TPTP Problem File: SLH0215^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_03165_128992__15929688_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1627 ( 736 unt; 350 typ; 0 def)
% Number of atoms : 3525 (1802 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 11012 ( 281 ~; 56 |; 298 &;9228 @)
% ( 0 <=>;1149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 50 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1389 (1389 >; 0 *; 0 +; 0 <<)
% Number of symbols : 315 ( 312 usr; 36 con; 0-5 aty)
% Number of variables : 3664 ( 433 ^;3152 !; 79 ?;3664 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:11:40.521
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Pr1021281732701644634_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Mapping__Omapping_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
option_Sum_sum_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (312)
thf(sy_c_Ailamazyan_Oad__agr__close__rec_001tf__a,type,
ad_agr_close_rec_a: nat > ( nat > option_Sum_sum_a_nat ) > set_a > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oad__agr__list_001t__Nat__Onat_001t__Nat__Onat,type,
ad_agr_list_nat_nat: set_nat > list_Sum_sum_nat_nat > list_Sum_sum_nat_nat > $o ).
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001t__Nat__Onat,type,
ad_agr_sets_a_nat: set_nat > set_nat > set_a > ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Ailamazyan_Oad__equiv__list_001tf__a_001t__Nat__Onat,type,
ad_equiv_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oeval__conj__tuple_001t__Nat__Onat,type,
eval_conj_tuple_nat: set_nat > list_nat > list_nat > list_Sum_sum_nat_nat > list_Sum_sum_nat_nat > set_li3402123583998030149at_nat ).
thf(sy_c_Ailamazyan_Oeval__conj__tuple_001tf__a,type,
eval_conj_tuple_a: set_a > list_nat > list_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oext__tuple_001t__Nat__Onat,type,
ext_tuple_nat: set_nat > list_nat > list_nat > list_Sum_sum_nat_nat > set_li3402123583998030149at_nat ).
thf(sy_c_Ailamazyan_Oext__tuple_001tf__a,type,
ext_tuple_a: set_a > list_nat > list_nat > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlz_001t__Nat__Onat,type,
fo_nmlz_nat: set_nat > list_Sum_sum_nat_nat > list_Sum_sum_nat_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlz_001tf__a,type,
fo_nmlz_a: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlz__rec_001tf__a,type,
fo_nmlz_rec_a: nat > ( sum_sum_a_nat > option_nat ) > set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlzd_001tf__a,type,
fo_nmlzd_a: set_a > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oid__map_001tf__a,type,
id_map_a: nat > sum_sum_a_nat > option_nat ).
thf(sy_c_Ailamazyan_Omerge_001t__Nat__Onat,type,
merge_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_Ailamazyan_Omerge_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
merge_5076080148444063375at_nat: list_P1835631256188631350at_nat > list_P1835631256188631350at_nat > list_P1835631256188631350at_nat ).
thf(sy_c_Ailamazyan_Omerge_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
merge_Sum_sum_a_nat: list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_Ailamazyan_Onall__tuples_001tf__a,type,
nall_tuples_a: set_a > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Onall__tuples__rec_001t__Nat__Onat,type,
nall_tuples_rec_nat: set_nat > nat > nat > set_li3402123583998030149at_nat ).
thf(sy_c_Ailamazyan_Onall__tuples__rec_001tf__a,type,
nall_tuples_rec_a: set_a > nat > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Onats,type,
nats: list_nat > $o ).
thf(sy_c_Ailamazyan_Oproj__tuple_001tf__a,type,
proj_tuple_a: list_nat > list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Nat__Onat,type,
proj_vals_nat: set_nat_nat > list_nat > set_list_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
proj_v1967724908790944216at_nat: set_na2019211207066354340at_nat > list_nat > set_li3402123583998030149at_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
proj_v3643391342904276326_a_nat: set_na3699693778330250182_a_nat > list_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Osp__equiv_001tf__a_001t__Nat__Onat,type,
sp_equiv_a_nat: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001t__Nat__Onat,type,
sp_equiv_list_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
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_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
finite4449111032884125436_a_nat: set_Pr1021281732701644634_a_nat > nat ).
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finite6080979521523705895_a_nat: set_Sum_sum_a_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Fun__Def_Ois__measure_001t__List__Olist_It__Nat__Onat_J,type,
fun_is1289455817882274531st_nat: ( list_nat > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
fun_is1110089413066145474_a_nat: ( list_Sum_sum_a_nat > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__Nat__Onat,type,
fun_is_measure_nat: ( nat > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
fun_is6612574661097730855_a_nat: ( produc5454855657571582244_a_nat > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__String__Ochar,type,
fun_is_measure_char: ( char > nat ) > $o ).
thf(sy_c_Fun__Def_Ois__measure_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
fun_is2216824623064540604_a_nat: ( sum_sum_a_nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
if_lis4685338526944683083_a_nat: $o > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
if_set7709265119413304363_a_nat: $o > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in6527859786845829079at_nat: set_li3402123583998030149at_nat > set_li3402123583998030149at_nat > set_li3402123583998030149at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inf_in3249246906714053971_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inf_in5539880675939433708_a_nat: set_Pr1021281732701644634_a_nat > set_Pr1021281732701644634_a_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inf_in7084830621192376909_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su4083067149120280889_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su6804446743777130803_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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append1996214168388709506_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Oappend_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
append_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odistinct_001_Eo,type,
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thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
vimage6136348398256869390_nat_o: ( list_Sum_sum_a_nat > $o ) > set_o > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage7538740559230589179_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
vimage4660099214859102426at_nat: ( list_Sum_sum_a_nat > nat ) > set_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001tf__a,type,
vimage7906019935533595764_nat_a: ( list_Sum_sum_a_nat > a ) > set_a > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage8387349081243083832_a_nat: ( nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
vimage1655988174671605260at_nat: ( nat > sum_sum_nat_nat ) > set_Sum_sum_nat_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage3040984495076556338_a_nat: ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
vimage7444966428737627988at_nat: ( sum_sum_a_nat > nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage6545432446589551483_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
vimage3056561457863698554_nat_a: ( sum_sum_a_nat > a ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001_Eo,type,
vimage_a_o: ( a > $o ) > set_o > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage1894578532518109122_a_nat: ( a > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage4607795094749595324_a_nat: ( a > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__Nat__Onat,type,
set_or9089632773640736191rt_nat: list_nat > list_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_Sum__Type_Osum_Oisl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_isl_nat_nat: sum_sum_nat_nat > $o ).
thf(sy_c_Sum__Type_Osum_Oisl_001tf__a_001t__Nat__Onat,type,
sum_isl_a_nat: sum_sum_a_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member322858896541912486at_nat: list_Sum_sum_nat_nat > set_li3402123583998030149at_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member3650934779329710139_a_nat: produc5454855657571582244_a_nat > set_Pr1021281732701644634_a_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_R_092_060phi_062,type,
r_phi: set_na3699693778330250182_a_nat ).
thf(sy_v_R_092_060psi_062,type,
r_psi: set_na3699693778330250182_a_nat ).
thf(sy_v_X_092_060phi_062,type,
x_phi: set_li6526943997496501093_a_nat ).
thf(sy_v_X_092_060psi_062,type,
x_psi: set_li6526943997496501093_a_nat ).
thf(sy_v__092_060sigma_062xs,type,
sigma_xs: nat > sum_sum_a_nat ).
thf(sy_v__092_060sigma_062ys,type,
sigma_ys: nat > sum_sum_a_nat ).
thf(sy_v_cxs,type,
cxs: list_P5056861408695629236_a_nat ).
thf(sy_v_cys,type,
cys: list_P5056861408695629236_a_nat ).
thf(sy_v_fs_092_060phi_062,type,
fs_phi: list_Sum_sum_a_nat ).
thf(sy_v_fs_092_060psi_062,type,
fs_psi: list_Sum_sum_a_nat ).
thf(sy_v_ns_092_060phi_062,type,
ns_phi: list_nat ).
thf(sy_v_ns_092_060phi_062_H,type,
ns_phi2: list_nat ).
thf(sy_v_ns_092_060psi_062,type,
ns_psi: list_nat ).
thf(sy_v_ns_092_060psi_062_H,type,
ns_psi2: list_nat ).
thf(sy_v_nxs,type,
nxs: list_nat ).
thf(sy_v_nys,type,
nys: list_nat ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
thf(sy_v_xs_092_060phi_062,type,
xs_phi: list_Sum_sum_a_nat ).
thf(sy_v_ys,type,
ys: list_Sum_sum_a_nat ).
thf(sy_v_ys_092_060psi_062,type,
ys_psi: list_Sum_sum_a_nat ).
% Relevant facts (1267)
thf(fact_0_xs__ys__def_I2_J,axiom,
member408289922725080238_a_nat @ ys @ x_psi ).
% xs_ys_def(2)
thf(fact_1_xs__ys__def_I1_J,axiom,
member408289922725080238_a_nat @ xs @ x_phi ).
% xs_ys_def(1)
thf(fact_2_fs_092_060phi_062__def,axiom,
( fs_phi
= ( map_na823391071729141993_a_nat @ sigma_xs @ ns_phi2 ) ) ).
% fs\<phi>_def
thf(fact_3_len__xs__ys_I2_J,axiom,
( ( size_s5283204784079214577_a_nat @ ys )
= ( size_size_list_nat @ ns_psi ) ) ).
% len_xs_ys(2)
thf(fact_4_len__xs__ys_I1_J,axiom,
( ( size_s5283204784079214577_a_nat @ xs )
= ( size_size_list_nat @ ns_phi ) ) ).
% len_xs_ys(1)
thf(fact_5_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_6_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_7_neq__if__length__neq,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_8_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_9_size__neq__size__imp__neq,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ X )
!= ( size_s5283204784079214577_a_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_10_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_11_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_12_ad__agr__list__length,axiom,
! [X2: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs2 @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_13_ad__agr__close__rec__length,axiom,
! [Ys: list_Sum_sum_a_nat,I: nat,M: nat > option_Sum_sum_a_nat,AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys @ ( ad_agr_close_rec_a @ I @ M @ AD @ Xs2 ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_close_rec_length
thf(fact_14_nall__tuples__rec__length,axiom,
! [Xs2: list_Sum_sum_a_nat,AD: set_a,I: nat,N: nat] :
( ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ I @ N ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs2 )
= N ) ) ).
% nall_tuples_rec_length
thf(fact_15_fo__nmlz__rec__length,axiom,
! [I: nat,M: sum_sum_a_nat > option_nat,AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_rec_a @ I @ M @ AD @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% fo_nmlz_rec_length
thf(fact_16_fo__nmlz__length,axiom,
! [AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_a @ AD @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% fo_nmlz_length
thf(fact_17_measure__size,axiom,
fun_is1110089413066145474_a_nat @ size_s5283204784079214577_a_nat ).
% measure_size
thf(fact_18_measure__size,axiom,
fun_is1289455817882274531st_nat @ size_size_list_nat ).
% measure_size
thf(fact_19_measure__size,axiom,
fun_is_measure_char @ size_size_char ).
% measure_size
thf(fact_20_assms_I13_J,axiom,
( xs
= ( map_na823391071729141993_a_nat @ sigma_xs @ ns_phi ) ) ).
% assms(13)
thf(fact_21_assms_I16_J,axiom,
( ys
= ( map_na823391071729141993_a_nat @ sigma_ys @ ns_psi ) ) ).
% assms(16)
thf(fact_22_length__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
= ( size_s4076174644546656840_a_nat @ Xs2 ) ) ).
% length_map
thf(fact_23_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% length_map
thf(fact_24_length__map,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_25_length__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( size_size_list_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( size_s4076174644546656840_a_nat @ Xs2 ) ) ).
% length_map
thf(fact_26_length__map,axiom,
! [F: sum_sum_a_nat > nat,Xs2: list_Sum_sum_a_nat] :
( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% length_map
thf(fact_27_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_28_assms_I15_J,axiom,
( ys_psi
= ( map_na823391071729141993_a_nat @ sigma_xs @ nys ) ) ).
% assms(15)
thf(fact_29_fo__nmlz__eq,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) )
= ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 ) ) ).
% fo_nmlz_eq
thf(fact_30_fo__nmlz__eqD,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) )
=> ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 ) ) ).
% fo_nmlz_eqD
thf(fact_31_fo__nmlz__eqI,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 )
=> ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) ) ) ).
% fo_nmlz_eqI
thf(fact_32_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_33_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Ns: list_P5056861408695629236_a_nat] :
? [Tau: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) )
= ( map_Pr2515564165894599445_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_34_fo__nmlz__ad__agr,axiom,
! [AD: set_a,Xs2: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ AD @ Xs2 @ ( fo_nmlz_a @ AD @ Xs2 ) ) ).
% fo_nmlz_ad_agr
thf(fact_35_ad__agr__list__comm,axiom,
! [X2: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs2 @ Ys )
=> ( ad_agr_list_a_nat @ X2 @ Ys @ Xs2 ) ) ).
% ad_agr_list_comm
thf(fact_36_ad__agr__list__refl,axiom,
! [X2: set_a,Xs2: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X2 @ Xs2 @ Xs2 ) ).
% ad_agr_list_refl
thf(fact_37_ad__agr__list__trans,axiom,
! [X2: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ Xs2 @ Ys )
=> ( ( ad_agr_list_a_nat @ X2 @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X2 @ Xs2 @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_38_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_39_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > nat,Xs2: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_40_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_41_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_42_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_43_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_44_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: produc5454855657571582244_a_nat > nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_45_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_46_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_47_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_48_fs_092_060psi_062__def,axiom,
( fs_psi
= ( map_na823391071729141993_a_nat @ sigma_ys @ ns_psi2 ) ) ).
% fs\<psi>_def
thf(fact_49_assms_I18_J,axiom,
( xs_phi
= ( map_na823391071729141993_a_nat @ sigma_ys @ nxs ) ) ).
% assms(18)
thf(fact_50_nall__tuples__rec__fo__nmlz__rec__complete,axiom,
! [J: nat,AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_rec_a @ J @ ( id_map_a @ J ) @ AD @ Xs2 )
= Xs2 )
=> ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ J @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) ) ) ).
% nall_tuples_rec_fo_nmlz_rec_complete
thf(fact_51_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_52_map__eq__map__tailrec,axiom,
map_Pr2058079554187054650at_nat = map_ta3945597131924107396at_nat ).
% map_eq_map_tailrec
thf(fact_53_map__eq__map__tailrec,axiom,
map_Pr2515564165894599445_a_nat = map_ta8991945714069663307_a_nat ).
% map_eq_map_tailrec
thf(fact_54_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_55_nall__tuples__rec__fo__nmlz,axiom,
! [Xs2: list_Sum_sum_a_nat,AD: set_a] :
( ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) )
= ( ( fo_nmlz_a @ AD @ Xs2 )
= Xs2 ) ) ).
% nall_tuples_rec_fo_nmlz
thf(fact_56_X_092_060psi_062__def,axiom,
( x_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r_psi @ ns_psi ) ) ) ).
% X\<psi>_def
thf(fact_57_X_092_060phi_062__def,axiom,
( x_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r_phi @ ns_phi ) ) ) ).
% X\<phi>_def
thf(fact_58_length__enumerate,axiom,
! [N: nat,Xs2: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% length_enumerate
thf(fact_59_length__enumerate,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_enumerate
thf(fact_60_exists__fo__nmlzd,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ F2 @ Xs2 ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_61_exists__fo__nmlzd,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F2: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs2 ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_62_exists__fo__nmlzd,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs2 )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F2: nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ F2 @ Xs2 ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_63_length__rotate1,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rotate2765497868024679250_a_nat @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_64_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_65_ad__agr__list__def,axiom,
( ad_agr_list_a_nat
= ( ^ [X3: set_a,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
& ( ad_equiv_list_a_nat @ X3 @ Xs3 @ Ys2 )
& ( sp_equiv_list_a_nat @ Xs3 @ Ys2 ) ) ) ) ).
% ad_agr_list_def
thf(fact_66_ns_092_060phi_062_H__def,axiom,
( ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
@ ns_psi ) ) ).
% ns\<phi>'_def
thf(fact_67_ns_092_060psi_062_H__def,axiom,
( ns_psi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
@ ns_phi ) ) ).
% ns\<psi>'_def
thf(fact_68_map__ident,axiom,
( ( map_nat_nat
@ ^ [X4: nat] : X4 )
= ( ^ [Xs3: list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_69_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X4: $o] : ( member_o @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_76_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs2: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs2 ) )
= ( filter_nat
@ ^ [X4: nat] :
( ( Q @ X4 )
& ( P @ X4 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_77_filter__filter,axiom,
! [P: produc5454855657571582244_a_nat > $o,Q: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] :
( ( filter1580576373184442087_a_nat @ P @ ( filter1580576373184442087_a_nat @ Q @ Xs2 ) )
= ( filter1580576373184442087_a_nat
@ ^ [X4: produc5454855657571582244_a_nat] :
( ( Q @ X4 )
& ( P @ X4 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_78_map__eq__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( map_Pr2058079554187054650at_nat @ G @ Xs2 ) )
= ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_79_map__eq__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs2 ) )
= ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_80_map__eq__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Xs2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_81_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_82_filter__True,axiom,
! [Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( filter1580576373184442087_a_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_83_filter__True,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( filter_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_84_filter__True,axiom,
! [Xs2: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( filter_Sum_sum_a_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_85_set__rotate1,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_86_set__rotate1,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( rotate2765497868024679250_a_nat @ Xs2 ) )
= ( set_Sum_sum_a_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_87_distinct1__rotate,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct1_rotate
thf(fact_88_set__filter,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs2: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs2 ) )
= ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_89_set__filter,axiom,
! [P: $o > $o,Xs2: list_o] :
( ( set_o2 @ ( filter_o @ P @ Xs2 ) )
= ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_90_set__filter,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] :
( ( set_Pr2196415342531872975_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) )
= ( collec2196941500952594553_a_nat
@ ^ [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_91_set__filter,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs2 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_92_set__filter,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs2 ) )
= ( collec7073057861543223018_a_nat
@ ^ [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_93_list_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ V ) )
= ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_94_list_Oset__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,V: list_P5056861408695629236_a_nat] :
( ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ F @ V ) )
= ( image_1677410150705809403at_nat @ F @ ( set_Pr2196415342531872975_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_95_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_96_list_Oset__map,axiom,
! [F: sum_sum_a_nat > nat,V: list_Sum_sum_a_nat] :
( ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ V ) )
= ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_97_list_Oset__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,V: list_P5056861408695629236_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ V ) )
= ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_98_list_Oset__map,axiom,
! [F: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_99_list_Oset__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_100_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ Xs2 ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_101_filter__cong,axiom,
! [Xs2: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter5373755100216644354_a_nat @ P @ Xs2 )
= ( filter5373755100216644354_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_102_filter__cong,axiom,
! [Xs2: list_o,Ys: list_o,P: $o > $o,Q: $o > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter_o @ P @ Xs2 )
= ( filter_o @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_103_filter__cong,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o,Q: produc5454855657571582244_a_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter1580576373184442087_a_nat @ P @ Xs2 )
= ( filter1580576373184442087_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_104_filter__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter_nat @ P @ Xs2 )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_105_filter__cong,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( filter_Sum_sum_a_nat @ P @ Xs2 )
= ( filter_Sum_sum_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_106_filter__id__conv,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] :
( ( ( filter1580576373184442087_a_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_107_filter__id__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_108_filter__id__conv,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] :
( ( ( filter_Sum_sum_a_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_109_distinct__filter,axiom,
! [Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( distin5548549501492119275_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ).
% distinct_filter
thf(fact_110_distinct__filter,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( filter_nat @ P @ Xs2 ) ) ) ).
% distinct_filter
thf(fact_111_ad__agr__list__fo__nmlzd,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( ( fo_nmlzd_a @ AD @ Vs2 )
=> ( Vs = Vs2 ) ) ) ) ).
% ad_agr_list_fo_nmlzd
thf(fact_112_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X4: nat] : X4
@ T )
= T ) ).
% list.map_ident
thf(fact_113_image__set,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
= ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_114_image__set,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( image_1677410150705809403at_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
= ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_115_image__set,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
= ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_116_image__set,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_117_image__set,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ Xs2 ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_118_image__set,axiom,
! [F: sum_sum_a_nat > nat,Xs2: list_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_119_image__set,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) ) ) ).
% image_set
thf(fact_120_ex__map__conv,axiom,
! [Ys: list_nat,F: produc5454855657571582244_a_nat > nat] :
( ( ? [Xs3: list_P5056861408695629236_a_nat] :
( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Xs3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ? [Y2: produc5454855657571582244_a_nat] :
( X4
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_121_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X4
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_122_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs3 ) ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X4
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_123_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ? [Xs3: list_P5056861408695629236_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Xs3 ) ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: produc5454855657571582244_a_nat] :
( X4
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_124_map__cong,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( map_Pr2058079554187054650at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_125_map__cong,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( Xs2 = Ys )
=> ( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_126_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs2 = Ys )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_127_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_128_map__idI,axiom,
! [Xs2: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_129_map__idI,axiom,
! [Xs2: list_o,F: $o > $o] :
( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_o_o @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_130_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_131_map__idI,axiom,
! [Xs2: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( ( F @ X5 )
= X5 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_132_map__ext,axiom,
! [Xs2: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( map_Pr2058079554187054650at_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_133_map__ext,axiom,
! [Xs2: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_134_map__ext,axiom,
! [Xs2: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_135_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_136_list_Omap__ident__strong,axiom,
! [T: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( set_li2392974972034027290_a_nat @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_137_list_Omap__ident__strong,axiom,
! [T: list_o,F: $o > $o] :
( ! [Z: $o] :
( ( member_o @ Z @ ( set_o2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_o_o @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_138_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_139_list_Omap__ident__strong,axiom,
! [T: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_140_list_Oinj__map__strong,axiom,
! [X: list_P5056861408695629236_a_nat,Xa: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,Fa: produc5454855657571582244_a_nat > nat] :
( ! [Z: produc5454855657571582244_a_nat,Za: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X ) )
=> ( ( member3650934779329710139_a_nat @ Za @ ( set_Pr2196415342531872975_a_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Pr2058079554187054650at_nat @ F @ X )
= ( map_Pr2058079554187054650at_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_141_list_Oinj__map__strong,axiom,
! [X: list_P5056861408695629236_a_nat,Xa: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Fa: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z: produc5454855657571582244_a_nat,Za: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X ) )
=> ( ( member3650934779329710139_a_nat @ Za @ ( set_Pr2196415342531872975_a_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ F @ X )
= ( map_Pr2515564165894599445_a_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_142_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > sum_sum_a_nat,Fa: nat > sum_sum_a_nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_143_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_144_list_Omap__cong0,axiom,
! [X: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ X )
= ( map_Pr2058079554187054650at_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_145_list_Omap__cong0,axiom,
! [X: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X )
= ( map_Pr2515564165894599445_a_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_146_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X )
= ( map_na823391071729141993_a_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_147_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_148_list_Omap__cong,axiom,
! [X: list_P5056861408695629236_a_nat,Ya: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > nat,G: produc5454855657571582244_a_nat > nat] :
( ( X = Ya )
=> ( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ F @ X )
= ( map_Pr2058079554187054650at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_149_list_Omap__cong,axiom,
! [X: list_P5056861408695629236_a_nat,Ya: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( X = Ya )
=> ( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X )
= ( map_Pr2515564165894599445_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_150_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_151_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_152_fo__nmlzd__code,axiom,
( fo_nmlzd_a
= ( ^ [AD2: set_a,Xs3: list_Sum_sum_a_nat] :
( ( fo_nmlz_a @ AD2 @ Xs3 )
= Xs3 ) ) ) ).
% fo_nmlzd_code
thf(fact_153_fo__nmlz__sound,axiom,
! [AD: set_a,Xs2: list_Sum_sum_a_nat] : ( fo_nmlzd_a @ AD @ ( fo_nmlz_a @ AD @ Xs2 ) ) ).
% fo_nmlz_sound
thf(fact_154_fo__nmlz__idem,axiom,
! [AD: set_a,Ys: list_Sum_sum_a_nat] :
( ( fo_nmlzd_a @ AD @ Ys )
=> ( ( fo_nmlz_a @ AD @ Ys )
= Ys ) ) ).
% fo_nmlz_idem
thf(fact_155_rotate1__map,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( rotate2765497868024679250_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
= ( map_na823391071729141993_a_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_156_rotate1__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( rotate1_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( map_Pr2058079554187054650at_nat @ F @ ( rotate86600233086810641_a_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_157_rotate1__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( rotate2765497868024679250_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
= ( map_Pr2515564165894599445_a_nat @ F @ ( rotate86600233086810641_a_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_158_rotate1__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_159_exists__map,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ? [F2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_160_exists__map,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ? [F2: produc5454855657571582244_a_nat > nat] :
( Ys
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_161_exists__map,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ? [F2: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_162_exists__map,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ? [F2: sum_sum_a_nat > nat] :
( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_163_exists__map,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs2 )
=> ? [F2: nat > sum_sum_a_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_164_exists__map,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs2 )
=> ? [F2: nat > nat] :
( Ys
= ( map_nat_nat @ F2 @ Xs2 ) ) ) ) ).
% exists_map
thf(fact_165_ad__agr,axiom,
( ad_agr_list_a_nat @ ad
@ ( map_na823391071729141993_a_nat @ sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ns_psi @ ns_psi2 ) ) )
@ ( map_na823391071729141993_a_nat @ sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ns_phi @ ns_phi2 ) ) ) ) ).
% ad_agr
thf(fact_166_image__ident,axiom,
! [Y3: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : X4
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_167_image__ident,axiom,
! [Y3: set_nat] :
( ( image_nat_nat
@ ^ [X4: nat] : X4
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_168_distinct__map__filterI,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,P: nat > $o] :
( ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
=> ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% distinct_map_filterI
thf(fact_169_distinct__map__filterI,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
=> ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ) ).
% distinct_map_filterI
thf(fact_170_distinct__map__filterI,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distin2701893636801681144_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
=> ( distin2701893636801681144_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ) ).
% distinct_map_filterI
thf(fact_171_distinct__map__filterI,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o] :
( ( distinct_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( distinct_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% distinct_map_filterI
thf(fact_172_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_173_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_174_image__eqI,axiom,
! [B: nat,F: list_Sum_sum_a_nat > nat,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_175_image__eqI,axiom,
! [B: $o,F: list_Sum_sum_a_nat > $o,X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X ) )
=> ( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_176_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_177_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_178_image__eqI,axiom,
! [B: $o,F: nat > $o,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_179_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_180_image__eqI,axiom,
! [B: nat,F: $o > nat,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B: $o,F: $o > $o,X: $o,A2: set_o] :
( ( B
= ( F @ X ) )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_182_distinct__union,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs2 @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_183_sp__equiv__list__link,axiom,
! [Sigma: nat > sum_sum_a_nat,Ns: list_nat,Tau2: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) )
= ( sp_equiv_a_nat @ Sigma @ Tau2 @ ( set_nat2 @ Ns ) ) ) ).
% sp_equiv_list_link
thf(fact_184_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma @ Tau2 )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_185_nall__tuples__set,axiom,
( nall_tuples_a
= ( ^ [AD2: set_a,N2: nat] :
( collec7555443234367654128_a_nat
@ ^ [Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= N2 )
& ( fo_nmlzd_a @ AD2 @ Ys2 ) ) ) ) ) ).
% nall_tuples_set
thf(fact_186_same__append__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_187_append__same__eq,axiom,
! [Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs2 )
= ( append_nat @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_188_append__assoc,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
= ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_189_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
= ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_190_append__eq__append__conv,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us )
= ( size_s5283204784079214577_a_nat @ Vs ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs2 @ Us )
= ( append_Sum_sum_a_nat @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_191_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_192_map__append,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,Ys: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_193_map__append,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( map_Pr2058079554187054650at_nat @ F @ ( append338925788367110473_a_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) @ ( map_Pr2058079554187054650at_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_194_map__append,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( append338925788367110473_a_nat @ Xs2 @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) @ ( map_Pr2515564165894599445_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_195_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_196_filter__append,axiom,
! [P: nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( filter_nat @ P @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( filter_nat @ P @ Xs2 ) @ ( filter_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_197_filter__append,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( filter1580576373184442087_a_nat @ P @ ( append338925788367110473_a_nat @ Xs2 @ Ys ) )
= ( append338925788367110473_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) @ ( filter1580576373184442087_a_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_198_set__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( set_nat2 @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_sort
thf(fact_199_length__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_sort
thf(fact_200_distinct__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( distinct_nat @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct_sort
thf(fact_201_stable__sort__key__sort__key,axiom,
linord227665693835759911at_nat @ linord738340561235409698at_nat ).
% stable_sort_key_sort_key
thf(fact_202_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_203_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_204_sort__key__const,axiom,
! [C2: nat,Xs2: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X4: nat] : C2
@ Xs2 )
= Xs2 ) ).
% sort_key_const
thf(fact_205_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_206_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( ? [Us2: list_P5056861408695629236_a_nat,Vs3: list_P5056861408695629236_a_nat] :
( ( Xs2
= ( append338925788367110473_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Us2 ) )
& ( Zs
= ( map_Pr2058079554187054650at_nat @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_207_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
= ( ? [Us2: list_P5056861408695629236_a_nat,Vs3: list_P5056861408695629236_a_nat] :
( ( Xs2
= ( append338925788367110473_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_Pr2515564165894599445_a_nat @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_208_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_209_map__eq__append__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_210_map__eq__append__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_P5056861408695629236_a_nat,Vs3: list_P5056861408695629236_a_nat] :
( ( Xs2
= ( append338925788367110473_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Pr2058079554187054650at_nat @ F @ Us2 ) )
& ( Zs
= ( map_Pr2058079554187054650at_nat @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_211_map__eq__append__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ? [Us2: list_P5056861408695629236_a_nat,Vs3: list_P5056861408695629236_a_nat] :
( ( Xs2
= ( append338925788367110473_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_Pr2515564165894599445_a_nat @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_212_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_213_filter__sort,axiom,
! [P: nat > $o,F: nat > nat,Xs2: list_nat] :
( ( filter_nat @ P @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( linord738340561235409698at_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% filter_sort
thf(fact_214_sort__key__stable,axiom,
! [F: nat > nat,K: nat,Xs2: list_nat] :
( ( filter_nat
@ ^ [Y2: nat] :
( ( F @ Y2 )
= K )
@ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( filter_nat
@ ^ [Y2: nat] :
( ( F @ Y2 )
= K )
@ Xs2 ) ) ).
% sort_key_stable
thf(fact_215_proj__vals__def,axiom,
( proj_vals_nat
= ( ^ [R: set_nat_nat,Ns2: list_nat] :
( image_2065990475722762790st_nat
@ ^ [Tau3: nat > nat] : ( map_nat_nat @ Tau3 @ Ns2 )
@ R ) ) ) ).
% proj_vals_def
thf(fact_216_proj__vals__def,axiom,
( proj_v3643391342904276326_a_nat
= ( ^ [R: set_na3699693778330250182_a_nat,Ns2: list_nat] :
( image_6721470456781115300_a_nat
@ ^ [Tau3: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau3 @ Ns2 )
@ R ) ) ) ).
% proj_vals_def
thf(fact_217_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_218_imageI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_219_imageI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_220_imageI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_3984946558445957976_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_221_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_222_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_223_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_224_imageI,axiom,
! [X: $o,A2: set_o,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ ( image_733135354807394034_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_225_imageI,axiom,
! [X: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_226_imageI,axiom,
! [X: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_227_image__iff,axiom,
! [Z2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_228_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_229_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X6 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_230_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ).
% bex_imageD
thf(fact_231_image__cong,axiom,
! [M2: set_li6526943997496501093_a_nat,N3: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( M2 = N3 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ N3 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M2 )
= ( image_5081948215111134021_a_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_232_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N3 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ N3 )
=> ( ( F @ X5 )
= ( G @ X5 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_233_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ( P @ X5 ) )
=> ! [X6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_234_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X5 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_235_rev__image__eqI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_236_rev__image__eqI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_237_rev__image__eqI,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: $o,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_238_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_239_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_240_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_241_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_242_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: nat,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_243_rev__image__eqI,axiom,
! [X: $o,A2: set_o,B: $o,F: $o > $o] :
( ( member_o @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_o @ B @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_244_nall__tuples__code,axiom,
( nall_tuples_a
= ( ^ [AD2: set_a] : ( nall_tuples_rec_a @ AD2 @ zero_zero_nat ) ) ) ).
% nall_tuples_code
thf(fact_245_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ~ ! [X5: list_Sum_sum_a_nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member408289922725080238_a_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_246_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
=> ~ ! [X5: nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_247_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,A2: set_o] :
( ( member408289922725080238_a_nat @ B @ ( image_733135354807394034_a_nat @ F @ A2 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_o @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_248_imageE,axiom,
! [B: nat,F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) )
=> ~ ! [X5: list_Sum_sum_a_nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member408289922725080238_a_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_249_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X5: nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_250_imageE,axiom,
! [B: nat,F: $o > nat,A2: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A2 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_o @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_251_imageE,axiom,
! [B: $o,F: list_Sum_sum_a_nat > $o,A2: set_li6526943997496501093_a_nat] :
( ( member_o @ B @ ( image_3984946558445957976_nat_o @ F @ A2 ) )
=> ~ ! [X5: list_Sum_sum_a_nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member408289922725080238_a_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_252_imageE,axiom,
! [B: $o,F: nat > $o,A2: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A2 ) )
=> ~ ! [X5: nat] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_nat @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_253_imageE,axiom,
! [B: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X5: $o] :
( ( B
= ( F @ X5 ) )
=> ~ ( member_o @ X5 @ A2 ) ) ) ).
% imageE
thf(fact_254_image__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A2 ) )
= ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_255_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A2 ) ) ).
% image_image
thf(fact_256_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_5081948215111134021_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_257_Compr__image__eq,axiom,
! [F: $o > list_Sum_sum_a_nat,A2: set_o,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_733135354807394034_a_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_733135354807394034_a_nat @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_258_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > $o,A2: set_li6526943997496501093_a_nat,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_3984946558445957976_nat_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_3984946558445957976_nat_o @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_259_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_260_Compr__image__eq,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_6262589752765146990_a_nat @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_261_Compr__image__eq,axiom,
! [F: nat > $o,A2: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_nat_o @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_262_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_2535339886381165584at_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_2535339886381165584at_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_263_Compr__image__eq,axiom,
! [F: $o > nat,A2: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_o_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_264_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_265_nall__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( member408289922725080238_a_nat @ Vs @ ( nall_tuples_a @ AD @ N ) ) ) ) ).
% nall_tuplesI
thf(fact_266_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_267_sort__append__swap2,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) ) )
= ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ys @ ( append_nat @ Xs2 @ Zs ) ) ) ) ).
% sort_append_swap2
thf(fact_268_sort__append__swap,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Xs2 @ Ys ) )
= ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ys @ Xs2 ) ) ) ).
% sort_append_swap
thf(fact_269_length__code,axiom,
( size_s5283204784079214577_a_nat
= ( gen_le1340941697924381074_a_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_270_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_271_ad__agr__list__fo__nmlzd__isl,axiom,
! [X2: set_a,F: list_Sum_sum_a_nat > sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,G: list_Sum_sum_a_nat > sum_sum_a_nat,X: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ ( map_li3106614296470775678_a_nat @ F @ Xs2 ) @ ( map_li3106614296470775678_a_nat @ G @ Xs2 ) )
=> ( ( fo_nmlzd_a @ X2 @ ( map_li3106614296470775678_a_nat @ F @ Xs2 ) )
=> ( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ( ( sum_isl_a_nat @ ( F @ X ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_272_ad__agr__list__fo__nmlzd__isl,axiom,
! [X2: set_a,F: $o > sum_sum_a_nat,Xs2: list_o,G: $o > sum_sum_a_nat,X: $o] :
( ( ad_agr_list_a_nat @ X2 @ ( map_o_Sum_sum_a_nat @ F @ Xs2 ) @ ( map_o_Sum_sum_a_nat @ G @ Xs2 ) )
=> ( ( fo_nmlzd_a @ X2 @ ( map_o_Sum_sum_a_nat @ F @ Xs2 ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ( ( sum_isl_a_nat @ ( F @ X ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_273_ad__agr__list__fo__nmlzd__isl,axiom,
! [X2: set_a,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,X: produc5454855657571582244_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) @ ( map_Pr2515564165894599445_a_nat @ G @ Xs2 ) )
=> ( ( fo_nmlzd_a @ X2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
=> ( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ( ( sum_isl_a_nat @ ( F @ X ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_274_ad__agr__list__fo__nmlzd__isl,axiom,
! [X2: set_a,F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > sum_sum_a_nat,X: nat] :
( ( ad_agr_list_a_nat @ X2 @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) )
=> ( ( fo_nmlzd_a @ X2 @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( sum_isl_a_nat @ ( F @ X ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_275_ad__agr__list__fo__nmlzd__isl,axiom,
! [X2: set_a,F: sum_sum_a_nat > sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) @ ( map_Su2790769393171190532_a_nat @ G @ Xs2 ) )
=> ( ( fo_nmlzd_a @ X2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) )
=> ( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( ( sum_isl_a_nat @ ( F @ X ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ) ).
% ad_agr_list_fo_nmlzd_isl
thf(fact_276_Inf_OINF__identity__eq,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( Inf
@ ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : X4
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_277_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_278_Sup_OSUP__identity__eq,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( Sup
@ ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : X4
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_279_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_280_distinct__product,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( distinct_nat @ Ys )
=> ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs2 @ Ys ) ) ) ) ).
% distinct_product
thf(fact_281_filter__in__nths,axiom,
! [Xs2: list_l4703314356710769291_a_nat,S: set_nat] :
( ( distin811021574259663358_a_nat @ Xs2 )
=> ( ( filter5373755100216644354_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ ( nths_l8399376204360867977_a_nat @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths_l8399376204360867977_a_nat @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_282_filter__in__nths,axiom,
! [Xs2: list_o,S: set_nat] :
( ( distinct_o @ Xs2 )
=> ( ( filter_o
@ ^ [X4: $o] : ( member_o @ X4 @ ( set_o2 @ ( nths_o @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths_o @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_283_filter__in__nths,axiom,
! [Xs2: list_P5056861408695629236_a_nat,S: set_nat] :
( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( filter1580576373184442087_a_nat
@ ^ [X4: produc5454855657571582244_a_nat] : ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ ( nths_P8319198327007079712_a_nat @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths_P8319198327007079712_a_nat @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_284_filter__in__nths,axiom,
! [Xs2: list_nat,S: set_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( filter_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ ( set_nat2 @ ( nths_nat @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths_nat @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_285_filter__in__nths,axiom,
! [Xs2: list_Sum_sum_a_nat,S: set_nat] :
( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( filter_Sum_sum_a_nat
@ ^ [X4: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ ( nths_Sum_sum_a_nat @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths_Sum_sum_a_nat @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_286_in__set__nthsD,axiom,
! [X: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,I2: set_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ ( nths_l8399376204360867977_a_nat @ Xs2 @ I2 ) ) )
=> ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_287_in__set__nthsD,axiom,
! [X: $o,Xs2: list_o,I2: set_nat] :
( ( member_o @ X @ ( set_o2 @ ( nths_o @ Xs2 @ I2 ) ) )
=> ( member_o @ X @ ( set_o2 @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_288_in__set__nthsD,axiom,
! [X: nat,Xs2: list_nat,I2: set_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( nths_nat @ Xs2 @ I2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_289_in__set__nthsD,axiom,
! [X: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,I2: set_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( nths_Sum_sum_a_nat @ Xs2 @ I2 ) ) )
=> ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) ).
% in_set_nthsD
thf(fact_290_notin__set__nthsI,axiom,
! [X: list_Sum_sum_a_nat,Xs2: list_l4703314356710769291_a_nat,I2: set_nat] :
( ~ ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ~ ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ ( nths_l8399376204360867977_a_nat @ Xs2 @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_291_notin__set__nthsI,axiom,
! [X: $o,Xs2: list_o,I2: set_nat] :
( ~ ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ~ ( member_o @ X @ ( set_o2 @ ( nths_o @ Xs2 @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_292_notin__set__nthsI,axiom,
! [X: nat,Xs2: list_nat,I2: set_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( nths_nat @ Xs2 @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_293_notin__set__nthsI,axiom,
! [X: sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,I2: set_nat] :
( ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ~ ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ ( nths_Sum_sum_a_nat @ Xs2 @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_294_nths__map,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat,I2: set_nat] :
( ( nths_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ I2 )
= ( map_na823391071729141993_a_nat @ F @ ( nths_nat @ Xs2 @ I2 ) ) ) ).
% nths_map
thf(fact_295_nths__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,I2: set_nat] :
( ( nths_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) @ I2 )
= ( map_Pr2058079554187054650at_nat @ F @ ( nths_P8319198327007079712_a_nat @ Xs2 @ I2 ) ) ) ).
% nths_map
thf(fact_296_nths__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,I2: set_nat] :
( ( nths_Sum_sum_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) @ I2 )
= ( map_Pr2515564165894599445_a_nat @ F @ ( nths_P8319198327007079712_a_nat @ Xs2 @ I2 ) ) ) ).
% nths_map
thf(fact_297_nths__map,axiom,
! [F: nat > nat,Xs2: list_nat,I2: set_nat] :
( ( nths_nat @ ( map_nat_nat @ F @ Xs2 ) @ I2 )
= ( map_nat_nat @ F @ ( nths_nat @ Xs2 @ I2 ) ) ) ).
% nths_map
thf(fact_298_distinct__nthsI,axiom,
! [Xs2: list_nat,I2: set_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( nths_nat @ Xs2 @ I2 ) ) ) ).
% distinct_nthsI
thf(fact_299_Sup_OSUP__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C3: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A2 = B2 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ B2 )
=> ( ( C3 @ X5 )
= ( D @ X5 ) ) )
=> ( ( Sup @ ( image_5081948215111134021_a_nat @ C3 @ A2 ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_300_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ B2 )
=> ( ( C3 @ X5 )
= ( D @ X5 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C3 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_301_Inf_OINF__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C3: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A2 = B2 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ B2 )
=> ( ( C3 @ X5 )
= ( D @ X5 ) ) )
=> ( ( Inf @ ( image_5081948215111134021_a_nat @ C3 @ A2 ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_302_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ B2 )
=> ( ( C3 @ X5 )
= ( D @ X5 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C3 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_303_nxs__def,axiom,
( nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
& ~ ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_phi @ xs ) ) ) ) ).
% nxs_def
thf(fact_304_nys__def,axiom,
( nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
& ~ ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_psi @ ys ) ) ) ) ).
% nys_def
thf(fact_305_cxs__def,axiom,
( cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_phi @ xs ) ) ) ).
% cxs_def
thf(fact_306_distinct__quicksort,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ ( set_or9089632773640736191rt_nat @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct_quicksort
thf(fact_307_cys__def,axiom,
( cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ ns_psi @ ys ) ) ) ).
% cys_def
thf(fact_308_set__quicksort,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( set_or9089632773640736191rt_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_quicksort
thf(fact_309_length__product,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s7247017532682008665_a_nat @ ( produc7134955936706270469_a_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs2 ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_product
thf(fact_310_length__product,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_nat] :
( ( size_s5183913381435988258at_nat @ ( produc2474788113070794826at_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s5283204784079214577_a_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_311_length__product,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( produc7294178216264498984_a_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_product
thf(fact_312_length__product,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_313_nall__tuples__rec__filter,axiom,
! [Xs2: list_Sum_sum_nat_nat,AD: set_nat,N: nat,Ys: list_Sum_sum_nat_nat] :
( ( member322858896541912486at_nat @ Xs2 @ ( nall_tuples_rec_nat @ AD @ N @ ( size_s5552450856825661347at_nat @ Xs2 ) ) )
=> ( ( Ys
= ( filter4040215016827425730at_nat
@ ^ [X4: sum_sum_nat_nat] :
~ ( sum_isl_nat_nat @ X4 )
@ Xs2 ) )
=> ( member322858896541912486at_nat @ Ys @ ( nall_tuples_rec_nat @ bot_bot_set_nat @ N @ ( size_s5552450856825661347at_nat @ Ys ) ) ) ) ) ).
% nall_tuples_rec_filter
thf(fact_314_nall__tuples__rec__filter,axiom,
! [Xs2: list_Sum_sum_a_nat,AD: set_a,N: nat,Ys: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ N @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) )
=> ( ( Ys
= ( filter_Sum_sum_a_nat
@ ^ [X4: sum_sum_a_nat] :
~ ( sum_isl_a_nat @ X4 )
@ Xs2 ) )
=> ( member408289922725080238_a_nat @ Ys @ ( nall_tuples_rec_a @ bot_bot_set_a @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ) ) ).
% nall_tuples_rec_filter
thf(fact_315_quicksort__conv__sort,axiom,
( set_or9089632773640736191rt_nat
= ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4 ) ) ).
% quicksort_conv_sort
thf(fact_316_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_317_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_318_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_319_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_320_all__not__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ! [X4: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X4 @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_321_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X4: $o] :
~ ( member_o @ X4 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_322_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_323_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_324_empty__iff,axiom,
! [C2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_325_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_326_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_327_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_328_image__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% image_empty
thf(fact_329_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_330_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_331_image__empty,axiom,
! [F: nat > a] :
( ( image_nat_a @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_332_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_333_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_334_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_335_empty__is__image,axiom,
! [F: nat > a,A2: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_336_empty__is__image,axiom,
! [F: a > nat,A2: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_337_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_338_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_339_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_340_image__is__empty,axiom,
! [F: nat > a,A2: set_nat] :
( ( ( image_nat_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_341_image__is__empty,axiom,
! [F: a > nat,A2: set_a] :
( ( ( image_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_342_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_343_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_344_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_345_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_346_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_347_zip__append,axiom,
! [Xs2: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs2 @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs2 @ Us ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_348_zip__append,axiom,
! [Xs2: list_Sum_sum_a_nat,Us: list_nat,Ys: list_Sum_sum_a_nat,Vs: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs2 @ Us ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_349_zip__append,axiom,
! [Xs2: list_nat,Us: list_Sum_sum_a_nat,Ys: list_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs2 @ Us ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_350_zip__append,axiom,
! [Xs2: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_351_map__fst__zip,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs2 @ Ys ) )
= Xs2 ) ) ).
% map_fst_zip
thf(fact_352_map__fst__zip,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ ( zip_Su6417478541797927256at_nat @ Xs2 @ Ys ) )
= Xs2 ) ) ).
% map_fst_zip
thf(fact_353_map__fst__zip,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
= Xs2 ) ) ).
% map_fst_zip
thf(fact_354_map__fst__zip,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys ) )
= Xs2 ) ) ).
% map_fst_zip
thf(fact_355_assms_I14_J,axiom,
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys )
= ( map_na823391071729141993_a_nat @ sigma_xs @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cys ) ) ) ).
% assms(14)
thf(fact_356_assms_I17_J,axiom,
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs )
= ( map_na823391071729141993_a_nat @ sigma_ys @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cxs ) ) ) ).
% assms(17)
thf(fact_357_ex__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ? [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A2 ) )
= ( A2 != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_358_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X4: $o] : ( member_o @ X4 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_359_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_360_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_361_equals0I,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [Y4: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y4 @ A2 )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_362_equals0I,axiom,
! [A2: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_363_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_364_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_365_equals0D,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( A2 = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_366_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_367_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_368_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_369_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_370_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_371_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_372_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_373_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_374_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_375_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_376_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_377_Set_Oempty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% Set.empty_def
thf(fact_378_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X4: nat] : $false ) ) ).
% Set.empty_def
thf(fact_379_map2__map__map,axiom,
! [H: nat > nat > nat,F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_380_map2__map__map,axiom,
! [H: sum_sum_a_nat > nat > nat,F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr8434518024483680788at_nat @ ( produc8957381051445135989at_nat @ H ) @ ( zip_Su6417478541797927256at_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_381_map2__map__map,axiom,
! [H: nat > nat > sum_sum_a_nat,F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr8085477292072239746_a_nat @ ( produc7425670176118603825_a_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_382_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > nat,F: nat > nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr2058079554187054650at_nat @ ( produc2606280072924899667at_nat @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_383_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > $o,F: nat > nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr4279255274729099950_nat_o @ ( produc8810490045567577941_nat_o @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) )
= ( map_nat_o
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_384_map2__map__map,axiom,
! [H: sum_sum_a_nat > sum_sum_a_nat > nat,F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr321637745973761139at_nat @ ( produc4523541316010391868at_nat @ H ) @ ( zip_Su7355543910597222519_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) )
= ( map_nat_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_385_map2__map__map,axiom,
! [H: sum_sum_a_nat > nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > nat] :
( ( map_Pr6349035483858946235_a_nat @ ( produc119559382349320218_a_nat @ H ) @ ( zip_Su6417478541797927256at_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_nat_nat @ G @ Xs2 ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_386_map2__map__map,axiom,
! [H: nat > nat > nat,F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) @ ( map_Pr2058079554187054650at_nat @ G @ Xs2 ) ) )
= ( map_Pr2058079554187054650at_nat
@ ^ [X4: produc5454855657571582244_a_nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_387_map2__map__map,axiom,
! [H: nat > sum_sum_a_nat > sum_sum_a_nat,F: nat > nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ ( produc2370069609027602876_a_nat @ H ) @ ( zip_na2013496608136855606_a_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_388_map2__map__map,axiom,
! [H: sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs2: list_nat,G: nat > sum_sum_a_nat] :
( ( map_Pr3527087583225239196_a_nat @ ( produc906896977214093203_a_nat @ H ) @ ( zip_Su7355543910597222519_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) @ ( map_na823391071729141993_a_nat @ G @ Xs2 ) ) )
= ( map_na823391071729141993_a_nat
@ ^ [X4: nat] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_389_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_390_measure__fst,axiom,
! [F: nat > nat] :
( ( fun_is_measure_nat @ F )
=> ( fun_is6612574661097730855_a_nat
@ ^ [P2: produc5454855657571582244_a_nat] : ( F @ ( produc1510075366785312874_a_nat @ P2 ) ) ) ) ).
% measure_fst
thf(fact_391_distinct__zipI2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_nat] :
( ( distin2701893636801681144_a_nat @ Ys )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_392_distinct__zipI1,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( distinct_nat @ Xs2 )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_393_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_394_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_395_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_396_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_397_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_398_folding__Map__graph_Odistinct__if__distinct__map,axiom,
! [Xs2: list_P5056861408695629236_a_nat] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs2 ) )
=> ( distin5548549501492119275_a_nat @ Xs2 ) ) ).
% folding_Map_graph.distinct_if_distinct_map
thf(fact_399_distinct__map__fst__filter,axiom,
! [Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs2 ) )
=> ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ) ).
% distinct_map_fst_filter
thf(fact_400_split__part,axiom,
! [P: $o,Q: nat > sum_sum_a_nat > $o] :
( ( produc8810490045567577941_nat_o
@ ^ [A3: nat,B3: sum_sum_a_nat] :
( P
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: produc5454855657571582244_a_nat] :
( P
& ( produc8810490045567577941_nat_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_401_merge__length,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( size_size_list_nat @ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ).
% merge_length
thf(fact_402_map__snd__enumerate,axiom,
! [N: nat,Xs2: list_Sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs2 ) )
= Xs2 ) ).
% map_snd_enumerate
thf(fact_403_map__snd__zip,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs2 @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_404_map__snd__zip,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ ( zip_Su6417478541797927256at_nat @ Xs2 @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_405_map__snd__zip,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs2 @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_406_map__snd__zip,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_407_prod_Oexpand,axiom,
! [Prod: produc5454855657571582244_a_nat,Prod2: produc5454855657571582244_a_nat] :
( ( ( ( produc1510075366785312874_a_nat @ Prod )
= ( produc1510075366785312874_a_nat @ Prod2 ) )
& ( ( produc1280857981117566888_a_nat @ Prod )
= ( produc1280857981117566888_a_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_408_prod__eqI,axiom,
! [P3: produc5454855657571582244_a_nat,Q2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ P3 )
= ( produc1510075366785312874_a_nat @ Q2 ) )
=> ( ( ( produc1280857981117566888_a_nat @ P3 )
= ( produc1280857981117566888_a_nat @ Q2 ) )
=> ( P3 = Q2 ) ) ) ).
% prod_eqI
thf(fact_409_prod__eq__iff,axiom,
( ( ^ [Y5: produc5454855657571582244_a_nat,Z3: produc5454855657571582244_a_nat] : ( Y5 = Z3 ) )
= ( ^ [S2: produc5454855657571582244_a_nat,T2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ S2 )
= ( produc1510075366785312874_a_nat @ T2 ) )
& ( ( produc1280857981117566888_a_nat @ S2 )
= ( produc1280857981117566888_a_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_410_snd__def,axiom,
( produc1280857981117566888_a_nat
= ( produc2370069609027602876_a_nat
@ ^ [X1: nat,X22: sum_sum_a_nat] : X22 ) ) ).
% snd_def
thf(fact_411_case__prod__beta,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F3: nat > sum_sum_a_nat > $o,P2: produc5454855657571582244_a_nat] : ( F3 @ ( produc1510075366785312874_a_nat @ P2 ) @ ( produc1280857981117566888_a_nat @ P2 ) ) ) ) ).
% case_prod_beta
thf(fact_412_split__beta,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F3: nat > sum_sum_a_nat > $o,Prod3: produc5454855657571582244_a_nat] : ( F3 @ ( produc1510075366785312874_a_nat @ Prod3 ) @ ( produc1280857981117566888_a_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_413_Product__Type_OCollect__case__prodD,axiom,
! [X: produc5454855657571582244_a_nat,A2: nat > sum_sum_a_nat > $o] :
( ( member3650934779329710139_a_nat @ X @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ A2 ) ) )
=> ( A2 @ ( produc1510075366785312874_a_nat @ X ) @ ( produc1280857981117566888_a_nat @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_414_case__prod__unfold,axiom,
( produc8810490045567577941_nat_o
= ( ^ [C: nat > sum_sum_a_nat > $o,P2: produc5454855657571582244_a_nat] : ( C @ ( produc1510075366785312874_a_nat @ P2 ) @ ( produc1280857981117566888_a_nat @ P2 ) ) ) ) ).
% case_prod_unfold
thf(fact_415_case__prod__beta_H,axiom,
( produc8810490045567577941_nat_o
= ( ^ [F3: nat > sum_sum_a_nat > $o,X4: produc5454855657571582244_a_nat] : ( F3 @ ( produc1510075366785312874_a_nat @ X4 ) @ ( produc1280857981117566888_a_nat @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_416_measure__snd,axiom,
! [F: sum_sum_a_nat > nat] :
( ( fun_is2216824623064540604_a_nat @ F )
=> ( fun_is6612574661097730855_a_nat
@ ^ [P2: produc5454855657571582244_a_nat] : ( F @ ( produc1280857981117566888_a_nat @ P2 ) ) ) ) ).
% measure_snd
thf(fact_417_pair__list__eqI,axiom,
! [Xs2: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs2 )
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Ys ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Xs2 )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Ys ) )
=> ( Xs2 = Ys ) ) ) ).
% pair_list_eqI
thf(fact_418_zip__map__fst__snd,axiom,
! [Zs: list_P5056861408695629236_a_nat] :
( ( zip_na2013496608136855606_a_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs ) @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_419_zip__eq__conv,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_P1195027771636113901_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_Su7355543910597222519_a_nat @ Xs2 @ Ys )
= Zs )
= ( ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Zs )
= Xs2 )
& ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_420_zip__eq__conv,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_Su6417478541797927256at_nat @ Xs2 @ Ys )
= Zs )
= ( ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ Zs )
= Xs2 )
& ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_421_zip__eq__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_nat_nat @ Xs2 @ Ys )
= Zs )
= ( ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Zs )
= Xs2 )
& ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_422_zip__eq__conv,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys )
= Zs )
= ( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs )
= Xs2 )
& ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_423_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: nat > sum_sum_a_nat > $o,Prod: produc5454855657571582244_a_nat] :
( ( H @ ( produc8810490045567577941_nat_o @ F @ Prod ) )
= ( produc8810490045567577941_nat_o
@ ^ [X1: nat,X22: sum_sum_a_nat] : ( H @ ( F @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_424_prod_Odisc__eq__case,axiom,
! [Prod: produc5454855657571582244_a_nat] :
( produc8810490045567577941_nat_o
@ ^ [Uu: nat,Uv: sum_sum_a_nat] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_425_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_426_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_427_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_428_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_429_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_430_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_431_fst__def,axiom,
( produc1510075366785312874_a_nat
= ( produc2606280072924899667at_nat
@ ^ [X1: nat,X22: sum_sum_a_nat] : X1 ) ) ).
% fst_def
thf(fact_432_exE__realizer,axiom,
! [P: sum_sum_a_nat > nat > $o,P3: produc5454855657571582244_a_nat,Q: $o > $o,F: nat > sum_sum_a_nat > $o] :
( ( P @ ( produc1280857981117566888_a_nat @ P3 ) @ ( produc1510075366785312874_a_nat @ P3 ) )
=> ( ! [X5: nat,Y4: sum_sum_a_nat] :
( ( P @ Y4 @ X5 )
=> ( Q @ ( F @ X5 @ Y4 ) ) )
=> ( Q @ ( produc8810490045567577941_nat_o @ F @ P3 ) ) ) ) ).
% exE_realizer
thf(fact_433_ys_092_060psi_062__def,axiom,
( member408289922725080238_a_nat @ ( map_na823391071729141993_a_nat @ sigma_xs @ nys )
@ ( nall_tuples_rec_a @ bot_bot_set_a
@ ( finite_card_nat
@ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a
@ ( set_Sum_sum_a_nat2
@ ( map_na823391071729141993_a_nat @ sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ns_phi @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ cys ) ) ) ) ) ) )
@ ( size_size_list_nat @ nys ) ) ) ).
% ys\<psi>_def
thf(fact_434_eval__conj__tuple__def,axiom,
( eval_conj_tuple_nat
= ( ^ [AD2: set_nat,Ns_phi: list_nat,Ns_psi: list_nat,Xs3: list_Sum_sum_nat_nat,Ys2: list_Sum_sum_nat_nat] :
( inf_in6527859786845829079at_nat
@ ( image_4446412052380048421at_nat @ ( fo_nmlz_nat @ AD2 )
@ ( ext_tuple_nat @ bot_bot_set_nat
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi
@ ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,Y2: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ( sum_isl_nat_nat @ Y2 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_psi @ Ys2 ) ) ) ) )
@ ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,Y2: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ~ ( sum_isl_nat_nat @ Y2 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_psi @ Ys2 ) ) )
@ ( map_Pr7935113432278004075at_nat @ produc1222775543808117526at_nat
@ ( merge_5076080148444063375at_nat @ ( zip_na3998243202614522632at_nat @ Ns_phi @ Xs3 )
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,Y2: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ( sum_isl_nat_nat @ Y2 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_psi @ Ys2 ) ) ) ) ) )
@ ( image_4446412052380048421at_nat @ ( fo_nmlz_nat @ AD2 )
@ ( ext_tuple_nat @ bot_bot_set_nat
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi
@ ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,X4: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_nat_nat @ X4 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_phi @ Xs3 ) ) ) ) )
@ ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,X4: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ~ ( sum_isl_nat_nat @ X4 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_phi @ Xs3 ) ) )
@ ( map_Pr7935113432278004075at_nat @ produc1222775543808117526at_nat
@ ( merge_5076080148444063375at_nat @ ( zip_na3998243202614522632at_nat @ Ns_psi @ Ys2 )
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,X4: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_nat_nat @ X4 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_phi @ Xs3 ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_def
thf(fact_435_eval__conj__tuple__def,axiom,
( eval_conj_tuple_a
= ( ^ [AD2: set_a,Ns_phi: list_nat,Ns_psi: list_nat,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( inf_in3249246906714053971_a_nat
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( ext_tuple_a @ bot_bot_set_a
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi
@ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys2 ) ) ) ) )
@ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ~ ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys2 ) ) )
@ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat
@ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_phi @ Xs3 )
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys2 ) ) ) ) ) )
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 )
@ ( ext_tuple_a @ bot_bot_set_a
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi
@ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi @ Xs3 ) ) ) ) )
@ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ~ ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi @ Xs3 ) ) )
@ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat
@ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_psi @ Ys2 )
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi @ Xs3 ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_def
thf(fact_436_exE__realizer_H,axiom,
! [P: sum_sum_a_nat > nat > $o,P3: produc5454855657571582244_a_nat] :
( ( P @ ( produc1280857981117566888_a_nat @ P3 ) @ ( produc1510075366785312874_a_nat @ P3 ) )
=> ~ ! [X5: nat,Y4: sum_sum_a_nat] :
~ ( P @ Y4 @ X5 ) ) ).
% exE_realizer'
thf(fact_437_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_438_proj__tuple__idle,axiom,
! [Nxs: list_P5056861408695629236_a_nat] :
( ( proj_tuple_a @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ Nxs )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs ) ) ).
% proj_tuple_idle
thf(fact_439_distinct__ordered__entries,axiom,
! [M: mappin2117621025242276154_a_nat] : ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( ordere406669810731199336_a_nat @ M ) ) ) ).
% distinct_ordered_entries
thf(fact_440_Int__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
= ( ( member408289922725080238_a_nat @ C2 @ A2 )
& ( member408289922725080238_a_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_441_Int__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_442_Int__iff,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B2 ) )
= ( ( member_o @ C2 @ A2 )
& ( member_o @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_443_IntI,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( ( member408289922725080238_a_nat @ C2 @ B2 )
=> ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_444_IntI,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_445_IntI,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ A2 )
=> ( ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_446_vimage__eq,axiom,
! [A: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_447_vimage__eq,axiom,
! [A: nat,F: nat > $o,B2: set_o] :
( ( member_nat @ A @ ( vimage_nat_o @ F @ B2 ) )
= ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_448_vimage__eq,axiom,
! [A: $o,F: $o > nat,B2: set_nat] :
( ( member_o @ A @ ( vimage_o_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_449_vimage__eq,axiom,
! [A: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
= ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_450_vimage__eq,axiom,
! [A: nat,F: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_451_vimage__eq,axiom,
! [A: a,F: a > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_452_vimage__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_453_vimage__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ( member408289922725080238_a_nat @ A @ ( vimage6136348398256869390_nat_o @ F @ B2 ) )
= ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_454_vimage__eq,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_455_vimage__eq,axiom,
! [A: $o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_o @ A @ ( vimage2884537194618305448_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_456_vimageI,axiom,
! [F: nat > nat,A: nat,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_457_vimageI,axiom,
! [F: $o > nat,A: $o,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member_o @ A @ ( vimage_o_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_458_vimageI,axiom,
! [F: nat > $o,A: nat,B: $o,B2: set_o] :
( ( ( F @ A )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member_nat @ A @ ( vimage_nat_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_459_vimageI,axiom,
! [F: $o > $o,A: $o,B: $o,B2: set_o] :
( ( ( F @ A )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member_o @ A @ ( vimage_o_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_460_vimageI,axiom,
! [F: nat > sum_sum_a_nat,A: nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_461_vimageI,axiom,
! [F: a > sum_sum_a_nat,A: a,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_462_vimageI,axiom,
! [F: nat > list_Sum_sum_a_nat,A: nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_463_vimageI,axiom,
! [F: $o > list_Sum_sum_a_nat,A: $o,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member_o @ A @ ( vimage2884537194618305448_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_464_vimageI,axiom,
! [F: list_Sum_sum_a_nat > nat,A: list_Sum_sum_a_nat,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_465_vimageI,axiom,
! [F: list_Sum_sum_a_nat > $o,A: list_Sum_sum_a_nat,B: $o,B2: set_o] :
( ( ( F @ A )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage6136348398256869390_nat_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_466_vimage__Collect__eq,axiom,
! [F: a > sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( vimage4607795094749595324_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_a
@ ^ [Y2: a] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_467_vimage__Collect__eq,axiom,
! [F: nat > sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat
@ ^ [Y2: nat] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_468_vimage__Collect__eq,axiom,
! [F: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y2: nat] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_469_vimage__empty,axiom,
! [F: a > sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_470_vimage__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_471_vimage__empty,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_472_vimage__empty,axiom,
! [F: nat > a] :
( ( vimage_nat_a @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_473_vimage__empty,axiom,
! [F: a > nat] :
( ( vimage_a_nat @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_474_vimage__empty,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_475_vimage__Int,axiom,
! [F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A2 ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_476_vimage__Int,axiom,
! [F: a > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( vimage4607795094749595324_a_nat @ F @ A2 ) @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_477_distinct__append,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Xs2 @ Ys ) )
= ( ( distin2701893636801681144_a_nat @ Xs2 )
& ( distin2701893636801681144_a_nat @ Ys )
& ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% distinct_append
thf(fact_478_distinct__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ Ys ) )
= ( ( distinct_a @ Xs2 )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_479_distinct__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( ( distinct_nat @ Xs2 )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_480_vimage__def,axiom,
( vimage4607795094749595324_a_nat
= ( ^ [F3: a > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collect_a
@ ^ [X4: a] : ( member_Sum_sum_a_nat @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_481_vimage__def,axiom,
( vimage8387349081243083832_a_nat
= ( ^ [F3: nat > list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
( collect_nat
@ ^ [X4: nat] : ( member408289922725080238_a_nat @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_482_vimage__def,axiom,
( vimage_nat_nat
= ( ^ [F3: nat > nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] : ( member_nat @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_483_vimage__def,axiom,
( vimage_nat_o
= ( ^ [F3: nat > $o,B4: set_o] :
( collect_nat
@ ^ [X4: nat] : ( member_o @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_484_vimage__def,axiom,
( vimage3040984495076556338_a_nat
= ( ^ [F3: nat > sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collect_nat
@ ^ [X4: nat] : ( member_Sum_sum_a_nat @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% vimage_def
thf(fact_485_Int__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A4 )
& ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_486_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( member_o @ X4 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_487_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_488_Int__Collect,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X @ ( inf_in3249246906714053971_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) ) )
= ( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_489_Int__Collect,axiom,
! [X: $o,A2: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_490_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_491_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_492_vimage__inter__cong,axiom,
! [S3: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S3 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ Y ) @ S3 )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ G @ Y ) @ S3 ) ) ) ).
% vimage_inter_cong
thf(fact_493_vimage__inter__cong,axiom,
! [S3: set_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [W: a] :
( ( member_a @ W @ S3 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_a @ ( vimage4607795094749595324_a_nat @ F @ Y ) @ S3 )
= ( inf_inf_set_a @ ( vimage4607795094749595324_a_nat @ G @ Y ) @ S3 ) ) ) ).
% vimage_inter_cong
thf(fact_494_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: a > sum_sum_a_nat,Q: a > $o] :
( ! [X5: a] :
( ( P @ ( F @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage4607795094749595324_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_495_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ ( F @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_496_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ ( F @ X5 ) )
= ( Q @ X5 ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_497_vimageI2,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_498_vimageI2,axiom,
! [F: $o > nat,A: $o,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member_o @ A @ ( vimage_o_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_499_vimageI2,axiom,
! [F: nat > $o,A: nat,A2: set_o] :
( ( member_o @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage_nat_o @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_500_vimageI2,axiom,
! [F: $o > $o,A: $o,A2: set_o] :
( ( member_o @ ( F @ A ) @ A2 )
=> ( member_o @ A @ ( vimage_o_o @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_501_vimageI2,axiom,
! [F: nat > sum_sum_a_nat,A: nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_502_vimageI2,axiom,
! [F: a > sum_sum_a_nat,A: a,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 )
=> ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_503_vimageI2,axiom,
! [F: nat > list_Sum_sum_a_nat,A: nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_504_vimageI2,axiom,
! [F: $o > list_Sum_sum_a_nat,A: $o,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A ) @ A2 )
=> ( member_o @ A @ ( vimage2884537194618305448_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_505_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > nat,A: list_Sum_sum_a_nat,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_506_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > $o,A: list_Sum_sum_a_nat,A2: set_o] :
( ( member_o @ ( F @ A ) @ A2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage6136348398256869390_nat_o @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_507_vimageE,axiom,
! [A: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_508_vimageE,axiom,
! [A: nat,F: nat > $o,B2: set_o] :
( ( member_nat @ A @ ( vimage_nat_o @ F @ B2 ) )
=> ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_509_vimageE,axiom,
! [A: $o,F: $o > nat,B2: set_nat] :
( ( member_o @ A @ ( vimage_o_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_510_vimageE,axiom,
! [A: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
=> ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_511_vimageE,axiom,
! [A: nat,F: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_512_vimageE,axiom,
! [A: a,F: a > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_513_vimageE,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_514_vimageE,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ( member408289922725080238_a_nat @ A @ ( vimage6136348398256869390_nat_o @ F @ B2 ) )
=> ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_515_vimageE,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_516_vimageE,axiom,
! [A: $o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_o @ A @ ( vimage2884537194618305448_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_517_vimageD,axiom,
! [A: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_518_vimageD,axiom,
! [A: nat,F: nat > $o,A2: set_o] :
( ( member_nat @ A @ ( vimage_nat_o @ F @ A2 ) )
=> ( member_o @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_519_vimageD,axiom,
! [A: $o,F: $o > nat,A2: set_nat] :
( ( member_o @ A @ ( vimage_o_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_520_vimageD,axiom,
! [A: $o,F: $o > $o,A2: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ A2 ) )
=> ( member_o @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_521_vimageD,axiom,
! [A: nat,F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ A2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_522_vimageD,axiom,
! [A: a,F: a > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ A2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_523_vimageD,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_524_vimageD,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,A2: set_o] :
( ( member408289922725080238_a_nat @ A @ ( vimage6136348398256869390_nat_o @ F @ A2 ) )
=> ( member_o @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_525_vimageD,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ A2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_526_vimageD,axiom,
! [A: $o,F: $o > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member_o @ A @ ( vimage2884537194618305448_a_nat @ F @ A2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_527_IntD2,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ( member408289922725080238_a_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_528_IntD2,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ B2 ) ) ).
% IntD2
thf(fact_529_IntD2,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B2 ) )
=> ( member_o @ C2 @ B2 ) ) ).
% IntD2
thf(fact_530_IntD1,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ( member408289922725080238_a_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_531_IntD1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% IntD1
thf(fact_532_IntD1,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B2 ) )
=> ( member_o @ C2 @ A2 ) ) ).
% IntD1
thf(fact_533_IntE,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ~ ( member408289922725080238_a_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_534_IntE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ~ ( member_nat @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_535_IntE,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A2 @ B2 ) )
=> ~ ( ( member_o @ C2 @ A2 )
=> ~ ( member_o @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_536_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_537_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_538_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_539_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_540_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_541_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_542_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_543_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_544_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_545_disjoint__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ~ ( member408289922725080238_a_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_546_disjoint__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A2 )
=> ~ ( member_o @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_547_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_548_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_549_Int__emptyI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ~ ( member408289922725080238_a_nat @ X5 @ B2 ) )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_550_Int__emptyI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ~ ( member_o @ X5 @ B2 ) )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_551_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ~ ( member_a @ X5 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_552_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ~ ( member_nat @ X5 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_553_distinct__length__filter,axiom,
! [Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( size_s4076174644546656840_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) )
= ( finite4449111032884125436_a_nat @ ( inf_in5539880675939433708_a_nat @ ( collec2196941500952594553_a_nat @ P ) @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_554_distinct__length__filter,axiom,
! [Xs2: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( size_s5283204784079214577_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs2 ) )
= ( finite6080979521523705895_a_nat @ ( inf_in7084830621192376909_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_555_distinct__length__filter,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( distinct_nat @ Xs2 )
=> ( ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) )
= ( finite_card_nat @ ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_556_ext__tuple__eq,axiom,
! [Fv_sub: list_nat,As: list_Sum_sum_a_nat,AD: set_a,Fv_sub_comp: list_nat] :
( ( ( size_size_list_nat @ Fv_sub )
= ( size_s5283204784079214577_a_nat @ As ) )
=> ( ( ext_tuple_a @ AD @ Fv_sub @ Fv_sub_comp @ As )
= ( image_5081948215111134021_a_nat
@ ^ [Fs: list_Sum_sum_a_nat] : ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Fv_sub @ As ) @ ( zip_na2013496608136855606_a_nat @ Fv_sub_comp @ Fs ) ) )
@ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As ) ) ) @ ( size_size_list_nat @ Fv_sub_comp ) ) ) ) ) ).
% ext_tuple_eq
thf(fact_557_inter__set__filter,axiom,
! [A2: set_li6526943997496501093_a_nat,Xs2: list_l4703314356710769291_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
= ( set_li2392974972034027290_a_nat
@ ( filter5373755100216644354_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A2 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_558_inter__set__filter,axiom,
! [A2: set_o,Xs2: list_o] :
( ( inf_inf_set_o @ A2 @ ( set_o2 @ Xs2 ) )
= ( set_o2
@ ( filter_o
@ ^ [X4: $o] : ( member_o @ X4 @ A2 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_559_inter__set__filter,axiom,
! [A2: set_Pr1021281732701644634_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( inf_in5539880675939433708_a_nat @ A2 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
= ( set_Pr2196415342531872975_a_nat
@ ( filter1580576373184442087_a_nat
@ ^ [X4: produc5454855657571582244_a_nat] : ( member3650934779329710139_a_nat @ X4 @ A2 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_560_inter__set__filter,axiom,
! [A2: set_nat,Xs2: list_nat] :
( ( inf_inf_set_nat @ A2 @ ( set_nat2 @ Xs2 ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_561_inter__set__filter,axiom,
! [A2: set_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( set_Sum_sum_a_nat2
@ ( filter_Sum_sum_a_nat
@ ^ [X4: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X4 @ A2 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_562_distinct__card,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_563_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_564_card__distinct,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( distin2701893636801681144_a_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_565_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_566_combine__map2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_P5056861408695629236_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s4076174644546656840_a_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_s4076174644546656840_a_nat @ Xs4 ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( distin5548549501492119275_a_nat @ Xs4 )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) )
= bot_bo2474679825254379502_a_nat )
=> ? [F2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( Ys
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_567_combine__map2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_s5283204784079214577_a_nat @ Xs4 ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( distin2701893636801681144_a_nat @ Xs4 )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs4 ) )
= bot_bo3438331934148233675_a_nat )
=> ? [F2: sum_sum_a_nat > sum_sum_a_nat] :
( ( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_568_combine__map2,axiom,
! [Ys: list_nat,Xs2: list_P5056861408695629236_a_nat,Ys3: list_nat,Xs4: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s4076174644546656840_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s4076174644546656840_a_nat @ Xs4 ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( distin5548549501492119275_a_nat @ Xs4 )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) )
= bot_bo2474679825254379502_a_nat )
=> ? [F2: produc5454855657571582244_a_nat > nat] :
( ( Ys
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_569_combine__map2,axiom,
! [Ys: list_nat,Xs2: list_Sum_sum_a_nat,Ys3: list_nat,Xs4: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s5283204784079214577_a_nat @ Xs4 ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( distin2701893636801681144_a_nat @ Xs4 )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs4 ) )
= bot_bo3438331934148233675_a_nat )
=> ? [F2: sum_sum_a_nat > nat] :
( ( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Su5227373005390213643at_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_570_combine__map2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_a,Ys3: list_Sum_sum_a_nat,Xs4: list_a] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_a @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_size_list_a @ Xs4 ) )
=> ( ( distinct_a @ Xs2 )
=> ( ( distinct_a @ Xs4 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs4 ) )
= bot_bot_set_a )
=> ? [F2: a > sum_sum_a_nat] :
( ( Ys
= ( map_a_Sum_sum_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_a_Sum_sum_a_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_571_combine__map2,axiom,
! [Ys: list_nat,Xs2: list_a,Ys3: list_nat,Xs4: list_a] :
( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Xs4 ) )
=> ( ( distinct_a @ Xs2 )
=> ( ( distinct_a @ Xs4 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs4 ) )
= bot_bot_set_a )
=> ? [F2: a > nat] :
( ( Ys
= ( map_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_a_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_572_combine__map2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_size_list_nat @ Xs4 ) )
=> ( ( distinct_nat @ Xs2 )
=> ( ( distinct_nat @ Xs4 )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs4 ) )
= bot_bot_set_nat )
=> ? [F2: nat > sum_sum_a_nat] :
( ( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_na823391071729141993_a_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_573_combine__map2,axiom,
! [Ys: list_nat,Xs2: list_nat,Ys3: list_nat,Xs4: list_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Xs4 ) )
=> ( ( distinct_nat @ Xs2 )
=> ( ( distinct_nat @ Xs4 )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs4 ) )
= bot_bot_set_nat )
=> ? [F2: nat > nat] :
( ( Ys
= ( map_nat_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_nat_nat @ F2 @ Xs4 ) ) ) ) ) ) ) ) ).
% combine_map2
thf(fact_574_combine__map3,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_P5056861408695629236_a_nat,Ys4: list_Sum_sum_a_nat,Xs5: list_P5056861408695629236_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s4076174644546656840_a_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_s4076174644546656840_a_nat @ Xs4 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
= ( size_s4076174644546656840_a_nat @ Xs5 ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( distin5548549501492119275_a_nat @ Xs4 )
=> ( ( distin5548549501492119275_a_nat @ Xs5 )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) )
= bot_bo2474679825254379502_a_nat )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs5 ) )
= bot_bo2474679825254379502_a_nat )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) @ ( set_Pr2196415342531872975_a_nat @ Xs5 ) )
= bot_bo2474679825254379502_a_nat )
=> ? [F2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( Ys
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_Pr2515564165894599445_a_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_575_combine__map3,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_Sum_sum_a_nat,Ys4: list_Sum_sum_a_nat,Xs5: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_s5283204784079214577_a_nat @ Xs4 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
= ( size_s5283204784079214577_a_nat @ Xs5 ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( distin2701893636801681144_a_nat @ Xs4 )
=> ( ( distin2701893636801681144_a_nat @ Xs5 )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs4 ) )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs5 ) )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs4 ) @ ( set_Sum_sum_a_nat2 @ Xs5 ) )
= bot_bo3438331934148233675_a_nat )
=> ? [F2: sum_sum_a_nat > sum_sum_a_nat] :
( ( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_576_combine__map3,axiom,
! [Ys: list_nat,Xs2: list_P5056861408695629236_a_nat,Ys3: list_nat,Xs4: list_P5056861408695629236_a_nat,Ys4: list_nat,Xs5: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s4076174644546656840_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s4076174644546656840_a_nat @ Xs4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s4076174644546656840_a_nat @ Xs5 ) )
=> ( ( distin5548549501492119275_a_nat @ Xs2 )
=> ( ( distin5548549501492119275_a_nat @ Xs4 )
=> ( ( distin5548549501492119275_a_nat @ Xs5 )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) )
= bot_bo2474679825254379502_a_nat )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) @ ( set_Pr2196415342531872975_a_nat @ Xs5 ) )
= bot_bo2474679825254379502_a_nat )
=> ( ( ( inf_in5539880675939433708_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs4 ) @ ( set_Pr2196415342531872975_a_nat @ Xs5 ) )
= bot_bo2474679825254379502_a_nat )
=> ? [F2: produc5454855657571582244_a_nat > nat] :
( ( Ys
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_Pr2058079554187054650at_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_577_combine__map3,axiom,
! [Ys: list_nat,Xs2: list_Sum_sum_a_nat,Ys3: list_nat,Xs4: list_Sum_sum_a_nat,Ys4: list_nat,Xs5: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_s5283204784079214577_a_nat @ Xs4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_s5283204784079214577_a_nat @ Xs5 ) )
=> ( ( distin2701893636801681144_a_nat @ Xs2 )
=> ( ( distin2701893636801681144_a_nat @ Xs4 )
=> ( ( distin2701893636801681144_a_nat @ Xs5 )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs4 ) )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Xs5 ) )
= bot_bo3438331934148233675_a_nat )
=> ( ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs4 ) @ ( set_Sum_sum_a_nat2 @ Xs5 ) )
= bot_bo3438331934148233675_a_nat )
=> ? [F2: sum_sum_a_nat > nat] :
( ( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_Su5227373005390213643at_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_Su5227373005390213643at_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_578_combine__map3,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_a,Ys3: list_Sum_sum_a_nat,Xs4: list_a,Ys4: list_Sum_sum_a_nat,Xs5: list_a] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_a @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_size_list_a @ Xs4 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
= ( size_size_list_a @ Xs5 ) )
=> ( ( distinct_a @ Xs2 )
=> ( ( distinct_a @ Xs4 )
=> ( ( distinct_a @ Xs5 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs4 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs5 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs4 ) @ ( set_a2 @ Xs5 ) )
= bot_bot_set_a )
=> ? [F2: a > sum_sum_a_nat] :
( ( Ys
= ( map_a_Sum_sum_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_a_Sum_sum_a_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_a_Sum_sum_a_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_579_combine__map3,axiom,
! [Ys: list_nat,Xs2: list_a,Ys3: list_nat,Xs4: list_a,Ys4: list_nat,Xs5: list_a] :
( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Xs4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Xs5 ) )
=> ( ( distinct_a @ Xs2 )
=> ( ( distinct_a @ Xs4 )
=> ( ( distinct_a @ Xs5 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs4 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Xs5 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs4 ) @ ( set_a2 @ Xs5 ) )
= bot_bot_set_a )
=> ? [F2: a > nat] :
( ( Ys
= ( map_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_a_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_a_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_580_combine__map3,axiom,
! [Ys: list_Sum_sum_a_nat,Xs2: list_nat,Ys3: list_Sum_sum_a_nat,Xs4: list_nat,Ys4: list_Sum_sum_a_nat,Xs5: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Xs2 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys3 )
= ( size_size_list_nat @ Xs4 ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Ys4 )
= ( size_size_list_nat @ Xs5 ) )
=> ( ( distinct_nat @ Xs2 )
=> ( ( distinct_nat @ Xs4 )
=> ( ( distinct_nat @ Xs5 )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs4 ) )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs5 ) )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs4 ) @ ( set_nat2 @ Xs5 ) )
= bot_bot_set_nat )
=> ? [F2: nat > sum_sum_a_nat] :
( ( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_na823391071729141993_a_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_na823391071729141993_a_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_581_combine__map3,axiom,
! [Ys: list_nat,Xs2: list_nat,Ys3: list_nat,Xs4: list_nat,Ys4: list_nat,Xs5: list_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Xs4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Xs5 ) )
=> ( ( distinct_nat @ Xs2 )
=> ( ( distinct_nat @ Xs4 )
=> ( ( distinct_nat @ Xs5 )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs4 ) )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Xs5 ) )
= bot_bot_set_nat )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs4 ) @ ( set_nat2 @ Xs5 ) )
= bot_bot_set_nat )
=> ? [F2: nat > nat] :
( ( Ys
= ( map_nat_nat @ F2 @ Xs2 ) )
& ( Ys3
= ( map_nat_nat @ F2 @ Xs4 ) )
& ( Ys4
= ( map_nat_nat @ F2 @ Xs5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% combine_map3
thf(fact_582_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_583_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_584_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_585_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_586_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_587_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_588_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_589_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_590_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_591_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_592_eval__conj__set__aux_I1_J,axiom,
! [Ns_phi2: list_nat,Ns_phi3: list_nat,Ns_psi2: list_nat,Ns_psi3: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs2: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
@ Ns_psi2 ) )
=> ( ( Ns_psi3
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
@ Ns_phi3 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi3 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi2 ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi3 )
& ( distinct_nat @ Ns_phi3 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi2 )
& ( distinct_nat @ Ns_psi2 ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ~ ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ~ ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs2 @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs2
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi3 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi3 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) @ ( size_size_list_nat @ Ns_phi2 ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi3 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi2 @ Ns_psi3 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ Ns_phi2 ) ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) @ ( zip_na2013496608136855606_a_nat @ Ns_phi2 @ Fs_phi ) ) )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat
@ ( merge_Sum_sum_a_nat
@ ( zip_na2013496608136855606_a_nat
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) )
@ ( zip_na2013496608136855606_a_nat @ Nys @ ( map_na823391071729141993_a_nat @ Sigma_xs @ Nys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(1)
thf(fact_593_eval__conj__set__aux_I2_J,axiom,
! [Ns_phi2: list_nat,Ns_phi3: list_nat,Ns_psi2: list_nat,Ns_psi3: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs2: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
@ Ns_psi2 ) )
=> ( ( Ns_psi3
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
@ Ns_phi3 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi3 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi2 ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi3 )
& ( distinct_nat @ Ns_phi3 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi2 )
& ( distinct_nat @ Ns_psi2 ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ~ ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ~ ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs2 @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs2
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi3 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi3 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) @ ( size_size_list_nat @ Ns_phi2 ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi3 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi2 @ Ns_psi3 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ Ns_phi2 ) ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) @ Cys ) )
= ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(2)
thf(fact_594_distinct_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_phi )
& ( distinct_nat @ ns_phi ) ) ).
% distinct(1)
thf(fact_595_distinct_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_psi )
& ( distinct_nat @ ns_psi ) ) ).
% distinct(2)
thf(fact_596_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_597_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_598_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_599_sorted__map,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X4: produc5454855657571582244_a_nat,Y2: produc5454855657571582244_a_nat] : ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_600_sorted__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_601_sorted__same,axiom,
! [G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( filter_nat
@ ^ [X4: nat] :
( X4
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_602_sorted__sort,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ Xs2 ) ) ).
% sorted_sort
thf(fact_603_sorted__sort__id,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_604_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_605_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_606_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_607_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_608_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_609_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_610_sorted__nths,axiom,
! [Xs2: list_nat,I2: set_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( nths_nat @ Xs2 @ I2 ) ) ) ).
% sorted_nths
thf(fact_611_sorted__wrt__true,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_612_sorted__quicksort,axiom,
! [Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or9089632773640736191rt_nat @ Xs2 ) ) ).
% sorted_quicksort
thf(fact_613_sorted__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ).
% sorted_append
thf(fact_614_sorted__distinct__set__unique,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( distinct_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs2 )
= ( set_nat2 @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_615_sorted__filter,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ) ).
% sorted_filter
thf(fact_616_sorted__filter,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% sorted_filter
thf(fact_617_sort__key__id__if__sorted,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
=> ( ( linord4234369348993689078at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_618_sort__key__id__if__sorted,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_619_sorted__sort__key,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ F @ ( linord4234369348993689078at_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_620_sorted__sort__key,axiom,
! [F: nat > nat,Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord738340561235409698at_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_621_sorted__wrt__mono__rel,axiom,
! [Xs2: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X5: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ( ( member408289922725080238_a_nat @ Y4 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted1521004414301201522_a_nat @ P @ Xs2 )
=> ( sorted1521004414301201522_a_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_622_sorted__wrt__mono__rel,axiom,
! [Xs2: list_o,P: $o > $o > $o,Q: $o > $o > $o] :
( ! [X5: $o,Y4: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
=> ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted_wrt_o @ P @ Xs2 )
=> ( sorted_wrt_o @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_623_sorted__wrt__mono__rel,axiom,
! [Xs2: list_Sum_sum_a_nat,P: sum_sum_a_nat > sum_sum_a_nat > $o,Q: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X5: sum_sum_a_nat,Y4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( ( member_Sum_sum_a_nat @ Y4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted6245805940552704876_a_nat @ P @ Xs2 )
=> ( sorted6245805940552704876_a_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_624_sorted__wrt__mono__rel,axiom,
! [Xs2: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X5: nat,Y4: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
=> ( ( P @ X5 @ Y4 )
=> ( Q @ X5 @ Y4 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs2 )
=> ( sorted_wrt_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_625_sorted__wrt__filter,axiom,
! [F: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ( sorted7320194442031503735_a_nat @ F @ Xs2 )
=> ( sorted7320194442031503735_a_nat @ F @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) ) ).
% sorted_wrt_filter
thf(fact_626_sorted__wrt__filter,axiom,
! [F: nat > nat > $o,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ F @ Xs2 )
=> ( sorted_wrt_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% sorted_wrt_filter
thf(fact_627_sorted__map__same,axiom,
! [F: produc5454855657571582244_a_nat > nat,G: list_P5056861408695629236_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_Pr2058079554187054650at_nat @ F
@ ( filter1580576373184442087_a_nat
@ ^ [X4: produc5454855657571582244_a_nat] :
( ( F @ X4 )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_map_same
thf(fact_628_sorted__map__same,axiom,
! [F: nat > nat,G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [X4: nat] :
( ( F @ X4 )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_map_same
thf(fact_629_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_630_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_631_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_632_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_633_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_634_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_635_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_636_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_637_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_638_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_639_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_640_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_641_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_642_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_643_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_644_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_645_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_646_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_647_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_648_sorted__wrt__map,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( sorted6245805940552704876_a_nat @ R2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X4: produc5454855657571582244_a_nat,Y2: produc5454855657571582244_a_nat] : ( R2 @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_649_sorted__wrt__map,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( sorted6245805940552704876_a_nat @ R2 @ ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X4: nat,Y2: nat] : ( R2 @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_650_sorted__wrt__map,axiom,
! [R2: nat > nat > $o,F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( sorted_wrt_nat @ R2 @ ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( sorted7320194442031503735_a_nat
@ ^ [X4: produc5454855657571582244_a_nat,Y2: produc5454855657571582244_a_nat] : ( R2 @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_651_sorted__wrt__map,axiom,
! [R2: nat > nat > $o,F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ R2 @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X4: nat,Y2: nat] : ( R2 @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_652_inf__set__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( inf_in954358986474102090_nat_o
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A4 )
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_653_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ A4 )
@ ^ [X4: $o] : ( member_o @ X4 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_654_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_655_sorted__ordered__entries,axiom,
! [M: mappin2117621025242276154_a_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( ordere406669810731199336_a_nat @ M ) ) ) ).
% sorted_ordered_entries
thf(fact_656_sorted__enumerate,axiom,
! [N: nat,Xs2: list_Sum_sum_a_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( enumer3164015132978342500_a_nat @ N @ Xs2 ) ) ) ).
% sorted_enumerate
thf(fact_657_sorted__wrt__append,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( sorted6245805940552704876_a_nat @ P @ ( append_Sum_sum_a_nat @ Xs2 @ Ys ) )
= ( ( sorted6245805940552704876_a_nat @ P @ Xs2 )
& ( sorted6245805940552704876_a_nat @ P @ Ys )
& ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ! [Y2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( P @ X4 @ Y2 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_658_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs2 @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs2 )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ( P @ X4 @ Y2 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_659_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_660_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_661_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_662_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_663_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_664_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_665_mult__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_right_mono
thf(fact_666_mult__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_left_mono
thf(fact_667_mult__mono_H,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_668_mult__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_669_length__filter__le,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] : ( ord_less_eq_nat @ ( size_s4076174644546656840_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) @ ( size_s4076174644546656840_a_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_670_length__filter__le,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( filter_Sum_sum_a_nat @ P @ Xs2 ) ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_671_length__filter__le,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_672_proj__tuple__merge,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
= bot_bot_set_nat )
=> ( ( proj_tuple_a @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs ) ) ) ) ) ).
% proj_tuple_merge
thf(fact_673_map__fst__merge,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ) ) ).
% map_fst_merge
thf(fact_674_merge__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
= bot_bot_set_nat )
=> ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( merge_nat @ ( zip_nat_nat @ Ns @ ( map_nat_nat @ Sigma @ Ns ) ) @ ( zip_nat_nat @ Ms @ ( map_nat_nat @ Sigma @ Ms ) ) ) )
= ( map_nat_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns @ Ms ) ) ) ) ) ) ) ).
% merge_map
thf(fact_675_merge__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
= bot_bot_set_nat )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Ns @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) ) @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) ) ) )
= ( map_na823391071729141993_a_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns @ Ms ) ) ) ) ) ) ) ).
% merge_map
thf(fact_676_merge__map_H,axiom,
! [Nxs: list_P6011104703257516679at_nat,Mys: list_P6011104703257516679at_nat,Sigma: nat > nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Mys ) ) )
=> ( ( ( inf_inf_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( set_Pr5648618587558075414at_nat @ Nxs ) ) @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( set_Pr5648618587558075414at_nat @ Mys ) ) )
= bot_bot_set_nat )
=> ( ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Nxs )
= ( map_nat_nat @ Sigma @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Nxs ) ) )
=> ( ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Mys )
= ( map_nat_nat @ Sigma @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Mys ) ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( merge_nat @ Nxs @ Mys ) )
= ( map_nat_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Nxs ) @ ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Mys ) ) ) ) ) ) ) ) ) ) ).
% merge_map'
thf(fact_677_merge__map_H,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( ( inf_inf_set_nat @ ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Nxs ) ) @ ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Mys ) ) )
= bot_bot_set_nat )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs )
= ( map_na823391071729141993_a_nat @ Sigma @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Mys )
= ( map_na823391071729141993_a_nat @ Sigma @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( map_na823391071729141993_a_nat @ Sigma
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ) ) ) ) ) ) ).
% merge_map'
thf(fact_678_card__length,axiom,
! [Xs2: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% card_length
thf(fact_679_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_680_nall__tuples__rec__fo__nmlz__rec__sound,axiom,
! [I: nat,J: nat,Xs2: list_Sum_sum_a_nat,AD: set_a,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ I @ N ) )
=> ( ( fo_nmlz_rec_a @ J @ ( id_map_a @ J ) @ AD @ Xs2 )
= Xs2 ) ) ) ).
% nall_tuples_rec_fo_nmlz_rec_sound
thf(fact_681_eval__conj__tuple__empty,axiom,
! [Z4: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Nsx: list_nat,Nsy: list_nat,Ns: list_nat] :
( ( fo_nmlzd_a @ Z4 @ Xs2 )
=> ( ( fo_nmlzd_a @ Z4 @ Ys )
=> ( ( ( size_size_list_nat @ Nsx )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Nsy )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsx )
& ( distinct_nat @ Nsx ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsy )
& ( distinct_nat @ Nsy ) )
=> ( ( Ns
= ( filter_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ ( set_nat2 @ Nsy ) )
@ Nsx ) )
=> ( ( ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs2 ) ) )
!= ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
=> ( ( eval_conj_tuple_a @ Z4 @ Nsx @ Nsy @ Xs2 @ Ys )
= bot_bo1033123847703346641_a_nat ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_empty
thf(fact_682_card__Inr__vimage__le__length,axiom,
! [Xs2: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) ).
% card_Inr_vimage_le_length
thf(fact_683_eval__conj__set__aux_I3_J,axiom,
! [Ns_phi2: list_nat,Ns_phi3: list_nat,Ns_psi2: list_nat,Ns_psi3: list_nat,X_phi: set_li3402123583998030149at_nat,AD: set_nat,R_phi: set_na2019211207066354340at_nat,X_psi: set_li3402123583998030149at_nat,R_psi: set_na2019211207066354340at_nat,Cxs: list_P1835631256188631350at_nat,Xs2: list_Sum_sum_nat_nat,Nxs: list_nat,Cys: list_P1835631256188631350at_nat,Ys: list_Sum_sum_nat_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_nat_nat,Fs_phi: list_Sum_sum_nat_nat,Sigma_ys: nat > sum_sum_nat_nat,Fs_psi: list_Sum_sum_nat_nat] :
( ( Ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
@ Ns_psi2 ) )
=> ( ( Ns_psi3
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
@ Ns_phi3 ) )
=> ( ( X_phi
= ( image_4446412052380048421at_nat @ ( fo_nmlz_nat @ AD ) @ ( proj_v1967724908790944216at_nat @ R_phi @ Ns_phi3 ) ) )
=> ( ( X_psi
= ( image_4446412052380048421at_nat @ ( fo_nmlz_nat @ AD ) @ ( proj_v1967724908790944216at_nat @ R_psi @ Ns_psi2 ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi3 )
& ( distinct_nat @ Ns_phi3 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi2 )
& ( distinct_nat @ Ns_psi2 ) )
=> ( ( Cxs
= ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,X4: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ( sum_isl_nat_nat @ X4 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_phi3 @ Xs2 ) ) )
=> ( ( Nxs
= ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,X4: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ~ ( sum_isl_nat_nat @ X4 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_phi3 @ Xs2 ) ) ) )
=> ( ( Cys
= ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,Y2: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ( sum_isl_nat_nat @ Y2 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_psi2 @ Ys ) ) )
=> ( ( Nys
= ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat
@ ( filter2628677162644168493at_nat
@ ( produc1742522513024853595_nat_o
@ ^ [N2: nat,Y2: sum_sum_nat_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ~ ( sum_isl_nat_nat @ Y2 ) ) )
@ ( zip_na3998243202614522632at_nat @ Ns_psi2 @ Ys ) ) ) )
=> ( ( member322858896541912486at_nat @ Xs2 @ X_phi )
=> ( ( member322858896541912486at_nat @ Ys @ X_psi )
=> ( ( Xs2
= ( map_na7095077453091140501at_nat @ Sigma_xs @ Ns_phi3 ) )
=> ( ( Fs_phi
= ( map_na7095077453091140501at_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Ys
= ( map_na7095077453091140501at_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( Fs_psi
= ( map_na7095077453091140501at_nat @ Sigma_ys @ Ns_psi3 ) )
=> ( ( member322858896541912486at_nat @ Fs_phi @ ( nall_tuples_rec_nat @ AD @ ( finite_card_nat @ ( vimage1655988174671605260at_nat @ sum_Inr_nat_nat @ ( set_Sum_sum_nat_nat2 @ Xs2 ) ) ) @ ( size_size_list_nat @ Ns_phi2 ) ) )
=> ( ( member322858896541912486at_nat @ Fs_psi @ ( nall_tuples_rec_nat @ AD @ ( finite_card_nat @ ( vimage1655988174671605260at_nat @ sum_Inr_nat_nat @ ( set_Sum_sum_nat_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi3 ) ) )
=> ( ( ad_agr_list_nat_nat @ AD
@ ( map_na7095077453091140501at_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi2 @ Ns_psi3 ) ) )
@ ( map_na7095077453091140501at_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ Ns_phi2 ) ) ) )
=> ( member322858896541912486at_nat @ ( map_na7095077453091140501at_nat @ Sigma_xs @ Nys )
@ ( nall_tuples_rec_nat @ bot_bot_set_nat
@ ( finite_card_nat
@ ( vimage1655988174671605260at_nat @ sum_Inr_nat_nat
@ ( set_Sum_sum_nat_nat2
@ ( map_na7095077453091140501at_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ ( map_Pr8178318585354329148at_nat @ produc2855310212150633940at_nat @ Cys ) ) ) ) ) ) )
@ ( size_size_list_nat @ Nys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(3)
thf(fact_684_eval__conj__set__aux_I3_J,axiom,
! [Ns_phi2: list_nat,Ns_phi3: list_nat,Ns_psi2: list_nat,Ns_psi3: list_nat,X_phi: set_li6526943997496501093_a_nat,AD: set_a,R_phi: set_na3699693778330250182_a_nat,X_psi: set_li6526943997496501093_a_nat,R_psi: set_na3699693778330250182_a_nat,Cxs: list_P5056861408695629236_a_nat,Xs2: list_Sum_sum_a_nat,Nxs: list_nat,Cys: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,Nys: list_nat,Sigma_xs: nat > sum_sum_a_nat,Fs_phi: list_Sum_sum_a_nat,Sigma_ys: nat > sum_sum_a_nat,Fs_psi: list_Sum_sum_a_nat] :
( ( Ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
@ Ns_psi2 ) )
=> ( ( Ns_psi3
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
@ Ns_phi3 ) )
=> ( ( X_phi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_phi @ Ns_phi3 ) ) )
=> ( ( X_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R_psi @ Ns_psi2 ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_phi3 )
& ( distinct_nat @ Ns_phi3 ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns_psi2 )
& ( distinct_nat @ Ns_psi2 ) )
=> ( ( Cxs
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) )
=> ( ( Nxs
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,X4: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_psi2 ) )
& ~ ( sum_isl_a_nat @ X4 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_phi3 @ Xs2 ) ) ) )
=> ( ( Cys
= ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) )
=> ( ( Nys
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat
@ ( filter1580576373184442087_a_nat
@ ( produc8810490045567577941_nat_o
@ ^ [N2: nat,Y2: sum_sum_a_nat] :
( ~ ( member_nat @ N2 @ ( set_nat2 @ Ns_phi3 ) )
& ~ ( sum_isl_a_nat @ Y2 ) ) )
@ ( zip_na2013496608136855606_a_nat @ Ns_psi2 @ Ys ) ) ) )
=> ( ( member408289922725080238_a_nat @ Xs2 @ X_phi )
=> ( ( member408289922725080238_a_nat @ Ys @ X_psi )
=> ( ( Xs2
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi3 ) )
=> ( ( Fs_phi
= ( map_na823391071729141993_a_nat @ Sigma_xs @ Ns_phi2 ) )
=> ( ( Ys
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi2 ) )
=> ( ( Fs_psi
= ( map_na823391071729141993_a_nat @ Sigma_ys @ Ns_psi3 ) )
=> ( ( member408289922725080238_a_nat @ Fs_phi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) @ ( size_size_list_nat @ Ns_phi2 ) ) )
=> ( ( member408289922725080238_a_nat @ Fs_psi @ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) @ ( size_size_list_nat @ Ns_psi3 ) ) )
=> ( ( ad_agr_list_a_nat @ AD
@ ( map_na823391071729141993_a_nat @ Sigma_ys
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_psi2 @ Ns_psi3 ) ) )
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ Ns_phi2 ) ) ) )
=> ( member408289922725080238_a_nat @ ( map_na823391071729141993_a_nat @ Sigma_xs @ Nys )
@ ( nall_tuples_rec_a @ bot_bot_set_a
@ ( finite_card_nat
@ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a
@ ( set_Sum_sum_a_nat2
@ ( map_na823391071729141993_a_nat @ Sigma_xs
@ ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( append_nat @ Ns_phi3 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Cys ) ) ) ) ) ) )
@ ( size_size_list_nat @ Nys ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_set_aux(3)
thf(fact_685_eval__conj__tuple__empty2,axiom,
! [Z4: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Nsx: list_nat,Nsy: list_nat,Ns: list_nat] :
( ( fo_nmlzd_a @ Z4 @ Xs2 )
=> ( ( fo_nmlzd_a @ Z4 @ Ys )
=> ( ( ( size_size_list_nat @ Nsx )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( ( size_size_list_nat @ Nsy )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsx )
& ( distinct_nat @ Nsx ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Nsy )
& ( distinct_nat @ Nsy ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( inf_inf_set_nat @ ( set_nat2 @ Nsx ) @ ( set_nat2 @ Nsy ) ) )
=> ( ( ( ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs2 ) ) )
!= ( fo_nmlz_a @ Z4 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
| ( ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs2 ) )
!= ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) )
& ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsx @ Xs2 ) ) ) )
=> ( sum_isl_a_nat @ X5 ) )
& ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Nsy @ Ys ) ) ) )
=> ( sum_isl_a_nat @ X5 ) ) ) )
=> ( ( eval_conj_tuple_a @ Z4 @ Nsx @ Nsy @ Xs2 @ Ys )
= bot_bo1033123847703346641_a_nat ) ) ) ) ) ) ) ) ) ) ).
% eval_conj_tuple_empty2
thf(fact_686_proj__tuple__length,axiom,
! [Ns: list_nat,Ms: list_nat,Xs2: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( ( size_size_list_nat @ Ms )
= ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( ( size_s5283204784079214577_a_nat @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ Xs2 ) ) )
= ( size_size_list_nat @ Ns ) ) ) ) ) ) ).
% proj_tuple_length
thf(fact_687_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_688_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_689_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_690_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_691_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_692_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_693_proj__tuple__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) ) )
= ( map_na823391071729141993_a_nat @ Sigma @ Ns ) ) ) ) ) ).
% proj_tuple_map
thf(fact_694_ext__tuple__def,axiom,
( ext_tuple_a
= ( ^ [AD2: set_a,Fv_sub2: list_nat,Fv_sub_comp2: list_nat,As2: list_Sum_sum_a_nat] :
( if_set7709265119413304363_a_nat @ ( Fv_sub_comp2 = nil_nat ) @ ( insert2950094090816004437_a_nat @ As2 @ bot_bo1033123847703346641_a_nat )
@ ( image_5081948215111134021_a_nat
@ ^ [Fs: list_Sum_sum_a_nat] : ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Fv_sub2 @ As2 ) @ ( zip_na2013496608136855606_a_nat @ Fv_sub_comp2 @ Fs ) ) )
@ ( nall_tuples_rec_a @ AD2 @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As2 ) ) ) @ ( size_size_list_nat @ Fv_sub_comp2 ) ) ) ) ) ) ).
% ext_tuple_def
thf(fact_695_subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( member408289922725080238_a_nat @ X5 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_696_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ X5 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_697_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( member_o @ X5 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_698_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_a @ X5 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_699_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_700_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_701_insert__iff,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ A2 ) )
= ( ( A = B )
| ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_702_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_703_insert__iff,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
= ( ( A = B )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_704_insertCI,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A @ B2 )
=> ( A = B ) )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_705_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_706_insertCI,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A @ B2 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_707_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_708_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_709_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_710_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_711_image__insert,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ ( F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_712_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_713_insert__image,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_714_insert__image,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( insert_nat @ ( F @ X ) @ ( image_2535339886381165584at_nat @ F @ A2 ) )
= ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_715_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_716_insert__image,axiom,
! [X: $o,A2: set_o,F: $o > nat] :
( ( member_o @ X @ A2 )
=> ( ( insert_nat @ ( F @ X ) @ ( image_o_nat @ F @ A2 ) )
= ( image_o_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_717_list_Omap__disc__iff,axiom,
! [F: nat > sum_sum_a_nat,A: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_718_list_Omap__disc__iff,axiom,
! [F: produc5454855657571582244_a_nat > nat,A: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ A )
= nil_nat )
= ( A = nil_Pr237480997409426078_a_nat ) ) ).
% list.map_disc_iff
thf(fact_719_list_Omap__disc__iff,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,A: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_Pr237480997409426078_a_nat ) ) ).
% list.map_disc_iff
thf(fact_720_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_721_Nil__is__map__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( nil_Sum_sum_a_nat
= ( map_na823391071729141993_a_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_722_Nil__is__map__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( nil_nat
= ( map_Pr2058079554187054650at_nat @ F @ Xs2 ) )
= ( Xs2 = nil_Pr237480997409426078_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_723_Nil__is__map__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( nil_Sum_sum_a_nat
= ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) )
= ( Xs2 = nil_Pr237480997409426078_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_724_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_725_map__is__Nil__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs2: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs2 )
= nil_Sum_sum_a_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_726_map__is__Nil__conv,axiom,
! [F: produc5454855657571582244_a_nat > nat,Xs2: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_Pr237480997409426078_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_727_map__is__Nil__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs2: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs2 )
= nil_Sum_sum_a_nat )
= ( Xs2 = nil_Pr237480997409426078_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_728_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_729_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_730_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_731_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_732_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_733_insert__subset,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X @ A2 ) @ B2 )
= ( ( member408289922725080238_a_nat @ X @ B2 )
& ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_734_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_735_insert__subset,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( ( member_o @ X @ B2 )
& ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_736_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_737_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_738_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_739_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_740_append__self__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_741_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_742_append__self__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_743_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_744_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_745_Int__subset__iff,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C3 @ A2 )
& ( ord_less_eq_set_a @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_746_Int__insert__right__if1,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_747_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_748_Int__insert__right__if1,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_749_Int__insert__right__if0,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_750_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_751_Int__insert__right__if0,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_752_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_753_Int__insert__left__if1,axiom,
! [A: list_Sum_sum_a_nat,C3: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ C3 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C3 )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_754_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_755_Int__insert__left__if1,axiom,
! [A: $o,C3: set_o,B2: set_o] :
( ( member_o @ A @ C3 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C3 )
= ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_756_Int__insert__left__if0,axiom,
! [A: list_Sum_sum_a_nat,C3: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ C3 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C3 )
= ( inf_in3249246906714053971_a_nat @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_757_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_758_Int__insert__left__if0,axiom,
! [A: $o,C3: set_o,B2: set_o] :
( ~ ( member_o @ A @ C3 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C3 )
= ( inf_inf_set_o @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_759_zip__eq__Nil__iff,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys )
= nil_Pr237480997409426078_a_nat )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_760_zip__eq__Nil__iff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( zip_nat_nat @ Xs2 @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_761_Nil__eq__zip__iff,axiom,
! [Xs2: list_nat,Ys: list_Sum_sum_a_nat] :
( ( nil_Pr237480997409426078_a_nat
= ( zip_na2013496608136855606_a_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_762_Nil__eq__zip__iff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_763_zip__Nil,axiom,
! [Ys: list_Sum_sum_a_nat] :
( ( zip_na2013496608136855606_a_nat @ nil_nat @ Ys )
= nil_Pr237480997409426078_a_nat ) ).
% zip_Nil
thf(fact_764_sort__key__simps_I1_J,axiom,
! [F: nat > nat] :
( ( linord738340561235409698at_nat @ F @ nil_nat )
= nil_nat ) ).
% sort_key_simps(1)
thf(fact_765_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_766_rotate1__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_767_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_768_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z3: a] : ( Y5 = Z3 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_769_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_770_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X4: a] : ( X4 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_771_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X4: nat] : ( X4 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_772_set__empty2,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( set_Sum_sum_a_nat2 @ Xs2 ) )
= ( Xs2 = nil_Sum_sum_a_nat ) ) ).
% set_empty2
thf(fact_773_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_774_set__empty2,axiom,
! [Xs2: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% set_empty2
thf(fact_775_List_Oset__empty,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( ( set_Sum_sum_a_nat2 @ Xs2 )
= bot_bo3438331934148233675_a_nat )
= ( Xs2 = nil_Sum_sum_a_nat ) ) ).
% List.set_empty
thf(fact_776_List_Oset__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% List.set_empty
thf(fact_777_List_Oset__empty,axiom,
! [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= bot_bot_set_nat )
= ( Xs2 = nil_nat ) ) ).
% List.set_empty
thf(fact_778_length__0__conv,axiom,
! [Xs2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Sum_sum_a_nat ) ) ).
% length_0_conv
thf(fact_779_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_780_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_781_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_782_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_783_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_784_disjoint__insert_I2_J,axiom,
! [A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) )
= ( ~ ( member408289922725080238_a_nat @ B @ A2 )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_785_disjoint__insert_I2_J,axiom,
! [A2: set_o,B: $o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ ( insert_o @ B @ B2 ) ) )
= ( ~ ( member_o @ B @ A2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_786_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_787_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_788_disjoint__insert_I1_J,axiom,
! [B2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ A2 ) )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ B2 @ A2 )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_789_disjoint__insert_I1_J,axiom,
! [B2: set_o,A: $o,A2: set_o] :
( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A @ A2 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B2 )
& ( ( inf_inf_set_o @ B2 @ A2 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_790_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_791_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_792_insert__disjoint_I2_J,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_793_insert__disjoint_I2_J,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 ) )
= ( ~ ( member_o @ A @ B2 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_794_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_795_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_796_insert__disjoint_I1_J,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ A2 ) @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_797_insert__disjoint_I1_J,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B2 )
& ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_798_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_799_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_800_filter__False,axiom,
! [Xs2: list_P5056861408695629236_a_nat,P: produc5454855657571582244_a_nat > $o] :
( ! [X5: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X5 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ~ ( P @ X5 ) )
=> ( ( filter1580576373184442087_a_nat @ P @ Xs2 )
= nil_Pr237480997409426078_a_nat ) ) ).
% filter_False
thf(fact_801_filter__False,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X5 ) )
=> ( ( filter_nat @ P @ Xs2 )
= nil_nat ) ) ).
% filter_False
thf(fact_802_filter__False,axiom,
! [Xs2: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ! [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ~ ( P @ X5 ) )
=> ( ( filter_Sum_sum_a_nat @ P @ Xs2 )
= nil_Sum_sum_a_nat ) ) ).
% filter_False
thf(fact_803_nths__empty,axiom,
! [Xs2: list_nat] :
( ( nths_nat @ Xs2 @ bot_bot_set_nat )
= nil_nat ) ).
% nths_empty
thf(fact_804_Collect__case__prod__mono,axiom,
! [A2: nat > sum_sum_a_nat > $o,B2: nat > sum_sum_a_nat > $o] :
( ( ord_le759605360153541901_nat_o @ A2 @ B2 )
=> ( ord_le8420040822840260538_a_nat @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ A2 ) ) @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_805_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_806_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_807_subset__singleton__iff,axiom,
! [X2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X2 = bot_bot_set_nat )
| ( X2
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_808_subset__singleton__iff,axiom,
! [X2: set_a,A: a] :
( ( ord_less_eq_set_a @ X2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X2 = bot_bot_set_a )
| ( X2
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_809_Collect__subset,axiom,
! [A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_810_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_811_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_812_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_813_in__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_814_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_815_in__mono,axiom,
! [A2: set_o,B2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ X @ A2 )
=> ( member_o @ X @ B2 ) ) ) ).
% in_mono
thf(fact_816_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_817_subsetD,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_818_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_819_subsetD,axiom,
! [A2: set_o,B2: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_820_subsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_821_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_822_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A4 )
=> ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_823_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_824_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( member_o @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_825_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_826_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_827_Set_OequalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% Set.equalityD2
thf(fact_828_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A4 )
=> ( member408289922725080238_a_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_829_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_830_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A4 )
=> ( member_o @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_831_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_832_insert__mono,axiom,
! [C3: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_833_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_834_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_835_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_836_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_837_subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_838_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_839_subset__insert,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X @ B2 ) )
= ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_840_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_841_subset__insert,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_842_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_843_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_844_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_845_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_846_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_847_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_848_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_849_pred__subset__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( ord_le8737610411969296920_nat_o
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ R2 )
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ S3 ) )
= ( ord_le1147066620699065093_a_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_850_pred__subset__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
= ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_851_pred__subset__eq,axiom,
! [R2: set_o,S3: set_o] :
( ( ord_less_eq_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ R2 )
@ ^ [X4: $o] : ( member_o @ X4 @ S3 ) )
= ( ord_less_eq_set_o @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_852_pred__subset__eq,axiom,
! [R2: set_a,S3: set_a] :
( ( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R2 )
@ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
= ( ord_less_eq_set_a @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_853_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ A4 )
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_854_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_855_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
( ord_less_eq_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ A4 )
@ ^ [X4: $o] : ( member_o @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_856_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A4 )
@ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_857_insert__compr,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A3: list_Sum_sum_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( X4 = A3 )
| ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_858_insert__compr,axiom,
( insert_o
= ( ^ [A3: $o,B4: set_o] :
( collect_o
@ ^ [X4: $o] :
( ( X4 = A3 )
| ( member_o @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_859_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( X4 = A3 )
| ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_860_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_861_mk__disjoint__insert,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ? [B5: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ A @ B5 ) )
& ~ ( member408289922725080238_a_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_862_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_863_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B5: set_o] :
( ( A2
= ( insert_o @ A @ B5 ) )
& ~ ( member_o @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_864_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_865_insert__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ B @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ A @ A2 )
= ( insert2950094090816004437_a_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ B @ C4 ) )
& ~ ( member408289922725080238_a_nat @ B @ C4 )
& ( B2
= ( insert2950094090816004437_a_nat @ A @ C4 ) )
& ~ ( member408289922725080238_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_866_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B2
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_867_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A = ~ B )
=> ? [C4: set_o] :
( ( A2
= ( insert_o @ B @ C4 ) )
& ~ ( member_o @ B @ C4 )
& ( B2
= ( insert_o @ A @ C4 ) )
& ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_868_insert__absorb,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_869_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_870_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_871_insert__ident,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ X @ A2 )
= ( insert2950094090816004437_a_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_872_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_873_insert__ident,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ~ ( member_o @ X @ B2 )
=> ( ( ( insert_o @ X @ A2 )
= ( insert_o @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_874_Set_Oset__insert,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ~ ! [B5: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ X @ B5 ) )
=> ( member408289922725080238_a_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_875_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_876_Set_Oset__insert,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ~ ! [B5: set_o] :
( ( A2
= ( insert_o @ X @ B5 ) )
=> ( member_o @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_877_insertI2,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_878_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_879_insertI2,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( member_o @ A @ B2 )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_880_insertI1,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_881_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_882_insertI1,axiom,
! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% insertI1
thf(fact_883_insertE,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_884_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_885_insertE,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_886_zip_Osimps_I1_J,axiom,
! [Xs2: list_nat] :
( ( zip_na2013496608136855606_a_nat @ Xs2 @ nil_Sum_sum_a_nat )
= nil_Pr237480997409426078_a_nat ) ).
% zip.simps(1)
thf(fact_887_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_888_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_889_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_890_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_891_doubleton__eq__iff,axiom,
! [A: a,B: a,C2: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_892_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_893_singleton__iff,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_894_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_895_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_896_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_897_singletonD,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_898_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_899_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_900_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_901_Int__insert__right,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_902_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_903_Int__insert__right,axiom,
! [A: $o,A2: set_o,B2: set_o] :
( ( ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_904_Int__insert__left,axiom,
! [A: list_Sum_sum_a_nat,C3: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ A @ C3 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C3 )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C3 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A @ C3 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C3 )
= ( inf_in3249246906714053971_a_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_905_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) )
& ( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_906_Int__insert__left,axiom,
! [A: $o,C3: set_o,B2: set_o] :
( ( ( member_o @ A @ C3 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C3 )
= ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C3 ) ) ) )
& ( ~ ( member_o @ A @ C3 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C3 )
= ( inf_inf_set_o @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_907_image__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_908_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_909_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_910_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_911_image__subsetI,axiom,
! [A2: set_nat,F: nat > $o,B2: set_o] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_o @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_912_image__subsetI,axiom,
! [A2: set_o,F: $o > nat,B2: set_nat] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( member_nat @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_913_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B2: set_o] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( member_o @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_914_image__subsetI,axiom,
! [A2: set_nat,F: nat > a,B2: set_a] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_a @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_915_image__subsetI,axiom,
! [A2: set_o,F: $o > a,B2: set_a] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( member_a @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_916_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,B2: set_nat] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( member_nat @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_2535339886381165584at_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_917_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,B2: set_o] :
( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( member_o @ ( F @ X5 ) @ B2 ) )
=> ( ord_less_eq_set_o @ ( image_3984946558445957976_nat_o @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_918_image__subsetI,axiom,
! [A2: set_nat,F: nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X5 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_919_image__subsetI,axiom,
! [A2: set_o,F: $o > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X5 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_733135354807394034_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_920_subset__imageE,axiom,
! [B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ~ ! [C5: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C5 @ A2 )
=> ( B2
!= ( image_5081948215111134021_a_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_921_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_922_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_923_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B2 )
= ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_924_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_925_subset__image__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A2 )
& ( B2
= ( image_5081948215111134021_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_926_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_927_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_928_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B4 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B4 @ A2 )
=> ( P @ ( image_5081948215111134021_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_929_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_930_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_931_subset__code_I1_J,axiom,
! [Xs2: list_l4703314356710769291_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs2 ) @ B2 )
= ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xs2 ) )
=> ( member408289922725080238_a_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_932_subset__code_I1_J,axiom,
! [Xs2: list_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ B2 )
= ( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
=> ( member_o @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_933_subset__code_I1_J,axiom,
! [Xs2: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_934_subset__code_I1_J,axiom,
! [Xs2: list_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ B2 )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ( member_Sum_sum_a_nat @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_935_subset__code_I1_J,axiom,
! [Xs2: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B2 )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_936_Int__Collect__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B2 @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_937_Int__Collect__mono,axiom,
! [A2: set_o,B2: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B2 @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_938_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_939_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_940_Int__greatest,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ( ord_less_eq_set_a @ C3 @ B2 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_941_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_942_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_943_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_944_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_945_Int__mono,axiom,
! [A2: set_a,C3: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_946_list_Osimps_I8_J,axiom,
! [F: nat > sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ nil_nat )
= nil_Sum_sum_a_nat ) ).
% list.simps(8)
thf(fact_947_list_Osimps_I8_J,axiom,
! [F: produc5454855657571582244_a_nat > nat] :
( ( map_Pr2058079554187054650at_nat @ F @ nil_Pr237480997409426078_a_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_948_list_Osimps_I8_J,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ nil_Pr237480997409426078_a_nat )
= nil_Sum_sum_a_nat ) ).
% list.simps(8)
thf(fact_949_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_950_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_951_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_952_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_953_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_954_subset__vimage__iff,axiom,
! [A2: set_nat,F: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_955_subset__vimage__iff,axiom,
! [A2: set_a,F: a > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_a @ A2 @ ( vimage1894578532518109122_a_nat @ F @ B2 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_956_subset__vimage__iff,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ( ord_less_eq_set_a @ A2 @ ( vimage_a_nat @ F @ B2 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_957_subset__vimage__iff,axiom,
! [A2: set_a,F: a > $o,B2: set_o] :
( ( ord_less_eq_set_a @ A2 @ ( vimage_a_o @ F @ B2 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_o @ ( F @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_958_subset__vimage__iff,axiom,
! [A2: set_a,F: a > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ B2 ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_959_vimage__mono,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A2 ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_mono
thf(fact_960_vimage__mono,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ F @ A2 ) @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ).
% vimage_mono
thf(fact_961_vimage__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A2 ) @ ( vimage_a_a @ F @ B2 ) ) ) ).
% vimage_mono
thf(fact_962_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_963_filter_Osimps_I1_J,axiom,
! [P: produc5454855657571582244_a_nat > $o] :
( ( filter1580576373184442087_a_nat @ P @ nil_Pr237480997409426078_a_nat )
= nil_Pr237480997409426078_a_nat ) ).
% filter.simps(1)
thf(fact_964_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_965_ad__agr__list__mono,axiom,
! [X2: set_a,Y3: set_a,Ys: list_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ad_agr_list_a_nat @ Y3 @ Ys @ Xs2 )
=> ( ad_agr_list_a_nat @ X2 @ Ys @ Xs2 ) ) ) ).
% ad_agr_list_mono
thf(fact_966_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_967_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_968_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_969_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_970_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_971_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_972_vimage__singleton__eq,axiom,
! [A: nat,F: nat > sum_sum_a_nat,B: sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ ( insert_Sum_sum_a_nat @ B @ bot_bo3438331934148233675_a_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_973_vimage__singleton__eq,axiom,
! [A: a,F: a > sum_sum_a_nat,B: sum_sum_a_nat] :
( ( member_a @ A @ ( vimage4607795094749595324_a_nat @ F @ ( insert_Sum_sum_a_nat @ B @ bot_bo3438331934148233675_a_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_974_vimage__singleton__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > a,B: a] :
( ( member408289922725080238_a_nat @ A @ ( vimage7906019935533595764_nat_a @ F @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_975_vimage__singleton__eq,axiom,
! [A: nat,F: nat > a,B: a] :
( ( member_nat @ A @ ( vimage_nat_a @ F @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_976_vimage__singleton__eq,axiom,
! [A: $o,F: $o > a,B: a] :
( ( member_o @ A @ ( vimage_o_a @ F @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_977_vimage__singleton__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B: nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_978_vimage__singleton__eq,axiom,
! [A: nat,F: nat > nat,B: nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_979_vimage__singleton__eq,axiom,
! [A: $o,F: $o > nat,B: nat] :
( ( member_o @ A @ ( vimage_o_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
thf(fact_980_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_981_image__Int__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_982_empty__set,axiom,
( bot_bo3438331934148233675_a_nat
= ( set_Sum_sum_a_nat2 @ nil_Sum_sum_a_nat ) ) ).
% empty_set
thf(fact_983_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_984_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_985_filter__is__subset,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] : ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ ( filter1580576373184442087_a_nat @ P @ Xs2 ) ) @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) ) ).
% filter_is_subset
thf(fact_986_filter__is__subset,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_987_filter__is__subset,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( filter_Sum_sum_a_nat @ P @ Xs2 ) ) @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_988_filter__is__subset,axiom,
! [P: a > $o,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( filter_a @ P @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_989_image__subset__iff__subset__vimage,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B2 )
= ( ord_le1147066620699065093_a_nat @ A2 @ ( vimage7538740559230589179_a_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_990_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
= ( ord_less_eq_set_nat @ A2 @ ( vimage_nat_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_991_image__subset__iff__subset__vimage,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ B2 )
= ( ord_less_eq_set_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_992_image__subset__iff__subset__vimage,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ B2 )
= ( ord_less_eq_set_a @ A2 @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_993_image__subset__iff__subset__vimage,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 )
= ( ord_less_eq_set_a @ A2 @ ( vimage_a_a @ F @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_994_image__vimage__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A2 ) ) @ A2 ) ).
% image_vimage_subset
thf(fact_995_image__vimage__subset,axiom,
! [F: nat > nat,A2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A2 ) ) @ A2 ) ).
% image_vimage_subset
thf(fact_996_image__vimage__subset,axiom,
! [F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( vimage3040984495076556338_a_nat @ F @ A2 ) ) @ A2 ) ).
% image_vimage_subset
thf(fact_997_image__vimage__subset,axiom,
! [F: a > sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( vimage4607795094749595324_a_nat @ F @ A2 ) ) @ A2 ) ).
% image_vimage_subset
thf(fact_998_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_999_filter__empty__conv,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] :
( ( ( filter1580576373184442087_a_nat @ P @ Xs2 )
= nil_Pr237480997409426078_a_nat )
= ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_1000_filter__empty__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= nil_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_1001_filter__empty__conv,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] :
( ( ( filter_Sum_sum_a_nat @ P @ Xs2 )
= nil_Sum_sum_a_nat )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_1002_set__nths__subset,axiom,
! [Xs2: list_nat,I2: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs2 @ I2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_nths_subset
thf(fact_1003_set__nths__subset,axiom,
! [Xs2: list_Sum_sum_a_nat,I2: set_nat] : ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( nths_Sum_sum_a_nat @ Xs2 @ I2 ) ) @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ).
% set_nths_subset
thf(fact_1004_set__nths__subset,axiom,
! [Xs2: list_a,I2: set_nat] : ( ord_less_eq_set_a @ ( set_a2 @ ( nths_a @ Xs2 @ I2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_nths_subset
thf(fact_1005_image__constant__conv,axiom,
! [A2: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
( ( ( A2 = bot_bo1033123847703346641_a_nat )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : C2
@ A2 )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert2950094090816004437_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1006_image__constant__conv,axiom,
! [A2: set_a,C2: a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X4: a] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X4: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1007_image__constant__conv,axiom,
! [A2: set_a,C2: nat] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X4: a] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X4: a] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1008_image__constant__conv,axiom,
! [A2: set_nat,C2: a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X4: nat] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X4: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_1009_image__constant__conv,axiom,
! [A2: set_nat,C2: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X4: nat] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X4: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1010_image__constant,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert2950094090816004437_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% image_constant
thf(fact_1011_image__constant,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: a] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( image_4685209397977471678_nat_a
@ ^ [X4: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1012_image__constant,axiom,
! [X: nat,A2: set_nat,C2: a] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_a
@ ^ [X4: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1013_image__constant,axiom,
! [X: $o,A2: set_o,C2: a] :
( ( member_o @ X @ A2 )
=> ( ( image_o_a
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_1014_image__constant,axiom,
! [X: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( image_2535339886381165584at_nat
@ ^ [X4: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1015_image__constant,axiom,
! [X: nat,A2: set_nat,C2: nat] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_nat
@ ^ [X4: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1016_image__constant,axiom,
! [X: $o,A2: set_o,C2: nat] :
( ( member_o @ X @ A2 )
=> ( ( image_o_nat
@ ^ [X4: $o] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1017_Collect__split__mono__strong,axiom,
! [X2: set_nat,A2: set_Pr1021281732701644634_a_nat,Y3: set_Sum_sum_a_nat,P: nat > sum_sum_a_nat > $o,Q: nat > sum_sum_a_nat > $o] :
( ( X2
= ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ A2 ) )
=> ( ( Y3
= ( image_5908522764169838868_a_nat @ produc1280857981117566888_a_nat @ A2 ) )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ X2 )
=> ! [Xa2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Xa2 @ Y3 )
=> ( ( P @ X5 @ Xa2 )
=> ( Q @ X5 @ Xa2 ) ) ) )
=> ( ( ord_le8420040822840260538_a_nat @ A2 @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ P ) ) )
=> ( ord_le8420040822840260538_a_nat @ A2 @ ( collec2196941500952594553_a_nat @ ( produc8810490045567577941_nat_o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_1018_ad__agr__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,X2: set_a,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X2 @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X2 @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1019_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X2: set_a,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X2 @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X2 @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1020_ad__agr__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,X2: set_a,Sigma: sum_sum_a_nat > sum_sum_a_nat,Sigma2: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X2 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ns ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X2 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1021_ad__agr__list__subset,axiom,
! [Ms: list_a,Ns: list_a,X2: set_a,Sigma: a > sum_sum_a_nat,Sigma2: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ms ) @ ( set_a2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X2 @ ( map_a_Sum_sum_a_nat @ Sigma @ Ns ) @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X2 @ ( map_a_Sum_sum_a_nat @ Sigma @ Ms ) @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_1022_sp__equiv__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,Sigma: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_1023_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_1024_inf__Int__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( inf_in954358986474102090_nat_o
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ R2 )
@ ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ S3 ) )
= ( ^ [X4: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X4 @ ( inf_in3249246906714053971_a_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_1025_inf__Int__eq,axiom,
! [R2: set_nat,S3: set_nat] :
( ( inf_inf_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( inf_inf_set_nat @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_1026_inf__Int__eq,axiom,
! [R2: set_o,S3: set_o] :
( ( inf_inf_o_o
@ ^ [X4: $o] : ( member_o @ X4 @ R2 )
@ ^ [X4: $o] : ( member_o @ X4 @ S3 ) )
= ( ^ [X4: $o] : ( member_o @ X4 @ ( inf_inf_set_o @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_1027_assms_I22_J,axiom,
ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cxs ) ) @ ( set_Sum_sum_a_nat2 @ xs_phi ) ) ) @ ad ).
% assms(22)
thf(fact_1028_assms_I23_J,axiom,
ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ cys ) ) @ ( set_Sum_sum_a_nat2 @ ys_psi ) ) ) @ ad ).
% assms(23)
thf(fact_1029_Containers__Auxiliary_Oempty__filter__conv,axiom,
! [P: produc5454855657571582244_a_nat > $o,Xs2: list_P5056861408695629236_a_nat] :
( ( nil_Pr237480997409426078_a_nat
= ( filter1580576373184442087_a_nat @ P @ Xs2 ) )
= ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% Containers_Auxiliary.empty_filter_conv
thf(fact_1030_Containers__Auxiliary_Oempty__filter__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% Containers_Auxiliary.empty_filter_conv
thf(fact_1031_Containers__Auxiliary_Oempty__filter__conv,axiom,
! [P: sum_sum_a_nat > $o,Xs2: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( filter_Sum_sum_a_nat @ P @ Xs2 ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs2 ) )
=> ~ ( P @ X4 ) ) ) ) ).
% Containers_Auxiliary.empty_filter_conv
thf(fact_1032_length__nth__simps_I1_J,axiom,
( ( size_s5283204784079214577_a_nat @ nil_Sum_sum_a_nat )
= zero_zero_nat ) ).
% length_nth_simps(1)
thf(fact_1033_length__nth__simps_I1_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% length_nth_simps(1)
thf(fact_1034_nall__tuples__rec__filter__rev,axiom,
! [Ys: list_Sum_sum_nat_nat,N: nat,Xs2: list_Sum_sum_nat_nat,AD: set_nat] :
( ( member322858896541912486at_nat @ Ys @ ( nall_tuples_rec_nat @ bot_bot_set_nat @ N @ ( size_s5552450856825661347at_nat @ Ys ) ) )
=> ( ( Ys
= ( filter4040215016827425730at_nat
@ ^ [X4: sum_sum_nat_nat] :
~ ( sum_isl_nat_nat @ X4 )
@ Xs2 ) )
=> ( ( ord_less_eq_set_nat @ ( vimage1655988174671605260at_nat @ sum_Inl_nat_nat @ ( set_Sum_sum_nat_nat2 @ Xs2 ) ) @ AD )
=> ( member322858896541912486at_nat @ Xs2 @ ( nall_tuples_rec_nat @ AD @ N @ ( size_s5552450856825661347at_nat @ Xs2 ) ) ) ) ) ) ).
% nall_tuples_rec_filter_rev
thf(fact_1035_nall__tuples__rec__filter__rev,axiom,
! [Ys: list_Sum_sum_a_nat,N: nat,Xs2: list_Sum_sum_a_nat,AD: set_a] :
( ( member408289922725080238_a_nat @ Ys @ ( nall_tuples_rec_a @ bot_bot_set_a @ N @ ( size_s5283204784079214577_a_nat @ Ys ) ) )
=> ( ( Ys
= ( filter_Sum_sum_a_nat
@ ^ [X4: sum_sum_a_nat] :
~ ( sum_isl_a_nat @ X4 )
@ Xs2 ) )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) @ AD )
=> ( member408289922725080238_a_nat @ Xs2 @ ( nall_tuples_rec_a @ AD @ N @ ( size_s5283204784079214577_a_nat @ Xs2 ) ) ) ) ) ) ).
% nall_tuples_rec_filter_rev
thf(fact_1036_listset_Osimps_I1_J,axiom,
( ( listset_nat @ nil_set_nat )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listset.simps(1)
thf(fact_1037_UnCI,axiom,
! [C2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C2 @ B2 )
=> ( member408289922725080238_a_nat @ C2 @ A2 ) )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1038_UnCI,axiom,
! [C2: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1039_UnCI,axiom,
! [C2: $o,B2: set_o,A2: set_o] :
( ( ~ ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ A2 ) )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1040_UnCI,axiom,
! [C2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C2 @ B2 )
=> ( member_Sum_sum_a_nat @ C2 @ A2 ) )
=> ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1041_Un__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( ( member408289922725080238_a_nat @ C2 @ A2 )
| ( member408289922725080238_a_nat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1042_Un__iff,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1043_Un__iff,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( member_o @ C2 @ A2 )
| ( member_o @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1044_Un__iff,axiom,
! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( ( member_Sum_sum_a_nat @ C2 @ A2 )
| ( member_Sum_sum_a_nat @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_1045_fo__nmlz__Nil,axiom,
! [AD: set_a] :
( ( fo_nmlz_a @ AD @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% fo_nmlz_Nil
thf(fact_1046_sup__bot_Oright__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1047_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1048_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1049_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1050_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1051_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1052_sup__bot_Oleft__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1053_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1054_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1055_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A @ B )
= bot_bo3438331934148233675_a_nat )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1056_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1057_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1058_sup__eq__bot__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ X @ Y )
= bot_bo3438331934148233675_a_nat )
= ( ( X = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1059_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1060_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1061_bot__eq__sup__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ X @ Y ) )
= ( ( X = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1062_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1063_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1064_sup__bot__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ bot_bo3438331934148233675_a_nat )
= X ) ).
% sup_bot_right
thf(fact_1065_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_1066_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_1067_sup__bot__left,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_1068_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1069_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_1070_Un__empty,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% Un_empty
thf(fact_1071_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1072_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1073_Un__subset__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ C3 )
= ( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
& ( ord_le1325389633284124927_a_nat @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_1074_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C3 )
= ( ( ord_less_eq_set_a @ A2 @ C3 )
& ( ord_less_eq_set_a @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_1075_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C3 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1076_Un__insert__left,axiom,
! [A: sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat @ A @ B2 ) @ C3 )
= ( insert_Sum_sum_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_1077_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1078_Un__insert__right,axiom,
! [A2: set_Sum_sum_a_nat,A: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ A @ B2 ) )
= ( insert_Sum_sum_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1079_Int__Un__eq_I4_J,axiom,
! [T3: set_Sum_sum_a_nat,S3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ T3 @ ( inf_in7084830621192376909_a_nat @ S3 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1080_Int__Un__eq_I3_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ S3 @ ( inf_in7084830621192376909_a_nat @ S3 @ T3 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1081_Int__Un__eq_I2_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1082_Int__Un__eq_I1_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1083_Un__Int__eq_I4_J,axiom,
! [T3: set_Sum_sum_a_nat,S3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ T3 @ ( sup_su6804446743777130803_a_nat @ S3 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1084_Un__Int__eq_I3_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ S3 @ ( sup_su6804446743777130803_a_nat @ S3 @ T3 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1085_Un__Int__eq_I2_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S3 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1086_Un__Int__eq_I1_J,axiom,
! [S3: set_Sum_sum_a_nat,T3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S3 @ T3 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1087_Inl__Inr__False,axiom,
! [X: a,Y: nat] :
( ( sum_Inl_a_nat @ X )
!= ( sum_Inr_nat_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_1088_Inr__Inl__False,axiom,
! [X: nat,Y: a] :
( ( sum_Inr_nat_a @ X )
!= ( sum_Inl_a_nat @ Y ) ) ).
% Inr_Inl_False
thf(fact_1089_vimage__Un,axiom,
! [F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A2 ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_1090_vimage__Un,axiom,
! [F: a > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_sup_set_a @ ( vimage4607795094749595324_a_nat @ F @ A2 ) @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_1091_vimage__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage6545432446589551483_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A2 ) @ ( vimage6545432446589551483_a_nat @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_1092_set__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1093_set__append,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( append_Sum_sum_a_nat @ Xs2 @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1094_if__image__distrib,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X4: list_Sum_sum_a_nat] : ( if_lis4685338526944683083_a_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S3 )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ S3 @ ( collec7555443234367654128_a_nat @ P ) ) )
@ ( image_5081948215111134021_a_nat @ G
@ ( inf_in3249246906714053971_a_nat @ S3
@ ( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1095_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S3: set_nat] :
( ( image_nat_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S3 )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S3 @ ( collect_nat @ P ) ) )
@ ( image_nat_nat @ G
@ ( inf_inf_set_nat @ S3
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1096_if__image__distrib,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,S3: set_nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X4: nat] : ( if_Sum_sum_a_nat @ ( P @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S3 )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ S3 @ ( collect_nat @ P ) ) )
@ ( image_7293268710728258664_a_nat @ G
@ ( inf_inf_set_nat @ S3
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1097_set__union,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_1098_set__union,axiom,
! [Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs2 @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_1099_UnE,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
=> ( ~ ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_1100_UnE,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_1101_UnE,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) )
=> ( ~ ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_1102_UnE,axiom,
! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
=> ( ~ ( member_Sum_sum_a_nat @ C2 @ A2 )
=> ( member_Sum_sum_a_nat @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_1103_UnI1,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1104_UnI1,axiom,
! [C2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1105_UnI1,axiom,
! [C2: $o,A2: set_o,B2: set_o] :
( ( member_o @ C2 @ A2 )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1106_UnI1,axiom,
! [C2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C2 @ A2 )
=> ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1107_UnI2,axiom,
! [C2: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ B2 )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1108_UnI2,axiom,
! [C2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1109_UnI2,axiom,
! [C2: $o,B2: set_o,A2: set_o] :
( ( member_o @ C2 @ B2 )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1110_UnI2,axiom,
! [C2: sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C2 @ B2 )
=> ( member_Sum_sum_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1111_bex__Un,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_1112_ball__Un,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_1113_Un__assoc,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ C3 )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_1114_Un__absorb,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1115_Un__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1116_Un__left__absorb,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_1117_Un__left__commute,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) )
= ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_1118_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1119_Collect__disj__eq,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X4: sum_sum_a_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_su6804446743777130803_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1120_Un__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A4: set_li6526943997496501093_a_nat,B4: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A4 )
| ( member408289922725080238_a_nat @ X4 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1121_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
| ( member_o @ X4 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1122_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
| ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1123_Un__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A4 )
| ( member_Sum_sum_a_nat @ X4 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1124_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ bot_bo3438331934148233675_a_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1125_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1126_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_1127_image__Un,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1128_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1129_image__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ ( image_7142520692256960453_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1130_Un__empty__left,axiom,
! [B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1131_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1132_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1133_Un__empty__right,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1134_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_1135_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1136_subset__Un__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A4: set_Sum_sum_a_nat,B4: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1137_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1138_subset__UnE,axiom,
! [C3: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
=> ~ ! [A5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A5 @ A2 )
=> ! [B6: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B6 @ B2 )
=> ( C3
!= ( sup_su6804446743777130803_a_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1139_subset__UnE,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B2 )
=> ( C3
!= ( sup_sup_set_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1140_Un__absorb2,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_1141_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_1142_Un__absorb1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1143_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1144_Un__upper2,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_1145_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_1146_Un__upper1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_1147_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_1148_Un__least,axiom,
! [A2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_1149_Un__least,axiom,
! [A2: set_a,C3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_1150_Un__mono,axiom,
! [A2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C3 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_1151_Un__mono,axiom,
! [A2: set_a,C3: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_1152_Un__Int__crazy,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ ( inf_in7084830621192376909_a_nat @ B2 @ C3 ) ) @ ( inf_in7084830621192376909_a_nat @ C3 @ A2 ) )
= ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) ) @ ( sup_su6804446743777130803_a_nat @ C3 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1153_Int__Un__distrib,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ ( inf_in7084830621192376909_a_nat @ A2 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_1154_Un__Int__distrib,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C3 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ A2 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_1155_Int__Un__distrib2,axiom,
! [B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) @ A2 )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B2 @ A2 ) @ ( inf_in7084830621192376909_a_nat @ C3 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1156_Un__Int__distrib2,axiom,
! [B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B2 @ C3 ) @ A2 )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ A2 ) @ ( sup_su6804446743777130803_a_nat @ C3 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1157_Set_Oinsert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X4: nat] : ( X4 = A3 ) ) ) ) ) ).
% Set.insert_def
thf(fact_1158_Set_Oinsert__def,axiom,
( insert_Sum_sum_a_nat
= ( ^ [A3: sum_sum_a_nat] :
( sup_su6804446743777130803_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X4: sum_sum_a_nat] : ( X4 = A3 ) ) ) ) ) ).
% Set.insert_def
thf(fact_1159_singleton__Un__iff,axiom,
! [X: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) )
& ( B2 = bot_bo3438331934148233675_a_nat ) )
| ( ( A2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) )
& ( B2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1160_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1161_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1162_Un__singleton__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X: sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) )
= ( ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) )
& ( B2 = bot_bo3438331934148233675_a_nat ) )
| ( ( A2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) )
& ( B2
= ( insert_Sum_sum_a_nat @ X @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1163_Un__singleton__iff,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1164_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1165_insert__is__Un,axiom,
( insert_Sum_sum_a_nat
= ( ^ [A3: sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat @ A3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1166_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1167_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1168_Un__Int__assoc__eq,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ C3 )
= ( inf_in7084830621192376909_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C3 ) ) )
= ( ord_le1325389633284124927_a_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1169_Un__Int__assoc__eq,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1170_fo__nmlz__set__rev,axiom,
! [AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_a @ AD @ Xs2 ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) ) ) ).
% fo_nmlz_set_rev
thf(fact_1171_ad__agr__list__set,axiom,
! [X2: set_a,Xs2: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: a] :
( ( ad_agr_list_a_nat @ X2 @ Xs2 @ Ys )
=> ( ( member_a @ Y @ X2 )
=> ( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) ) ) ) ).
% ad_agr_list_set
thf(fact_1172_fo__nmlz__rec__set__rev,axiom,
! [I: nat,M: sum_sum_a_nat > option_nat,AD: set_a,Xs2: list_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ ( fo_nmlz_rec_a @ I @ M @ AD @ Xs2 ) ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) ) ) ).
% fo_nmlz_rec_set_rev
thf(fact_1173_fo__nmlzd__all__AD,axiom,
! [Xs2: list_Sum_sum_a_nat,AD: set_a] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( fo_nmlzd_a @ AD @ Xs2 ) ) ).
% fo_nmlzd_all_AD
thf(fact_1174_vimage__insert,axiom,
! [F: nat > sum_sum_a_nat,A: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( insert_Sum_sum_a_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1175_vimage__insert,axiom,
! [F: a > sum_sum_a_nat,A: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage4607795094749595324_a_nat @ F @ ( insert_Sum_sum_a_nat @ A @ B2 ) )
= ( sup_sup_set_a @ ( vimage4607795094749595324_a_nat @ F @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ) @ ( vimage4607795094749595324_a_nat @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1176_vimage__insert,axiom,
! [F: sum_sum_a_nat > a,A: a,B2: set_a] :
( ( vimage3056561457863698554_nat_a @ F @ ( insert_a @ A @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage3056561457863698554_nat_a @ F @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( vimage3056561457863698554_nat_a @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1177_vimage__insert,axiom,
! [F: sum_sum_a_nat > nat,A: nat,B2: set_nat] :
( ( vimage7444966428737627988at_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage7444966428737627988at_nat @ F @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( vimage7444966428737627988at_nat @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1178_ad__agr__list__eq,axiom,
! [Ys: list_a,AD: set_a,Xs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ AD )
=> ( ( ad_agr_list_a_nat @ AD @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Xs2 ) @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Ys ) )
=> ( Xs2 = Ys ) ) ) ).
% ad_agr_list_eq
thf(fact_1179_ad__agr__close__rec_Osimps_I1_J,axiom,
! [I: nat,M: nat > option_Sum_sum_a_nat,AD: set_a] :
( ( ad_agr_close_rec_a @ I @ M @ AD @ nil_Sum_sum_a_nat )
= ( insert2950094090816004437_a_nat @ nil_Sum_sum_a_nat @ bot_bo1033123847703346641_a_nat ) ) ).
% ad_agr_close_rec.simps(1)
thf(fact_1180_ad__agr__list__rev__mono,axiom,
! [Y3: set_a,X2: set_a,Ys: list_Sum_sum_a_nat,Xs2: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ad_agr_list_a_nat @ Y3 @ Ys @ Xs2 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) @ Y3 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Ys ) ) @ Y3 )
=> ( ad_agr_list_a_nat @ X2 @ Ys @ Xs2 ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_1181_fo__nmlz__mono,axiom,
! [AD: set_a,AD3: set_a,Xs2: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ AD @ AD3 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs2 ) ) @ AD )
=> ( ( fo_nmlz_a @ AD3 @ Xs2 )
= ( fo_nmlz_a @ AD @ Xs2 ) ) ) ) ).
% fo_nmlz_mono
thf(fact_1182_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1183_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1184_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1185_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1186_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1187_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1188_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1189_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1190_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1191_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1192_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1193_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1194_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1195_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1196_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1197_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1198_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1199_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1200_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1201_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1202_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1203_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1204_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1205_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_1206_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_1207_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_1208_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_1209_nat_Oinject,axiom,
! [X23: nat,Y22: nat] :
( ( ( suc @ X23 )
= ( suc @ Y22 ) )
= ( X23 = Y22 ) ) ).
% nat.inject
thf(fact_1210_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1211_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1212_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1213_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1214_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_1215_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1216_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1217_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1218_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1219_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1220_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1221_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R2 @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z: nat] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ Y4 @ Z )
=> ( R2 @ X5 @ Z ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1222_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1223_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1224_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1225_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1226_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1227_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1228_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1229_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1230_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1231_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1232_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1233_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1234_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1235_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1236_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1237_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1238_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1239_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1240_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ X5 @ Y4 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1241_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1242_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1243_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1244_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1245_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1246_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1247_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1248_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1249_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X4: nat] : X4
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_1250_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1251_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1252_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1253_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1254_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1255_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1256_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_1257_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_1258_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_1259_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1260_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1261_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1262_nats__def,axiom,
( nats
= ( ^ [Ns2: list_nat] :
( Ns2
= ( upt @ zero_zero_nat @ ( size_size_list_nat @ Ns2 ) ) ) ) ) ).
% nats_def
thf(fact_1263_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1264_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1265_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1266_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: set_li6526943997496501093_a_nat,Y: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( size_s5283204784079214577_a_nat @ fs_phi )
= ( size_size_list_nat @ ns_phi2 ) ) ).
%------------------------------------------------------------------------------