TPTP Problem File: SLH0968^1.p
View Solutions
- 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 : Safe_Range_RC/0021_Relational_Calculus/prob_01635_061457__17475502_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1468 ( 432 unt; 185 typ; 0 def)
% Number of atoms : 4636 (1425 equ; 0 cnn)
% Maximal formula atoms : 45 ( 3 avg)
% Number of connectives : 14954 ( 763 ~; 94 |; 389 &;11266 @)
% ( 0 <=>;2442 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1069 (1069 >; 0 *; 0 +; 0 <<)
% Number of symbols : 168 ( 165 usr; 11 con; 0-4 aty)
% Number of variables : 4481 ( 390 ^;3878 !; 213 ?;4481 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:26:20.226
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
produc1132964494702330949_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
produc5835360497134304175_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Oterm_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc6058688428250151583at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__a_J_Mt__Relational____Calculus__Oterm_Itf__a_J_J,type,
produc8608687409264118859term_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
produc7366699395886430672_b_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
set_se6865892389300016395la_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
set_Re381260168593705685la_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Relational____Calculus__Oterm_Itf__a_J_J,type,
list_R6823256787227418703term_a: $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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relational_fmla_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Relational____Calculus__Oterm_Itf__a_J,type,
relational_term_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (165)
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
finite5600759454172676150la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
fun_up9185980720990446la_a_b: ( nat > relational_fmla_a_b ) > nat > relational_fmla_a_b > nat > relational_fmla_a_b ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
fun_upd_nat_a: ( nat > a ) > nat > a > nat > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
minus_9215201808853403479_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
minus_4077726661957047470la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
if_Rel1279876242545935705la_a_b: $o > relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
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_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
inf_in2474466416471573982_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > relational_fmla_a_b > $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_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__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
inf_in8483230781156617063la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
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_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
sup_su1471977682094119364_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
sup_su5130108678486352897la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
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_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
lattic5380700691367270794_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
lattic7900345719479732037la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ) > ( relational_fmla_a_b > relational_fmla_a_b > $o ) > ( relational_fmla_a_b > relational_fmla_a_b > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3109210760196336428et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
lattic4423677692272362385la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
lattic5961991414251573132_set_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
lattic6555223568391141957la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(sy_c_List_Olist_Omap_001t__Relational____Calculus__Oterm_Itf__a_J_001t__Relational____Calculus__Oterm_Itf__a_J,type,
map_Re5736185711816362116term_a: ( relational_term_a > relational_term_a ) > list_R6823256787227418703term_a > list_R6823256787227418703term_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
bot_bo8852203127187332700_a_b_o: relational_fmla_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
bot_bo4495933725496725865la_a_b: set_Re381260168593705685la_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
ord_le6021219098528097948_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
ord_le8964290712037365747la_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
ord_le7152733262289451305la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_M_Eo_J,type,
ord_le7191224889845164944_a_b_o: ( relational_fmla_a_b > $o ) > ( relational_fmla_a_b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
ord_le4236940698060306943la_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
ord_le4112832032246704949la_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_J,type,
ord_le1577343677690852715la_a_b: set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mtf__a_J_001t__Relational____Calculus__Oterm_Itf__a_J,type,
produc8917778089171359291term_a: ( nat > a ) > relational_term_a > produc8608687409264118859term_a ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_001_062_It__Nat__Onat_Mtf__a_J,type,
produc2895298938842563487_nat_a: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > produc5835360497134304175_nat_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
produc4282057684358614024_b_nat: relational_fmla_a_b > nat > produc7366699395886430672_b_nat ).
thf(sy_c_Product__Type_OPair_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
produc6598558901832717687_nat_a: relational_fmla_a_b > produc5835360497134304175_nat_a > produc1132964494702330949_nat_a ).
thf(sy_c_Product__Type_OPair_001t__Relational____Calculus__Oterm_Itf__a_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc2180204704594896271at_nat: relational_term_a > product_prod_nat_nat > produc6058688428250151583at_nat ).
thf(sy_c_Relational__Calculus_ODISJ_001tf__a_001tf__b,type,
relational_DISJ_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oadom_001tf__b_001tf__a,type,
relational_adom_b_a: ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Relational__Calculus_Oap_001tf__a_001tf__b,type,
relational_ap_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_001tf__a_001tf__b,type,
relational_cov_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocov_H_001tf__a_001tf__b,type,
relational_cov_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocp_001tf__a_001tf__b,type,
relational_cp_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ocpropagated_001tf__a_001tf__b,type,
relati1591879772219623554ed_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ocsts_001tf__a_001tf__b,type,
relational_csts_a_b: relational_fmla_a_b > set_a ).
thf(sy_c_Relational__Calculus_Oeqs_001tf__a_001tf__b,type,
relational_eqs_a_b: nat > set_Re381260168593705685la_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Oequiv_001tf__a_001tf__b,type,
relational_equiv_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oerase_001tf__a_001tf__b,type,
relational_erase_a_b: relational_fmla_a_b > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Oerase__rel_001tf__a_001tf__b,type,
relati5987653437628155313el_a_b: produc7366699395886430672_b_nat > produc7366699395886430672_b_nat > $o ).
thf(sy_c_Relational__Calculus_Oeval_001tf__a_001tf__b,type,
relational_eval_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oeval__on_001tf__a_001tf__b,type,
relati8814510239606734169on_a_b: set_nat > relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_list_a ).
thf(sy_c_Relational__Calculus_Oexists_001tf__a_001tf__b,type,
relati3989891337220013914ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OBool_001tf__a_001tf__b,type,
relational_Bool_a_b: $o > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OConj_001tf__a_001tf__b,type,
relational_Conj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ODisj_001tf__a_001tf__b,type,
relational_Disj_a_b: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OEq_001tf__a_001tf__b,type,
relational_Eq_a_b: nat > relational_term_a > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OExists_001tf__a_001tf__b,type,
relati591517084277583526ts_a_b: nat > relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_ONeg_001tf__a_001tf__b,type,
relational_Neg_a_b: relational_fmla_a_b > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofmla_OPred_001tf__b_001tf__a,type,
relational_Pred_b_a: b > list_R6823256787227418703term_a > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Ofresh2_001tf__a_001tf__b,type,
relati2677767559083392098h2_a_b: nat > nat > relational_fmla_a_b > nat ).
thf(sy_c_Relational__Calculus_Ofresh__val_001tf__a_001tf__b,type,
relati2318939533276802993al_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_a > a ).
thf(sy_c_Relational__Calculus_Ofv_001tf__a_001tf__b,type,
relational_fv_a_b: relational_fmla_a_b > set_nat ).
thf(sy_c_Relational__Calculus_Ofv__rel_001tf__a_001tf__b,type,
relati5703530512245835757el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Ofv__term__set_001tf__a,type,
relati6004689760767320788_set_a: relational_term_a > set_nat ).
thf(sy_c_Relational__Calculus_Ofv__terms__set_001tf__a,type,
relati4569515538964159125_set_a: list_R6823256787227418703term_a > set_nat ).
thf(sy_c_Relational__Calculus_Ogen_001tf__a_001tf__b,type,
relational_gen_a_b: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogen_H_001tf__a_001tf__b,type,
relational_gen_a_b2: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Relational__Calculus_Ogenempty_001tf__a_001tf__b,type,
relati5999705594545617851ty_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onocp_001tf__a_001tf__b,type,
relational_nocp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Onocp__rel_001tf__a_001tf__b,type,
relati3149960101488570543el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqp_001tf__a_001tf__b,type,
relational_qp_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqp__impl_001tf__a_001tf__b,type,
relati3725921752842749053pl_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqp__impl__rel_001tf__a_001tf__b,type,
relati7364465619720499582el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Oqps_001tf__a_001tf__b,type,
relational_qps_a_b: set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Relational__Calculus_Orrb_001tf__a_001tf__b,type,
relational_rrb_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osat_001tf__a_001tf__b,type,
relational_sat_a_b: relational_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o ).
thf(sy_c_Relational__Calculus_Osimplification_001tf__a_001tf__b,type,
relati2910603115655104169on_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > ( relational_fmla_a_b > $o ) > $o ).
thf(sy_c_Relational__Calculus_Osr_001tf__a_001tf__b,type,
relational_sr_a_b: relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osub_001tf__a_001tf__b,type,
relational_sub_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Relational__Calculus_Osub__rel_001tf__a_001tf__b,type,
relati7309537865537208983el_a_b: relational_fmla_a_b > relational_fmla_a_b > $o ).
thf(sy_c_Relational__Calculus_Osubst_001tf__a_001tf__b,type,
relational_subst_a_b: relational_fmla_a_b > nat > nat > relational_fmla_a_b ).
thf(sy_c_Relational__Calculus_Osubst__term_001tf__a,type,
relati7175845559408349773term_a: relational_term_a > nat > nat > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_OConst_001tf__a,type,
relational_Const_a: a > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_OVar_001tf__a,type,
relational_Var_a: nat > relational_term_a ).
thf(sy_c_Relational__Calculus_Oterm_Ocase__term_001tf__a_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
relati582353067970734056la_a_b: ( a > relational_fmla_a_b ) > ( nat > relational_fmla_a_b ) > relational_term_a > relational_fmla_a_b ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
collec3419995626248312948la_a_b: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_4386371547000553590la_a_b: ( nat > relational_fmla_a_b ) > set_nat > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
image_341122591648980342_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > set_nat ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001tf__a,type,
image_4722625287770715864_a_b_a: ( relational_fmla_a_b > a ) > set_Re381260168593705685la_a_b > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
image_7051608999182166449la_a_b: ( set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ) > set_se6865892389300016395la_a_b > set_se6865892389300016395la_a_b ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
is_sin6594375743535830443la_a_b: set_Re381260168593705685la_a_b > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
remove4261432235257513082la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).
thf(sy_c_Set_Othe__elem_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
the_el6350558617753882986la_a_b: set_Re381260168593705685la_a_b > relational_fmla_a_b ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
accp_P4351966040938400857_b_nat: ( produc7366699395886430672_b_nat > produc7366699395886430672_b_nat > $o ) > produc7366699395886430672_b_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
accp_R989495437599811158la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
member3481406638322139244la_a_b: set_Re381260168593705685la_a_b > set_se6865892389300016395la_a_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: set_Re381260168593705685la_a_b ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_Q,type,
q: relational_fmla_a_b ).
thf(sy_v_Q3____,type,
q3: relational_fmla_a_b ).
thf(sy_v__092_060sigma_062,type,
sigma: nat > a ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (1277)
thf(fact_0_assms_I1_J,axiom,
relational_cov_a_b @ x @ q @ g ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
member_nat @ x @ ( relational_fv_a_b @ q ) ).
% assms(2)
thf(fact_2_fmla_Oinject_I5_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
( ( ( relational_Conj_a_b @ X51 @ X52 )
= ( relational_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fmla.inject(5)
thf(fact_3_qps__in,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( member4680049679412964150la_a_b @ Q @ G ) ) ).
% qps_in
thf(fact_4_cov__fv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% cov_fv
thf(fact_5_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_6_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_7_subsetI,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b] :
( ! [X2: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X2 @ A )
=> ( member3481406638322139244la_a_b @ X2 @ B ) )
=> ( ord_le1577343677690852715la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_8_subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ( member4680049679412964150la_a_b @ X2 @ B ) )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ).
% subsetI
thf(fact_9_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_10_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_11_order__refl,axiom,
! [X: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ X @ X ) ).
% order_refl
thf(fact_12_order__refl,axiom,
! [X: $o > set_nat] : ( ord_le7022414076629706543et_nat @ X @ X ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X: $o > nat] : ( ord_less_eq_o_nat @ X @ X ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X: nat > $o] : ( ord_less_eq_nat_o @ X @ X ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_17_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_18_dual__order_Orefl,axiom,
! [A2: relational_fmla_a_b > $o] : ( ord_le7191224889845164944_a_b_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A2: $o > set_nat] : ( ord_le7022414076629706543et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_20_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A2: nat > $o] : ( ord_less_eq_nat_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_24_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_25_cov__Gen__qps,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) ) ) @ X_1 ) ) ) ).
% cov_Gen_qps
thf(fact_26_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_27_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_28_in__mono,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,X: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ X @ A )
=> ( member3481406638322139244la_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_29_in__mono,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_30_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_31_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_32_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_33_subsetD,axiom,
! [A: set_se6865892389300016395la_a_b,B: set_se6865892389300016395la_a_b,C: set_Re381260168593705685la_a_b] :
( ( ord_le1577343677690852715la_a_b @ A @ B )
=> ( ( member3481406638322139244la_a_b @ C @ A )
=> ( member3481406638322139244la_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_34_subsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_35_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_36_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_37_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_38_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_39_subset__eq,axiom,
( ord_le1577343677690852715la_a_b
= ( ^ [A3: set_se6865892389300016395la_a_b,B2: set_se6865892389300016395la_a_b] :
! [X3: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ X3 @ A3 )
=> ( member3481406638322139244la_a_b @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_40_subset__eq,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
=> ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_41_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_42_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_43_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_44_gen__qps,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_qps_a_b @ G )
= G ) ) ).
% gen_qps
thf(fact_45_gen_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_gen_a_b @ X @ Q1 @ G )
| ( relational_gen_a_b @ X @ Q2 @ G ) )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).
% gen.intros(7)
thf(fact_46_Gen__Conj_I1_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q1 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_1 ) ) ).
% Gen_Conj(1)
thf(fact_47_Gen__Conj_I2_J,axiom,
! [X: nat,Q2: relational_fmla_a_b,Q1: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_1 ) ) ).
% Gen_Conj(2)
thf(fact_48_gen__fv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% gen_fv
thf(fact_49_order__antisym__conv,axiom,
! [Y: relational_fmla_a_b > $o,X: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ Y @ X )
=> ( ( ord_le7191224889845164944_a_b_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_50_order__antisym__conv,axiom,
! [Y: $o > set_nat,X: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ Y @ X )
=> ( ( ord_le7022414076629706543et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_51_order__antisym__conv,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_52_order__antisym__conv,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_53_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_54_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_55_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_56_linorder__le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_57_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_58_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A2: a,B3: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A2: a,B3: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__le__eq__subst,axiom,
! [A2: a,B3: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_ord__eq__le__subst,axiom,
! [A2: a,F: nat > a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A2: nat,F: a > nat,B3: a,C: a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A2: a,F: a > a,B3: a,C: a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_ord__eq__le__subst,axiom,
! [A2: a,F: set_nat > a,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A2: set_nat,F: a > set_nat,B3: a,C: a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A2: $o > nat,F: nat > $o > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_linorder__linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linorder_linear
thf(fact_79_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_80_order__eq__refl,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o] :
( ( X = Y )
=> ( ord_le7191224889845164944_a_b_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_81_order__eq__refl,axiom,
! [X: $o > set_nat,Y: $o > set_nat] :
( ( X = Y )
=> ( ord_le7022414076629706543et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_82_order__eq__refl,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( X = Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_83_order__eq__refl,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X = Y )
=> ( ord_less_eq_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_84_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_85_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_86_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_87_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_a @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A2: a,B3: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A2: a,B3: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ord_less_eq_a @ ( F @ B3 ) @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A2: a,B3: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_o_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A2: nat,F: a > nat,B3: a,C: a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_order__subst1,axiom,
! [A2: a,F: nat > a,B3: nat,C: nat] :
( ( ord_less_eq_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_order__subst1,axiom,
! [A2: a,F: a > a,B3: a,C: a] :
( ( ord_less_eq_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A2: set_nat,F: a > set_nat,B3: a,C: a] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_order__subst1,axiom,
! [A2: a,F: set_nat > a,B3: set_nat,C: set_nat] :
( ( ord_less_eq_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B3: $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_o_nat @ B3 @ C )
=> ( ! [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] : ( Y3 = Z ) )
= ( ^ [A4: relational_fmla_a_b > $o,B4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A4 @ B4 )
& ( ord_le7191224889845164944_a_b_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_108_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > set_nat,Z: $o > set_nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > set_nat,B4: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ A4 @ B4 )
& ( ord_le7022414076629706543et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > nat,B4: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B4 )
& ( ord_less_eq_o_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat > $o,Z: nat > $o] : ( Y3 = Z ) )
= ( ^ [A4: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A4 @ B4 )
& ( ord_less_eq_nat_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_111_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ A4 @ B4 )
& ( ord_less_eq_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_112_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_113_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_114_le__fun__def,axiom,
( ord_le7191224889845164944_a_b_o
= ( ^ [F2: relational_fmla_a_b > $o,G2: relational_fmla_a_b > $o] :
! [X3: relational_fmla_a_b] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_115_le__fun__def,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [F2: $o > set_nat,G2: $o > set_nat] :
! [X3: $o] : ( ord_less_eq_set_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_116_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G2: $o > nat] :
! [X3: $o] : ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_117_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o] :
! [X3: nat] : ( ord_less_eq_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_118_le__funI,axiom,
! [F: relational_fmla_a_b > $o,G3: relational_fmla_a_b > $o] :
( ! [X2: relational_fmla_a_b] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G3 @ X2 ) )
=> ( ord_le7191224889845164944_a_b_o @ F @ G3 ) ) ).
% le_funI
thf(fact_119_le__funI,axiom,
! [F: nat > $o,G3: nat > $o] :
( ! [X2: nat] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G3 @ X2 ) )
=> ( ord_less_eq_nat_o @ F @ G3 ) ) ).
% le_funI
thf(fact_120_le__funI,axiom,
! [F: $o > set_nat,G3: $o > set_nat] :
( ! [X2: $o] : ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G3 @ X2 ) )
=> ( ord_le7022414076629706543et_nat @ F @ G3 ) ) ).
% le_funI
thf(fact_121_le__funI,axiom,
! [F: $o > nat,G3: $o > nat] :
( ! [X2: $o] : ( ord_less_eq_nat @ ( F @ X2 ) @ ( G3 @ X2 ) )
=> ( ord_less_eq_o_nat @ F @ G3 ) ) ).
% le_funI
thf(fact_122_le__funE,axiom,
! [F: relational_fmla_a_b > $o,G3: relational_fmla_a_b > $o,X: relational_fmla_a_b] :
( ( ord_le7191224889845164944_a_b_o @ F @ G3 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_123_le__funE,axiom,
! [F: $o > set_nat,G3: $o > set_nat,X: $o] :
( ( ord_le7022414076629706543et_nat @ F @ G3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_124_le__funE,axiom,
! [F: $o > nat,G3: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_125_le__funE,axiom,
! [F: nat > $o,G3: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G3 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funE
thf(fact_126_le__funD,axiom,
! [F: relational_fmla_a_b > $o,G3: relational_fmla_a_b > $o,X: relational_fmla_a_b] :
( ( ord_le7191224889845164944_a_b_o @ F @ G3 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_127_le__funD,axiom,
! [F: $o > set_nat,G3: $o > set_nat,X: $o] :
( ( ord_le7022414076629706543et_nat @ F @ G3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_128_le__funD,axiom,
! [F: $o > nat,G3: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_129_le__funD,axiom,
! [F: nat > $o,G3: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F @ G3 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G3 @ X ) ) ) ).
% le_funD
thf(fact_130_antisym,axiom,
! [A2: relational_fmla_a_b > $o,B3: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ A2 @ B3 )
=> ( ( ord_le7191224889845164944_a_b_o @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_131_antisym,axiom,
! [A2: $o > set_nat,B3: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ A2 @ B3 )
=> ( ( ord_le7022414076629706543et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_132_antisym,axiom,
! [A2: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_133_antisym,axiom,
! [A2: nat > $o,B3: nat > $o] :
( ( ord_less_eq_nat_o @ A2 @ B3 )
=> ( ( ord_less_eq_nat_o @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_134_antisym,axiom,
! [A2: a,B3: a] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ord_less_eq_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_137_dual__order_Otrans,axiom,
! [B3: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o,C: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B3 @ A2 )
=> ( ( ord_le7191224889845164944_a_b_o @ C @ B3 )
=> ( ord_le7191224889845164944_a_b_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_138_dual__order_Otrans,axiom,
! [B3: $o > set_nat,A2: $o > set_nat,C: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ B3 @ A2 )
=> ( ( ord_le7022414076629706543et_nat @ C @ B3 )
=> ( ord_le7022414076629706543et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_139_dual__order_Otrans,axiom,
! [B3: $o > nat,A2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( ( ord_less_eq_o_nat @ C @ B3 )
=> ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Otrans,axiom,
! [B3: nat > $o,A2: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B3 @ A2 )
=> ( ( ord_less_eq_nat_o @ C @ B3 )
=> ( ord_less_eq_nat_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Otrans,axiom,
! [B3: a,A2: a,C: a] :
( ( ord_less_eq_a @ B3 @ A2 )
=> ( ( ord_less_eq_a @ C @ B3 )
=> ( ord_less_eq_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_143_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Oantisym,axiom,
! [B3: relational_fmla_a_b > $o,A2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B3 @ A2 )
=> ( ( ord_le7191224889845164944_a_b_o @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_145_dual__order_Oantisym,axiom,
! [B3: $o > set_nat,A2: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ B3 @ A2 )
=> ( ( ord_le7022414076629706543et_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_146_dual__order_Oantisym,axiom,
! [B3: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B3: nat > $o,A2: nat > $o] :
( ( ord_less_eq_nat_o @ B3 @ A2 )
=> ( ( ord_less_eq_nat_o @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B3: a,A2: a] :
( ( ord_less_eq_a @ B3 @ A2 )
=> ( ( ord_less_eq_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: relational_fmla_a_b > $o,Z: relational_fmla_a_b > $o] : ( Y3 = Z ) )
= ( ^ [A4: relational_fmla_a_b > $o,B4: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ B4 @ A4 )
& ( ord_le7191224889845164944_a_b_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > set_nat,Z: $o > set_nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > set_nat,B4: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ B4 @ A4 )
& ( ord_le7022414076629706543et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A4: $o > nat,B4: $o > nat] :
( ( ord_less_eq_o_nat @ B4 @ A4 )
& ( ord_less_eq_o_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat > $o,Z: nat > $o] : ( Y3 = Z ) )
= ( ^ [A4: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A4 )
& ( ord_less_eq_nat_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [A4: a,B4: a] :
( ( ord_less_eq_a @ B4 @ A4 )
& ( ord_less_eq_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_linorder__wlog,axiom,
! [P: a > a > $o,A2: a,B3: a] :
( ! [A5: a,B5: a] :
( ( ord_less_eq_a @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: a,B5: a] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_159_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_160_order__trans,axiom,
! [X: relational_fmla_a_b > $o,Y: relational_fmla_a_b > $o,Z2: relational_fmla_a_b > $o] :
( ( ord_le7191224889845164944_a_b_o @ X @ Y )
=> ( ( ord_le7191224889845164944_a_b_o @ Y @ Z2 )
=> ( ord_le7191224889845164944_a_b_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_161_order__trans,axiom,
! [X: $o > set_nat,Y: $o > set_nat,Z2: $o > set_nat] :
( ( ord_le7022414076629706543et_nat @ X @ Y )
=> ( ( ord_le7022414076629706543et_nat @ Y @ Z2 )
=> ( ord_le7022414076629706543et_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_162_order__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z2 )
=> ( ord_less_eq_o_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X: nat > $o,Y: nat > $o,Z2: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ Z2 )
=> ( ord_less_eq_nat_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_164_order__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_165_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_166_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_167_order_Otrans,axiom,
! [A2: a,B3: a,C: a] :
( ( ord_less_eq_a @ A2 @ B3 )
=> ( ( ord_less_eq_a @ B3 @ C )
=> ( ord_less_eq_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_168_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_169_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_170_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_171_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_172_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_173_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_174_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_175_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_176_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_177_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_178_mem__Collect__eq,axiom,
! [A2: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ A2 @ ( collec3419995626248312948la_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_179_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_180_Collect__mem__eq,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_181_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_182_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_183_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_184_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_185_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_186_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_187_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_188_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_189_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_190_subset__iff,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
! [T: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ T @ A3 )
=> ( member4680049679412964150la_a_b @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_192_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_193_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_194_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_195_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_196_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_197_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_198_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_199_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_200_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_201_simplification_Ofv__simp,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q: relational_fmla_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( Simp @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ) ).
% simplification.fv_simp
thf(fact_202_assms_I4_J,axiom,
relational_sat_a_b @ ( relational_erase_a_b @ q @ x ) @ i @ sigma ).
% assms(4)
thf(fact_203_ex__cov,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_rrb_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_cov_a_b @ X @ Q @ X_1 ) ) ) ).
% ex_cov
thf(fact_204_cov__csts,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% cov_csts
thf(fact_205_gen__csts,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ord_less_eq_set_a @ ( relational_csts_a_b @ Qqp ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% gen_csts
thf(fact_206_cov_Ononfree,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).
% cov.nonfree
thf(fact_207_qp__Gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_1 ) ) ) ).
% qp_Gen
thf(fact_208_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_209_empty__iff,axiom,
! [C: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).
% empty_iff
thf(fact_210_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_211_all__not__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ! [X3: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X3 @ A ) )
= ( A = bot_bo4495933725496725865la_a_b ) ) ).
% all_not_in_conv
thf(fact_212_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_213_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_214_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_215_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_216_rrb__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q2 ) ) ) ).
% rrb_simps(6)
thf(fact_217_qps__empty,axiom,
( ( relational_qps_a_b @ bot_bo4495933725496725865la_a_b )
= bot_bo4495933725496725865la_a_b ) ).
% qps_empty
thf(fact_218_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_219_emptyE,axiom,
! [A2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ).
% emptyE
thf(fact_220_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_221_equals0D,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( A = bot_bo4495933725496725865la_a_b )
=> ~ ( member4680049679412964150la_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_222_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_223_equals0I,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [Y2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ Y2 @ A )
=> ( A = bot_bo4495933725496725865la_a_b ) ) ).
% equals0I
thf(fact_224_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_225_ex__in__conv,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( ? [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A ) )
= ( A != bot_bo4495933725496725865la_a_b ) ) ).
% ex_in_conv
thf(fact_226_rrb__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_rrb_a_b @ Q )
=> ( relational_rrb_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% rrb_erase
thf(fact_227_simplification_Osat__simp,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( relational_sat_a_b @ ( Simp @ Q ) @ I @ Sigma )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).
% simplification.sat_simp
thf(fact_228_gen__sat__erase,axiom,
! [Y: nat,Q: relational_fmla_a_b,Gy: set_Re381260168593705685la_a_b,X: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I @ Sigma )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Gy )
& ( relational_sat_a_b @ X2 @ I @ Sigma ) ) ) ) ).
% gen_sat_erase
thf(fact_229_sat_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
& ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(5)
thf(fact_230_sat__fv__cong,axiom,
! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
( ! [N: nat] :
( ( member_nat @ N @ ( relational_fv_a_b @ Phi ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( relational_sat_a_b @ Phi @ I @ Sigma )
= ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).
% sat_fv_cong
thf(fact_231_gen__sat,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ G )
& ( relational_sat_a_b @ X2 @ I @ Sigma ) ) ) ) ).
% gen_sat
thf(fact_232_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_233_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_234_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_235_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_236_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_237_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_238_erase_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X )
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% erase.simps(5)
thf(fact_239_gen__Gen__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z2: nat] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_erase_a_b @ Q @ Z2 ) @ X_1 ) ) ).
% gen_Gen_erase
thf(fact_240_Gen__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,Z2: nat] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_erase_a_b @ Q @ Z2 ) @ X_1 ) ) ).
% Gen_erase
thf(fact_241_qp__Conj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ).
% qp_Conj
thf(fact_242_gen__qp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen_qp
thf(fact_243_qps__qp,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( relational_qp_a_b @ Q ) ) ).
% qps_qp
thf(fact_244_simplification_Osimplified__fv__simp,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q: relational_fmla_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( Simplified @ Q )
=> ( ( relational_fv_a_b @ ( Simp @ Q ) )
= ( relational_fv_a_b @ Q ) ) ) ) ).
% simplification.simplified_fv_simp
thf(fact_245_simplification_Ogen__Gen__simp,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( Simp @ Q ) @ X_1 ) ) ) ).
% simplification.gen_Gen_simp
thf(fact_246_simplification_OGen__simp,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,X: nat,Q: relational_fmla_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( Simp @ Q ) @ X_1 ) ) ) ).
% simplification.Gen_simp
thf(fact_247_qps__rrb,axiom,
! [Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ Q @ ( relational_qps_a_b @ G ) )
=> ( relational_rrb_a_b @ Q ) ) ).
% qps_rrb
thf(fact_248_subset__emptyI,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ! [X2: relational_fmla_a_b] :
~ ( member4680049679412964150la_a_b @ X2 @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).
% subset_emptyI
thf(fact_249_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_250_qp__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_qp_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% qp_fresh_val
thf(fact_251_assms_I3_J,axiom,
finite_finite_a @ ( relational_adom_b_a @ i ) ).
% assms(3)
thf(fact_252_Gen__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( ( relational_qp_a_b @ X2 )
& ( member_nat @ X @ ( relational_fv_a_b @ X2 ) ) ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_DISJ_a_b @ Q3 ) @ X_1 ) ) ) ).
% Gen_DISJ
thf(fact_253_qp__gen,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% qp_gen
thf(fact_254_rrb__simps_I8_J,axiom,
! [Y: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y @ Qy ) )
= ( ( ( member_nat @ Y @ ( relational_fv_a_b @ Qy ) )
=> ? [X4: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy @ X4 ) )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(8)
thf(fact_255_sat__DISJ,axiom,
! [G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( finite5600759454172676150la_a_b @ G )
=> ( ( relational_sat_a_b @ ( relational_DISJ_a_b @ G ) @ I @ Sigma )
= ( ? [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ G )
& ( relational_sat_a_b @ X3 @ I @ Sigma ) ) ) ) ) ).
% sat_DISJ
thf(fact_256_Disj__single,axiom,
! [X: relational_fmla_a_b] :
( ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= X ) ).
% Disj_single
thf(fact_257_insert__iff,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_258_insert__iff,axiom,
! [A2: relational_fmla_a_b,B3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_259_insertCI,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_260_insertCI,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B3 @ B ) ) ) ).
% insertCI
thf(fact_261_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_262_singletonI,axiom,
! [A2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) ).
% singletonI
thf(fact_263_insert__subset,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B )
= ( ( member4680049679412964150la_a_b @ X @ B )
& ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ).
% insert_subset
thf(fact_264_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_265_singleton__insert__inj__eq,axiom,
! [B3: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B3 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_266_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B3: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_267_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat @ A2 @ B6 ) )
& ~ ( member_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_268_mk__disjoint__insert,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [B6: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ A2 @ B6 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_269_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B3: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B3 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B3 @ C3 ) )
& ~ ( member_nat @ B3 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_270_insert__eq__iff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ B3 @ B )
=> ( ( ( insert7010464514620295119la_a_b @ A2 @ A )
= ( insert7010464514620295119la_a_b @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ B3 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ B3 @ C3 )
& ( B
= ( insert7010464514620295119la_a_b @ A2 @ C3 ) )
& ~ ( member4680049679412964150la_a_b @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_271_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_272_insert__absorb,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_273_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_274_insert__ident,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( insert7010464514620295119la_a_b @ X @ A )
= ( insert7010464514620295119la_a_b @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_275_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_276_Set_Oset__insert,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ~ ! [B6: set_Re381260168593705685la_a_b] :
( ( A
= ( insert7010464514620295119la_a_b @ X @ B6 ) )
=> ( member4680049679412964150la_a_b @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_277_insertI2,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_278_insertI2,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ B )
=> ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B3 @ B ) ) ) ).
% insertI2
thf(fact_279_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_280_insertI1,axiom,
! [A2: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] : ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A2 @ B ) ) ).
% insertI1
thf(fact_281_insertE,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_282_insertE,axiom,
! [A2: relational_fmla_a_b,B3: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member4680049679412964150la_a_b @ A2 @ A ) ) ) ).
% insertE
thf(fact_283_insert__subsetI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( ord_le4112832032246704949la_a_b @ X5 @ A )
=> ( ord_le4112832032246704949la_a_b @ ( insert7010464514620295119la_a_b @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_284_insert__subsetI,axiom,
! [X: nat,A: set_nat,X5: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_285_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_286_singleton__iff,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_287_singleton__iff,axiom,
! [B3: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B3 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_288_singletonD,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_289_singletonD,axiom,
! [B3: relational_fmla_a_b,A2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ B3 @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_290_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_291_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_292_subset__insert,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ).
% subset_insert
thf(fact_293_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_294_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_295_gen__finite,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% gen_finite
thf(fact_296_cov__finite,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( finite5600759454172676150la_a_b @ G ) ) ).
% cov_finite
thf(fact_297_finite__qps,axiom,
! [G: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ G )
=> ( finite5600759454172676150la_a_b @ ( relational_qps_a_b @ G ) ) ) ).
% finite_qps
thf(fact_298_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_299_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_300_rrb__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( relational_rrb_a_b @ X2 ) )
=> ( relational_rrb_a_b @ ( relational_DISJ_a_b @ Q3 ) ) ) ) ).
% rrb_DISJ
thf(fact_301_qps__insert,axiom,
! [Q: relational_fmla_a_b,Qs: set_Re381260168593705685la_a_b] :
( ( ( relational_qp_a_b @ Q )
=> ( ( relational_qps_a_b @ ( insert7010464514620295119la_a_b @ Q @ Qs ) )
= ( insert7010464514620295119la_a_b @ Q @ ( relational_qps_a_b @ Qs ) ) ) )
& ( ~ ( relational_qp_a_b @ Q )
=> ( ( relational_qps_a_b @ ( insert7010464514620295119la_a_b @ Q @ Qs ) )
= ( relational_qps_a_b @ Qs ) ) ) ) ).
% qps_insert
thf(fact_302_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_303_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_304_finite__subset__induct_H,axiom,
! [F3: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A5: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A )
=> ( ( ord_le4112832032246704949la_a_b @ F4 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_305_finite__subset__induct_H,axiom,
! [F3: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ~ ( member_a @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_306_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_307_finite__subset__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( ord_le4112832032246704949la_a_b @ F3 @ A )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A5: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A )
=> ( ~ ( member4680049679412964150la_a_b @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_308_finite__subset__induct,axiom,
! [F3: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A5 @ A )
=> ( ~ ( member_a @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_309_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_310_finite__ranking__induct,axiom,
! [S: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b,S2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ S2 )
=> ( ! [Y5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert7010464514620295119la_a_b @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_311_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_312_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_313_infinite__finite__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,A: set_Re381260168593705685la_a_b] :
( ! [A6: set_Re381260168593705685la_a_b] :
( ~ ( finite5600759454172676150la_a_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_314_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_315_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_316_finite__ne__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( F3 != bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b] : ( P @ ( insert7010464514620295119la_a_b @ X2 @ bot_bo4495933725496725865la_a_b ) )
=> ( ! [X2: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ( F4 != bot_bo4495933725496725865la_a_b )
=> ( ~ ( member4680049679412964150la_a_b @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_317_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_318_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_319_finite__induct,axiom,
! [F3: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ F3 )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b,F4: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ F4 )
=> ( ~ ( member4680049679412964150la_a_b @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert7010464514620295119la_a_b @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_320_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_321_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_322_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_323_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_324_finite__fv,axiom,
! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).
% finite_fv
thf(fact_325_finite__has__maximal2,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
& ( ord_le4236940698060306943la_a_b @ A2 @ X2 )
& ! [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ A )
=> ( ( ord_le4236940698060306943la_a_b @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_326_finite__has__maximal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ A2 @ X2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_327_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A2 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_328_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_329_finite__has__minimal2,axiom,
! [A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
& ( ord_le4236940698060306943la_a_b @ X2 @ A2 )
& ! [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ A )
=> ( ( ord_le4236940698060306943la_a_b @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_330_finite__has__minimal2,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ord_less_eq_a @ X2 @ A2 )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_331_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_332_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_333_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_334_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_335_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_336_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_337_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_338_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_339_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_340_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_341_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_342_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_343_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_344_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_345_finite__has__maximal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_346_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_347_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_348_finite__has__minimal,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_349_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_350_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_351_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A2
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_352_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A2
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_353_fresh__val_I2_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ ( relational_csts_a_b @ Q ) ) ) ) ).
% fresh_val(2)
thf(fact_354_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_355_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_356_bot__empty__eq,axiom,
( bot_bo8852203127187332700_a_b_o
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ).
% bot_empty_eq
thf(fact_357_arg__min__least,axiom,
! [S: set_Re381260168593705685la_a_b,Y: relational_fmla_a_b,F: relational_fmla_a_b > nat] :
( ( finite5600759454172676150la_a_b @ S )
=> ( ( S != bot_bo4495933725496725865la_a_b )
=> ( ( member4680049679412964150la_a_b @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5380700691367270794_b_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_358_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_359_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_360_fresh__val_I1_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ ( relational_adom_b_a @ I ) ) ) ) ).
% fresh_val(1)
thf(fact_361_fresh__val_I3_J,axiom,
! [I: product_prod_b_nat > set_list_a,A: set_a,Q: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( finite_finite_a @ A )
=> ~ ( member_a @ ( relati2318939533276802993al_a_b @ Q @ I @ A ) @ A ) ) ) ).
% fresh_val(3)
thf(fact_362_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_363_is__singletonI_H,axiom,
! [A: set_Re381260168593705685la_a_b] :
( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ( ( member4680049679412964150la_a_b @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_sin6594375743535830443la_a_b @ A ) ) ) ).
% is_singletonI'
thf(fact_364_ap__fresh__val,axiom,
! [Q: relational_fmla_a_b,Sigma: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
( ( relational_ap_a_b @ Q )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_adom_b_a @ I ) )
=> ( ~ ( member_a @ ( Sigma @ X ) @ ( relational_csts_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).
% ap_fresh_val
thf(fact_365_Gen__cp__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( ( relational_qp_a_b @ X2 )
& ( member_nat @ X @ ( relational_fv_a_b @ X2 ) ) ) )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ ( relational_DISJ_a_b @ Q3 ) ) @ X_1 ) ) ) ).
% Gen_cp_DISJ
thf(fact_366_cov_Oap,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov.ap
thf(fact_367_gen_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen.intros(2)
thf(fact_368_sr__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b,X5: set_nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( ( relational_fv_a_b @ X2 )
= X5 ) )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( relational_sr_a_b @ X2 ) )
=> ( relational_sr_a_b @ ( relational_DISJ_a_b @ Q3 ) ) ) ) ) ).
% sr_DISJ
thf(fact_369_simplification_Ofv__simp__DISJ__eq,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q3: set_Re381260168593705685la_a_b,A: set_nat] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ( ( Simp @ ( relational_DISJ_a_b @ Q3 ) )
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( ( Simplified @ X2 )
& ( ( relational_fv_a_b @ X2 )
= A ) ) )
=> ( ( relational_fv_a_b @ ( Simp @ ( relational_DISJ_a_b @ Q3 ) ) )
= A ) ) ) ) ) ).
% simplification.fv_simp_DISJ_eq
thf(fact_370_Sup__fin_Osubset__imp,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_371_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_372_Inf__fin_Osubset__imp,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_373_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_374_gen__Bool__True,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
~ ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $true ) @ G ) ).
% gen_Bool_True
thf(fact_375_sat__cp,axiom,
! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_cp_a_b @ Q ) @ I @ Sigma )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ).
% sat_cp
thf(fact_376_gen__Bool__False,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ G )
= ( G = bot_bo4495933725496725865la_a_b ) ) ).
% gen_Bool_False
thf(fact_377_Disj__empty,axiom,
( ( relational_DISJ_a_b @ bot_bo4495933725496725865la_a_b )
= ( relational_Bool_a_b @ $false ) ) ).
% Disj_empty
thf(fact_378_ap__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% ap_cp_erase
thf(fact_379_gen__empty__cp,axiom,
! [Z2: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ Q @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( ( relational_cp_a_b @ Q )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_empty_cp
thf(fact_380_gen__cp__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Qqp @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% gen_cp_erase
thf(fact_381_gen__Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).
% gen_Gen_cp
thf(fact_382_Gen__cp,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).
% Gen_cp
thf(fact_383_fmla_Odistinct_I17_J,axiom,
! [X22: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(17)
thf(fact_384_fv_Osimps_I2_J,axiom,
! [B3: $o] :
( ( relational_fv_a_b @ ( relational_Bool_a_b @ B3 ) )
= bot_bot_set_nat ) ).
% fv.simps(2)
thf(fact_385_sat_Osimps_I2_J,axiom,
! [B3: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Bool_a_b @ B3 ) @ I @ Sigma )
= B3 ) ).
% sat.simps(2)
thf(fact_386_erase_Osimps_I1_J,axiom,
! [T3: $o,X: nat] :
( ( relational_erase_a_b @ ( relational_Bool_a_b @ T3 ) @ X )
= ( relational_Bool_a_b @ T3 ) ) ).
% erase.simps(1)
thf(fact_387_cov_OConjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.ConjR
thf(fact_388_cov_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.ConjL
thf(fact_389_qp__cp__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
= ( relational_Bool_a_b @ $false ) ) ) ) ).
% qp_cp_erase
thf(fact_390_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_391_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_392_gen_Ointros_I1_J,axiom,
! [X: nat] : ( relational_gen_a_b @ X @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).
% gen.intros(1)
thf(fact_393_exists__cp__erase,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati3989891337220013914ts_a_b @ X @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) )
= ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% exists_cp_erase
thf(fact_394_fv__cp,axiom,
! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ).
% fv_cp
thf(fact_395_Sup__fin_OcoboundedI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ A2 @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_396_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_397_Inf__fin_OcoboundedI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_398_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_399_Inf__fin_OboundedE,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
=> ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A )
=> ( ord_less_eq_set_nat @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_400_Inf__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_401_Inf__fin_OboundedI,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ( ord_less_eq_set_nat @ X @ A5 ) )
=> ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_402_Inf__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( ord_less_eq_nat @ X @ A5 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_403_Sup__fin_OboundedE,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
=> ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A )
=> ( ord_less_eq_set_nat @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_404_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_405_Sup__fin_OboundedI,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ( ord_less_eq_set_nat @ A5 @ X ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_406_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( ord_less_eq_nat @ A5 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_407_Inf__fin_Obounded__iff,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_408_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_409_Sup__fin_Obounded__iff,axiom,
! [A: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_410_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_411_fv__cp__DISJ__eq,axiom,
! [Q3: set_Re381260168593705685la_a_b,A: set_nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ( ( relational_cp_a_b @ ( relational_DISJ_a_b @ Q3 ) )
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
=> ( ( relati1591879772219623554ed_a_b @ X2 )
& ( ( relational_fv_a_b @ X2 )
= A ) ) )
=> ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( relational_DISJ_a_b @ Q3 ) ) )
= A ) ) ) ) ).
% fv_cp_DISJ_eq
thf(fact_412_Inf__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_413_Sup__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_414_gen_H_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_gen_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen'.intros(2)
thf(fact_415_cov_H_Oap,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_ap_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% cov'.ap
thf(fact_416_cov_H_OConjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.ConjR
thf(fact_417_cov_H_OConjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $false ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.ConjL
thf(fact_418_cov_ODisjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.DisjR
thf(fact_419_finite__Un,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) )
= ( ( finite_finite_a @ F3 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_420_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_421_finite__Int,axiom,
! [F3: set_a,G: set_a] :
( ( ( finite_finite_a @ F3 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_422_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_423_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_424_Int__insert__right__if1,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( inf_in8483230781156617063la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_425_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_426_Int__insert__right__if0,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( inf_in8483230781156617063la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_427_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_428_Int__insert__left__if1,axiom,
! [A2: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B ) @ C2 )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_429_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_430_Int__insert__left__if0,axiom,
! [A2: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B ) @ C2 )
= ( inf_in8483230781156617063la_a_b @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_431_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_432_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_433_fmla_Oinject_I6_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
( ( ( relational_Disj_a_b @ X61 @ X62 )
= ( relational_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fmla.inject(6)
thf(fact_434_qps__union,axiom,
! [X5: set_Re381260168593705685la_a_b,Y7: set_Re381260168593705685la_a_b] :
( ( relational_qps_a_b @ ( sup_su5130108678486352897la_a_b @ X5 @ Y7 ) )
= ( sup_su5130108678486352897la_a_b @ ( relational_qps_a_b @ X5 ) @ ( relational_qps_a_b @ Y7 ) ) ) ).
% qps_union
thf(fact_435_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_436_insert__disjoint_I1_J,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A ) @ B )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
& ( ( inf_in8483230781156617063la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% insert_disjoint(1)
thf(fact_437_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_438_insert__disjoint_I2_J,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ A ) @ B ) )
= ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_439_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_440_disjoint__insert_I1_J,axiom,
! [B: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ B @ ( insert7010464514620295119la_a_b @ A2 @ A ) )
= bot_bo4495933725496725865la_a_b )
= ( ~ ( member4680049679412964150la_a_b @ A2 @ B )
& ( ( inf_in8483230781156617063la_a_b @ B @ A )
= bot_bo4495933725496725865la_a_b ) ) ) ).
% disjoint_insert(1)
thf(fact_441_disjoint__insert_I2_J,axiom,
! [A: set_nat,B3: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B3 @ B ) ) )
= ( ~ ( member_nat @ B3 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_442_disjoint__insert_I2_J,axiom,
! [A: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A @ ( insert7010464514620295119la_a_b @ B3 @ B ) ) )
= ( ~ ( member4680049679412964150la_a_b @ B3 @ A )
& ( bot_bo4495933725496725865la_a_b
= ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_443_rrb__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_rrb_a_b @ Q1 )
& ( relational_rrb_a_b @ Q2 ) ) ) ).
% rrb_simps(5)
thf(fact_444_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_445_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_446_csts_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q2 ) ) ) ).
% csts.simps(6)
thf(fact_447_gen_H_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b2 @ X @ Q2 @ G22 )
=> ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen'.intros(6)
thf(fact_448_cov_H_ODisj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q2 @ G22 )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov'.Disj
thf(fact_449_cov_ODisj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q2 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Disj
thf(fact_450_gen_Ointros_I6_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q1 @ G1 )
=> ( ( relational_gen_a_b @ X @ Q2 @ G22 )
=> ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% gen.intros(6)
thf(fact_451_cov_H_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b2 @ X @ Q2 @ G22 )
=> ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov'.Conj
thf(fact_452_fv_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(6)
thf(fact_453_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_454_finite__UnI,axiom,
! [F3: set_a,G: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_455_finite__UnI,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_456_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_457_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_458_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_459_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_460_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_461_disjoint__iff,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( inf_in8483230781156617063la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b )
= ( ! [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
=> ~ ( member4680049679412964150la_a_b @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_462_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_463_Int__emptyI,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ~ ( member4680049679412964150la_a_b @ X2 @ B ) )
=> ( ( inf_in8483230781156617063la_a_b @ A @ B )
= bot_bo4495933725496725865la_a_b ) ) ).
% Int_emptyI
thf(fact_464_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_465_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A8: set_nat] :
( ( ord_less_eq_set_nat @ A8 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_466_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_467_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_468_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_469_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_470_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_471_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_472_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_473_Int__insert__right,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( inf_in8483230781156617063la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( inf_in8483230781156617063la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ B ) )
= ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_474_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_475_Int__insert__left,axiom,
! [A2: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B ) @ C2 )
= ( insert7010464514620295119la_a_b @ A2 @ ( inf_in8483230781156617063la_a_b @ B @ C2 ) ) ) )
& ( ~ ( member4680049679412964150la_a_b @ A2 @ C2 )
=> ( ( inf_in8483230781156617063la_a_b @ ( insert7010464514620295119la_a_b @ A2 @ B ) @ C2 )
= ( inf_in8483230781156617063la_a_b @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_476_Int__Collect__mono,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o,Q: relational_fmla_a_b > $o] :
( ( ord_le4112832032246704949la_a_b @ A @ B )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le4112832032246704949la_a_b @ ( inf_in8483230781156617063la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) ) @ ( inf_in8483230781156617063la_a_b @ B @ ( collec3419995626248312948la_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_477_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_478_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_479_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_480_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_481_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_482_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_483_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_484_fmla_Odistinct_I37_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(37)
thf(fact_485_fmla_Odistinct_I19_J,axiom,
! [X22: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(19)
thf(fact_486_sat_Osimps_I6_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
= ( ( relational_sat_a_b @ Phi @ I @ Sigma )
| ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).
% sat.simps(6)
thf(fact_487_gen_H_Ointros_I7_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_gen_a_b2 @ X @ Q1 @ G )
| ( relational_gen_a_b2 @ X @ Q2 @ G ) )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).
% gen'.intros(7)
thf(fact_488_erase_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ X )
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).
% erase.simps(6)
thf(fact_489_gen_H__gen,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ Q @ G ) ) ).
% gen'_gen
thf(fact_490_gen__gen_H,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ Q @ G ) ) ).
% gen_gen'
thf(fact_491_gen__eq__gen_H,axiom,
relational_gen_a_b = relational_gen_a_b2 ).
% gen_eq_gen'
thf(fact_492_qp__Disj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ).
% qp_Disj
thf(fact_493_cov_H__cov,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( relational_cov_a_b @ X @ Q @ G ) ) ).
% cov'_cov
thf(fact_494_cov__cov_H,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( relational_cov_a_b2 @ X @ Q @ G ) ) ).
% cov_cov'
thf(fact_495_cov__eq__cov_H,axiom,
relational_cov_a_b = relational_cov_a_b2 ).
% cov_eq_cov'
thf(fact_496_gen_H__qp,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( relational_qp_a_b @ Qqp ) ) ) ).
% gen'_qp
thf(fact_497_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_498_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_499_cov_H_ODisjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.DisjL
thf(fact_500_cov_H_ODisjR,axiom,
! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
( ( relational_cov_a_b2 @ X @ Q2 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b2 @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov'.DisjR
thf(fact_501_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_502_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_503_fv_Osimps_I5_J,axiom,
! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).
% fv.simps(5)
thf(fact_504_cov_OConj,axiom,
! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G1 )
=> ( ( relational_cov_a_b @ X @ Q2 @ G22 )
=> ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ).
% cov.Conj
thf(fact_505_cov_H_Ononfree,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_cov_a_b2 @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).
% cov'.nonfree
thf(fact_506_csts_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q2 ) ) ) ).
% csts.simps(5)
thf(fact_507_simplification_Ofv__simp__Disj__same,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q1: relational_fmla_a_b,X5: set_nat,Q2: relational_fmla_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( ( relational_fv_a_b @ ( Simp @ Q1 ) )
= X5 )
=> ( ( ( relational_fv_a_b @ ( Simp @ Q2 ) )
= X5 )
=> ( ( relational_fv_a_b @ ( Simp @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) )
= X5 ) ) ) ) ).
% simplification.fv_simp_Disj_same
thf(fact_508_sr__Disj,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ Q1 )
= ( relational_fv_a_b @ Q2 ) )
=> ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_sr_a_b @ Q1 )
& ( relational_sr_a_b @ Q2 ) ) ) ) ).
% sr_Disj
thf(fact_509_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_510_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_511_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_512_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_513_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_514_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_515_cov_ODisjL,axiom,
! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q1 @ G )
=> ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
= ( relational_Bool_a_b @ $true ) )
=> ( relational_cov_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).
% cov.DisjL
thf(fact_516_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_517_Int__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( inf_in8483230781156617063la_a_b @ A @ B ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
& ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_518_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_519_IntI,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ ( inf_in8483230781156617063la_a_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_520_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_521_Un__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
| ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% Un_iff
thf(fact_522_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_523_UnCI,axiom,
! [C: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ~ ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ A ) )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnCI
thf(fact_524_sup_Obounded__iff,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B3 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_525_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_526_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_527_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_528_inf_Obounded__iff,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B3 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_529_inf_Obounded__iff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_530_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_531_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_532_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_533_IntD2,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( inf_in8483230781156617063la_a_b @ A @ B ) )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ).
% IntD2
thf(fact_534_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_535_IntD1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( inf_in8483230781156617063la_a_b @ A @ B ) )
=> ( member4680049679412964150la_a_b @ C @ A ) ) ).
% IntD1
thf(fact_536_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_537_UnI2,axiom,
! [C: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnI2
thf(fact_538_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_539_UnI1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) ) ) ).
% UnI1
thf(fact_540_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_541_IntE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( inf_in8483230781156617063la_a_b @ A @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C @ A )
=> ~ ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% IntE
thf(fact_542_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_543_UnE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( sup_su5130108678486352897la_a_b @ A @ B ) )
=> ( ~ ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% UnE
thf(fact_544_sup_OcoboundedI2,axiom,
! [C: set_nat,B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_545_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_546_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_547_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_548_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_549_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_550_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_551_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_552_sup_Ocobounded2,axiom,
! [B3: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_553_sup_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_554_sup_Ocobounded1,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_555_sup_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_556_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_557_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_558_sup_OboundedI,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_559_sup_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_560_sup_OboundedE,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_561_sup_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_562_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_563_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_564_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_565_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_566_sup_Oabsorb2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( sup_sup_set_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_567_sup_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_568_sup_Oabsorb1,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_569_sup_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_570_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_571_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_572_sup_OorderI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B3 ) )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_573_sup_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_574_sup_OorderE,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_575_sup_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_576_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_577_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_578_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_579_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_580_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_581_sup__mono,axiom,
! [A2: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_582_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_583_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_584_le__supI2,axiom,
! [X: set_nat,B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B3 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_585_le__supI2,axiom,
! [X: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_586_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_587_le__supI1,axiom,
! [X: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_588_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_589_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_590_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_591_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_592_le__supI,axiom,
! [A2: set_nat,X: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B3 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_593_le__supI,axiom,
! [A2: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_594_le__supE,axiom,
! [A2: set_nat,B3: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B3 @ X ) ) ) ).
% le_supE
thf(fact_595_le__supE,axiom,
! [A2: nat,B3: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B3 @ X ) ) ) ).
% le_supE
thf(fact_596_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_597_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_598_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_599_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_600_inf_OcoboundedI2,axiom,
! [B3: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_601_inf_OcoboundedI2,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_602_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_603_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_604_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_605_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_606_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_607_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_608_inf_Ocobounded2,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_609_inf_Ocobounded2,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_610_inf_Ocobounded1,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_611_inf_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_612_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_613_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_614_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_615_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_616_inf_OboundedI,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_617_inf_OboundedI,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_618_inf_OboundedE,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_619_inf_OboundedE,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_620_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_621_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_622_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_623_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_624_inf_Oabsorb2,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_625_inf_Oabsorb2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_626_inf_Oabsorb1,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_627_inf_Oabsorb1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_628_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_629_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_630_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_631_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_632_inf_OorderI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_633_inf_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_634_inf_OorderE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_635_inf_OorderE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2
= ( inf_inf_nat @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_636_le__infI2,axiom,
! [B3: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_637_le__infI2,axiom,
! [B3: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_638_le__infI1,axiom,
! [A2: set_nat,X: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_639_le__infI1,axiom,
! [A2: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_640_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B3: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B3 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_641_inf__mono,axiom,
! [A2: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_642_le__infI,axiom,
! [X: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ B3 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_643_le__infI,axiom,
! [X: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_644_le__infE,axiom,
! [X: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A2 )
=> ~ ( ord_less_eq_set_nat @ X @ B3 ) ) ) ).
% le_infE
thf(fact_645_le__infE,axiom,
! [X: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B3 ) ) ) ).
% le_infE
thf(fact_646_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_647_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_648_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_649_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_650_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_651_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_652_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_653_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_654_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_655_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_656_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z2 ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_657_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_658_DISJ__insert__reorder_H,axiom,
! [Q3: set_Re381260168593705685la_a_b,Q4: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ( finite5600759454172676150la_a_b @ Q4 )
=> ( relational_equiv_a_b @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ ( relational_DISJ_a_b @ Q4 ) @ Q2 ) @ Q3 ) ) @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ Q2 @ ( sup_su5130108678486352897la_a_b @ Q4 @ Q3 ) ) ) ) ) ) ).
% DISJ_insert_reorder'
thf(fact_659_Inf__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_660_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_661_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_662_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_663_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_664_Diff__iff,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
= ( ( member4680049679412964150la_a_b @ C @ A )
& ~ ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_665_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_666_DiffI,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ A )
=> ( ~ ( member4680049679412964150la_a_b @ C @ B )
=> ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ) ).
% DiffI
thf(fact_667_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_668_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_669_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_670_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_671_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_672_insert__Diff1,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_673_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_674_Diff__insert0,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_675_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_676_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_677_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_678_fv__exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_exists
thf(fact_679_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_680_DiffD2,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ~ ( member4680049679412964150la_a_b @ C @ B ) ) ).
% DiffD2
thf(fact_681_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_682_DiffD1,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ( member4680049679412964150la_a_b @ C @ A ) ) ).
% DiffD1
thf(fact_683_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_684_DiffE,axiom,
! [C: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ C @ ( minus_4077726661957047470la_a_b @ A @ B ) )
=> ~ ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% DiffE
thf(fact_685_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_686_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_687_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_688_insert__Diff__if,axiom,
! [X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b,A: set_Re381260168593705685la_a_b] :
( ( ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B )
= ( minus_4077726661957047470la_a_b @ A @ B ) ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ B )
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_689_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_690_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_691_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_692_equiv__Disj__cong,axiom,
! [Q1: relational_fmla_a_b,Q12: relational_fmla_a_b,Q2: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q12 )
=> ( ( relational_equiv_a_b @ Q2 @ Q22 )
=> ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) ) ).
% equiv_Disj_cong
thf(fact_693_equiv__Disj__Assoc,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,Q32: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ Q32 ) @ ( relational_Disj_a_b @ Q1 @ ( relational_Disj_a_b @ Q2 @ Q32 ) ) ) ).
% equiv_Disj_Assoc
thf(fact_694_equiv__Conj__cong,axiom,
! [Q1: relational_fmla_a_b,Q12: relational_fmla_a_b,Q2: relational_fmla_a_b,Q22: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q1 @ Q12 )
=> ( ( relational_equiv_a_b @ Q2 @ Q22 )
=> ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) ) ).
% equiv_Conj_cong
thf(fact_695_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_696_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_697_Diff__insert__absorb,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( minus_4077726661957047470la_a_b @ ( insert7010464514620295119la_a_b @ X @ A ) @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_698_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_699_insert__Diff,axiom,
! [A2: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( ( insert7010464514620295119la_a_b @ A2 @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ A2 @ bot_bo4495933725496725865la_a_b ) ) )
= A ) ) ).
% insert_Diff
thf(fact_700_subset__Diff__insert,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,C2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B @ ( insert7010464514620295119la_a_b @ X @ C2 ) ) )
= ( ( ord_le4112832032246704949la_a_b @ A @ ( minus_4077726661957047470la_a_b @ B @ C2 ) )
& ~ ( member4680049679412964150la_a_b @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_701_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_702_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_703_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_704_finite__empty__induct,axiom,
! [A: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ A )
=> ( ( P @ A )
=> ( ! [A5: relational_fmla_a_b,A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( member4680049679412964150la_a_b @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ A5 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) )
=> ( P @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% finite_empty_induct
thf(fact_705_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_706_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_707_infinite__coinduct,axiom,
! [X5: set_a > $o,A: set_a] :
( ( X5 @ A )
=> ( ! [A6: set_a] :
( ( X5 @ A6 )
=> ? [X6: a] :
( ( member_a @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_708_infinite__coinduct,axiom,
! [X5: set_nat > $o,A: set_nat] :
( ( X5 @ A )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_709_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_710_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_711_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_712_subset__insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_713_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_714_Relational__Calculus_Oequiv__def,axiom,
( relational_equiv_a_b
= ( ^ [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
! [I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( ( finite_finite_a @ ( relational_adom_b_a @ I2 ) )
=> ( ( relational_sat_a_b @ Q13 @ I2 @ Sigma3 )
= ( relational_sat_a_b @ Q23 @ I2 @ Sigma3 ) ) ) ) ) ).
% Relational_Calculus.equiv_def
thf(fact_715_remove__induct,axiom,
! [P: set_Re381260168593705685la_a_b > $o,B: set_Re381260168593705685la_a_b] :
( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ( ~ ( finite5600759454172676150la_a_b @ B )
=> ( P @ B ) )
=> ( ! [A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( A6 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A6 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_716_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_717_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_718_finite__remove__induct,axiom,
! [B: set_Re381260168593705685la_a_b,P: set_Re381260168593705685la_a_b > $o] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( P @ bot_bo4495933725496725865la_a_b )
=> ( ! [A6: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ A6 )
=> ( ( A6 != bot_bo4495933725496725865la_a_b )
=> ( ( ord_le4112832032246704949la_a_b @ A6 @ B )
=> ( ! [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ A6 )
=> ( P @ ( minus_4077726661957047470la_a_b @ A6 @ ( insert7010464514620295119la_a_b @ X6 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_719_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X6 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_720_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_721_cov__fv__aux,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member4680049679412964150la_a_b @ Qqp @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
& ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).
% cov_fv_aux
thf(fact_722_fv__erase,axiom,
! [Q: relational_fmla_a_b,X: nat] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_erase_a_b @ Q @ X ) ) @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% fv_erase
thf(fact_723_DISJ__insert__reorder,axiom,
! [Q3: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( relational_equiv_a_b @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ Q3 ) ) @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ Q2 @ ( insert7010464514620295119la_a_b @ Q1 @ Q3 ) ) ) ) ) ).
% DISJ_insert_reorder
thf(fact_724_DISJ__push__in,axiom,
! [Q3: set_Re381260168593705685la_a_b,Q: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q @ ( relational_DISJ_a_b @ Q3 ) ) @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ Q @ Q3 ) ) ) ) ).
% DISJ_push_in
thf(fact_725_DISJ__insert,axiom,
! [X5: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( finite5600759454172676150la_a_b @ X5 )
=> ( relational_equiv_a_b @ ( relational_DISJ_a_b @ ( insert7010464514620295119la_a_b @ X @ X5 ) ) @ ( relational_Disj_a_b @ X @ ( relational_DISJ_a_b @ X5 ) ) ) ) ).
% DISJ_insert
thf(fact_726_DISJ__union,axiom,
! [X5: set_Re381260168593705685la_a_b,Y7: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ X5 )
=> ( ( finite5600759454172676150la_a_b @ Y7 )
=> ( relational_equiv_a_b @ ( relational_DISJ_a_b @ ( sup_su5130108678486352897la_a_b @ X5 @ Y7 ) ) @ ( relational_Disj_a_b @ ( relational_DISJ_a_b @ X5 ) @ ( relational_DISJ_a_b @ Y7 ) ) ) ) ) ).
% DISJ_union
thf(fact_727_insert__remove__id,axiom,
! [X: nat,X5: set_nat] :
( ( member_nat @ X @ X5 )
=> ( X5
= ( insert_nat @ X @ ( minus_minus_set_nat @ X5 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% insert_remove_id
thf(fact_728_insert__remove__id,axiom,
! [X: relational_fmla_a_b,X5: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ X5 )
=> ( X5
= ( insert7010464514620295119la_a_b @ X @ ( minus_4077726661957047470la_a_b @ X5 @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% insert_remove_id
thf(fact_729_equiv__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ( relational_fv_a_b @ Q )
= ( relational_fv_a_b @ Q5 ) )
=> ( ( relational_equiv_a_b @ Q @ Q5 )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).
% equiv_eval_eqI
thf(fact_730_finite__eval__Disj2D,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) )
=> ( finite_finite_list_a @ ( relational_eval_a_b @ Q2 @ I ) ) ) ).
% finite_eval_Disj2D
thf(fact_731_equiv__eval__on__eval__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q5 ) )
=> ( ( relational_equiv_a_b @ Q @ Q5 )
=> ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q5 ) @ Q @ I )
= ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).
% equiv_eval_on_eval_eqI
thf(fact_732_equiv__eval__on__eqI,axiom,
! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b,X5: set_nat] :
( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ( relational_equiv_a_b @ Q @ Q5 )
=> ( ( relati8814510239606734169on_a_b @ X5 @ Q @ I )
= ( relati8814510239606734169on_a_b @ X5 @ Q5 @ I ) ) ) ) ).
% equiv_eval_on_eqI
thf(fact_733_infinite__eval__Disj2,axiom,
! [Q2: relational_fmla_a_b,Q1: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( ord_less_set_nat @ ( relational_fv_a_b @ Q2 ) @ ( relational_fv_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) )
=> ( ( relational_sat_a_b @ Q2 @ I @ Sigma )
=> ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) ) ) ) ).
% infinite_eval_Disj2
thf(fact_734_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_735_member__remove,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ X @ ( remove4261432235257513082la_a_b @ Y @ A ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_736_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_737_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_738_psubset__imp__ex__mem,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B )
=> ? [B5: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ B5 @ ( minus_4077726661957047470la_a_b @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_739_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_740_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_741_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_742_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_743_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_744_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_745_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_746_order__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_747_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_748_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_749_ord__less__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_750_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_751_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_752_order__less__asym_H,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order_less_asym'
thf(fact_753_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_754_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_755_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_756_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_757_psubsetD,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_Re381260168593705685la_a_b,C: relational_fmla_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ B )
=> ( ( member4680049679412964150la_a_b @ C @ A )
=> ( member4680049679412964150la_a_b @ C @ B ) ) ) ).
% psubsetD
thf(fact_758_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( A2 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_759_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( A2 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_760_dual__order_Ostrict__trans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_761_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_762_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_763_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_764_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_765_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_766_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_767_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_768_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_769_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_770_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_771_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_772_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_773_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_774_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_775_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_776_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_777_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_778_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_779_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_780_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_781_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_782_finite__psubset__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B8: set_a] :
( ( ord_less_set_a @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_783_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A6 )
=> ( P @ B8 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_784_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_785_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_786_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A9: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A9 ) )
= ( ord_less_nat @ A9 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_787_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_788_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_789_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_790_nless__le,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_791_nless__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A2 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ).
% nless_le
thf(fact_792_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_793_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_794_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_795_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_796_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_797_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_798_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_799_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_800_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_801_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_802_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_803_order_Ostrict__trans1,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_804_order_Ostrict__trans1,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_805_order_Ostrict__trans2,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_806_order_Ostrict__trans2,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_807_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_808_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_809_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_810_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_811_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_812_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_813_dual__order_Ostrict__trans1,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_814_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_815_dual__order_Ostrict__trans2,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_816_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_817_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_818_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_819_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_820_order_Ostrict__implies__order,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_821_dual__order_Ostrict__implies__order,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B3 @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_822_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_823_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_824_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_825_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_826_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_827_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_828_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_829_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_830_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_831_order__le__neq__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_832_order__le__neq__trans,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_833_order__neq__le__trans,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_834_order__neq__le__trans,axiom,
! [A2: nat,B3: nat] :
( ( A2 != B3 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ord_less_nat @ A2 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_835_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_836_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_837_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_838_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_839_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_840_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_841_order__le__less__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_842_order__le__less__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_843_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_844_order__le__less__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_845_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_846_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_847_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_848_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_849_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_850_order__less__le__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_851_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_852_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_853_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_854_ex__min__if__finite,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ S )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ S )
& ( ord_less_a @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_855_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_856_infinite__growing,axiom,
! [X5: set_Re381260168593705685la_a_b] :
( ( X5 != bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ X5 )
=> ? [Xa: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Xa @ X5 )
& ( ord_le8964290712037365747la_a_b @ X2 @ Xa ) ) )
=> ~ ( finite5600759454172676150la_a_b @ X5 ) ) ) ).
% infinite_growing
thf(fact_857_infinite__growing,axiom,
! [X5: set_a] :
( ( X5 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X5 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X5 )
& ( ord_less_a @ X2 @ Xa ) ) )
=> ~ ( finite_finite_a @ X5 ) ) ) ).
% infinite_growing
thf(fact_858_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_859_finite__linorder__min__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( ord_less_a @ B5 @ X6 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_a @ B5 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_860_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( ord_less_nat @ B5 @ X6 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_861_finite__linorder__max__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [B5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [X6: a] :
( ( member_a @ X6 @ A6 )
=> ( ord_less_a @ X6 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_a @ B5 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_862_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( ord_less_nat @ X6 @ B5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B5 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_863_infinite__eval__on__Disj2,axiom,
! [Q2: relational_fmla_a_b,X5: set_nat,Q1: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( ord_less_set_nat @ ( relational_fv_a_b @ Q2 ) @ X5 )
=> ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q1 ) @ X5 )
=> ( ( finite_finite_nat @ X5 )
=> ( ( relational_sat_a_b @ Q2 @ I @ Sigma )
=> ~ ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) ) ) ) ) ) ).
% infinite_eval_on_Disj2
thf(fact_864_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X6: a] :
( ( member_a @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_865_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_866_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T4: set_a] :
( ( ord_less_set_a @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X6: a] :
( ( member_a @ X6 @ ( minus_minus_set_a @ S @ T4 ) )
& ( P @ ( insert_a @ X6 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_867_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T4 ) )
& ( P @ ( insert_nat @ X6 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_868_psubset__insert__iff,axiom,
! [A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ( ord_le7152733262289451305la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ B ) )
= ( ( ( member4680049679412964150la_a_b @ X @ B )
=> ( ord_le7152733262289451305la_a_b @ A @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ B )
=> ( ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le7152733262289451305la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) )
& ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ord_le4112832032246704949la_a_b @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_869_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_870_finite__eval__on__Disj2D,axiom,
! [X5: set_nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) )
=> ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X5 @ Q2 @ I ) ) ) ) ).
% finite_eval_on_Disj2D
thf(fact_871_Relational__Calculus_Oeval__def,axiom,
( relational_eval_a_b
= ( ^ [Q6: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q6 ) @ Q6 ) ) ) ).
% Relational_Calculus.eval_def
thf(fact_872_minf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% minf(8)
thf(fact_873_minf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% minf(6)
thf(fact_874_pinf_I8_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T3 @ X6 ) ) ).
% pinf(8)
thf(fact_875_pinf_I6_J,axiom,
! [T3: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T3 ) ) ).
% pinf(6)
thf(fact_876_complete__interval,axiom,
! [A2: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B3 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A2 @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_877_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_878_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_879_fv_Osimps_I7_J,axiom,
! [Z2: nat,Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Phi ) )
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) ).
% fv.simps(7)
thf(fact_880_rrb__simps_I7_J,axiom,
! [Y: nat,Qy: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relati591517084277583526ts_a_b @ Y @ Qy ) )
= ( ? [X4: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy @ X4 )
& ( relational_rrb_a_b @ Qy ) ) ) ).
% rrb_simps(7)
thf(fact_881_erase_Osimps_I7_J,axiom,
! [X: nat,Z2: nat,Q: relational_fmla_a_b] :
( ( ( X = Z2 )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ X @ Q ) ) )
& ( ( X != Z2 )
=> ( ( relational_erase_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q ) @ X )
= ( relati591517084277583526ts_a_b @ Z2 @ ( relational_erase_a_b @ Q @ X ) ) ) ) ) ).
% erase.simps(7)
thf(fact_882_fmla_Odistinct_I39_J,axiom,
! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Conj_a_b @ X51 @ X52 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(39)
thf(fact_883_fmla_Odistinct_I41_J,axiom,
! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Disj_a_b @ X61 @ X62 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(41)
thf(fact_884_equiv__Exists__Disj,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ Q1 ) @ ( relati591517084277583526ts_a_b @ X @ Q2 ) ) ) ).
% equiv_Exists_Disj
thf(fact_885_exists__Exists,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relati3989891337220013914ts_a_b @ X @ Q )
= ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ).
% exists_Exists
thf(fact_886_exists__def,axiom,
( relati3989891337220013914ts_a_b
= ( ^ [X3: nat,Q6: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) ) @ ( relati591517084277583526ts_a_b @ X3 @ Q6 ) @ Q6 ) ) ) ).
% exists_def
thf(fact_887_Exists__nonfree__equiv,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ).
% Exists_nonfree_equiv
thf(fact_888_qp__ExistsE,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
=> ~ ( ( relational_qp_a_b @ Q )
=> ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).
% qp_ExistsE
thf(fact_889_qp__Exists,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_qp_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ) ).
% qp_Exists
thf(fact_890_DISJ__exists__pull__out,axiom,
! [Q3: set_Re381260168593705685la_a_b,Q: relational_fmla_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( ( member4680049679412964150la_a_b @ Q @ Q3 )
=> ( relational_equiv_a_b @ ( relational_DISJ_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ Q3 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ ( relational_DISJ_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ ( minus_4077726661957047470la_a_b @ Q3 @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ).
% DISJ_exists_pull_out
thf(fact_891_Exists__cp__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_cp_a_b @ ( relational_DISJ_a_b @ Q3 ) ) ) @ ( relational_DISJ_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ Q3 ) ) ) ) ).
% Exists_cp_DISJ
thf(fact_892_Exists__DISJ,axiom,
! [Q3: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ Q3 )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ ( relational_DISJ_a_b @ Q3 ) ) @ ( relational_DISJ_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ X ) @ Q3 ) ) ) ) ).
% Exists_DISJ
thf(fact_893_image__eqI,axiom,
! [B3: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B3
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_894_image__eqI,axiom,
! [B3: relational_fmla_a_b,F: nat > relational_fmla_a_b,X: nat,A: set_nat] :
( ( B3
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member4680049679412964150la_a_b @ B3 @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_895_image__eqI,axiom,
! [B3: nat,F: relational_fmla_a_b > nat,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B3
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member_nat @ B3 @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_896_image__eqI,axiom,
! [B3: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( B3
= ( F @ X ) )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ B3 @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_897_image__is__empty,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat] :
( ( ( image_4386371547000553590la_a_b @ F @ A )
= bot_bo4495933725496725865la_a_b )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_898_empty__is__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat] :
( ( bot_bo4495933725496725865la_a_b
= ( image_4386371547000553590la_a_b @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_899_image__empty,axiom,
! [F: nat > relational_fmla_a_b] :
( ( image_4386371547000553590la_a_b @ F @ bot_bot_set_nat )
= bot_bo4495933725496725865la_a_b ) ).
% image_empty
thf(fact_900_finite__imageI,axiom,
! [F3: set_a,H: a > a] :
( ( finite_finite_a @ F3 )
=> ( finite_finite_a @ ( image_a_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_901_finite__imageI,axiom,
! [F3: set_a,H: a > nat] :
( ( finite_finite_a @ F3 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_902_finite__imageI,axiom,
! [F3: set_nat,H: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ F3 )
=> ( finite5600759454172676150la_a_b @ ( image_4386371547000553590la_a_b @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_903_finite__imageI,axiom,
! [F3: set_nat,H: nat > a] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_904_finite__imageI,axiom,
! [F3: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_905_image__insert,axiom,
! [F: nat > relational_fmla_a_b,A2: nat,B: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( insert_nat @ A2 @ B ) )
= ( insert7010464514620295119la_a_b @ ( F @ A2 ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_906_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( ( insert7010464514620295119la_a_b @ ( F @ X ) @ ( image_4386371547000553590la_a_b @ F @ A ) )
= ( image_4386371547000553590la_a_b @ F @ A ) ) ) ).
% insert_image
thf(fact_907_qps__exists,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_qps_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) )
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( relational_qps_a_b @ G ) ) ) ) ).
% qps_exists
thf(fact_908_subset__image__iff,axiom,
! [B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_4386371547000553590la_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_909_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_910_image__subset__iff,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,B: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_911_subset__imageE,axiom,
! [B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_4386371547000553590la_a_b @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_912_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_913_image__subsetI,axiom,
! [A: set_nat,F: nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_914_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_6790371041703824709la_a_b @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_915_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_916_image__subsetI,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat,B: set_nat] :
( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_341122591648980342_b_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_917_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > relational_fmla_a_b] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_918_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_919_all__subset__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [B2: set_Re381260168593705685la_a_b] :
( ( ord_le4112832032246704949la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_4386371547000553590la_a_b @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_920_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_921_image__Un,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,B: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su5130108678486352897la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_Un
thf(fact_922_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_923_imageI,axiom,
! [X: nat,A: set_nat,F: nat > relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_924_imageI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ).
% imageI
thf(fact_925_imageI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( member4680049679412964150la_a_b @ ( F @ X ) @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ).
% imageI
thf(fact_926_image__iff,axiom,
! [Z2: relational_fmla_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( member4680049679412964150la_a_b @ Z2 @ ( image_4386371547000553590la_a_b @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_927_bex__imageD,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: relational_fmla_a_b > $o] :
( ? [X6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X6 @ ( image_4386371547000553590la_a_b @ F @ A ) )
& ( P @ X6 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_928_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > relational_fmla_a_b,G3: nat > relational_fmla_a_b] :
( ( M2 = N3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) )
=> ( ( image_4386371547000553590la_a_b @ F @ M2 )
= ( image_4386371547000553590la_a_b @ G3 @ N3 ) ) ) ) ).
% image_cong
thf(fact_929_ball__imageD,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: relational_fmla_a_b > $o] :
( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_930_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B3: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B3
= ( F @ X ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_931_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B3: relational_fmla_a_b,F: nat > relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( ( B3
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B3 @ ( image_4386371547000553590la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_932_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B3: nat,F: relational_fmla_a_b > nat] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( B3
= ( F @ X ) )
=> ( member_nat @ B3 @ ( image_341122591648980342_b_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_933_rev__image__eqI,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,B3: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( B3
= ( F @ X ) )
=> ( member4680049679412964150la_a_b @ B3 @ ( image_6790371041703824709la_a_b @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_934_infinite__surj,axiom,
! [A: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,B: set_nat] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( ord_le4112832032246704949la_a_b @ A @ ( image_4386371547000553590la_a_b @ F @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_935_infinite__surj,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_a_a @ F @ B ) )
=> ~ ( finite_finite_a @ B ) ) ) ).
% infinite_surj
thf(fact_936_infinite__surj,axiom,
! [A: set_a,F: nat > a,B: set_nat] :
( ~ ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( image_nat_a @ F @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_937_infinite__surj,axiom,
! [A: set_nat,F: a > nat,B: set_a] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_a_nat @ F @ B ) )
=> ~ ( finite_finite_a @ B ) ) ) ).
% infinite_surj
thf(fact_938_infinite__surj,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ B ) )
=> ~ ( finite_finite_nat @ B ) ) ) ).
% infinite_surj
thf(fact_939_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) )
=> ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_940_all__finite__subset__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: set_Re381260168593705685la_a_b > $o] :
( ( ! [B2: set_Re381260168593705685la_a_b] :
( ( ( finite5600759454172676150la_a_b @ B2 )
& ( ord_le4112832032246704949la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_4386371547000553590la_a_b @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_941_all__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_942_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) )
=> ( P @ ( image_a_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_943_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_944_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A )
& ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_945_ex__finite__subset__image,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: set_Re381260168593705685la_a_b > $o] :
( ( ? [B2: set_Re381260168593705685la_a_b] :
( ( finite5600759454172676150la_a_b @ B2 )
& ( ord_le4112832032246704949la_a_b @ B2 @ ( image_4386371547000553590la_a_b @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_4386371547000553590la_a_b @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_946_ex__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_947_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A )
& ( P @ ( image_a_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_948_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_949_finite__subset__image,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_950_finite__subset__image,axiom,
! [B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( finite5600759454172676150la_a_b @ B )
=> ( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_4386371547000553590la_a_b @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_951_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_a @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_952_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ? [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A )
& ( finite_finite_a @ C5 )
& ( B
= ( image_a_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_953_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_954_finite__surj,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_955_finite__surj,axiom,
! [A: set_nat,B: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b] :
( ( finite_finite_nat @ A )
=> ( ( ord_le4112832032246704949la_a_b @ B @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ( finite5600759454172676150la_a_b @ B ) ) ) ).
% finite_surj
thf(fact_956_finite__surj,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_957_finite__surj,axiom,
! [A: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_958_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_959_image__Int__subset,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,B: set_nat] : ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_in8483230781156617063la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_960_image__diff__subset,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,B: set_nat] : ( ord_le4112832032246704949la_a_b @ ( minus_4077726661957047470la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) @ ( image_4386371547000553590la_a_b @ F @ B ) ) @ ( image_4386371547000553590la_a_b @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_961_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > relational_fmla_a_b,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_el6350558617753882986la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_962_gen_Ointros_I10_J,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).
% gen.intros(10)
thf(fact_963_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X2: nat,Y2: nat] :
( ( H @ ( sup_sup_nat @ X2 @ Y2 ) )
= ( sup_sup_nat @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N3 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_964_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X2: nat,Y2: nat] :
( ( H @ ( inf_inf_nat @ X2 @ Y2 ) )
= ( inf_inf_nat @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N3 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_965_eqs__exists,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_eqs_a_b @ X @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) )
= ( minus_minus_set_nat @ ( relational_eqs_a_b @ X @ G ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ).
% eqs_exists
thf(fact_966_qp__impl_Osimps_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_qp_a_b @ Q ) ) ) ).
% qp_impl.simps(3)
thf(fact_967_Exists__in__sub__cp__DISJ,axiom,
! [X: nat,Q5: relational_fmla_a_b,Q3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( relational_DISJ_a_b @ Q3 ) ) ) )
=> ( ( finite5600759454172676150la_a_b @ Q3 )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X2 ) ) ) ) ) ) ).
% Exists_in_sub_cp_DISJ
thf(fact_968_gen__eqs,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Z2: nat] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( ( relational_eqs_a_b @ Z2 @ G )
= bot_bot_set_nat ) ) ).
% gen_eqs
thf(fact_969_eqs__empty,axiom,
! [X: nat] :
( ( relational_eqs_a_b @ X @ bot_bo4495933725496725865la_a_b )
= bot_bot_set_nat ) ).
% eqs_empty
thf(fact_970_eqs__union,axiom,
! [X: nat,X5: set_Re381260168593705685la_a_b,Y7: set_Re381260168593705685la_a_b] :
( ( relational_eqs_a_b @ X @ ( sup_su5130108678486352897la_a_b @ X5 @ Y7 ) )
= ( sup_sup_set_nat @ ( relational_eqs_a_b @ X @ X5 ) @ ( relational_eqs_a_b @ X @ Y7 ) ) ) ).
% eqs_union
thf(fact_971_cpropagated__sub,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member4680049679412964150la_a_b @ Q5 @ ( relational_sub_a_b @ Q ) )
=> ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).
% cpropagated_sub
thf(fact_972_qp__impl_Osimps_I8_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V @ Va ) ) ).
% qp_impl.simps(8)
thf(fact_973_qp__impl_Osimps_I7_J,axiom,
! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).
% qp_impl.simps(7)
thf(fact_974_simplification_Osimplified__sub,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( Simplified @ Q )
=> ( ( member4680049679412964150la_a_b @ Q5 @ ( relational_sub_a_b @ Q ) )
=> ( Simplified @ Q5 ) ) ) ) ).
% simplification.simplified_sub
thf(fact_975_eqs__noteq,axiom,
! [Y: nat,X: nat,Q: set_Re381260168593705685la_a_b] :
( ( member_nat @ Y @ ( relational_eqs_a_b @ X @ Q ) )
=> ( X != Y ) ) ).
% eqs_noteq
thf(fact_976_not__self__eqs,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b] :
~ ( member_nat @ X @ ( relational_eqs_a_b @ X @ G ) ) ).
% not_self_eqs
thf(fact_977_finite__eqs,axiom,
! [G: set_Re381260168593705685la_a_b,X: nat] :
( ( finite5600759454172676150la_a_b @ G )
=> ( finite_finite_nat @ ( relational_eqs_a_b @ X @ G ) ) ) ).
% finite_eqs
thf(fact_978_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B )
=> ~ ( member_nat @ X @ C5 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_979_in__image__insert__iff,axiom,
! [B: set_se6865892389300016395la_a_b,X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ! [C5: set_Re381260168593705685la_a_b] :
( ( member3481406638322139244la_a_b @ C5 @ B )
=> ~ ( member4680049679412964150la_a_b @ X @ C5 ) )
=> ( ( member3481406638322139244la_a_b @ A @ ( image_7051608999182166449la_a_b @ ( insert7010464514620295119la_a_b @ X ) @ B ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( member3481406638322139244la_a_b @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_980_sub_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ).
% sub.simps(6)
thf(fact_981_sub_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ).
% sub.simps(5)
thf(fact_982_rrb__def,axiom,
( relational_rrb_a_b
= ( ^ [Q6: relational_fmla_a_b] :
! [Y4: nat,Qy2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ Y4 @ Qy2 ) @ ( relational_sub_a_b @ Q6 ) )
=> ? [X4: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y4 @ Qy2 @ X4 ) ) ) ) ).
% rrb_def
thf(fact_983_Exists__in__sub__DISJ,axiom,
! [X: nat,Q5: relational_fmla_a_b,Q3: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ ( relational_DISJ_a_b @ Q3 ) ) )
=> ( ( finite5600759454172676150la_a_b @ Q3 )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ Q3 )
& ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X @ Q5 ) @ ( relational_sub_a_b @ X2 ) ) ) ) ) ).
% Exists_in_sub_DISJ
thf(fact_984_simplification__def,axiom,
( relati2910603115655104169on_a_b
= ( ^ [Simp2: relational_fmla_a_b > relational_fmla_a_b,Simplified2: relational_fmla_a_b > $o] :
( ! [Q6: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( ( relational_sat_a_b @ ( Simp2 @ Q6 ) @ I2 @ Sigma3 )
= ( relational_sat_a_b @ Q6 @ I2 @ Sigma3 ) )
& ! [Q6: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( Simp2 @ Q6 ) ) @ ( relational_fv_a_b @ Q6 ) )
& ! [Q6: relational_fmla_a_b] :
( ( relational_rrb_a_b @ Q6 )
=> ( relational_rrb_a_b @ ( Simp2 @ Q6 ) ) )
& ! [X3: nat,Q6: relational_fmla_a_b] :
( ? [X4: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ Q6 @ X4 )
=> ? [X4: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ ( Simp2 @ Q6 ) @ X4 ) )
& ! [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ ( Simp2 @ Q23 ) )
= ( relational_fv_a_b @ ( Simp2 @ Q13 ) ) )
=> ( ( relational_fv_a_b @ ( Simp2 @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
= ( relational_fv_a_b @ ( Simp2 @ Q13 ) ) ) )
& ( ( Simp2 @ ( relational_Bool_a_b @ $false ) )
= ( relational_Bool_a_b @ $false ) )
& ! [Q6: relational_fmla_a_b] :
( ( Simplified2 @ Q6 )
=> ! [Q7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q7 @ ( relational_sub_a_b @ Q6 ) )
=> ( Simplified2 @ Q7 ) ) )
& ! [Q6: relational_fmla_a_b] :
( ( Simplified2 @ Q6 )
=> ! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q6 ) )
| ( member_nat @ Y4 @ ( relational_fv_a_b @ Q6 ) ) )
=> ( Simplified2 @ ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y4 ) ) ) ) ) ) )
& ! [Q6: relational_fmla_a_b] :
( ( Simplified2 @ Q6 )
=> ( ( relational_fv_a_b @ ( Simp2 @ Q6 ) )
= ( relational_fv_a_b @ Q6 ) ) )
& ! [Q6: relational_fmla_a_b] : ( Simplified2 @ ( Simp2 @ Q6 ) )
& ! [Q6: relational_fmla_a_b] : ( Simplified2 @ ( relational_cp_a_b @ Q6 ) ) ) ) ) ).
% simplification_def
thf(fact_985_simplification_Ointro,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o] :
( ! [Q8: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( ( relational_sat_a_b @ ( Simp @ Q8 ) @ I3 @ Sigma4 )
= ( relational_sat_a_b @ Q8 @ I3 @ Sigma4 ) )
=> ( ! [Q8: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( Simp @ Q8 ) ) @ ( relational_fv_a_b @ Q8 ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( relational_rrb_a_b @ Q8 )
=> ( relational_rrb_a_b @ ( Simp @ Q8 ) ) )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X2 @ Q8 @ X_1 )
=> ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X2 @ ( Simp @ Q8 ) @ X_12 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( ( relational_fv_a_b @ ( Simp @ Q24 ) )
= ( relational_fv_a_b @ ( Simp @ Q14 ) ) )
=> ( ( relational_fv_a_b @ ( Simp @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) )
= ( relational_fv_a_b @ ( Simp @ Q14 ) ) ) )
=> ( ( ( Simp @ ( relational_Bool_a_b @ $false ) )
= ( relational_Bool_a_b @ $false ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( Simplified @ Q8 )
=> ! [Q9: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q9 @ ( relational_sub_a_b @ Q8 ) )
=> ( Simplified @ Q9 ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( Simplified @ Q8 )
=> ! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
| ( member_nat @ Y2 @ ( relational_fv_a_b @ Q8 ) ) )
=> ( Simplified @ ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y2 ) ) ) ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( Simplified @ Q8 )
=> ( ( relational_fv_a_b @ ( Simp @ Q8 ) )
= ( relational_fv_a_b @ Q8 ) ) )
=> ( ! [Q8: relational_fmla_a_b] : ( Simplified @ ( Simp @ Q8 ) )
=> ( ! [Q8: relational_fmla_a_b] : ( Simplified @ ( relational_cp_a_b @ Q8 ) )
=> ( relati2910603115655104169on_a_b @ Simp @ Simplified ) ) ) ) ) ) ) ) ) ) ) ) ).
% simplification.intro
thf(fact_986_semilattice__order__set_Osubset__imp,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,B: set_a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != bot_bot_set_a )
=> ( ( finite_finite_a @ B )
=> ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_987_semilattice__order__set_Osubset__imp,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,B: set_nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_988_fmla_Oinject_I3_J,axiom,
! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
( ( ( relational_Eq_a_b @ X31 @ X32 )
= ( relational_Eq_a_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fmla.inject(3)
thf(fact_989_term_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( relational_Var_a @ X22 )
= ( relational_Var_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% term.inject(2)
thf(fact_990_rrb__simps_I3_J,axiom,
! [X: nat,T3: relational_term_a] : ( relational_rrb_a_b @ ( relational_Eq_a_b @ X @ T3 ) ) ).
% rrb_simps(3)
thf(fact_991_cpropagated__simps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( T3
!= ( relational_Var_a @ X ) ) ) ).
% cpropagated_simps(3)
thf(fact_992_qps__minus,axiom,
! [G: set_Re381260168593705685la_a_b,X: nat,Y: nat] :
( ( relational_qps_a_b @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
= ( relational_qps_a_b @ G ) ) ).
% qps_minus
thf(fact_993_eqs__minus,axiom,
! [X: nat,G: set_Re381260168593705685la_a_b,Y: nat] :
( ( relational_eqs_a_b @ X @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
= ( minus_minus_set_nat @ ( relational_eqs_a_b @ X @ G ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ).
% eqs_minus
thf(fact_994_eqs__insert_H,axiom,
! [Y: nat,X: nat,Qs: set_Re381260168593705685la_a_b] :
( ( Y != X )
=> ( ( relational_eqs_a_b @ X @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ Qs ) )
= ( insert_nat @ Y @ ( relational_eqs_a_b @ X @ Qs ) ) ) ) ).
% eqs_insert'
thf(fact_995_qp__impl_Osimps_I5_J,axiom,
! [V: nat,Vb: nat] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ V @ ( relational_Var_a @ Vb ) ) ) ).
% qp_impl.simps(5)
thf(fact_996_eqs__in,axiom,
! [Y: nat,X: nat,G: set_Re381260168593705685la_a_b] :
( ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) )
=> ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G ) ) ).
% eqs_in
thf(fact_997_notin__eqs,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ~ ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) ) ) ).
% notin_eqs
thf(fact_998_cov_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov.Exists
thf(fact_999_cov_H_OExists,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).
% cov'.Exists
thf(fact_1000_fmla_Odistinct_I29_J,axiom,
! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(29)
thf(fact_1001_cov_H_OEqL,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqL
thf(fact_1002_cov_H_OEqR,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov'.EqR
thf(fact_1003_simplification_Osimplified__Conj__eq,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( Simplified @ Q )
=> ( ( X != Y )
=> ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
| ( member_nat @ Y @ ( relational_fv_a_b @ Q ) ) )
=> ( Simplified @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ) ) ) ).
% simplification.simplified_Conj_eq
thf(fact_1004_fmla_Odistinct_I27_J,axiom,
! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(27)
thf(fact_1005_fmla_Odistinct_I25_J,axiom,
! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(25)
thf(fact_1006_fmla_Odistinct_I13_J,axiom,
! [X22: $o,X31: nat,X32: relational_term_a] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(13)
thf(fact_1007_qp__eq,axiom,
! [X: nat,Y: nat] :
~ ( relational_qp_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ).
% qp_eq
thf(fact_1008_cov_H_OEq__self,axiom,
! [X: nat] : ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov'.Eq_self
thf(fact_1009_cov_OEq__self,axiom,
! [X: nat] : ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).
% cov.Eq_self
thf(fact_1010_cov_OEqR,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqR
thf(fact_1011_cov_OEqL,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).
% cov.EqL
thf(fact_1012_sr__Conj__eq,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_sr_a_b @ Q )
=> ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
| ( member_nat @ Y @ ( relational_fv_a_b @ Q ) ) )
=> ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ) ).
% sr_Conj_eq
thf(fact_1013_semilattice__order__set_OcoboundedI,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,Less_eq: relational_fmla_a_b > relational_fmla_a_b > $o,Less: relational_fmla_a_b > relational_fmla_a_b > $o,A: set_Re381260168593705685la_a_b,A2: relational_fmla_a_b] :
( ( lattic7900345719479732037la_a_b @ F @ Less_eq @ Less )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( member4680049679412964150la_a_b @ A2 @ A )
=> ( Less_eq @ ( lattic6555223568391141957la_a_b @ F @ A ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_1014_semilattice__order__set_OcoboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,A2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_1015_semilattice__order__set_OcoboundedI,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,A2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ A ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_1016_sub_Osimps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relational_sub_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ).
% sub.simps(3)
thf(fact_1017_semilattice__order__set_OboundedE,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,Less_eq: relational_fmla_a_b > relational_fmla_a_b > $o,Less: relational_fmla_a_b > relational_fmla_a_b > $o,A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( lattic7900345719479732037la_a_b @ F @ Less_eq @ Less )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ( Less_eq @ X @ ( lattic6555223568391141957la_a_b @ F @ A ) )
=> ! [A7: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A7 @ A )
=> ( Less_eq @ X @ A7 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1018_semilattice__order__set_OboundedE,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
=> ! [A7: a] :
( ( member_a @ A7 @ A )
=> ( Less_eq @ X @ A7 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1019_semilattice__order__set_OboundedE,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( Less_eq @ X @ A7 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1020_semilattice__order__set_OboundedI,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,Less_eq: relational_fmla_a_b > relational_fmla_a_b > $o,Less: relational_fmla_a_b > relational_fmla_a_b > $o,A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( lattic7900345719479732037la_a_b @ F @ Less_eq @ Less )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [A5: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A5 @ A )
=> ( Less_eq @ X @ A5 ) )
=> ( Less_eq @ X @ ( lattic6555223568391141957la_a_b @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1021_semilattice__order__set_OboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( Less_eq @ X @ A5 ) )
=> ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1022_semilattice__order__set_OboundedI,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( Less_eq @ X @ A5 ) )
=> ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1023_semilattice__order__set_Obounded__iff,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( Less_eq @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1024_semilattice__order__set_Obounded__iff,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1025_qp__impl_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb2: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% qp_impl.elims(3)
thf(fact_1026_fv_Osimps_I3_J,axiom,
! [X: nat,T5: relational_term_a] :
( ( relational_fv_a_b @ ( relational_Eq_a_b @ X @ T5 ) )
= ( sup_sup_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) ) ).
% fv.simps(3)
thf(fact_1027_semilattice__set_Oinsert__remove,axiom,
! [F: a > a > a,A: set_a,X: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( ( ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
= ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1028_semilattice__set_Oinsert__remove,axiom,
! [F: nat > nat > nat,A: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1029_fmla_Oinject_I4_J,axiom,
! [X42: relational_fmla_a_b,Y42: relational_fmla_a_b] :
( ( ( relational_Neg_a_b @ X42 )
= ( relational_Neg_a_b @ Y42 ) )
= ( X42 = Y42 ) ) ).
% fmla.inject(4)
thf(fact_1030_rrb__simps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_rrb_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_rrb_a_b @ Q ) ) ).
% rrb_simps(4)
thf(fact_1031_fv__term__setD,axiom,
! [N4: nat,T3: relational_term_a] :
( ( member_nat @ N4 @ ( relati6004689760767320788_set_a @ T3 ) )
=> ( T3
= ( relational_Var_a @ N4 ) ) ) ).
% fv_term_setD
thf(fact_1032_fmla_Odistinct_I23_J,axiom,
! [X31: nat,X32: relational_term_a,X42: relational_fmla_a_b] :
( ( relational_Eq_a_b @ X31 @ X32 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(23)
thf(fact_1033_csts_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_csts_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_csts_a_b @ Q ) ) ).
% csts.simps(4)
thf(fact_1034_qp__Neg,axiom,
! [Q: relational_fmla_a_b] :
~ ( relational_qp_a_b @ ( relational_Neg_a_b @ Q ) ) ).
% qp_Neg
thf(fact_1035_gen_H_Ointros_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).
% gen'.intros(3)
thf(fact_1036_cov_H_ONeg,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( relational_cov_a_b2 @ X @ ( relational_Neg_a_b @ Q ) @ G ) ) ).
% cov'.Neg
thf(fact_1037_erase_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relational_erase_a_b @ ( relational_Neg_a_b @ Q ) @ X )
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ).
% erase.simps(4)
thf(fact_1038_cov_ONeg,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( relational_cov_a_b @ X @ ( relational_Neg_a_b @ Q ) @ G ) ) ).
% cov.Neg
thf(fact_1039_sat_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi ) @ I @ Sigma )
= ( ~ ( relational_sat_a_b @ Phi @ I @ Sigma ) ) ) ).
% sat.simps(4)
thf(fact_1040_gen_Ointros_I3_J,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).
% gen.intros(3)
thf(fact_1041_fv_Osimps_I4_J,axiom,
! [Phi: relational_fmla_a_b] :
( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
= ( relational_fv_a_b @ Phi ) ) ).
% fv.simps(4)
thf(fact_1042_fmla_Odistinct_I15_J,axiom,
! [X22: $o,X42: relational_fmla_a_b] :
( ( relational_Bool_a_b @ X22 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(15)
thf(fact_1043_fmla_Odistinct_I31_J,axiom,
! [X42: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X42 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(31)
thf(fact_1044_fmla_Odistinct_I33_J,axiom,
! [X42: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X42 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(33)
thf(fact_1045_equiv__Neg__cong,axiom,
! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
( ( relational_equiv_a_b @ Q @ Q5 )
=> ( relational_equiv_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_Neg_a_b @ Q5 ) ) ) ).
% equiv_Neg_cong
thf(fact_1046_fmla_Odistinct_I35_J,axiom,
! [X42: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
( ( relational_Neg_a_b @ X42 )
!= ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).
% fmla.distinct(35)
thf(fact_1047_qp__impl_Osimps_I6_J,axiom,
! [V: relational_fmla_a_b] :
~ ( relati3725921752842749053pl_a_b @ ( relational_Neg_a_b @ V ) ) ).
% qp_impl.simps(6)
thf(fact_1048_sub_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q ) )
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).
% sub.simps(4)
thf(fact_1049_semilattice__set_Oin__idem,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( lattic4423677692272362385la_a_b @ F )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( F @ X @ ( lattic6555223568391141957la_a_b @ F @ A ) )
= ( lattic6555223568391141957la_a_b @ F @ A ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_1050_semilattice__set_Oin__idem,axiom,
! [F: a > a > a,A: set_a,X: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
= ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_1051_semilattice__set_Oin__idem,axiom,
! [F: nat > nat > nat,A: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) )
= ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_1052_gen_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen.intros(5)
thf(fact_1053_gen_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen.intros(4)
thf(fact_1054_gen_H_Ointros_I4_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen'.intros(4)
thf(fact_1055_gen_H_Ointros_I5_J,axiom,
! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).
% gen'.intros(5)
thf(fact_1056_fv__term__set_Osimps_I1_J,axiom,
! [N4: nat] :
( ( relati6004689760767320788_set_a @ ( relational_Var_a @ N4 ) )
= ( insert_nat @ N4 @ bot_bot_set_nat ) ) ).
% fv_term_set.simps(1)
thf(fact_1057_semilattice__set_Ohom__commute,axiom,
! [F: a > a > a,H: a > a,N3: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ! [X2: a,Y2: a] :
( ( H @ ( F @ X2 @ Y2 ) )
= ( F @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_a @ N3 )
=> ( ( N3 != bot_bot_set_a )
=> ( ( H @ ( lattic5116578512385870296ce_F_a @ F @ N3 ) )
= ( lattic5116578512385870296ce_F_a @ F @ ( image_a_a @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1058_semilattice__set_Ohom__commute,axiom,
! [F: nat > nat > nat,H: nat > nat,N3: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ! [X2: nat,Y2: nat] :
( ( H @ ( F @ X2 @ Y2 ) )
= ( F @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic7742739596368939638_F_nat @ F @ N3 ) )
= ( lattic7742739596368939638_F_nat @ F @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1059_semilattice__set_Oclosed,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( lattic4423677692272362385la_a_b @ F )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ ( F @ X2 @ Y2 ) @ ( insert7010464514620295119la_a_b @ X2 @ ( insert7010464514620295119la_a_b @ Y2 @ bot_bo4495933725496725865la_a_b ) ) )
=> ( member4680049679412964150la_a_b @ ( lattic6555223568391141957la_a_b @ F @ A ) @ A ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1060_semilattice__set_Oclosed,axiom,
! [F: a > a > a,A: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] : ( member_a @ ( F @ X2 @ Y2 ) @ ( insert_a @ X2 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
=> ( member_a @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ A ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1061_semilattice__set_Oclosed,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( F @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic7742739596368939638_F_nat @ F @ A ) @ A ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1062_semilattice__set_Oinsert,axiom,
! [F: a > a > a,A: set_a,X: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
= ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1063_semilattice__set_Oinsert,axiom,
! [F: nat > nat > nat,A: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1064_semilattice__set_Oinsert__not__elem,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( lattic4423677692272362385la_a_b @ F )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ~ ( member4680049679412964150la_a_b @ X @ A )
=> ( ( A != bot_bo4495933725496725865la_a_b )
=> ( ( lattic6555223568391141957la_a_b @ F @ ( insert7010464514620295119la_a_b @ X @ A ) )
= ( F @ X @ ( lattic6555223568391141957la_a_b @ F @ A ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1065_semilattice__set_Oinsert__not__elem,axiom,
! [F: a > a > a,A: set_a,X: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ~ ( member_a @ X @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X @ A ) )
= ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1066_semilattice__set_Oinsert__not__elem,axiom,
! [F: nat > nat > nat,A: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X @ A ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1067_semilattice__set_Osubset,axiom,
! [F: a > a > a,A: set_a,B: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( B != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( F @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
= ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1068_semilattice__set_Osubset,axiom,
! [F: nat > nat > nat,A: set_nat,B: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A ) )
= ( lattic7742739596368939638_F_nat @ F @ A ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1069_semilattice__set_Ounion,axiom,
! [F: a > a > a,A: set_a,B: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( finite_finite_a @ B )
=> ( ( B != bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( sup_sup_set_a @ A @ B ) )
= ( F @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ ( lattic5116578512385870296ce_F_a @ F @ B ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1070_semilattice__set_Ounion,axiom,
! [F: nat > nat > nat,A: set_nat,B: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( F @ ( lattic7742739596368939638_F_nat @ F @ A ) @ ( lattic7742739596368939638_F_nat @ F @ B ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1071_erase_Osimps_I3_J,axiom,
! [T3: relational_term_a,Z2: nat,X: nat] :
( ( ( T3
= ( relational_Var_a @ Z2 ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X )
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T3
!= ( relational_Var_a @ Z2 ) )
=> ( ( ( ( X = Z2 )
| ( member_nat @ X @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X )
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( X = Z2 )
| ( member_nat @ X @ ( relati6004689760767320788_set_a @ T3 ) ) )
=> ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X )
= ( relational_Eq_a_b @ Z2 @ T3 ) ) ) ) ) ) ).
% erase.simps(3)
thf(fact_1072_semilattice__set_Oremove,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,X: relational_fmla_a_b] :
( ( lattic4423677692272362385la_a_b @ F )
=> ( ( finite5600759454172676150la_a_b @ A )
=> ( ( member4680049679412964150la_a_b @ X @ A )
=> ( ( ( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
= bot_bo4495933725496725865la_a_b )
=> ( ( lattic6555223568391141957la_a_b @ F @ A )
= X ) )
& ( ( ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) )
!= bot_bo4495933725496725865la_a_b )
=> ( ( lattic6555223568391141957la_a_b @ F @ A )
= ( F @ X @ ( lattic6555223568391141957la_a_b @ F @ ( minus_4077726661957047470la_a_b @ A @ ( insert7010464514620295119la_a_b @ X @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1073_semilattice__set_Oremove,axiom,
! [F: a > a > a,A: set_a,X: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ X @ A )
=> ( ( ( ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A )
= X ) )
& ( ( ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A )
= ( F @ X @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1074_semilattice__set_Oremove,axiom,
! [F: nat > nat > nat,A: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1075_sub_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y )
=> ( ! [T6: $o] :
( ( X
= ( relational_Bool_a_b @ T6 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T6 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) @ bot_bo4495933725496725865la_a_b ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q8 ) @ ( relational_sub_a_b @ Q8 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q14 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q14 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) )
=> ( Y
!= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) @ ( relational_sub_a_b @ Q8 ) ) ) ) ) ) ) ) ) ) ) ).
% sub.elims
thf(fact_1076_Q3__def,axiom,
( q3
= ( relational_Conj_a_b @ ( relational_erase_a_b @ q @ x )
@ ( relational_Neg_a_b
@ ( relational_Disj_a_b @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) )
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Eq_a_b @ x @ ( relational_Var_a @ Y4 ) )
@ ( relational_eqs_a_b @ x @ g ) ) ) ) ) ) ) ).
% Q3_def
thf(fact_1077_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1078_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1079_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1080_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1081_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1082_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1083_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1084_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1085_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1086_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1087_if__image__distrib,axiom,
! [P: nat > $o,F: nat > relational_fmla_a_b,G3: nat > relational_fmla_a_b,S: set_nat] :
( ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : ( if_Rel1279876242545935705la_a_b @ ( P @ X3 ) @ ( F @ X3 ) @ ( G3 @ X3 ) )
@ S )
= ( sup_su5130108678486352897la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_4386371547000553590la_a_b @ G3
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X3: nat] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1088_fmla_Odistinct_I5_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X42: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Neg_a_b @ X42 ) ) ).
% fmla.distinct(5)
thf(fact_1089_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic3109210760196336428et_nat @ sup_sup_set_nat
@ ^ [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ X3 )
@ ^ [X3: set_nat,Y4: set_nat] : ( ord_less_set_nat @ Y4 @ X3 ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_1090_Sup__fin_Osemilattice__order__set__axioms,axiom,
( lattic6009151579333465974et_nat @ sup_sup_nat
@ ^ [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X3 )
@ ^ [X3: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X3 ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_1091_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_1092_less__set__def,axiom,
( ord_le7152733262289451305la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ord_le6021219098528097948_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ).
% less_set_def
thf(fact_1093_pigeonhole__infinite__rel,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_a,R: relational_fmla_a_b > a > $o] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1094_pigeonhole__infinite__rel,axiom,
! [A: set_Re381260168593705685la_a_b,B: set_nat,R: relational_fmla_a_b > nat > $o] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1095_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1096_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1097_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1098_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1099_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1100_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1101_fmla_Odistinct_I9_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Disj_a_b @ X61 @ X62 ) ) ).
% fmla.distinct(9)
thf(fact_1102_fmla_Odistinct_I7_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Conj_a_b @ X51 @ X52 ) ) ).
% fmla.distinct(7)
thf(fact_1103_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_1104_pred__subset__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( ord_le7191224889845164944_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ord_le4112832032246704949la_a_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1105_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1106_prop__restrict,axiom,
! [X: relational_fmla_a_b,Z4: set_Re381260168593705685la_a_b,X5: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ Z4 )
=> ( ( ord_le4112832032246704949la_a_b @ Z4
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1107_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_1108_Collect__restrict,axiom,
! [X5: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1109_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( P @ X3 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1110_Collect__subset,axiom,
! [A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ord_le4112832032246704949la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1111_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1112_less__eq__set__def,axiom,
( ord_le4112832032246704949la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( ord_le7191224889845164944_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_1113_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_1114_qps__def,axiom,
( relational_qps_a_b
= ( ^ [G4: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [Q6: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ Q6 @ G4 )
& ( relational_qp_a_b @ Q6 ) ) ) ) ) ).
% qps_def
thf(fact_1115_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1116_sup__Un__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( sup_su5130108678486352897la_a_b @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1117_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ^ [X3: nat] : ( member_nat @ X3 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1118_inf__Int__eq,axiom,
! [R: set_Re381260168593705685la_a_b,S: set_Re381260168593705685la_a_b] :
( ( inf_in2474466416471573982_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ R )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ S ) )
= ( ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ ( inf_in8483230781156617063la_a_b @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1119_Un__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
| ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_1120_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
| ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_1121_Int__def,axiom,
( inf_in8483230781156617063la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
& ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_1122_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% Int_def
thf(fact_1123_Int__Collect,axiom,
! [X: relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( member4680049679412964150la_a_b @ X @ ( inf_in8483230781156617063la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) ) )
= ( ( member4680049679412964150la_a_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1124_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1125_inf__set__def,axiom,
( inf_in8483230781156617063la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( inf_in2474466416471573982_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1126_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% inf_set_def
thf(fact_1127_sup__set__def,axiom,
( sup_su5130108678486352897la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( sup_su1471977682094119364_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1128_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_1129_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1130_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1131_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1132_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A2 = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1133_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1134_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_1135_insert__compr,axiom,
( insert7010464514620295119la_a_b
= ( ^ [A4: relational_fmla_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( X3 = A4 )
| ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_1136_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A4 )
| ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_1137_minus__set__def,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ( minus_9215201808853403479_a_b_o
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ A3 )
@ ^ [X3: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1138_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1139_set__diff__eq,axiom,
( minus_4077726661957047470la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b,B2: set_Re381260168593705685la_a_b] :
( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A3 )
& ~ ( member4680049679412964150la_a_b @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1140_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ~ ( member_nat @ X3 @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1141_fmla_Odistinct_I3_J,axiom,
! [X11: b,X12: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
( ( relational_Pred_b_a @ X11 @ X12 )
!= ( relational_Eq_a_b @ X31 @ X32 ) ) ).
% fmla.distinct(3)
thf(fact_1142_eqs__def,axiom,
( relational_eqs_a_b
= ( ^ [X3: nat,G4: set_Re381260168593705685la_a_b] :
( collect_nat
@ ^ [Y4: nat] :
( ( X3 != Y4 )
& ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y4 ) ) @ G4 ) ) ) ) ) ).
% eqs_def
thf(fact_1143_image__constant,axiom,
! [X: nat,A: set_nat,C: relational_fmla_a_b] :
( ( member_nat @ X @ A )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ).
% image_constant
thf(fact_1144_image__constant__conv,axiom,
! [A: set_nat,C: relational_fmla_a_b] :
( ( ( A = bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : C
@ A )
= bot_bo4495933725496725865la_a_b ) )
& ( ( A != bot_bot_set_nat )
=> ( ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : C
@ A )
= ( insert7010464514620295119la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% image_constant_conv
thf(fact_1145_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_6790371041703824709la_a_b @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_6790371041703824709la_a_b @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1146_Compr__image__eq,axiom,
! [F: nat > relational_fmla_a_b,A: set_nat,P: relational_fmla_a_b > $o] :
( ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ ( image_4386371547000553590la_a_b @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_4386371547000553590la_a_b @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1147_Compr__image__eq,axiom,
! [F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_341122591648980342_b_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_341122591648980342_b_nat @ F
@ ( collec3419995626248312948la_a_b
@ ^ [X3: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1148_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1149_image__image,axiom,
! [F: relational_fmla_a_b > relational_fmla_a_b,G3: nat > relational_fmla_a_b,A: set_nat] :
( ( image_6790371041703824709la_a_b @ F @ ( image_4386371547000553590la_a_b @ G3 @ A ) )
= ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : ( F @ ( G3 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1150_image__image,axiom,
! [F: nat > relational_fmla_a_b,G3: nat > nat,A: set_nat] :
( ( image_4386371547000553590la_a_b @ F @ ( image_nat_nat @ G3 @ A ) )
= ( image_4386371547000553590la_a_b
@ ^ [X3: nat] : ( F @ ( G3 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1151_imageE,axiom,
! [B3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X2: nat] :
( ( B3
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% imageE
thf(fact_1152_imageE,axiom,
! [B3: nat,F: relational_fmla_a_b > nat,A: set_Re381260168593705685la_a_b] :
( ( member_nat @ B3 @ ( image_341122591648980342_b_nat @ F @ A ) )
=> ~ ! [X2: relational_fmla_a_b] :
( ( B3
= ( F @ X2 ) )
=> ~ ( member4680049679412964150la_a_b @ X2 @ A ) ) ) ).
% imageE
thf(fact_1153_imageE,axiom,
! [B3: relational_fmla_a_b,F: nat > relational_fmla_a_b,A: set_nat] :
( ( member4680049679412964150la_a_b @ B3 @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ~ ! [X2: nat] :
( ( B3
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% imageE
thf(fact_1154_imageE,axiom,
! [B3: relational_fmla_a_b,F: relational_fmla_a_b > relational_fmla_a_b,A: set_Re381260168593705685la_a_b] :
( ( member4680049679412964150la_a_b @ B3 @ ( image_6790371041703824709la_a_b @ F @ A ) )
=> ~ ! [X2: relational_fmla_a_b] :
( ( B3
= ( F @ X2 ) )
=> ~ ( member4680049679412964150la_a_b @ X2 @ A ) ) ) ).
% imageE
thf(fact_1155_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > relational_fmla_a_b,B: set_Re381260168593705685la_a_b] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member4680049679412964150la_a_b @ ( F @ X2 ) @ B ) )
=> ( ord_le4112832032246704949la_a_b @ ( image_4386371547000553590la_a_b @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1156_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1157_pigeonhole__infinite,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > a] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_a @ ( image_4722625287770715864_a_b_a @ F @ A ) )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1158_pigeonhole__infinite,axiom,
! [A: set_Re381260168593705685la_a_b,F: relational_fmla_a_b > nat] :
( ~ ( finite5600759454172676150la_a_b @ A )
=> ( ( finite_finite_nat @ ( image_341122591648980342_b_nat @ F @ A ) )
=> ? [X2: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ X2 @ A )
& ~ ( finite5600759454172676150la_a_b
@ ( collec3419995626248312948la_a_b
@ ^ [A4: relational_fmla_a_b] :
( ( member4680049679412964150la_a_b @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1159_pigeonhole__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1160_pigeonhole__infinite,axiom,
! [A: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1161_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > relational_fmla_a_b] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite5600759454172676150la_a_b @ ( image_4386371547000553590la_a_b @ F @ A ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1162_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1163_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1164_nat__seg__image__imp__finite,axiom,
! [A: set_Re381260168593705685la_a_b,F: nat > relational_fmla_a_b,N4: nat] :
( ( A
= ( image_4386371547000553590la_a_b @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N4 ) ) ) )
=> ( finite5600759454172676150la_a_b @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1165_nat__seg__image__imp__finite,axiom,
! [A: set_a,F: nat > a,N4: nat] :
( ( A
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N4 ) ) ) )
=> ( finite_finite_a @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1166_nat__seg__image__imp__finite,axiom,
! [A: set_nat,F: nat > nat,N4: nat] :
( ( A
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N4 ) ) ) )
=> ( finite_finite_nat @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1167_finite__conv__nat__seg__image,axiom,
( finite5600759454172676150la_a_b
= ( ^ [A3: set_Re381260168593705685la_a_b] :
? [N2: nat,F2: nat > relational_fmla_a_b] :
( A3
= ( image_4386371547000553590la_a_b @ F2
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1168_finite__conv__nat__seg__image,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
? [N2: nat,F2: nat > a] :
( A3
= ( image_nat_a @ F2
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1169_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
? [N2: nat,F2: nat > nat] :
( A3
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1170_cov__sat__erase,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_sat_a_b
@ ( relational_Neg_a_b
@ ( relational_Disj_a_b @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) )
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) )
@ ( relational_eqs_a_b @ X @ G ) ) ) ) )
@ I
@ Sigma )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
= ( relational_sat_a_b @ ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) @ I @ Sigma ) ) ) ) ).
% cov_sat_erase
thf(fact_1171_fmla_Oexhaust,axiom,
! [Y: relational_fmla_a_b] :
( ! [X112: b,X122: list_R6823256787227418703term_a] :
( Y
!= ( relational_Pred_b_a @ X112 @ X122 ) )
=> ( ! [X23: $o] :
( Y
!= ( relational_Bool_a_b @ X23 ) )
=> ( ! [X312: nat,X322: relational_term_a] :
( Y
!= ( relational_Eq_a_b @ X312 @ X322 ) )
=> ( ! [X43: relational_fmla_a_b] :
( Y
!= ( relational_Neg_a_b @ X43 ) )
=> ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
( Y
!= ( relational_Conj_a_b @ X512 @ X522 ) )
=> ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
( Y
!= ( relational_Disj_a_b @ X612 @ X622 ) )
=> ~ ! [X712: nat,X722: relational_fmla_a_b] :
( Y
!= ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).
% fmla.exhaust
thf(fact_1172_nocp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [B5: $o] :
( X
!= ( relational_Bool_a_b @ B5 ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ! [Q8: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q8 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) ) ) ) ) ) ) ) ).
% nocp.cases
thf(fact_1173_fv_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( ! [B5: $o] :
( X
!= ( relational_Bool_a_b @ B5 ) )
=> ( ! [X2: nat,T7: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X2 @ T7 ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Phi2 ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) ) ) ) ) ) ) ) ).
% fv.cases
thf(fact_1174_cp_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X2: nat,T6: relational_term_a] :
( X
!= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ! [Q8: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ Q8 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ V2 @ Va2 ) )
=> ~ ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).
% cp.cases
thf(fact_1175_fv_Oelims,axiom,
! [X: relational_fmla_a_b,Y: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y )
=> ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( Y
!= ( relati4569515538964159125_set_a @ Ts ) ) )
=> ( ( ? [B5: $o] :
( X
= ( relational_Bool_a_b @ B5 ) )
=> ( Y != bot_bot_set_nat ) )
=> ( ! [X2: nat,T7: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T7 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T7 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi2 ) )
=> ( Y
!= ( relational_fv_a_b @ Phi2 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) ) )
=> ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).
% fv.elims
thf(fact_1176_simplification_Ocov__eval__fin,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ! [Sigma4: nat > a] :
~ ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I @ Sigma4 )
=> ( ( relational_eval_a_b @ Q @ I )
= ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q )
@ ( relational_Disj_a_b @ ( Simp @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) ) ) )
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Conj_a_b @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y4 ) ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) ) )
@ ( relational_eqs_a_b @ X @ G ) ) ) )
@ I ) ) ) ) ) ) ) ).
% simplification.cov_eval_fin
thf(fact_1177_qps__subst,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ( relational_qps_a_b
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ X @ Y )
@ G ) )
= ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ X @ Y )
@ ( relational_qps_a_b @ G ) ) ) ).
% qps_subst
thf(fact_1178_gen__Pred,axiom,
! [Z2: nat,P5: b,Ts2: list_R6823256787227418703term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ ( relational_Pred_b_a @ P5 @ Ts2 ) @ G )
= ( ( member_nat @ Z2 @ ( relati4569515538964159125_set_a @ Ts2 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P5 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ).
% gen_Pred
thf(fact_1179_eqs__subst,axiom,
! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_eqs_a_b @ Y
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ X @ Y )
@ G ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ ( relational_eqs_a_b @ Y @ G ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( minus_minus_set_nat @ ( relational_eqs_a_b @ X @ G ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).
% eqs_subst
thf(fact_1180_subst_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X @ Y )
= ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ).
% subst.simps(4)
thf(fact_1181_subst_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ X @ Y )
= ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q2 @ X @ Y ) ) ) ).
% subst.simps(6)
thf(fact_1182_Gen__cp__subst,axiom,
! [Z2: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ Q @ X_12 )
=> ( ( Z2 != X )
=> ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Z2 @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) ) @ X_1 ) ) ) ).
% Gen_cp_subst
thf(fact_1183_subst_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X @ Y )
= ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q2 @ X @ Y ) ) ) ).
% subst.simps(5)
thf(fact_1184_fv_Osimps_I1_J,axiom,
! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
= ( relati4569515538964159125_set_a @ Ts2 ) ) ).
% fv.simps(1)
thf(fact_1185_Eq__eq__subst__iff,axiom,
! [Y: nat,Z2: nat,Q: relational_fmla_a_b,X: nat] :
( ( ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z2 ) )
= ( relational_subst_a_b @ Q @ X @ Y ) )
= ( ( ( Z2 = X )
=> ( ( X = Y )
& ( Q
= ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) ) ) )
& ( ( Z2 != X )
=> ( ( Q
= ( relational_Eq_a_b @ X @ ( relational_Var_a @ Z2 ) ) )
| ( Q
= ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z2 ) ) )
| ( ( Z2 = Y )
& ( member4680049679412964150la_a_b @ Q @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ).
% Eq_eq_subst_iff
thf(fact_1186_fv__subst,axiom,
! [X: nat,Q: relational_fmla_a_b,Y: nat] :
( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( insert_nat @ Y @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) )
& ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
= ( relational_fv_a_b @ Q ) ) ) ) ).
% fv_subst
thf(fact_1187_erase_Osimps_I2_J,axiom,
! [X: nat,Ts2: list_R6823256787227418703term_a,P5: b] :
( ( ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P5 @ Ts2 ) @ X )
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ X @ ( relati4569515538964159125_set_a @ Ts2 ) )
=> ( ( relational_erase_a_b @ ( relational_Pred_b_a @ P5 @ Ts2 ) @ X )
= ( relational_Pred_b_a @ P5 @ Ts2 ) ) ) ) ).
% erase.simps(2)
thf(fact_1188_gen__nocp__intros_I2_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ G ) ) ) ).
% gen_nocp_intros(2)
thf(fact_1189_gen__nocp__intros_I1_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ G ) ) ) ).
% gen_nocp_intros(1)
thf(fact_1190_gen_H_Ointros_I8_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ G ) ) ) ).
% gen'.intros(8)
thf(fact_1191_gen_H_Ointros_I9_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ G ) ) ) ).
% gen'.intros(9)
thf(fact_1192_gen_Ointros_I8_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(8)
thf(fact_1193_gen_Ointros_I9_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b @ Y @ Q @ G )
=> ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ G ) ) ) ).
% gen.intros(9)
thf(fact_1194_gen_H__cp__intros_I2_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(2)
thf(fact_1195_gen_H__cp__intros_I1_J,axiom,
! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
( ( relational_gen_a_b2 @ Y @ Q @ G )
=> ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ G ) ) ) ).
% gen'_cp_intros(1)
thf(fact_1196_cov__nocp__intros,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ Gy ) ) ) ) ) ) ).
% cov_nocp_intros
thf(fact_1197_cov_H_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_subst_a_b @ Q6 @ Y @ X )
@ Gy ) ) ) ) ) ) ).
% cov'.Exists_gen
thf(fact_1198_cov_OExists__gen,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov.Exists_gen
thf(fact_1199_cov_H__cp__intros,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
( ( X != Y )
=> ( ( relational_cov_a_b2 @ X @ Q @ G )
=> ( ( relational_gen_a_b @ Y @ Q @ Gy )
=> ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
@ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Q6: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q6 @ Y @ X ) )
@ Gy ) ) ) ) ) ) ).
% cov'_cp_intros
thf(fact_1200_simplification_Ocov__Exists__equiv,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q )
@ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X @ ( Simp @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) ) ) ) )
@ ( relational_Disj_a_b
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y4 ) )
@ ( relational_eqs_a_b @ X @ G ) ) )
@ ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) ) ) ) ) ) ) ) ).
% simplification.cov_Exists_equiv
thf(fact_1201_cov__equiv,axiom,
! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Simp: relational_fmla_a_b > relational_fmla_a_b] :
( ( relational_cov_a_b @ X @ Q @ G )
=> ( ! [Q8: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( ( relational_sat_a_b @ ( Simp @ Q8 ) @ I3 @ Sigma4 )
= ( relational_sat_a_b @ Q8 @ I3 @ Sigma4 ) )
=> ( relational_equiv_a_b @ Q
@ ( relational_Disj_a_b @ ( Simp @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) ) ) )
@ ( relational_Disj_a_b
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Conj_a_b @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y4 ) ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) ) )
@ ( relational_eqs_a_b @ X @ G ) ) )
@ ( relational_Conj_a_b @ ( relational_erase_a_b @ Q @ X )
@ ( relational_Neg_a_b
@ ( relational_Disj_a_b @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) )
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) )
@ ( relational_eqs_a_b @ X @ G ) ) ) ) ) ) ) ) ) ) ) ).
% cov_equiv
thf(fact_1202_gen__induct,axiom,
! [X1: nat,X22: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
( ( relational_gen_a_b @ X1 @ X22 @ X33 )
=> ( ! [X2: nat] : ( P @ X2 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
=> ( ! [Q8: relational_fmla_a_b] :
( ( relational_ap_a_b @ Q8 )
=> ! [X2: nat] :
( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
=> ( P @ X2 @ Q8 @ ( insert7010464514620295119la_a_b @ Q8 @ bot_bo4495933725496725865la_a_b ) ) ) )
=> ( ! [X2: nat,Q8: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ Q8 @ G5 )
=> ( ( P @ X2 @ Q8 @ G5 )
=> ( P @ X2 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q8 ) ) @ G5 ) ) )
=> ( ! [X2: nat,Q14: relational_fmla_a_b,Q24: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G5 )
=> ( ( P @ X2 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G5 )
=> ( P @ X2 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) @ G5 ) ) )
=> ( ! [X2: nat,Q14: relational_fmla_a_b,Q24: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G5 )
=> ( ( P @ X2 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G5 )
=> ( P @ X2 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) @ G5 ) ) )
=> ( ! [X2: nat,Q14: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ Q14 @ G12 )
=> ( ( P @ X2 @ Q14 @ G12 )
=> ! [Q24: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ Q24 @ G23 )
=> ( ( P @ X2 @ Q24 @ G23 )
=> ( P @ X2 @ ( relational_Disj_a_b @ Q14 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) ) ) ) ) )
=> ( ! [X2: nat,Q14: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( ( ( relational_gen_a_b @ X2 @ Q14 @ G5 )
& ( P @ X2 @ Q14 @ G5 ) )
| ( ( relational_gen_a_b @ X2 @ Q24 @ G5 )
& ( P @ X2 @ Q24 @ G5 ) ) )
=> ( P @ X2 @ ( relational_Conj_a_b @ Q14 @ Q24 ) @ G5 ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y2 @ Q8 @ G5 )
=> ( ( P @ Y2 @ Q8 @ G5 )
=> ! [X2: nat] :
( P @ X2 @ ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y2 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ X2 )
@ G5 ) ) ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Y2 @ Q8 @ G5 )
=> ( ( P @ Y2 @ Q8 @ G5 )
=> ! [X2: nat] :
( P @ X2 @ ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ X2 ) ) )
@ ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ X2 )
@ G5 ) ) ) )
=> ( ! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ! [Q8: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ X2 @ Q8 @ G5 )
=> ( ( P @ X2 @ Q8 @ G5 )
=> ( P @ X2 @ ( relati591517084277583526ts_a_b @ Y2 @ Q8 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y2 ) @ G5 ) ) ) ) )
=> ( P @ X1 @ X22 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen_induct
thf(fact_1203_gen_H_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b2 @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q8 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ Q8 @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b2 @ A1 @ Q14 @ G12 )
=> ~ ( relational_gen_a_b2 @ A1 @ Q24 @ G23 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( A32 = G5 )
=> ~ ( ( relational_gen_a_b2 @ A1 @ Q14 @ G5 )
| ( relational_gen_a_b2 @ A1 @ Q24 @ G5 ) ) ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y2 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ A1 )
@ G5 ) )
=> ~ ( relational_gen_a_b2 @ Y2 @ Q8 @ G5 ) ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ A1 )
@ G5 ) )
=> ~ ( relational_gen_a_b2 @ Y2 @ Q8 @ G5 ) ) )
=> ~ ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y2 @ Q8 ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y2 ) @ G5 ) )
=> ( ( A1 != Y2 )
=> ~ ( relational_gen_a_b2 @ A1 @ Q8 @ G5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen'.cases
thf(fact_1204_gen_H_Osimps,axiom,
( relational_gen_a_b2
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) )
& ( relational_gen_a_b2 @ A12 @ Q6 @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q13 @ Q23 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b2 @ A12 @ Q13 @ G13 )
& ( relational_gen_a_b2 @ A12 @ Q23 @ G24 ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q13 @ Q23 ) )
& ( ( relational_gen_a_b2 @ A12 @ Q13 @ A33 )
| ( relational_gen_a_b2 @ A12 @ Q23 @ A33 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y4 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y4 @ A12 )
@ G4 ) )
& ( relational_gen_a_b2 @ Y4 @ Q6 @ G4 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ Y4 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y4 @ A12 )
@ G4 ) )
& ( relational_gen_a_b2 @ Y4 @ Q6 @ G4 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y4 @ Q6 ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y4 ) @ G4 ) )
& ( A12 != Y4 )
& ( relational_gen_a_b2 @ A12 @ Q6 @ G4 ) ) ) ) ) ) ).
% gen'.simps
thf(fact_1205_erase_Oelims,axiom,
! [X: relational_fmla_a_b,Xa2: nat,Y: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa2 )
= Y )
=> ( ! [T6: $o] :
( ( X
= ( relational_Bool_a_b @ T6 ) )
=> ( Y
!= ( relational_Bool_a_b @ T6 ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ ( ( ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y
= ( relational_Pred_b_a @ P4 @ Ts ) ) ) ) )
=> ( ! [Z3: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z3 @ T6 ) )
=> ~ ( ( ( T6
= ( relational_Var_a @ Z3 ) )
=> ( Y
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T6
!= ( relational_Var_a @ Z3 ) )
=> ( ( ( ( Xa2 = Z3 )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T6 ) ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa2 = Z3 )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T6 ) ) )
=> ( Y
= ( relational_Eq_a_b @ Z3 @ T6 ) ) ) ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( Y
!= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q8 @ Xa2 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q14 @ Xa2 ) @ ( relational_erase_a_b @ Q24 @ Xa2 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q14 @ Xa2 ) @ ( relational_erase_a_b @ Q24 @ Xa2 ) ) ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) )
=> ~ ( ( ( Xa2 = Z3 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa2 @ Q8 ) ) )
& ( ( Xa2 != Z3 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z3 @ ( relational_erase_a_b @ Q8 @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.elims
thf(fact_1206_gen_Ocases,axiom,
! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( relational_Bool_a_b @ $false ) )
=> ( A32 != bot_bo4495933725496725865la_a_b ) )
=> ( ( ( A32
= ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
=> ( ( relational_ap_a_b @ A22 )
=> ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q8 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ Q8 @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
=> ( ! [Q14: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ! [G23: set_Re381260168593705685la_a_b] :
( ( A32
= ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
=> ( ( relational_gen_a_b @ A1 @ Q14 @ G12 )
=> ~ ( relational_gen_a_b @ A1 @ Q24 @ G23 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,G5: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( A32 = G5 )
=> ~ ( ( relational_gen_a_b @ A1 @ Q14 @ G5 )
| ( relational_gen_a_b @ A1 @ Q24 @ G5 ) ) ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y2 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y2 @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y2 @ Q8 @ G5 ) ) )
=> ( ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ A1 ) ) ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y2 @ A1 ) )
@ G5 ) )
=> ~ ( relational_gen_a_b @ Y2 @ Q8 @ G5 ) ) )
=> ~ ! [Y2: nat,Q8: relational_fmla_a_b] :
( ( A22
= ( relati591517084277583526ts_a_b @ Y2 @ Q8 ) )
=> ! [G5: set_Re381260168593705685la_a_b] :
( ( A32
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y2 ) @ G5 ) )
=> ( ( A1 != Y2 )
=> ~ ( relational_gen_a_b @ A1 @ Q8 @ G5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% gen.cases
thf(fact_1207_gen_Osimps,axiom,
( relational_gen_a_b
= ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
( ( ( A23
= ( relational_Bool_a_b @ $false ) )
& ( A33 = bot_bo4495933725496725865la_a_b ) )
| ( ( A33
= ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
& ( relational_ap_a_b @ A23 )
& ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
| ? [Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) )
& ( relational_gen_a_b @ A12 @ Q6 @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
& ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
| ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Disj_a_b @ Q13 @ Q23 ) )
& ? [G24: set_Re381260168593705685la_a_b] :
( ( A33
= ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) )
& ( relational_gen_a_b @ A12 @ Q13 @ G13 )
& ( relational_gen_a_b @ A12 @ Q23 @ G24 ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q13 @ Q23 ) )
& ( ( relational_gen_a_b @ A12 @ Q13 @ A33 )
| ( relational_gen_a_b @ A12 @ Q23 @ A33 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y4 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y4 @ A12 ) )
@ G4 ) )
& ( relational_gen_a_b @ Y4 @ Q6 @ G4 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ Y4 @ ( relational_Var_a @ A12 ) ) ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b
@ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y4 @ A12 ) )
@ G4 ) )
& ( relational_gen_a_b @ Y4 @ Q6 @ G4 ) ) )
| ? [Y4: nat,Q6: relational_fmla_a_b] :
( ( A23
= ( relati591517084277583526ts_a_b @ Y4 @ Q6 ) )
& ? [G4: set_Re381260168593705685la_a_b] :
( ( A33
= ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y4 ) @ G4 ) )
& ( A12 != Y4 )
& ( relational_gen_a_b @ A12 @ Q6 @ G4 ) ) ) ) ) ) ).
% gen.simps
thf(fact_1208_simplification_Ocov__sat__fin,axiom,
! [Simp: relational_fmla_a_b > relational_fmla_a_b,Simplified: relational_fmla_a_b > $o,X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relati2910603115655104169on_a_b @ Simp @ Simplified )
=> ( ( relational_cov_a_b @ X @ Q @ G )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
=> ( ! [Sigma4: nat > a] :
~ ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I @ Sigma4 )
=> ( ( relational_sat_a_b @ Q @ I @ Sigma )
= ( relational_sat_a_b
@ ( relational_Disj_a_b @ ( Simp @ ( relational_Conj_a_b @ Q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ G ) ) ) )
@ ( relational_DISJ_a_b
@ ( image_4386371547000553590la_a_b
@ ^ [Y4: nat] : ( relational_Conj_a_b @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X @ Y4 ) ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) ) )
@ ( relational_eqs_a_b @ X @ G ) ) ) )
@ I
@ Sigma ) ) ) ) ) ) ) ).
% simplification.cov_sat_fin
thf(fact_1209_qp__impl_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y )
=> ( ( ? [X2: nat,C4: a] :
( X
= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) )
=> ~ Y )
=> ( ( ? [X2: b,Ts: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ X2 @ Ts ) )
=> ~ Y )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( Y
= ( ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) ) ) )
=> ( ( ? [V2: $o] :
( X
= ( relational_Bool_a_b @ V2 ) )
=> Y )
=> ( ( ? [V2: nat,Vb2: nat] :
( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> Y )
=> ( ( ? [V2: relational_fmla_a_b] :
( X
= ( relational_Neg_a_b @ V2 ) )
=> Y )
=> ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> Y )
=> ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> Y ) ) ) ) ) ) ) ) ) ).
% qp_impl.elims(1)
thf(fact_1210_gen__Eq,axiom,
! [Z2: nat,A2: nat,T3: relational_term_a,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ ( relational_Eq_a_b @ A2 @ T3 ) @ G )
= ( ( Z2 = A2 )
& ? [C6: a] :
( ( T3
= ( relational_Const_a @ C6 ) )
& ( G
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A2 @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).
% gen_Eq
thf(fact_1211_fresh2__gt_I3_J,axiom,
! [Z2: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( member_nat @ Z2 @ ( relational_fv_a_b @ Q ) )
=> ( ord_less_nat @ Z2 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) ) ).
% fresh2_gt(3)
thf(fact_1212_fresh2_I3_J,axiom,
! [X: nat,Y: nat,Q: relational_fmla_a_b] :
~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_fv_a_b @ Q ) ) ).
% fresh2(3)
thf(fact_1213_qp__impl_Osimps_I1_J,axiom,
! [X: nat,C: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).
% qp_impl.simps(1)
thf(fact_1214_fv__fo__term__list_Ocases,axiom,
! [X: relational_term_a] :
( ! [N: nat] :
( X
!= ( relational_Var_a @ N ) )
=> ~ ! [V2: a] :
( X
!= ( relational_Const_a @ V2 ) ) ) ).
% fv_fo_term_list.cases
thf(fact_1215_Relational__Calculus_Oterm_Oexhaust,axiom,
! [Y: relational_term_a] :
( ! [X13: a] :
( Y
!= ( relational_Const_a @ X13 ) )
=> ~ ! [X23: nat] :
( Y
!= ( relational_Var_a @ X23 ) ) ) ).
% Relational_Calculus.term.exhaust
thf(fact_1216_term_Odistinct_I1_J,axiom,
! [X1: a,X22: nat] :
( ( relational_Const_a @ X1 )
!= ( relational_Var_a @ X22 ) ) ).
% term.distinct(1)
thf(fact_1217_Eqc,axiom,
! [X: nat,C: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C ) ) ) ).
% Eqc
thf(fact_1218_ap_Osimps,axiom,
( relational_ap_a_b
= ( ^ [A4: relational_fmla_a_b] :
( ? [P6: b,Ts3: list_R6823256787227418703term_a] :
( A4
= ( relational_Pred_b_a @ P6 @ Ts3 ) )
| ? [X3: nat,C6: a] :
( A4
= ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C6 ) ) ) ) ) ) ).
% ap.simps
thf(fact_1219_ap_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relational_ap_a_b @ A2 )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( A2
!= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ ! [X2: nat,C4: a] :
( A2
!= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) ) ) ) ).
% ap.cases
thf(fact_1220_subst__exists,axiom,
! [Z2: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( ( member_nat @ Z2 @ ( relational_fv_a_b @ Q ) )
=> ( ( ( X = Z2 )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z2 @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ X @ Q ) ) )
& ( ( X != Z2 )
=> ( ( ( Z2 = Y )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z2 @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z2 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
& ( ( Z2 != Y )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z2 @ Q ) @ X @ Y )
= ( relati3989891337220013914ts_a_b @ Z2 @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) )
& ( ~ ( member_nat @ Z2 @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z2 @ Q ) @ X @ Y )
= ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).
% subst_exists
thf(fact_1221_fv__term__set_Oelims,axiom,
! [X: relational_term_a,Y: set_nat] :
( ( ( relati6004689760767320788_set_a @ X )
= Y )
=> ( ! [N: nat] :
( ( X
= ( relational_Var_a @ N ) )
=> ( Y
!= ( insert_nat @ N @ bot_bot_set_nat ) ) )
=> ~ ( ? [V2: a] :
( X
= ( relational_Const_a @ V2 ) )
=> ( Y != bot_bot_set_nat ) ) ) ) ).
% fv_term_set.elims
thf(fact_1222_qp__impl_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ! [X2: nat,C4: a] :
( X
!= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) )
=> ( ! [X2: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X2 @ Ts ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) ) ) ) ) ).
% qp_impl.elims(2)
thf(fact_1223_qp__impl_Ocases,axiom,
! [X: relational_fmla_a_b] :
( ! [X2: nat,C4: a] :
( X
!= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) )
=> ( ! [X2: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ X2 @ Ts ) )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( X
!= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ! [V2: $o] :
( X
!= ( relational_Bool_a_b @ V2 ) )
=> ( ! [V2: nat,Vb2: nat] :
( X
!= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( X
!= ( relational_Neg_a_b @ V2 ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( X
!= ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).
% qp_impl.cases
thf(fact_1224_nocp_Oelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relational_nocp_a_b @ X )
= Y )
=> ( ( ? [B5: $o] :
( X
= ( relational_Bool_a_b @ B5 ) )
=> Y )
=> ( ( ? [P4: b,Ts: list_R6823256787227418703term_a] :
( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ Y )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( Y
= ( T6
= ( relational_Var_a @ X2 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( Y
= ( ~ ( relational_nocp_a_b @ Q8 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( Y
= ( ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( Y
= ( ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( Y
= ( ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.elims(1)
thf(fact_1225_fv_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_nat] :
( ( ( relational_fv_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X )
=> ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ Uu @ Ts ) )
=> ( ( Y
= ( relati4569515538964159125_set_a @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu @ Ts ) ) ) )
=> ( ! [B5: $o] :
( ( X
= ( relational_Bool_a_b @ B5 ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B5 ) ) ) )
=> ( ! [X2: nat,T7: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T7 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T7 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X2 @ T7 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Phi2 ) )
=> ( ( Y
= ( relational_fv_a_b @ Phi2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv.pelims
thf(fact_1226_cpropagated__simps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% cpropagated_simps(6)
thf(fact_1227_cpropagated__simps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% cpropagated_simps(4)
thf(fact_1228_cpropagated__simps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% cpropagated_simps(5)
thf(fact_1229_cpropagated__simps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_simps(7)
thf(fact_1230_nocp_Osimps_I4_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Neg_a_b @ Q ) )
= ( relational_nocp_a_b @ Q ) ) ).
% nocp.simps(4)
thf(fact_1231_nocp_Osimps_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% nocp.simps(6)
thf(fact_1232_nocp_Osimps_I3_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relational_nocp_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( T3
!= ( relational_Var_a @ X ) ) ) ).
% nocp.simps(3)
thf(fact_1233_cpropagated__nocp,axiom,
! [Q: relational_fmla_a_b,X: nat] :
( ( relati1591879772219623554ed_a_b @ Q )
=> ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
=> ( relational_nocp_a_b @ Q ) ) ) ).
% cpropagated_nocp
thf(fact_1234_nocp_Osimps_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
= ( ( relational_nocp_a_b @ Q1 )
& ( relational_nocp_a_b @ Q2 ) ) ) ).
% nocp.simps(5)
thf(fact_1235_nocp_Osimps_I7_J,axiom,
! [X: nat,Q: relational_fmla_a_b] :
( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
= ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
& ( relational_nocp_a_b @ Q ) ) ) ).
% nocp.simps(7)
thf(fact_1236_nocp_Oelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ! [B5: $o] :
( X
!= ( relational_Bool_a_b @ B5 ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( T6
!= ( relational_Var_a @ X2 ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( relational_nocp_a_b @ Q8 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(3)
thf(fact_1237_nocp_Oelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( X
!= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( T6
= ( relational_Var_a @ X2 ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ~ ( relational_nocp_a_b @ Q8 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) ) ) ) ) ) ) ) ).
% nocp.elims(2)
thf(fact_1238_sub_Opelims,axiom,
! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
( ( ( relational_sub_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X )
=> ( ! [T6: $o] :
( ( X
= ( relational_Bool_a_b @ T6 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T6 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T6 ) ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) @ bot_bo4495933725496725865la_a_b ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q8 ) @ ( relational_sub_a_b @ Q8 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q8 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q14 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q14 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) )
=> ( ( Y
= ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) @ ( relational_sub_a_b @ Q8 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sub.pelims
thf(fact_1239_qp__impl_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relati3725921752842749053pl_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X2: nat,C4: a] :
( ( X
= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) ) ) )
=> ( ! [X2: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X2 @ Ts ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X2 @ Ts ) ) ) )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( Y
= ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
=> ( ! [V2: nat,Vb2: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(1)
thf(fact_1240_qp__impl_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X2: nat,C4: a] :
( ( X
= ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X2 @ ( relational_Const_a @ C4 ) ) ) )
=> ( ! [X2: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ X2 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X2 @ Ts ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(2)
thf(fact_1241_qp__impl_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relati3725921752842749053pl_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
=> ( ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_qp_a_b @ Q8 ) ) ) )
=> ( ! [V2: $o] :
( ( X
= ( relational_Bool_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
=> ( ! [V2: nat,Vb2: nat] :
( ( X
= ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) )
=> ( ! [V2: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ V2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
=> ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
=> ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ V2 @ Va2 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% qp_impl.pelims(3)
thf(fact_1242_nocp_Opelims_I1_J,axiom,
! [X: relational_fmla_a_b,Y: $o] :
( ( ( relational_nocp_a_b @ X )
= Y )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B5: $o] :
( ( X
= ( relational_Bool_a_b @ B5 ) )
=> ( ~ Y
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B5 ) ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( Y
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ( Y
= ( T6
!= ( relational_Var_a @ X2 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( ( Y
= ( relational_nocp_a_b @ Q8 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q8 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( Y
= ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(1)
thf(fact_1243_nocp_Opelims_I2_J,axiom,
! [X: relational_fmla_a_b] :
( ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts ) ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( T6
= ( relational_Var_a @ X2 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q8 ) )
=> ~ ( relational_nocp_a_b @ Q8 ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ~ ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ~ ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(2)
thf(fact_1244_nocp_Opelims_I3_J,axiom,
! [X: relational_fmla_a_b] :
( ~ ( relational_nocp_a_b @ X )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
=> ( ! [B5: $o] :
( ( X
= ( relational_Bool_a_b @ B5 ) )
=> ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B5 ) ) )
=> ( ! [X2: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X2 @ T6 ) )
=> ( T6
!= ( relational_Var_a @ X2 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q8 ) )
=> ( relational_nocp_a_b @ Q8 ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( relational_nocp_a_b @ Q14 )
& ( relational_nocp_a_b @ Q24 ) ) ) )
=> ~ ! [X2: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q8 ) )
=> ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q8 ) )
& ( relational_nocp_a_b @ Q8 ) ) ) ) ) ) ) ) ) ) ) ).
% nocp.pelims(3)
thf(fact_1245_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1246_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I5: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I5 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1247_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1248_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N4: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N3 )
=> ( ord_less_nat @ X2 @ N4 ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1249_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1250_cp_Osimps_I1_J,axiom,
! [X: nat,T3: relational_term_a] :
( ( relational_cp_a_b @ ( relational_Eq_a_b @ X @ T3 ) )
= ( relati582353067970734056la_a_b
@ ^ [A4: a] : ( relational_Eq_a_b @ X @ T3 )
@ ^ [Y4: nat] : ( if_Rel1279876242545935705la_a_b @ ( X = Y4 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) ) )
@ T3 ) ) ).
% cp.simps(1)
thf(fact_1251_subst__term_Osimps_I1_J,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( relati7175845559408349773term_a @ ( relational_Var_a @ Z2 ) @ X @ Y )
= ( relational_Var_a @ ( if_nat @ ( X = Z2 ) @ Y @ Z2 ) ) ) ).
% subst_term.simps(1)
thf(fact_1252_Var__eq__subst__iff,axiom,
! [Z2: nat,T3: relational_term_a,X: nat,Y: nat] :
( ( ( relational_Var_a @ Z2 )
= ( relati7175845559408349773term_a @ T3 @ X @ Y ) )
= ( ( ( Z2 = X )
=> ( ( X = Y )
& ( T3
= ( relational_Var_a @ X ) ) ) )
& ( ( Z2 != X )
=> ( ( ( Z2 = Y )
=> ( ( T3
= ( relational_Var_a @ X ) )
| ( T3
= ( relational_Var_a @ Y ) ) ) )
& ( ( Z2 != Y )
=> ( T3
= ( relational_Var_a @ Z2 ) ) ) ) ) ) ) ).
% Var_eq_subst_iff
thf(fact_1253_subst__term_Oelims,axiom,
! [X: relational_term_a,Xa2: nat,Xb: nat,Y: relational_term_a] :
( ( ( relati7175845559408349773term_a @ X @ Xa2 @ Xb )
= Y )
=> ( ! [Z3: nat] :
( ( X
= ( relational_Var_a @ Z3 ) )
=> ( Y
!= ( relational_Var_a @ ( if_nat @ ( Xa2 = Z3 ) @ Xb @ Z3 ) ) ) )
=> ~ ! [C4: a] :
( ( X
= ( relational_Const_a @ C4 ) )
=> ( Y
!= ( relational_Const_a @ C4 ) ) ) ) ) ).
% subst_term.elims
thf(fact_1254_subst_Osimps_I3_J,axiom,
! [Z2: nat,T3: relational_term_a,X: nat,Y: nat] :
( ( relational_subst_a_b @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X @ Y )
= ( relational_Eq_a_b @ ( if_nat @ ( Z2 = X ) @ Y @ Z2 ) @ ( relati7175845559408349773term_a @ T3 @ X @ Y ) ) ) ).
% subst.simps(3)
thf(fact_1255_subst_Oelims,axiom,
! [X: relational_fmla_a_b,Xa2: nat,Xb: nat,Y: relational_fmla_a_b] :
( ( ( relational_subst_a_b @ X @ Xa2 @ Xb )
= Y )
=> ( ! [T6: $o] :
( ( X
= ( relational_Bool_a_b @ T6 ) )
=> ( Y
!= ( relational_Bool_a_b @ T6 ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( Y
!= ( relational_Pred_b_a @ P4
@ ( map_Re5736185711816362116term_a
@ ^ [T: relational_term_a] : ( relati7175845559408349773term_a @ T @ Xa2 @ Xb )
@ Ts ) ) ) )
=> ( ! [Z3: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z3 @ T6 ) )
=> ( Y
!= ( relational_Eq_a_b @ ( if_nat @ ( Z3 = Xa2 ) @ Xb @ Z3 ) @ ( relati7175845559408349773term_a @ T6 @ Xa2 @ Xb ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( Y
!= ( relational_Neg_a_b @ ( relational_subst_a_b @ Q8 @ Xa2 @ Xb ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( relational_Conj_a_b @ ( relational_subst_a_b @ Q14 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( Y
!= ( relational_Disj_a_b @ ( relational_subst_a_b @ Q14 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) )
=> ~ ( ( ( Xa2 = Z3 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa2 @ Q8 ) ) )
& ( ( Xa2 != Z3 )
=> ( ( ( Z3 = Xb )
=> ( Y
= ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q8 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q8 @ Z3 @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q8 ) ) @ Xa2 @ Xb ) ) ) )
& ( ( Z3 != Xb )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z3 @ ( relational_subst_a_b @ Q8 @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% subst.elims
thf(fact_1256_genempty_Ocases,axiom,
! [A2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ A2 )
=> ( ( A2
!= ( relational_Bool_a_b @ $false ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q8 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q8 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q14 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( relati5999705594545617851ty_a_b @ Q14 )
=> ~ ( relati5999705594545617851ty_a_b @ Q24 ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( A2
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ~ ( ( relati5999705594545617851ty_a_b @ Q14 )
| ( relati5999705594545617851ty_a_b @ Q24 ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ? [X2: nat,Y2: nat] :
( A2
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y2 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q8 ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ? [Y2: nat,X2: nat] :
( A2
= ( relational_Conj_a_b @ Q8 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ X2 ) ) ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q8 ) )
=> ~ ! [Q8: relational_fmla_a_b] :
( ? [Y2: nat] :
( A2
= ( relati591517084277583526ts_a_b @ Y2 @ Q8 ) )
=> ~ ( relati5999705594545617851ty_a_b @ Q8 ) ) ) ) ) ) ) ) ) ) ) ).
% genempty.cases
thf(fact_1257_genempty_Ointros_I6_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( ( relati5999705594545617851ty_a_b @ Q1 )
| ( relati5999705594545617851ty_a_b @ Q2 ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ).
% genempty.intros(6)
thf(fact_1258_gen__genempty,axiom,
! [Z2: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
( ( relational_gen_a_b @ Z2 @ Q @ G )
=> ( ( G = bot_bo4495933725496725865la_a_b )
=> ( relati5999705594545617851ty_a_b @ Q ) ) ) ).
% gen_genempty
thf(fact_1259_genempty_Ointros_I5_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q1 )
=> ( ( relati5999705594545617851ty_a_b @ Q2 )
=> ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(5)
thf(fact_1260_genempty_Ointros_I2_J,axiom,
! [Q: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) ) ) ).
% genempty.intros(2)
thf(fact_1261_genempty_Ointros_I7_J,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ).
% genempty.intros(7)
thf(fact_1262_genempty_Ointros_I8_J,axiom,
! [Q: relational_fmla_a_b,Y: nat,X: nat] :
( ( relati5999705594545617851ty_a_b @ Q )
=> ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) ) ) ) ).
% genempty.intros(8)
thf(fact_1263_genempty_Ointros_I4_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(4)
thf(fact_1264_genempty_Ointros_I3_J,axiom,
! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
=> ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).
% genempty.intros(3)
thf(fact_1265_genempty_Osimps,axiom,
( relati5999705594545617851ty_a_b
= ( ^ [A4: relational_fmla_a_b] :
( ( A4
= ( relational_Bool_a_b @ $false ) )
| ? [Q6: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q6 ) ) )
& ( relati5999705594545617851ty_a_b @ Q6 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
& ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Disj_a_b @ Q13 @ Q23 ) )
& ( relati5999705594545617851ty_a_b @ Q13 )
& ( relati5999705594545617851ty_a_b @ Q23 ) )
| ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
( ( A4
= ( relational_Conj_a_b @ Q13 @ Q23 ) )
& ( ( relati5999705594545617851ty_a_b @ Q13 )
| ( relati5999705594545617851ty_a_b @ Q23 ) ) )
| ? [Q6: relational_fmla_a_b] :
( ? [X3: nat,Y4: nat] :
( A4
= ( relational_Conj_a_b @ Q6 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y4 ) ) ) )
& ( relati5999705594545617851ty_a_b @ Q6 ) )
| ? [Q6: relational_fmla_a_b] :
( ? [Y4: nat] :
( A4
= ( relati591517084277583526ts_a_b @ Y4 @ Q6 ) )
& ( relati5999705594545617851ty_a_b @ Q6 ) ) ) ) ) ).
% genempty.simps
thf(fact_1266_erase_Ocases,axiom,
! [X: produc7366699395886430672_b_nat] :
( ! [T6: $o,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T6 ) @ X2 ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P4 @ Ts ) @ X2 ) )
=> ( ! [Z3: nat,T6: relational_term_a,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z3 @ T6 ) @ X2 ) )
=> ( ! [Q8: relational_fmla_a_b,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q8 ) @ X2 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q14 @ Q24 ) @ X2 ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q14 @ Q24 ) @ X2 ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b,X2: nat] :
( X
!= ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) @ X2 ) ) ) ) ) ) ) ) ).
% erase.cases
thf(fact_1267_sat__subst,axiom,
! [Q: relational_fmla_a_b,X: nat,Y: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) @ I @ Sigma )
= ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ X @ ( Sigma @ Y ) ) ) ) ).
% sat_subst
thf(fact_1268_sat__exists,axiom,
! [N4: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relati3989891337220013914ts_a_b @ N4 @ Q ) @ I @ Sigma )
= ( ? [X3: a] : ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N4 @ X3 ) ) ) ) ).
% sat_exists
thf(fact_1269_sat__erase,axiom,
! [Q: relational_fmla_a_b,X: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z2: a] :
( ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I @ ( fun_upd_nat_a @ Sigma @ X @ Z2 ) )
= ( relational_sat_a_b @ ( relational_erase_a_b @ Q @ X ) @ I @ Sigma ) ) ).
% sat_erase
thf(fact_1270_sat__fun__upd,axiom,
! [N4: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z2: a] :
( ~ ( member_nat @ N4 @ ( relational_fv_a_b @ Q ) )
=> ( ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N4 @ Z2 ) )
= ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).
% sat_fun_upd
thf(fact_1271_sat_Osimps_I7_J,axiom,
! [Z2: nat,Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
( ( relational_sat_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Phi ) @ I @ Sigma )
= ( ? [X3: a] : ( relational_sat_a_b @ Phi @ I @ ( fun_upd_nat_a @ Sigma @ Z2 @ X3 ) ) ) ) ).
% sat.simps(7)
thf(fact_1272_fun__upd__image,axiom,
! [X: nat,A: set_nat,F: nat > relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( ( member_nat @ X @ A )
=> ( ( image_4386371547000553590la_a_b @ ( fun_up9185980720990446la_a_b @ F @ X @ Y ) @ A )
= ( insert7010464514620295119la_a_b @ Y @ ( image_4386371547000553590la_a_b @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( image_4386371547000553590la_a_b @ ( fun_up9185980720990446la_a_b @ F @ X @ Y ) @ A )
= ( image_4386371547000553590la_a_b @ F @ A ) ) ) ) ).
% fun_upd_image
thf(fact_1273_erase_Opelims,axiom,
! [X: relational_fmla_a_b,Xa2: nat,Y: relational_fmla_a_b] :
( ( ( relational_erase_a_b @ X @ Xa2 )
= Y )
=> ( ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ X @ Xa2 ) )
=> ( ! [T6: $o] :
( ( X
= ( relational_Bool_a_b @ T6 ) )
=> ( ( Y
= ( relational_Bool_a_b @ T6 ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T6 ) @ Xa2 ) ) ) )
=> ( ! [P4: b,Ts: list_R6823256787227418703term_a] :
( ( X
= ( relational_Pred_b_a @ P4 @ Ts ) )
=> ( ( ( ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( member_nat @ Xa2 @ ( relati4569515538964159125_set_a @ Ts ) )
=> ( Y
= ( relational_Pred_b_a @ P4 @ Ts ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P4 @ Ts ) @ Xa2 ) ) ) )
=> ( ! [Z3: nat,T6: relational_term_a] :
( ( X
= ( relational_Eq_a_b @ Z3 @ T6 ) )
=> ( ( ( ( T6
= ( relational_Var_a @ Z3 ) )
=> ( Y
= ( relational_Bool_a_b @ $true ) ) )
& ( ( T6
!= ( relational_Var_a @ Z3 ) )
=> ( ( ( ( Xa2 = Z3 )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T6 ) ) )
=> ( Y
= ( relational_Bool_a_b @ $false ) ) )
& ( ~ ( ( Xa2 = Z3 )
| ( member_nat @ Xa2 @ ( relati6004689760767320788_set_a @ T6 ) ) )
=> ( Y
= ( relational_Eq_a_b @ Z3 @ T6 ) ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z3 @ T6 ) @ Xa2 ) ) ) )
=> ( ! [Q8: relational_fmla_a_b] :
( ( X
= ( relational_Neg_a_b @ Q8 ) )
=> ( ( Y
= ( relational_Neg_a_b @ ( relational_erase_a_b @ Q8 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q8 ) @ Xa2 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Conj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( relational_Conj_a_b @ ( relational_erase_a_b @ Q14 @ Xa2 ) @ ( relational_erase_a_b @ Q24 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q14 @ Q24 ) @ Xa2 ) ) ) )
=> ( ! [Q14: relational_fmla_a_b,Q24: relational_fmla_a_b] :
( ( X
= ( relational_Disj_a_b @ Q14 @ Q24 ) )
=> ( ( Y
= ( relational_Disj_a_b @ ( relational_erase_a_b @ Q14 @ Xa2 ) @ ( relational_erase_a_b @ Q24 @ Xa2 ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q14 @ Q24 ) @ Xa2 ) ) ) )
=> ~ ! [Z3: nat,Q8: relational_fmla_a_b] :
( ( X
= ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) )
=> ( ( ( ( Xa2 = Z3 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Xa2 @ Q8 ) ) )
& ( ( Xa2 != Z3 )
=> ( Y
= ( relati591517084277583526ts_a_b @ Z3 @ ( relational_erase_a_b @ Q8 @ Xa2 ) ) ) ) )
=> ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q8 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% erase.pelims
thf(fact_1274_sat_Ocases,axiom,
! [X: produc1132964494702330949_nat_a] :
( ! [R2: b,Ts: list_R6823256787227418703term_a,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ( ! [B5: $o,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B5 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ( ! [X2: nat,T7: relational_term_a,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X2 @ T7 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) )
=> ~ ! [Z3: nat,Phi2: relational_fmla_a_b,I3: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
( X
!= ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I3 @ Sigma4 ) ) ) ) ) ) ) ) ) ).
% sat.cases
thf(fact_1275_subst__term_Ocases,axiom,
! [X: produc6058688428250151583at_nat] :
( ! [Z3: nat,X2: nat,Y2: nat] :
( X
!= ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z3 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
=> ~ ! [C4: a,X2: nat,Y2: nat] :
( X
!= ( produc2180204704594896271at_nat @ ( relational_Const_a @ C4 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ).
% subst_term.cases
thf(fact_1276_eval__term_Ocases,axiom,
! [X: produc8608687409264118859term_a] :
( ! [Sigma4: nat > a,C4: a] :
( X
!= ( produc8917778089171359291term_a @ Sigma4 @ ( relational_Const_a @ C4 ) ) )
=> ~ ! [Sigma4: nat > a,N: nat] :
( X
!= ( produc8917778089171359291term_a @ Sigma4 @ ( relational_Var_a @ N ) ) ) ) ).
% eval_term.cases
% Helper facts (5)
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_3_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
( ( if_Rel1279876242545935705la_a_b @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_Conj_a_b @ q @ ( relational_DISJ_a_b @ ( relational_qps_a_b @ g ) ) ) ) @ ( relational_fv_a_b @ q ) ).
%------------------------------------------------------------------------------