X3: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_list_tm @ F @ X3 )
= ( map_fm_list_tm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_263_list_Omap__cong0,axiom,
! [X3: list_fm,F: fm > fm,G: fm > fm] :
( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_fm @ F @ X3 )
= ( map_fm_fm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_264_list_Omap__cong0,axiom,
! [X3: list_fm,F: fm > tm,G: fm > tm] :
( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_tm @ F @ X3 )
= ( map_fm_tm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_265_list_Omap__cong0,axiom,
! [X3: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_set_nat @ F @ X3 )
= ( map_tm_set_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_266_list_Omap__cong0,axiom,
! [X3: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_list_tm @ F @ X3 )
= ( map_tm_list_tm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_267_list_Omap__cong0,axiom,
! [X3: list_tm,F: tm > fm,G: tm > fm] :
( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_fm @ F @ X3 )
= ( map_tm_fm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_268_list_Omap__cong0,axiom,
! [X3: list_tm,F: tm > tm,G: tm > tm] :
( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_tm @ F @ X3 )
= ( map_tm_tm @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_269_list_Omap__cong,axiom,
! [X3: list_fm,Ya: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ( X3 = Ya )
=> ( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_list_tm @ F @ X3 )
= ( map_fm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_270_list_Omap__cong,axiom,
! [X3: list_fm,Ya: list_fm,F: fm > fm,G: fm > fm] :
( ( X3 = Ya )
=> ( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_fm @ F @ X3 )
= ( map_fm_fm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_271_list_Omap__cong,axiom,
! [X3: list_fm,Ya: list_fm,F: fm > tm,G: fm > tm] :
( ( X3 = Ya )
=> ( ! [Z4: fm] :
( ( member_fm2 @ Z4 @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_fm_tm @ F @ X3 )
= ( map_fm_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_272_list_Omap__cong,axiom,
! [X3: list_tm,Ya: list_tm,F: tm > set_nat,G: tm > set_nat] :
( ( X3 = Ya )
=> ( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_set_nat @ F @ X3 )
= ( map_tm_set_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_273_list_Omap__cong,axiom,
! [X3: list_tm,Ya: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ( X3 = Ya )
=> ( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_list_tm @ F @ X3 )
= ( map_tm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_274_list_Omap__cong,axiom,
! [X3: list_tm,Ya: list_tm,F: tm > fm,G: tm > fm] :
( ( X3 = Ya )
=> ( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_fm @ F @ X3 )
= ( map_tm_fm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_275_list_Omap__cong,axiom,
! [X3: list_tm,Ya: list_tm,F: tm > tm,G: tm > tm] :
( ( X3 = Ya )
=> ( ! [Z4: tm] :
( ( member_tm2 @ Z4 @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_tm_tm @ F @ X3 )
= ( map_tm_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_276_map__eq__append__conv,axiom,
! [F: fm > fm,Xs: list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs2
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_277_map__eq__append__conv,axiom,
! [F: fm > tm,Xs: list_fm,Ys: list_tm,Zs2: list_tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs2
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_278_map__eq__append__conv,axiom,
! [F: tm > fm,Xs: list_tm,Ys: list_fm,Zs2: list_fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs2
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_279_map__eq__append__conv,axiom,
! [F: tm > tm,Xs: list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs2
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_280_map__eq__append__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm,Ys: list_tm,Zs2: list_tm] :
( ( ( map_list_fm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ? [Us: list_list_fm,Vs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us @ Vs ) )
& ( Ys
= ( map_list_fm_tm @ F @ Us ) )
& ( Zs2
= ( map_list_fm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_281_map__eq__append__conv,axiom,
! [F: list_tm > tm,Xs: list_list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( map_list_tm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_tm @ F @ Us ) )
& ( Zs2
= ( map_list_tm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_282_map__eq__append__conv,axiom,
! [F: set_nat > tm,Xs: list_set_nat,Ys: list_tm,Zs2: list_tm] :
( ( ( map_set_nat_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ? [Us: list_set_nat,Vs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Us @ Vs ) )
& ( Ys
= ( map_set_nat_tm @ F @ Us ) )
& ( Zs2
= ( map_set_nat_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_283_map__eq__append__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( map_list_fm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ? [Us: list_list_fm,Vs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us @ Vs ) )
& ( Ys
= ( map_list_fm_fm @ F @ Us ) )
& ( Zs2
= ( map_list_fm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_284_map__eq__append__conv,axiom,
! [F: list_tm > fm,Xs: list_list_tm,Ys: list_fm,Zs2: list_fm] :
( ( ( map_list_tm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_fm @ F @ Us ) )
& ( Zs2
= ( map_list_tm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_285_map__eq__append__conv,axiom,
! [F: set_nat > fm,Xs: list_set_nat,Ys: list_fm,Zs2: list_fm] :
( ( ( map_set_nat_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ? [Us: list_set_nat,Vs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Us @ Vs ) )
& ( Ys
= ( map_set_nat_fm @ F @ Us ) )
& ( Zs2
= ( map_set_nat_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_286_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,F: fm > fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs2 )
= ( map_fm_fm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs2
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_287_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,F: fm > tm,Xs: list_fm] :
( ( ( append_tm @ Ys @ Zs2 )
= ( map_fm_tm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs2
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_288_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,F: tm > fm,Xs: list_tm] :
( ( ( append_fm @ Ys @ Zs2 )
= ( map_tm_fm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs2
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_289_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,F: tm > tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs2 )
= ( map_tm_tm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs2
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_290_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,F: list_fm > tm,Xs: list_list_fm] :
( ( ( append_tm @ Ys @ Zs2 )
= ( map_list_fm_tm @ F @ Xs ) )
= ( ? [Us: list_list_fm,Vs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us @ Vs ) )
& ( Ys
= ( map_list_fm_tm @ F @ Us ) )
& ( Zs2
= ( map_list_fm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_291_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,F: list_tm > tm,Xs: list_list_tm] :
( ( ( append_tm @ Ys @ Zs2 )
= ( map_list_tm_tm @ F @ Xs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_tm @ F @ Us ) )
& ( Zs2
= ( map_list_tm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_292_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,F: set_nat > tm,Xs: list_set_nat] :
( ( ( append_tm @ Ys @ Zs2 )
= ( map_set_nat_tm @ F @ Xs ) )
= ( ? [Us: list_set_nat,Vs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Us @ Vs ) )
& ( Ys
= ( map_set_nat_tm @ F @ Us ) )
& ( Zs2
= ( map_set_nat_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_293_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,F: list_fm > fm,Xs: list_list_fm] :
( ( ( append_fm @ Ys @ Zs2 )
= ( map_list_fm_fm @ F @ Xs ) )
= ( ? [Us: list_list_fm,Vs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us @ Vs ) )
& ( Ys
= ( map_list_fm_fm @ F @ Us ) )
& ( Zs2
= ( map_list_fm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_294_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,F: list_tm > fm,Xs: list_list_tm] :
( ( ( append_fm @ Ys @ Zs2 )
= ( map_list_tm_fm @ F @ Xs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_fm @ F @ Us ) )
& ( Zs2
= ( map_list_tm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_295_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,F: set_nat > fm,Xs: list_set_nat] :
( ( ( append_fm @ Ys @ Zs2 )
= ( map_set_nat_fm @ F @ Xs ) )
= ( ? [Us: list_set_nat,Vs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Us @ Vs ) )
& ( Ys
= ( map_set_nat_fm @ F @ Us ) )
& ( Zs2
= ( map_set_nat_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_remdups__append2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( remdups_list_tm @ ( append_list_tm @ Xs @ ( remdups_list_tm @ Ys ) ) )
= ( remdups_list_tm @ ( append_list_tm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_304_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_305_remdups__append2,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( remdups_set_nat @ ( append_set_nat @ Xs @ ( remdups_set_nat @ Ys ) ) )
= ( remdups_set_nat @ ( append_set_nat @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_306_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_307_remdups__append2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( remdups_list_fm @ ( append_list_fm @ Xs @ ( remdups_list_fm @ Ys ) ) )
= ( remdups_list_fm @ ( append_list_fm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_308_remdups__map__remdups,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( remdups_fm @ ( map_fm_fm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_fm @ ( map_fm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_309_remdups__map__remdups,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( remdups_tm @ ( map_fm_tm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_tm @ ( map_fm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_310_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_311_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_312_remdups__map__remdups,axiom,
! [F: set_nat > tm,Xs: list_set_nat] :
( ( remdups_tm @ ( map_set_nat_tm @ F @ ( remdups_set_nat @ Xs ) ) )
= ( remdups_tm @ ( map_set_nat_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_313_remdups__map__remdups,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( remdups_tm @ ( map_list_fm_tm @ F @ ( remdups_list_fm @ Xs ) ) )
= ( remdups_tm @ ( map_list_fm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_314_remdups__map__remdups,axiom,
! [F: fm > set_nat,Xs: list_fm] :
( ( remdups_set_nat @ ( map_fm_set_nat @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_set_nat @ ( map_fm_set_nat @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_315_remdups__map__remdups,axiom,
! [F: set_nat > fm,Xs: list_set_nat] :
( ( remdups_fm @ ( map_set_nat_fm @ F @ ( remdups_set_nat @ Xs ) ) )
= ( remdups_fm @ ( map_set_nat_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_316_remdups__map__remdups,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( remdups_fm @ ( map_list_fm_fm @ F @ ( remdups_list_fm @ Xs ) ) )
= ( remdups_fm @ ( map_list_fm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_317_remdups__map__remdups,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( remdups_list_fm @ ( map_tm_list_fm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_list_fm @ ( map_tm_list_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_318_split__list__first__prop__iff,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ( ? [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_list_tm,X2: list_tm] :
( ? [Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys4 @ ( cons_list_tm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_tm] :
( ( member_list_tm @ Y2 @ ( set_list_tm2 @ Ys4 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_319_split__list__first__prop__iff,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_set_nat,X2: set_nat] :
( ? [Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys4 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: set_nat] :
( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Ys4 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_320_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_fm,X2: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys4 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: fm] :
( ( member_fm2 @ Y2 @ ( set_fm2 @ Ys4 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_321_split__list__first__prop__iff,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_list_fm,X2: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys4 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_fm] :
( ( member_list_fm2 @ Y2 @ ( set_list_fm2 @ Ys4 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_322_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_tm,X2: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys4 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( set_tm2 @ Ys4 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_323_split__list__last__prop__iff,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ( ? [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_list_tm,X2: list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys4 @ ( cons_list_tm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_tm] :
( ( member_list_tm @ Y2 @ ( set_list_tm2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_324_split__list__last__prop__iff,axiom,
! [Xs: list_set_nat,P2: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_set_nat,X2: set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys4 @ ( cons_set_nat @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: set_nat] :
( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_325_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P2: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_fm,X2: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys4 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: fm] :
( ( member_fm2 @ Y2 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_326_split__list__last__prop__iff,axiom,
! [Xs: list_list_fm,P2: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_list_fm,X2: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys4 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: list_fm] :
( ( member_list_fm2 @ Y2 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_327_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys4: list_tm,X2: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys4 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P2 @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_328_in__set__conv__decomp__first,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_329_in__set__conv__decomp__first,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
= ( ? [Ys4: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys4 @ ( cons_o @ X3 @ Zs3 ) ) )
& ~ ( member_o2 @ X3 @ ( set_o2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_330_in__set__conv__decomp__first,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys4: list_list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys4 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
& ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_331_in__set__conv__decomp__first,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys4: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys4 @ ( cons_set_nat @ X3 @ Zs3 ) ) )
& ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_332_in__set__conv__decomp__first,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys4: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys4 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ~ ( member_fm2 @ X3 @ ( set_fm2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_333_in__set__conv__decomp__first,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys4: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys4 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_334_in__set__conv__decomp__first,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys4: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys4 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ~ ( member_tm2 @ X3 @ ( set_tm2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_335_in__set__conv__decomp__last,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_336_in__set__conv__decomp__last,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
= ( ? [Ys4: list_o,Zs3: list_o] :
( ( Xs
= ( append_o @ Ys4 @ ( cons_o @ X3 @ Zs3 ) ) )
& ~ ( member_o2 @ X3 @ ( set_o2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_337_in__set__conv__decomp__last,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys4: list_list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys4 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
& ~ ( member_list_tm @ X3 @ ( set_list_tm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_338_in__set__conv__decomp__last,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys4: list_set_nat,Zs3: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys4 @ ( cons_set_nat @ X3 @ Zs3 ) ) )
& ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_339_in__set__conv__decomp__last,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys4: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys4 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ~ ( member_fm2 @ X3 @ ( set_fm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_340_in__set__conv__decomp__last,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys4: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys4 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_341_in__set__conv__decomp__last,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys4: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys4 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ~ ( member_tm2 @ X3 @ ( set_tm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_342_split__list__first__propE,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_tm,X4: list_tm] :
( ? [Zs: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_tm] :
( ( member_list_tm @ Xa @ ( set_list_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_343_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] :
( ? [Zs: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_344_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] :
( ? [Zs: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_345_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] :
( ? [Zs: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_346_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] :
( ? [Zs: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_347_split__list__last__propE,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ~ ! [Ys2: list_list_tm,X4: list_tm,Zs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_tm] :
( ( member_list_tm @ Xa @ ( set_list_tm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_348_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,Zs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_349_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,Zs: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_350_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,Zs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_351_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,Zs: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_352_split__list__first__prop,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_tm,X4: list_tm] :
( ? [Zs: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: list_tm] :
( ( member_list_tm @ Xa @ ( set_list_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_353_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] :
( ? [Zs: list_set_nat] :
( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_354_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] :
( ? [Zs: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_355_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] :
( ? [Zs: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_356_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] :
( ? [Zs: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Ys2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_357_split__list__last__prop,axiom,
! [Xs: list_list_tm,P2: list_tm > $o] :
( ? [X: list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
& ( P2 @ X ) )
=> ? [Ys2: list_list_tm,X4: list_tm,Zs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: list_tm] :
( ( member_list_tm @ Xa @ ( set_list_tm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_358_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,Zs: list_set_nat] :
( ( Xs
= ( append_set_nat @ Ys2 @ ( cons_set_nat @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ ( set_set_nat2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_359_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,Zs: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: fm] :
( ( member_fm2 @ Xa @ ( set_fm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_360_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,Zs: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: list_fm] :
( ( member_list_fm2 @ Xa @ ( set_list_fm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_361_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,Zs: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X4 @ Zs ) ) )
& ( P2 @ X4 )
& ! [Xa: tm] :
( ( member_tm2 @ Xa @ ( set_tm2 @ Zs ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_362_in__set__conv__decomp,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys4: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_363_in__set__conv__decomp,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
= ( ? [Ys4: list_o,Zs3: list_o] :
( Xs
= ( append_o @ Ys4 @ ( cons_o @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_364_in__set__conv__decomp,axiom,
! [X3: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X3 @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys4: list_list_tm,Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys4 @ ( cons_list_tm @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_365_in__set__conv__decomp,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
= ( ? [Ys4: list_set_nat,Zs3: list_set_nat] :
( Xs
= ( append_set_nat @ Ys4 @ ( cons_set_nat @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_366_in__set__conv__decomp,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
= ( ? [Ys4: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys4 @ ( cons_fm @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_367_in__set__conv__decomp,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys4: list_list_fm,Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys4 @ ( cons_list_fm @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_368_in__set__conv__decomp,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
= ( ? [Ys4: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys4 @ ( cons_tm @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_369_subtermFm_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( dis @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(3)
thf(fact_370__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_371_subtermFm_Osimps_I7_J,axiom,
! [P: fm] :
( ( subtermFm @ ( neg @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(7)
thf(fact_372_bind__simps_I2_J,axiom,
! [X3: fm,Xs: list_fm,F: fm > list_tm] :
( ( bind_fm_tm @ ( cons_fm @ X3 @ Xs ) @ F )
= ( append_tm @ ( F @ X3 ) @ ( bind_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_373_bind__simps_I2_J,axiom,
! [X3: fm,Xs: list_fm,F: fm > list_fm] :
( ( bind_fm_fm @ ( cons_fm @ X3 @ Xs ) @ F )
= ( append_fm @ ( F @ X3 ) @ ( bind_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_374_bind__simps_I2_J,axiom,
! [X3: tm,Xs: list_tm,F: tm > list_tm] :
( ( bind_tm_tm @ ( cons_tm @ X3 @ Xs ) @ F )
= ( append_tm @ ( F @ X3 ) @ ( bind_tm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_375_bind__simps_I2_J,axiom,
! [X3: tm,Xs: list_tm,F: tm > list_fm] :
( ( bind_tm_fm @ ( cons_tm @ X3 @ Xs ) @ F )
= ( append_fm @ ( F @ X3 ) @ ( bind_tm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_376_bind__simps_I2_J,axiom,
! [X3: fm,Xs: list_fm,F: fm > list_list_fm] :
( ( bind_fm_list_fm @ ( cons_fm @ X3 @ Xs ) @ F )
= ( append_list_fm @ ( F @ X3 ) @ ( bind_fm_list_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_377_bind__simps_I2_J,axiom,
! [X3: fm,Xs: list_fm,F: fm > list_list_tm] :
( ( bind_fm_list_tm @ ( cons_fm @ X3 @ Xs ) @ F )
= ( append_list_tm @ ( F @ X3 ) @ ( bind_fm_list_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_378_bind__simps_I2_J,axiom,
! [X3: fm,Xs: list_fm,F: fm > list_set_nat] :
( ( bind_fm_set_nat @ ( cons_fm @ X3 @ Xs ) @ F )
= ( append_set_nat @ ( F @ X3 ) @ ( bind_fm_set_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_379_bind__simps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm,F: list_fm > list_tm] :
( ( bind_list_fm_tm @ ( cons_list_fm @ X3 @ Xs ) @ F )
= ( append_tm @ ( F @ X3 ) @ ( bind_list_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_380_bind__simps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm,F: list_fm > list_fm] :
( ( bind_list_fm_fm @ ( cons_list_fm @ X3 @ Xs ) @ F )
= ( append_fm @ ( F @ X3 ) @ ( bind_list_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_381_bind__simps_I2_J,axiom,
! [X3: tm,Xs: list_tm,F: tm > list_list_fm] :
( ( bind_tm_list_fm @ ( cons_tm @ X3 @ Xs ) @ F )
= ( append_list_fm @ ( F @ X3 ) @ ( bind_tm_list_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_382_Basic,axiom,
! [P: fm,Z: list_fm] :
( ( member_fm @ ( neg @ P ) @ Z )
=> ( sequent_calculus @ ( cons_fm @ P @ Z ) ) ) ).
% Basic
thf(fact_383_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_384_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_385_maps__def,axiom,
( maps_fm_tm
= ( ^ [F2: fm > list_tm,Xs3: list_fm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_386_maps__def,axiom,
( maps_tm_tm
= ( ^ [F2: tm > list_tm,Xs3: list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_387_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_388_maps__simps_I1_J,axiom,
! [F: fm > list_tm,X3: fm,Xs: list_fm] :
( ( maps_fm_tm @ F @ ( cons_fm @ X3 @ Xs ) )
= ( append_tm @ ( F @ X3 ) @ ( maps_fm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_389_maps__simps_I1_J,axiom,
! [F: fm > list_fm,X3: fm,Xs: list_fm] :
( ( maps_fm_fm @ F @ ( cons_fm @ X3 @ Xs ) )
= ( append_fm @ ( F @ X3 ) @ ( maps_fm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_390_maps__simps_I1_J,axiom,
! [F: tm > list_tm,X3: tm,Xs: list_tm] :
( ( maps_tm_tm @ F @ ( cons_tm @ X3 @ Xs ) )
= ( append_tm @ ( F @ X3 ) @ ( maps_tm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_391_maps__simps_I1_J,axiom,
! [F: tm > list_fm,X3: tm,Xs: list_tm] :
( ( maps_tm_fm @ F @ ( cons_tm @ X3 @ Xs ) )
= ( append_fm @ ( F @ X3 ) @ ( maps_tm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_392_maps__simps_I1_J,axiom,
! [F: fm > list_list_fm,X3: fm,Xs: list_fm] :
( ( maps_fm_list_fm @ F @ ( cons_fm @ X3 @ Xs ) )
= ( append_list_fm @ ( F @ X3 ) @ ( maps_fm_list_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_393_maps__simps_I1_J,axiom,
! [F: fm > list_list_tm,X3: fm,Xs: list_fm] :
( ( maps_fm_list_tm @ F @ ( cons_fm @ X3 @ Xs ) )
= ( append_list_tm @ ( F @ X3 ) @ ( maps_fm_list_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_394_maps__simps_I1_J,axiom,
! [F: fm > list_set_nat,X3: fm,Xs: list_fm] :
( ( maps_fm_set_nat @ F @ ( cons_fm @ X3 @ Xs ) )
= ( append_set_nat @ ( F @ X3 ) @ ( maps_fm_set_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_395_maps__simps_I1_J,axiom,
! [F: list_fm > list_tm,X3: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_tm @ F @ ( cons_list_fm @ X3 @ Xs ) )
= ( append_tm @ ( F @ X3 ) @ ( maps_list_fm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_396_maps__simps_I1_J,axiom,
! [F: list_fm > list_fm,X3: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_fm @ F @ ( cons_list_fm @ X3 @ Xs ) )
= ( append_fm @ ( F @ X3 ) @ ( maps_list_fm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_397_maps__simps_I1_J,axiom,
! [F: tm > list_list_fm,X3: tm,Xs: list_tm] :
( ( maps_tm_list_fm @ F @ ( cons_tm @ X3 @ Xs ) )
= ( append_list_fm @ ( F @ X3 ) @ ( maps_tm_list_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_398_not__in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= ( cons_nat @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_399_not__in__set__insert,axiom,
! [X3: $o,Xs: list_o] :
( ~ ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( ( insert_o @ X3 @ Xs )
= ( cons_o @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_400_not__in__set__insert,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X3 @ Xs )
= ( cons_set_nat @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_401_not__in__set__insert,axiom,
! [X3: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X3 @ Xs )
= ( cons_fm @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_402_not__in__set__insert,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ~ ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X3 @ Xs )
= ( cons_list_fm @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_403_not__in__set__insert,axiom,
! [X3: tm,Xs: list_tm] :
( ~ ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X3 @ Xs )
= ( cons_tm @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_404_member,axiom,
( member_nat
= ( ^ [P3: nat,Z3: list_nat] : ( member_nat2 @ P3 @ ( set_nat2 @ Z3 ) ) ) ) ).
% member
thf(fact_405_member,axiom,
( member_o
= ( ^ [P3: $o,Z3: list_o] : ( member_o2 @ P3 @ ( set_o2 @ Z3 ) ) ) ) ).
% member
thf(fact_406_member,axiom,
( member_list_fm
= ( ^ [P3: list_fm,Z3: list_list_fm] : ( member_list_fm2 @ P3 @ ( set_list_fm2 @ Z3 ) ) ) ) ).
% member
thf(fact_407_member,axiom,
( member_fm
= ( ^ [P3: fm,Z3: list_fm] : ( member_fm2 @ P3 @ ( set_fm2 @ Z3 ) ) ) ) ).
% member
thf(fact_408_member,axiom,
( member_tm
= ( ^ [P3: tm,Z3: list_tm] : ( member_tm2 @ P3 @ ( set_tm2 @ Z3 ) ) ) ) ).
% member
thf(fact_409_member,axiom,
( member_set_nat
= ( ^ [P3: set_nat,Z3: list_set_nat] : ( member_set_nat2 @ P3 @ ( set_set_nat2 @ Z3 ) ) ) ) ).
% member
thf(fact_410_map__eq__map__tailrec,axiom,
map_fm_list_tm = map_ta7425747110069464646ist_tm ).
% map_eq_map_tailrec
thf(fact_411_map__eq__map__tailrec,axiom,
map_tm_set_nat = map_ta6199207329629434205et_nat ).
% map_eq_map_tailrec
thf(fact_412_map__eq__map__tailrec,axiom,
map_tm_list_tm = map_ta7538967730773405780ist_tm ).
% map_eq_map_tailrec
thf(fact_413_map__eq__map__tailrec,axiom,
map_fm_fm = map_tailrec_fm_fm ).
% map_eq_map_tailrec
thf(fact_414_map__eq__map__tailrec,axiom,
map_fm_tm = map_tailrec_fm_tm ).
% map_eq_map_tailrec
thf(fact_415_map__eq__map__tailrec,axiom,
map_tm_fm = map_tailrec_tm_fm ).
% map_eq_map_tailrec
thf(fact_416_map__eq__map__tailrec,axiom,
map_tm_tm = map_tailrec_tm_tm ).
% map_eq_map_tailrec
thf(fact_417_map__ident,axiom,
( ( map_fm_fm
@ ^ [X2: fm] : X2 )
= ( ^ [Xs3: list_fm] : Xs3 ) ) ).
% map_ident
thf(fact_418_map__ident,axiom,
( ( map_tm_tm
@ ^ [X2: tm] : X2 )
= ( ^ [Xs3: list_tm] : Xs3 ) ) ).
% map_ident
thf(fact_419_in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_420_in__set__insert,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o2 @ X3 @ ( set_o2 @ Xs ) )
=> ( ( insert_o @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_421_in__set__insert,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_422_in__set__insert,axiom,
! [X3: fm,Xs: list_fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_423_in__set__insert,axiom,
! [X3: tm,Xs: list_tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_424_in__set__insert,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat2 @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_425_list_Omap__ident,axiom,
! [T: list_fm] :
( ( map_fm_fm
@ ^ [X2: fm] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_426_list_Omap__ident,axiom,
! [T: list_tm] :
( ( map_tm_tm
@ ^ [X2: tm] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_427_list__prod_Osimps_I2_J,axiom,
! [Hs2: list_list_list_fm,T: list_list_fm,Ts: list_list_list_fm] :
( ( list_prod_list_fm @ Hs2 @ ( cons_list_list_fm @ T @ Ts ) )
= ( append_list_list_fm
@ ( map_li4351931137408529412ist_fm
@ ^ [H: list_list_fm] : ( append_list_fm @ H @ T )
@ Hs2 )
@ ( list_prod_list_fm @ Hs2 @ Ts ) ) ) ).
% list_prod.simps(2)
thf(fact_428_list__prod_Osimps_I2_J,axiom,
! [Hs2: list_list_list_tm,T: list_list_tm,Ts: list_list_list_tm] :
( ( list_prod_list_tm @ Hs2 @ ( cons_list_list_tm @ T @ Ts ) )
= ( append_list_list_tm
@ ( map_li4116520515254442628ist_tm
@ ^ [H: list_list_tm] : ( append_list_tm @ H @ T )
@ Hs2 )
@ ( list_prod_list_tm @ Hs2 @ Ts ) ) ) ).
% list_prod.simps(2)
thf(fact_429_list__prod_Osimps_I2_J,axiom,
! [Hs2: list_list_set_nat,T: list_set_nat,Ts: list_list_set_nat] :
( ( list_prod_set_nat @ Hs2 @ ( cons_list_set_nat @ T @ Ts ) )
= ( append_list_set_nat
@ ( map_li9190756048810191986et_nat
@ ^ [H: list_set_nat] : ( append_set_nat @ H @ T )
@ Hs2 )
@ ( list_prod_set_nat @ Hs2 @ Ts ) ) ) ).
% list_prod.simps(2)
thf(fact_430_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_431_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_432_insert__remdups,axiom,
! [X3: tm,Xs: list_tm] :
( ( insert_tm @ X3 @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ ( insert_tm @ X3 @ Xs ) ) ) ).
% insert_remdups
thf(fact_433_insert__remdups,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( insert_set_nat @ X3 @ ( remdups_set_nat @ Xs ) )
= ( remdups_set_nat @ ( insert_set_nat @ X3 @ Xs ) ) ) ).
% insert_remdups
thf(fact_434_insert__remdups,axiom,
! [X3: fm,Xs: list_fm] :
( ( insert_fm @ X3 @ ( remdups_fm @ Xs ) )
= ( remdups_fm @ ( insert_fm @ X3 @ Xs ) ) ) ).
% insert_remdups
thf(fact_435_insert__remdups,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( insert_list_fm @ X3 @ ( remdups_list_fm @ Xs ) )
= ( remdups_list_fm @ ( insert_list_fm @ X3 @ Xs ) ) ) ).
% insert_remdups
thf(fact_436_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( member_fm @ P @ ( cons_fm @ Q @ Z ) )
= ( ( P != Q )
=> ( member_fm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_437_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: list_fm,Q: list_fm,Z: list_list_fm] :
( ( member_list_fm @ P @ ( cons_list_fm @ Q @ Z ) )
= ( ( P != Q )
=> ( member_list_fm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_438_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: tm,Q: tm,Z: list_tm] :
( ( member_tm @ P @ ( cons_tm @ Q @ Z ) )
= ( ( P != Q )
=> ( member_tm @ P @ Z ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_439_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_440_List_Oinsert__def,axiom,
( insert_o
= ( ^ [X2: $o,Xs3: list_o] : ( if_list_o @ ( member_o2 @ X2 @ ( set_o2 @ Xs3 ) ) @ Xs3 @ ( cons_o @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_441_List_Oinsert__def,axiom,
( insert_set_nat
= ( ^ [X2: set_nat,Xs3: list_set_nat] : ( if_list_set_nat @ ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_set_nat @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_442_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm2 @ X2 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_443_List_Oinsert__def,axiom,
( insert_list_fm
= ( ^ [X2: list_fm,Xs3: list_list_fm] : ( if_list_list_fm @ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_list_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_444_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X2: tm,Xs3: list_tm] : ( if_list_tm @ ( member_tm2 @ X2 @ ( set_tm2 @ Xs3 ) ) @ Xs3 @ ( cons_tm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_445_List_Obind__def,axiom,
( bind_fm_tm
= ( ^ [Xs3: list_fm,F2: fm > list_tm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_446_List_Obind__def,axiom,
( bind_tm_tm
= ( ^ [Xs3: list_tm,F2: tm > list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_447_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_448_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_449_list__prod__is__cartesian,axiom,
! [Hs2: list_list_list_fm,Ts: list_list_list_fm] :
( ( set_list_list_fm2 @ ( list_prod_list_fm @ Hs2 @ Ts ) )
= ( collect_list_list_fm
@ ^ [Uu: list_list_fm] :
? [H: list_list_fm,T2: list_list_fm] :
( ( Uu
= ( append_list_fm @ H @ T2 ) )
& ( member_list_list_fm @ H @ ( set_list_list_fm2 @ Hs2 ) )
& ( member_list_list_fm @ T2 @ ( set_list_list_fm2 @ Ts ) ) ) ) ) ).
% list_prod_is_cartesian
thf(fact_450_list__prod__is__cartesian,axiom,
! [Hs2: list_list_list_tm,Ts: list_list_list_tm] :
( ( set_list_list_tm2 @ ( list_prod_list_tm @ Hs2 @ Ts ) )
= ( collect_list_list_tm
@ ^ [Uu: list_list_tm] :
? [H: list_list_tm,T2: list_list_tm] :
( ( Uu
= ( append_list_tm @ H @ T2 ) )
& ( member_list_list_tm @ H @ ( set_list_list_tm2 @ Hs2 ) )
& ( member_list_list_tm @ T2 @ ( set_list_list_tm2 @ Ts ) ) ) ) ) ).
% list_prod_is_cartesian
thf(fact_451_list__prod__is__cartesian,axiom,
! [Hs2: list_list_set_nat,Ts: list_list_set_nat] :
( ( set_list_set_nat2 @ ( list_prod_set_nat @ Hs2 @ Ts ) )
= ( collect_list_set_nat
@ ^ [Uu: list_set_nat] :
? [H: list_set_nat,T2: list_set_nat] :
( ( Uu
= ( append_set_nat @ H @ T2 ) )
& ( member_list_set_nat @ H @ ( set_list_set_nat2 @ Hs2 ) )
& ( member_list_set_nat @ T2 @ ( set_list_set_nat2 @ Ts ) ) ) ) ) ).
% list_prod_is_cartesian
thf(fact_452_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_453_set__Cons__def,axiom,
( set_Cons_nat
= ( ^ [A3: set_nat,XS: set_list_nat] :
( collect_list_nat
@ ^ [Z3: list_nat] :
? [X2: nat,Xs3: list_nat] :
( ( Z3
= ( cons_nat @ X2 @ Xs3 ) )
& ( member_nat2 @ X2 @ A3 )
& ( member_list_nat @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_454_set__Cons__def,axiom,
( set_Cons_o
= ( ^ [A3: set_o,XS: set_list_o] :
( collect_list_o
@ ^ [Z3: list_o] :
? [X2: $o,Xs3: list_o] :
( ( Z3
= ( cons_o @ X2 @ Xs3 ) )
& ( member_o2 @ X2 @ A3 )
& ( member_list_o @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_455_set__Cons__def,axiom,
( set_Cons_fm
= ( ^ [A3: set_fm,XS: set_list_fm] :
( collect_list_fm
@ ^ [Z3: list_fm] :
? [X2: fm,Xs3: list_fm] :
( ( Z3
= ( cons_fm @ X2 @ Xs3 ) )
& ( member_fm2 @ X2 @ A3 )
& ( member_list_fm2 @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_456_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] :
? [X2: list_fm,Xs3: list_list_fm] :
( ( Z3
= ( cons_list_fm @ X2 @ Xs3 ) )
& ( member_list_fm2 @ X2 @ A3 )
& ( member_list_list_fm @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_457_set__Cons__def,axiom,
( set_Cons_tm
= ( ^ [A3: set_tm,XS: set_list_tm] :
( collect_list_tm
@ ^ [Z3: list_tm] :
? [X2: tm,Xs3: list_tm] :
( ( Z3
= ( cons_tm @ X2 @ Xs3 ) )
& ( member_tm2 @ X2 @ A3 )
& ( member_list_tm @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_458_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_459_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_460_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_461_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
@ ^ [X2: list_fm] : ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X2 ) @ ( produc373462945560358120ist_fm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_462_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
@ ^ [X2: tm] : ( map_list_tm_list_tm @ ( cons_tm @ X2 ) @ ( product_lists_tm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_463_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
@ ^ [X2: fm] : ( map_list_fm_list_fm @ ( cons_fm @ X2 ) @ ( product_lists_fm @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_464_concat__map__singleton,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( concat_list_tm
@ ( map_fm_list_list_tm
@ ^ [X2: fm] : ( cons_list_tm @ ( F @ X2 ) @ nil_list_tm )
@ Xs ) )
= ( map_fm_list_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_465_concat__map__singleton,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( concat_set_nat
@ ( map_tm_list_set_nat
@ ^ [X2: tm] : ( cons_set_nat @ ( F @ X2 ) @ nil_set_nat )
@ Xs ) )
= ( map_tm_set_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_466_concat__map__singleton,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( concat_list_tm
@ ( map_tm_list_list_tm
@ ^ [X2: tm] : ( cons_list_tm @ ( F @ X2 ) @ nil_list_tm )
@ Xs ) )
= ( map_tm_list_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_467_concat__map__singleton,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( concat_fm
@ ( map_fm_list_fm
@ ^ [X2: fm] : ( cons_fm @ ( F @ X2 ) @ nil_fm )
@ Xs ) )
= ( map_fm_fm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_468_concat__map__singleton,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( concat_fm
@ ( map_tm_list_fm
@ ^ [X2: tm] : ( cons_fm @ ( F @ X2 ) @ nil_fm )
@ Xs ) )
= ( map_tm_fm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_469_concat__map__singleton,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( concat_tm
@ ( map_fm_list_tm
@ ^ [X2: fm] : ( cons_tm @ ( F @ X2 ) @ nil_tm )
@ Xs ) )
= ( map_fm_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_470_concat__map__singleton,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( concat_tm
@ ( map_tm_list_tm
@ ^ [X2: tm] : ( cons_tm @ ( F @ X2 ) @ nil_tm )
@ Xs ) )
= ( map_tm_tm @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_471_subseqs_Osimps_I2_J,axiom,
! [X3: fm,Xs: list_fm] :
( ( subseqs_fm @ ( cons_fm @ X3 @ Xs ) )
= ( append_list_fm @ ( map_list_fm_list_fm @ ( cons_fm @ X3 ) @ ( subseqs_fm @ Xs ) ) @ ( subseqs_fm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_472_subseqs_Osimps_I2_J,axiom,
! [X3: list_fm,Xs: list_list_fm] :
( ( subseqs_list_fm @ ( cons_list_fm @ X3 @ Xs ) )
= ( append_list_list_fm @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X3 ) @ ( subseqs_list_fm @ Xs ) ) @ ( subseqs_list_fm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_473_subseqs_Osimps_I2_J,axiom,
! [X3: tm,Xs: list_tm] :
( ( subseqs_tm @ ( cons_tm @ X3 @ Xs ) )
= ( append_list_tm @ ( map_list_tm_list_tm @ ( cons_tm @ X3 ) @ ( subseqs_tm @ Xs ) ) @ ( subseqs_tm @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_474_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_475_concat__eq__appendD,axiom,
! [Xss: list_list_list_fm,Ys: list_list_fm,Zs2: list_list_fm] :
( ( ( concat_list_fm @ Xss )
= ( append_list_fm @ Ys @ Zs2 ) )
=> ( ( Xss != nil_list_list_fm )
=> ? [Xss1: list_list_list_fm,Xs4: list_list_fm,Xs5: list_list_fm,Xss2: list_list_list_fm] :
( ( Xss
= ( append_list_list_fm @ Xss1 @ ( cons_list_list_fm @ ( append_list_fm @ Xs4 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_list_fm @ ( concat_list_fm @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_list_fm @ Xs5 @ ( concat_list_fm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_476_concat__eq__appendD,axiom,
! [Xss: list_list_list_tm,Ys: list_list_tm,Zs2: list_list_tm] :
( ( ( concat_list_tm @ Xss )
= ( append_list_tm @ Ys @ Zs2 ) )
=> ( ( Xss != nil_list_list_tm )
=> ? [Xss1: list_list_list_tm,Xs4: list_list_tm,Xs5: list_list_tm,Xss2: list_list_list_tm] :
( ( Xss
= ( append_list_list_tm @ Xss1 @ ( cons_list_list_tm @ ( append_list_tm @ Xs4 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_list_tm @ ( concat_list_tm @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_list_tm @ Xs5 @ ( concat_list_tm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_477_concat__eq__appendD,axiom,
! [Xss: list_list_set_nat,Ys: list_set_nat,Zs2: list_set_nat] :
( ( ( concat_set_nat @ Xss )
= ( append_set_nat @ Ys @ Zs2 ) )
=> ( ( Xss != nil_list_set_nat )
=> ? [Xss1: list_list_set_nat,Xs4: list_set_nat,Xs5: list_set_nat,Xss2: list_list_set_nat] :
( ( Xss
= ( append_list_set_nat @ Xss1 @ ( cons_list_set_nat @ ( append_set_nat @ Xs4 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_set_nat @ ( concat_set_nat @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_set_nat @ Xs5 @ ( concat_set_nat @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_478_concat__eq__appendD,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs2 ) )
=> ( ( Xss != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs4: list_tm,Xs5: list_tm,Xss2: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs4 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_tm @ Xs5 @ ( concat_tm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_479_concat__eq__appendD,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs2 ) )
=> ( ( Xss != nil_list_fm )
=> ? [Xss1: list_list_fm,Xs4: list_fm,Xs5: list_fm,Xss2: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss1 @ ( cons_list_fm @ ( append_fm @ Xs4 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append_fm @ Xs5 @ ( concat_fm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_480_parts__preserves__unaffected,axiom,
! [R: rule,P: fm,Z5: list_fm,A2: list_tm] :
( ~ ( affects @ R @ P )
=> ( ( member_list_fm2 @ Z5 @ ( set_list_fm2 @ ( parts @ A2 @ R @ P ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ).
% parts_preserves_unaffected
thf(fact_481_append__is__Nil__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= nil_list_tm )
= ( ( Xs = nil_list_tm )
& ( Ys = nil_list_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_482_append__is__Nil__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= nil_set_nat )
= ( ( Xs = nil_set_nat )
& ( Ys = nil_set_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_483_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_484_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_485_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_486_Nil__is__append__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( nil_list_tm
= ( append_list_tm @ Xs @ Ys ) )
= ( ( Xs = nil_list_tm )
& ( Ys = nil_list_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_487_Nil__is__append__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( nil_set_nat
= ( append_set_nat @ Xs @ Ys ) )
= ( ( Xs = nil_set_nat )
& ( Ys = nil_set_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_488_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_489_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_490_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_491_self__append__conv2,axiom,
! [Y: list_list_tm,Xs: list_list_tm] :
( ( Y
= ( append_list_tm @ Xs @ Y ) )
= ( Xs = nil_list_tm ) ) ).
% self_append_conv2
thf(fact_492_self__append__conv2,axiom,
! [Y: list_set_nat,Xs: list_set_nat] :
( ( Y
= ( append_set_nat @ Xs @ Y ) )
= ( Xs = nil_set_nat ) ) ).
% self_append_conv2
thf(fact_493_self__append__conv2,axiom,
! [Y: list_fm,Xs: list_fm] :
( ( Y
= ( append_fm @ Xs @ Y ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_494_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_495_self__append__conv2,axiom,
! [Y: list_tm,Xs: list_tm] :
( ( Y
= ( append_tm @ Xs @ Y ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_496_append__self__conv2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_tm ) ) ).
% append_self_conv2
thf(fact_497_append__self__conv2,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_set_nat ) ) ).
% append_self_conv2
thf(fact_498_append__self__conv2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_499_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_500_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_501_self__append__conv,axiom,
! [Y: list_list_tm,Ys: list_list_tm] :
( ( Y
= ( append_list_tm @ Y @ Ys ) )
= ( Ys = nil_list_tm ) ) ).
% self_append_conv
thf(fact_502_self__append__conv,axiom,
! [Y: list_set_nat,Ys: list_set_nat] :
( ( Y
= ( append_set_nat @ Y @ Ys ) )
= ( Ys = nil_set_nat ) ) ).
% self_append_conv
thf(fact_503_self__append__conv,axiom,
! [Y: list_fm,Ys: list_fm] :
( ( Y
= ( append_fm @ Y @ Ys ) )
= ( Ys = nil_fm ) ) ).
% self_append_conv
thf(fact_504_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_505_self__append__conv,axiom,
! [Y: list_tm,Ys: list_tm] :
( ( Y
= ( append_tm @ Y @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_506_append__self__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_tm ) ) ).
% append_self_conv
thf(fact_507_append__self__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_set_nat ) ) ).
% append_self_conv
thf(fact_508_append__self__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_fm ) ) ).
% append_self_conv
thf(fact_509_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_510_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_511_append__Nil2,axiom,
! [Xs: list_list_tm] :
( ( append_list_tm @ Xs @ nil_list_tm )
= Xs ) ).
% append_Nil2
thf(fact_512_append__Nil2,axiom,
! [Xs: list_set_nat] :
( ( append_set_nat @ Xs @ nil_set_nat )
= Xs ) ).
% append_Nil2
thf(fact_513_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_514_append__Nil2,axiom,
! [Xs: list_list_fm] :
( ( append_list_fm @ Xs @ nil_list_fm )
= Xs ) ).
% append_Nil2
thf(fact_515_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_516_append_Oright__neutral,axiom,
! [A: list_list_tm] :
( ( append_list_tm @ A @ nil_list_tm )
= A ) ).
% append.right_neutral
thf(fact_517_append_Oright__neutral,axiom,
! [A: list_set_nat] :
( ( append_set_nat @ A @ nil_set_nat )
= A ) ).
% append.right_neutral
thf(fact_518_append_Oright__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ A @ nil_fm )
= A ) ).
% append.right_neutral
thf(fact_519_append_Oright__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ A @ nil_list_fm )
= A ) ).
% append.right_neutral
thf(fact_520_append_Oright__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ A @ nil_tm )
= A ) ).
% append.right_neutral
thf(fact_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_remdups__eq__nil__right__iff,axiom,
! [X3: list_set_nat] :
( ( nil_set_nat
= ( remdups_set_nat @ X3 ) )
= ( X3 = nil_set_nat ) ) ).
% remdups_eq_nil_right_iff
thf(fact_552_remdups__eq__nil__right__iff,axiom,
! [X3: list_fm] :
( ( nil_fm
= ( remdups_fm @ X3 ) )
= ( X3 = nil_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_553_remdups__eq__nil__right__iff,axiom,
! [X3: list_list_fm] :
( ( nil_list_fm
= ( remdups_list_fm @ X3 ) )
= ( X3 = nil_list_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_554_remdups__eq__nil__right__iff,axiom,
! [X3: list_tm] :
( ( nil_tm
= ( remdups_tm @ X3 ) )
= ( X3 = nil_tm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_555_remdups__eq__nil__iff,axiom,
! [X3: list_set_nat] :
( ( ( remdups_set_nat @ X3 )
= nil_set_nat )
= ( X3 = nil_set_nat ) ) ).
% remdups_eq_nil_iff
thf(fact_556_remdups__eq__nil__iff,axiom,
! [X3: list_fm] :
( ( ( remdups_fm @ X3 )
= nil_fm )
= ( X3 = nil_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_557_remdups__eq__nil__iff,axiom,
! [X3: list_list_fm] :
( ( ( remdups_list_fm @ X3 )
= nil_list_fm )
= ( X3 = nil_list_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_558_remdups__eq__nil__iff,axiom,
! [X3: list_tm] :
( ( ( remdups_tm @ X3 )
= nil_tm )
= ( X3 = nil_tm ) ) ).
% remdups_eq_nil_iff
thf(fact_559_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_560_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_561_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_562_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_563_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_564_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_565_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_566_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_567_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_568_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_569_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_570_append1__eq__conv,axiom,
! [Xs: list_list_tm,X3: list_tm,Ys: list_list_tm,Y: list_tm] :
( ( ( append_list_tm @ Xs @ ( cons_list_tm @ X3 @ nil_list_tm ) )
= ( append_list_tm @ Ys @ ( cons_list_tm @ Y @ nil_list_tm ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_571_append1__eq__conv,axiom,
! [Xs: list_set_nat,X3: set_nat,Ys: list_set_nat,Y: set_nat] :
( ( ( append_set_nat @ Xs @ ( cons_set_nat @ X3 @ nil_set_nat ) )
= ( append_set_nat @ Ys @ ( cons_set_nat @ Y @ nil_set_nat ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_572_append1__eq__conv,axiom,
! [Xs: list_fm,X3: fm,Ys: list_fm,Y: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X3 @ nil_fm ) )
= ( append_fm @ Ys @ ( cons_fm @ Y @ nil_fm ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_573_append1__eq__conv,axiom,
! [Xs: list_list_fm,X3: list_fm,Ys: list_list_fm,Y: list_fm] :
( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X3 @ nil_list_fm ) )
= ( append_list_fm @ Ys @ ( cons_list_fm @ Y @ nil_list_fm ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_574_append1__eq__conv,axiom,
! [Xs: list_tm,X3: tm,Ys: list_tm,Y: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X3 @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_concat__eq__Nil__conv,axiom,
! [Xss: list_list_list_fm] :
( ( ( concat_list_fm @ Xss )
= nil_list_fm )
= ( ! [X2: list_list_fm] :
( ( member_list_list_fm @ X2 @ ( set_list_list_fm2 @ Xss ) )
=> ( X2 = nil_list_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_586_concat__eq__Nil__conv,axiom,
! [Xss: list_list_tm] :
( ( ( concat_tm @ Xss )
= nil_tm )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss ) )
=> ( X2 = nil_tm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_587_concat__eq__Nil__conv,axiom,
! [Xss: list_list_fm] :
( ( ( concat_fm @ Xss )
= nil_fm )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xss ) )
=> ( X2 = nil_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_588_Nil__eq__concat__conv,axiom,
! [Xss: list_list_list_fm] :
( ( nil_list_fm
= ( concat_list_fm @ Xss ) )
= ( ! [X2: list_list_fm] :
( ( member_list_list_fm @ X2 @ ( set_list_list_fm2 @ Xss ) )
=> ( X2 = nil_list_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_589_Nil__eq__concat__conv,axiom,
! [Xss: list_list_tm] :
( ( nil_tm
= ( concat_tm @ Xss ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss ) )
=> ( X2 = nil_tm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_590_Nil__eq__concat__conv,axiom,
! [Xss: list_list_fm] :
( ( nil_fm
= ( concat_fm @ Xss ) )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xss ) )
=> ( X2 = nil_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_591_insert__Nil,axiom,
! [X3: fm] :
( ( insert_fm @ X3 @ nil_fm )
= ( cons_fm @ X3 @ nil_fm ) ) ).
% insert_Nil
thf(fact_592_insert__Nil,axiom,
! [X3: list_fm] :
( ( insert_list_fm @ X3 @ nil_list_fm )
= ( cons_list_fm @ X3 @ nil_list_fm ) ) ).
% insert_Nil
thf(fact_593_insert__Nil,axiom,
! [X3: tm] :
( ( insert_tm @ X3 @ nil_tm )
= ( cons_tm @ X3 @ nil_tm ) ) ).
% insert_Nil
thf(fact_594_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_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_603_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_604_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_605_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_606_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_607_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_608_ext_Osimps_I1_J,axiom,
! [Y: list_list_fm] : ( ext_list_fm @ Y @ nil_list_fm ) ).
% ext.simps(1)
thf(fact_609_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_610_concat_Osimps_I1_J,axiom,
( ( concat_list_fm @ nil_list_list_fm )
= nil_list_fm ) ).
% concat.simps(1)
thf(fact_611_concat_Osimps_I1_J,axiom,
( ( concat_tm @ nil_list_tm )
= nil_tm ) ).
% concat.simps(1)
thf(fact_612_concat_Osimps_I1_J,axiom,
( ( concat_fm @ nil_list_fm )
= nil_fm ) ).
% concat.simps(1)
thf(fact_613_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_614_transpose_Ocases,axiom,
! [X3: list_list_list_fm] :
( ( X3 != nil_list_list_fm )
=> ( ! [Xss3: list_list_list_fm] :
( X3
!= ( cons_list_list_fm @ nil_list_fm @ Xss3 ) )
=> ~ ! [X4: list_fm,Xs4: list_list_fm,Xss3: list_list_list_fm] :
( X3
!= ( cons_list_list_fm @ ( cons_list_fm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_615_transpose_Ocases,axiom,
! [X3: list_list_tm] :
( ( X3 != nil_list_tm )
=> ( ! [Xss3: list_list_tm] :
( X3
!= ( cons_list_tm @ nil_tm @ Xss3 ) )
=> ~ ! [X4: tm,Xs4: list_tm,Xss3: list_list_tm] :
( X3
!= ( cons_list_tm @ ( cons_tm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_616_transpose_Ocases,axiom,
! [X3: list_list_fm] :
( ( X3 != nil_list_fm )
=> ( ! [Xss3: list_list_fm] :
( X3
!= ( cons_list_fm @ nil_fm @ Xss3 ) )
=> ~ ! [X4: fm,Xs4: list_fm,Xss3: list_list_fm] :
( X3
!= ( cons_list_fm @ ( cons_fm @ X4 @ Xs4 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_617_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_618_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_619_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_620_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_621_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_622_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_623_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_624_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_625_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_626_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_627_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_628_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_629_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y2: fm,Ys4: list_fm] :
( Xs
= ( cons_fm @ Y2 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_630_neq__Nil__conv,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
= ( ? [Y2: list_fm,Ys4: list_list_fm] :
( Xs
= ( cons_list_fm @ Y2 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_631_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y2: tm,Ys4: list_tm] :
( Xs
= ( cons_tm @ Y2 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_632_remdups__adj_Ocases,axiom,
! [X3: list_fm] :
( ( X3 != nil_fm )
=> ( ! [X4: fm] :
( X3
!= ( cons_fm @ X4 @ nil_fm ) )
=> ~ ! [X4: fm,Y3: fm,Xs4: list_fm] :
( X3
!= ( cons_fm @ X4 @ ( cons_fm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_633_remdups__adj_Ocases,axiom,
! [X3: list_list_fm] :
( ( X3 != nil_list_fm )
=> ( ! [X4: list_fm] :
( X3
!= ( cons_list_fm @ X4 @ nil_list_fm ) )
=> ~ ! [X4: list_fm,Y3: list_fm,Xs4: list_list_fm] :
( X3
!= ( cons_list_fm @ X4 @ ( cons_list_fm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_634_remdups__adj_Ocases,axiom,
! [X3: list_tm] :
( ( X3 != nil_tm )
=> ( ! [X4: tm] :
( X3
!= ( cons_tm @ X4 @ nil_tm ) )
=> ~ ! [X4: tm,Y3: tm,Xs4: list_tm] :
( X3
!= ( cons_tm @ X4 @ ( cons_tm @ Y3 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_635_list_Oexhaust,axiom,
! [Y: list_fm] :
( ( Y != nil_fm )
=> ~ ! [X212: fm,X222: list_fm] :
( Y
!= ( cons_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_636_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_637_list_Oexhaust,axiom,
! [Y: list_tm] :
( ( Y != nil_tm )
=> ~ ! [X212: tm,X222: list_tm] :
( Y
!= ( cons_tm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_638_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X22: list_fm] :
( ( List
= ( cons_fm @ X21 @ X22 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_639_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_640_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X22: list_tm] :
( ( List
= ( cons_tm @ X21 @ X22 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_641_list_Odistinct_I1_J,axiom,
! [X21: fm,X22: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_642_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_643_list_Odistinct_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_644_subset__code_I1_J,axiom,
! [Xs: list_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
= ( ! [X2: $o] :
( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
=> ( member_o2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_645_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B2: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B2 )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_646_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_647_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_648_subset__code_I1_J,axiom,
! [Xs: list_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B2 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_649_subset__code_I1_J,axiom,
! [Xs: list_fm,B2: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B2 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_650_eq__Nil__appendI,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_tm @ nil_list_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_651_eq__Nil__appendI,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_set_nat @ nil_set_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_652_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_fm @ nil_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_653_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_654_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_655_append_Oleft__neutral,axiom,
! [A: list_list_tm] :
( ( append_list_tm @ nil_list_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_656_append_Oleft__neutral,axiom,
! [A: list_set_nat] :
( ( append_set_nat @ nil_set_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_657_append_Oleft__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ nil_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_658_append_Oleft__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_659_append_Oleft__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ nil_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_660_append__Nil,axiom,
! [Ys: list_list_tm] :
( ( append_list_tm @ nil_list_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_661_append__Nil,axiom,
! [Ys: list_set_nat] :
( ( append_set_nat @ nil_set_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_662_append__Nil,axiom,
! [Ys: list_fm] :
( ( append_fm @ nil_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_663_append__Nil,axiom,
! [Ys: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_664_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_665_list_Osimps_I8_J,axiom,
! [F: fm > fm] :
( ( map_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_666_list_Osimps_I8_J,axiom,
! [F: fm > tm] :
( ( map_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_667_list_Osimps_I8_J,axiom,
! [F: tm > fm] :
( ( map_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% list.simps(8)
thf(fact_668_list_Osimps_I8_J,axiom,
! [F: tm > tm] :
( ( map_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% list.simps(8)
thf(fact_669_list_Osimps_I8_J,axiom,
! [F: fm > list_fm] :
( ( map_fm_list_fm @ F @ nil_fm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_670_list_Osimps_I8_J,axiom,
! [F: list_fm > fm] :
( ( map_list_fm_fm @ F @ nil_list_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_671_list_Osimps_I8_J,axiom,
! [F: list_fm > tm] :
( ( map_list_fm_tm @ F @ nil_list_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_672_list_Osimps_I8_J,axiom,
! [F: tm > list_fm] :
( ( map_tm_list_fm @ F @ nil_tm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_673_list_Osimps_I8_J,axiom,
! [F: fm > list_tm] :
( ( map_fm_list_tm @ F @ nil_fm )
= nil_list_tm ) ).
% list.simps(8)
thf(fact_674_list_Osimps_I8_J,axiom,
! [F: tm > set_nat] :
( ( map_tm_set_nat @ F @ nil_tm )
= nil_set_nat ) ).
% list.simps(8)
thf(fact_675_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_fm] :
( ( set_fm2 @ ( bind_fm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_fm_set_fm
@ ^ [X2: fm] : ( set_fm2 @ ( F @ X2 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_676_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_fm] :
( ( set_fm2 @ ( bind_tm_fm @ Xs @ F ) )
= ( comple2134933779557159616set_fm
@ ( image_tm_set_fm
@ ^ [X2: tm] : ( set_fm2 @ ( F @ X2 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_677_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_nat] :
( ( set_nat2 @ ( bind_fm_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X2: fm] : ( set_nat2 @ ( F @ X2 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_678_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_nat] :
( ( set_nat2 @ ( bind_tm_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X2: tm] : ( set_nat2 @ ( F @ X2 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_679_set__list__bind,axiom,
! [Xs: list_fm,F: fm > list_tm] :
( ( set_tm2 @ ( bind_fm_tm @ Xs @ F ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X2: fm] : ( set_tm2 @ ( F @ X2 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_680_set__list__bind,axiom,
! [Xs: list_tm,F: tm > list_tm] :
( ( set_tm2 @ ( bind_tm_tm @ Xs @ F ) )
= ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X2: tm] : ( set_tm2 @ ( F @ X2 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_681_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
@ ^ [X2: fm] : ( set_list_fm2 @ ( F @ X2 ) )
@ ( set_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_682_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
@ ^ [X2: tm] : ( set_list_fm2 @ ( F @ X2 ) )
@ ( set_tm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_683_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
@ ^ [X2: list_fm] : ( set_fm2 @ ( F @ X2 ) )
@ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_684_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
@ ^ [X2: set_nat] : ( set_fm2 @ ( F @ X2 ) )
@ ( set_set_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_685_remdups_Osimps_I1_J,axiom,
( ( remdups_set_nat @ nil_set_nat )
= nil_set_nat ) ).
% remdups.simps(1)
thf(fact_686_remdups_Osimps_I1_J,axiom,
( ( remdups_fm @ nil_fm )
= nil_fm ) ).
% remdups.simps(1)
thf(fact_687_remdups_Osimps_I1_J,axiom,
( ( remdups_list_fm @ nil_list_fm )
= nil_list_fm ) ).
% remdups.simps(1)
thf(fact_688_remdups_Osimps_I1_J,axiom,
( ( remdups_tm @ nil_tm )
= nil_tm ) ).
% remdups.simps(1)
thf(fact_689_params_Osimps_I7_J,axiom,
! [P: fm] :
( ( params @ ( neg @ P ) )
= ( params @ P ) ) ).
% params.simps(7)
thf(fact_690_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_691_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: list_fm] :
~ ( member_list_fm @ P @ nil_list_fm ) ).
% SeCaV.member.simps(1)
thf(fact_692_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_693_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_694_maps__simps_I2_J,axiom,
! [F: fm > list_fm] :
( ( maps_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_695_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_696_maps__simps_I2_J,axiom,
! [F: fm > list_tm] :
( ( maps_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_697_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_698_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_699_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_700_maps__simps_I2_J,axiom,
! [F: tm > list_fm] :
( ( maps_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_701_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_702_maps__simps_I2_J,axiom,
! [F: tm > list_tm] :
( ( maps_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_703_subseqs__refl,axiom,
! [Xs: list_fm] : ( member_list_fm2 @ Xs @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ).
% subseqs_refl
thf(fact_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_set__subset__Cons,axiom,
! [Xs: list_set_nat,X3: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ ( cons_set_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_715_set__subset__Cons,axiom,
! [Xs: list_list_fm,X3: list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ ( cons_list_fm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_716_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_717_set__subset__Cons,axiom,
! [Xs: list_tm,X3: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_718_set__subset__Cons,axiom,
! [Xs: list_fm,X3: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_719_concat__eq__append__conv,axiom,
! [Xss: list_list_list_tm,Ys: list_list_tm,Zs2: list_list_tm] :
( ( ( concat_list_tm @ Xss )
= ( append_list_tm @ Ys @ Zs2 ) )
= ( ( ( Xss = nil_list_list_tm )
=> ( ( Ys = nil_list_tm )
& ( Zs2 = nil_list_tm ) ) )
& ( ( Xss != nil_list_list_tm )
=> ? [Xss12: list_list_list_tm,Xs3: list_list_tm,Xs6: list_list_tm,Xss22: list_list_list_tm] :
( ( Xss
= ( append_list_list_tm @ Xss12 @ ( cons_list_list_tm @ ( append_list_tm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_tm @ ( concat_list_tm @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_list_tm @ Xs6 @ ( concat_list_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_720_concat__eq__append__conv,axiom,
! [Xss: list_list_set_nat,Ys: list_set_nat,Zs2: list_set_nat] :
( ( ( concat_set_nat @ Xss )
= ( append_set_nat @ Ys @ Zs2 ) )
= ( ( ( Xss = nil_list_set_nat )
=> ( ( Ys = nil_set_nat )
& ( Zs2 = nil_set_nat ) ) )
& ( ( Xss != nil_list_set_nat )
=> ? [Xss12: list_list_set_nat,Xs3: list_set_nat,Xs6: list_set_nat,Xss22: list_list_set_nat] :
( ( Xss
= ( append_list_set_nat @ Xss12 @ ( cons_list_set_nat @ ( append_set_nat @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_set_nat @ ( concat_set_nat @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_set_nat @ Xs6 @ ( concat_set_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_721_concat__eq__append__conv,axiom,
! [Xss: list_list_list_fm,Ys: list_list_fm,Zs2: list_list_fm] :
( ( ( concat_list_fm @ Xss )
= ( append_list_fm @ Ys @ Zs2 ) )
= ( ( ( Xss = nil_list_list_fm )
=> ( ( Ys = nil_list_fm )
& ( Zs2 = nil_list_fm ) ) )
& ( ( Xss != nil_list_list_fm )
=> ? [Xss12: list_list_list_fm,Xs3: list_list_fm,Xs6: list_list_fm,Xss22: list_list_list_fm] :
( ( Xss
= ( append_list_list_fm @ Xss12 @ ( cons_list_list_fm @ ( append_list_fm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_fm @ ( concat_list_fm @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_list_fm @ Xs6 @ ( concat_list_fm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_722_concat__eq__append__conv,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs2 ) )
= ( ( ( Xss = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs2 = nil_tm ) ) )
& ( ( Xss != nil_list_tm )
=> ? [Xss12: list_list_tm,Xs3: list_tm,Xs6: list_tm,Xss22: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss12 @ ( cons_list_tm @ ( append_tm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_tm @ Xs6 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_723_concat__eq__append__conv,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs2 ) )
= ( ( ( Xss = nil_list_fm )
=> ( ( Ys = nil_fm )
& ( Zs2 = nil_fm ) ) )
& ( ( Xss != nil_list_fm )
=> ? [Xss12: list_list_fm,Xs3: list_fm,Xs6: list_fm,Xss22: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss12 @ ( cons_list_fm @ ( append_fm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_fm @ Xs6 @ ( concat_fm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_724_rev__nonempty__induct,axiom,
! [Xs: list_list_tm,P2: list_list_tm > $o] :
( ( Xs != nil_list_tm )
=> ( ! [X4: list_tm] : ( P2 @ ( cons_list_tm @ X4 @ nil_list_tm ) )
=> ( ! [X4: list_tm,Xs4: list_list_tm] :
( ( Xs4 != nil_list_tm )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append_list_tm @ Xs4 @ ( cons_list_tm @ X4 @ nil_list_tm ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_725_rev__nonempty__induct,axiom,
! [Xs: list_set_nat,P2: list_set_nat > $o] :
( ( Xs != nil_set_nat )
=> ( ! [X4: set_nat] : ( P2 @ ( cons_set_nat @ X4 @ nil_set_nat ) )
=> ( ! [X4: set_nat,Xs4: list_set_nat] :
( ( Xs4 != nil_set_nat )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append_set_nat @ Xs4 @ ( cons_set_nat @ X4 @ nil_set_nat ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_726_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_727_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_728_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_729_append__eq__Cons__conv,axiom,
! [Ys: list_list_tm,Zs2: list_list_tm,X3: list_tm,Xs: list_list_tm] :
( ( ( append_list_tm @ Ys @ Zs2 )
= ( cons_list_tm @ X3 @ Xs ) )
= ( ( ( Ys = nil_list_tm )
& ( Zs2
= ( cons_list_tm @ X3 @ Xs ) ) )
| ? [Ys5: list_list_tm] :
( ( Ys
= ( cons_list_tm @ X3 @ Ys5 ) )
& ( ( append_list_tm @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_730_append__eq__Cons__conv,axiom,
! [Ys: list_set_nat,Zs2: list_set_nat,X3: set_nat,Xs: list_set_nat] :
( ( ( append_set_nat @ Ys @ Zs2 )
= ( cons_set_nat @ X3 @ Xs ) )
= ( ( ( Ys = nil_set_nat )
& ( Zs2
= ( cons_set_nat @ X3 @ Xs ) ) )
| ? [Ys5: list_set_nat] :
( ( Ys
= ( cons_set_nat @ X3 @ Ys5 ) )
& ( ( append_set_nat @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_731_append__eq__Cons__conv,axiom,
! [Ys: list_fm,Zs2: list_fm,X3: fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs2 )
= ( cons_fm @ X3 @ Xs ) )
= ( ( ( Ys = nil_fm )
& ( Zs2
= ( cons_fm @ X3 @ Xs ) ) )
| ? [Ys5: list_fm] :
( ( Ys
= ( cons_fm @ X3 @ Ys5 ) )
& ( ( append_fm @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_732_append__eq__Cons__conv,axiom,
! [Ys: list_list_fm,Zs2: list_list_fm,X3: list_fm,Xs: list_list_fm] :
( ( ( append_list_fm @ Ys @ Zs2 )
= ( cons_list_fm @ X3 @ Xs ) )
= ( ( ( Ys = nil_list_fm )
& ( Zs2
= ( cons_list_fm @ X3 @ Xs ) ) )
| ? [Ys5: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X3 @ Ys5 ) )
& ( ( append_list_fm @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_733_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs2: list_tm,X3: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs2 )
= ( cons_tm @ X3 @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs2
= ( cons_tm @ X3 @ Xs ) ) )
| ? [Ys5: list_tm] :
( ( Ys
= ( cons_tm @ X3 @ Ys5 ) )
& ( ( append_tm @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_734_Cons__eq__append__conv,axiom,
! [X3: list_tm,Xs: list_list_tm,Ys: list_list_tm,Zs2: list_list_tm] :
( ( ( cons_list_tm @ X3 @ Xs )
= ( append_list_tm @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_list_tm )
& ( ( cons_list_tm @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list_list_tm] :
( ( ( cons_list_tm @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_tm @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_735_Cons__eq__append__conv,axiom,
! [X3: set_nat,Xs: list_set_nat,Ys: list_set_nat,Zs2: list_set_nat] :
( ( ( cons_set_nat @ X3 @ Xs )
= ( append_set_nat @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_set_nat )
& ( ( cons_set_nat @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list_set_nat] :
( ( ( cons_set_nat @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_set_nat @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_736_Cons__eq__append__conv,axiom,
! [X3: fm,Xs: list_fm,Ys: list_fm,Zs2: list_fm] :
( ( ( cons_fm @ X3 @ Xs )
= ( append_fm @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_fm )
& ( ( cons_fm @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list_fm] :
( ( ( cons_fm @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_fm @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_737_Cons__eq__append__conv,axiom,
! [X3: list_fm,Xs: list_list_fm,Ys: list_list_fm,Zs2: list_list_fm] :
( ( ( cons_list_fm @ X3 @ Xs )
= ( append_list_fm @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_list_fm )
& ( ( cons_list_fm @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list_list_fm] :
( ( ( cons_list_fm @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_fm @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_738_Cons__eq__append__conv,axiom,
! [X3: tm,Xs: list_tm,Ys: list_tm,Zs2: list_tm] :
( ( ( cons_tm @ X3 @ Xs )
= ( append_tm @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list_tm] :
( ( ( cons_tm @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_tm @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_739_rev__exhaust,axiom,
! [Xs: list_list_tm] :
( ( Xs != nil_list_tm )
=> ~ ! [Ys2: list_list_tm,Y3: list_tm] :
( Xs
!= ( append_list_tm @ Ys2 @ ( cons_list_tm @ Y3 @ nil_list_tm ) ) ) ) ).
% rev_exhaust
thf(fact_740_rev__exhaust,axiom,
! [Xs: list_set_nat] :
( ( Xs != nil_set_nat )
=> ~ ! [Ys2: list_set_nat,Y3: set_nat] :
( Xs
!= ( append_set_nat @ Ys2 @ ( cons_set_nat @ Y3 @ nil_set_nat ) ) ) ) ).
% rev_exhaust
thf(fact_741_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_742_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_743_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_744_rev__induct,axiom,
! [P2: list_list_tm > $o,Xs: list_list_tm] :
( ( P2 @ nil_list_tm )
=> ( ! [X4: list_tm,Xs4: list_list_tm] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append_list_tm @ Xs4 @ ( cons_list_tm @ X4 @ nil_list_tm ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_745_rev__induct,axiom,
! [P2: list_set_nat > $o,Xs: list_set_nat] :
( ( P2 @ nil_set_nat )
=> ( ! [X4: set_nat,Xs4: list_set_nat] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append_set_nat @ Xs4 @ ( cons_set_nat @ X4 @ nil_set_nat ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_746_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_747_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_748_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_749_Ext,axiom,
! [Z: list_fm,Y: list_fm] :
( ( sequent_calculus @ Z )
=> ( ( ext_fm @ Y @ Z )
=> ( sequent_calculus @ Y ) ) ) ).
% Ext
thf(fact_750_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_751_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_752_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_753_children__preserves__unaffected,axiom,
! [P: fm,Z: list_fm,R: rule,Z5: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ~ ( affects @ R @ P )
=> ( ( member_list_fm2 @ Z5 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% children_preserves_unaffected
thf(fact_754_UN__iff,axiom,
! [B: nat,B2: fm > set_nat,A2: set_fm] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ( member_nat2 @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_755_UN__iff,axiom,
! [B: nat,B2: tm > set_nat,A2: set_tm] :
( ( member_nat2 @ B @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ( member_nat2 @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_756_UN__iff,axiom,
! [B: tm,B2: tm > set_tm,A2: set_tm] :
( ( member_tm2 @ B @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ( member_tm2 @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_757_UN__iff,axiom,
! [B: tm,B2: fm > set_tm,A2: set_fm] :
( ( member_tm2 @ B @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ( member_tm2 @ B @ ( B2 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_769_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_770_SUP__identity__eq,axiom,
! [A2: set_set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_set_tm_set_tm
@ ^ [X2: set_tm] : X2
@ A2 ) )
= ( comple2138885804642794802set_tm @ A2 ) ) ).
% SUP_identity_eq
thf(fact_771_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_772_UN__ball__bex__simps_I4_J,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_773_UN__ball__bex__simps_I4_J,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_774_UN__ball__bex__simps_I4_J,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_775_UN__ball__bex__simps_I4_J,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_776_UN__ball__bex__simps_I2_J,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_777_UN__ball__bex__simps_I2_J,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_778_UN__ball__bex__simps_I2_J,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_779_UN__ball__bex__simps_I2_J,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_780_bex__UN,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_781_bex__UN,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_782_bex__UN,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_783_bex__UN,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_784_ball__UN,axiom,
! [B2: fm > set_nat,A2: set_fm,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_785_ball__UN,axiom,
! [B2: tm > set_nat,A2: set_tm,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_786_ball__UN,axiom,
! [B2: tm > set_tm,A2: set_tm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_787_ball__UN,axiom,
! [B2: fm > set_tm,A2: set_fm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ ( B2 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_Union__iff,axiom,
! [A2: list_fm,C2: set_set_list_fm] :
( ( member_list_fm2 @ A2 @ ( comple8784269564784259782ist_fm @ C2 ) )
= ( ? [X2: set_list_fm] :
( ( member_set_list_fm @ X2 @ C2 )
& ( member_list_fm2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_809_Union__iff,axiom,
! [A2: fm,C2: set_set_fm] :
( ( member_fm2 @ A2 @ ( comple2134933779557159616set_fm @ C2 ) )
= ( ? [X2: set_fm] :
( ( member_set_fm @ X2 @ C2 )
& ( member_fm2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_810_Union__iff,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o2 @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
& ( member_o2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_811_Union__iff,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat2 @ A2 @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ C2 )
& ( member_nat2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_812_Union__iff,axiom,
! [A2: tm,C2: set_set_tm] :
( ( member_tm2 @ A2 @ ( comple2138885804642794802set_tm @ C2 ) )
= ( ? [X2: set_tm] :
( ( member_set_tm @ X2 @ C2 )
& ( member_tm2 @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_813_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_814_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_815_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_816_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_817_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_818_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A2 )
=> ! [Y2: nat] :
( ( member_nat2 @ Y2 @ X2 )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_819_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_tm,P2: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ A2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: set_tm] :
( ( member_set_tm @ X2 @ A2 )
=> ! [Y2: tm] :
( ( member_tm2 @ Y2 @ X2 )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_820_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A2 )
& ? [Y2: nat] :
( ( member_nat2 @ Y2 @ X2 )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_821_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_tm,P2: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( comple2138885804642794802set_tm @ A2 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: set_tm] :
( ( member_set_tm @ X2 @ A2 )
& ? [Y2: tm] :
( ( member_tm2 @ Y2 @ X2 )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_822_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_823_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_824_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_825_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_826_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_827_Sup__set__def,axiom,
( comple2134933779557159616set_fm
= ( ^ [A3: set_set_fm] :
( collect_fm
@ ^ [X2: fm] : ( complete_Sup_Sup_o @ ( image_set_fm_o @ ( member_fm2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_828_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_829_Sup__set__def,axiom,
( comple8784269564784259782ist_fm
= ( ^ [A3: set_set_list_fm] :
( collect_list_fm
@ ^ [X2: list_fm] : ( complete_Sup_Sup_o @ ( image_set_list_fm_o @ ( member_list_fm2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_830_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_831_Sup__set__def,axiom,
( comple2138885804642794802set_tm
= ( ^ [A3: set_set_tm] :
( collect_tm
@ ^ [X2: tm] : ( complete_Sup_Sup_o @ ( image_set_tm_o @ ( member_tm2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_832_parts__in__children,axiom,
! [P: fm,Z: list_fm,Z5: list_fm,A2: list_tm,R: rule] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ( member_list_fm2 @ Z5 @ ( 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 @ Z5 ) ) ) ) ) ).
% parts_in_children
thf(fact_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_855_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_856_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_857_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_858_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_859_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_860_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_861_Sup__le__iff,axiom,
! [A2: set_set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( comple2134933779557159616set_fm @ A2 ) @ B )
= ( ! [X2: set_fm] :
( ( member_set_fm @ X2 @ A2 )
=> ( ord_less_eq_set_fm @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_862_Sup__le__iff,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B )
= ( ! [X2: set_nat] :
( ( member_set_nat2 @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_863_Sup__le__iff,axiom,
! [A2: set_o,B: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B )
= ( ! [X2: $o] :
( ( member_o2 @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_864_Sup__le__iff,axiom,
! [A2: set_set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( comple2138885804642794802set_tm @ A2 ) @ B )
= ( ! [X2: set_tm] :
( ( member_set_tm @ X2 @ A2 )
=> ( ord_less_eq_set_tm @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_865_Sup__upper,axiom,
! [X3: set_fm,A2: set_set_fm] :
( ( member_set_fm @ X3 @ A2 )
=> ( ord_less_eq_set_fm @ X3 @ ( comple2134933779557159616set_fm @ A2 ) ) ) ).
% Sup_upper
thf(fact_866_Sup__upper,axiom,
! [X3: set_nat,A2: set_set_nat] :
( ( member_set_nat2 @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_867_Sup__upper,axiom,
! [X3: $o,A2: set_o] :
( ( member_o2 @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_868_Sup__upper,axiom,
! [X3: set_tm,A2: set_set_tm] :
( ( member_set_tm @ X3 @ A2 )
=> ( ord_less_eq_set_tm @ X3 @ ( comple2138885804642794802set_tm @ A2 ) ) ) ).
% Sup_upper
thf(fact_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_Sup__eqI,axiom,
! [A2: set_set_fm,X3: set_fm] :
( ! [Y3: set_fm] :
( ( member_set_fm @ Y3 @ A2 )
=> ( ord_less_eq_set_fm @ Y3 @ X3 ) )
=> ( ! [Y3: set_fm] :
( ! [Z6: set_fm] :
( ( member_set_fm @ Z6 @ A2 )
=> ( ord_less_eq_set_fm @ Z6 @ Y3 ) )
=> ( ord_less_eq_set_fm @ X3 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_878_Sup__eqI,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat2 @ Y3 @ A2 )
=> ( ord_less_eq_set_nat @ Y3 @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [Z6: set_nat] :
( ( member_set_nat2 @ Z6 @ A2 )
=> ( ord_less_eq_set_nat @ Z6 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_879_Sup__eqI,axiom,
! [A2: set_o,X3: $o] :
( ! [Y3: $o] :
( ( member_o2 @ Y3 @ A2 )
=> ( ord_less_eq_o @ Y3 @ X3 ) )
=> ( ! [Y3: $o] :
( ! [Z6: $o] :
( ( member_o2 @ Z6 @ A2 )
=> ( ord_less_eq_o @ Z6 @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_880_Sup__eqI,axiom,
! [A2: set_set_tm,X3: set_tm] :
( ! [Y3: set_tm] :
( ( member_set_tm @ Y3 @ A2 )
=> ( ord_less_eq_set_tm @ Y3 @ X3 ) )
=> ( ! [Y3: set_tm] :
( ! [Z6: set_tm] :
( ( member_set_tm @ Z6 @ A2 )
=> ( ord_less_eq_set_tm @ Z6 @ Y3 ) )
=> ( ord_less_eq_set_tm @ X3 @ Y3 ) )
=> ( ( comple2138885804642794802set_tm @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_nat,B2: fm > set_fm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X2: fm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_920_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_nat,B2: tm > set_fm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X2: tm] : ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_921_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B2: fm > set_nat,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X2: fm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_922_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B2: tm > set_nat,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X2: tm] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_923_UN__extend__simps_I9_J,axiom,
! [C2: tm > set_nat,B2: fm > set_tm,A2: set_fm] :
( ( comple7399068483239264473et_nat
@ ( image_fm_set_nat
@ ^ [X2: fm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_924_UN__extend__simps_I9_J,axiom,
! [C2: tm > set_nat,B2: tm > set_tm,A2: set_tm] :
( ( comple7399068483239264473et_nat
@ ( image_tm_set_nat
@ ^ [X2: tm] : ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ C2 @ ( comple2138885804642794802set_tm @ ( image_tm_set_tm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_925_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_tm,B2: tm > set_fm,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X2: tm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_926_UN__extend__simps_I9_J,axiom,
! [C2: fm > set_tm,B2: fm > set_fm,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X2: fm] : ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_fm_set_tm @ C2 @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_927_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_tm,B2: tm > set_nat,A2: set_tm] :
( ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [X2: tm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_928_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_tm,B2: fm > set_nat,A2: set_fm] :
( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [X2: fm] : ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( B2 @ X2 ) ) )
@ A2 ) )
= ( comple2138885804642794802set_tm @ ( image_nat_set_tm @ C2 @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_SUP__eqI,axiom,
! [A2: set_fm,F: fm > $o,X3: $o] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: fm] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_970_SUP__eqI,axiom,
! [A2: set_nat,F: nat > $o,X3: $o] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_971_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X3: $o] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: $o] :
( ( member_o2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_972_SUP__eqI,axiom,
! [A2: set_tm,F: tm > $o,X3: $o] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: tm] :
( ( member_tm2 @ I3 @ A2 )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_973_SUP__eqI,axiom,
! [A2: set_fm,F: fm > set_fm,X3: set_fm] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_fm] :
( ! [I3: fm] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X3 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_974_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_fm,X3: set_fm] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_fm] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X3 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_975_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_fm,X3: set_fm] :
( ! [I2: $o] :
( ( member_o2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_fm] :
( ! [I3: $o] :
( ( member_o2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X3 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_976_SUP__eqI,axiom,
! [A2: set_tm,F: tm > set_fm,X3: set_fm] :
( ! [I2: tm] :
( ( member_tm2 @ I2 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_fm] :
( ! [I3: tm] :
( ( member_tm2 @ I3 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_fm @ X3 @ Y3 ) )
=> ( ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_977_SUP__eqI,axiom,
! [A2: set_fm,F: fm > set_nat,X3: set_nat] :
( ! [I2: fm] :
( ( member_fm2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I3: fm] :
( ( member_fm2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_978_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X3: set_nat] :
( ! [I2: nat] :
( ( member_nat2 @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_SUP__upper,axiom,
! [I4: fm,A2: set_fm,F: fm > $o] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1004_SUP__upper,axiom,
! [I4: nat,A2: set_nat,F: nat > $o] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1005_SUP__upper,axiom,
! [I4: $o,A2: set_o,F: $o > $o] :
( ( member_o2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1006_SUP__upper,axiom,
! [I4: tm,A2: set_tm,F: tm > $o] :
( ( member_tm2 @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1007_SUP__upper,axiom,
! [I4: fm,A2: set_fm,F: fm > set_fm] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1008_SUP__upper,axiom,
! [I4: nat,A2: set_nat,F: nat > set_fm] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1009_SUP__upper,axiom,
! [I4: $o,A2: set_o,F: $o > set_fm] :
( ( member_o2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1010_SUP__upper,axiom,
! [I4: tm,A2: set_tm,F: tm > set_fm] :
( ( member_tm2 @ I4 @ A2 )
=> ( ord_less_eq_set_fm @ ( F @ I4 ) @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1011_SUP__upper,axiom,
! [I4: fm,A2: set_fm,F: fm > set_nat] :
( ( member_fm2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1012_SUP__upper,axiom,
! [I4: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_1013_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 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_1014_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 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_1015_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 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_1016_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 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ A2 )
=> ( ord_less_eq_set_tm @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_1017_SUP__upper2,axiom,
! [I4: fm,A2: set_fm,U: $o,F: fm > $o] :
( ( member_fm2 @ I4 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_fm_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1018_SUP__upper2,axiom,
! [I4: nat,A2: set_nat,U: $o,F: nat > $o] :
( ( member_nat2 @ I4 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1019_SUP__upper2,axiom,
! [I4: $o,A2: set_o,U: $o,F: $o > $o] :
( ( member_o2 @ I4 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1020_SUP__upper2,axiom,
! [I4: tm,A2: set_tm,U: $o,F: tm > $o] :
( ( member_tm2 @ I4 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I4 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_tm_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1021_SUP__upper2,axiom,
! [I4: fm,A2: set_fm,U: set_fm,F: fm > set_fm] :
( ( member_fm2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_fm_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1022_SUP__upper2,axiom,
! [I4: nat,A2: set_nat,U: set_fm,F: nat > set_fm] :
( ( member_nat2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_nat_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1023_SUP__upper2,axiom,
! [I4: $o,A2: set_o,U: set_fm,F: $o > set_fm] :
( ( member_o2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_o_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1024_SUP__upper2,axiom,
! [I4: tm,A2: set_tm,U: set_fm,F: tm > set_fm] :
( ( member_tm2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_fm @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_fm @ U @ ( comple2134933779557159616set_fm @ ( image_tm_set_fm @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1025_SUP__upper2,axiom,
! [I4: fm,A2: set_fm,U: set_nat,F: fm > set_nat] :
( ( member_fm2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1026_SUP__upper2,axiom,
! [I4: nat,A2: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I4 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I4 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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
@ ^ [X2: fm] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1038_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
@ ^ [X2: tm] : ( image_nat_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1039_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
@ ^ [X2: fm] : ( image_nat_tm @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1040_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
@ ^ [X2: tm] : ( image_nat_tm @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1041_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
@ ^ [X2: tm] : ( image_tm_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1042_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
@ ^ [X2: fm] : ( image_tm_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1043_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
@ ^ [X2: tm] : ( image_tm_tm @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1044_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
@ ^ [X2: fm] : ( image_tm_tm @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1045_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
@ ^ [X2: tm] : ( image_tm_set_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1046_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
@ ^ [X2: fm] : ( image_tm_set_nat @ F @ ( B2 @ X2 ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_1078_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 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ I5 )
=> ( ord_less_eq_set_nat @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_1079_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 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ I5 )
=> ( ord_less_eq_set_tm @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_1080_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 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ I5 )
=> ( ord_less_eq_set_tm @ ( A2 @ X2 ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_1081_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_image__eqI,axiom,
! [B: set_tm,F: tm > set_tm,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_set_tm @ B @ ( image_tm_set_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1092_image__eqI,axiom,
! [B: list_fm,F: tm > list_fm,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_list_fm2 @ B @ ( image_tm_list_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1093_image__eqI,axiom,
! [B: fm,F: tm > fm,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_fm2 @ B @ ( image_tm_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1094_image__eqI,axiom,
! [B: nat,F: tm > nat,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_nat2 @ B @ ( image_tm_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1095_image__eqI,axiom,
! [B: $o,F: tm > $o,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_o2 @ B @ ( image_tm_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1096_image__eqI,axiom,
! [B: tm,F: tm > tm,X3: tm,A2: set_tm] :
( ( B
= ( F @ X3 ) )
=> ( ( member_tm2 @ X3 @ A2 )
=> ( member_tm2 @ B @ ( image_tm_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_1097_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o2 @ $true ) ) ).
% Sup_bool_def
thf(fact_1098_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_1099_p0,axiom,
( paramsts
= ( ^ [Ts2: list_tm] : ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst @ Ts2 ) ) ) ) ) ).
% p0
thf(fact_1100_s1_I1_J,axiom,
( new_term
= ( ^ [C3: nat,T2: tm] :
~ ( member_nat2 @ C3 @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_1101_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_1102_s1_I2_J,axiom,
( new_list
= ( ^ [C3: nat,L: list_tm] :
~ ( member_nat2 @ C3 @ ( paramsts @ L ) ) ) ) ).
% s1(2)
thf(fact_1103_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_1104_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_1105_params_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( dis @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(3)
thf(fact_1106_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_1107_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_1108_sub__list_Osimps_I1_J,axiom,
! [V: nat,S2: tm] :
( ( sub_list @ V @ S2 @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_1109_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_1110_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_1111_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_1112_sub_Osimps_I3_J,axiom,
! [V: nat,S2: tm,P: fm,Q: fm] :
( ( sub @ V @ S2 @ ( dis @ P @ Q ) )
= ( dis @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q ) ) ) ).
% sub.simps(3)
thf(fact_1113_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_1114_s5_I1_J,axiom,
( sub_term
= ( ^ [V2: nat,S3: tm,T2: tm] : ( substt @ T2 @ S3 @ V2 ) ) ) ).
% s5(1)
thf(fact_1115_s5_I2_J,axiom,
( sub_list
= ( ^ [V2: nat,S3: tm,L: list_tm] : ( substts @ L @ S3 @ V2 ) ) ) ).
% s5(2)
thf(fact_1116_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_1117_substts_Osimps_I1_J,axiom,
! [S2: tm,K: nat] :
( ( substts @ nil_tm @ S2 @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_1118_paramsts_Osimps_I1_J,axiom,
( ( paramsts @ nil_tm )
= bot_bot_set_nat ) ).
% paramsts.simps(1)
thf(fact_1119_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_1120_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_1121_sub__term_Osimps_I2_J,axiom,
! [V: nat,S2: tm,I4: nat,L2: list_tm] :
( ( sub_term @ V @ S2 @ ( fun @ I4 @ L2 ) )
= ( fun @ I4 @ ( sub_list @ V @ S2 @ L2 ) ) ) ).
% sub_term.simps(2)
thf(fact_1122_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_1123_new__term_Osimps_I2_J,axiom,
! [C: nat,I4: nat,L2: list_tm] :
( ( new_term @ C @ ( fun @ I4 @ L2 ) )
= ( ( I4 != C )
& ( ( I4 != C )
=> ( new_list @ C @ L2 ) ) ) ) ).
% new_term.simps(2)
thf(fact_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_p1_H,axiom,
paramst3 = paramst ).
% p1'
thf(fact_1132_subtermTm__refl,axiom,
! [T: tm] : ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_1133_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_1134_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_1135_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_1136_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_1137_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_1138_paramst_H_H_Oelims,axiom,
! [X3: tm,Y: set_nat] :
( ( ( paramst3 @ X3 )
= Y )
=> ( ( ? [N: nat] :
( X3
= ( var @ N ) )
=> ( Y != bot_bot_set_nat ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( ( X3
= ( 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_1139_paramst_H_H_Opelims,axiom,
! [X3: tm,Y: set_nat] :
( ( ( paramst3 @ X3 )
= Y )
=> ( ( accp_tm @ paramst_rel @ X3 )
=> ( ! [N: nat] :
( ( X3
= ( var @ N ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_tm @ paramst_rel @ ( var @ N ) ) ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( ( X3
= ( 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_1140_tm_Oinject_I2_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( var @ X23 )
= ( var @ Y23 ) )
= ( X23 = Y23 ) ) ).
% tm.inject(2)
thf(fact_1141_paramst_H_H_Ocases,axiom,
! [X3: tm] :
( ! [N: nat] :
( X3
!= ( var @ N ) )
=> ~ ! [A4: nat,Ts3: list_tm] :
( X3
!= ( fun @ A4 @ Ts3 ) ) ) ).
% paramst''.cases
thf(fact_1142_tm_Oexhaust,axiom,
! [Y: tm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X24: nat] :
( Y
!= ( var @ X24 ) ) ) ).
% tm.exhaust
thf(fact_1143_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X23: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X23 ) ) ).
% tm.distinct(1)
thf(fact_1144_new__term_Osimps_I1_J,axiom,
! [C: nat,N2: nat] : ( new_term @ C @ ( var @ N2 ) ) ).
% new_term.simps(1)
thf(fact_1145_paramst_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst.simps(1)
thf(fact_1146_paramst_H_H_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst3 @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst''.simps(1)
thf(fact_1147_paramst_H_Osimps_I1_J,axiom,
! [N2: nat] :
( ( paramst2 @ ( var @ N2 ) )
= bot_bot_set_nat ) ).
% paramst'.simps(1)
thf(fact_1148_subtermTm_Osimps_I2_J,axiom,
! [N2: nat] :
( ( subtermTm @ ( var @ N2 ) )
= ( cons_tm @ ( var @ N2 ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_1149_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_1150_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_1151_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_1152_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1153_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_1154_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_1155_params_Osimps_I1_J,axiom,
! [B: nat,Ts: list_tm] :
( ( params @ ( pre @ B @ Ts ) )
= ( paramsts @ Ts ) ) ).
% params.simps(1)
thf(fact_1156_sub_Osimps_I1_J,axiom,
! [V: nat,S2: tm,I4: nat,L2: list_tm] :
( ( sub @ V @ S2 @ ( pre @ I4 @ L2 ) )
= ( pre @ I4 @ ( sub_list @ V @ S2 @ L2 ) ) ) ).
% sub.simps(1)
thf(fact_1157_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_1158_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_1159_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_1160_p2,axiom,
params2 = params ).
% p2
thf(fact_1161_params_H_Osimps_I7_J,axiom,
! [P: fm] :
( ( params2 @ ( neg @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(7)
thf(fact_1162_params_H_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( params2 @ ( dis @ P @ Q ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q ) ) ) ).
% params'.simps(3)
thf(fact_1163_DeltaExi,axiom,
! [I4: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P ) ) @ Z ) )
=> ( ( news @ I4 @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P ) ) @ Z ) ) ) ) ).
% DeltaExi
thf(fact_1164_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1165_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1166_fm_Oinject_I5_J,axiom,
! [X52: fm,Y52: fm] :
( ( ( exi @ X52 )
= ( exi @ Y52 ) )
= ( X52 = Y52 ) ) ).
% fm.inject(5)
thf(fact_1167_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_1168_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_1169_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_1170_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1171_le__trans,axiom,
! [I4: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_1172_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1173_params_H_Osimps_I5_J,axiom,
! [P: fm] :
( ( params2 @ ( exi @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(5)
thf(fact_1174_fm_Odistinct_I25_J,axiom,
! [X31: fm,X32: fm,X52: fm] :
( ( dis @ X31 @ X32 )
!= ( exi @ X52 ) ) ).
% fm.distinct(25)
thf(fact_1175_subtermFm_Osimps_I5_J,axiom,
! [P: fm] :
( ( subtermFm @ ( exi @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(5)
thf(fact_1176_params_Osimps_I5_J,axiom,
! [P: fm] :
( ( params @ ( exi @ P ) )
= ( params @ P ) ) ).
% params.simps(5)
thf(fact_1177_fm_Odistinct_I39_J,axiom,
! [X52: fm,X7: fm] :
( ( exi @ X52 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_1178_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X52: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X52 ) ) ).
% fm.distinct(7)
thf(fact_1179_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1180_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_1181_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_1182_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1183_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_1184_DeltaUni,axiom,
! [I4: nat,P: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P ) @ Z ) )
=> ( ( news @ I4 @ ( cons_fm @ P @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P ) @ Z ) ) ) ) ).
% DeltaUni
thf(fact_1185_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_1186_fm_Oinject_I6_J,axiom,
! [X62: fm,Y6: fm] :
( ( ( uni @ X62 )
= ( uni @ Y6 ) )
= ( X62 = Y6 ) ) ).
% fm.inject(6)
thf(fact_1187_p2_H,axiom,
params3 = params ).
% p2'
thf(fact_1188_params_H_H_Osimps_I5_J,axiom,
! [P: fm] :
( ( params3 @ ( exi @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(5)
thf(fact_1189_fm_Odistinct_I37_J,axiom,
! [X52: fm,X62: fm] :
( ( exi @ X52 )
!= ( uni @ X62 ) ) ).
% fm.distinct(37)
thf(fact_1190_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X62: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X62 ) ) ).
% fm.distinct(9)
thf(fact_1191_fm_Odistinct_I41_J,axiom,
! [X62: fm,X7: fm] :
( ( uni @ X62 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_1192_params_H_H_Osimps_I6_J,axiom,
! [P: fm] :
( ( params3 @ ( uni @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(6)
thf(fact_1193_params_H_H_Osimps_I7_J,axiom,
! [P: fm] :
( ( params3 @ ( neg @ P ) )
= ( params3 @ P ) ) ).
% params''.simps(7)
thf(fact_1194_params_Osimps_I6_J,axiom,
! [P: fm] :
( ( params @ ( uni @ P ) )
= ( params @ P ) ) ).
% params.simps(6)
thf(fact_1195_subtermFm_Osimps_I6_J,axiom,
! [P: fm] :
( ( subtermFm @ ( uni @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(6)
thf(fact_1196_fm_Odistinct_I27_J,axiom,
! [X31: fm,X32: fm,X62: fm] :
( ( dis @ X31 @ X32 )
!= ( uni @ X62 ) ) ).
% fm.distinct(27)
thf(fact_1197_params_H_Osimps_I6_J,axiom,
! [P: fm] :
( ( params2 @ ( uni @ P ) )
= ( params2 @ P ) ) ).
% params'.simps(6)
thf(fact_1198_params_H_H_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( params3 @ ( dis @ P @ Q ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q ) ) ) ).
% params''.simps(3)
thf(fact_1199_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_1200_sequent__calculus_Ocases,axiom,
! [A: list_fm] :
( ( sequent_calculus @ A )
=> ( ! [P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ P4 @ Z4 ) )
=> ~ ( member_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ! [P4: fm,Q3: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( dis @ P4 @ Q3 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ ( cons_fm @ Q3 @ Z4 ) ) ) )
=> ( ! [P4: fm,Q3: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( imp @ P4 @ Q3 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ Q3 @ Z4 ) ) ) )
=> ( ! [P4: fm,Q3: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( con @ P4 @ Q3 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( con @ P4 @ Q3 ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ Q3 @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( neg @ ( imp @ P4 @ Q3 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q3: fm] :
( ( A
= ( cons_fm @ ( neg @ ( dis @ P4 @ Q3 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z4 ) ) ) )
=> ( ! [T3: tm,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( exi @ P4 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T3 @ P4 ) @ Z4 ) ) )
=> ( ! [T3: tm,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( uni @ P4 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T3 @ P4 ) ) @ Z4 ) ) )
=> ( ! [I2: nat,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( uni @ P4 ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P4 ) @ Z4 ) )
=> ~ ( news @ I2 @ ( cons_fm @ P4 @ Z4 ) ) ) )
=> ( ! [I2: nat,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( exi @ P4 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P4 ) ) @ Z4 ) )
=> ~ ( news @ I2 @ ( cons_fm @ P4 @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( neg @ P4 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) ) )
=> ~ ! [Z4: list_fm] :
( ( sequent_calculus @ Z4 )
=> ~ ( ext_fm @ A @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sequent_calculus.cases
thf(fact_1201_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_1202_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_1203_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_1204_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X62: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X62 ) ) ).
% fm.distinct(33)
thf(fact_1205_fm_Odistinct_I19_J,axiom,
! [X21: fm,X22: fm,X62: fm] :
( ( imp @ X21 @ X22 )
!= ( uni @ X62 ) ) ).
% fm.distinct(19)
thf(fact_1206_BetaCon,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P @ Q ) @ Z ) ) ) ) ).
% BetaCon
thf(fact_1207_fm_Odistinct_I13_J,axiom,
! [X21: fm,X22: fm,X31: fm,X32: fm] :
( ( imp @ X21 @ X22 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(13)
thf(fact_1208_fm_Odistinct_I23_J,axiom,
! [X31: fm,X32: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X32 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_1209_fm_Odistinct_I15_J,axiom,
! [X21: fm,X22: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X22 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_1210_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_1211_fm_Odistinct_I21_J,axiom,
! [X21: fm,X22: fm,X7: fm] :
( ( imp @ X21 @ X22 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_1212_sub_Osimps_I4_J,axiom,
! [V: nat,S2: tm,P: fm,Q: fm] :
( ( sub @ V @ S2 @ ( con @ P @ Q ) )
= ( con @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q ) ) ) ).
% sub.simps(4)
thf(fact_1213_sub_Osimps_I2_J,axiom,
! [V: nat,S2: tm,P: fm,Q: fm] :
( ( sub @ V @ S2 @ ( imp @ P @ Q ) )
= ( imp @ ( sub @ V @ S2 @ P ) @ ( sub @ V @ S2 @ Q ) ) ) ).
% sub.simps(2)
thf(fact_1214_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_1215_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_1216_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X52: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X52 ) ) ).
% fm.distinct(31)
thf(fact_1217_fm_Odistinct_I17_J,axiom,
! [X21: fm,X22: fm,X52: fm] :
( ( imp @ X21 @ X22 )
!= ( exi @ X52 ) ) ).
% fm.distinct(17)
thf(fact_1218_params_H_H_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( params3 @ ( con @ P @ Q ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q ) ) ) ).
% params''.simps(4)
thf(fact_1219_params_H_H_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( params3 @ ( imp @ P @ Q ) )
= ( sup_sup_set_nat @ ( params3 @ P ) @ ( params3 @ Q ) ) ) ).
% params''.simps(2)
thf(fact_1220_params_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( imp @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(2)
thf(fact_1221_params_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( con @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(4)
thf(fact_1222_params_H_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( params2 @ ( con @ P @ Q ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q ) ) ) ).
% params'.simps(4)
thf(fact_1223_params_H_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( params2 @ ( imp @ P @ Q ) )
= ( sup_sup_set_nat @ ( params2 @ P ) @ ( params2 @ Q ) ) ) ).
% params'.simps(2)
thf(fact_1224_subtermFm_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( imp @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(2)
thf(fact_1225_subtermFm_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( con @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(4)
thf(fact_1226_params_H_H_Ocases,axiom,
! [X3: fm] :
( ! [B4: nat,Ts3: list_tm] :
( X3
!= ( pre @ B4 @ Ts3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( imp @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( dis @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( con @ P4 @ Q3 ) )
=> ( ! [P4: fm] :
( X3
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X3
!= ( uni @ P4 ) )
=> ~ ! [P4: fm] :
( X3
!= ( neg @ P4 ) ) ) ) ) ) ) ) ).
% params''.cases
thf(fact_1227_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_1228_Neg__exhaust,axiom,
! [X3: fm] :
( ! [I2: nat,Ts3: list_tm] :
( X3
!= ( pre @ I2 @ Ts3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( imp @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( dis @ P4 @ Q3 ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( con @ P4 @ Q3 ) )
=> ( ! [P4: fm] :
( X3
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X3
!= ( uni @ P4 ) )
=> ( ! [I2: nat,Ts3: list_tm] :
( X3
!= ( neg @ ( pre @ I2 @ Ts3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( neg @ ( imp @ P4 @ Q3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( neg @ ( dis @ P4 @ Q3 ) ) )
=> ( ! [P4: fm,Q3: fm] :
( X3
!= ( neg @ ( con @ P4 @ Q3 ) ) )
=> ( ! [P4: fm] :
( X3
!= ( neg @ ( exi @ P4 ) ) )
=> ( ! [P4: fm] :
( X3
!= ( neg @ ( uni @ P4 ) ) )
=> ~ ! [P4: fm] :
( X3
!= ( neg @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Neg_exhaust
thf(fact_1229_AlphaImp,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ Q @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P @ Q ) @ Z ) ) ) ).
% AlphaImp
thf(fact_1230_AlphaCon,axiom,
! [P: fm,Q: fm,Z: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ ( neg @ Q ) @ Z ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P @ Q ) ) @ Z ) ) ) ).
% AlphaCon
thf(fact_1231_BetaImp,axiom,
! [P: fm,Z: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P @ Q ) ) @ Z ) ) ) ) ).
% BetaImp
thf(fact_1232_branchDone_Ocases,axiom,
! [X3: list_fm] :
( ( X3 != nil_fm )
=> ( ! [P4: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ! [V3: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( X3
!= ( cons_fm @ ( uni @ V3 ) @ Z4 ) ) ) ) ) ) ) ) ) ).
% branchDone.cases
thf(fact_1233_params_H_H_Oelims,axiom,
! [X3: fm,Y: set_nat] :
( ( ( params3 @ X3 )
= Y )
=> ( ! [B4: nat,Ts3: list_tm] :
( ( X3
= ( pre @ B4 @ Ts3 ) )
=> ( Y
!= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( imp @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( dis @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( con @ P4 @ Q3 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) ) )
=> ( ! [P4: fm] :
( ( X3
= ( exi @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) )
=> ( ! [P4: fm] :
( ( X3
= ( uni @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) )
=> ~ ! [P4: fm] :
( ( X3
= ( neg @ P4 ) )
=> ( Y
!= ( params3 @ P4 ) ) ) ) ) ) ) ) ) ) ).
% params''.elims
thf(fact_1234_params_H_H_Opelims,axiom,
! [X3: fm,Y: set_nat] :
( ( ( params3 @ X3 )
= Y )
=> ( ( accp_fm @ params_rel @ X3 )
=> ( ! [B4: nat,Ts3: list_tm] :
( ( X3
= ( pre @ B4 @ Ts3 ) )
=> ( ( Y
= ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts3 ) ) ) )
=> ~ ( accp_fm @ params_rel @ ( pre @ B4 @ Ts3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( imp @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( imp @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( dis @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( dis @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm,Q3: fm] :
( ( X3
= ( con @ P4 @ Q3 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( params3 @ P4 ) @ ( params3 @ Q3 ) ) )
=> ~ ( accp_fm @ params_rel @ ( con @ P4 @ Q3 ) ) ) )
=> ( ! [P4: fm] :
( ( X3
= ( exi @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( exi @ P4 ) ) ) )
=> ( ! [P4: fm] :
( ( X3
= ( uni @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( uni @ P4 ) ) ) )
=> ~ ! [P4: fm] :
( ( X3
= ( neg @ P4 ) )
=> ( ( Y
= ( params3 @ P4 ) )
=> ~ ( accp_fm @ params_rel @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% params''.pelims
thf(fact_1235_branchDone_Oelims_I1_J,axiom,
! [X3: list_fm,Y: $o] :
( ( ( branchDone @ X3 )
= Y )
=> ( ( ( X3 = nil_fm )
=> Y )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(1)
thf(fact_1236_branchDone_Osimps_I1_J,axiom,
~ ( branchDone @ nil_fm ) ).
% branchDone.simps(1)
thf(fact_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_branchDone_Oelims_I2_J,axiom,
! [X3: list_fm] :
( ( branchDone @ X3 )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(2)
thf(fact_1246_branchDone_Oelims_I3_J,axiom,
! [X3: list_fm] :
( ~ ( branchDone @ X3 )
=> ( ( X3 != nil_fm )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(3)
thf(fact_1247_branchDone_Opelims_I1_J,axiom,
! [X3: list_fm,Y: $o] :
( ( ( branchDone @ X3 )
= Y )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ( ( X3 = nil_fm )
=> ( ~ Y
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z4 ) ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(1)
thf(fact_1248_branchDone_Opelims_I3_J,axiom,
! [X3: list_fm] :
( ~ ( branchDone @ X3 )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ( ( X3 = nil_fm )
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(3)
thf(fact_1249_branchDone_Opelims_I2_J,axiom,
! [X3: list_fm] :
( ( branchDone @ X3 )
=> ( ( accp_list_fm @ branchDone_rel @ X3 )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm2 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: nat,Va: list_tm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( pre @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( imp @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( dis @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Va: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V3 @ Va ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( con @ V3 @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V3 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( exi @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ~ ! [V3: fm,Z4: list_fm] :
( ( X3
= ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V3 ) @ Z4 ) )
=> ~ ( ( member_fm2 @ ( neg @ ( uni @ V3 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(2)
thf(fact_1250_tm_Osize__gen_I2_J,axiom,
! [X23: nat] :
( ( size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_1251_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_1252_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,Z4: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) )
=> ( ! [N: nat] : ( R2 @ N @ ( suc @ N ) )
=> ( R2 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1253_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_1254_full__nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% full_nat_induct
thf(fact_1255_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_1256_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1257_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_1258_Suc__le__D,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1259_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1260_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_1261_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1262_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1263_bounded__Max__nat,axiom,
! [P2: nat > $o,X3: nat,M5: nat] :
( ( P2 @ X3 )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1264_tm_Osize_I4_J,axiom,
! [X23: nat] :
( ( size_size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_1265_parts__in__effect,axiom,
! [P: fm,Z: list_fm,B2: list_tm,Z5: list_fm,R: rule,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( 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 @ Z5 ) ) ) ) ) ).
% parts_in_effect
thf(fact_1266_ne__effect__not__branchDone,axiom,
! [B2: list_tm,Z5: list_fm,R: rule,A2: list_tm,Z: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
=> ~ ( branchDone @ Z ) ) ).
% ne_effect_not_branchDone
thf(fact_1267_effect__preserves__unaffected,axiom,
! [P: fm,Z: list_fm,R: rule,B2: list_tm,Z5: list_fm,A2: list_tm] :
( ( member_fm2 @ P @ ( set_fm2 @ Z ) )
=> ( ~ ( affects @ R @ P )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B2 @ Z5 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z ) ) )
=> ( member_fm2 @ P @ ( set_fm2 @ Z5 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_1268_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_1269_eff__def,axiom,
( eff
= ( ^ [R3: rule,S3: produc6018962875968178549ist_fm] :
( ^ [Y8: fset_P8989946509869081563ist_fm,Z7: fset_P8989946509869081563ist_fm] : ( Y8 = Z7 )
@ ( effect @ R3 @ S3 ) ) ) ) ).
% eff_def
thf(fact_1270_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
@ ^ [Z8: list_fm] : ( produc1414352766439514085ist_fm @ ( remdups_tm @ ( append_tm @ A2 @ ( append_tm @ ( subterms @ Z ) @ ( subterms @ Z8 ) ) ) ) @ Z8 )
@ ( fset_of_list_list_fm @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z ) ) ) ) @ R @ Z ) ) ) ) ) ) ).
% effect.simps
% Helper facts (13)
thf(help_If_2_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X3: list_o,Y: list_o] :
( ( if_list_o @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_I_Eo_J_T,axiom,
! [X3: list_o,Y: list_o] :
( ( if_list_o @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X3: list_fm,Y: list_fm] :
( ( if_list_fm @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X3: list_fm,Y: list_fm] :
( ( if_list_fm @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y: list_tm] :
( ( if_list_tm @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X3: list_tm,Y: list_tm] :
( ( if_list_tm @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X3: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
! [X3: list_set_nat,Y: list_set_nat] :
( ( if_list_set_nat @ $true @ X3 @ Y )
= X3 ) ).
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,
! [X3: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X3: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( 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 ) @ X4 ) ) ) ) ).
%------------------------------------------------------------------------------