TPTP Problem File: SLH0724^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Separation_Logic_Unbounded/0003_FixedPoint/prob_00151_003852__6816346_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1351 ( 382 unt; 288 typ; 0 def)
% Number of atoms : 3390 (1301 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13664 ( 271 ~; 17 |; 203 &;11855 @)
% ( 0 <=>;1318 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 3855 (3855 >; 0 *; 0 +; 0 <<)
% Number of symbols : 259 ( 256 usr; 22 con; 0-7 aty)
% Number of variables : 4646 ( 182 ^;4387 !; 77 ?;4646 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:15.677
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
set_Pr1216123688828223200_a_c_d: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
produc5105196854009589546_a_c_d: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J_J,type,
set_Pr1275464188344874039_a_c_d: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
produc5278197477302038359_a_c_d: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
option3890169911263941780_a_c_d: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
set_Pr336584576397674490_a_c_d: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
produc5213381314664832452_a_c_d: $tType ).
thf(ty_n_t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mt__Option__Ooption_Itf__a_J_Mtf__a_J,type,
assert1556940916145061938on_a_a: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
option6413918287372586467_a_c_d: $tType ).
thf(ty_n_t__Set__Oset_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_J,type,
set_as909545710669178647_b_d_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_c_d_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
set_option_option_a: $tType ).
thf(ty_n_t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J,type,
assertion_a_b_d_c: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
product_prod_a_c_d: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
option_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__b_J_J,type,
set_option_b: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Option__Ooption_I_062_Itf__c_Mtf__d_J_J,type,
option_c_d: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Option__Ooption_Itf__b_J,type,
option_b: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $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__d,type,
d: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (256)
thf(sy_c_Combinability_Ologic_Ocombinable_001tf__a_001tf__b_001tf__c_001tf__d,type,
combinable_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( b > b > b ) > ( a > $o ) > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).
thf(sy_c_Combinability_Ologic_Ounambiguous_001tf__a_001tf__b_001tf__a_001t__Option__Ooption_Itf__a_J,type,
unambi704529886615442436tion_a: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > ( ( a > option_a ) > set_a ) > assert1556940916145061938on_a_a > a > $o ).
thf(sy_c_Combinability_Ologic_Ounambiguous_001tf__a_001tf__b_001tf__c_001tf__d,type,
unambiguous_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > ( ( c > d ) > set_a ) > assertion_a_b_d_c > c > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
finite8114217219359860531tion_a: set_option_option_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_Itf__a_J,type,
finite1674126218327898605tion_a: set_option_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Option__Ooption_Itf__a_J,type,
finite6501707464432451470tion_a: ( a > option_a > option_a ) > option_a > set_a > option_a ).
thf(sy_c_FixedPoint_Ologic_Oapplies__eq_001tf__a_001tf__b_001tf__d_001tf__c,type,
applies_eq_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > ( ( c > d ) > set_a ) > ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oapplies__eq__rel_001tf__a_001tf__b_001tf__d_001tf__c,type,
applie8886407701077375079_b_d_c: produc5105196854009589546_a_c_d > produc5105196854009589546_a_c_d > $o ).
thf(sy_c_FixedPoint_Ologic_Oempty__interp_001_062_Itf__c_Mtf__d_J_001tf__a,type,
empty_interp_c_d_a: ( c > d ) > set_a ).
thf(sy_c_FixedPoint_Ologic_Oindep__interp_001tf__a_001tf__b_001tf__d_001tf__c,type,
indep_interp_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > $o ).
thf(sy_c_FixedPoint_Ologic_Omonotonic_001tf__c_001tf__d_001tf__a,type,
monotonic_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Onon__increasing_001tf__c_001tf__d_001tf__a,type,
non_increasing_c_d_a: ( ( ( c > d ) > set_a ) > ( c > d ) > set_a ) > $o ).
thf(sy_c_FixedPoint_Ologic_Osmaller__interp_001tf__c_001tf__d_001tf__a,type,
smaller_interp_c_d_a: ( ( c > d ) > set_a ) > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
bij_be5431266891817924854tion_a: ( option_a > option_a ) > set_option_a > set_option_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001tf__a,type,
bij_betw_option_a_a: ( option_a > a ) > set_option_a > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Option__Ooption_Itf__a_J_001tf__b,type,
bij_betw_option_a_b: ( option_a > b ) > set_option_a > set_b > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Option__Ooption_Itf__a_J,type,
bij_betw_a_option_a: ( a > option_a ) > set_a > set_option_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__b,type,
bij_betw_a_b: ( a > b ) > set_a > set_b > $o ).
thf(sy_c_Fun_Obij__betw_001tf__b_001t__Option__Ooption_Itf__a_J,type,
bij_betw_b_option_a: ( b > option_a ) > set_b > set_option_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__b_001tf__a,type,
bij_betw_b_a: ( b > a ) > set_b > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__b_001tf__b,type,
bij_betw_b_b: ( b > b ) > set_b > set_b > $o ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
comp_o1254687777855551975tion_a: ( option_option_a > option_a ) > ( option_a > option_option_a ) > option_a > option_a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Nat__Onat_001t__Option__Ooption_Itf__a_J,type,
comp_o8583038678572498833tion_a: ( option_a > nat ) > ( option_a > option_a ) > option_a > nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
comp_o3154387707078715297tion_a: ( option_a > option_a ) > ( option_a > option_a ) > option_a > option_a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J_001tf__a,type,
comp_o6087033147929006299on_a_a: ( option_a > option_a ) > ( a > option_a ) > a > option_a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001t__Set__Oset_Itf__a_J_001tf__a,type,
comp_o6143895765626710849et_a_a: ( option_a > set_a ) > ( a > option_a ) > a > set_a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001t__Option__Ooption_Itf__a_J,type,
comp_o3864519266390211175tion_a: ( option_a > a ) > ( option_a > option_a ) > option_a > a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001tf__a,type,
comp_option_a_a_a: ( option_a > a ) > ( a > option_a ) > a > a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__a_001tf__b,type,
comp_option_a_a_b: ( option_a > a ) > ( b > option_a ) > b > a ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_Itf__a_J_001tf__b_001tf__a,type,
comp_option_a_b_a: ( option_a > b ) > ( a > option_a ) > a > b ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
comp_s1419921648917501825_set_a: ( set_option_a > set_a ) > ( set_a > set_option_a ) > set_a > set_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001_Eo_001tf__a,type,
comp_a_o_a: ( a > $o ) > ( a > a ) > a > $o ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001tf__a,type,
comp_a_nat_a: ( a > nat ) > ( a > a ) > a > nat ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
comp_a6249931511552232923tion_a: ( a > option_a ) > ( option_a > a ) > option_a > option_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001tf__a,type,
comp_a_option_a_a: ( a > option_a ) > ( a > a ) > a > option_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Option__Ooption_Itf__a_J_001tf__b,type,
comp_a_option_a_b: ( a > option_a ) > ( b > a ) > b > option_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Option__Ooption_Itf__a_J,type,
comp_a_a_option_a: ( a > a ) > ( option_a > a ) > option_a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__b,type,
comp_a_a_b: ( a > a ) > ( b > a ) > b > a ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Option__Ooption_Itf__a_J,type,
comp_a_b_option_a: ( a > b ) > ( option_a > a ) > option_a > b ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
thf(sy_c_Fun_Ofcomp_001tf__a_001t__Option__Ooption_Itf__a_J_001tf__a,type,
fcomp_a_option_a_a: ( a > option_a ) > ( option_a > a ) > a > a ).
thf(sy_c_Fun_Ofun__upd_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001t__Option__Ooption_I_062_Itf__c_Mtf__d_J_J,type,
fun_up2820008246124789800on_c_d: ( ( ( c > d ) > set_a ) > option_c_d ) > ( ( c > d ) > set_a ) > option_c_d > ( ( c > d ) > set_a ) > option_c_d ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
fun_up1079276522633388797tion_a: ( option_a > option_a ) > option_a > option_a > option_a > option_a ).
thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_Itf__a_J_001tf__a,type,
fun_upd_option_a_a: ( option_a > a ) > option_a > a > option_a > a ).
thf(sy_c_Fun_Ofun__upd_001t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_001t__Option__Ooption_It__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
fun_up8563802042059451790_a_c_d: ( assertion_a_b_d_c > option3890169911263941780_a_c_d ) > assertion_a_b_d_c > option3890169911263941780_a_c_d > assertion_a_b_d_c > option3890169911263941780_a_c_d ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Option__Ooption_Itf__a_J,type,
fun_upd_a_option_a: ( a > option_a ) > a > option_a > a > option_a ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
fun_upd_a_a: ( a > a ) > a > a > a > a ).
thf(sy_c_Fun_Ofun__upd_001tf__b_001t__Option__Ooption_Itf__a_J,type,
fun_upd_b_option_a: ( b > option_a ) > b > option_a > b > option_a ).
thf(sy_c_Fun_Ofun__upd_001tf__b_001tf__a,type,
fun_upd_b_a: ( b > a ) > b > a > b > a ).
thf(sy_c_Fun_Ofun__upd_001tf__c_001tf__d,type,
fun_upd_c_d: ( c > d ) > c > d > c > d ).
thf(sy_c_Fun_Oid_001t__Option__Ooption_Itf__a_J,type,
id_option_a: option_a > option_a ).
thf(sy_c_Fun_Oid_001t__Set__Oset_Itf__a_J,type,
id_set_a: set_a > set_a ).
thf(sy_c_Fun_Oid_001tf__a,type,
id_a: a > a ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
inj_on2224753519991154999tion_a: ( option_a > option_option_a ) > set_option_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
inj_on8559383841115902449tion_a: ( option_a > option_a ) > set_option_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
inj_on7881382345526841553tion_a: ( option_a > set_option_a ) > set_option_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001tf__a,type,
inj_on_option_a_a: ( option_a > a ) > set_option_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_Itf__a_J_001tf__b,type,
inj_on_option_a_b: ( option_a > b ) > set_option_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Option__Ooption_Itf__a_J,type,
inj_on_a_option_a: ( a > option_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
inj_on_a_set_a: ( a > set_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
inj_on_a_b: ( a > b ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001t__Option__Ooption_Itf__a_J,type,
inj_on_b_option_a: ( b > option_a ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
inj_on_b_a: ( b > a ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Option__Ooption_Itf__a_J_001tf__a,type,
monoto4483592047695070497on_a_a: set_option_a > ( option_a > option_a > $o ) > ( a > a > $o ) > ( option_a > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__b_J,type,
monoto7172710147596598632_set_b: set_set_a > ( set_a > set_a > $o ) > ( set_b > set_b > $o ) > ( set_a > set_b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
monoto8660672180588168358_set_a: set_set_b > ( set_b > set_b > $o ) > ( set_a > set_a > $o ) > ( set_b > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
monoto8660672184891397159_set_b: set_set_b > ( set_b > set_b > $o ) > ( set_b > set_b > $o ) > ( set_b > set_b ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Option__Ooption_Itf__a_J,type,
monoto2261078166156275373tion_a: set_a > ( a > a > $o ) > ( option_a > option_a > $o ) > ( a > option_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__a,type,
monotone_on_a_a: set_a > ( a > a > $o ) > ( a > a > $o ) > ( a > a ) > $o ).
thf(sy_c_Fun_Ooverride__on_001tf__a_001t__Option__Ooption_Itf__a_J,type,
overri633547075744967556tion_a: ( a > option_a ) > ( a > option_a ) > set_a > a > option_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
the_in2538339130118444083tion_a: set_option_a > ( option_a > option_a ) > option_a > option_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001tf__a,type,
the_in1757154643552616557on_a_a: set_option_a > ( option_a > a ) > a > option_a ).
thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_Itf__a_J_001tf__b,type,
the_in1757154643552616558on_a_b: set_option_a > ( option_a > b ) > b > option_a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Option__Ooption_Itf__a_J,type,
the_in8758012798868597241tion_a: set_a > ( a > option_a ) > option_a > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__b,type,
the_inv_into_a_b: set_a > ( a > b ) > b > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__b_001t__Option__Ooption_Itf__a_J,type,
the_in5672256556878602680tion_a: set_b > ( b > option_a ) > option_a > b ).
thf(sy_c_Fun_Othe__inv__into_001tf__b_001tf__a,type,
the_inv_into_b_a: set_b > ( b > a ) > a > b ).
thf(sy_c_Fun_Othe__inv__into_001tf__b_001tf__b,type,
the_inv_into_b_b: set_b > ( b > b ) > b > b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
minus_1574173051537231627tion_a: set_option_a > set_option_a > set_option_a ).
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_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
uminus6205308855922866075tion_a: set_option_a > set_option_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_HOL_OThe_001t__Option__Ooption_Itf__a_J,type,
the_option_a: ( option_a > $o ) > option_a ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_OThe_001tf__b,type,
the_b: ( b > $o ) > b ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_If_001t__Option__Ooption_Itf__a_J,type,
if_option_a: $o > option_a > option_a > option_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
inf_inf_set_option_a: set_option_a > set_option_a > set_option_a ).
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_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
sup_sup_set_option_a: set_option_a > set_option_a > set_option_a ).
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_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
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_001tf__a,type,
lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).
thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001tf__a,type,
dom_option_a_a: ( option_a > option_a ) > set_option_a ).
thf(sy_c_Map_Odom_001tf__a_001tf__a,type,
dom_a_a: ( a > option_a ) > set_a ).
thf(sy_c_Map_Odom_001tf__b_001tf__a,type,
dom_b_a: ( b > option_a ) > set_b ).
thf(sy_c_Map_Ograph_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_Itf__c_Mtf__d_J,type,
graph_c_d_set_a_c_d: ( ( ( c > d ) > set_a ) > option_c_d ) > set_Pr336584576397674490_a_c_d ).
thf(sy_c_Map_Ograph_001t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
graph_7603009230766167293_a_c_d: ( assertion_a_b_d_c > option3890169911263941780_a_c_d ) > set_Pr1216123688828223200_a_c_d ).
thf(sy_c_Map_Ograph_001tf__a_001tf__a,type,
graph_a_a: ( a > option_a ) > set_Product_prod_a_a ).
thf(sy_c_Map_Omap__add_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_add_option_a_a: ( option_a > option_a ) > ( option_a > option_a ) > option_a > option_a ).
thf(sy_c_Map_Omap__add_001tf__a_001tf__a,type,
map_add_a_a: ( a > option_a ) > ( a > option_a ) > a > option_a ).
thf(sy_c_Map_Omap__add_001tf__b_001tf__a,type,
map_add_b_a: ( b > option_a ) > ( b > option_a ) > b > option_a ).
thf(sy_c_Map_Omap__le_001tf__a_001tf__a,type,
map_le_a_a: ( a > option_a ) > ( a > option_a ) > $o ).
thf(sy_c_Map_Oran_001tf__a_001tf__a,type,
ran_a_a: ( a > option_a ) > set_a ).
thf(sy_c_Map_Orestrict__map_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_Itf__c_Mtf__d_J,type,
restri4474245042709046629_a_c_d: ( ( ( c > d ) > set_a ) > option_c_d ) > set_c_d_set_a > ( ( c > d ) > set_a ) > option_c_d ).
thf(sy_c_Map_Orestrict__map_001t__Option__Ooption_Itf__a_J_001tf__a,type,
restri3984065703976872170on_a_a: ( option_a > option_a ) > set_option_a > option_a > option_a ).
thf(sy_c_Map_Orestrict__map_001t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
restri3968632621895983051_a_c_d: ( assertion_a_b_d_c > option3890169911263941780_a_c_d ) > set_as909545710669178647_b_d_c > assertion_a_b_d_c > option3890169911263941780_a_c_d ).
thf(sy_c_Map_Orestrict__map_001tf__a_001tf__a,type,
restrict_map_a_a: ( a > option_a ) > set_a > a > option_a ).
thf(sy_c_Option_Obind_001tf__a_001tf__a,type,
bind_a_a: option_a > ( a > option_a ) > option_a ).
thf(sy_c_Option_Ocombine__options_001tf__a,type,
combine_options_a: ( a > a > a ) > option_a > option_a > option_a ).
thf(sy_c_Option_Ois__none_001tf__a,type,
is_none_a: option_a > $o ).
thf(sy_c_Option_Ooption_ONone_001_062_Itf__c_Mtf__d_J,type,
none_c_d: option_c_d ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_Itf__a_J,type,
none_option_a: option_option_a ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
none_P4438893274231186595_a_c_d: option3890169911263941780_a_c_d ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_ONone_001tf__b,type,
none_b: option_b ).
thf(sy_c_Option_Ooption_OSome_001_062_Itf__c_Mtf__d_J,type,
some_c_d: ( c > d ) > option_c_d ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_Itf__a_J,type,
some_option_a: option_a > option_option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
some_P3194730542479778335_a_c_d: produc5213381314664832452_a_c_d > option3890169911263941780_a_c_d ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
some_P1084500821511757806_a_c_d: product_prod_a_c_d > option6413918287372586467_a_c_d ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Ooption_OSome_001tf__b,type,
some_b: b > option_b ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001tf__a,type,
case_option_o_a: $o > ( a > $o ) > option_a > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_Itf__a_J_001tf__a,type,
case_o3148979394504432965on_a_a: option_a > ( a > option_a ) > option_a > option_a ).
thf(sy_c_Option_Ooption_Ocase__option_001tf__a_001t__Option__Ooption_Itf__a_J,type,
case_o926465512965637841tion_a: a > ( option_a > a ) > option_option_a > a ).
thf(sy_c_Option_Ooption_Ocase__option_001tf__a_001tf__a,type,
case_option_a_a: a > ( a > a ) > option_a > a ).
thf(sy_c_Option_Ooption_Omap__option_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
map_op788413144570152203tion_a: ( option_a > option_a ) > option_option_a > option_option_a ).
thf(sy_c_Option_Ooption_Omap__option_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_op4563205767754224965on_a_a: ( option_a > a ) > option_option_a > option_a ).
thf(sy_c_Option_Ooption_Omap__option_001tf__a_001t__Option__Ooption_Itf__a_J,type,
map_op2340691886215429841tion_a: ( a > option_a ) > option_a > option_option_a ).
thf(sy_c_Option_Ooption_Omap__option_001tf__a_001tf__a,type,
map_option_a_a: ( a > a ) > option_a > option_a ).
thf(sy_c_Option_Ooption_Omap__option_001tf__b_001tf__b,type,
map_option_b_b: ( b > b ) > option_b > option_b ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_Itf__a_J,type,
set_option_option_a2: option_option_a > set_option_a ).
thf(sy_c_Option_Ooption_Oset__option_001tf__a,type,
set_option_a2: option_a > set_a ).
thf(sy_c_Option_Ooption_Oset__option_001tf__b,type,
set_option_b2: option_b > set_b ).
thf(sy_c_Option_Ooption_Osize__option_001tf__a,type,
size_option_a: ( a > nat ) > option_a > nat ).
thf(sy_c_Option_Ooption_Othe_001t__Option__Ooption_Itf__a_J,type,
the_option_a2: option_option_a > option_a ).
thf(sy_c_Option_Ooption_Othe_001tf__a,type,
the_a2: option_a > a ).
thf(sy_c_Option_Ooption_Othe_001tf__b,type,
the_b2: option_b > b ).
thf(sy_c_Option_Othese_001t__Option__Ooption_Itf__a_J,type,
these_option_a: set_option_option_a > set_option_a ).
thf(sy_c_Option_Othese_001tf__a,type,
these_a: set_option_a > set_a ).
thf(sy_c_Option_Othese_001tf__b,type,
these_b: set_option_b > set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
bot_bo4163488203964334806tion_a: set_option_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
bot_bot_set_option_a: set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
ord_le1955136853071979460tion_a: set_option_a > set_option_a > $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_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
top_to1659475022456381882tion_a: set_option_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
top_top_set_option_a: set_option_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
top_top_set_set_b: set_set_b ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
top_top_set_b: set_b ).
thf(sy_c_Product__Type_OPair_001_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_001_062_Itf__c_Mtf__d_J,type,
produc7376592049607813182_a_c_d: ( ( c > d ) > set_a ) > ( c > d ) > produc5213381314664832452_a_c_d ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
produc8093790510458973071_a_c_d: product_prod_a_c_d > option6413918287372586467_a_c_d > produc5278197477302038359_a_c_d ).
thf(sy_c_Product__Type_OPair_001t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
produc8894421531525210148_a_c_d: assertion_a_b_d_c > produc5213381314664832452_a_c_d > produc5105196854009589546_a_c_d ).
thf(sy_c_Product__Type_OPair_001tf__a_001_062_Itf__c_Mtf__d_J,type,
product_Pair_a_c_d: a > ( c > d ) > product_prod_a_c_d ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_Itf__a_J,type,
collect_option_a: ( option_a > $o ) > set_option_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
collec2771355035510247705_a_c_d: ( produc5213381314664832452_a_c_d > $o ) > set_Pr336584576397674490_a_c_d ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
collec4762846013371775487_a_c_d: ( produc5105196854009589546_a_c_d > $o ) > set_Pr1216123688828223200_a_c_d ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
image_2132136900116418507tion_a: ( option_a > option_option_a ) > set_option_a > set_option_option_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
image_7439109396645324421tion_a: ( option_a > option_a ) > set_option_a > set_option_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
image_option_a_a: ( option_a > a ) > set_option_a > set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001tf__b,type,
image_option_a_b: ( option_a > b ) > set_option_a > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Option__Ooption_Itf__a_J,type,
image_b_option_a: ( b > option_a ) > set_b > set_option_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
insert605063979879581146tion_a: option_option_a > set_option_option_a > set_option_option_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
insert9214331609911559156_a_c_d: produc5213381314664832452_a_c_d > set_Pr336584576397674490_a_c_d > set_Pr336584576397674490_a_c_d ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
insert1952503980790619482_a_c_d: produc5105196854009589546_a_c_d > set_Pr1216123688828223200_a_c_d > set_Pr1216123688828223200_a_c_d ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ovimage_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
vimage1562710927270423099tion_a: ( option_a > option_a ) > set_option_a > set_option_a ).
thf(sy_c_Set_Ovimage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
vimage_option_a_a: ( option_a > a ) > set_a > set_option_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
vimage_a_option_a: ( a > option_a ) > set_option_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__b_001t__Option__Ooption_Itf__a_J,type,
vimage_b_option_a: ( b > option_a ) > set_option_a > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__a,type,
vimage_b_a: ( b > a ) > set_a > set_b ).
thf(sy_c_UnboundedLogic_Oassertion_OAnd_001tf__a_001tf__b_001tf__d_001tf__c,type,
and_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OBounded_001tf__a_001tf__b_001tf__d_001tf__c,type,
bounded_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OExists_001tf__a_001tf__a_001tf__b_001t__Option__Ooption_Itf__a_J,type,
exists7165000112504185261tion_a: a > assert1556940916145061938on_a_a > assert1556940916145061938on_a_a ).
thf(sy_c_UnboundedLogic_Oassertion_OExists_001tf__c_001tf__a_001tf__b_001tf__d,type,
exists_c_a_b_d: c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OForall_001tf__a_001tf__a_001tf__b_001t__Option__Ooption_Itf__a_J,type,
forall5484998627543102345tion_a: a > assert1556940916145061938on_a_a > assert1556940916145061938on_a_a ).
thf(sy_c_UnboundedLogic_Oassertion_OForall_001tf__c_001tf__a_001tf__b_001tf__d,type,
forall_c_a_b_d: c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OImp_001tf__a_001tf__b_001tf__d_001tf__c,type,
imp_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OMult_001tf__b_001tf__a_001tf__d_001tf__c,type,
mult_b_a_d_c: b > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OOr_001tf__a_001tf__b_001tf__d_001tf__c,type,
or_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OPred_001tf__a_001tf__b_001tf__d_001tf__c,type,
pred_a_b_d_c: assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OSem_001tf__c_001tf__d_001tf__a_001tf__b,type,
sem_c_d_a_b: ( ( c > d ) > a > $o ) > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OStar_001tf__a_001tf__b_001tf__d_001tf__c,type,
star_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OWand_001tf__a_001tf__b_001tf__d_001tf__c,type,
wand_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_OWildcard_001tf__a_001tf__b_001tf__d_001tf__c,type,
wildcard_a_b_d_c: assertion_a_b_d_c > assertion_a_b_d_c ).
thf(sy_c_UnboundedLogic_Oassertion_Oset__assertion_001tf__a_001tf__b_001tf__d_001tf__c,type,
set_as7232682317586342732_b_d_c: assertion_a_b_d_c > set_b ).
thf(sy_c_UnboundedLogic_Ologic_001tf__a_001tf__b,type,
logic_a_b: ( a > a > option_a ) > ( b > a > a ) > ( b > b > b ) > ( b > b > b ) > ( b > b ) > b > ( a > $o ) > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oentails_001tf__a_001tf__b_001tf__d_001tf__c,type,
entails_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oequal__outside_001tf__c_001tf__d,type,
equal_outside_c_d: ( c > d ) > ( c > d ) > set_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oequivalent_001tf__a_001tf__b_001tf__d_001tf__c,type,
equivalent_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oframe__property_001tf__a_001tf__c_001tf__d,type,
frame_property_a_c_d: ( a > a > option_a ) > ( a > $o ) > set_Pr1275464188344874039_a_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ointuitionistic_001tf__a_001tf__b_001tf__c_001tf__d,type,
intuit7508411120625971703_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > ( c > d ) > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Omodified_001tf__a_001tf__c_001tf__d,type,
modified_a_c_d: set_Pr1275464188344874039_a_c_d > set_c ).
thf(sy_c_UnboundedLogic_Ologic_Onot__in__fv_001tf__a_001tf__b_001tf__d_001tf__c,type,
not_in_fv_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > set_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Opure_001tf__a_001tf__b_001tf__d_001tf__c,type,
pure_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osafe_001tf__a_001tf__c_001tf__d,type,
safe_a_c_d: set_Pr1275464188344874039_a_c_d > product_prod_a_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osafety__monotonicity_001tf__a_001tf__c_001tf__d,type,
safety844553430189520448_a_c_d: ( a > a > option_a ) > ( a > $o ) > set_Pr1275464188344874039_a_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osat_001tf__a_001tf__b_001tf__a_001t__Option__Ooption_Itf__a_J,type,
sat_a_b_a_option_a: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > a > ( a > option_a ) > ( ( a > option_a ) > set_a ) > assert1556940916145061938on_a_a > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osat_001tf__a_001tf__b_001tf__c_001tf__d,type,
sat_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > a > ( c > d ) > ( ( c > d ) > set_a ) > assertion_a_b_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ovalid__command_001tf__a_001tf__c_001tf__d,type,
valid_command_a_c_d: ( a > $o ) > set_Pr1275464188344874039_a_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ovalid__hoare__triple_001tf__a_001tf__b_001tf__d_001tf__c,type,
valid_6037315502795721655_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_d_c > set_Pr1275464188344874039_a_c_d > assertion_a_b_d_c > ( ( c > d ) > set_a ) > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Ocompatible_001tf__a,type,
pre_compatible_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_UnboundedLogic_Opre__logic_Olarger_001tf__a,type,
pre_larger_a: ( a > a > option_a ) > a > a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
accp_P5381461700908302305_a_c_d: ( produc5105196854009589546_a_c_d > produc5105196854009589546_a_c_d > $o ) > produc5105196854009589546_a_c_d > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
member5113800082084363315tion_a: option_option_a > set_option_option_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__b_J,type,
member_option_b: option_b > set_option_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
member1642667639779969243_a_c_d: produc5213381314664832452_a_c_d > set_Pr336584576397674490_a_c_d > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
member1180172933830803072_a_c_d: produc5278197477302038359_a_c_d > set_Pr1275464188344874039_a_c_d > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__d_Mtf__c_J_Mt__Product____Type__Oprod_I_062_I_062_Itf__c_Mtf__d_J_Mt__Set__Oset_Itf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
member537768723423446209_a_c_d: produc5105196854009589546_a_c_d > set_Pr1216123688828223200_a_c_d > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_A,type,
a2: assertion_a_b_d_c ).
thf(sy_v_B,type,
b2: assertion_a_b_d_c ).
thf(sy_v__092_060Delta_062_H____,type,
delta: ( c > d ) > set_a ).
thf(sy_v__092_060Delta_062____,type,
delta2: ( c > d ) > set_a ).
thf(sy_v_mult,type,
mult: b > a > a ).
thf(sy_v_one,type,
one: b ).
thf(sy_v_plus,type,
plus: a > a > option_a ).
thf(sy_v_sadd,type,
sadd: b > b > b ).
thf(sy_v_sinv,type,
sinv: b > b ).
thf(sy_v_smult,type,
smult: b > b > b ).
thf(sy_v_valid,type,
valid: a > $o ).
% Relevant facts (1055)
thf(fact_0_smaller__interpI,axiom,
! [Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a] :
( ! [S: c > d,X: a] :
( ( member_a @ X @ ( Delta @ S ) )
=> ( member_a @ X @ ( Delta2 @ S ) ) )
=> ( smaller_interp_c_d_a @ Delta @ Delta2 ) ) ).
% smaller_interpI
thf(fact_1_smaller__interp__refl,axiom,
! [Delta: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ Delta @ Delta ) ).
% smaller_interp_refl
thf(fact_2_smaller__interp__trans,axiom,
! [A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ).
% smaller_interp_trans
thf(fact_3_commutative,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% commutative
thf(fact_4_can__divide,axiom,
! [P: b,A: a,B: a] :
( ( ( mult @ P @ A )
= ( mult @ P @ B ) )
=> ( A = B ) ) ).
% can_divide
thf(fact_5_smaller__empty,axiom,
! [X2: ( c > d ) > set_a] : ( smaller_interp_c_d_a @ empty_interp_c_d_a @ X2 ) ).
% smaller_empty
thf(fact_6_asm0,axiom,
smaller_interp_c_d_a @ delta2 @ delta ).
% asm0
thf(fact_7_unique__inv,axiom,
! [A: a,P: b,B: a] :
( ( A
= ( mult @ P @ B ) )
= ( B
= ( mult @ ( sinv @ P ) @ A ) ) ) ).
% unique_inv
thf(fact_8_assms_I2_J,axiom,
monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ b2 ) ).
% assms(2)
thf(fact_9_assms_I1_J,axiom,
monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ a2 ) ).
% assms(1)
thf(fact_10_monotonicI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F @ Delta3 ) @ ( F @ Delta4 ) ) )
=> ( monotonic_c_d_a @ F ) ) ).
% monotonicI
thf(fact_11_monotonic__def,axiom,
( monotonic_c_d_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F2 @ Delta5 ) @ ( F2 @ Delta6 ) ) ) ) ) ).
% monotonic_def
thf(fact_12_non__increasingI,axiom,
! [F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F @ Delta4 ) @ ( F @ Delta3 ) ) )
=> ( non_increasing_c_d_a @ F ) ) ).
% non_increasingI
thf(fact_13_non__increasing__def,axiom,
( non_increasing_c_d_a
= ( ^ [F2: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F2 @ Delta6 ) @ ( F2 @ Delta5 ) ) ) ) ) ).
% non_increasing_def
thf(fact_14_mono__instantiate,axiom,
! [A2: assertion_a_b_d_c,X2: a,Delta: ( c > d ) > set_a,S2: c > d,Delta2: ( c > d ) > set_a] :
( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 ) )
=> ( ( member_a @ X2 @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ S2 ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta2 )
=> ( member_a @ X2 @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta2 @ S2 ) ) ) ) ) ).
% mono_instantiate
thf(fact_15_logic_Oapplies__eq_Ocong,axiom,
applies_eq_a_b_d_c = applies_eq_a_b_d_c ).
% logic.applies_eq.cong
thf(fact_16_unambiguous__star,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c,B2: assertion_a_b_d_c] :
( ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A2 @ X2 )
=> ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) @ X2 ) ) ).
% unambiguous_star
thf(fact_17_indep__implies__non__increasing,axiom,
! [A2: assertion_a_b_d_c] :
( ( indep_interp_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 ) ) ) ).
% indep_implies_non_increasing
thf(fact_18_smaller__interp__applies__cons,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a,A: a,S2: c > d] :
( ( smaller_interp_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta ) @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta2 ) )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S2 @ Delta2 @ A2 ) ) ) ).
% smaller_interp_applies_cons
thf(fact_19_valid__mono,axiom,
! [A: a,B: a] :
( ( ( valid @ A )
& ( pre_larger_a @ plus @ A @ B ) )
=> ( valid @ B ) ) ).
% valid_mono
thf(fact_20_larger__same,axiom,
! [A: a,B: a,P: b] :
( ( pre_larger_a @ plus @ A @ B )
= ( pre_larger_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% larger_same
thf(fact_21_compatible__iff,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
= ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_iff
thf(fact_22_compatible__imp,axiom,
! [A: a,B: a,P: b] :
( ( pre_compatible_a @ plus @ A @ B )
=> ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ P @ B ) ) ) ).
% compatible_imp
thf(fact_23_compatible__multiples,axiom,
! [P: b,A: a,Q: b,B: a] :
( ( pre_compatible_a @ plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) )
=> ( pre_compatible_a @ plus @ A @ B ) ) ).
% compatible_multiples
thf(fact_24_one__neutral,axiom,
! [A: a] :
( ( mult @ one @ A )
= A ) ).
% one_neutral
thf(fact_25_assertion_Oinject_I3_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,Y31: assertion_a_b_d_c,Y32: assertion_a_b_d_c] :
( ( ( star_a_b_d_c @ X31 @ X32 )
= ( star_a_b_d_c @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% assertion.inject(3)
thf(fact_26_plus__mult,axiom,
! [A: a,B: a,C: a,P: b] :
( ( ( some_a @ A )
= ( plus @ B @ C ) )
=> ( ( some_a @ ( mult @ P @ A ) )
= ( plus @ ( mult @ P @ B ) @ ( mult @ P @ C ) ) ) ) ).
% plus_mult
thf(fact_27_indep__interp__def,axiom,
! [A2: assertion_a_b_d_c] :
( ( indep_interp_a_b_d_c @ plus @ mult @ valid @ A2 )
= ( ! [X3: a,S3: c > d,Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ X3 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ X3 @ S3 @ Delta6 @ A2 ) ) ) ) ).
% indep_interp_def
thf(fact_28_move__sum,axiom,
! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X2: a,X1: a,X22: a] :
( ( ( some_a @ A )
= ( plus @ A1 @ A22 ) )
=> ( ( ( some_a @ B )
= ( plus @ B1 @ B22 ) )
=> ( ( ( some_a @ X2 )
= ( plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X22 )
= ( plus @ A22 @ B22 ) )
=> ( ( some_a @ X2 )
= ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).
% move_sum
thf(fact_29_asso1,axiom,
! [A: a,B: a,Ab: a,C: a,Bc: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ( ( plus @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( plus @ Ab @ C )
= ( plus @ A @ Bc ) ) ) ).
% asso1
thf(fact_30_asso3,axiom,
! [A: a,B: a,C: a,Bc: a] :
( ~ ( pre_compatible_a @ plus @ A @ B )
=> ( ( ( plus @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ plus @ A @ Bc ) ) ) ).
% asso3
thf(fact_31_asso2,axiom,
! [A: a,B: a,Ab: a,C: a] :
( ( ( ( plus @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ plus @ B @ C ) )
=> ~ ( pre_compatible_a @ plus @ Ab @ C ) ) ).
% asso2
thf(fact_32_sum__both__larger,axiom,
! [X4: a,A3: a,B3: a,X2: a,A: a,B: a] :
( ( ( some_a @ X4 )
= ( plus @ A3 @ B3 ) )
=> ( ( ( some_a @ X2 )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ A3 @ A )
=> ( ( pre_larger_a @ plus @ B3 @ B )
=> ( pre_larger_a @ plus @ X4 @ X2 ) ) ) ) ) ).
% sum_both_larger
thf(fact_33_larger__first__sum,axiom,
! [Y: a,A: a,B: a,X2: a] :
( ( ( some_a @ Y )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ X2 @ Y )
=> ? [A4: a] :
( ( ( some_a @ X2 )
= ( plus @ A4 @ B ) )
& ( pre_larger_a @ plus @ A4 @ A ) ) ) ) ).
% larger_first_sum
thf(fact_34_larger__def,axiom,
! [A: a,B: a] :
( ( pre_larger_a @ plus @ A @ B )
= ( ? [C2: a] :
( ( some_a @ A )
= ( plus @ B @ C2 ) ) ) ) ).
% larger_def
thf(fact_35_larger__implies__compatible,axiom,
! [X2: a,Y: a] :
( ( pre_larger_a @ plus @ X2 @ Y )
=> ( pre_compatible_a @ plus @ X2 @ Y ) ) ).
% larger_implies_compatible
thf(fact_36_compatible__smaller,axiom,
! [A: a,B: a,X2: a] :
( ( pre_larger_a @ plus @ A @ B )
=> ( ( pre_compatible_a @ plus @ X2 @ A )
=> ( pre_compatible_a @ plus @ X2 @ B ) ) ) ).
% compatible_smaller
thf(fact_37_sat_Osimps_I2_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) )
= ( ? [A5: a,B4: a] :
( ( ( some_a @ Sigma )
= ( plus @ A5 @ B4 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S2 @ Delta @ B2 ) ) ) ) ).
% sat.simps(2)
thf(fact_38_intuitionistic__def,axiom,
! [S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S2 @ Delta @ A2 )
= ( ! [A5: a,B4: a] :
( ( ( pre_larger_a @ plus @ A5 @ B4 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B4 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ).
% intuitionistic_def
thf(fact_39_intuitionisticI,axiom,
! [S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ! [A6: a,B5: a] :
( ( ( pre_larger_a @ plus @ A6 @ B5 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B5 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A6 @ S2 @ Delta @ A2 ) )
=> ( intuit7508411120625971703_b_c_d @ plus @ mult @ valid @ S2 @ Delta @ A2 ) ) ).
% intuitionisticI
thf(fact_40_not__in__fv__def,axiom,
! [A2: assertion_a_b_d_c,S4: set_c] :
( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A2 @ S4 )
= ( ! [Sigma2: a,S3: c > d,Delta5: ( c > d ) > set_a,S5: c > d] :
( ( equal_outside_c_d @ S3 @ S5 @ S4 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S5 @ Delta5 @ A2 ) ) ) ) ) ).
% not_in_fv_def
thf(fact_41_logic_Osat_Ocong,axiom,
sat_a_b_c_d = sat_a_b_c_d ).
% logic.sat.cong
thf(fact_42_mem__Collect__eq,axiom,
! [A: b,P2: b > $o] :
( ( member_b @ A @ ( collect_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: option_a,P2: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_option_a] :
( ( collect_option_a
@ ^ [X3: option_a] : ( member_option_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X: a] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_49_Collect__cong,axiom,
! [P2: option_a > $o,Q2: option_a > $o] :
( ! [X: option_a] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_option_a @ P2 )
= ( collect_option_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_50_pre__logic_Olarger__def,axiom,
( pre_larger_a
= ( ^ [Plus: a > a > option_a,A5: a,B4: a] :
? [C2: a] :
( ( some_a @ A5 )
= ( Plus @ B4 @ C2 ) ) ) ) ).
% pre_logic.larger_def
thf(fact_51_pre__logic_Olarger_Ocong,axiom,
pre_larger_a = pre_larger_a ).
% pre_logic.larger.cong
thf(fact_52_pre__logic_Ocompatible_Ocong,axiom,
pre_compatible_a = pre_compatible_a ).
% pre_logic.compatible.cong
thf(fact_53_logic_Oindep__interp_Ocong,axiom,
indep_interp_a_b_d_c = indep_interp_a_b_d_c ).
% logic.indep_interp.cong
thf(fact_54_pure__def,axiom,
! [A2: assertion_a_b_d_c] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
= ( ! [Sigma2: a,Sigma3: a,S3: c > d,Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta6 @ A2 ) ) ) ) ).
% pure_def
thf(fact_55_unambiguous__def,axiom,
! [Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a,X2: a] :
( ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A2 @ X2 )
= ( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V2: option_a,S3: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_1 @ ( fun_upd_a_option_a @ S3 @ X2 @ V1 ) @ Delta @ A2 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_2 @ ( fun_upd_a_option_a @ S3 @ X2 @ V2 ) @ Delta @ A2 ) )
=> ( V1 = V2 ) ) ) ) ).
% unambiguous_def
thf(fact_56_unambiguous__def,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A2 @ X2 )
= ( ! [Sigma_1: a,Sigma_2: a,V1: d,V2: d,S3: c > d] :
( ( ( pre_compatible_a @ plus @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_1 @ ( fun_upd_c_d @ S3 @ X2 @ V1 ) @ Delta @ A2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_2 @ ( fun_upd_c_d @ S3 @ X2 @ V2 ) @ Delta @ A2 ) )
=> ( V1 = V2 ) ) ) ) ).
% unambiguous_def
thf(fact_57_unambiguousI,axiom,
! [X2: a,Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a] :
( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V22: option_a,S: a > option_a] :
( ( ( pre_compatible_a @ plus @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_12 @ ( fun_upd_a_option_a @ S @ X2 @ V12 ) @ Delta @ A2 )
& ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma_22 @ ( fun_upd_a_option_a @ S @ X2 @ V22 ) @ Delta @ A2 ) )
=> ( V12 = V22 ) )
=> ( unambi704529886615442436tion_a @ plus @ mult @ valid @ Delta @ A2 @ X2 ) ) ).
% unambiguousI
thf(fact_58_unambiguousI,axiom,
! [X2: c,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ! [Sigma_12: a,Sigma_22: a,V12: d,V22: d,S: c > d] :
( ( ( pre_compatible_a @ plus @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_12 @ ( fun_upd_c_d @ S @ X2 @ V12 ) @ Delta @ A2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma_22 @ ( fun_upd_c_d @ S @ X2 @ V22 ) @ Delta @ A2 ) )
=> ( V12 = V22 ) )
=> ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A2 @ X2 ) ) ).
% unambiguousI
thf(fact_59_sat_Osimps_I10_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ pred_a_b_d_c )
= ( member_a @ Sigma @ ( Delta @ S2 ) ) ) ).
% sat.simps(10)
thf(fact_60_sat__wand,axiom,
! [S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,Sigma: a,B2: assertion_a_b_d_c] :
( ! [A6: a,Sigma4: a] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A6 @ S2 @ Delta @ A2 )
& ( ( some_a @ Sigma4 )
= ( plus @ Sigma @ A6 ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S2 @ Delta @ B2 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ).
% sat_wand
thf(fact_61_sat_Osimps_I3_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) )
= ( ! [A5: a,Sigma3: a] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 )
& ( ( some_a @ Sigma3 )
= ( plus @ Sigma @ A5 ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S2 @ Delta @ B2 ) ) ) ) ).
% sat.simps(3)
thf(fact_62_equivalentI,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A2 ) )
=> ( equivalent_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ) ).
% equivalentI
thf(fact_63_sat_Osimps_I11_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( bounded_a_b_d_c @ A2 ) )
= ( ( valid @ Sigma )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ A2 ) ) ) ).
% sat.simps(11)
thf(fact_64_sat__imp,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ B2 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ).
% sat_imp
thf(fact_65_sat_Osimps_I5_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ).
% sat.simps(5)
thf(fact_66_sat_Osimps_I6_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( or_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ A2 )
| ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ).
% sat.simps(6)
thf(fact_67_assertion_Oinject_I7_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,Y71: assertion_a_b_d_c,Y72: assertion_a_b_d_c] :
( ( ( imp_a_b_d_c @ X71 @ X72 )
= ( imp_a_b_d_c @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% assertion.inject(7)
thf(fact_68_assertion_Oinject_I5_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,Y51: assertion_a_b_d_c,Y52: assertion_a_b_d_c] :
( ( ( or_a_b_d_c @ X51 @ X52 )
= ( or_a_b_d_c @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% assertion.inject(5)
thf(fact_69_assertion_Oinject_I4_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,Y41: assertion_a_b_d_c,Y42: assertion_a_b_d_c] :
( ( ( wand_a_b_d_c @ X41 @ X42 )
= ( wand_a_b_d_c @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% assertion.inject(4)
thf(fact_70_assertion_Oinject_I10_J,axiom,
! [X11: assertion_a_b_d_c,Y11: assertion_a_b_d_c] :
( ( ( bounded_a_b_d_c @ X11 )
= ( bounded_a_b_d_c @ Y11 ) )
= ( X11 = Y11 ) ) ).
% assertion.inject(10)
thf(fact_71_assertion_Odistinct_I127_J,axiom,
! [X11: assertion_a_b_d_c] :
( pred_a_b_d_c
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(127)
thf(fact_72_assertion_Odistinct_I109_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( imp_a_b_d_c @ X71 @ X72 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(109)
thf(fact_73_assertion_Odistinct_I107_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( imp_a_b_d_c @ X71 @ X72 )
!= pred_a_b_d_c ) ).
% assertion.distinct(107)
thf(fact_74_assertion_Odistinct_I87_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(87)
thf(fact_75_assertion_Odistinct_I85_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= pred_a_b_d_c ) ).
% assertion.distinct(85)
thf(fact_76_assertion_Odistinct_I79_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(79)
thf(fact_77_assertion_Odistinct_I73_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(73)
thf(fact_78_assertion_Odistinct_I71_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= pred_a_b_d_c ) ).
% assertion.distinct(71)
thf(fact_79_assertion_Odistinct_I65_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(65)
thf(fact_80_assertion_Odistinct_I61_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( or_a_b_d_c @ X51 @ X52 ) ) ).
% assertion.distinct(61)
thf(fact_81_logic_Oequivalent_Ocong,axiom,
equivalent_a_b_d_c = equivalent_a_b_d_c ).
% logic.equivalent.cong
thf(fact_82_logic_Onot__in__fv_Ocong,axiom,
not_in_fv_a_b_d_c = not_in_fv_a_b_d_c ).
% logic.not_in_fv.cong
thf(fact_83_assertion_Odistinct_I49_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(49)
thf(fact_84_assertion_Odistinct_I45_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( or_a_b_d_c @ X51 @ X52 ) ) ).
% assertion.distinct(45)
thf(fact_85_assertion_Odistinct_I43_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( wand_a_b_d_c @ X41 @ X42 ) ) ).
% assertion.distinct(43)
thf(fact_86_assertion_Odistinct_I57_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(57)
thf(fact_87_assertion_Odistinct_I55_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= pred_a_b_d_c ) ).
% assertion.distinct(55)
thf(fact_88_hoare__triple__input,axiom,
! [P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
= ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ ( bounded_a_b_d_c @ P2 ) @ C @ Q2 @ Delta ) ) ).
% hoare_triple_input
thf(fact_89_DotFull,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ one @ A2 ) @ Delta @ A2 ) ).
% DotFull
thf(fact_90_DotPure,axiom,
! [A2: assertion_a_b_d_c,P: b,Delta: ( c > d ) > set_a] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ A2 ) @ Delta @ A2 ) ) ).
% DotPure
thf(fact_91_DotImp,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% DotImp
thf(fact_92_DotOr,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% DotOr
thf(fact_93_DotWand,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% DotWand
thf(fact_94_sat_Osimps_I8_J,axiom,
! [Sigma: a,S2: a > option_a,Delta: ( a > option_a ) > set_a,X2: a,A2: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( exists7165000112504185261tion_a @ X2 @ A2 ) )
= ( ? [V: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(8)
thf(fact_95_sat_Osimps_I8_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,X2: c,A2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( exists_c_a_b_d @ X2 @ A2 ) )
= ( ? [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(8)
thf(fact_96_DotStar,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% DotStar
thf(fact_97_sat_Osimps_I9_J,axiom,
! [Sigma: a,S2: a > option_a,Delta: ( a > option_a ) > set_a,X2: a,A2: assert1556940916145061938on_a_a] :
( ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( forall5484998627543102345tion_a @ X2 @ A2 ) )
= ( ! [V: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(9)
thf(fact_98_sat_Osimps_I9_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,X2: c,A2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) )
= ( ! [V: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(9)
thf(fact_99_sat__forall,axiom,
! [Sigma: a,S2: a > option_a,X2: a,Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a] :
( ! [V3: option_a] : ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V3 ) @ Delta @ A2 )
=> ( sat_a_b_a_option_a @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( forall5484998627543102345tion_a @ X2 @ A2 ) ) ) ).
% sat_forall
thf(fact_100_sat__forall,axiom,
! [Sigma: a,S2: c > d,X2: c,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ! [V3: d] : ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V3 ) @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ).
% sat_forall
thf(fact_101_assertion_Oinject_I2_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,Y21: b,Y22: assertion_a_b_d_c] :
( ( ( mult_b_a_d_c @ X21 @ X222 )
= ( mult_b_a_d_c @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% assertion.inject(2)
thf(fact_102_assertion_Oinject_I9_J,axiom,
! [X91: c,X92: assertion_a_b_d_c,Y91: c,Y92: assertion_a_b_d_c] :
( ( ( forall_c_a_b_d @ X91 @ X92 )
= ( forall_c_a_b_d @ Y91 @ Y92 ) )
= ( ( X91 = Y91 )
& ( X92 = Y92 ) ) ) ).
% assertion.inject(9)
thf(fact_103_assertion_Oinject_I8_J,axiom,
! [X81: c,X82: assertion_a_b_d_c,Y81: c,Y82: assertion_a_b_d_c] :
( ( ( exists_c_a_b_d @ X81 @ X82 )
= ( exists_c_a_b_d @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 ) ) ) ).
% assertion.inject(8)
thf(fact_104_sat__mult,axiom,
! [Sigma: a,P: b,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ! [A6: a] :
( ( Sigma
= ( mult @ P @ A6 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ A6 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% sat_mult
thf(fact_105_sat_Osimps_I1_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,P: b,A2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) )
= ( ? [A5: a] :
( ( Sigma
= ( mult @ P @ A5 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ).
% sat.simps(1)
thf(fact_106_DotForall,axiom,
! [P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% DotForall
thf(fact_107_DotExists,axiom,
! [P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% DotExists
thf(fact_108_hoare__triple__output,axiom,
! [C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( valid_command_a_c_d @ valid @ C )
=> ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
= ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ ( bounded_a_b_d_c @ Q2 ) @ Delta ) ) ) ).
% hoare_triple_output
thf(fact_109_logic_Ovalid__hoare__triple_Ocong,axiom,
valid_6037315502795721655_b_d_c = valid_6037315502795721655_b_d_c ).
% logic.valid_hoare_triple.cong
thf(fact_110_assertion_Odistinct_I33_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(33)
thf(fact_111_assertion_Odistinct_I35_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(35)
thf(fact_112_assertion_Odistinct_I113_J,axiom,
! [X81: c,X82: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( exists_c_a_b_d @ X81 @ X82 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(113)
thf(fact_113_assertion_Odistinct_I23_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( star_a_b_d_c @ X31 @ X32 ) ) ).
% assertion.distinct(23)
thf(fact_114_assertion_Odistinct_I25_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( wand_a_b_d_c @ X41 @ X42 ) ) ).
% assertion.distinct(25)
thf(fact_115_assertion_Odistinct_I27_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( or_a_b_d_c @ X51 @ X52 ) ) ).
% assertion.distinct(27)
thf(fact_116_assertion_Odistinct_I31_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(31)
thf(fact_117_assertion_Odistinct_I39_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(39)
thf(fact_118_assertion_Odistinct_I37_J,axiom,
! [X21: b,X222: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= pred_a_b_d_c ) ).
% assertion.distinct(37)
thf(fact_119_assertion_Odistinct_I53_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(53)
thf(fact_120_assertion_Odistinct_I51_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(51)
thf(fact_121_assertion_Odistinct_I69_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(69)
thf(fact_122_assertion_Odistinct_I83_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(83)
thf(fact_123_assertion_Odistinct_I105_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( imp_a_b_d_c @ X71 @ X72 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(105)
thf(fact_124_assertion_Odistinct_I67_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(67)
thf(fact_125_assertion_Odistinct_I81_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(81)
thf(fact_126_assertion_Odistinct_I103_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( imp_a_b_d_c @ X71 @ X72 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(103)
thf(fact_127_assertion_Odistinct_I123_J,axiom,
! [X91: c,X92: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( forall_c_a_b_d @ X91 @ X92 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(123)
thf(fact_128_assertion_Odistinct_I117_J,axiom,
! [X81: c,X82: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( exists_c_a_b_d @ X81 @ X82 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(117)
thf(fact_129_assertion_Odistinct_I121_J,axiom,
! [X91: c,X92: assertion_a_b_d_c] :
( ( forall_c_a_b_d @ X91 @ X92 )
!= pred_a_b_d_c ) ).
% assertion.distinct(121)
thf(fact_130_assertion_Odistinct_I115_J,axiom,
! [X81: c,X82: assertion_a_b_d_c] :
( ( exists_c_a_b_d @ X81 @ X82 )
!= pred_a_b_d_c ) ).
% assertion.distinct(115)
thf(fact_131_DotAnd,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ).
% DotAnd
thf(fact_132_DotDot,axiom,
! [P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( smult @ P @ Q ) @ A2 ) ) ).
% DotDot
thf(fact_133_mult__one__same2,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ ( mult_b_a_d_c @ one @ A2 ) ) ).
% mult_one_same2
thf(fact_134_mult__one__same1,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ one @ A2 ) @ Delta @ A2 ) ).
% mult_one_same1
thf(fact_135_WildPure,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ A2 ) @ Delta @ A2 ) ) ).
% WildPure
thf(fact_136_WildOr,axiom,
! [A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ).
% WildOr
thf(fact_137_WildExists,axiom,
! [X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% WildExists
thf(fact_138_pure__mult2,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,P: b] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% pure_mult2
thf(fact_139_pure__mult1,axiom,
! [A2: assertion_a_b_d_c,P: b,Delta: ( c > d ) > set_a] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ A2 ) @ Delta @ A2 ) ) ).
% pure_mult1
thf(fact_140_dot__wand2,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ).
% dot_wand2
thf(fact_141_can__factorize,axiom,
! [Q: b,P: b] :
? [R: b] :
( Q
= ( smult @ R @ P ) ) ).
% can_factorize
thf(fact_142_smult__asso,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ ( smult @ P @ Q ) @ R2 )
= ( smult @ P @ ( smult @ Q @ R2 ) ) ) ).
% smult_asso
thf(fact_143_smult__comm,axiom,
! [P: b,Q: b] :
( ( smult @ P @ Q )
= ( smult @ Q @ P ) ) ).
% smult_comm
thf(fact_144_double__mult,axiom,
! [P: b,Q: b,A: a] :
( ( mult @ P @ ( mult @ Q @ A ) )
= ( mult @ ( smult @ P @ Q ) @ A ) ) ).
% double_mult
thf(fact_145_sone__neutral,axiom,
! [P: b] :
( ( smult @ one @ P )
= P ) ).
% sone_neutral
thf(fact_146_assertion_Oinject_I11_J,axiom,
! [X12: assertion_a_b_d_c,Y12: assertion_a_b_d_c] :
( ( ( wildcard_a_b_d_c @ X12 )
= ( wildcard_a_b_d_c @ Y12 ) )
= ( X12 = Y12 ) ) ).
% assertion.inject(11)
thf(fact_147_assertion_Oinject_I6_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,Y61: assertion_a_b_d_c,Y62: assertion_a_b_d_c] :
( ( ( and_a_b_d_c @ X61 @ X62 )
= ( and_a_b_d_c @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% assertion.inject(6)
thf(fact_148_sinv__inverse,axiom,
! [P: b] :
( ( smult @ P @ ( sinv @ P ) )
= one ) ).
% sinv_inverse
thf(fact_149_entailsI,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ).
% entailsI
thf(fact_150_entails__def,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( ! [Sigma2: a,S3: c > d] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta @ A2 )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta @ B2 ) ) ) ) ).
% entails_def
thf(fact_151_DotPos,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c,Pi: b] :
( ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ Pi @ A2 ) @ Delta @ ( mult_b_a_d_c @ Pi @ B2 ) ) ) ).
% DotPos
thf(fact_152_sat_Osimps_I12_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( wildcard_a_b_d_c @ A2 ) )
= ( ? [A5: a,P3: b] :
( ( Sigma
= ( mult @ P3 @ A5 ) )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ).
% sat.simps(12)
thf(fact_153_WildPos,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 )
=> ( entails_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ A2 ) @ Delta @ ( wildcard_a_b_d_c @ B2 ) ) ) ).
% WildPos
thf(fact_154_equivalent__def,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( equivalent_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( ( entails_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ B2 )
& ( entails_a_b_d_c @ plus @ mult @ valid @ B2 @ Delta @ A2 ) ) ) ).
% equivalent_def
thf(fact_155_sat_Osimps_I7_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( and_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ A2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ).
% sat.simps(7)
thf(fact_156_WildWild,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ).
% WildWild
thf(fact_157_DotWild,axiom,
! [P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( wildcard_a_b_d_c @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ).
% DotWild
thf(fact_158_WildDot,axiom,
! [P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( equivalent_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ).
% WildDot
thf(fact_159_dot__star1,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% dot_star1
thf(fact_160_dot__star2,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) ) ).
% dot_star2
thf(fact_161_dot__forall1,axiom,
! [P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% dot_forall1
thf(fact_162_dot__forall2,axiom,
! [X2: c,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ).
% dot_forall2
thf(fact_163_dot__and1,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% dot_and1
thf(fact_164_dot__and2,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ).
% dot_and2
thf(fact_165_dot__exists1,axiom,
! [P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ).
% dot_exists1
thf(fact_166_dot__exists2,axiom,
! [X2: c,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) ) ).
% dot_exists2
thf(fact_167_dot__imp1,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ).
% dot_imp1
thf(fact_168_dot__imp2,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% dot_imp2
thf(fact_169_dot__or1,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% dot_or1
thf(fact_170_dot__or2,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) ) ).
% dot_or2
thf(fact_171_dot__wand1,axiom,
! [P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ).
% dot_wand1
thf(fact_172_WildStar1,axiom,
! [A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ).
% WildStar1
thf(fact_173_WildForall,axiom,
! [X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% WildForall
thf(fact_174_WildAnd,axiom,
! [A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( wildcard_a_b_d_c @ ( and_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( and_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ).
% WildAnd
thf(fact_175_dot__mult1,axiom,
! [P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( smult @ P @ Q ) @ A2 ) ) ).
% dot_mult1
thf(fact_176_dot__mult2,axiom,
! [P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( mult_b_a_d_c @ ( smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) ) ).
% dot_mult2
thf(fact_177_assertion_Odistinct_I101_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(101)
thf(fact_178_logic_Oentails_Ocong,axiom,
entails_a_b_d_c = entails_a_b_d_c ).
% logic.entails.cong
thf(fact_179_logic_Ovalid__command_Ocong,axiom,
valid_command_a_c_d = valid_command_a_c_d ).
% logic.valid_command.cong
thf(fact_180_assertion_Odistinct_I41_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(41)
thf(fact_181_assertion_Odistinct_I59_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(59)
thf(fact_182_assertion_Odistinct_I29_J,axiom,
! [X21: b,X222: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( mult_b_a_d_c @ X21 @ X222 )
!= ( and_a_b_d_c @ X61 @ X62 ) ) ).
% assertion.distinct(29)
thf(fact_183_assertion_Odistinct_I125_J,axiom,
! [X91: c,X92: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( forall_c_a_b_d @ X91 @ X92 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(125)
thf(fact_184_assertion_Odistinct_I119_J,axiom,
! [X81: c,X82: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( exists_c_a_b_d @ X81 @ X82 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(119)
thf(fact_185_assertion_Odistinct_I111_J,axiom,
! [X71: assertion_a_b_d_c,X72: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( imp_a_b_d_c @ X71 @ X72 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(111)
thf(fact_186_assertion_Odistinct_I89_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(89)
thf(fact_187_assertion_Odistinct_I75_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(75)
thf(fact_188_assertion_Odistinct_I131_J,axiom,
! [X11: assertion_a_b_d_c,X12: assertion_a_b_d_c] :
( ( bounded_a_b_d_c @ X11 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(131)
thf(fact_189_assertion_Odistinct_I129_J,axiom,
! [X12: assertion_a_b_d_c] :
( pred_a_b_d_c
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(129)
thf(fact_190_assertion_Odistinct_I47_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( star_a_b_d_c @ X31 @ X32 )
!= ( and_a_b_d_c @ X61 @ X62 ) ) ).
% assertion.distinct(47)
thf(fact_191_assertion_Odistinct_I95_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X91: c,X92: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(95)
thf(fact_192_assertion_Odistinct_I93_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X81: c,X82: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(93)
thf(fact_193_assertion_Odistinct_I91_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(91)
thf(fact_194_assertion_Odistinct_I77_J,axiom,
! [X51: assertion_a_b_d_c,X52: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( or_a_b_d_c @ X51 @ X52 )
!= ( and_a_b_d_c @ X61 @ X62 ) ) ).
% assertion.distinct(77)
thf(fact_195_assertion_Odistinct_I63_J,axiom,
! [X41: assertion_a_b_d_c,X42: assertion_a_b_d_c,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( wand_a_b_d_c @ X41 @ X42 )
!= ( and_a_b_d_c @ X61 @ X62 ) ) ).
% assertion.distinct(63)
thf(fact_196_assertion_Odistinct_I99_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c,X11: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(99)
thf(fact_197_assertion_Odistinct_I97_J,axiom,
! [X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( and_a_b_d_c @ X61 @ X62 )
!= pred_a_b_d_c ) ).
% assertion.distinct(97)
thf(fact_198_sat_Osimps_I4_J,axiom,
! [Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
( ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma @ S2 @ Delta @ ( sem_c_d_a_b @ B ) )
= ( B @ S2 @ Sigma ) ) ).
% sat.simps(4)
thf(fact_199_frame__rule,axiom,
! [C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,R3: assertion_a_b_d_c] :
( ( valid_command_a_c_d @ valid @ C )
=> ( ( safety844553430189520448_a_c_d @ plus @ valid @ C )
=> ( ( frame_property_a_c_d @ plus @ valid @ C )
=> ( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
=> ( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ R3 @ ( modified_a_c_d @ C ) )
=> ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ P2 @ R3 ) @ C @ ( star_a_b_d_c @ Q2 @ R3 ) @ Delta ) ) ) ) ) ) ).
% frame_rule
thf(fact_200_assertion_Oexhaust,axiom,
! [Y: assertion_a_b_d_c] :
( ! [X13: ( c > d ) > a > $o] :
( Y
!= ( sem_c_d_a_b @ X13 ) )
=> ( ! [X212: b,X223: assertion_a_b_d_c] :
( Y
!= ( mult_b_a_d_c @ X212 @ X223 ) )
=> ( ! [X312: assertion_a_b_d_c,X322: assertion_a_b_d_c] :
( Y
!= ( star_a_b_d_c @ X312 @ X322 ) )
=> ( ! [X412: assertion_a_b_d_c,X422: assertion_a_b_d_c] :
( Y
!= ( wand_a_b_d_c @ X412 @ X422 ) )
=> ( ! [X512: assertion_a_b_d_c,X522: assertion_a_b_d_c] :
( Y
!= ( or_a_b_d_c @ X512 @ X522 ) )
=> ( ! [X612: assertion_a_b_d_c,X622: assertion_a_b_d_c] :
( Y
!= ( and_a_b_d_c @ X612 @ X622 ) )
=> ( ! [X712: assertion_a_b_d_c,X722: assertion_a_b_d_c] :
( Y
!= ( imp_a_b_d_c @ X712 @ X722 ) )
=> ( ! [X812: c,X822: assertion_a_b_d_c] :
( Y
!= ( exists_c_a_b_d @ X812 @ X822 ) )
=> ( ! [X912: c,X922: assertion_a_b_d_c] :
( Y
!= ( forall_c_a_b_d @ X912 @ X922 ) )
=> ( ( Y != pred_a_b_d_c )
=> ( ! [X112: assertion_a_b_d_c] :
( Y
!= ( bounded_a_b_d_c @ X112 ) )
=> ~ ! [X122: assertion_a_b_d_c] :
( Y
!= ( wildcard_a_b_d_c @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% assertion.exhaust
thf(fact_201_fun__upd__apply,axiom,
( fun_upd_a_option_a
= ( ^ [F2: a > option_a,X3: a,Y2: option_a,Z: a] : ( if_option_a @ ( Z = X3 ) @ Y2 @ ( F2 @ Z ) ) ) ) ).
% fun_upd_apply
thf(fact_202_fun__upd__triv,axiom,
! [F: a > option_a,X2: a] :
( ( fun_upd_a_option_a @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_203_fun__upd__upd,axiom,
! [F: a > option_a,X2: a,Y: option_a,Z2: option_a] :
( ( fun_upd_a_option_a @ ( fun_upd_a_option_a @ F @ X2 @ Y ) @ X2 @ Z2 )
= ( fun_upd_a_option_a @ F @ X2 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_204_logic__axioms,axiom,
logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).
% logic_axioms
thf(fact_205_option_Oinject,axiom,
! [X22: a,Y23: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y23 ) )
= ( X22 = Y23 ) ) ).
% option.inject
thf(fact_206_sadd__comm,axiom,
! [P: b,Q: b] :
( ( sadd @ P @ Q )
= ( sadd @ Q @ P ) ) ).
% sadd_comm
thf(fact_207_smult__distrib,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ P @ ( sadd @ Q @ R2 ) )
= ( sadd @ ( smult @ P @ Q ) @ ( smult @ P @ R2 ) ) ) ).
% smult_distrib
thf(fact_208_assertion_Oinject_I1_J,axiom,
! [X1: ( c > d ) > a > $o,Y1: ( c > d ) > a > $o] :
( ( ( sem_c_d_a_b @ X1 )
= ( sem_c_d_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% assertion.inject(1)
thf(fact_209_distrib__mult,axiom,
! [P: b,Q: b,X2: a] :
( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X2 ) )
= ( plus @ ( mult @ P @ X2 ) @ ( mult @ Q @ X2 ) ) ) ).
% distrib_mult
thf(fact_210_logic_Osat_Osimps_I4_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,B: ( c > d ) > a > $o] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( sem_c_d_a_b @ B ) )
= ( B @ S2 @ Sigma ) ) ) ).
% logic.sat.simps(4)
thf(fact_211_logic_Osmult__distrib,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sadd @ Q @ R2 ) )
= ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R2 ) ) ) ) ).
% logic.smult_distrib
thf(fact_212_logic_Osone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ One @ P )
= P ) ) ).
% logic.sone_neutral
thf(fact_213_logic_Osinv__inverse,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sinv @ P ) )
= One ) ) ).
% logic.sinv_inverse
thf(fact_214_logic_Oone__neutral,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ One @ A )
= A ) ) ).
% logic.one_neutral
thf(fact_215_logic_Odouble__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ P @ ( Mult @ Q @ A ) )
= ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).
% logic.double_mult
thf(fact_216_logic_Ocommutative,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Plus2 @ A @ B )
= ( Plus2 @ B @ A ) ) ) ).
% logic.commutative
thf(fact_217_logic_Ounique__inv,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,P: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( A
= ( Mult @ P @ B ) )
= ( B
= ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).
% logic.unique_inv
thf(fact_218_logic_Osmult__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ Q )
= ( Smult @ Q @ P ) ) ) ).
% logic.smult_comm
thf(fact_219_logic_Osmult__asso,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,R2: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ ( Smult @ P @ Q ) @ R2 )
= ( Smult @ P @ ( Smult @ Q @ R2 ) ) ) ) ).
% logic.smult_asso
thf(fact_220_logic_Ocan__divide,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Mult @ P @ A )
= ( Mult @ P @ B ) )
=> ( A = B ) ) ) ).
% logic.can_divide
thf(fact_221_logic_Osadd__comm,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Sadd @ P @ Q )
= ( Sadd @ Q @ P ) ) ) ).
% logic.sadd_comm
thf(fact_222_logic_Oframe__property_Ocong,axiom,
frame_property_a_c_d = frame_property_a_c_d ).
% logic.frame_property.cong
thf(fact_223_logic_Osafety__monotonicity_Ocong,axiom,
safety844553430189520448_a_c_d = safety844553430189520448_a_c_d ).
% logic.safety_monotonicity.cong
thf(fact_224_logic_Oasso1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( Plus2 @ Ab @ C )
= ( Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso1
thf(fact_225_logic_Omove__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,A1: a,A22: a,B: a,B1: a,B22: a,X2: a,X1: a,X22: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ A1 @ A22 ) )
=> ( ( ( some_a @ B )
= ( Plus2 @ B1 @ B22 ) )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( Plus2 @ A1 @ B1 ) )
=> ( ( ( some_a @ X22 )
= ( Plus2 @ A22 @ B22 ) )
=> ( ( some_a @ X2 )
= ( Plus2 @ X1 @ X22 ) ) ) ) ) ) ) ) ).
% logic.move_sum
thf(fact_226_logic_Oplus__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus2 @ B @ C ) )
=> ( ( some_a @ ( Mult @ P @ A ) )
= ( Plus2 @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).
% logic.plus_mult
thf(fact_227_logic_Odistrib__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X2 ) )
= ( Plus2 @ ( Mult @ P @ X2 ) @ ( Mult @ Q @ X2 ) ) ) ) ).
% logic.distrib_mult
thf(fact_228_assertion_Odistinct_I1_J,axiom,
! [X1: ( c > d ) > a > $o,X21: b,X222: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( mult_b_a_d_c @ X21 @ X222 ) ) ).
% assertion.distinct(1)
thf(fact_229_logic_Ocompatible__iff,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
= ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_iff
thf(fact_230_logic_Ocompatible__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ A @ B )
=> ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_imp
thf(fact_231_logic_Ocompatible__multiples,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A: a,Q: b,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
=> ( pre_compatible_a @ Plus2 @ A @ B ) ) ) ).
% logic.compatible_multiples
thf(fact_232_logic_Osmaller__interp__trans,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: ( c > d ) > set_a,B: ( c > d ) > set_a,C: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( smaller_interp_c_d_a @ A @ B )
=> ( ( smaller_interp_c_d_a @ B @ C )
=> ( smaller_interp_c_d_a @ A @ C ) ) ) ) ).
% logic.smaller_interp_trans
thf(fact_233_logic_Osmaller__interp__refl,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( smaller_interp_c_d_a @ Delta @ Delta ) ) ).
% logic.smaller_interp_refl
thf(fact_234_logic_Osmaller__interpI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [S: c > d,X: a] :
( ( member_a @ X @ ( Delta @ S ) )
=> ( member_a @ X @ ( Delta2 @ S ) ) )
=> ( smaller_interp_c_d_a @ Delta @ Delta2 ) ) ) ).
% logic.smaller_interpI
thf(fact_235_logic_Ovalid__mono,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Valid @ A )
& ( pre_larger_a @ Plus2 @ A @ B ) )
=> ( Valid @ B ) ) ) ).
% logic.valid_mono
thf(fact_236_logic_Olarger__same,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
= ( pre_larger_a @ Plus2 @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.larger_same
thf(fact_237_assertion_Odistinct_I21_J,axiom,
! [X1: ( c > d ) > a > $o,X12: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( wildcard_a_b_d_c @ X12 ) ) ).
% assertion.distinct(21)
thf(fact_238_assertion_Odistinct_I3_J,axiom,
! [X1: ( c > d ) > a > $o,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( star_a_b_d_c @ X31 @ X32 ) ) ).
% assertion.distinct(3)
thf(fact_239_assertion_Odistinct_I15_J,axiom,
! [X1: ( c > d ) > a > $o,X91: c,X92: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( forall_c_a_b_d @ X91 @ X92 ) ) ).
% assertion.distinct(15)
thf(fact_240_assertion_Odistinct_I9_J,axiom,
! [X1: ( c > d ) > a > $o,X61: assertion_a_b_d_c,X62: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( and_a_b_d_c @ X61 @ X62 ) ) ).
% assertion.distinct(9)
thf(fact_241_assertion_Odistinct_I13_J,axiom,
! [X1: ( c > d ) > a > $o,X81: c,X82: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( exists_c_a_b_d @ X81 @ X82 ) ) ).
% assertion.distinct(13)
thf(fact_242_assertion_Odistinct_I5_J,axiom,
! [X1: ( c > d ) > a > $o,X41: assertion_a_b_d_c,X42: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( wand_a_b_d_c @ X41 @ X42 ) ) ).
% assertion.distinct(5)
thf(fact_243_assertion_Odistinct_I7_J,axiom,
! [X1: ( c > d ) > a > $o,X51: assertion_a_b_d_c,X52: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( or_a_b_d_c @ X51 @ X52 ) ) ).
% assertion.distinct(7)
thf(fact_244_assertion_Odistinct_I11_J,axiom,
! [X1: ( c > d ) > a > $o,X71: assertion_a_b_d_c,X72: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( imp_a_b_d_c @ X71 @ X72 ) ) ).
% assertion.distinct(11)
thf(fact_245_assertion_Odistinct_I19_J,axiom,
! [X1: ( c > d ) > a > $o,X11: assertion_a_b_d_c] :
( ( sem_c_d_a_b @ X1 )
!= ( bounded_a_b_d_c @ X11 ) ) ).
% assertion.distinct(19)
thf(fact_246_assertion_Odistinct_I17_J,axiom,
! [X1: ( c > d ) > a > $o] :
( ( sem_c_d_a_b @ X1 )
!= pred_a_b_d_c ) ).
% assertion.distinct(17)
thf(fact_247_logic_Osat__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,P: b,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A6: a] :
( ( Sigma
= ( Mult @ P @ A6 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A6 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.sat_mult
thf(fact_248_logic_Osat_Osimps_I1_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,P: b,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) )
= ( ? [A5: a] :
( ( Sigma
= ( Mult @ P @ A5 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ) ).
% logic.sat.simps(1)
thf(fact_249_logic_Osat_Osimps_I12_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( wildcard_a_b_d_c @ A2 ) )
= ( ? [A5: a,P3: b] :
( ( Sigma
= ( Mult @ P3 @ A5 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ) ).
% logic.sat.simps(12)
thf(fact_250_logic_Osat_Osimps_I7_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( and_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ A2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(7)
thf(fact_251_logic_Osat_Osimps_I6_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( or_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ A2 )
| ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(6)
thf(fact_252_logic_Osat_Osimps_I5_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) )
= ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(5)
thf(fact_253_logic_Osat__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ B2 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.sat_imp
thf(fact_254_logic_Oentails__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( ! [Sigma2: a,S3: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta @ B2 ) ) ) ) ) ).
% logic.entails_def
thf(fact_255_logic_OentailsI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ).
% logic.entailsI
thf(fact_256_logic_Osat_Osimps_I11_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( bounded_a_b_d_c @ A2 ) )
= ( ( Valid @ Sigma )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(11)
thf(fact_257_logic_Osat_Osimps_I10_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ pred_a_b_d_c )
= ( member_a @ Sigma @ ( Delta @ S2 ) ) ) ) ).
% logic.sat.simps(10)
thf(fact_258_logic_Oasso2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,Ab: a,C: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus2 @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ Plus2 @ B @ C ) )
=> ~ ( pre_compatible_a @ Plus2 @ Ab @ C ) ) ) ).
% logic.asso2
thf(fact_259_logic_Oasso3,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,C: a,Bc: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ~ ( pre_compatible_a @ Plus2 @ A @ B )
=> ( ( ( Plus2 @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ Plus2 @ A @ Bc ) ) ) ) ).
% logic.asso3
thf(fact_260_logic_Osum__both__larger,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X4: a,A3: a,B3: a,X2: a,A: a,B: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ X4 )
= ( Plus2 @ A3 @ B3 ) )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ A3 @ A )
=> ( ( pre_larger_a @ Plus2 @ B3 @ B )
=> ( pre_larger_a @ Plus2 @ X4 @ X2 ) ) ) ) ) ) ).
% logic.sum_both_larger
thf(fact_261_logic_Olarger__first__sum,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Y: a,A: a,B: a,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ Y )
= ( Plus2 @ A @ B ) )
=> ( ( pre_larger_a @ Plus2 @ X2 @ Y )
=> ? [A4: a] :
( ( ( some_a @ X2 )
= ( Plus2 @ A4 @ B ) )
& ( pre_larger_a @ Plus2 @ A4 @ A ) ) ) ) ) ).
% logic.larger_first_sum
thf(fact_262_logic_OequivalentI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( ! [Sigma5: a,S: c > d] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ B2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ A2 ) )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ) ).
% logic.equivalentI
thf(fact_263_logic_Oequivalent__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 )
& ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ B2 @ Delta @ A2 ) ) ) ) ).
% logic.equivalent_def
thf(fact_264_logic_Opure__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
= ( ! [Sigma2: a,Sigma3: a,S3: c > d,Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S3 @ Delta6 @ A2 ) ) ) ) ) ).
% logic.pure_def
thf(fact_265_logic_Ocompatible__smaller,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A: a,B: a,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ A @ B )
=> ( ( pre_compatible_a @ Plus2 @ X2 @ A )
=> ( pre_compatible_a @ Plus2 @ X2 @ B ) ) ) ) ).
% logic.compatible_smaller
thf(fact_266_logic_Olarger__implies__compatible,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: a,Y: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus2 @ X2 @ Y )
=> ( pre_compatible_a @ Plus2 @ X2 @ Y ) ) ) ).
% logic.larger_implies_compatible
thf(fact_267_logic_Ohoare__triple__input,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
= ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ ( bounded_a_b_d_c @ P2 ) @ C @ Q2 @ Delta ) ) ) ).
% logic.hoare_triple_input
thf(fact_268_logic_Oframe__rule,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,R3: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_c_d @ Valid @ C )
=> ( ( safety844553430189520448_a_c_d @ Plus2 @ Valid @ C )
=> ( ( frame_property_a_c_d @ Plus2 @ Valid @ C )
=> ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
=> ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ R3 @ ( modified_a_c_d @ C ) )
=> ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ P2 @ R3 ) @ C @ ( star_a_b_d_c @ Q2 @ R3 ) @ Delta ) ) ) ) ) ) ) ).
% logic.frame_rule
thf(fact_269_logic_Oindep__interp__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( indep_interp_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
= ( ! [X3: a,S3: c > d,Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X3 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X3 @ S3 @ Delta6 @ A2 ) ) ) ) ) ).
% logic.indep_interp_def
thf(fact_270_logic_Omonotonic__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ F )
= ( ! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F @ Delta5 ) @ ( F @ Delta6 ) ) ) ) ) ) ).
% logic.monotonic_def
thf(fact_271_logic_OmonotonicI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F @ Delta3 ) @ ( F @ Delta4 ) ) )
=> ( monotonic_c_d_a @ F ) ) ) ).
% logic.monotonicI
thf(fact_272_logic_Onon__increasing__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( non_increasing_c_d_a @ F )
= ( ! [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta5 @ Delta6 )
=> ( smaller_interp_c_d_a @ ( F @ Delta6 ) @ ( F @ Delta5 ) ) ) ) ) ) ).
% logic.non_increasing_def
thf(fact_273_logic_Onon__increasingI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,F: ( ( c > d ) > set_a ) > ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Delta3: ( c > d ) > set_a,Delta4: ( c > d ) > set_a] :
( ( smaller_interp_c_d_a @ Delta3 @ Delta4 )
=> ( smaller_interp_c_d_a @ ( F @ Delta4 ) @ ( F @ Delta3 ) ) )
=> ( non_increasing_c_d_a @ F ) ) ) ).
% logic.non_increasingI
thf(fact_274_logic_Osat_Osimps_I2_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) )
= ( ? [A5: a,B4: a] :
( ( ( some_a @ Sigma )
= ( Plus2 @ A5 @ B4 ) )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S2 @ Delta @ B2 ) ) ) ) ) ).
% logic.sat.simps(2)
thf(fact_275_logic_Osat_Osimps_I3_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) )
= ( ! [A5: a,Sigma3: a] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 )
& ( ( some_a @ Sigma3 )
= ( Plus2 @ Sigma @ A5 ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S2 @ Delta @ B2 ) ) ) ) ) ).
% logic.sat.simps(3)
thf(fact_276_logic_Osat__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,Sigma: a,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A6: a,Sigma4: a] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A6 @ S2 @ Delta @ A2 )
& ( ( some_a @ Sigma4 )
= ( Plus2 @ Sigma @ A6 ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S2 @ Delta @ B2 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.sat_wand
thf(fact_277_logic_Osat__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: a > option_a,X2: a,Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V3: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V3 ) @ Delta @ A2 )
=> ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( forall5484998627543102345tion_a @ X2 @ A2 ) ) ) ) ).
% logic.sat_forall
thf(fact_278_logic_Osat__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,X2: c,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V3: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V3 ) @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ) ).
% logic.sat_forall
thf(fact_279_logic_Osat_Osimps_I9_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: a > option_a,Delta: ( a > option_a ) > set_a,X2: a,A2: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( forall5484998627543102345tion_a @ X2 @ A2 ) )
= ( ! [V: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_280_logic_Osat_Osimps_I9_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,X2: c,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) )
= ( ! [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_281_logic_Osat_Osimps_I8_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: a > option_a,Delta: ( a > option_a ) > set_a,X2: a,A2: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( exists7165000112504185261tion_a @ X2 @ A2 ) )
= ( ? [V: option_a] : ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_a_option_a @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_282_logic_Osat_Osimps_I8_J,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Sigma: a,S2: c > d,Delta: ( c > d ) > set_a,X2: c,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ S2 @ Delta @ ( exists_c_a_b_d @ X2 @ A2 ) )
= ( ? [V: d] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma @ ( fun_upd_c_d @ S2 @ X2 @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_283_logic_Osmaller__empty,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( smaller_interp_c_d_a @ empty_interp_c_d_a @ X2 ) ) ).
% logic.smaller_empty
thf(fact_284_logic__def,axiom,
( logic_a_b
= ( ^ [Plus: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
( ! [A5: a,B4: a] :
( ( Plus @ A5 @ B4 )
= ( Plus @ B4 @ A5 ) )
& ! [A5: a,B4: a,Ab2: a,C2: a,Bc2: a] :
( ( ( ( Plus @ A5 @ B4 )
= ( some_a @ Ab2 ) )
& ( ( Plus @ B4 @ C2 )
= ( some_a @ Bc2 ) ) )
=> ( ( Plus @ Ab2 @ C2 )
= ( Plus @ A5 @ Bc2 ) ) )
& ! [A5: a,B4: a,Ab2: a,C2: a] :
( ( ( ( Plus @ A5 @ B4 )
= ( some_a @ Ab2 ) )
& ~ ( pre_compatible_a @ Plus @ B4 @ C2 ) )
=> ~ ( pre_compatible_a @ Plus @ Ab2 @ C2 ) )
& ! [P3: b] :
( ( Smult2 @ P3 @ ( Sinv2 @ P3 ) )
= One2 )
& ! [P3: b] :
( ( Smult2 @ One2 @ P3 )
= P3 )
& ! [P3: b,Q3: b] :
( ( Sadd2 @ P3 @ Q3 )
= ( Sadd2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b] :
( ( Smult2 @ P3 @ Q3 )
= ( Smult2 @ Q3 @ P3 ) )
& ! [P3: b,Q3: b,R4: b] :
( ( Smult2 @ P3 @ ( Sadd2 @ Q3 @ R4 ) )
= ( Sadd2 @ ( Smult2 @ P3 @ Q3 ) @ ( Smult2 @ P3 @ R4 ) ) )
& ! [P3: b,Q3: b,R4: b] :
( ( Smult2 @ ( Smult2 @ P3 @ Q3 ) @ R4 )
= ( Smult2 @ P3 @ ( Smult2 @ Q3 @ R4 ) ) )
& ! [P3: b,Q3: b,A5: a] :
( ( Mult2 @ P3 @ ( Mult2 @ Q3 @ A5 ) )
= ( Mult2 @ ( Smult2 @ P3 @ Q3 ) @ A5 ) )
& ! [A5: a,B4: a,C2: a,P3: b] :
( ( ( some_a @ A5 )
= ( Plus @ B4 @ C2 ) )
=> ( ( some_a @ ( Mult2 @ P3 @ A5 ) )
= ( Plus @ ( Mult2 @ P3 @ B4 ) @ ( Mult2 @ P3 @ C2 ) ) ) )
& ! [P3: b,Q3: b,X3: a] :
( ( some_a @ ( Mult2 @ ( Sadd2 @ P3 @ Q3 ) @ X3 ) )
= ( Plus @ ( Mult2 @ P3 @ X3 ) @ ( Mult2 @ Q3 @ X3 ) ) )
& ! [A5: a] :
( ( Mult2 @ One2 @ A5 )
= A5 )
& ! [A5: a,B4: a] :
( ( ( Valid2 @ A5 )
& ( pre_larger_a @ Plus @ A5 @ B4 ) )
=> ( Valid2 @ B4 ) ) ) ) ) ).
% logic_def
thf(fact_285_logic_Osmaller__interp__applies__cons,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a,A: a,S2: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( smaller_interp_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta ) @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta2 ) )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S2 @ Delta @ A2 )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S2 @ Delta2 @ A2 ) ) ) ) ).
% logic.smaller_interp_applies_cons
thf(fact_286_logic_OintuitionisticI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A6: a,B5: a] :
( ( ( pre_larger_a @ Plus2 @ A6 @ B5 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B5 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A6 @ S2 @ Delta @ A2 ) )
=> ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S2 @ Delta @ A2 ) ) ) ).
% logic.intuitionisticI
thf(fact_287_logic_Ointuitionistic__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S2: c > d,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( intuit7508411120625971703_b_c_d @ Plus2 @ Mult @ Valid @ S2 @ Delta @ A2 )
= ( ! [A5: a,B4: a] :
( ( ( pre_larger_a @ Plus2 @ A5 @ B4 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B4 @ S2 @ Delta @ A2 ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ) ).
% logic.intuitionistic_def
thf(fact_288_logic_Ohoare__triple__output,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d,P2: assertion_a_b_d_c,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_c_d @ Valid @ C )
=> ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
= ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ ( bounded_a_b_d_c @ Q2 ) @ Delta ) ) ) ) ).
% logic.hoare_triple_output
thf(fact_289_logic_Omono__instantiate,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,X2: a,Delta: ( c > d ) > set_a,S2: c > d,Delta2: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( monotonic_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 ) )
=> ( ( member_a @ X2 @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ S2 ) )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta2 )
=> ( member_a @ X2 @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta2 @ S2 ) ) ) ) ) ) ).
% logic.mono_instantiate
thf(fact_290_fun__upd__def,axiom,
( fun_upd_a_option_a
= ( ^ [F2: a > option_a,A5: a,B4: option_a,X3: a] : ( if_option_a @ ( X3 = A5 ) @ B4 @ ( F2 @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_291_fun__upd__eqD,axiom,
! [F: a > option_a,X2: a,Y: option_a,G: a > option_a,Z2: option_a] :
( ( ( fun_upd_a_option_a @ F @ X2 @ Y )
= ( fun_upd_a_option_a @ G @ X2 @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_292_fun__upd__idem,axiom,
! [F: a > option_a,X2: a,Y: option_a] :
( ( ( F @ X2 )
= Y )
=> ( ( fun_upd_a_option_a @ F @ X2 @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_293_fun__upd__same,axiom,
! [F: a > option_a,X2: a,Y: option_a] :
( ( fun_upd_a_option_a @ F @ X2 @ Y @ X2 )
= Y ) ).
% fun_upd_same
thf(fact_294_fun__upd__other,axiom,
! [Z2: a,X2: a,F: a > option_a,Y: option_a] :
( ( Z2 != X2 )
=> ( ( fun_upd_a_option_a @ F @ X2 @ Y @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_295_fun__upd__twist,axiom,
! [A: a,C: a,M: a > option_a,B: option_a,D: option_a] :
( ( A != C )
=> ( ( fun_upd_a_option_a @ ( fun_upd_a_option_a @ M @ A @ B ) @ C @ D )
= ( fun_upd_a_option_a @ ( fun_upd_a_option_a @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_296_fun__upd__idem__iff,axiom,
! [F: a > option_a,X2: a,Y: option_a] :
( ( ( fun_upd_a_option_a @ F @ X2 @ Y )
= F )
= ( ( F @ X2 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_297_logic_Onot__in__fv__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,S4: set_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ S4 )
= ( ! [Sigma2: a,S3: c > d,Delta5: ( c > d ) > set_a,S5: c > d] :
( ( equal_outside_c_d @ S3 @ S5 @ S4 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta5 @ A2 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S5 @ Delta5 @ A2 ) ) ) ) ) ) ).
% logic.not_in_fv_def
thf(fact_298_logic_Oindep__implies__non__increasing,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( indep_interp_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( non_increasing_c_d_a @ ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 ) ) ) ) ).
% logic.indep_implies_non_increasing
thf(fact_299_combinable__instantiate__one,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,A: a,S2: c > d,B: a,X2: a,P: b,Q: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S2 @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S2 @ Delta @ A2 )
=> ( ( ( some_a @ X2 )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( ( ( sadd @ P @ Q )
= one )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S2 @ Delta @ A2 ) ) ) ) ) ) ).
% combinable_instantiate_one
thf(fact_300_combinable__def,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
= ( ! [P3: b,Q3: b] : ( entails_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P3 @ A2 ) @ ( mult_b_a_d_c @ Q3 @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( sadd @ P3 @ Q3 ) @ A2 ) ) ) ) ).
% combinable_def
thf(fact_301_combinable__instantiate,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,A: a,S2: c > d,B: a,X2: a,P: b,Q: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ A @ S2 @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ B @ S2 @ Delta @ A2 )
=> ( ( ( some_a @ X2 )
= ( plus @ ( mult @ P @ A ) @ ( mult @ Q @ B ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S2 @ Delta @ ( mult_b_a_d_c @ ( sadd @ P @ Q ) @ A2 ) ) ) ) ) ) ).
% combinable_instantiate
thf(fact_302_combinable__exists,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( ( unambiguous_a_b_c_d @ plus @ mult @ valid @ Delta @ A2 @ X2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( exists_c_a_b_d @ X2 @ A2 ) ) ) ) ).
% combinable_exists
thf(fact_303_combinable__imp,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% combinable_imp
thf(fact_304_combinableI__old,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ! [A6: a,B5: a,P4: b,Q4: b,X: a,Sigma5: a,S: c > d] :
( ( ( sat_a_b_c_d @ plus @ mult @ valid @ A6 @ S @ Delta @ A2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ B5 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma5 )
= ( plus @ ( mult @ P4 @ A6 ) @ ( mult @ Q4 @ B5 ) ) )
& ( Sigma5
= ( mult @ ( sadd @ P4 @ Q4 ) @ X ) ) )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ X @ S @ Delta @ A2 ) )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 ) ) ).
% combinableI_old
thf(fact_305_logic_OunambiguousI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: a,Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma_12: a,Sigma_22: a,V12: option_a,V22: option_a,S: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_12 @ ( fun_upd_a_option_a @ S @ X2 @ V12 ) @ Delta @ A2 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_22 @ ( fun_upd_a_option_a @ S @ X2 @ V22 ) @ Delta @ A2 ) )
=> ( V12 = V22 ) )
=> ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 ) ) ) ).
% logic.unambiguousI
thf(fact_306_logic_OunambiguousI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: c,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma_12: a,Sigma_22: a,V12: d,V22: d,S: c > d] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_12 @ Sigma_22 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_12 @ ( fun_upd_c_d @ S @ X2 @ V12 ) @ Delta @ A2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_22 @ ( fun_upd_c_d @ S @ X2 @ V22 ) @ Delta @ A2 ) )
=> ( V12 = V22 ) )
=> ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 ) ) ) ).
% logic.unambiguousI
thf(fact_307_combinable__mult,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,Pi: b] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( mult_b_a_d_c @ Pi @ A2 ) ) ) ).
% combinable_mult
thf(fact_308_combinable__wildcard,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% combinable_wildcard
thf(fact_309_combinable__star,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% combinable_star
thf(fact_310_combinable__forall,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ).
% combinable_forall
thf(fact_311_combinable__and,axiom,
! [Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 )
=> ( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% combinable_and
thf(fact_312_combinable__wand,axiom,
! [Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c,A2: assertion_a_b_d_c] :
( ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ).
% combinable_wand
thf(fact_313_combinable__pure,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( pure_a_b_d_c @ plus @ mult @ valid @ A2 )
=> ( combinable_a_b_c_d @ plus @ mult @ sadd @ valid @ Delta @ A2 ) ) ).
% combinable_pure
thf(fact_314_logic_Ocombinable__mult,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,Pi: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( mult_b_a_d_c @ Pi @ A2 ) ) ) ) ).
% logic.combinable_mult
thf(fact_315_logic_Ocombinable__wildcard,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ) ) ).
% logic.combinable_wildcard
thf(fact_316_logic_Ocombinable__star,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) ) ) ) ) ).
% logic.combinable_star
thf(fact_317_logic_Ocombinable__forall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ) ).
% logic.combinable_forall
thf(fact_318_logic_Ocombinable__and,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ) ) ).
% logic.combinable_and
thf(fact_319_logic_Ocombinable__wand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.combinable_wand
thf(fact_320_logic_Ocombinable__pure,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 ) ) ) ).
% logic.combinable_pure
thf(fact_321_logic_Ocombinable__instantiate__one,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,A: a,S2: c > d,B: a,X2: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S2 @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S2 @ Delta @ A2 )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( ( ( Sadd @ P @ Q )
= One )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S2 @ Delta @ A2 ) ) ) ) ) ) ) ).
% logic.combinable_instantiate_one
thf(fact_322_logic_OcombinableI__old,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A6: a,B5: a,P4: b,Q4: b,X: a,Sigma5: a,S: c > d] :
( ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A6 @ S @ Delta @ A2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B5 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma5 )
= ( Plus2 @ ( Mult @ P4 @ A6 ) @ ( Mult @ Q4 @ B5 ) ) )
& ( Sigma5
= ( Mult @ ( Sadd @ P4 @ Q4 ) @ X ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X @ S @ Delta @ A2 ) )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 ) ) ) ).
% logic.combinableI_old
thf(fact_323_logic_Ocombinable__imp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ B2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ) ) ).
% logic.combinable_imp
thf(fact_324_logic_Ocombinable__exists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 )
=> ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ ( exists_c_a_b_d @ X2 @ A2 ) ) ) ) ) ).
% logic.combinable_exists
thf(fact_325_logic_Ocombinable__instantiate,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,A: a,S2: c > d,B: a,X2: a,P: b,Q: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A @ S2 @ Delta @ A2 )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ B @ S2 @ Delta @ A2 )
=> ( ( ( some_a @ X2 )
= ( Plus2 @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) ) )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S2 @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P @ Q ) @ A2 ) ) ) ) ) ) ) ).
% logic.combinable_instantiate
thf(fact_326_logic_Ocombinable__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( combinable_a_b_c_d @ Plus2 @ Mult @ Sadd @ Valid @ Delta @ A2 )
= ( ! [P3: b,Q3: b] : ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P3 @ A2 ) @ ( mult_b_a_d_c @ Q3 @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( Sadd @ P3 @ Q3 ) @ A2 ) ) ) ) ) ).
% logic.combinable_def
thf(fact_327_logic_Ounambiguous_Ocong,axiom,
unambiguous_a_b_c_d = unambiguous_a_b_c_d ).
% logic.unambiguous.cong
thf(fact_328_logic_Ounambiguous__star,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 )
=> ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ ( star_a_b_d_c @ A2 @ B2 ) @ X2 ) ) ) ).
% logic.unambiguous_star
thf(fact_329_logic_Ounambiguous__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( a > option_a ) > set_a,A2: assert1556940916145061938on_a_a,X2: a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambi704529886615442436tion_a @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 )
= ( ! [Sigma_1: a,Sigma_2: a,V1: option_a,V2: option_a,S3: a > option_a] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_1 @ ( fun_upd_a_option_a @ S3 @ X2 @ V1 ) @ Delta @ A2 )
& ( sat_a_b_a_option_a @ Plus2 @ Mult @ Valid @ Sigma_2 @ ( fun_upd_a_option_a @ S3 @ X2 @ V2 ) @ Delta @ A2 ) )
=> ( V1 = V2 ) ) ) ) ) ).
% logic.unambiguous_def
thf(fact_330_logic_Ounambiguous__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,A2: assertion_a_b_d_c,X2: c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( unambiguous_a_b_c_d @ Plus2 @ Mult @ Valid @ Delta @ A2 @ X2 )
= ( ! [Sigma_1: a,Sigma_2: a,V1: d,V2: d,S3: c > d] :
( ( ( pre_compatible_a @ Plus2 @ Sigma_1 @ Sigma_2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_1 @ ( fun_upd_c_d @ S3 @ X2 @ V1 ) @ Delta @ A2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma_2 @ ( fun_upd_c_d @ S3 @ X2 @ V2 ) @ Delta @ A2 ) )
=> ( V1 = V2 ) ) ) ) ) ).
% logic.unambiguous_def
thf(fact_331_logic_OWildPure,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ A2 ) @ Delta @ A2 ) ) ) ).
% logic.WildPure
thf(fact_332_logic_ODotPure,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,P: b,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ A2 ) @ Delta @ A2 ) ) ) ).
% logic.DotPure
thf(fact_333_logic_Opure__mult1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,P: b,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ A2 ) @ Delta @ A2 ) ) ) ).
% logic.pure_mult1
thf(fact_334_logic_Opure__mult2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.pure_mult2
thf(fact_335_logic_OWildOr,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ) ).
% logic.WildOr
thf(fact_336_logic_OWildExists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( wildcard_a_b_d_c @ A2 ) ) ) ) ).
% logic.WildExists
thf(fact_337_logic_Ocan__factorize,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Q: b,P: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ? [R: b] :
( Q
= ( Smult @ R @ P ) ) ) ).
% logic.can_factorize
thf(fact_338_logic_Omult__one__same2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ ( mult_b_a_d_c @ One @ A2 ) ) ) ).
% logic.mult_one_same2
thf(fact_339_logic_Omult__one__same1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ One @ A2 ) @ Delta @ A2 ) ) ).
% logic.mult_one_same1
thf(fact_340_logic_Odot__mult2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ ( Smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) ) ) ).
% logic.dot_mult2
thf(fact_341_logic_Odot__mult1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( Smult @ P @ Q ) @ A2 ) ) ) ).
% logic.dot_mult1
thf(fact_342_logic_ODotPos,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c,Pi: b] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ Pi @ A2 ) @ Delta @ ( mult_b_a_d_c @ Pi @ B2 ) ) ) ) ).
% logic.DotPos
thf(fact_343_logic_OWildPos,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,B2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ B2 )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ A2 ) @ Delta @ ( wildcard_a_b_d_c @ B2 ) ) ) ) ).
% logic.WildPos
thf(fact_344_logic_ODotFull,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ One @ A2 ) @ Delta @ A2 ) ) ).
% logic.DotFull
thf(fact_345_logic_ODotDot,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,Q: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( mult_b_a_d_c @ Q @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ ( Smult @ P @ Q ) @ A2 ) ) ) ).
% logic.DotDot
thf(fact_346_logic_OWildWild,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% logic.WildWild
thf(fact_347_logic_Odot__star2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.dot_star2
thf(fact_348_logic_Odot__star1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.dot_star1
thf(fact_349_logic_Odot__forall1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.dot_forall1
thf(fact_350_logic_Odot__forall2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: c,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) ) ) ).
% logic.dot_forall2
thf(fact_351_logic_Odot__and2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.dot_and2
thf(fact_352_logic_Odot__and1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.dot_and1
thf(fact_353_logic_Odot__exists1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.dot_exists1
thf(fact_354_logic_Odot__exists2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: c,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) ) ) ).
% logic.dot_exists2
thf(fact_355_logic_Odot__wand2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.dot_wand2
thf(fact_356_logic_Odot__wand1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.dot_wand1
thf(fact_357_logic_Odot__imp2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.dot_imp2
thf(fact_358_logic_Odot__imp1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.dot_imp1
thf(fact_359_logic_Odot__or2,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.dot_or2
thf(fact_360_logic_Odot__or1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.dot_or1
thf(fact_361_logic_OWildDot,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% logic.WildDot
thf(fact_362_logic_ODotWild,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( wildcard_a_b_d_c @ A2 ) ) @ Delta @ ( wildcard_a_b_d_c @ A2 ) ) ) ).
% logic.DotWild
thf(fact_363_logic_OWildStar1,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ) ).
% logic.WildStar1
thf(fact_364_logic_OWildForall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( wildcard_a_b_d_c @ A2 ) ) ) ) ).
% logic.WildForall
thf(fact_365_logic_ODotStar,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( star_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( star_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.DotStar
thf(fact_366_logic_OWildAnd,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_d_c @ Plus2 @ Mult @ Valid @ ( wildcard_a_b_d_c @ ( and_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( and_a_b_d_c @ ( wildcard_a_b_d_c @ A2 ) @ ( wildcard_a_b_d_c @ B2 ) ) ) ) ).
% logic.WildAnd
thf(fact_367_logic_ODotForall,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( forall_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( forall_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.DotForall
thf(fact_368_logic_ODotAnd,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( and_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) @ Delta @ ( mult_b_a_d_c @ P @ ( and_a_b_d_c @ A2 @ B2 ) ) ) ) ).
% logic.DotAnd
thf(fact_369_logic_ODotExists,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,X2: c,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( exists_c_a_b_d @ X2 @ A2 ) ) @ Delta @ ( exists_c_a_b_d @ X2 @ ( mult_b_a_d_c @ P @ A2 ) ) ) ) ).
% logic.DotExists
thf(fact_370_logic_ODotOr,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( or_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( or_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.DotOr
thf(fact_371_logic_ODotImp,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( imp_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( imp_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.DotImp
thf(fact_372_logic_ODotWand,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P: b,A2: assertion_a_b_d_c,B2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_d_c @ Plus2 @ Mult @ Valid @ ( mult_b_a_d_c @ P @ ( wand_a_b_d_c @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_d_c @ ( mult_b_a_d_c @ P @ A2 ) @ ( mult_b_a_d_c @ P @ B2 ) ) ) ) ).
% logic.DotWand
thf(fact_373_compatible__def,axiom,
! [A: a,B: a] :
( ( pre_compatible_a @ plus @ A @ B )
= ( ( plus @ A @ B )
!= none_a ) ) ).
% compatible_def
thf(fact_374_not__in__fv__mod,axiom,
! [A2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S2: c > d,Sigma6: a,S6: c > d,X2: a,Delta: ( c > d ) > set_a] :
( ( not_in_fv_a_b_d_c @ plus @ mult @ valid @ A2 @ ( modified_a_c_d @ C ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S2 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S2 @ Delta @ A2 )
= ( sat_a_b_c_d @ plus @ mult @ valid @ X2 @ S6 @ Delta @ A2 ) ) ) ) ).
% not_in_fv_mod
thf(fact_375_safety__monotonicity__def,axiom,
! [C: set_Pr1275464188344874039_a_c_d] :
( ( safety844553430189520448_a_c_d @ plus @ valid @ C )
= ( ! [Sigma2: a,Sigma3: a,S3: c > d] :
( ( ( valid @ Sigma3 )
& ( pre_larger_a @ plus @ Sigma3 @ Sigma2 )
& ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S3 ) ) ) ) ) ).
% safety_monotonicity_def
thf(fact_376_empty__interp__def,axiom,
( empty_interp_c_d_a
= ( ^ [S3: c > d] : bot_bot_set_a ) ) ).
% empty_interp_def
thf(fact_377_smaller__interp__def,axiom,
( smaller_interp_c_d_a
= ( ^ [Delta5: ( c > d ) > set_a,Delta6: ( c > d ) > set_a] :
! [S3: c > d] : ( ord_less_eq_set_a @ ( Delta5 @ S3 ) @ ( Delta6 @ S3 ) ) ) ) ).
% smaller_interp_def
thf(fact_378_valid__command__def,axiom,
! [C: set_Pr1275464188344874039_a_c_d] :
( ( valid_command_a_c_d @ valid @ C )
= ( ! [A5: a,B4: a,Sa: c > d,Sb: c > d] :
( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A5 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B4 @ Sb ) ) ) @ C )
& ( valid @ A5 ) )
=> ( valid @ B4 ) ) ) ) ).
% valid_command_def
thf(fact_379_not__Some__eq,axiom,
! [X2: option_a] :
( ( ! [Y2: a] :
( X2
!= ( some_a @ Y2 ) ) )
= ( X2 = none_a ) ) ).
% not_Some_eq
thf(fact_380_not__None__eq,axiom,
! [X2: option_a] :
( ( X2 != none_a )
= ( ? [Y2: a] :
( X2
= ( some_a @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_381_frame__property__def,axiom,
! [C: set_Pr1275464188344874039_a_c_d] :
( ( frame_property_a_c_d @ plus @ valid @ C )
= ( ! [Sigma2: a,Sigma_0: a,R4: a,Sigma3: a,S3: c > d,S5: c > d] :
( ( ( valid @ Sigma2 )
& ( valid @ Sigma3 )
& ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) )
& ( ( some_a @ Sigma2 )
= ( plus @ Sigma_0 @ R4 ) )
& ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C ) )
=> ? [Sigma_02: a] :
( ( ( some_a @ Sigma3 )
= ( plus @ Sigma_02 @ R4 ) )
& ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S5 ) ) ) @ C ) ) ) ) ) ).
% frame_property_def
thf(fact_382_valid__hoare__tripleI,axiom,
! [Delta: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c] :
( ! [Sigma5: a,S: c > d] :
( ( ( valid @ Sigma5 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ P2 ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S ) ) )
=> ( ! [Sigma5: a,S: c > d,Sigma4: a,S7: c > d] :
( ( ( valid @ Sigma5 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ P2 ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S7 ) ) ) @ C )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma4 @ S7 @ Delta @ Q2 ) ) )
=> ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta ) ) ) ).
% valid_hoare_tripleI
thf(fact_383_valid__hoare__triple__def,axiom,
! [P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( valid_6037315502795721655_b_d_c @ plus @ mult @ valid @ P2 @ C @ Q2 @ Delta )
= ( ! [Sigma2: a,S3: c > d] :
( ( ( valid @ Sigma2 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma2 @ S3 @ Delta @ P2 ) )
=> ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) )
& ! [Sigma3: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C )
=> ( sat_a_b_c_d @ plus @ mult @ valid @ Sigma3 @ S5 @ Delta @ Q2 ) ) ) ) ) ) ).
% valid_hoare_triple_def
thf(fact_384_logic_Ovalid__command__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_c_d @ Valid @ C )
= ( ! [A5: a,B4: a,Sa: c > d,Sb: c > d] :
( ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ A5 @ Sa ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ B4 @ Sb ) ) ) @ C )
& ( Valid @ A5 ) )
=> ( Valid @ B4 ) ) ) ) ) ).
% logic.valid_command_def
thf(fact_385_combine__options__cases,axiom,
! [X2: option_a,P2: option_a > option_a > $o,Y: option_a] :
( ( ( X2 = none_a )
=> ( P2 @ X2 @ Y ) )
=> ( ( ( Y = none_a )
=> ( P2 @ X2 @ Y ) )
=> ( ! [A6: a,B5: a] :
( ( X2
= ( some_a @ A6 ) )
=> ( ( Y
= ( some_a @ B5 ) )
=> ( P2 @ X2 @ Y ) ) )
=> ( P2 @ X2 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_386_split__option__all,axiom,
( ( ^ [P5: option_a > $o] :
! [X5: option_a] : ( P5 @ X5 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
& ! [X3: a] : ( P6 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_387_split__option__ex,axiom,
( ( ^ [P5: option_a > $o] :
? [X5: option_a] : ( P5 @ X5 ) )
= ( ^ [P6: option_a > $o] :
( ( P6 @ none_a )
| ? [X3: a] : ( P6 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_388_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X23: a] :
( Y
!= ( some_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_389_option_OdiscI,axiom,
! [Option: option_a,X22: a] :
( ( Option
= ( some_a @ X22 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_390_option_Odistinct_I1_J,axiom,
! [X22: a] :
( none_a
!= ( some_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_391_logic_Ovalid__hoare__tripleI,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma5: a,S: c > d] :
( ( ( Valid @ Sigma5 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ P2 ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma5 @ S ) ) )
=> ( ! [Sigma5: a,S: c > d,Sigma4: a,S7: c > d] :
( ( ( Valid @ Sigma5 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma5 @ S @ Delta @ P2 ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma5 @ S ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma4 @ S7 ) ) ) @ C )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma4 @ S7 @ Delta @ Q2 ) ) )
=> ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta ) ) ) ) ).
% logic.valid_hoare_tripleI
thf(fact_392_logic_Ovalid__hoare__triple__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Q2: assertion_a_b_d_c,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_6037315502795721655_b_d_c @ Plus2 @ Mult @ Valid @ P2 @ C @ Q2 @ Delta )
= ( ! [Sigma2: a,S3: c > d] :
( ( ( Valid @ Sigma2 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma2 @ S3 @ Delta @ P2 ) )
=> ( ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) )
& ! [Sigma3: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C )
=> ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Sigma3 @ S5 @ Delta @ Q2 ) ) ) ) ) ) ) ).
% logic.valid_hoare_triple_def
thf(fact_393_pre__logic_Ocompatible__def,axiom,
( pre_compatible_a
= ( ^ [Plus: a > a > option_a,A5: a,B4: a] :
( ( Plus @ A5 @ B4 )
!= none_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_394_logic_Oframe__property__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( frame_property_a_c_d @ Plus2 @ Valid @ C )
= ( ! [Sigma2: a,Sigma_0: a,R4: a,Sigma3: a,S3: c > d,S5: c > d] :
( ( ( Valid @ Sigma2 )
& ( Valid @ Sigma3 )
& ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) )
& ( ( some_a @ Sigma2 )
= ( Plus2 @ Sigma_0 @ R4 ) )
& ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C ) )
=> ? [Sigma_02: a] :
( ( ( some_a @ Sigma3 )
= ( Plus2 @ Sigma_02 @ R4 ) )
& ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma_0 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma_02 @ S5 ) ) ) @ C ) ) ) ) ) ) ).
% logic.frame_property_def
thf(fact_395_logic_Onot__in__fv__mod,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,C: set_Pr1275464188344874039_a_c_d,Sigma: a,S2: c > d,Sigma6: a,S6: c > d,X2: a,Delta: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ ( modified_a_c_d @ C ) )
=> ( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma @ S2 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma6 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S2 @ Delta @ A2 )
= ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ X2 @ S6 @ Delta @ A2 ) ) ) ) ) ).
% logic.not_in_fv_mod
thf(fact_396_logic_Osmaller__interp__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,Delta: ( c > d ) > set_a,Delta2: ( c > d ) > set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( smaller_interp_c_d_a @ Delta @ Delta2 )
= ( ! [S3: c > d] : ( ord_less_eq_set_a @ ( Delta @ S3 ) @ ( Delta2 @ S3 ) ) ) ) ) ).
% logic.smaller_interp_def
thf(fact_397_logic_Osafety__monotonicity__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( safety844553430189520448_a_c_d @ Plus2 @ Valid @ C )
= ( ! [Sigma2: a,Sigma3: a,S3: c > d] :
( ( ( Valid @ Sigma3 )
& ( pre_larger_a @ Plus2 @ Sigma3 @ Sigma2 )
& ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) ) )
=> ( safe_a_c_d @ C @ ( product_Pair_a_c_d @ Sigma3 @ S3 ) ) ) ) ) ) ).
% logic.safety_monotonicity_def
thf(fact_398_logic_Oempty__interp__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S2: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( empty_interp_c_d_a @ S2 )
= bot_bot_set_a ) ) ).
% logic.empty_interp_def
thf(fact_399_map__upd__nonempty,axiom,
! [T: a > option_a,K: a,X2: a] :
( ( fun_upd_a_option_a @ T @ K @ ( some_a @ X2 ) )
!= ( ^ [X3: a] : none_a ) ) ).
% map_upd_nonempty
thf(fact_400_map__upd__Some__unfold,axiom,
! [M: a > option_a,A: a,B: a,X2: a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) @ X2 )
= ( some_a @ Y ) )
= ( ( ( X2 = A )
& ( B = Y ) )
| ( ( X2 != A )
& ( ( M @ X2 )
= ( some_a @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_401_map__upd__triv,axiom,
! [T: a > option_a,K: a,X2: a] :
( ( ( T @ K )
= ( some_a @ X2 ) )
=> ( ( fun_upd_a_option_a @ T @ K @ ( some_a @ X2 ) )
= T ) ) ).
% map_upd_triv
thf(fact_402_map__upd__eqD1,axiom,
! [M: a > option_a,A: a,X2: a,N: a > option_a,Y: a] :
( ( ( fun_upd_a_option_a @ M @ A @ ( some_a @ X2 ) )
= ( fun_upd_a_option_a @ N @ A @ ( some_a @ Y ) ) )
=> ( X2 = Y ) ) ).
% map_upd_eqD1
thf(fact_403_map__le__imp__upd__le,axiom,
! [M1: a > option_a,M2: a > option_a,X2: a,Y: a] :
( ( map_le_a_a @ M1 @ M2 )
=> ( map_le_a_a @ ( fun_upd_a_option_a @ M1 @ X2 @ none_a ) @ ( fun_upd_a_option_a @ M2 @ X2 @ ( some_a @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_404_set__empty__eq,axiom,
! [Xo: option_option_a] :
( ( ( set_option_option_a2 @ Xo )
= bot_bot_set_option_a )
= ( Xo = none_option_a ) ) ).
% set_empty_eq
thf(fact_405_set__empty__eq,axiom,
! [Xo: option_a] :
( ( ( set_option_a2 @ Xo )
= bot_bot_set_a )
= ( Xo = none_a ) ) ).
% set_empty_eq
thf(fact_406_elem__set,axiom,
! [X2: option_a,Xo: option_option_a] :
( ( member_option_a @ X2 @ ( set_option_option_a2 @ Xo ) )
= ( Xo
= ( some_option_a @ X2 ) ) ) ).
% elem_set
thf(fact_407_elem__set,axiom,
! [X2: b,Xo: option_b] :
( ( member_b @ X2 @ ( set_option_b2 @ Xo ) )
= ( Xo
= ( some_b @ X2 ) ) ) ).
% elem_set
thf(fact_408_elem__set,axiom,
! [X2: a,Xo: option_a] :
( ( member_a @ X2 @ ( set_option_a2 @ Xo ) )
= ( Xo
= ( some_a @ X2 ) ) ) ).
% elem_set
thf(fact_409_ospec,axiom,
! [A2: option_a,P2: a > $o,X2: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_option_a2 @ A2 ) )
=> ( P2 @ X ) )
=> ( ( A2
= ( some_a @ X2 ) )
=> ( P2 @ X2 ) ) ) ).
% ospec
thf(fact_410_option_Oset__intros,axiom,
! [X22: option_a] : ( member_option_a @ X22 @ ( set_option_option_a2 @ ( some_option_a @ X22 ) ) ) ).
% option.set_intros
thf(fact_411_option_Oset__intros,axiom,
! [X22: b] : ( member_b @ X22 @ ( set_option_b2 @ ( some_b @ X22 ) ) ) ).
% option.set_intros
thf(fact_412_option_Oset__intros,axiom,
! [X22: a] : ( member_a @ X22 @ ( set_option_a2 @ ( some_a @ X22 ) ) ) ).
% option.set_intros
thf(fact_413_option_Oset__cases,axiom,
! [E: option_a,A: option_option_a] :
( ( member_option_a @ E @ ( set_option_option_a2 @ A ) )
=> ( A
= ( some_option_a @ E ) ) ) ).
% option.set_cases
thf(fact_414_option_Oset__cases,axiom,
! [E: b,A: option_b] :
( ( member_b @ E @ ( set_option_b2 @ A ) )
=> ( A
= ( some_b @ E ) ) ) ).
% option.set_cases
thf(fact_415_option_Oset__cases,axiom,
! [E: a,A: option_a] :
( ( member_a @ E @ ( set_option_a2 @ A ) )
=> ( A
= ( some_a @ E ) ) ) ).
% option.set_cases
thf(fact_416_option_Osimps_I14_J,axiom,
( ( set_option_option_a2 @ none_option_a )
= bot_bot_set_option_a ) ).
% option.simps(14)
thf(fact_417_option_Osimps_I14_J,axiom,
( ( set_option_a2 @ none_a )
= bot_bot_set_a ) ).
% option.simps(14)
thf(fact_418_option_Osimps_I15_J,axiom,
! [X22: option_a] :
( ( set_option_option_a2 @ ( some_option_a @ X22 ) )
= ( insert_option_a @ X22 @ bot_bot_set_option_a ) ) ).
% option.simps(15)
thf(fact_419_option_Osimps_I15_J,axiom,
! [X22: a] :
( ( set_option_a2 @ ( some_a @ X22 ) )
= ( insert_a @ X22 @ bot_bot_set_a ) ) ).
% option.simps(15)
thf(fact_420_option_Ocollapse,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( ( some_a @ ( the_a2 @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_421_map__add__upd,axiom,
! [F: a > option_a,G: a > option_a,X2: a,Y: a] :
( ( map_add_a_a @ F @ ( fun_upd_a_option_a @ G @ X2 @ ( some_a @ Y ) ) )
= ( fun_upd_a_option_a @ ( map_add_a_a @ F @ G ) @ X2 @ ( some_a @ Y ) ) ) ).
% map_add_upd
thf(fact_422_these__empty,axiom,
( ( these_option_a @ bot_bo4163488203964334806tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_423_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_424_option_Oset__sel,axiom,
! [A: option_option_a] :
( ( A != none_option_a )
=> ( member_option_a @ ( the_option_a2 @ A ) @ ( set_option_option_a2 @ A ) ) ) ).
% option.set_sel
thf(fact_425_option_Oset__sel,axiom,
! [A: option_b] :
( ( A != none_b )
=> ( member_b @ ( the_b2 @ A ) @ ( set_option_b2 @ A ) ) ) ).
% option.set_sel
thf(fact_426_option_Oset__sel,axiom,
! [A: option_a] :
( ( A != none_a )
=> ( member_a @ ( the_a2 @ A ) @ ( set_option_a2 @ A ) ) ) ).
% option.set_sel
thf(fact_427_map__add__find__right,axiom,
! [N: a > option_a,K: a,Xx: a,M: a > option_a] :
( ( ( N @ K )
= ( some_a @ Xx ) )
=> ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ Xx ) ) ) ).
% map_add_find_right
thf(fact_428_these__insert__None,axiom,
! [A2: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
= ( these_a @ A2 ) ) ).
% these_insert_None
thf(fact_429_these__insert__Some,axiom,
! [X2: option_a,A2: set_option_option_a] :
( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X2 ) @ A2 ) )
= ( insert_option_a @ X2 @ ( these_option_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_430_these__insert__Some,axiom,
! [X2: a,A2: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X2 ) @ A2 ) )
= ( insert_a @ X2 @ ( these_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_431_these__not__empty__eq,axiom,
! [B2: set_option_option_a] :
( ( ( these_option_a @ B2 )
!= bot_bot_set_option_a )
= ( ( B2 != bot_bo4163488203964334806tion_a )
& ( B2
!= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_432_these__not__empty__eq,axiom,
! [B2: set_option_a] :
( ( ( these_a @ B2 )
!= bot_bot_set_a )
= ( ( B2 != bot_bot_set_option_a )
& ( B2
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_433_these__empty__eq,axiom,
! [B2: set_option_option_a] :
( ( ( these_option_a @ B2 )
= bot_bot_set_option_a )
= ( ( B2 = bot_bo4163488203964334806tion_a )
| ( B2
= ( insert605063979879581146tion_a @ none_option_a @ bot_bo4163488203964334806tion_a ) ) ) ) ).
% these_empty_eq
thf(fact_434_these__empty__eq,axiom,
! [B2: set_option_a] :
( ( ( these_a @ B2 )
= bot_bot_set_a )
= ( ( B2 = bot_bot_set_option_a )
| ( B2
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_435_option_Osel,axiom,
! [X22: a] :
( ( the_a2 @ ( some_a @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_436_option_Oexpand,axiom,
! [Option: option_a,Option2: option_a] :
( ( ( Option = none_a )
= ( Option2 = none_a ) )
=> ( ( ( Option != none_a )
=> ( ( Option2 != none_a )
=> ( ( the_a2 @ Option )
= ( the_a2 @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_437_in__these__eq,axiom,
! [X2: option_a,A2: set_option_option_a] :
( ( member_option_a @ X2 @ ( these_option_a @ A2 ) )
= ( member5113800082084363315tion_a @ ( some_option_a @ X2 ) @ A2 ) ) ).
% in_these_eq
thf(fact_438_in__these__eq,axiom,
! [X2: b,A2: set_option_b] :
( ( member_b @ X2 @ ( these_b @ A2 ) )
= ( member_option_b @ ( some_b @ X2 ) @ A2 ) ) ).
% in_these_eq
thf(fact_439_in__these__eq,axiom,
! [X2: a,A2: set_option_a] :
( ( member_a @ X2 @ ( these_a @ A2 ) )
= ( member_option_a @ ( some_a @ X2 ) @ A2 ) ) ).
% in_these_eq
thf(fact_440_map__add__SomeD,axiom,
! [M: a > option_a,N: a > option_a,K: a,X2: a] :
( ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ X2 ) )
=> ( ( ( N @ K )
= ( some_a @ X2 ) )
| ( ( ( N @ K )
= none_a )
& ( ( M @ K )
= ( some_a @ X2 ) ) ) ) ) ).
% map_add_SomeD
thf(fact_441_map__add__Some__iff,axiom,
! [M: a > option_a,N: a > option_a,K: a,X2: a] :
( ( ( map_add_a_a @ M @ N @ K )
= ( some_a @ X2 ) )
= ( ( ( N @ K )
= ( some_a @ X2 ) )
| ( ( ( N @ K )
= none_a )
& ( ( M @ K )
= ( some_a @ X2 ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_442_option_Oexhaust__sel,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( Option
= ( some_a @ ( the_a2 @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_443_override__on__insert_H,axiom,
! [F: a > option_a,G: a > option_a,X2: a,X6: set_a] :
( ( overri633547075744967556tion_a @ F @ G @ ( insert_a @ X2 @ X6 ) )
= ( overri633547075744967556tion_a @ ( fun_upd_a_option_a @ F @ X2 @ ( G @ X2 ) ) @ G @ X6 ) ) ).
% override_on_insert'
thf(fact_444_override__on__insert,axiom,
! [F: a > option_a,G: a > option_a,X2: a,X6: set_a] :
( ( overri633547075744967556tion_a @ F @ G @ ( insert_a @ X2 @ X6 ) )
= ( fun_upd_a_option_a @ ( overri633547075744967556tion_a @ F @ G @ X6 ) @ X2 @ ( G @ X2 ) ) ) ).
% override_on_insert
thf(fact_445_ran__map__upd,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= none_a )
=> ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) )
= ( insert_a @ B @ ( ran_a_a @ M ) ) ) ) ).
% ran_map_upd
thf(fact_446_graph__map__upd,axiom,
! [M: assertion_a_b_d_c > option3890169911263941780_a_c_d,K: assertion_a_b_d_c,V4: produc5213381314664832452_a_c_d] :
( ( graph_7603009230766167293_a_c_d @ ( fun_up8563802042059451790_a_c_d @ M @ K @ ( some_P3194730542479778335_a_c_d @ V4 ) ) )
= ( insert1952503980790619482_a_c_d @ ( produc8894421531525210148_a_c_d @ K @ V4 ) @ ( graph_7603009230766167293_a_c_d @ ( fun_up8563802042059451790_a_c_d @ M @ K @ none_P4438893274231186595_a_c_d ) ) ) ) ).
% graph_map_upd
thf(fact_447_graph__map__upd,axiom,
! [M: ( ( c > d ) > set_a ) > option_c_d,K: ( c > d ) > set_a,V4: c > d] :
( ( graph_c_d_set_a_c_d @ ( fun_up2820008246124789800on_c_d @ M @ K @ ( some_c_d @ V4 ) ) )
= ( insert9214331609911559156_a_c_d @ ( produc7376592049607813182_a_c_d @ K @ V4 ) @ ( graph_c_d_set_a_c_d @ ( fun_up2820008246124789800on_c_d @ M @ K @ none_c_d ) ) ) ) ).
% graph_map_upd
thf(fact_448_graph__map__upd,axiom,
! [M: a > option_a,K: a,V4: a] :
( ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ ( some_a @ V4 ) ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ K @ V4 ) @ ( graph_a_a @ ( fun_upd_a_option_a @ M @ K @ none_a ) ) ) ) ).
% graph_map_upd
thf(fact_449_option_Osplit__sel__asm,axiom,
! [P2: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
( ( P2 @ ( case_option_o_a @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_a )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_450_option_Osplit__sel__asm,axiom,
! [P2: option_a > $o,F1: option_a,F22: a > option_a,Option: option_a] :
( ( P2 @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_a )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_451_option_Osplit__sel__asm,axiom,
! [P2: a > $o,F1: a,F22: a > a,Option: option_a] :
( ( P2 @ ( case_option_a_a @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_a )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_452_option_Osplit__sel,axiom,
! [P2: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
( ( P2 @ ( case_option_o_a @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_a )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_453_option_Osplit__sel,axiom,
! [P2: option_a > $o,F1: option_a,F22: a > option_a,Option: option_a] :
( ( P2 @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_a )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_454_option_Osplit__sel,axiom,
! [P2: a > $o,F1: a,F22: a > a,Option: option_a] :
( ( P2 @ ( case_option_a_a @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_a )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_a2 @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_455_map__add__upd__left,axiom,
! [M: option_a,E2: option_a > option_a,E1: option_a > option_a,U1: a] :
( ~ ( member_option_a @ M @ ( dom_option_a_a @ E2 ) )
=> ( ( map_add_option_a_a @ ( fun_up1079276522633388797tion_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_up1079276522633388797tion_a @ ( map_add_option_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_456_map__add__upd__left,axiom,
! [M: b,E2: b > option_a,E1: b > option_a,U1: a] :
( ~ ( member_b @ M @ ( dom_b_a @ E2 ) )
=> ( ( map_add_b_a @ ( fun_upd_b_option_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_upd_b_option_a @ ( map_add_b_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_457_map__add__upd__left,axiom,
! [M: a,E2: a > option_a,E1: a > option_a,U1: a] :
( ~ ( member_a @ M @ ( dom_a_a @ E2 ) )
=> ( ( map_add_a_a @ ( fun_upd_a_option_a @ E1 @ M @ ( some_a @ U1 ) ) @ E2 )
= ( fun_upd_a_option_a @ ( map_add_a_a @ E1 @ E2 ) @ M @ ( some_a @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_458_domD,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_459_domD,axiom,
! [A: b,M: b > option_a] :
( ( member_b @ A @ ( dom_b_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_460_domD,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
=> ? [B5: a] :
( ( M @ A )
= ( some_a @ B5 ) ) ) ).
% domD
thf(fact_461_domI,axiom,
! [M: option_a > option_a,A: option_a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_option_a @ A @ ( dom_option_a_a @ M ) ) ) ).
% domI
thf(fact_462_domI,axiom,
! [M: b > option_a,A: b,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_b @ A @ ( dom_b_a @ M ) ) ) ).
% domI
thf(fact_463_domI,axiom,
! [M: a > option_a,A: a,B: a] :
( ( ( M @ A )
= ( some_a @ B ) )
=> ( member_a @ A @ ( dom_a_a @ M ) ) ) ).
% domI
thf(fact_464_logic_Oapplies__eq_Ocases,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: produc5105196854009589546_a_c_d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ~ ! [A7: assertion_a_b_d_c,Delta3: ( c > d ) > set_a,S: c > d] :
( X2
!= ( produc8894421531525210148_a_c_d @ A7 @ ( produc7376592049607813182_a_c_d @ Delta3 @ S ) ) ) ) ).
% logic.applies_eq.cases
thf(fact_465_option_Osimps_I5_J,axiom,
! [F1: $o,F22: a > $o,X22: a] :
( ( case_option_o_a @ F1 @ F22 @ ( some_a @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_466_option_Osimps_I5_J,axiom,
! [F1: option_a,F22: a > option_a,X22: a] :
( ( case_o3148979394504432965on_a_a @ F1 @ F22 @ ( some_a @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_467_option_Osimps_I5_J,axiom,
! [F1: a,F22: a > a,X22: a] :
( ( case_option_a_a @ F1 @ F22 @ ( some_a @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_468_option_Osimps_I4_J,axiom,
! [F1: $o,F22: a > $o] :
( ( case_option_o_a @ F1 @ F22 @ none_a )
= F1 ) ).
% option.simps(4)
thf(fact_469_option_Osimps_I4_J,axiom,
! [F1: option_a,F22: a > option_a] :
( ( case_o3148979394504432965on_a_a @ F1 @ F22 @ none_a )
= F1 ) ).
% option.simps(4)
thf(fact_470_option_Osimps_I4_J,axiom,
! [F1: a,F22: a > a] :
( ( case_option_a_a @ F1 @ F22 @ none_a )
= F1 ) ).
% option.simps(4)
thf(fact_471_insert__dom,axiom,
! [F: a > option_a,X2: a,Y: a] :
( ( ( F @ X2 )
= ( some_a @ Y ) )
=> ( ( insert_a @ X2 @ ( dom_a_a @ F ) )
= ( dom_a_a @ F ) ) ) ).
% insert_dom
thf(fact_472_insert__dom,axiom,
! [F: option_a > option_a,X2: option_a,Y: a] :
( ( ( F @ X2 )
= ( some_a @ Y ) )
=> ( ( insert_option_a @ X2 @ ( dom_option_a_a @ F ) )
= ( dom_option_a_a @ F ) ) ) ).
% insert_dom
thf(fact_473_in__graphI,axiom,
! [M: assertion_a_b_d_c > option3890169911263941780_a_c_d,K: assertion_a_b_d_c,V4: produc5213381314664832452_a_c_d] :
( ( ( M @ K )
= ( some_P3194730542479778335_a_c_d @ V4 ) )
=> ( member537768723423446209_a_c_d @ ( produc8894421531525210148_a_c_d @ K @ V4 ) @ ( graph_7603009230766167293_a_c_d @ M ) ) ) ).
% in_graphI
thf(fact_474_in__graphI,axiom,
! [M: ( ( c > d ) > set_a ) > option_c_d,K: ( c > d ) > set_a,V4: c > d] :
( ( ( M @ K )
= ( some_c_d @ V4 ) )
=> ( member1642667639779969243_a_c_d @ ( produc7376592049607813182_a_c_d @ K @ V4 ) @ ( graph_c_d_set_a_c_d @ M ) ) ) ).
% in_graphI
thf(fact_475_in__graphD,axiom,
! [K: assertion_a_b_d_c,V4: produc5213381314664832452_a_c_d,M: assertion_a_b_d_c > option3890169911263941780_a_c_d] :
( ( member537768723423446209_a_c_d @ ( produc8894421531525210148_a_c_d @ K @ V4 ) @ ( graph_7603009230766167293_a_c_d @ M ) )
=> ( ( M @ K )
= ( some_P3194730542479778335_a_c_d @ V4 ) ) ) ).
% in_graphD
thf(fact_476_in__graphD,axiom,
! [K: ( c > d ) > set_a,V4: c > d,M: ( ( c > d ) > set_a ) > option_c_d] :
( ( member1642667639779969243_a_c_d @ ( produc7376592049607813182_a_c_d @ K @ V4 ) @ ( graph_c_d_set_a_c_d @ M ) )
=> ( ( M @ K )
= ( some_c_d @ V4 ) ) ) ).
% in_graphD
thf(fact_477_option_Ocase__eq__if,axiom,
( case_option_o_a
= ( ^ [F12: $o,F23: a > $o,Option3: option_a] :
( ( ( Option3 = none_a )
=> F12 )
& ( ( Option3 != none_a )
=> ( F23 @ ( the_a2 @ Option3 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_478_option_Ocase__eq__if,axiom,
( case_o3148979394504432965on_a_a
= ( ^ [F12: option_a,F23: a > option_a,Option3: option_a] : ( if_option_a @ ( Option3 = none_a ) @ F12 @ ( F23 @ ( the_a2 @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_479_option_Ocase__eq__if,axiom,
( case_option_a_a
= ( ^ [F12: a,F23: a > a,Option3: option_a] : ( if_a @ ( Option3 = none_a ) @ F12 @ ( F23 @ ( the_a2 @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_480_graph__restrictD_I2_J,axiom,
! [K: assertion_a_b_d_c,V4: produc5213381314664832452_a_c_d,M: assertion_a_b_d_c > option3890169911263941780_a_c_d,A2: set_as909545710669178647_b_d_c] :
( ( member537768723423446209_a_c_d @ ( produc8894421531525210148_a_c_d @ K @ V4 ) @ ( graph_7603009230766167293_a_c_d @ ( restri3968632621895983051_a_c_d @ M @ A2 ) ) )
=> ( ( M @ K )
= ( some_P3194730542479778335_a_c_d @ V4 ) ) ) ).
% graph_restrictD(2)
thf(fact_481_graph__restrictD_I2_J,axiom,
! [K: ( c > d ) > set_a,V4: c > d,M: ( ( c > d ) > set_a ) > option_c_d,A2: set_c_d_set_a] :
( ( member1642667639779969243_a_c_d @ ( produc7376592049607813182_a_c_d @ K @ V4 ) @ ( graph_c_d_set_a_c_d @ ( restri4474245042709046629_a_c_d @ M @ A2 ) ) )
=> ( ( M @ K )
= ( some_c_d @ V4 ) ) ) ).
% graph_restrictD(2)
thf(fact_482_these__image__Some__eq,axiom,
! [A2: set_a] :
( ( these_a @ ( image_a_option_a @ some_a @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_483_bind__eq__None__conv,axiom,
! [A: option_a,F: a > option_a] :
( ( ( bind_a_a @ A @ F )
= none_a )
= ( ( A = none_a )
| ( ( F @ ( the_a2 @ A ) )
= none_a ) ) ) ).
% bind_eq_None_conv
thf(fact_484_dom__eq__singleton__conv,axiom,
! [F: a > option_a,X2: a] :
( ( ( dom_a_a @ F )
= ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ? [V: a] :
( F
= ( fun_upd_a_option_a
@ ^ [X3: a] : none_a
@ X2
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_485_dom__eq__singleton__conv,axiom,
! [F: option_a > option_a,X2: option_a] :
( ( ( dom_option_a_a @ F )
= ( insert_option_a @ X2 @ bot_bot_set_option_a ) )
= ( ? [V: a] :
( F
= ( fun_up1079276522633388797tion_a
@ ^ [X3: option_a] : none_a
@ X2
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_486_bind__assoc,axiom,
! [X2: option_a,F: a > option_a,G: a > option_a] :
( ( bind_a_a @ ( bind_a_a @ X2 @ F ) @ G )
= ( bind_a_a @ X2
@ ^ [Y2: a] : ( bind_a_a @ ( F @ Y2 ) @ G ) ) ) ).
% bind_assoc
thf(fact_487_bind__runit,axiom,
! [X2: option_a] :
( ( bind_a_a @ X2 @ some_a )
= X2 ) ).
% bind_runit
thf(fact_488_bind__rzero,axiom,
! [X2: option_a] :
( ( bind_a_a @ X2
@ ^ [X3: a] : none_a )
= none_a ) ).
% bind_rzero
thf(fact_489_image__map__upd,axiom,
! [X2: option_a,A2: set_option_a,M: option_a > option_a,Y: a] :
( ~ ( member_option_a @ X2 @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
= ( image_7439109396645324421tion_a @ M @ A2 ) ) ) ).
% image_map_upd
thf(fact_490_image__map__upd,axiom,
! [X2: b,A2: set_b,M: b > option_a,Y: a] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( image_b_option_a @ ( fun_upd_b_option_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
= ( image_b_option_a @ M @ A2 ) ) ) ).
% image_map_upd
thf(fact_491_image__map__upd,axiom,
! [X2: a,A2: set_a,M: a > option_a,Y: a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ M @ X2 @ ( some_a @ Y ) ) @ A2 )
= ( image_a_option_a @ M @ A2 ) ) ) ).
% image_map_upd
thf(fact_492_map__add__def,axiom,
( map_add_a_a
= ( ^ [M12: a > option_a,M22: a > option_a,X3: a] : ( case_o3148979394504432965on_a_a @ ( M12 @ X3 ) @ some_a @ ( M22 @ X3 ) ) ) ) ).
% map_add_def
thf(fact_493_case__optionE,axiom,
! [P2: $o,Q2: a > $o,X2: option_a] :
( ( case_option_o_a @ P2 @ Q2 @ X2 )
=> ( ( ( X2 = none_a )
=> ~ P2 )
=> ~ ! [Y3: a] :
( ( X2
= ( some_a @ Y3 ) )
=> ~ ( Q2 @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_494_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_a] :
( ( Option != none_a )
= ( case_option_o_a @ $false
@ ^ [Uu: a] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_495_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_a] :
( ( Option = none_a )
= ( case_option_o_a @ $true
@ ^ [Uu: a] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_496_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
= ( case_option_o_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_497_option_Ocase__distrib,axiom,
! [H: $o > option_a,F1: $o,F22: a > $o,Option: option_a] :
( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
= ( case_o3148979394504432965on_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_498_option_Ocase__distrib,axiom,
! [H: $o > a,F1: $o,F22: a > $o,Option: option_a] :
( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
= ( case_option_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_499_option_Ocase__distrib,axiom,
! [H: option_a > $o,F1: option_a,F22: a > option_a,Option: option_a] :
( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
= ( case_option_o_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_500_option_Ocase__distrib,axiom,
! [H: option_a > option_a,F1: option_a,F22: a > option_a,Option: option_a] :
( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
= ( case_o3148979394504432965on_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_501_option_Ocase__distrib,axiom,
! [H: option_a > a,F1: option_a,F22: a > option_a,Option: option_a] :
( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
= ( case_option_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_502_option_Ocase__distrib,axiom,
! [H: a > $o,F1: a,F22: a > a,Option: option_a] :
( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
= ( case_option_o_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_503_option_Ocase__distrib,axiom,
! [H: a > option_a,F1: a,F22: a > a,Option: option_a] :
( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
= ( case_o3148979394504432965on_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_504_option_Ocase__distrib,axiom,
! [H: a > a,F1: a,F22: a > a,Option: option_a] :
( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
= ( case_option_a_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_505_bind__option__cong__code,axiom,
! [X2: option_a,Y: option_a,F: a > option_a] :
( ( X2 = Y )
=> ( ( bind_a_a @ X2 @ F )
= ( bind_a_a @ Y @ F ) ) ) ).
% bind_option_cong_code
thf(fact_506_Some__image__these__eq,axiom,
! [A2: set_option_a] :
( ( image_a_option_a @ some_a @ ( these_a @ A2 ) )
= ( collect_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
& ( X3 != none_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_507_bind_Obind__lunit,axiom,
! [X2: a,F: a > option_a] :
( ( bind_a_a @ ( some_a @ X2 ) @ F )
= ( F @ X2 ) ) ).
% bind.bind_lunit
thf(fact_508_Option_Obind__cong,axiom,
! [X2: option_a,Y: option_a,F: a > option_a,G: a > option_a] :
( ( X2 = Y )
=> ( ! [A6: a] :
( ( Y
= ( some_a @ A6 ) )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bind_a_a @ X2 @ F )
= ( bind_a_a @ Y @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_509_bind__eq__Some__conv,axiom,
! [F: option_a,G: a > option_a,X2: a] :
( ( ( bind_a_a @ F @ G )
= ( some_a @ X2 ) )
= ( ? [Y2: a] :
( ( F
= ( some_a @ Y2 ) )
& ( ( G @ Y2 )
= ( some_a @ X2 ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_510_bind_Obind__lzero,axiom,
! [F: a > option_a] :
( ( bind_a_a @ none_a @ F )
= none_a ) ).
% bind.bind_lzero
thf(fact_511_bind__option__cong,axiom,
! [X2: option_a,Y: option_a,F: a > option_a,G: a > option_a] :
( ( X2 = Y )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_option_a2 @ Y ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( bind_a_a @ X2 @ F )
= ( bind_a_a @ Y @ G ) ) ) ) ).
% bind_option_cong
thf(fact_512_None__notin__image__Some,axiom,
! [A2: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A2 ) ) ).
% None_notin_image_Some
thf(fact_513_bind__split__asm,axiom,
! [P2: option_a > $o,M: option_a,F: a > option_a] :
( ( P2 @ ( bind_a_a @ M @ F ) )
= ( ~ ( ( ( M = none_a )
& ~ ( P2 @ none_a ) )
| ? [X3: a] :
( ( M
= ( some_a @ X3 ) )
& ~ ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_514_bind__split,axiom,
! [P2: option_a > $o,M: option_a,F: a > option_a] :
( ( P2 @ ( bind_a_a @ M @ F ) )
= ( ( ( M = none_a )
=> ( P2 @ none_a ) )
& ! [V: a] :
( ( M
= ( some_a @ V ) )
=> ( P2 @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_515_applies__eq_Osimps,axiom,
! [A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S2: c > d] :
( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ A2 @ Delta @ S2 )
= ( collect_a
@ ^ [Uu: a] :
? [A5: a] :
( ( Uu = A5 )
& ( sat_a_b_c_d @ plus @ mult @ valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ).
% applies_eq.simps
thf(fact_516_applies__eq_Oelims,axiom,
! [X2: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ X2 @ Xa @ Xb )
= Y )
=> ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ plus @ mult @ valid @ Uu @ Xb @ Xa @ X2 ) ) ) ) ).
% applies_eq.elims
thf(fact_517_restrict__upd__same,axiom,
! [M: a > option_a,X2: a,Y: a] :
( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X2 @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% restrict_upd_same
thf(fact_518_restrict__upd__same,axiom,
! [M: option_a > option_a,X2: option_a,Y: a] :
( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X2 @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) )
= ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ).
% restrict_upd_same
thf(fact_519_finite__Map__induct,axiom,
! [M: b > option_a,P2: ( b > option_a ) > $o] :
( ( finite_finite_b @ ( dom_b_a @ M ) )
=> ( ( P2
@ ^ [X3: b] : none_a )
=> ( ! [K2: b,V3: a,M3: b > option_a] :
( ( finite_finite_b @ ( dom_b_a @ M3 ) )
=> ( ~ ( member_b @ K2 @ ( dom_b_a @ M3 ) )
=> ( ( P2 @ M3 )
=> ( P2 @ ( fun_upd_b_option_a @ M3 @ K2 @ ( some_a @ V3 ) ) ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_520_finite__Map__induct,axiom,
! [M: option_a > option_a,P2: ( option_a > option_a ) > $o] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M ) )
=> ( ( P2
@ ^ [X3: option_a] : none_a )
=> ( ! [K2: option_a,V3: a,M3: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M3 ) )
=> ( ~ ( member_option_a @ K2 @ ( dom_option_a_a @ M3 ) )
=> ( ( P2 @ M3 )
=> ( P2 @ ( fun_up1079276522633388797tion_a @ M3 @ K2 @ ( some_a @ V3 ) ) ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_521_finite__Map__induct,axiom,
! [M: a > option_a,P2: ( a > option_a ) > $o] :
( ( finite_finite_a @ ( dom_a_a @ M ) )
=> ( ( P2
@ ^ [X3: a] : none_a )
=> ( ! [K2: a,V3: a,M3: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ M3 ) )
=> ( ~ ( member_a @ K2 @ ( dom_a_a @ M3 ) )
=> ( ( P2 @ M3 )
=> ( P2 @ ( fun_upd_a_option_a @ M3 @ K2 @ ( some_a @ V3 ) ) ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_522_modified__def,axiom,
( modified_a_c_d
= ( ^ [C2: set_Pr1275464188344874039_a_c_d] :
( collect_c
@ ^ [Uu: c] :
? [X3: c] :
( ( Uu = X3 )
& ? [Sigma2: a,S3: c > d,Sigma3: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C2 )
& ( ( S3 @ X3 )
!= ( S5 @ X3 ) ) ) ) ) ) ) ).
% modified_def
thf(fact_523_logic_Omodified__def,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,C: set_Pr1275464188344874039_a_c_d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( modified_a_c_d @ C )
= ( collect_c
@ ^ [Uu: c] :
? [X3: c] :
( ( Uu = X3 )
& ? [Sigma2: a,S3: c > d,Sigma3: a,S5: c > d] :
( ( member1180172933830803072_a_c_d @ ( produc8093790510458973071_a_c_d @ ( product_Pair_a_c_d @ Sigma2 @ S3 ) @ ( some_P1084500821511757806_a_c_d @ ( product_Pair_a_c_d @ Sigma3 @ S5 ) ) ) @ C )
& ( ( S3 @ X3 )
!= ( S5 @ X3 ) ) ) ) ) ) ) ).
% logic.modified_def
thf(fact_524_graph__def,axiom,
( graph_7603009230766167293_a_c_d
= ( ^ [M4: assertion_a_b_d_c > option3890169911263941780_a_c_d] :
( collec4762846013371775487_a_c_d
@ ^ [Uu: produc5105196854009589546_a_c_d] :
? [A5: assertion_a_b_d_c,B4: produc5213381314664832452_a_c_d] :
( ( Uu
= ( produc8894421531525210148_a_c_d @ A5 @ B4 ) )
& ( ( M4 @ A5 )
= ( some_P3194730542479778335_a_c_d @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_525_graph__def,axiom,
( graph_c_d_set_a_c_d
= ( ^ [M4: ( ( c > d ) > set_a ) > option_c_d] :
( collec2771355035510247705_a_c_d
@ ^ [Uu: produc5213381314664832452_a_c_d] :
? [A5: ( c > d ) > set_a,B4: c > d] :
( ( Uu
= ( produc7376592049607813182_a_c_d @ A5 @ B4 ) )
& ( ( M4 @ A5 )
= ( some_c_d @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_526_logic_Oapplies__eq_Oelims,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ X2 @ Xa @ Xb )
= Y )
=> ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Uu @ Xb @ Xa @ X2 ) ) ) ) ) ).
% logic.applies_eq.elims
thf(fact_527_logic_Oapplies__eq_Osimps,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,A2: assertion_a_b_d_c,Delta: ( c > d ) > set_a,S2: c > d] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ A2 @ Delta @ S2 )
= ( collect_a
@ ^ [Uu: a] :
? [A5: a] :
( ( Uu = A5 )
& ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ A5 @ S2 @ Delta @ A2 ) ) ) ) ) ).
% logic.applies_eq.simps
thf(fact_528_applies__eq_Opelims,axiom,
! [X2: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( ( applies_eq_a_b_d_c @ plus @ mult @ valid @ X2 @ Xa @ Xb )
= Y )
=> ( ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X2 @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ plus @ mult @ valid @ Uu @ Xb @ Xa @ X2 ) ) )
=> ~ ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X2 @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) ) ) ) ) ).
% applies_eq.pelims
thf(fact_529_finite__update__induct,axiom,
! [F: a > option_a,C: option_a,P2: ( a > option_a ) > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( F @ A5 )
!= C ) ) )
=> ( ( P2
@ ^ [A5: a] : C )
=> ( ! [A6: a,B5: option_a,F3: a > option_a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [C2: a] :
( ( F3 @ C2 )
!= C ) ) )
=> ( ( ( F3 @ A6 )
= C )
=> ( ( B5 != C )
=> ( ( P2 @ F3 )
=> ( P2 @ ( fun_upd_a_option_a @ F3 @ A6 @ B5 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_update_induct
thf(fact_530_logic_Oapplies__eq_Opelims,axiom,
! [Plus2: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X2: assertion_a_b_d_c,Xa: ( c > d ) > set_a,Xb: c > d,Y: set_a] :
( ( logic_a_b @ Plus2 @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( applies_eq_a_b_d_c @ Plus2 @ Mult @ Valid @ X2 @ Xa @ Xb )
= Y )
=> ( ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X2 @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( collect_a
@ ^ [Uu: a] : ( sat_a_b_c_d @ Plus2 @ Mult @ Valid @ Uu @ Xb @ Xa @ X2 ) ) )
=> ~ ( accp_P5381461700908302305_a_c_d @ applie8886407701077375079_b_d_c @ ( produc8894421531525210148_a_c_d @ X2 @ ( produc7376592049607813182_a_c_d @ Xa @ Xb ) ) ) ) ) ) ) ).
% logic.applies_eq.pelims
thf(fact_531_combine__options__def,axiom,
( combine_options_a
= ( ^ [F2: a > a > a,X3: option_a,Y2: option_a] :
( case_o3148979394504432965on_a_a @ Y2
@ ^ [Z: a] :
( case_o3148979394504432965on_a_a @ ( some_a @ Z )
@ ^ [Aa: a] : ( some_a @ ( F2 @ Z @ Aa ) )
@ Y2 )
@ X3 ) ) ) ).
% combine_options_def
thf(fact_532_applies__eq_Ocases,axiom,
! [X2: produc5105196854009589546_a_c_d] :
~ ! [A7: assertion_a_b_d_c,Delta3: ( c > d ) > set_a,S: c > d] :
( X2
!= ( produc8894421531525210148_a_c_d @ A7 @ ( produc7376592049607813182_a_c_d @ Delta3 @ S ) ) ) ).
% applies_eq.cases
thf(fact_533_combine__options__simps_I3_J,axiom,
! [F: a > a > a,A: a,B: a] :
( ( combine_options_a @ F @ ( some_a @ A ) @ ( some_a @ B ) )
= ( some_a @ ( F @ A @ B ) ) ) ).
% combine_options_simps(3)
thf(fact_534_combine__options__simps_I1_J,axiom,
! [F: a > a > a,Y: option_a] :
( ( combine_options_a @ F @ none_a @ Y )
= Y ) ).
% combine_options_simps(1)
thf(fact_535_combine__options__simps_I2_J,axiom,
! [F: a > a > a,X2: option_a] :
( ( combine_options_a @ F @ X2 @ none_a )
= X2 ) ).
% combine_options_simps(2)
thf(fact_536_combine__options__left__commute,axiom,
! [F: a > a > a,Y: option_a,X2: option_a,Z2: option_a] :
( ! [X: a,Y3: a] :
( ( F @ X @ Y3 )
= ( F @ Y3 @ X ) )
=> ( ! [X: a,Y3: a,Z3: a] :
( ( F @ ( F @ X @ Y3 ) @ Z3 )
= ( F @ X @ ( F @ Y3 @ Z3 ) ) )
=> ( ( combine_options_a @ F @ Y @ ( combine_options_a @ F @ X2 @ Z2 ) )
= ( combine_options_a @ F @ X2 @ ( combine_options_a @ F @ Y @ Z2 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_537_combine__options__commute,axiom,
! [F: a > a > a,X2: option_a,Y: option_a] :
( ! [X: a,Y3: a] :
( ( F @ X @ Y3 )
= ( F @ Y3 @ X ) )
=> ( ( combine_options_a @ F @ X2 @ Y )
= ( combine_options_a @ F @ Y @ X2 ) ) ) ).
% combine_options_commute
thf(fact_538_combine__options__assoc,axiom,
! [F: a > a > a,X2: option_a,Y: option_a,Z2: option_a] :
( ! [X: a,Y3: a,Z3: a] :
( ( F @ ( F @ X @ Y3 ) @ Z3 )
= ( F @ X @ ( F @ Y3 @ Z3 ) ) )
=> ( ( combine_options_a @ F @ ( combine_options_a @ F @ X2 @ Y ) @ Z2 )
= ( combine_options_a @ F @ X2 @ ( combine_options_a @ F @ Y @ Z2 ) ) ) ) ).
% combine_options_assoc
thf(fact_539_fun__upd__image,axiom,
! [X2: b,A2: set_b,F: b > a,Y: a] :
( ( ( member_b @ X2 @ A2 )
=> ( ( image_b_a @ ( fun_upd_b_a @ F @ X2 @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( image_b_a @ ( fun_upd_b_a @ F @ X2 @ Y ) @ A2 )
= ( image_b_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_540_fun__upd__image,axiom,
! [X2: b,A2: set_b,F: b > option_a,Y: option_a] :
( ( ( member_b @ X2 @ A2 )
=> ( ( image_b_option_a @ ( fun_upd_b_option_a @ F @ X2 @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_b_option_a @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( image_b_option_a @ ( fun_upd_b_option_a @ F @ X2 @ Y ) @ A2 )
= ( image_b_option_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_541_fun__upd__image,axiom,
! [X2: a,A2: set_a,F: a > a,Y: a] :
( ( ( member_a @ X2 @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X2 @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( image_a_a @ ( fun_upd_a_a @ F @ X2 @ Y ) @ A2 )
= ( image_a_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_542_fun__upd__image,axiom,
! [X2: a,A2: set_a,F: a > option_a,Y: option_a] :
( ( ( member_a @ X2 @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ F @ X2 @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( image_a_option_a @ ( fun_upd_a_option_a @ F @ X2 @ Y ) @ A2 )
= ( image_a_option_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_543_fun__upd__image,axiom,
! [X2: option_a,A2: set_option_a,F: option_a > a,Y: a] :
( ( ( member_option_a @ X2 @ A2 )
=> ( ( image_option_a_a @ ( fun_upd_option_a_a @ F @ X2 @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X2 @ A2 )
=> ( ( image_option_a_a @ ( fun_upd_option_a_a @ F @ X2 @ Y ) @ A2 )
= ( image_option_a_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_544_fun__upd__image,axiom,
! [X2: option_a,A2: set_option_a,F: option_a > option_a,Y: option_a] :
( ( ( member_option_a @ X2 @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X2 @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X2 @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X2 @ Y ) @ A2 )
= ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_545_Option_Othese__def,axiom,
( these_a
= ( ^ [A8: set_option_a] :
( image_option_a_a @ the_a2
@ ( collect_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A8 )
& ( X3 != none_a ) ) ) ) ) ) ).
% Option.these_def
thf(fact_546_finite__range__updI,axiom,
! [F: option_a > option_a,A: option_a,B: a] :
( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) )
=> ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ A @ ( some_a @ B ) ) @ top_top_set_option_a ) ) ) ).
% finite_range_updI
thf(fact_547_finite__range__updI,axiom,
! [F: a > option_a,A: a,B: a] :
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ F @ top_top_set_a ) )
=> ( finite1674126218327898605tion_a @ ( image_a_option_a @ ( fun_upd_a_option_a @ F @ A @ ( some_a @ B ) ) @ top_top_set_a ) ) ) ).
% finite_range_updI
thf(fact_548_map__option__case,axiom,
( map_option_a_a
= ( ^ [F2: a > a] :
( case_o3148979394504432965on_a_a @ none_a
@ ^ [X3: a] : ( some_a @ ( F2 @ X3 ) ) ) ) ) ).
% map_option_case
thf(fact_549_map__option__eq__Some,axiom,
! [F: a > a,Xo: option_a,Y: a] :
( ( ( map_option_a_a @ F @ Xo )
= ( some_a @ Y ) )
= ( ? [Z: a] :
( ( Xo
= ( some_a @ Z ) )
& ( ( F @ Z )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_550_option_Omap__disc__iff,axiom,
! [F: a > a,A: option_a] :
( ( ( map_option_a_a @ F @ A )
= none_a )
= ( A = none_a ) ) ).
% option.map_disc_iff
thf(fact_551_map__option__is__None,axiom,
! [F: a > a,Opt: option_a] :
( ( ( map_option_a_a @ F @ Opt )
= none_a )
= ( Opt = none_a ) ) ).
% map_option_is_None
thf(fact_552_None__eq__map__option__iff,axiom,
! [F: a > a,X2: option_a] :
( ( none_a
= ( map_option_a_a @ F @ X2 ) )
= ( X2 = none_a ) ) ).
% None_eq_map_option_iff
thf(fact_553_finite__option__UNIV,axiom,
( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_option_UNIV
thf(fact_554_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_555_dom__const,axiom,
! [F: option_a > a] :
( ( dom_option_a_a
@ ^ [X3: option_a] : ( some_a @ ( F @ X3 ) ) )
= top_top_set_option_a ) ).
% dom_const
thf(fact_556_dom__const,axiom,
! [F: a > a] :
( ( dom_a_a
@ ^ [X3: a] : ( some_a @ ( F @ X3 ) ) )
= top_top_set_a ) ).
% dom_const
thf(fact_557_option_Omap__ident,axiom,
! [T: option_a] :
( ( map_option_a_a
@ ^ [X3: a] : X3
@ T )
= T ) ).
% option.map_ident
thf(fact_558_surj__def,axiom,
! [F: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
= ( ! [Y2: option_a] :
? [X3: option_a] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_559_surj__def,axiom,
! [F: option_a > a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
= ( ! [Y2: a] :
? [X3: option_a] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_560_surj__def,axiom,
! [F: a > option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
= ( ! [Y2: option_a] :
? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_561_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y2: a] :
? [X3: a] :
( Y2
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_562_surjI,axiom,
! [G: option_a > option_a,F: option_a > option_a] :
( ! [X: option_a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_563_surjI,axiom,
! [G: option_a > a,F: a > option_a] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_564_surjI,axiom,
! [G: a > option_a,F: option_a > a] :
( ! [X: option_a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_565_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X: a] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_566_surjE,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ~ ! [X: option_a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_567_surjE,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ~ ! [X: option_a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_568_surjE,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ~ ! [X: a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_569_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X: a] :
( Y
!= ( F @ X ) ) ) ).
% surjE
thf(fact_570_surjD,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ? [X: option_a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_571_surjD,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ? [X: option_a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_572_surjD,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ? [X: a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_573_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X: a] :
( Y
= ( F @ X ) ) ) ).
% surjD
thf(fact_574_map__option__idI,axiom,
! [X2: option_option_a,F: option_a > option_a] :
( ! [Y3: option_a] :
( ( member_option_a @ Y3 @ ( set_option_option_a2 @ X2 ) )
=> ( ( F @ Y3 )
= Y3 ) )
=> ( ( map_op788413144570152203tion_a @ F @ X2 )
= X2 ) ) ).
% map_option_idI
thf(fact_575_map__option__idI,axiom,
! [X2: option_b,F: b > b] :
( ! [Y3: b] :
( ( member_b @ Y3 @ ( set_option_b2 @ X2 ) )
=> ( ( F @ Y3 )
= Y3 ) )
=> ( ( map_option_b_b @ F @ X2 )
= X2 ) ) ).
% map_option_idI
thf(fact_576_map__option__idI,axiom,
! [X2: option_a,F: a > a] :
( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_option_a2 @ X2 ) )
=> ( ( F @ Y3 )
= Y3 ) )
=> ( ( map_option_a_a @ F @ X2 )
= X2 ) ) ).
% map_option_idI
thf(fact_577_option_Omap__ident__strong,axiom,
! [T: option_option_a,F: option_a > option_a] :
( ! [Z3: option_a] :
( ( member_option_a @ Z3 @ ( set_option_option_a2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_op788413144570152203tion_a @ F @ T )
= T ) ) ).
% option.map_ident_strong
thf(fact_578_option_Omap__ident__strong,axiom,
! [T: option_b,F: b > b] :
( ! [Z3: b] :
( ( member_b @ Z3 @ ( set_option_b2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_option_b_b @ F @ T )
= T ) ) ).
% option.map_ident_strong
thf(fact_579_option_Omap__ident__strong,axiom,
! [T: option_a,F: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_option_a2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_option_a_a @ F @ T )
= T ) ) ).
% option.map_ident_strong
thf(fact_580_option_Oinj__map__strong,axiom,
! [X2: option_a,Xa: option_a,F: a > a,Fa: a > a] :
( ! [Z3: a,Za: a] :
( ( member_a @ Z3 @ ( set_option_a2 @ X2 ) )
=> ( ( member_a @ Za @ ( set_option_a2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_option_a_a @ F @ X2 )
= ( map_option_a_a @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% option.inj_map_strong
thf(fact_581_option_Omap__cong0,axiom,
! [X2: option_a,F: a > a,G: a > a] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_option_a2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option_a_a @ F @ X2 )
= ( map_option_a_a @ G @ X2 ) ) ) ).
% option.map_cong0
thf(fact_582_option_Omap__cong,axiom,
! [X2: option_a,Ya: option_a,F: a > a,G: a > a] :
( ( X2 = Ya )
=> ( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_option_a2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option_a_a @ F @ X2 )
= ( map_option_a_a @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_583_map__option__cong,axiom,
! [X2: option_a,Y: option_a,F: a > a,G: a > a] :
( ( X2 = Y )
=> ( ! [A6: a] :
( ( Y
= ( some_a @ A6 ) )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( map_option_a_a @ F @ X2 )
= ( map_option_a_a @ G @ Y ) ) ) ) ).
% map_option_cong
thf(fact_584_option_Osimps_I9_J,axiom,
! [F: a > a,X22: a] :
( ( map_option_a_a @ F @ ( some_a @ X22 ) )
= ( some_a @ ( F @ X22 ) ) ) ).
% option.simps(9)
thf(fact_585_option_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_option_a_a @ F @ none_a )
= none_a ) ).
% option.simps(8)
thf(fact_586_option_Oset__map,axiom,
! [F: a > option_a,V4: option_a] :
( ( set_option_option_a2 @ ( map_op2340691886215429841tion_a @ F @ V4 ) )
= ( image_a_option_a @ F @ ( set_option_a2 @ V4 ) ) ) ).
% option.set_map
thf(fact_587_option_Oset__map,axiom,
! [F: option_a > a,V4: option_option_a] :
( ( set_option_a2 @ ( map_op4563205767754224965on_a_a @ F @ V4 ) )
= ( image_option_a_a @ F @ ( set_option_option_a2 @ V4 ) ) ) ).
% option.set_map
thf(fact_588_option_Oset__map,axiom,
! [F: a > a,V4: option_a] :
( ( set_option_a2 @ ( map_option_a_a @ F @ V4 ) )
= ( image_a_a @ F @ ( set_option_a2 @ V4 ) ) ) ).
% option.set_map
thf(fact_589_option_Omap__sel,axiom,
! [A: option_a,F: a > a] :
( ( A != none_a )
=> ( ( the_a2 @ ( map_option_a_a @ F @ A ) )
= ( F @ ( the_a2 @ A ) ) ) ) ).
% option.map_sel
thf(fact_590_UNIV__option__conv,axiom,
( top_to1659475022456381882tion_a
= ( insert605063979879581146tion_a @ none_option_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) ) ).
% UNIV_option_conv
thf(fact_591_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_592_surj__Compl__image__subset,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) ) @ ( image_7439109396645324421tion_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_593_surj__Compl__image__subset,axiom,
! [F: a > option_a,A2: set_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_a_option_a @ F @ A2 ) ) @ ( image_a_option_a @ F @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_594_surj__Compl__image__subset,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_option_a_a @ F @ A2 ) ) @ ( image_option_a_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_595_surj__Compl__image__subset,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A2 ) ) @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_596_finite__range__Some,axiom,
( ( finite8114217219359860531tion_a @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_range_Some
thf(fact_597_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_598_notin__range__Some,axiom,
! [X2: option_option_a] :
( ( ~ ( member5113800082084363315tion_a @ X2 @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) )
= ( X2 = none_option_a ) ) ).
% notin_range_Some
thf(fact_599_notin__range__Some,axiom,
! [X2: option_a] :
( ( ~ ( member_option_a @ X2 @ ( image_a_option_a @ some_a @ top_top_set_a ) ) )
= ( X2 = none_a ) ) ).
% notin_range_Some
thf(fact_600_inj__on__insert,axiom,
! [F: a > a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_601_inj__on__insert,axiom,
! [F: a > option_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_option_a @ F @ A2 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_602_inj__on__insert,axiom,
! [F: a > b,A: a,A2: set_a] :
( ( inj_on_a_b @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_b @ F @ A2 )
& ~ ( member_b @ ( F @ A ) @ ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_603_inj__on__insert,axiom,
! [F: option_a > a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ ( insert_option_a @ A @ A2 ) )
= ( ( inj_on_option_a_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_604_inj__on__insert,axiom,
! [F: option_a > option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( insert_option_a @ A @ A2 ) )
= ( ( inj_on8559383841115902449tion_a @ F @ A2 )
& ~ ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_605_inj__on__insert,axiom,
! [F: option_a > b,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_b @ F @ ( insert_option_a @ A @ A2 ) )
= ( ( inj_on_option_a_b @ F @ A2 )
& ~ ( member_b @ ( F @ A ) @ ( image_option_a_b @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_606_inj__image__Compl__subset,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ ( uminus_uminus_set_a @ A2 ) ) @ ( uminus6205308855922866075tion_a @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_607_inj__image__Compl__subset,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ord_less_eq_set_a @ ( image_option_a_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) @ ( uminus_uminus_set_a @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_608_inj__image__Compl__subset,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A2 ) ) @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_609_map__option__o__map__upd,axiom,
! [F: a > a,M: a > option_a,A: a,B: a] :
( ( comp_o6087033147929006299on_a_a @ ( map_option_a_a @ F ) @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) )
= ( fun_upd_a_option_a @ ( comp_o6087033147929006299on_a_a @ ( map_option_a_a @ F ) @ M ) @ A @ ( some_a @ ( F @ B ) ) ) ) ).
% map_option_o_map_upd
thf(fact_610_comp__apply,axiom,
( comp_option_a_a_a
= ( ^ [F2: option_a > a,G2: a > option_a,X3: a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_611_inj__on__empty,axiom,
! [F: a > option_a] : ( inj_on_a_option_a @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_612_inj__on__empty,axiom,
! [F: a > a] : ( inj_on_a_a @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_613_case__map__option,axiom,
! [G: a,H: option_a > a,F: a > option_a,X2: option_a] :
( ( case_o926465512965637841tion_a @ G @ H @ ( map_op2340691886215429841tion_a @ F @ X2 ) )
= ( case_option_a_a @ G @ ( comp_option_a_a_a @ H @ F ) @ X2 ) ) ).
% case_map_option
thf(fact_614_case__map__option,axiom,
! [G: $o,H: a > $o,F: a > a,X2: option_a] :
( ( case_option_o_a @ G @ H @ ( map_option_a_a @ F @ X2 ) )
= ( case_option_o_a @ G @ ( comp_a_o_a @ H @ F ) @ X2 ) ) ).
% case_map_option
thf(fact_615_case__map__option,axiom,
! [G: option_a,H: a > option_a,F: a > a,X2: option_a] :
( ( case_o3148979394504432965on_a_a @ G @ H @ ( map_option_a_a @ F @ X2 ) )
= ( case_o3148979394504432965on_a_a @ G @ ( comp_a_option_a_a @ H @ F ) @ X2 ) ) ).
% case_map_option
thf(fact_616_case__map__option,axiom,
! [G: a,H: a > a,F: a > a,X2: option_a] :
( ( case_option_a_a @ G @ H @ ( map_option_a_a @ F @ X2 ) )
= ( case_option_a_a @ G @ ( comp_a_a_a @ H @ F ) @ X2 ) ) ).
% case_map_option
thf(fact_617_image__eq__imp__comp,axiom,
! [F: a > a,A2: set_a,G: a > a,B2: set_a,H: a > a] :
( ( ( image_a_a @ F @ A2 )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_a_a @ ( comp_a_a_a @ H @ F ) @ A2 )
= ( image_a_a @ ( comp_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_618_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: a > option_a,B2: set_a,H: option_a > a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ G @ B2 ) )
=> ( ( image_a_a @ ( comp_option_a_a_a @ H @ F ) @ A2 )
= ( image_a_a @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_619_image__eq__imp__comp,axiom,
! [F: option_a > a,A2: set_option_a,G: a > a,B2: set_a,H: a > a] :
( ( ( image_option_a_a @ F @ A2 )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_option_a_a @ ( comp_a_a_option_a @ H @ F ) @ A2 )
= ( image_a_a @ ( comp_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_620_image__eq__imp__comp,axiom,
! [F: a > a,A2: set_a,G: option_a > a,B2: set_option_a,H: a > a] :
( ( ( image_a_a @ F @ A2 )
= ( image_option_a_a @ G @ B2 ) )
=> ( ( image_a_a @ ( comp_a_a_a @ H @ F ) @ A2 )
= ( image_option_a_a @ ( comp_a_a_option_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_621_image__eq__imp__comp,axiom,
! [F: a > a,A2: set_a,G: a > a,B2: set_a,H: a > option_a] :
( ( ( image_a_a @ F @ A2 )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_a_option_a @ ( comp_a_option_a_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_a_option_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_622_image__eq__imp__comp,axiom,
! [F: option_a > option_a,A2: set_option_a,G: a > option_a,B2: set_a,H: option_a > a] :
( ( ( image_7439109396645324421tion_a @ F @ A2 )
= ( image_a_option_a @ G @ B2 ) )
=> ( ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ H @ F ) @ A2 )
= ( image_a_a @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_623_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: option_a > option_a,B2: set_option_a,H: option_a > a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_7439109396645324421tion_a @ G @ B2 ) )
=> ( ( image_a_a @ ( comp_option_a_a_a @ H @ F ) @ A2 )
= ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_624_image__eq__imp__comp,axiom,
! [F: a > option_a,A2: set_a,G: a > option_a,B2: set_a,H: option_a > option_a] :
( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ G @ B2 ) )
=> ( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_625_image__eq__imp__comp,axiom,
! [F: option_a > a,A2: set_option_a,G: option_a > a,B2: set_option_a,H: a > a] :
( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ G @ B2 ) )
=> ( ( image_option_a_a @ ( comp_a_a_option_a @ H @ F ) @ A2 )
= ( image_option_a_a @ ( comp_a_a_option_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_626_image__eq__imp__comp,axiom,
! [F: option_a > a,A2: set_option_a,G: a > a,B2: set_a,H: a > option_a] :
( ( ( image_option_a_a @ F @ A2 )
= ( image_a_a @ G @ B2 ) )
=> ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ F ) @ A2 )
= ( image_a_option_a @ ( comp_a_option_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_627_inj__on__image__iff,axiom,
! [A2: set_a,G: a > option_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ G @ ( image_a_a @ F @ A2 ) )
= ( inj_on_a_option_a @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_628_inj__on__image__iff,axiom,
! [A2: set_a,G: a > a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A2 ) )
= ( inj_on_a_a @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_629_comp__inj__on__iff,axiom,
! [F: option_a > a,A2: set_option_a,F4: a > option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ F4 @ ( image_option_a_a @ F @ A2 ) )
= ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_630_comp__inj__on__iff,axiom,
! [F: option_a > a,A2: set_option_a,F4: a > a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ F4 @ ( image_option_a_a @ F @ A2 ) )
= ( inj_on_option_a_a @ ( comp_a_a_option_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_631_comp__inj__on__iff,axiom,
! [F: a > option_a,A2: set_a,F4: option_a > option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ F4 @ ( image_a_option_a @ F @ A2 ) )
= ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_632_comp__inj__on__iff,axiom,
! [F: a > option_a,A2: set_a,F4: option_a > a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( inj_on_option_a_a @ F4 @ ( image_a_option_a @ F @ A2 ) )
= ( inj_on_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_633_comp__inj__on__iff,axiom,
! [F: a > a,A2: set_a,F4: a > option_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ F4 @ ( image_a_a @ F @ A2 ) )
= ( inj_on_a_option_a @ ( comp_a_option_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_634_comp__inj__on__iff,axiom,
! [F: a > a,A2: set_a,F4: a > a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ F4 @ ( image_a_a @ F @ A2 ) )
= ( inj_on_a_a @ ( comp_a_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_635_inj__on__imageI,axiom,
! [G: a > option_a,F: option_a > a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ A2 )
=> ( inj_on_a_option_a @ G @ ( image_option_a_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_636_inj__on__imageI,axiom,
! [G: a > a,F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ A2 )
=> ( inj_on_a_a @ G @ ( image_option_a_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_637_inj__on__imageI,axiom,
! [G: option_a > option_a,F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ A2 )
=> ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_638_inj__on__imageI,axiom,
! [G: a > option_a,F: a > a,A2: set_a] :
( ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ A2 )
=> ( inj_on_a_option_a @ G @ ( image_a_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_639_inj__on__imageI,axiom,
! [G: option_a > a,F: a > option_a,A2: set_a] :
( ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 )
=> ( inj_on_option_a_a @ G @ ( image_a_option_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_640_inj__on__imageI,axiom,
! [G: a > a,F: a > a,A2: set_a] :
( ( inj_on_a_a @ ( comp_a_a_a @ G @ F ) @ A2 )
=> ( inj_on_a_a @ G @ ( image_a_a @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_641_comp__inj__on,axiom,
! [F: option_a > a,A2: set_option_a,G: a > option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ G @ ( image_option_a_a @ F @ A2 ) )
=> ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_642_comp__inj__on,axiom,
! [F: option_a > a,A2: set_option_a,G: a > a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ G @ ( image_option_a_a @ F @ A2 ) )
=> ( inj_on_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_643_comp__inj__on,axiom,
! [F: a > option_a,A2: set_a,G: option_a > option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ G @ ( image_a_option_a @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_644_comp__inj__on,axiom,
! [F: a > option_a,A2: set_a,G: option_a > a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( inj_on_option_a_a @ G @ ( image_a_option_a @ F @ A2 ) )
=> ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_645_comp__inj__on,axiom,
! [F: a > a,A2: set_a,G: a > option_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ G @ ( image_a_a @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_646_comp__inj__on,axiom,
! [F: a > a,A2: set_a,G: a > a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A2 ) )
=> ( inj_on_a_a @ ( comp_a_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_647_image__comp,axiom,
! [F: option_a > option_a,G: a > option_a,R2: set_a] :
( ( image_7439109396645324421tion_a @ F @ ( image_a_option_a @ G @ R2 ) )
= ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_648_image__comp,axiom,
! [F: a > option_a,G: option_a > a,R2: set_option_a] :
( ( image_a_option_a @ F @ ( image_option_a_a @ G @ R2 ) )
= ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_649_image__comp,axiom,
! [F: a > option_a,G: a > a,R2: set_a] :
( ( image_a_option_a @ F @ ( image_a_a @ G @ R2 ) )
= ( image_a_option_a @ ( comp_a_option_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_650_image__comp,axiom,
! [F: option_a > a,G: option_a > option_a,R2: set_option_a] :
( ( image_option_a_a @ F @ ( image_7439109396645324421tion_a @ G @ R2 ) )
= ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_651_image__comp,axiom,
! [F: option_a > a,G: a > option_a,R2: set_a] :
( ( image_option_a_a @ F @ ( image_a_option_a @ G @ R2 ) )
= ( image_a_a @ ( comp_option_a_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_652_image__comp,axiom,
! [F: a > a,G: option_a > a,R2: set_option_a] :
( ( image_a_a @ F @ ( image_option_a_a @ G @ R2 ) )
= ( image_option_a_a @ ( comp_a_a_option_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_653_image__comp,axiom,
! [F: a > a,G: a > a,R2: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G @ R2 ) )
= ( image_a_a @ ( comp_a_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_654_map__option_Ocomp,axiom,
! [F: option_a > a,G: a > option_a] :
( ( comp_o1254687777855551975tion_a @ ( map_op4563205767754224965on_a_a @ F ) @ ( map_op2340691886215429841tion_a @ G ) )
= ( map_option_a_a @ ( comp_option_a_a_a @ F @ G ) ) ) ).
% map_option.comp
thf(fact_655_map__option_Ocomp,axiom,
! [F: a > a,G: a > a] :
( ( comp_o3154387707078715297tion_a @ ( map_option_a_a @ F ) @ ( map_option_a_a @ G ) )
= ( map_option_a_a @ ( comp_a_a_a @ F @ G ) ) ) ).
% map_option.comp
thf(fact_656_option_Oinj__map,axiom,
! [F: a > option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( inj_on2224753519991154999tion_a @ ( map_op2340691886215429841tion_a @ F ) @ top_top_set_option_a ) ) ).
% option.inj_map
thf(fact_657_option_Oinj__map,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( inj_on8559383841115902449tion_a @ ( map_option_a_a @ F ) @ top_top_set_option_a ) ) ).
% option.inj_map
thf(fact_658_option_Omap__comp,axiom,
! [G: option_a > a,F: a > option_a,V4: option_a] :
( ( map_op4563205767754224965on_a_a @ G @ ( map_op2340691886215429841tion_a @ F @ V4 ) )
= ( map_option_a_a @ ( comp_option_a_a_a @ G @ F ) @ V4 ) ) ).
% option.map_comp
thf(fact_659_option_Omap__comp,axiom,
! [G: a > a,F: a > a,V4: option_a] :
( ( map_option_a_a @ G @ ( map_option_a_a @ F @ V4 ) )
= ( map_option_a_a @ ( comp_a_a_a @ G @ F ) @ V4 ) ) ).
% option.map_comp
thf(fact_660_map__option_Ocompositionality,axiom,
! [F: option_a > a,G: a > option_a,Option: option_a] :
( ( map_op4563205767754224965on_a_a @ F @ ( map_op2340691886215429841tion_a @ G @ Option ) )
= ( map_option_a_a @ ( comp_option_a_a_a @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_661_map__option_Ocompositionality,axiom,
! [F: a > a,G: a > a,Option: option_a] :
( ( map_option_a_a @ F @ ( map_option_a_a @ G @ Option ) )
= ( map_option_a_a @ ( comp_a_a_a @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_662_injD,axiom,
! [F: a > option_a,X2: a,Y: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ).
% injD
thf(fact_663_injD,axiom,
! [F: a > a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ).
% injD
thf(fact_664_injI,axiom,
! [F: a > option_a] :
( ! [X: a,Y3: a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_a_option_a @ F @ top_top_set_a ) ) ).
% injI
thf(fact_665_injI,axiom,
! [F: a > a] :
( ! [X: a,Y3: a] :
( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) )
=> ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% injI
thf(fact_666_inj__eq,axiom,
! [F: a > option_a,X2: a,Y: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ).
% inj_eq
thf(fact_667_inj__eq,axiom,
! [F: a > a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ).
% inj_eq
thf(fact_668_inj__def,axiom,
! [F: a > option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
= ( ! [X3: a,Y2: a] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ).
% inj_def
thf(fact_669_inj__def,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
= ( ! [X3: a,Y2: a] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ).
% inj_def
thf(fact_670_inj__compose,axiom,
! [F: option_a > option_a,G: a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( inj_on_a_option_a @ G @ top_top_set_a )
=> ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_671_inj__compose,axiom,
! [F: option_a > a,G: a > option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( inj_on_a_option_a @ G @ top_top_set_a )
=> ( inj_on_a_a @ ( comp_option_a_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_672_inj__compose,axiom,
! [F: a > option_a,G: option_a > a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( inj_on_option_a_a @ G @ top_top_set_option_a )
=> ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F @ G ) @ top_top_set_option_a ) ) ) ).
% inj_compose
thf(fact_673_inj__compose,axiom,
! [F: a > a,G: option_a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( inj_on_option_a_a @ G @ top_top_set_option_a )
=> ( inj_on_option_a_a @ ( comp_a_a_option_a @ F @ G ) @ top_top_set_option_a ) ) ) ).
% inj_compose
thf(fact_674_inj__compose,axiom,
! [F: a > option_a,G: a > a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( inj_on_a_a @ G @ top_top_set_a )
=> ( inj_on_a_option_a @ ( comp_a_option_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_675_inj__compose,axiom,
! [F: a > a,G: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( inj_on_a_a @ G @ top_top_set_a )
=> ( inj_on_a_a @ ( comp_a_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% inj_compose
thf(fact_676_inj__on__diff,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( inj_on_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_677_inj__on__diff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( inj_on_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_678_comp__eq__dest__lhs,axiom,
! [A: option_a > a,B: a > option_a,C: a > a,V4: a] :
( ( ( comp_option_a_a_a @ A @ B )
= C )
=> ( ( A @ ( B @ V4 ) )
= ( C @ V4 ) ) ) ).
% comp_eq_dest_lhs
thf(fact_679_inj__on__inverseI,axiom,
! [A2: set_a,G: option_a > a,F: a > option_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_680_inj__on__inverseI,axiom,
! [A2: set_a,G: a > a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( G @ ( F @ X ) )
= X ) )
=> ( inj_on_a_a @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_681_inj__on__imageI2,axiom,
! [F4: option_a > option_a,F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F4 @ F ) @ A2 )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_682_inj__on__imageI2,axiom,
! [F4: a > option_a,F: a > a,A2: set_a] :
( ( inj_on_a_option_a @ ( comp_a_option_a_a @ F4 @ F ) @ A2 )
=> ( inj_on_a_a @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_683_inj__on__imageI2,axiom,
! [F4: option_a > a,F: a > option_a,A2: set_a] :
( ( inj_on_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_684_inj__on__imageI2,axiom,
! [F4: a > a,F: a > a,A2: set_a] :
( ( inj_on_a_a @ ( comp_a_a_a @ F4 @ F ) @ A2 )
=> ( inj_on_a_a @ F @ A2 ) ) ).
% inj_on_imageI2
thf(fact_685_inj__on__contraD,axiom,
! [F: a > option_a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_686_inj__on__contraD,axiom,
! [F: a > a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_687_inj__on__eq__iff,axiom,
! [F: a > option_a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_688_inj__on__eq__iff,axiom,
! [F: a > a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_689_comp__eq__elim,axiom,
! [A: option_a > a,B: a > option_a,C: option_a > a,D: a > option_a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_option_a_a_a @ C @ D ) )
=> ! [V5: a] :
( ( A @ ( B @ V5 ) )
= ( C @ ( D @ V5 ) ) ) ) ).
% comp_eq_elim
thf(fact_690_comp__eq__dest,axiom,
! [A: option_a > a,B: a > option_a,C: option_a > a,D: a > option_a,V4: a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_option_a_a_a @ C @ D ) )
=> ( ( A @ ( B @ V4 ) )
= ( C @ ( D @ V4 ) ) ) ) ).
% comp_eq_dest
thf(fact_691_inj__on__cong,axiom,
! [A2: set_a,F: a > option_a,G: a > option_a] :
( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_a_option_a @ F @ A2 )
= ( inj_on_a_option_a @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_692_inj__on__cong,axiom,
! [A2: set_a,F: a > a,G: a > a] :
( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_a_a @ F @ A2 )
= ( inj_on_a_a @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_693_inj__on__def,axiom,
( inj_on_a_option_a
= ( ^ [F2: a > option_a,A8: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A8 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_694_inj__on__def,axiom,
( inj_on_a_a
= ( ^ [F2: a > a,A8: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A8 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y2 ) )
=> ( X3 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_695_comp__assoc,axiom,
! [F: option_a > a,G: a > option_a,H: a > a] :
( ( comp_a_a_a @ ( comp_option_a_a_a @ F @ G ) @ H )
= ( comp_option_a_a_a @ F @ ( comp_a_option_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_696_comp__assoc,axiom,
! [F: a > a,G: option_a > a,H: a > option_a] :
( ( comp_option_a_a_a @ ( comp_a_a_option_a @ F @ G ) @ H )
= ( comp_a_a_a @ F @ ( comp_option_a_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_697_comp__assoc,axiom,
! [F: option_a > a,G: option_a > option_a,H: a > option_a] :
( ( comp_option_a_a_a @ ( comp_o3864519266390211175tion_a @ F @ G ) @ H )
= ( comp_option_a_a_a @ F @ ( comp_o6087033147929006299on_a_a @ G @ H ) ) ) ).
% comp_assoc
thf(fact_698_comp__def,axiom,
( comp_option_a_a_a
= ( ^ [F2: option_a > a,G2: a > option_a,X3: a] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_699_inj__onI,axiom,
! [A2: set_a,F: a > option_a] :
( ! [X: a,Y3: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_onI
thf(fact_700_inj__onI,axiom,
! [A2: set_a,F: a > a] :
( ! [X: a,Y3: a] :
( ( member_a @ X @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) )
=> ( inj_on_a_a @ F @ A2 ) ) ).
% inj_onI
thf(fact_701_inj__onD,axiom,
! [F: a > option_a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_702_inj__onD,axiom,
! [F: a > a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_703_fun__upd__comp,axiom,
! [F: option_a > a,G: a > option_a,X2: a,Y: option_a] :
( ( comp_option_a_a_a @ F @ ( fun_upd_a_option_a @ G @ X2 @ Y ) )
= ( fun_upd_a_a @ ( comp_option_a_a_a @ F @ G ) @ X2 @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_704_fun__upd__comp,axiom,
! [F: option_a > option_a,G: a > option_a,X2: a,Y: option_a] :
( ( comp_o6087033147929006299on_a_a @ F @ ( fun_upd_a_option_a @ G @ X2 @ Y ) )
= ( fun_upd_a_option_a @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ X2 @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_705_inj__Some,axiom,
! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).
% inj_Some
thf(fact_706_subset__inj__on,axiom,
! [F: a > option_a,B2: set_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( inj_on_a_option_a @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_707_subset__inj__on,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( inj_on_a_a @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_708_inj__on__subset,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( inj_on_a_option_a @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_709_inj__on__subset,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( inj_on_a_a @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_710_inj__on__id2,axiom,
! [A2: set_a] :
( inj_on_a_a
@ ^ [X3: a] : X3
@ A2 ) ).
% inj_on_id2
thf(fact_711_range__ex1__eq,axiom,
! [F: option_a > a,B: a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( member_a @ B @ ( image_option_a_a @ F @ top_top_set_option_a ) )
= ( ? [X3: option_a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: option_a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_712_range__ex1__eq,axiom,
! [F: option_a > option_a,B: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( member_option_a @ B @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) )
= ( ? [X3: option_a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: option_a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_713_range__ex1__eq,axiom,
! [F: option_a > b,B: b] :
( ( inj_on_option_a_b @ F @ top_top_set_option_a )
=> ( ( member_b @ B @ ( image_option_a_b @ F @ top_top_set_option_a ) )
= ( ? [X3: option_a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: option_a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_714_range__ex1__eq,axiom,
! [F: a > a,B: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ B @ ( image_a_a @ F @ top_top_set_a ) )
= ( ? [X3: a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_715_range__ex1__eq,axiom,
! [F: a > option_a,B: option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( member_option_a @ B @ ( image_a_option_a @ F @ top_top_set_a ) )
= ( ? [X3: a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_716_range__ex1__eq,axiom,
! [F: a > b,B: b] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( member_b @ B @ ( image_a_b @ F @ top_top_set_a ) )
= ( ? [X3: a] :
( ( B
= ( F @ X3 ) )
& ! [Y2: a] :
( ( B
= ( F @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_717_inj__image__eq__iff,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_718_inj__image__eq__iff,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_719_inj__image__eq__iff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a @ F @ A2 )
= ( image_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_720_inj__image__mem__iff,axiom,
! [F: b > a,A: b,A2: set_b] :
( ( inj_on_b_a @ F @ top_top_set_b )
=> ( ( member_a @ ( F @ A ) @ ( image_b_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_721_inj__image__mem__iff,axiom,
! [F: b > option_a,A: b,A2: set_b] :
( ( inj_on_b_option_a @ F @ top_top_set_b )
=> ( ( member_option_a @ ( F @ A ) @ ( image_b_option_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_722_inj__image__mem__iff,axiom,
! [F: b > b,A: b,A2: set_b] :
( ( inj_on_b_b @ F @ top_top_set_b )
=> ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_723_inj__image__mem__iff,axiom,
! [F: option_a > a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_724_inj__image__mem__iff,axiom,
! [F: option_a > option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_725_inj__image__mem__iff,axiom,
! [F: option_a > b,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_b @ F @ top_top_set_option_a )
=> ( ( member_b @ ( F @ A ) @ ( image_option_a_b @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_726_inj__image__mem__iff,axiom,
! [F: a > option_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_727_inj__image__mem__iff,axiom,
! [F: a > b,A: a,A2: set_a] :
( ( inj_on_a_b @ F @ top_top_set_a )
=> ( ( member_b @ ( F @ A ) @ ( image_a_b @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_728_inj__image__mem__iff,axiom,
! [F: a > a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_729_inj__on__image__mem__iff,axiom,
! [F: option_a > a,B2: set_option_a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ B2 )
=> ( ( member_option_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_option_a_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_730_inj__on__image__mem__iff,axiom,
! [F: option_a > option_a,B2: set_option_a,A: option_a,A2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ B2 )
=> ( ( member_option_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_7439109396645324421tion_a @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_731_inj__on__image__mem__iff,axiom,
! [F: option_a > b,B2: set_option_a,A: option_a,A2: set_option_a] :
( ( inj_on_option_a_b @ F @ B2 )
=> ( ( member_option_a @ A @ B2 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_option_a_b @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_732_inj__on__image__mem__iff,axiom,
! [F: b > a,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_a @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_b_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_733_inj__on__image__mem__iff,axiom,
! [F: b > option_a,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_option_a @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_b_option_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_734_inj__on__image__mem__iff,axiom,
! [F: b > b,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_b @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_735_inj__on__image__mem__iff,axiom,
! [F: a > option_a,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_option_a @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_736_inj__on__image__mem__iff,axiom,
! [F: a > b,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_b @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_a_b @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_737_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_738_inj__on__image__eq__iff,axiom,
! [F: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_739_inj__on__image__eq__iff,axiom,
! [F: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_740_inj__on__image__eq__iff,axiom,
! [F: a > a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( ( image_a_a @ F @ A2 )
= ( image_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_741_inj__img__insertE,axiom,
! [F: option_a > a,A2: set_option_a,X2: a,B2: set_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ B2 )
= ( image_option_a_a @ F @ A2 ) )
=> ~ ! [X7: option_a,A9: set_option_a] :
( ~ ( member_option_a @ X7 @ A9 )
=> ( ( A2
= ( insert_option_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_option_a_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_742_inj__img__insertE,axiom,
! [F: b > a,A2: set_b,X2: a,B2: set_a] :
( ( inj_on_b_a @ F @ A2 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ B2 )
= ( image_b_a @ F @ A2 ) )
=> ~ ! [X7: b,A9: set_b] :
( ~ ( member_b @ X7 @ A9 )
=> ( ( A2
= ( insert_b @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_b_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_743_inj__img__insertE,axiom,
! [F: a > option_a,A2: set_a,X2: option_a,B2: set_option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ X2 @ B2 )
=> ( ( ( insert_option_a @ X2 @ B2 )
= ( image_a_option_a @ F @ A2 ) )
=> ~ ! [X7: a,A9: set_a] :
( ~ ( member_a @ X7 @ A9 )
=> ( ( A2
= ( insert_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_a_option_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_744_inj__img__insertE,axiom,
! [F: option_a > option_a,A2: set_option_a,X2: option_a,B2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ~ ( member_option_a @ X2 @ B2 )
=> ( ( ( insert_option_a @ X2 @ B2 )
= ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ~ ! [X7: option_a,A9: set_option_a] :
( ~ ( member_option_a @ X7 @ A9 )
=> ( ( A2
= ( insert_option_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_7439109396645324421tion_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_745_inj__img__insertE,axiom,
! [F: b > option_a,A2: set_b,X2: option_a,B2: set_option_a] :
( ( inj_on_b_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ X2 @ B2 )
=> ( ( ( insert_option_a @ X2 @ B2 )
= ( image_b_option_a @ F @ A2 ) )
=> ~ ! [X7: b,A9: set_b] :
( ~ ( member_b @ X7 @ A9 )
=> ( ( A2
= ( insert_b @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_b_option_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_746_inj__img__insertE,axiom,
! [F: a > b,A2: set_a,X2: b,B2: set_b] :
( ( inj_on_a_b @ F @ A2 )
=> ( ~ ( member_b @ X2 @ B2 )
=> ( ( ( insert_b @ X2 @ B2 )
= ( image_a_b @ F @ A2 ) )
=> ~ ! [X7: a,A9: set_a] :
( ~ ( member_a @ X7 @ A9 )
=> ( ( A2
= ( insert_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_a_b @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_747_inj__img__insertE,axiom,
! [F: option_a > b,A2: set_option_a,X2: b,B2: set_b] :
( ( inj_on_option_a_b @ F @ A2 )
=> ( ~ ( member_b @ X2 @ B2 )
=> ( ( ( insert_b @ X2 @ B2 )
= ( image_option_a_b @ F @ A2 ) )
=> ~ ! [X7: option_a,A9: set_option_a] :
( ~ ( member_option_a @ X7 @ A9 )
=> ( ( A2
= ( insert_option_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_option_a_b @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_748_inj__img__insertE,axiom,
! [F: b > b,A2: set_b,X2: b,B2: set_b] :
( ( inj_on_b_b @ F @ A2 )
=> ( ~ ( member_b @ X2 @ B2 )
=> ( ( ( insert_b @ X2 @ B2 )
= ( image_b_b @ F @ A2 ) )
=> ~ ! [X7: b,A9: set_b] :
( ~ ( member_b @ X7 @ A9 )
=> ( ( A2
= ( insert_b @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_b_b @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_749_inj__img__insertE,axiom,
! [F: a > a,A2: set_a,X2: a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ B2 )
= ( image_a_a @ F @ A2 ) )
=> ~ ! [X7: a,A9: set_a] :
( ~ ( member_a @ X7 @ A9 )
=> ( ( A2
= ( insert_a @ X7 @ A9 ) )
=> ( ( X2
= ( F @ X7 ) )
=> ( B2
!= ( image_a_a @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_750_comp__surj,axiom,
! [F: option_a > option_a,G: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_751_comp__surj,axiom,
! [F: option_a > option_a,G: option_a > a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_752_comp__surj,axiom,
! [F: option_a > a,G: a > option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_753_comp__surj,axiom,
! [F: option_a > a,G: a > a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_754_comp__surj,axiom,
! [F: a > option_a,G: option_a > option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( image_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_755_comp__surj,axiom,
! [F: a > option_a,G: option_a > a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( image_option_a_a @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_option_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_756_comp__surj,axiom,
! [F: a > a,G: a > option_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_option_a @ G @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_757_comp__surj,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ ( comp_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_758_inj__on__fun__updI,axiom,
! [F: option_a > a,A2: set_option_a,Y: a,X2: option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ Y @ ( image_option_a_a @ F @ A2 ) )
=> ( inj_on_option_a_a @ ( fun_upd_option_a_a @ F @ X2 @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_759_inj__on__fun__updI,axiom,
! [F: a > a,A2: set_a,Y: a,X2: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ~ ( member_a @ Y @ ( image_a_a @ F @ A2 ) )
=> ( inj_on_a_a @ ( fun_upd_a_a @ F @ X2 @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_760_inj__on__fun__updI,axiom,
! [F: a > option_a,A2: set_a,Y: option_a,X2: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ Y @ ( image_a_option_a @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( fun_upd_a_option_a @ F @ X2 @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_761_inj__singleton,axiom,
! [A2: set_a] :
( inj_on_a_set_a
@ ^ [X3: a] : ( insert_a @ X3 @ bot_bot_set_a )
@ A2 ) ).
% inj_singleton
thf(fact_762_inj__singleton,axiom,
! [A2: set_option_a] :
( inj_on7881382345526841553tion_a
@ ^ [X3: option_a] : ( insert_option_a @ X3 @ bot_bot_set_option_a )
@ A2 ) ).
% inj_singleton
thf(fact_763_bind__map__option,axiom,
! [F: a > a,X2: option_a,G: a > option_a] :
( ( bind_a_a @ ( map_option_a_a @ F @ X2 ) @ G )
= ( bind_a_a @ X2 @ ( comp_a_option_a_a @ G @ F ) ) ) ).
% bind_map_option
thf(fact_764_map__option__bind,axiom,
! [F: a > a,X2: option_a,G: a > option_a] :
( ( map_option_a_a @ F @ ( bind_a_a @ X2 @ G ) )
= ( bind_a_a @ X2 @ ( comp_o6087033147929006299on_a_a @ ( map_option_a_a @ F ) @ G ) ) ) ).
% map_option_bind
thf(fact_765_inj__image__subset__iff,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_766_inj__image__subset__iff,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B2 ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_767_inj__image__subset__iff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_768_image__set__diff,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_769_image__set__diff,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_770_image__set__diff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_771_inj__on__image__set__diff,axiom,
! [F: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
=> ( ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_772_inj__on__image__set__diff,axiom,
! [F: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_773_inj__on__image__set__diff,axiom,
! [F: a > a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_774_map__conv__bind__option,axiom,
( map_option_a_a
= ( ^ [F2: a > a,X3: option_a] : ( bind_a_a @ X3 @ ( comp_a_option_a_a @ some_a @ F2 ) ) ) ) ).
% map_conv_bind_option
thf(fact_775_ran__map__upd__Some,axiom,
! [M: a > option_a,X2: a,Y: a,Z2: a] :
( ( ( M @ X2 )
= ( some_a @ Y ) )
=> ( ( inj_on_a_option_a @ M @ ( dom_a_a @ M ) )
=> ( ~ ( member_a @ Z2 @ ( ran_a_a @ M ) )
=> ( ( ran_a_a @ ( fun_upd_a_option_a @ M @ X2 @ ( some_a @ Z2 ) ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( ran_a_a @ M ) @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( insert_a @ Z2 @ bot_bot_set_a ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_776_the__inv__into__comp,axiom,
! [F: option_a > option_a,G: a > option_a,A2: set_a,X2: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( image_a_option_a @ G @ A2 ) )
=> ( ( inj_on_a_option_a @ G @ A2 )
=> ( ( member_option_a @ X2 @ ( image_7439109396645324421tion_a @ F @ ( image_a_option_a @ G @ A2 ) ) )
=> ( ( the_in8758012798868597241tion_a @ A2 @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ X2 )
= ( comp_o3864519266390211175tion_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in2538339130118444083tion_a @ ( image_a_option_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_777_the__inv__into__comp,axiom,
! [F: a > option_a,G: option_a > a,A2: set_option_a,X2: option_a] :
( ( inj_on_a_option_a @ F @ ( image_option_a_a @ G @ A2 ) )
=> ( ( inj_on_option_a_a @ G @ A2 )
=> ( ( member_option_a @ X2 @ ( image_a_option_a @ F @ ( image_option_a_a @ G @ A2 ) ) )
=> ( ( the_in2538339130118444083tion_a @ A2 @ ( comp_a6249931511552232923tion_a @ F @ G ) @ X2 )
= ( comp_a6249931511552232923tion_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_in8758012798868597241tion_a @ ( image_option_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_778_the__inv__into__comp,axiom,
! [F: a > b,G: option_a > a,A2: set_option_a,X2: b] :
( ( inj_on_a_b @ F @ ( image_option_a_a @ G @ A2 ) )
=> ( ( inj_on_option_a_a @ G @ A2 )
=> ( ( member_b @ X2 @ ( image_a_b @ F @ ( image_option_a_a @ G @ A2 ) ) )
=> ( ( the_in1757154643552616558on_a_b @ A2 @ ( comp_a_b_option_a @ F @ G ) @ X2 )
= ( comp_a_option_a_b @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_inv_into_a_b @ ( image_option_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_779_the__inv__into__comp,axiom,
! [F: option_a > b,G: a > option_a,A2: set_a,X2: b] :
( ( inj_on_option_a_b @ F @ ( image_a_option_a @ G @ A2 ) )
=> ( ( inj_on_a_option_a @ G @ A2 )
=> ( ( member_b @ X2 @ ( image_option_a_b @ F @ ( image_a_option_a @ G @ A2 ) ) )
=> ( ( the_inv_into_a_b @ A2 @ ( comp_option_a_b_a @ F @ G ) @ X2 )
= ( comp_option_a_a_b @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in1757154643552616558on_a_b @ ( image_a_option_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_780_the__inv__into__comp,axiom,
! [F: a > a,G: option_a > a,A2: set_option_a,X2: a] :
( ( inj_on_a_a @ F @ ( image_option_a_a @ G @ A2 ) )
=> ( ( inj_on_option_a_a @ G @ A2 )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ ( image_option_a_a @ G @ A2 ) ) )
=> ( ( the_in1757154643552616557on_a_a @ A2 @ ( comp_a_a_option_a @ F @ G ) @ X2 )
= ( comp_a_option_a_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_inv_into_a_a @ ( image_option_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_781_the__inv__into__comp,axiom,
! [F: a > option_a,G: a > a,A2: set_a,X2: option_a] :
( ( inj_on_a_option_a @ F @ ( image_a_a @ G @ A2 ) )
=> ( ( inj_on_a_a @ G @ A2 )
=> ( ( member_option_a @ X2 @ ( image_a_option_a @ F @ ( image_a_a @ G @ A2 ) ) )
=> ( ( the_in8758012798868597241tion_a @ A2 @ ( comp_a_option_a_a @ F @ G ) @ X2 )
= ( comp_a_a_option_a @ ( the_inv_into_a_a @ A2 @ G ) @ ( the_in8758012798868597241tion_a @ ( image_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_782_the__inv__into__comp,axiom,
! [F: a > b,G: a > a,A2: set_a,X2: b] :
( ( inj_on_a_b @ F @ ( image_a_a @ G @ A2 ) )
=> ( ( inj_on_a_a @ G @ A2 )
=> ( ( member_b @ X2 @ ( image_a_b @ F @ ( image_a_a @ G @ A2 ) ) )
=> ( ( the_inv_into_a_b @ A2 @ ( comp_a_b_a @ F @ G ) @ X2 )
= ( comp_a_a_b @ ( the_inv_into_a_a @ A2 @ G ) @ ( the_inv_into_a_b @ ( image_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_783_the__inv__into__comp,axiom,
! [F: option_a > a,G: a > option_a,A2: set_a,X2: a] :
( ( inj_on_option_a_a @ F @ ( image_a_option_a @ G @ A2 ) )
=> ( ( inj_on_a_option_a @ G @ A2 )
=> ( ( member_a @ X2 @ ( image_option_a_a @ F @ ( image_a_option_a @ G @ A2 ) ) )
=> ( ( the_inv_into_a_a @ A2 @ ( comp_option_a_a_a @ F @ G ) @ X2 )
= ( comp_option_a_a_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in1757154643552616557on_a_a @ ( image_a_option_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_784_the__inv__into__comp,axiom,
! [F: a > a,G: a > a,A2: set_a,X2: a] :
( ( inj_on_a_a @ F @ ( image_a_a @ G @ A2 ) )
=> ( ( inj_on_a_a @ G @ A2 )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ ( image_a_a @ G @ A2 ) ) )
=> ( ( the_inv_into_a_a @ A2 @ ( comp_a_a_a @ F @ G ) @ X2 )
= ( comp_a_a_a @ ( the_inv_into_a_a @ A2 @ G ) @ ( the_inv_into_a_a @ ( image_a_a @ G @ A2 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_785_the__inv__into__into,axiom,
! [F: option_a > a,A2: set_option_a,X2: a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( member_a @ X2 @ ( image_option_a_a @ F @ A2 ) )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( member_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_786_the__inv__into__into,axiom,
! [F: b > a,A2: set_b,X2: a,B2: set_b] :
( ( inj_on_b_a @ F @ A2 )
=> ( ( member_a @ X2 @ ( image_b_a @ F @ A2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b @ ( the_inv_into_b_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_787_the__inv__into__into,axiom,
! [F: option_a > option_a,A2: set_option_a,X2: option_a,B2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( member_option_a @ X2 @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( member_option_a @ ( the_in2538339130118444083tion_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_788_the__inv__into__into,axiom,
! [F: b > option_a,A2: set_b,X2: option_a,B2: set_b] :
( ( inj_on_b_option_a @ F @ A2 )
=> ( ( member_option_a @ X2 @ ( image_b_option_a @ F @ A2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b @ ( the_in5672256556878602680tion_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_789_the__inv__into__into,axiom,
! [F: option_a > b,A2: set_option_a,X2: b,B2: set_option_a] :
( ( inj_on_option_a_b @ F @ A2 )
=> ( ( member_b @ X2 @ ( image_option_a_b @ F @ A2 ) )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( member_option_a @ ( the_in1757154643552616558on_a_b @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_790_the__inv__into__into,axiom,
! [F: b > b,A2: set_b,X2: b,B2: set_b] :
( ( inj_on_b_b @ F @ A2 )
=> ( ( member_b @ X2 @ ( image_b_b @ F @ A2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b @ ( the_inv_into_b_b @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_791_the__inv__into__into,axiom,
! [F: a > option_a,A2: set_a,X2: option_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_option_a @ X2 @ ( image_a_option_a @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a @ ( the_in8758012798868597241tion_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_792_the__inv__into__into,axiom,
! [F: a > b,A2: set_a,X2: b,B2: set_a] :
( ( inj_on_a_b @ F @ A2 )
=> ( ( member_b @ X2 @ ( image_a_b @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a @ ( the_inv_into_a_b @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_793_the__inv__into__into,axiom,
! [F: a > a,A2: set_a,X2: a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a @ ( the_inv_into_a_a @ A2 @ F @ X2 ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_794_the__inv__into__onto,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( image_a_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F ) @ ( image_option_a_a @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_795_the__inv__into__onto,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( image_option_a_a @ ( the_in8758012798868597241tion_a @ A2 @ F ) @ ( image_a_option_a @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_796_the__inv__into__onto,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( image_a_a @ ( the_inv_into_a_a @ A2 @ F ) @ ( image_a_a @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_797_the__inv__into__f__f,axiom,
! [F: a > option_a,A2: set_a,X2: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( the_in8758012798868597241tion_a @ A2 @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_798_the__inv__into__f__f,axiom,
! [F: a > a,A2: set_a,X2: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( the_inv_into_a_a @ A2 @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_799_the__inv__into__f__eq,axiom,
! [F: a > option_a,A2: set_a,X2: a,Y: option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ( F @ X2 )
= Y )
=> ( ( member_a @ X2 @ A2 )
=> ( ( the_in8758012798868597241tion_a @ A2 @ F @ Y )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_800_the__inv__into__f__eq,axiom,
! [F: a > a,A2: set_a,X2: a,Y: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( ( F @ X2 )
= Y )
=> ( ( member_a @ X2 @ A2 )
=> ( ( the_inv_into_a_a @ A2 @ F @ Y )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_801_inj__on__Un__image__eq__iff,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ B2 ) )
=> ( ( ( image_option_a_a @ F @ A2 )
= ( image_option_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_802_inj__on__Un__image__eq__iff,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ( ( image_a_option_a @ F @ A2 )
= ( image_a_option_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_803_inj__on__Un__image__eq__iff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ( ( image_a_a @ F @ A2 )
= ( image_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_804_f__the__inv__into__f,axiom,
! [F: option_a > a,A2: set_option_a,Y: a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( member_a @ Y @ ( image_option_a_a @ F @ A2 ) )
=> ( ( F @ ( the_in1757154643552616557on_a_a @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_805_f__the__inv__into__f,axiom,
! [F: a > option_a,A2: set_a,Y: option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_option_a @ Y @ ( image_a_option_a @ F @ A2 ) )
=> ( ( F @ ( the_in8758012798868597241tion_a @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_806_f__the__inv__into__f,axiom,
! [F: a > a,A2: set_a,Y: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( member_a @ Y @ ( image_a_a @ F @ A2 ) )
=> ( ( F @ ( the_inv_into_a_a @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_807_inj__on__the__inv__into,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( inj_on_a_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F ) @ ( image_option_a_a @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_808_inj__on__the__inv__into,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( inj_on_option_a_a @ ( the_in8758012798868597241tion_a @ A2 @ F ) @ ( image_a_option_a @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_809_inj__on__the__inv__into,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( inj_on_a_a @ ( the_inv_into_a_a @ A2 @ F ) @ ( image_a_a @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_810_the__inv__f__f,axiom,
! [F: a > option_a,X2: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( the_in8758012798868597241tion_a @ top_top_set_a @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_811_the__inv__f__f,axiom,
! [F: a > a,X2: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( the_inv_into_a_a @ top_top_set_a @ F @ ( F @ X2 ) )
= X2 ) ) ).
% the_inv_f_f
thf(fact_812_inj__on__Un,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ B2 ) )
= ( ( inj_on_option_a_a @ F @ A2 )
& ( inj_on_option_a_a @ F @ B2 )
& ( ( inf_inf_set_a @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) ) @ ( image_option_a_a @ F @ ( minus_1574173051537231627tion_a @ B2 @ A2 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_813_inj__on__Un,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( inj_on_a_a @ F @ A2 )
& ( inj_on_a_a @ F @ B2 )
& ( ( inf_inf_set_a @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ B2 @ A2 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_814_inj__on__Un,axiom,
! [F: b > a,A2: set_b,B2: set_b] :
( ( inj_on_b_a @ F @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( inj_on_b_a @ F @ A2 )
& ( inj_on_b_a @ F @ B2 )
& ( ( inf_inf_set_a @ ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) @ ( image_b_a @ F @ ( minus_minus_set_b @ B2 @ A2 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_815_inj__on__Un,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( inj_on_a_option_a @ F @ A2 )
& ( inj_on_a_option_a @ F @ B2 )
& ( ( inf_inf_set_option_a @ ( image_a_option_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( image_a_option_a @ F @ ( minus_minus_set_a @ B2 @ A2 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_816_inj__on__Un,axiom,
! [F: b > option_a,A2: set_b,B2: set_b] :
( ( inj_on_b_option_a @ F @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( inj_on_b_option_a @ F @ A2 )
& ( inj_on_b_option_a @ F @ B2 )
& ( ( inf_inf_set_option_a @ ( image_b_option_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) @ ( image_b_option_a @ F @ ( minus_minus_set_b @ B2 @ A2 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_817_inj__on__disjoint__Un,axiom,
! [F: option_a > a,A2: set_option_a,G: option_a > a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( inj_on_option_a_a @ G @ B2 )
=> ( ( ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ G @ B2 ) )
= bot_bot_set_a )
=> ( inj_on_option_a_a
@ ^ [X3: option_a] : ( if_a @ ( member_option_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_818_inj__on__disjoint__Un,axiom,
! [F: a > a,A2: set_a,G: a > a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_a @ G @ B2 )
=> ( ( ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ G @ B2 ) )
= bot_bot_set_a )
=> ( inj_on_a_a
@ ^ [X3: a] : ( if_a @ ( member_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_819_inj__on__disjoint__Un,axiom,
! [F: b > a,A2: set_b,G: b > a,B2: set_b] :
( ( inj_on_b_a @ F @ A2 )
=> ( ( inj_on_b_a @ G @ B2 )
=> ( ( ( inf_inf_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ G @ B2 ) )
= bot_bot_set_a )
=> ( inj_on_b_a
@ ^ [X3: b] : ( if_a @ ( member_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_820_inj__on__disjoint__Un,axiom,
! [F: option_a > option_a,A2: set_option_a,G: option_a > option_a,B2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( inj_on8559383841115902449tion_a @ G @ B2 )
=> ( ( ( inf_inf_set_option_a @ ( image_7439109396645324421tion_a @ F @ A2 ) @ ( image_7439109396645324421tion_a @ G @ B2 ) )
= bot_bot_set_option_a )
=> ( inj_on8559383841115902449tion_a
@ ^ [X3: option_a] : ( if_option_a @ ( member_option_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_821_inj__on__disjoint__Un,axiom,
! [F: a > option_a,A2: set_a,G: a > option_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ G @ B2 )
=> ( ( ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ G @ B2 ) )
= bot_bot_set_option_a )
=> ( inj_on_a_option_a
@ ^ [X3: a] : ( if_option_a @ ( member_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_822_inj__on__disjoint__Un,axiom,
! [F: b > option_a,A2: set_b,G: b > option_a,B2: set_b] :
( ( inj_on_b_option_a @ F @ A2 )
=> ( ( inj_on_b_option_a @ G @ B2 )
=> ( ( ( inf_inf_set_option_a @ ( image_b_option_a @ F @ A2 ) @ ( image_b_option_a @ G @ B2 ) )
= bot_bot_set_option_a )
=> ( inj_on_b_option_a
@ ^ [X3: b] : ( if_option_a @ ( member_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_823_inj__on__Int,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( ( inj_on_a_option_a @ F @ A2 )
| ( inj_on_a_option_a @ F @ B2 ) )
=> ( inj_on_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inj_on_Int
thf(fact_824_inj__on__Int,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( ( inj_on_a_a @ F @ A2 )
| ( inj_on_a_a @ F @ B2 ) )
=> ( inj_on_a_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inj_on_Int
thf(fact_825_image__Int,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( image_option_a_a @ F @ ( inf_inf_set_option_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B2 ) ) ) ) ).
% image_Int
thf(fact_826_image__Int,axiom,
! [F: a > option_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( image_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B2 ) ) ) ) ).
% image_Int
thf(fact_827_image__Int,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ).
% image_Int
thf(fact_828_inj__on__image__Int,axiom,
! [F: option_a > a,C3: set_option_a,A2: set_option_a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ C3 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ C3 )
=> ( ( image_option_a_a @ F @ ( inf_inf_set_option_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_option_a_a @ F @ A2 ) @ ( image_option_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_829_inj__on__image__Int,axiom,
! [F: a > option_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_option_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_option_a @ ( image_a_option_a @ F @ A2 ) @ ( image_a_option_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_830_inj__on__image__Int,axiom,
! [F: a > a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_831_comp__the__Some,axiom,
( ( comp_option_a_a_a @ the_a2 @ some_a )
= id_a ) ).
% comp_the_Some
thf(fact_832_vimage__subsetI,axiom,
! [F: a > option_a,B2: set_option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a @ F @ A2 ) )
=> ( ord_less_eq_set_a @ ( vimage_a_option_a @ F @ B2 ) @ A2 ) ) ) ).
% vimage_subsetI
thf(fact_833_vimage__subsetI,axiom,
! [F: option_a > a,B2: set_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_option_a_a @ F @ A2 ) )
=> ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F @ B2 ) @ A2 ) ) ) ).
% vimage_subsetI
thf(fact_834_vimage__subsetI,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B2 ) @ A2 ) ) ) ).
% vimage_subsetI
thf(fact_835_id__apply,axiom,
( id_a
= ( ^ [X3: a] : X3 ) ) ).
% id_apply
thf(fact_836_image__id,axiom,
( ( image_a_a @ id_a )
= id_set_a ) ).
% image_id
thf(fact_837_vimage__id,axiom,
( ( vimage_a_a @ id_a )
= id_set_a ) ).
% vimage_id
thf(fact_838_comp__eq__id__dest,axiom,
! [A: option_a > a,B: a > option_a,C: a > a,V4: a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_a_a_a @ id_a @ C ) )
=> ( ( A @ ( B @ V4 ) )
= ( C @ V4 ) ) ) ).
% comp_eq_id_dest
thf(fact_839_vimage__comp,axiom,
! [F: a > option_a,G: option_a > a,X2: set_a] :
( ( vimage_a_option_a @ F @ ( vimage_option_a_a @ G @ X2 ) )
= ( vimage_a_a @ ( comp_option_a_a_a @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_840_set_Ocompositionality,axiom,
! [F: a > option_a,G: option_a > a,Set: set_a] :
( ( vimage_a_option_a @ F @ ( vimage_option_a_a @ G @ Set ) )
= ( vimage_a_a @ ( comp_option_a_a_a @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_841_set_Ocomp,axiom,
! [F: a > option_a,G: option_a > a] :
( ( comp_s1419921648917501825_set_a @ ( vimage_a_option_a @ F ) @ ( vimage_option_a_a @ G ) )
= ( vimage_a_a @ ( comp_option_a_a_a @ G @ F ) ) ) ).
% set.comp
thf(fact_842_map__option_Oidentity,axiom,
( ( map_option_a_a
@ ^ [X3: a] : X3 )
= id_option_a ) ).
% map_option.identity
thf(fact_843_id__def,axiom,
( id_a
= ( ^ [X3: a] : X3 ) ) ).
% id_def
thf(fact_844_eq__id__iff,axiom,
! [F: a > a] :
( ( ! [X3: a] :
( ( F @ X3 )
= X3 ) )
= ( F = id_a ) ) ).
% eq_id_iff
thf(fact_845_option_Omap__id0,axiom,
( ( map_option_a_a @ id_a )
= id_option_a ) ).
% option.map_id0
thf(fact_846_option_Omap__id,axiom,
! [T: option_a] :
( ( map_option_a_a @ id_a @ T )
= T ) ).
% option.map_id
thf(fact_847_inj__on__id,axiom,
! [A2: set_a] : ( inj_on_a_a @ id_a @ A2 ) ).
% inj_on_id
thf(fact_848_surj__image__vimage__eq,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ ( vimage1562710927270423099tion_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_849_surj__image__vimage__eq,axiom,
! [F: option_a > a,A2: set_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_option_a_a @ F @ ( vimage_option_a_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_850_surj__image__vimage__eq,axiom,
! [F: a > option_a,A2: set_option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_a_option_a @ F @ ( vimage_a_option_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_851_surj__image__vimage__eq,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ F @ ( vimage_a_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_852_surj__id,axiom,
( ( image_7439109396645324421tion_a @ id_option_a @ top_top_set_option_a )
= top_top_set_option_a ) ).
% surj_id
thf(fact_853_surj__id,axiom,
( ( image_a_a @ id_a @ top_top_set_a )
= top_top_set_a ) ).
% surj_id
thf(fact_854_surj__vimage__empty,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ( vimage1562710927270423099tion_a @ F @ A2 )
= bot_bot_set_option_a )
= ( A2 = bot_bot_set_option_a ) ) ) ).
% surj_vimage_empty
thf(fact_855_surj__vimage__empty,axiom,
! [F: option_a > a,A2: set_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( vimage_option_a_a @ F @ A2 )
= bot_bot_set_option_a )
= ( A2 = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_856_surj__vimage__empty,axiom,
! [F: a > option_a,A2: set_option_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( vimage_a_option_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_option_a ) ) ) ).
% surj_vimage_empty
thf(fact_857_surj__vimage__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( vimage_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_858_vimage__subsetD,axiom,
! [F: option_a > option_a,B2: set_option_a,A2: set_option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F @ B2 ) @ A2 )
=> ( ord_le1955136853071979460tion_a @ B2 @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_859_vimage__subsetD,axiom,
! [F: option_a > a,B2: set_a,A2: set_option_a] :
( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F @ B2 ) @ A2 )
=> ( ord_less_eq_set_a @ B2 @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_860_vimage__subsetD,axiom,
! [F: a > option_a,B2: set_option_a,A2: set_a] :
( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_option_a @ F @ B2 ) @ A2 )
=> ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_861_vimage__subsetD,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B2 ) @ A2 )
=> ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_862_inj__vimage__image__eq,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( vimage_option_a_a @ F @ ( image_option_a_a @ F @ A2 ) )
= A2 ) ) ).
% inj_vimage_image_eq
thf(fact_863_inj__vimage__image__eq,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( vimage_a_option_a @ F @ ( image_a_option_a @ F @ A2 ) )
= A2 ) ) ).
% inj_vimage_image_eq
thf(fact_864_inj__vimage__image__eq,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( vimage_a_a @ F @ ( image_a_a @ F @ A2 ) )
= A2 ) ) ).
% inj_vimage_image_eq
thf(fact_865_inj__on__vimage__singleton,axiom,
! [F: b > a,A2: set_b,A: a] :
( ( inj_on_b_a @ F @ A2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ ( vimage_b_a @ F @ ( insert_a @ A @ bot_bot_set_a ) ) @ A2 )
@ ( insert_b
@ ( the_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_b ) ) ) ).
% inj_on_vimage_singleton
thf(fact_866_inj__on__vimage__singleton,axiom,
! [F: option_a > a,A2: set_option_a,A: a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ ( vimage_option_a_a @ F @ ( insert_a @ A @ bot_bot_set_a ) ) @ A2 )
@ ( insert_option_a
@ ( the_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_option_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_867_inj__on__vimage__singleton,axiom,
! [F: b > option_a,A2: set_b,A: option_a] :
( ( inj_on_b_option_a @ F @ A2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ ( vimage_b_option_a @ F @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ A2 )
@ ( insert_b
@ ( the_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_b ) ) ) ).
% inj_on_vimage_singleton
thf(fact_868_inj__on__vimage__singleton,axiom,
! [F: option_a > option_a,A2: set_option_a,A: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ord_le1955136853071979460tion_a @ ( inf_inf_set_option_a @ ( vimage1562710927270423099tion_a @ F @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ A2 )
@ ( insert_option_a
@ ( the_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_option_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_869_inj__on__vimage__singleton,axiom,
! [F: a > a,A2: set_a,A: a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ A @ bot_bot_set_a ) ) @ A2 )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_870_inj__on__vimage__singleton,axiom,
! [F: a > option_a,A2: set_a,A: option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( vimage_a_option_a @ F @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ A2 )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( ( F @ X3 )
= A ) ) )
@ bot_bot_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_871_inj__vimage__singleton,axiom,
! [F: option_a > a,A: a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F @ ( insert_a @ A @ bot_bot_set_a ) )
@ ( insert_option_a
@ ( the_option_a
@ ^ [X3: option_a] :
( ( F @ X3 )
= A ) )
@ bot_bot_set_option_a ) ) ) ).
% inj_vimage_singleton
thf(fact_872_inj__vimage__singleton,axiom,
! [F: option_a > option_a,A: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
@ ( insert_option_a
@ ( the_option_a
@ ^ [X3: option_a] :
( ( F @ X3 )
= A ) )
@ bot_bot_set_option_a ) ) ) ).
% inj_vimage_singleton
thf(fact_873_inj__vimage__singleton,axiom,
! [F: a > a,A: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ ( insert_a @ A @ bot_bot_set_a ) )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( F @ X3 )
= A ) )
@ bot_bot_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_874_inj__vimage__singleton,axiom,
! [F: a > option_a,A: option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( vimage_a_option_a @ F @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( F @ X3 )
= A ) )
@ bot_bot_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_875_vimage__subset__eq,axiom,
! [F: option_a > option_a,B2: set_option_a,A2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( ( ord_le1955136853071979460tion_a @ ( vimage1562710927270423099tion_a @ F @ B2 ) @ A2 )
= ( ord_le1955136853071979460tion_a @ B2 @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_876_vimage__subset__eq,axiom,
! [F: option_a > a,B2: set_a,A2: set_option_a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F @ B2 ) @ A2 )
= ( ord_less_eq_set_a @ B2 @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_877_vimage__subset__eq,axiom,
! [F: a > option_a,B2: set_option_a,A2: set_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_option_a @ F @ B2 ) @ A2 )
= ( ord_le1955136853071979460tion_a @ B2 @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_878_vimage__subset__eq,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( ord_less_eq_set_a @ ( vimage_a_a @ F @ B2 ) @ A2 )
= ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_879_bij__betw__id,axiom,
! [A2: set_a] : ( bij_betw_a_a @ id_a @ A2 @ A2 ) ).
% bij_betw_id
thf(fact_880_bij__betw__comp__iff,axiom,
! [F: a > option_a,A2: set_a,A10: set_option_a,F4: option_a > a,A11: set_a] :
( ( bij_betw_a_option_a @ F @ A2 @ A10 )
=> ( ( bij_betw_option_a_a @ F4 @ A10 @ A11 )
= ( bij_betw_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 @ A11 ) ) ) ).
% bij_betw_comp_iff
thf(fact_881_bij__betw__comp__iff,axiom,
! [F: a > a,A2: set_a,A10: set_a,F4: a > a,A11: set_a] :
( ( bij_betw_a_a @ F @ A2 @ A10 )
=> ( ( bij_betw_a_a @ F4 @ A10 @ A11 )
= ( bij_betw_a_a @ ( comp_a_a_a @ F4 @ F ) @ A2 @ A11 ) ) ) ).
% bij_betw_comp_iff
thf(fact_882_bij__betw__trans,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a,G: option_a > a,C3: set_a] :
( ( bij_betw_a_option_a @ F @ A2 @ B2 )
=> ( ( bij_betw_option_a_a @ G @ B2 @ C3 )
=> ( bij_betw_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 @ C3 ) ) ) ).
% bij_betw_trans
thf(fact_883_bij__betw__trans,axiom,
! [F: a > a,A2: set_a,B2: set_a,G: a > a,C3: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( bij_betw_a_a @ G @ B2 @ C3 )
=> ( bij_betw_a_a @ ( comp_a_a_a @ G @ F ) @ A2 @ C3 ) ) ) ).
% bij_betw_trans
thf(fact_884_bij__betw__id__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ id_a @ A2 @ B2 )
= ( A2 = B2 ) ) ).
% bij_betw_id_iff
thf(fact_885_bij__betw__imp__inj__on,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ B2 )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_886_bij__betw__imp__inj__on,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( inj_on_a_a @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_887_bij__betw__the__inv__into,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( bij_betw_a_a @ ( the_inv_into_a_a @ A2 @ F ) @ B2 @ A2 ) ) ).
% bij_betw_the_inv_into
thf(fact_888_f__the__inv__into__f__bij__betw,axiom,
! [F: a > a,A2: set_a,B2: set_a,X2: a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ( F @ ( the_inv_into_a_a @ A2 @ F @ X2 ) )
= X2 ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_889_bij__betw__imp__surj__on,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ B2 )
=> ( ( image_a_option_a @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_890_bij__betw__imp__surj__on,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_a] :
( ( bij_betw_option_a_a @ F @ A2 @ B2 )
=> ( ( image_option_a_a @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_891_bij__betw__imp__surj__on,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( image_a_a @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_892_bij__betw__iff__bijections,axiom,
( bij_betw_option_a_a
= ( ^ [F2: option_a > a,A8: set_option_a,B6: set_a] :
? [G2: a > option_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ A8 )
=> ( ( member_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ B6 )
=> ( ( member_option_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_893_bij__betw__iff__bijections,axiom,
( bij_betw_b_a
= ( ^ [F2: b > a,A8: set_b,B6: set_a] :
? [G2: a > b] :
( ! [X3: b] :
( ( member_b @ X3 @ A8 )
=> ( ( member_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ B6 )
=> ( ( member_b @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_894_bij__betw__iff__bijections,axiom,
( bij_betw_a_option_a
= ( ^ [F2: a > option_a,A8: set_a,B6: set_option_a] :
? [G2: option_a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( ( member_option_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: option_a] :
( ( member_option_a @ X3 @ B6 )
=> ( ( member_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_895_bij__betw__iff__bijections,axiom,
( bij_be5431266891817924854tion_a
= ( ^ [F2: option_a > option_a,A8: set_option_a,B6: set_option_a] :
? [G2: option_a > option_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ A8 )
=> ( ( member_option_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: option_a] :
( ( member_option_a @ X3 @ B6 )
=> ( ( member_option_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_896_bij__betw__iff__bijections,axiom,
( bij_betw_b_option_a
= ( ^ [F2: b > option_a,A8: set_b,B6: set_option_a] :
? [G2: option_a > b] :
( ! [X3: b] :
( ( member_b @ X3 @ A8 )
=> ( ( member_option_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: option_a] :
( ( member_option_a @ X3 @ B6 )
=> ( ( member_b @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_897_bij__betw__iff__bijections,axiom,
( bij_betw_a_b
= ( ^ [F2: a > b,A8: set_a,B6: set_b] :
? [G2: b > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( ( member_b @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: b] :
( ( member_b @ X3 @ B6 )
=> ( ( member_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_898_bij__betw__iff__bijections,axiom,
( bij_betw_option_a_b
= ( ^ [F2: option_a > b,A8: set_option_a,B6: set_b] :
? [G2: b > option_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ A8 )
=> ( ( member_b @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: b] :
( ( member_b @ X3 @ B6 )
=> ( ( member_option_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_899_bij__betw__iff__bijections,axiom,
( bij_betw_b_b
= ( ^ [F2: b > b,A8: set_b,B6: set_b] :
? [G2: b > b] :
( ! [X3: b] :
( ( member_b @ X3 @ A8 )
=> ( ( member_b @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: b] :
( ( member_b @ X3 @ B6 )
=> ( ( member_b @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_900_bij__betw__iff__bijections,axiom,
( bij_betw_a_a
= ( ^ [F2: a > a,A8: set_a,B6: set_a] :
? [G2: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A8 )
=> ( ( member_a @ ( F2 @ X3 ) @ B6 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ B6 )
=> ( ( member_a @ ( G2 @ X3 ) @ A8 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_901_bij__betw__apply,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a,A: a] :
( ( bij_betw_a_option_a @ F @ A2 @ B2 )
=> ( ( member_a @ A @ A2 )
=> ( member_option_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_902_bij__betw__apply,axiom,
! [F: a > b,A2: set_a,B2: set_b,A: a] :
( ( bij_betw_a_b @ F @ A2 @ B2 )
=> ( ( member_a @ A @ A2 )
=> ( member_b @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_903_bij__betw__apply,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_a,A: option_a] :
( ( bij_betw_option_a_a @ F @ A2 @ B2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_904_bij__betw__apply,axiom,
! [F: option_a > option_a,A2: set_option_a,B2: set_option_a,A: option_a] :
( ( bij_be5431266891817924854tion_a @ F @ A2 @ B2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member_option_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_905_bij__betw__apply,axiom,
! [F: option_a > b,A2: set_option_a,B2: set_b,A: option_a] :
( ( bij_betw_option_a_b @ F @ A2 @ B2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member_b @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_906_bij__betw__apply,axiom,
! [F: b > a,A2: set_b,B2: set_a,A: b] :
( ( bij_betw_b_a @ F @ A2 @ B2 )
=> ( ( member_b @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_907_bij__betw__apply,axiom,
! [F: b > option_a,A2: set_b,B2: set_option_a,A: b] :
( ( bij_betw_b_option_a @ F @ A2 @ B2 )
=> ( ( member_b @ A @ A2 )
=> ( member_option_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_908_bij__betw__apply,axiom,
! [F: b > b,A2: set_b,B2: set_b,A: b] :
( ( bij_betw_b_b @ F @ A2 @ B2 )
=> ( ( member_b @ A @ A2 )
=> ( member_b @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_909_bij__betw__apply,axiom,
! [F: a > a,A2: set_a,B2: set_a,A: a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( member_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_910_bij__betw__cong,axiom,
! [A2: set_a,F: a > a,G: a > a,A10: set_a] :
( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( bij_betw_a_a @ F @ A2 @ A10 )
= ( bij_betw_a_a @ G @ A2 @ A10 ) ) ) ).
% bij_betw_cong
thf(fact_911_bij__betw__ball,axiom,
! [F: a > a,A2: set_a,B2: set_a,Phi: a > $o] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( Phi @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( Phi @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_912_bij__betw__inv,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ? [G3: a > a] : ( bij_betw_a_a @ G3 @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_913_bij__betwE,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ! [X8: a] :
( ( member_a @ X8 @ A2 )
=> ( member_a @ ( F @ X8 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_914_bij__betw__empty2,axiom,
! [F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bij_betw_empty2
thf(fact_915_bij__betw__empty2,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( bij_betw_option_a_a @ F @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_option_a ) ) ).
% bij_betw_empty2
thf(fact_916_bij__betw__empty2,axiom,
! [F: a > option_a,A2: set_a] :
( ( bij_betw_a_option_a @ F @ A2 @ bot_bot_set_option_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bij_betw_empty2
thf(fact_917_bij__betw__empty2,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ A2 @ bot_bot_set_option_a )
=> ( A2 = bot_bot_set_option_a ) ) ).
% bij_betw_empty2
thf(fact_918_bij__betw__empty1,axiom,
! [F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F @ bot_bot_set_a @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% bij_betw_empty1
thf(fact_919_bij__betw__empty1,axiom,
! [F: a > option_a,A2: set_option_a] :
( ( bij_betw_a_option_a @ F @ bot_bot_set_a @ A2 )
=> ( A2 = bot_bot_set_option_a ) ) ).
% bij_betw_empty1
thf(fact_920_bij__betw__empty1,axiom,
! [F: option_a > a,A2: set_a] :
( ( bij_betw_option_a_a @ F @ bot_bot_set_option_a @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% bij_betw_empty1
thf(fact_921_bij__betw__empty1,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ bot_bot_set_option_a @ A2 )
=> ( A2 = bot_bot_set_option_a ) ) ).
% bij_betw_empty1
thf(fact_922_involuntory__imp__bij,axiom,
! [F: option_a > option_a] :
( ! [X: option_a] :
( ( F @ ( F @ X ) )
= X )
=> ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a ) ) ).
% involuntory_imp_bij
thf(fact_923_involuntory__imp__bij,axiom,
! [F: a > a] :
( ! [X: a] :
( ( F @ ( F @ X ) )
= X )
=> ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a ) ) ).
% involuntory_imp_bij
thf(fact_924_bij__pointE,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ~ ! [X: option_a] :
( ( Y
= ( F @ X ) )
=> ~ ! [X9: option_a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X ) ) ) ) ).
% bij_pointE
thf(fact_925_bij__pointE,axiom,
! [F: option_a > a,Y: a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ~ ! [X: option_a] :
( ( Y
= ( F @ X ) )
=> ~ ! [X9: option_a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X ) ) ) ) ).
% bij_pointE
thf(fact_926_bij__pointE,axiom,
! [F: a > option_a,Y: option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ~ ! [X: a] :
( ( Y
= ( F @ X ) )
=> ~ ! [X9: a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X ) ) ) ) ).
% bij_pointE
thf(fact_927_bij__pointE,axiom,
! [F: a > a,Y: a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ~ ! [X: a] :
( ( Y
= ( F @ X ) )
=> ~ ! [X9: a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X ) ) ) ) ).
% bij_pointE
thf(fact_928_bij__iff,axiom,
! [F: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
= ( ! [X3: option_a] :
? [Y2: option_a] :
( ( ( F @ Y2 )
= X3 )
& ! [Z: option_a] :
( ( ( F @ Z )
= X3 )
=> ( Z = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_929_bij__iff,axiom,
! [F: option_a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
= ( ! [X3: a] :
? [Y2: option_a] :
( ( ( F @ Y2 )
= X3 )
& ! [Z: option_a] :
( ( ( F @ Z )
= X3 )
=> ( Z = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_930_bij__iff,axiom,
! [F: a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
= ( ! [X3: option_a] :
? [Y2: a] :
( ( ( F @ Y2 )
= X3 )
& ! [Z: a] :
( ( ( F @ Z )
= X3 )
=> ( Z = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_931_bij__iff,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
= ( ! [X3: a] :
? [Y2: a] :
( ( ( F @ Y2 )
= X3 )
& ! [Z: a] :
( ( ( F @ Z )
= X3 )
=> ( Z = Y2 ) ) ) ) ) ).
% bij_iff
thf(fact_932_bij__is__surj,axiom,
! [F: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a ) ) ).
% bij_is_surj
thf(fact_933_bij__is__surj,axiom,
! [F: option_a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a ) ) ).
% bij_is_surj
thf(fact_934_bij__is__surj,axiom,
! [F: a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a ) ) ).
% bij_is_surj
thf(fact_935_bij__is__surj,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a ) ) ).
% bij_is_surj
thf(fact_936_bij__betw__imp__surj,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ A2 @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a ) ) ).
% bij_betw_imp_surj
thf(fact_937_bij__betw__imp__surj,axiom,
! [F: a > option_a,A2: set_a] :
( ( bij_betw_a_option_a @ F @ A2 @ top_top_set_option_a )
=> ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a ) ) ).
% bij_betw_imp_surj
thf(fact_938_bij__betw__imp__surj,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( bij_betw_option_a_a @ F @ A2 @ top_top_set_a )
=> ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a ) ) ).
% bij_betw_imp_surj
thf(fact_939_bij__betw__imp__surj,axiom,
! [F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ top_top_set_a )
=> ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a ) ) ).
% bij_betw_imp_surj
thf(fact_940_bij__betw__byWitness,axiom,
! [A2: set_a,F4: option_a > a,F: a > option_a,A10: set_option_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: option_a] :
( ( member_option_a @ X @ A10 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ A10 )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F4 @ A10 ) @ A2 )
=> ( bij_betw_a_option_a @ F @ A2 @ A10 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_941_bij__betw__byWitness,axiom,
! [A2: set_option_a,F4: a > option_a,F: option_a > a,A10: set_a] :
( ! [X: option_a] :
( ( member_option_a @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ A10 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ A10 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F4 @ A10 ) @ A2 )
=> ( bij_betw_option_a_a @ F @ A2 @ A10 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_942_bij__betw__byWitness,axiom,
! [A2: set_a,F4: a > a,F: a > a,A10: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( F4 @ ( F @ X ) )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ A10 )
=> ( ( F @ ( F4 @ X ) )
= X ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ A10 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F4 @ A10 ) @ A2 )
=> ( bij_betw_a_a @ F @ A2 @ A10 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_943_bij__betw__subset,axiom,
! [F: option_a > a,A2: set_option_a,A10: set_a,B2: set_option_a,B7: set_a] :
( ( bij_betw_option_a_a @ F @ A2 @ A10 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ A2 )
=> ( ( ( image_option_a_a @ F @ B2 )
= B7 )
=> ( bij_betw_option_a_a @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_944_bij__betw__subset,axiom,
! [F: a > option_a,A2: set_a,A10: set_option_a,B2: set_a,B7: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ A10 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ( image_a_option_a @ F @ B2 )
= B7 )
=> ( bij_betw_a_option_a @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_945_bij__betw__subset,axiom,
! [F: a > a,A2: set_a,A10: set_a,B2: set_a,B7: set_a] :
( ( bij_betw_a_a @ F @ A2 @ A10 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ( image_a_a @ F @ B2 )
= B7 )
=> ( bij_betw_a_a @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_946_bij__comp,axiom,
! [F: option_a > option_a,G: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( ( bij_be5431266891817924854tion_a @ G @ top_top_set_option_a @ top_top_set_option_a )
=> ( bij_be5431266891817924854tion_a @ ( comp_o3154387707078715297tion_a @ G @ F ) @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).
% bij_comp
thf(fact_947_bij__comp,axiom,
! [F: option_a > option_a,G: option_a > a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( ( bij_betw_option_a_a @ G @ top_top_set_option_a @ top_top_set_a )
=> ( bij_betw_option_a_a @ ( comp_o3864519266390211175tion_a @ G @ F ) @ top_top_set_option_a @ top_top_set_a ) ) ) ).
% bij_comp
thf(fact_948_bij__comp,axiom,
! [F: option_a > a,G: a > option_a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( ( bij_betw_a_option_a @ G @ top_top_set_a @ top_top_set_option_a )
=> ( bij_be5431266891817924854tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).
% bij_comp
thf(fact_949_bij__comp,axiom,
! [F: option_a > a,G: a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( ( bij_betw_a_a @ G @ top_top_set_a @ top_top_set_a )
=> ( bij_betw_option_a_a @ ( comp_a_a_option_a @ G @ F ) @ top_top_set_option_a @ top_top_set_a ) ) ) ).
% bij_comp
thf(fact_950_bij__comp,axiom,
! [F: a > option_a,G: option_a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( ( bij_be5431266891817924854tion_a @ G @ top_top_set_option_a @ top_top_set_option_a )
=> ( bij_betw_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ top_top_set_a @ top_top_set_option_a ) ) ) ).
% bij_comp
thf(fact_951_bij__comp,axiom,
! [F: a > option_a,G: option_a > a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( ( bij_betw_option_a_a @ G @ top_top_set_option_a @ top_top_set_a )
=> ( bij_betw_a_a @ ( comp_option_a_a_a @ G @ F ) @ top_top_set_a @ top_top_set_a ) ) ) ).
% bij_comp
thf(fact_952_bij__comp,axiom,
! [F: a > a,G: a > option_a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( bij_betw_a_option_a @ G @ top_top_set_a @ top_top_set_option_a )
=> ( bij_betw_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ top_top_set_a @ top_top_set_option_a ) ) ) ).
% bij_comp
thf(fact_953_bij__comp,axiom,
! [F: a > a,G: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( bij_betw_a_a @ G @ top_top_set_a @ top_top_set_a )
=> ( bij_betw_a_a @ ( comp_a_a_a @ G @ F ) @ top_top_set_a @ top_top_set_a ) ) ) ).
% bij_comp
thf(fact_954_inj__on__imp__bij__betw,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( bij_betw_option_a_a @ F @ A2 @ ( image_option_a_a @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_955_inj__on__imp__bij__betw,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( bij_betw_a_option_a @ F @ A2 @ ( image_a_option_a @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_956_inj__on__imp__bij__betw,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( bij_betw_a_a @ F @ A2 @ ( image_a_a @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_957_bij__betw__imageI,axiom,
! [F: option_a > a,A2: set_option_a,B2: set_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( ( image_option_a_a @ F @ A2 )
= B2 )
=> ( bij_betw_option_a_a @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_958_bij__betw__imageI,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ( image_a_option_a @ F @ A2 )
= B2 )
=> ( bij_betw_a_option_a @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_959_bij__betw__imageI,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( ( image_a_a @ F @ A2 )
= B2 )
=> ( bij_betw_a_a @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_960_bij__betw__def,axiom,
( bij_betw_option_a_a
= ( ^ [F2: option_a > a,A8: set_option_a,B6: set_a] :
( ( inj_on_option_a_a @ F2 @ A8 )
& ( ( image_option_a_a @ F2 @ A8 )
= B6 ) ) ) ) ).
% bij_betw_def
thf(fact_961_bij__betw__def,axiom,
( bij_betw_a_option_a
= ( ^ [F2: a > option_a,A8: set_a,B6: set_option_a] :
( ( inj_on_a_option_a @ F2 @ A8 )
& ( ( image_a_option_a @ F2 @ A8 )
= B6 ) ) ) ) ).
% bij_betw_def
thf(fact_962_bij__betw__def,axiom,
( bij_betw_a_a
= ( ^ [F2: a > a,A8: set_a,B6: set_a] :
( ( inj_on_a_a @ F2 @ A8 )
& ( ( image_a_a @ F2 @ A8 )
= B6 ) ) ) ) ).
% bij_betw_def
thf(fact_963_bij__is__inj,axiom,
! [F: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a ) ) ).
% bij_is_inj
thf(fact_964_bij__is__inj,axiom,
! [F: option_a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( inj_on_option_a_a @ F @ top_top_set_option_a ) ) ).
% bij_is_inj
thf(fact_965_bij__is__inj,axiom,
! [F: a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( inj_on_a_option_a @ F @ top_top_set_a ) ) ).
% bij_is_inj
thf(fact_966_bij__is__inj,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% bij_is_inj
thf(fact_967_bij__id,axiom,
bij_be5431266891817924854tion_a @ id_option_a @ top_top_set_option_a @ top_top_set_option_a ).
% bij_id
thf(fact_968_bij__id,axiom,
bij_betw_a_a @ id_a @ top_top_set_a @ top_top_set_a ).
% bij_id
thf(fact_969_the__inv__into__def,axiom,
( the_inv_into_a_a
= ( ^ [A8: set_a,F2: a > a,X3: a] :
( the_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ A8 )
& ( ( F2 @ Y2 )
= X3 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_970_bij__betw__comp__iff2,axiom,
! [F4: option_a > a,A10: set_option_a,A11: set_a,F: a > option_a,A2: set_a] :
( ( bij_betw_option_a_a @ F4 @ A10 @ A11 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ F @ A2 ) @ A10 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A10 )
= ( bij_betw_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_971_bij__betw__comp__iff2,axiom,
! [F4: a > a,A10: set_a,A11: set_a,F: option_a > a,A2: set_option_a] :
( ( bij_betw_a_a @ F4 @ A10 @ A11 )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a @ F @ A2 ) @ A10 )
=> ( ( bij_betw_option_a_a @ F @ A2 @ A10 )
= ( bij_betw_option_a_a @ ( comp_a_a_option_a @ F4 @ F ) @ A2 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_972_bij__betw__comp__iff2,axiom,
! [F4: a > a,A10: set_a,A11: set_a,F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F4 @ A10 @ A11 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ A10 )
=> ( ( bij_betw_a_a @ F @ A2 @ A10 )
= ( bij_betw_a_a @ ( comp_a_a_a @ F4 @ F ) @ A2 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_973_bij__def,axiom,
! [F: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
= ( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
& ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a ) ) ) ).
% bij_def
thf(fact_974_bij__def,axiom,
! [F: option_a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
= ( ( inj_on_option_a_a @ F @ top_top_set_option_a )
& ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% bij_def
thf(fact_975_bij__def,axiom,
! [F: a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
= ( ( inj_on_a_option_a @ F @ top_top_set_a )
& ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% bij_def
thf(fact_976_bij__def,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
= ( ( inj_on_a_a @ F @ top_top_set_a )
& ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a ) ) ) ).
% bij_def
thf(fact_977_bijI,axiom,
! [F: option_a > option_a] :
( ( inj_on8559383841115902449tion_a @ F @ top_top_set_option_a )
=> ( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a ) ) ) ).
% bijI
thf(fact_978_bijI,axiom,
! [F: option_a > a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ( image_option_a_a @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a ) ) ) ).
% bijI
thf(fact_979_bijI,axiom,
! [F: a > option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( image_a_option_a @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a ) ) ) ).
% bijI
thf(fact_980_bijI,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a ) ) ) ).
% bijI
thf(fact_981_notIn__Un__bij__betw,axiom,
! [B: b,A2: set_b,F: b > b,A10: set_b] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_b @ F @ A2 @ A10 )
=> ( bij_betw_b_b @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_982_notIn__Un__bij__betw,axiom,
! [B: b,A2: set_b,F: b > a,A10: set_a] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_a @ F @ A2 @ A10 )
=> ( bij_betw_b_a @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_983_notIn__Un__bij__betw,axiom,
! [B: b,A2: set_b,F: b > option_a,A10: set_option_a] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_option_a @ F @ A2 @ A10 )
=> ( bij_betw_b_option_a @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_984_notIn__Un__bij__betw,axiom,
! [B: a,A2: set_a,F: a > b,A10: set_b] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_b @ F @ A2 @ A10 )
=> ( bij_betw_a_b @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_985_notIn__Un__bij__betw,axiom,
! [B: a,A2: set_a,F: a > option_a,A10: set_option_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A10 )
=> ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_986_notIn__Un__bij__betw,axiom,
! [B: option_a,A2: set_option_a,F: option_a > b,A10: set_b] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_option_a_b @ F @ A2 @ A10 )
=> ( bij_betw_option_a_b @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_987_notIn__Un__bij__betw,axiom,
! [B: option_a,A2: set_option_a,F: option_a > a,A10: set_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_option_a_a @ F @ A2 @ A10 )
=> ( bij_betw_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_988_notIn__Un__bij__betw,axiom,
! [B: option_a,A2: set_option_a,F: option_a > option_a,A10: set_option_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_be5431266891817924854tion_a @ F @ A2 @ A10 )
=> ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_989_notIn__Un__bij__betw,axiom,
! [B: a,A2: set_a,F: a > a,A10: set_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_a @ F @ A2 @ A10 )
=> ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_990_notIn__Un__bij__betw3,axiom,
! [B: b,A2: set_b,F: b > b,A10: set_b] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_b @ F @ A2 @ A10 )
= ( bij_betw_b_b @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_991_notIn__Un__bij__betw3,axiom,
! [B: b,A2: set_b,F: b > a,A10: set_a] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_a @ F @ A2 @ A10 )
= ( bij_betw_b_a @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_992_notIn__Un__bij__betw3,axiom,
! [B: b,A2: set_b,F: b > option_a,A10: set_option_a] :
( ~ ( member_b @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_b_option_a @ F @ A2 @ A10 )
= ( bij_betw_b_option_a @ F @ ( sup_sup_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_993_notIn__Un__bij__betw3,axiom,
! [B: a,A2: set_a,F: a > b,A10: set_b] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_b @ F @ A2 @ A10 )
= ( bij_betw_a_b @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_994_notIn__Un__bij__betw3,axiom,
! [B: a,A2: set_a,F: a > option_a,A10: set_option_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A10 )
= ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_995_notIn__Un__bij__betw3,axiom,
! [B: option_a,A2: set_option_a,F: option_a > b,A10: set_b] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_b @ ( F @ B ) @ A10 )
=> ( ( bij_betw_option_a_b @ F @ A2 @ A10 )
= ( bij_betw_option_a_b @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_b @ A10 @ ( insert_b @ ( F @ B ) @ bot_bot_set_b ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_996_notIn__Un__bij__betw3,axiom,
! [B: option_a,A2: set_option_a,F: option_a > a,A10: set_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_option_a_a @ F @ A2 @ A10 )
= ( bij_betw_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_997_notIn__Un__bij__betw3,axiom,
! [B: option_a,A2: set_option_a,F: option_a > option_a,A10: set_option_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A10 )
=> ( ( bij_be5431266891817924854tion_a @ F @ A2 @ A10 )
= ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A10 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_998_notIn__Un__bij__betw3,axiom,
! [B: a,A2: set_a,F: a > a,A10: set_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A10 )
=> ( ( bij_betw_a_a @ F @ A2 @ A10 )
= ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A10 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_999_bij__betw__combine,axiom,
! [F: a > b,A2: set_a,B2: set_b,C3: set_a,D2: set_b] :
( ( bij_betw_a_b @ F @ A2 @ B2 )
=> ( ( bij_betw_a_b @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_b @ B2 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_a_b @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_b @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1000_bij__betw__combine,axiom,
! [F: b > b,A2: set_b,B2: set_b,C3: set_b,D2: set_b] :
( ( bij_betw_b_b @ F @ A2 @ B2 )
=> ( ( bij_betw_b_b @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_b @ B2 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_b_b @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_b @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1001_bij__betw__combine,axiom,
! [F: a > a,A2: set_a,B2: set_a,C3: set_a,D2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ B2 )
=> ( ( bij_betw_a_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_a @ B2 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1002_bij__betw__combine,axiom,
! [F: b > a,A2: set_b,B2: set_a,C3: set_b,D2: set_a] :
( ( bij_betw_b_a @ F @ A2 @ B2 )
=> ( ( bij_betw_b_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_a @ B2 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_b_a @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1003_bij__betw__combine,axiom,
! [F: a > option_a,A2: set_a,B2: set_option_a,C3: set_a,D2: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ B2 )
=> ( ( bij_betw_a_option_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_option_a @ B2 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1004_bij__betw__combine,axiom,
! [F: b > option_a,A2: set_b,B2: set_option_a,C3: set_b,D2: set_option_a] :
( ( bij_betw_b_option_a @ F @ A2 @ B2 )
=> ( ( bij_betw_b_option_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_option_a @ B2 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_b_option_a @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_1005_bij__betw__partition,axiom,
! [F: b > b,A2: set_b,C3: set_b,B2: set_b,D2: set_b] :
( ( bij_betw_b_b @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_b @ B2 @ D2 ) )
=> ( ( bij_betw_b_b @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ C3 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ B2 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_b_b @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1006_bij__betw__partition,axiom,
! [F: b > a,A2: set_b,C3: set_b,B2: set_a,D2: set_a] :
( ( bij_betw_b_a @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) )
=> ( ( bij_betw_b_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ C3 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_a @ B2 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_b_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1007_bij__betw__partition,axiom,
! [F: b > option_a,A2: set_b,C3: set_b,B2: set_option_a,D2: set_option_a] :
( ( bij_betw_b_option_a @ F @ ( sup_sup_set_b @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) )
=> ( ( bij_betw_b_option_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ C3 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_option_a @ B2 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_b_option_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1008_bij__betw__partition,axiom,
! [F: a > b,A2: set_a,C3: set_a,B2: set_b,D2: set_b] :
( ( bij_betw_a_b @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_b @ B2 @ D2 ) )
=> ( ( bij_betw_a_b @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ C3 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_b @ B2 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_a_b @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1009_bij__betw__partition,axiom,
! [F: a > a,A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
( ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) )
=> ( ( bij_betw_a_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ C3 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B2 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_a_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1010_bij__betw__partition,axiom,
! [F: a > option_a,A2: set_a,C3: set_a,B2: set_option_a,D2: set_option_a] :
( ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) )
=> ( ( bij_betw_a_option_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ C3 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_option_a @ B2 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_a_option_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1011_bij__betw__partition,axiom,
! [F: option_a > b,A2: set_option_a,C3: set_option_a,B2: set_b,D2: set_b] :
( ( bij_betw_option_a_b @ F @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_b @ B2 @ D2 ) )
=> ( ( bij_betw_option_a_b @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ C3 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_b @ B2 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_option_a_b @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1012_bij__betw__partition,axiom,
! [F: option_a > a,A2: set_option_a,C3: set_option_a,B2: set_a,D2: set_a] :
( ( bij_betw_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D2 ) )
=> ( ( bij_betw_option_a_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ C3 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_a @ B2 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_option_a_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1013_bij__betw__partition,axiom,
! [F: option_a > option_a,A2: set_option_a,C3: set_option_a,B2: set_option_a,D2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D2 ) )
=> ( ( bij_be5431266891817924854tion_a @ F @ C3 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ C3 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_option_a @ B2 @ D2 )
= bot_bot_set_option_a )
=> ( bij_be5431266891817924854tion_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_1014_bij__image__Compl__eq,axiom,
! [F: option_a > option_a,A2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
=> ( ( image_7439109396645324421tion_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( uminus6205308855922866075tion_a @ ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1015_bij__image__Compl__eq,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ( ( image_option_a_a @ F @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( uminus_uminus_set_a @ ( image_option_a_a @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1016_bij__image__Compl__eq,axiom,
! [F: a > option_a,A2: set_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
=> ( ( image_a_option_a @ F @ ( uminus_uminus_set_a @ A2 ) )
= ( uminus6205308855922866075tion_a @ ( image_a_option_a @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1017_bij__image__Compl__eq,axiom,
! [F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( image_a_a @ F @ ( uminus_uminus_set_a @ A2 ) )
= ( uminus_uminus_set_a @ ( image_a_a @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1018_bij__betw__disjoint__Un,axiom,
! [F: b > b,A2: set_b,C3: set_b,G: b > b,B2: set_b,D2: set_b] :
( ( bij_betw_b_b @ F @ A2 @ C3 )
=> ( ( bij_betw_b_b @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ C3 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_b_b
@ ^ [X3: b] : ( if_b @ ( member_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_b @ A2 @ B2 )
@ ( sup_sup_set_b @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1019_bij__betw__disjoint__Un,axiom,
! [F: b > a,A2: set_b,C3: set_a,G: b > a,B2: set_b,D2: set_a] :
( ( bij_betw_b_a @ F @ A2 @ C3 )
=> ( ( bij_betw_b_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_a @ C3 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_b_a
@ ^ [X3: b] : ( if_a @ ( member_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_b @ A2 @ B2 )
@ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1020_bij__betw__disjoint__Un,axiom,
! [F: b > option_a,A2: set_b,C3: set_option_a,G: b > option_a,B2: set_b,D2: set_option_a] :
( ( bij_betw_b_option_a @ F @ A2 @ C3 )
=> ( ( bij_betw_b_option_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_option_a @ C3 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_b_option_a
@ ^ [X3: b] : ( if_option_a @ ( member_b @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_b @ A2 @ B2 )
@ ( sup_sup_set_option_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1021_bij__betw__disjoint__Un,axiom,
! [F: a > b,A2: set_a,C3: set_b,G: a > b,B2: set_a,D2: set_b] :
( ( bij_betw_a_b @ F @ A2 @ C3 )
=> ( ( bij_betw_a_b @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_b @ C3 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_a_b
@ ^ [X3: a] : ( if_b @ ( member_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A2 @ B2 )
@ ( sup_sup_set_b @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1022_bij__betw__disjoint__Un,axiom,
! [F: a > a,A2: set_a,C3: set_a,G: a > a,B2: set_a,D2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ C3 )
=> ( ( bij_betw_a_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ C3 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_a_a
@ ^ [X3: a] : ( if_a @ ( member_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A2 @ B2 )
@ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1023_bij__betw__disjoint__Un,axiom,
! [F: a > option_a,A2: set_a,C3: set_option_a,G: a > option_a,B2: set_a,D2: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ C3 )
=> ( ( bij_betw_a_option_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_option_a @ C3 @ D2 )
= bot_bot_set_option_a )
=> ( bij_betw_a_option_a
@ ^ [X3: a] : ( if_option_a @ ( member_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A2 @ B2 )
@ ( sup_sup_set_option_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1024_bij__betw__disjoint__Un,axiom,
! [F: option_a > b,A2: set_option_a,C3: set_b,G: option_a > b,B2: set_option_a,D2: set_b] :
( ( bij_betw_option_a_b @ F @ A2 @ C3 )
=> ( ( bij_betw_option_a_b @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ B2 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_b @ C3 @ D2 )
= bot_bot_set_b )
=> ( bij_betw_option_a_b
@ ^ [X3: option_a] : ( if_b @ ( member_option_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 )
@ ( sup_sup_set_b @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1025_bij__betw__disjoint__Un,axiom,
! [F: option_a > a,A2: set_option_a,C3: set_a,G: option_a > a,B2: set_option_a,D2: set_a] :
( ( bij_betw_option_a_a @ F @ A2 @ C3 )
=> ( ( bij_betw_option_a_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ B2 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_a @ C3 @ D2 )
= bot_bot_set_a )
=> ( bij_betw_option_a_a
@ ^ [X3: option_a] : ( if_a @ ( member_option_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 )
@ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1026_bij__betw__disjoint__Un,axiom,
! [F: option_a > option_a,A2: set_option_a,C3: set_option_a,G: option_a > option_a,B2: set_option_a,D2: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ A2 @ C3 )
=> ( ( bij_be5431266891817924854tion_a @ G @ B2 @ D2 )
=> ( ( ( inf_inf_set_option_a @ A2 @ B2 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_option_a @ C3 @ D2 )
= bot_bot_set_option_a )
=> ( bij_be5431266891817924854tion_a
@ ^ [X3: option_a] : ( if_option_a @ ( member_option_a @ X3 @ A2 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 )
@ ( sup_sup_set_option_a @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1027_is__none__bind,axiom,
! [F: option_a,G: a > option_a] :
( ( is_none_a @ ( bind_a_a @ F @ G ) )
= ( ( is_none_a @ F )
| ( is_none_a @ ( G @ ( the_a2 @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_1028_the__map__option,axiom,
! [X2: option_a,F: a > a] :
( ~ ( is_none_a @ X2 )
=> ( ( the_a2 @ ( map_option_a_a @ F @ X2 ) )
= ( F @ ( the_a2 @ X2 ) ) ) ) ).
% the_map_option
thf(fact_1029_option_Osize__gen__o__map,axiom,
! [F: a > nat,G: a > a] :
( ( comp_o8583038678572498833tion_a @ ( size_option_a @ F ) @ ( map_option_a_a @ G ) )
= ( size_option_a @ ( comp_a_nat_a @ F @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_1030_is__none__code_I2_J,axiom,
! [X2: a] :
~ ( is_none_a @ ( some_a @ X2 ) ) ).
% is_none_code(2)
thf(fact_1031_is__none__code_I1_J,axiom,
is_none_a @ none_a ).
% is_none_code(1)
thf(fact_1032_is__none__map__option,axiom,
! [F: a > a,X2: option_a] :
( ( is_none_a @ ( map_option_a_a @ F @ X2 ) )
= ( is_none_a @ X2 ) ) ).
% is_none_map_option
thf(fact_1033_is__none__simps_I2_J,axiom,
! [X2: a] :
~ ( is_none_a @ ( some_a @ X2 ) ) ).
% is_none_simps(2)
thf(fact_1034_Option_Ois__none__def,axiom,
( is_none_a
= ( ^ [X3: option_a] : ( X3 = none_a ) ) ) ).
% Option.is_none_def
thf(fact_1035_is__none__simps_I1_J,axiom,
is_none_a @ none_a ).
% is_none_simps(1)
thf(fact_1036_option_Othe__def,axiom,
( the_a2
= ( case_option_a_a @ undefined_a
@ ^ [X24: a] : X24 ) ) ).
% option.the_def
thf(fact_1037_set__bind__option,axiom,
! [X2: option_a,F: a > option_a] :
( ( set_option_a2 @ ( bind_a_a @ X2 @ F ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( comp_o6143895765626710849et_a_a @ set_option_a2 @ F ) @ ( set_option_a2 @ X2 ) ) ) ) ).
% set_bind_option
thf(fact_1038_fcomp__comp,axiom,
( fcomp_a_option_a_a
= ( ^ [F2: a > option_a,G2: option_a > a] : ( comp_option_a_a_a @ G2 @ F2 ) ) ) ).
% fcomp_comp
thf(fact_1039_monotone__on__o,axiom,
! [A2: set_option_a,Orda: option_a > option_a > $o,Ordb: a > a > $o,F: option_a > a,B2: set_a,Ordc: a > a > $o,G: a > option_a] :
( ( monoto4483592047695070497on_a_a @ A2 @ Orda @ Ordb @ F )
=> ( ( monoto2261078166156275373tion_a @ B2 @ Ordc @ Orda @ G )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a @ G @ B2 ) @ A2 )
=> ( monotone_on_a_a @ B2 @ Ordc @ Ordb @ ( comp_option_a_a_a @ F @ G ) ) ) ) ) ).
% monotone_on_o
thf(fact_1040_mono__sup,axiom,
! [F: set_b > set_b,A2: set_b,B2: set_b] :
( ( monoto8660672184891397159_set_b @ top_top_set_set_b @ ord_less_eq_set_b @ ord_less_eq_set_b @ F )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% mono_sup
thf(fact_1041_mono__sup,axiom,
! [F: set_b > set_a,A2: set_b,B2: set_b] :
( ( monoto8660672180588168358_set_a @ top_top_set_set_b @ ord_less_eq_set_b @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% mono_sup
thf(fact_1042_mono__sup,axiom,
! [F: set_a > set_b,A2: set_a,B2: set_a] :
( ( monoto7172710147596598632_set_b @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_b @ F )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% mono_sup
thf(fact_1043_mono__sup,axiom,
! [F: set_a > set_a,A2: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% mono_sup
thf(fact_1044_mono__inf,axiom,
! [F: set_a > set_a,A2: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_inf_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ).
% mono_inf
thf(fact_1045_mono__Int,axiom,
! [F: set_a > set_a,A2: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( F @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_inf_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ).
% mono_Int
thf(fact_1046_mono__Un,axiom,
! [F: set_b > set_b,A2: set_b,B2: set_b] :
( ( monoto8660672184891397159_set_b @ top_top_set_set_b @ ord_less_eq_set_b @ ord_less_eq_set_b @ F )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_1047_mono__Un,axiom,
! [F: set_b > set_a,A2: set_b,B2: set_b] :
( ( monoto8660672180588168358_set_a @ top_top_set_set_b @ ord_less_eq_set_b @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_1048_mono__Un,axiom,
! [F: set_a > set_b,A2: set_a,B2: set_a] :
( ( monoto7172710147596598632_set_b @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_b @ F )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_1049_mono__Un,axiom,
! [F: set_a > set_a,A2: set_a,B2: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( F @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_1050_mono__onI,axiom,
! [A2: set_set_a,F: set_a > set_a] :
( ! [R: set_a,S: set_a] :
( ( member_set_a @ R @ A2 )
=> ( ( member_set_a @ S @ A2 )
=> ( ( ord_less_eq_set_a @ R @ S )
=> ( ord_less_eq_set_a @ ( F @ R ) @ ( F @ S ) ) ) ) )
=> ( monoto7172710143293369831_set_a @ A2 @ ord_less_eq_set_a @ ord_less_eq_set_a @ F ) ) ).
% mono_onI
thf(fact_1051_assertion_Osimps_I326_J,axiom,
! [X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
( ( set_as7232682317586342732_b_d_c @ ( star_a_b_d_c @ X31 @ X32 ) )
= ( sup_sup_set_b @ ( set_as7232682317586342732_b_d_c @ X31 ) @ ( set_as7232682317586342732_b_d_c @ X32 ) ) ) ).
% assertion.simps(326)
thf(fact_1052_assertion_Oset__intros_I4_J,axiom,
! [Yd: b,X32: assertion_a_b_d_c,X31: assertion_a_b_d_c] :
( ( member_b @ Yd @ ( set_as7232682317586342732_b_d_c @ X32 ) )
=> ( member_b @ Yd @ ( set_as7232682317586342732_b_d_c @ ( star_a_b_d_c @ X31 @ X32 ) ) ) ) ).
% assertion.set_intros(4)
thf(fact_1053_assertion_Oset__intros_I3_J,axiom,
! [Yc: b,X31: assertion_a_b_d_c,X32: assertion_a_b_d_c] :
( ( member_b @ Yc @ ( set_as7232682317586342732_b_d_c @ X31 ) )
=> ( member_b @ Yc @ ( set_as7232682317586342732_b_d_c @ ( star_a_b_d_c @ X31 @ X32 ) ) ) ) ).
% assertion.set_intros(3)
thf(fact_1054_semilattice__set_Oeq__fold_H,axiom,
! [F: a > a > a,A2: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A2 )
= ( the_a2
@ ( finite6501707464432451470tion_a
@ ^ [X3: a,Y2: option_a] : ( some_a @ ( case_option_a_a @ X3 @ ( F @ X3 ) @ Y2 ) )
@ none_a
@ A2 ) ) ) ) ).
% semilattice_set.eq_fold'
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X2: b,Y: b] :
( ( if_b @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X2: b,Y: b] :
( ( if_b @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X2: option_a,Y: option_a] :
( ( if_option_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X2: option_a,Y: option_a] :
( ( if_option_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
smaller_interp_c_d_a @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ a2 @ b2 ) @ delta2 ) @ ( applies_eq_a_b_d_c @ plus @ mult @ valid @ ( star_a_b_d_c @ a2 @ b2 ) @ delta ) ).
%------------------------------------------------------------------------------