TPTP Problem File: SLH0496^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Separation_Logic_Unbounded/0002_Distributivity/prob_00318_008909__6765724_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1455 ( 390 unt; 384 typ; 0 def)
% Number of atoms : 3483 (1294 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13818 ( 404 ~; 30 |; 241 &;11635 @)
% ( 0 <=>;1508 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 51 ( 50 usr)
% Number of type conns : 2604 (2604 >; 0 *; 0 +; 0 <<)
% Number of symbols : 337 ( 334 usr; 24 con; 0-7 aty)
% Number of variables : 4218 ( 148 ^;3971 !; 99 ?;4218 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:05:06.465
%------------------------------------------------------------------------------
% Could-be-implicit typings (50)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J_J,type,
set_Pr7868159745199425715_a_d_c: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
produc6150846815813599699_a_d_c: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J_J,type,
set_Pr2064693230030669831tion_a: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_J,type,
option788170273582809878list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__a_J_J_J_J,type,
set_Pr452613198074451719tion_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J,type,
produc8176476770652931111tion_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
produc8652252815484796455tion_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
produc3854044251064639184list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
list_P2261792721279755821tion_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J_J,type,
set_op7164657265554968289tion_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
set_Pr7585778909603769095tion_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
set_Pr5845495582615845127_set_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
produc5032551385658279741list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
produc3509355604313844263tion_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J_J,type,
list_P3363167923603265779tion_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J_J,type,
list_P6260409590414597735on_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J_J,type,
set_a_2282238091077557671tion_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
set_op7810783462584961947tion_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J_J,type,
set_Pr3411724424142761165tion_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J_J,type,
set_Pr6308966090954093121on_a_a: $tType ).
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
option2362401199305441953_a_d_c: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
produc1703568184450464039_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__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
produc3964210925746912109tion_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
produc3083010940779526881on_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Option__Ooption_Itf__a_J_J_J,type,
set_list_option_a: $tType ).
thf(ty_n_t__UnboundedLogic__Oassertion_Itf__a_Mtf__b_Mtf__c_Mtf__d_J,type,
assertion_a_b_c_d: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
set_set_option_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
produc8685980395799941037list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Option__Ooption_Itf__a_J_J_J,type,
set_a_option_a: $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__d_Mtf__c_J_J,type,
product_prod_a_d_c: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
option_option_a: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_Itf__a_J_J,type,
list_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $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__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__d_J,type,
set_d: $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 (334)
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Option__Ooption_Itf__a_J,type,
bNF_We8432232079604507440tion_a: set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001tf__a,type,
bNF_We1162827675446709994_rel_a: set_Product_prod_a_a > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Option__Ooption_Itf__a_J,type,
bNF_We2467337426749329402tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001tf__a,type,
bNF_We6697304935525757620inim_a: set_Product_prod_a_a > set_a > a > $o ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Option__Ooption_Itf__a_J,type,
bNF_We4567742444881707410tion_a: set_Pr7585778909603769095tion_a > option_a > option_a > option_a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001tf__a,type,
bNF_We3763454674811381836max2_a: set_Product_prod_a_a > a > a > a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Option__Ooption_Itf__a_J,type,
bNF_We6579146059749918992tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001tf__a,type,
bNF_We5615626441682584778inim_a: set_Product_prod_a_a > set_a > a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001t__Option__Ooption_Itf__a_J,type,
bNF_We5356091070762920229tion_a: set_Pr7585778909603769095tion_a > set_option_a > option_a ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc_001tf__a,type,
bNF_We6154283375207884895_suc_a: set_Product_prod_a_a > set_a > a ).
thf(sy_c_Equiv__Relations_Oequiv_001t__Option__Ooption_Itf__a_J,type,
equiv_equiv_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Equiv__Relations_Oequiv_001tf__a,type,
equiv_equiv_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Equiv__Relations_Oproj_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
equiv_6865337221296424970tion_a: set_Pr7585778909603769095tion_a > option_a > set_option_a ).
thf(sy_c_Equiv__Relations_Oproj_001tf__a_001tf__a,type,
equiv_proj_a_a: set_Product_prod_a_a > a > set_a ).
thf(sy_c_Equiv__Relations_Oquotient_001t__Option__Ooption_Itf__a_J,type,
equiv_2859340374733651339tion_a: set_option_a > set_Pr7585778909603769095tion_a > set_set_option_a ).
thf(sy_c_Equiv__Relations_Oquotient_001tf__a,type,
equiv_quotient_a: set_a > set_Product_prod_a_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
finite3089212293327181890tion_a: set_op7164657265554968289tion_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
finite8942580144290239484tion_a: set_op7810783462584961947tion_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
finite595534299322690568tion_a: set_a_2282238091077557671tion_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
finite5998080203967203522tion_a: set_a_option_a > $o ).
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_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_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_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_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__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001tf__a,type,
comp_P5977721380588955012_a_a_a: ( product_prod_a_a > a ) > ( a > product_prod_a_a ) > a > a ).
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_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_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_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__d_001tf__c,type,
fun_upd_d_c: ( d > c ) > d > c > d > c ).
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_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_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_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_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_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
minus_6512073291116468334tion_a: set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
minus_6817036919807184750od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_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_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_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_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_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_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
sup_su1214438497309894875tion_a: set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J_J,type,
sup_su3065098059273993749on_a_a: set_Pr6308966090954093121on_a_a > set_Pr6308966090954093121on_a_a > set_Pr6308966090954093121on_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J_J,type,
sup_su167856392462661793tion_a: set_Pr3411724424142761165tion_a > set_Pr3411724424142761165tion_a > set_Pr3411724424142761165tion_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
sup_su3048258781599657691od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_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_List_Oappend_001t__Option__Ooption_Itf__a_J,type,
append_option_a: list_option_a > list_option_a > list_option_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Oextract_001tf__a,type,
extract_a: ( a > $o ) > list_a > option788170273582809878list_a ).
thf(sy_c_List_Ofind_001t__Option__Ooption_Itf__a_J,type,
find_option_a: ( option_a > $o ) > list_option_a > option_option_a ).
thf(sy_c_List_Ofind_001tf__a,type,
find_a: ( a > $o ) > list_a > option_a ).
thf(sy_c_List_Olenlex_001t__Option__Ooption_Itf__a_J,type,
lenlex_option_a: set_Pr7585778909603769095tion_a > set_Pr2064693230030669831tion_a ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__Option__Ooption_Itf__a_J,type,
lex_option_a: set_Pr7585778909603769095tion_a > set_Pr2064693230030669831tion_a ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexn_001tf__a,type,
lexn_a: set_Product_prod_a_a > nat > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olexord_001t__Option__Ooption_Itf__a_J,type,
lexord_option_a: set_Pr7585778909603769095tion_a > set_Pr2064693230030669831tion_a ).
thf(sy_c_List_Olexord_001tf__a,type,
lexord_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
cons_P7316939126706565853od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_option_a_a: ( option_a > a ) > list_option_a > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
map_a_7860052162900579309od_a_a: ( a > product_prod_a_a ) > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__Option__Ooption_Itf__a_J,type,
set_option_a2: list_option_a > set_option_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
set_Pr948796958549772220tion_a: list_P2261792721279755821tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
set_Pr1233600038994746358on_a_a: list_P6260409590414597735on_a_a > set_Pr6308966090954093121on_a_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
set_Pr2114800023962131586tion_a: list_P3363167923603265779tion_a > set_Pr3411724424142761165tion_a ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
set_Product_prod_a_a2: list_P1396940483166286381od_a_a > set_Product_prod_a_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
nth_Product_prod_a_a: list_P1396940483166286381od_a_a > nat > product_prod_a_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Opartition_001t__Option__Ooption_Itf__a_J,type,
partition_option_a: ( option_a > $o ) > list_option_a > produc8176476770652931111tion_a ).
thf(sy_c_List_Opartition_001tf__a,type,
partition_a: ( a > $o ) > list_a > produc9164743771328383783list_a ).
thf(sy_c_List_Oshuffles_001t__Option__Ooption_Itf__a_J,type,
shuffles_option_a: list_option_a > list_option_a > set_list_option_a ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Otake_001t__Option__Ooption_Itf__a_J,type,
take_option_a: nat > list_option_a > list_option_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Ozip_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
zip_op6411647709037274935tion_a: list_option_a > list_option_a > list_P2261792721279755821tion_a ).
thf(sy_c_List_Ozip_001t__Option__Ooption_Itf__a_J_001tf__a,type,
zip_option_a_a: list_option_a > list_a > list_P6260409590414597735on_a_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Option__Ooption_Itf__a_J,type,
zip_a_option_a: list_a > list_option_a > list_P3363167923603265779tion_a ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_Map_Odom_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
dom_op4724496951392727122tion_a: ( option_a > option_option_a ) > set_option_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_001t__Option__Ooption_Itf__a_J,type,
dom_a_option_a: ( a > option_option_a ) > set_a ).
thf(sy_c_Map_Odom_001tf__a_001tf__a,type,
dom_a_a: ( a > option_a ) > set_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__of_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_of_option_a_a: list_P6260409590414597735on_a_a > option_a > option_a ).
thf(sy_c_Map_Omap__of_001tf__a_001tf__a,type,
map_of_a_a: list_P1396940483166286381od_a_a > a > option_a ).
thf(sy_c_Map_Omap__upds_001t__Option__Ooption_Itf__a_J_001tf__a,type,
map_upds_option_a_a: ( option_a > option_a ) > list_option_a > list_a > option_a > option_a ).
thf(sy_c_Map_Omap__upds_001tf__a_001tf__a,type,
map_upds_a_a: ( a > option_a ) > list_a > list_a > a > option_a ).
thf(sy_c_Map_Oran_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
ran_op6317565877353657455tion_a: ( option_a > option_option_a ) > set_option_a ).
thf(sy_c_Map_Oran_001t__Option__Ooption_Itf__a_J_001tf__a,type,
ran_option_a_a: ( option_a > option_a ) > set_a ).
thf(sy_c_Map_Oran_001tf__a_001t__Option__Ooption_Itf__a_J,type,
ran_a_option_a: ( a > option_option_a ) > set_option_a ).
thf(sy_c_Map_Oran_001tf__a_001tf__a,type,
ran_a_a: ( a > option_a ) > set_a ).
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_001tf__a_001tf__a,type,
restrict_map_a_a: ( a > option_a ) > set_a > a > option_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_Itf__a_J_J,type,
size_s3078493964004954806tion_a: list_option_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Option_Obind_001tf__a_001tf__a,type,
bind_a_a: option_a > ( 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_001t__Option__Ooption_Itf__a_J,type,
none_option_a: option_option_a ).
thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
none_P5893993846586699057list_a: option788170273582809878list_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
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_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
some_P5354654743593010357list_a: produc3854044251064639184list_a > option788170273582809878list_a ).
thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J,type,
some_P377817780860425132_a_d_c: product_prod_a_d_c > option2362401199305441953_a_d_c ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
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_a2: ( a > a ) > option_a > option_a ).
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_a3: option_a > set_a ).
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_a: option_option_a > option_a ).
thf(sy_c_Option_Ooption_Othe_001tf__a,type,
the_a: option_a > a ).
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_Order__Relation_OAboveS_001t__Option__Ooption_Itf__a_J,type,
order_6500638856667293583tion_a: set_Pr7585778909603769095tion_a > set_option_a > set_option_a ).
thf(sy_c_Order__Relation_OAboveS_001tf__a,type,
order_AboveS_a: set_Product_prod_a_a > set_a > set_a ).
thf(sy_c_Order__Relation_Olinear__order__on_001t__Option__Ooption_Itf__a_J,type,
order_7850372301378808569tion_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Order__Relation_Olinear__order__on_001tf__a,type,
order_8768733634509060147r_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Order__Relation_Oofilter_001t__Option__Ooption_Itf__a_J,type,
order_6420974439381506266tion_a: set_Pr7585778909603769095tion_a > set_option_a > $o ).
thf(sy_c_Order__Relation_Oofilter_001tf__a,type,
order_ofilter_a: set_Product_prod_a_a > set_a > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001t__Option__Ooption_Itf__a_J,type,
order_4134995541221112539tion_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Order__Relation_Opreorder__on_001tf__a,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Order__Relation_OunderS_001t__Option__Ooption_Itf__a_J,type,
order_8525669848891258378tion_a: set_Pr7585778909603769095tion_a > option_a > set_option_a ).
thf(sy_c_Order__Relation_OunderS_001tf__a,type,
order_underS_a: set_Product_prod_a_a > a > set_a ).
thf(sy_c_Order__Relation_Ounder_001t__Option__Ooption_Itf__a_J,type,
order_under_option_a: set_Pr7585778909603769095tion_a > option_a > set_option_a ).
thf(sy_c_Order__Relation_Ounder_001tf__a,type,
order_under_a: set_Product_prod_a_a > a > set_a ).
thf(sy_c_Order__Relation_Owell__order__on_001t__Option__Ooption_Itf__a_J,type,
order_4821795997958563554tion_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Order__Relation_Owell__order__on_001tf__a,type,
order_6972113574731384220r_on_a: set_a > set_Product_prod_a_a > $o ).
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_It__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J_J,type,
bot_bo6007253611978100339tion_a: set_Pr2064693230030669831tion_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
bot_bo2955605580254355571list_a: set_Pr4048851178543822343list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
bot_bo235252021745139059tion_a: set_Pr7585778909603769095tion_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo3357376287454694259od_a_a: set_Product_prod_a_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_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
ord_le5631237216984945872tion_a: set_option_a > set_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
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_It__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J_J,type,
ord_le4471550158292877991tion_a: set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_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_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
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_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Option__Ooption_Itf__a_J_J_001t__List__Olist_It__Option__Ooption_Itf__a_J_J,type,
produc6071813090954082327tion_a: list_option_a > list_option_a > produc8176476770652931111tion_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
produc3204708664006668352list_a: list_a > produc8685980395799941037list_a > produc3854044251064639184list_a ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
produc9011544418120257559tion_a: option_a > option_a > produc3509355604313844263tion_a ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_Itf__a_J_001tf__a,type,
produc3446707977624461905on_a_a: option_a > a > produc3083010940779526881on_a_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J,type,
produc5208860900648697099_a_d_c: product_prod_a_d_c > option2362401199305441953_a_d_c > produc6150846815813599699_a_d_c ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
produc8179951581375851543tion_a: set_option_a > set_option_a > produc8652252815484796455tion_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
produc9088192753505129239_set_a: set_a > set_a > produc1703568184450464039_set_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001_062_Itf__d_Mtf__c_J,type,
product_Pair_a_d_c: a > ( d > c ) > product_prod_a_d_c ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_Itf__a_J,type,
produc6670463072477821725list_a: a > list_a > produc8685980395799941037list_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Option__Ooption_Itf__a_J,type,
produc1224194096085666781tion_a: a > option_a > produc3964210925746912109tion_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OSigma_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
produc269287337874323144tion_a: set_option_a > ( option_a > set_option_a ) > set_Pr7585778909603769095tion_a ).
thf(sy_c_Product__Type_OSigma_001t__Option__Ooption_Itf__a_J_001tf__a,type,
produc3962846827955709570on_a_a: set_option_a > ( option_a > set_a ) > set_Pr6308966090954093121on_a_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001t__Option__Ooption_Itf__a_J,type,
produc1740332946416914446tion_a: set_a > ( a > set_option_a ) > set_Pr3411724424142761165tion_a ).
thf(sy_c_Product__Type_OSigma_001tf__a_001tf__a,type,
product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Option__Ooption_Itf__a_J_001tf__a,type,
produc8941638570267940413on_a_a: produc3083010940779526881on_a_a > option_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
product_fst_a_a: product_prod_a_a > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
product_snd_a_a: product_prod_a_a > a ).
thf(sy_c_Relation_ODomain_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
domain5649462347324568460tion_a: set_Pr7585778909603769095tion_a > set_option_a ).
thf(sy_c_Relation_ODomain_001tf__a_001tf__a,type,
domain_a_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_OField_001t__Option__Ooption_Itf__a_J,type,
field_option_a: set_Pr7585778909603769095tion_a > set_option_a ).
thf(sy_c_Relation_OField_001tf__a,type,
field_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_OId_001t__Option__Ooption_Itf__a_J,type,
id_option_a2: set_Pr7585778909603769095tion_a ).
thf(sy_c_Relation_OId_001tf__a,type,
id_a2: set_Product_prod_a_a ).
thf(sy_c_Relation_OImage_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
image_4442594622209975379tion_a: set_Pr7585778909603769095tion_a > set_option_a > set_option_a ).
thf(sy_c_Relation_OImage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
image_option_a_a: set_Pr6308966090954093121on_a_a > set_option_a > set_a ).
thf(sy_c_Relation_OImage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: set_Pr3411724424142761165tion_a > set_a > set_option_a ).
thf(sy_c_Relation_OImage_001tf__a_001tf__a,type,
image_a_a: set_Product_prod_a_a > set_a > set_a ).
thf(sy_c_Relation_Oantisym__on_001t__Option__Ooption_Itf__a_J,type,
antisym_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Oantisym__on_001tf__a,type,
antisym_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Oasym__on_001t__Option__Ooption_Itf__a_J,type,
asym_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Oasym__on_001tf__a,type,
asym_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Oirrefl__on_001t__Option__Ooption_Itf__a_J,type,
irrefl_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Oirrefl__on_001tf__a,type,
irrefl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Orefl__on_001t__Option__Ooption_Itf__a_J,type,
refl_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Orefl__on_001tf__a,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Ototal__on_001t__Option__Ooption_Itf__a_J,type,
total_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Ototal__on_001tf__a,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Otrans__on_001t__Option__Ooption_Itf__a_J,type,
trans_on_option_a: set_option_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Relation_Otrans__on_001tf__a,type,
trans_on_a: set_a > set_Product_prod_a_a > $o ).
thf(sy_c_Set_OCollect_001_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
collec2458836999851688832tion_a: ( ( option_a > option_option_a ) > $o ) > set_op7164657265554968289tion_a ).
thf(sy_c_Set_OCollect_001_062_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
collec5803847638788715578tion_a: ( ( option_a > option_a ) > $o ) > set_op7810783462584961947tion_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
collec6680173830675942470tion_a: ( ( a > option_option_a ) > $o ) > set_a_2282238091077557671tion_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
collect_a_option_a: ( ( a > option_a ) > $o ) > set_a_option_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_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__d,type,
collect_d: ( d > $o ) > set_d ).
thf(sy_c_Set_Odisjnt_001t__Option__Ooption_Itf__a_J,type,
disjnt_option_a: set_option_a > set_option_a > $o ).
thf(sy_c_Set_Odisjnt_001tf__a,type,
disjnt_a: set_a > set_a > $o ).
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_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
image_2176871512121339682on_a_a: ( option_a > produc3083010940779526881on_a_a ) > set_option_a > set_Pr6308966090954093121on_a_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_7456799861883459304od_a_a: ( option_a > product_prod_a_a ) > set_option_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
image_option_a_a2: ( option_a > a ) > set_option_a > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J_001t__Option__Ooption_Itf__a_J,type,
image_3098826861768462248tion_a: ( produc3083010940779526881on_a_a > option_a ) > set_Pr6308966090954093121on_a_a > set_option_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
image_3437945252899457948_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a2: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
image_7468189554597481134tion_a: ( a > produc3509355604313844263tion_a ) > set_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
image_1318920958782419572tion_a: ( a > produc3964210925746912109tion_a ) > set_a > set_Pr3411724424142761165tion_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a2: ( a > a ) > set_a > set_a ).
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_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J,type,
insert3212945957549752823tion_a: produc8176476770652931111tion_a > set_Pr2064693230030669831tion_a > set_Pr2064693230030669831tion_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
insert1856800524785285367list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
insert1246254401036548087tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
insert6058462421052899505on_a_a: produc3083010940779526881on_a_a > set_Pr6308966090954093121on_a_a > set_Pr6308966090954093121on_a_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
insert6939662406020284733tion_a: produc3964210925746912109tion_a > set_Pr3411724424142761165tion_a > set_Pr3411724424142761165tion_a ).
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_Ovimage_001t__List__Olist_It__Option__Ooption_Itf__a_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J,type,
vimage3571068517928849688tion_a: ( list_option_a > produc8176476770652931111tion_a ) > set_Pr2064693230030669831tion_a > set_list_option_a ).
thf(sy_c_Set_Ovimage_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
vimage4558233055442567774list_a: ( list_a > produc9164743771328383783list_a ) > set_Pr4048851178543822343list_a > set_list_a ).
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_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
vimage6204385730559098270tion_a: ( option_a > produc3509355604313844263tion_a ) > set_Pr7585778909603769095tion_a > set_option_a ).
thf(sy_c_Set_Ovimage_001t__Option__Ooption_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
vimage1039097674533424484tion_a: ( option_a > produc3964210925746912109tion_a ) > set_Pr3411724424142761165tion_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_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
vimage7079528361448986782on_a_a: ( a > produc3083010940779526881on_a_a ) > set_Pr6308966090954093121on_a_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
vimage5143925195038468708od_a_a: ( a > product_prod_a_a ) > set_Product_prod_a_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_Transitive__Closure_Ortrancl_001t__List__Olist_Itf__a_J,type,
transi7631188966963710983list_a: set_Pr4048851178543822343list_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Option__Ooption_Itf__a_J,type,
transi330218190764880583tion_a: set_Pr7585778909603769095tion_a > set_Pr7585778909603769095tion_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_UnboundedLogic_Oassertion_OAnd_001tf__a_001tf__b_001tf__c_001tf__d,type,
and_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OBounded_001tf__a_001tf__b_001tf__c_001tf__d,type,
bounded_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OExists_001tf__d_001tf__a_001tf__b_001tf__c,type,
exists_d_a_b_c: d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OForall_001tf__d_001tf__a_001tf__b_001tf__c,type,
forall_d_a_b_c: d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OImp_001tf__a_001tf__b_001tf__c_001tf__d,type,
imp_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OMult_001tf__b_001tf__a_001tf__c_001tf__d,type,
mult_b_a_c_d: b > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OOr_001tf__a_001tf__b_001tf__c_001tf__d,type,
or_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OPred_001tf__a_001tf__b_001tf__c_001tf__d,type,
pred_a_b_c_d: assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OSem_001tf__d_001tf__c_001tf__a_001tf__b,type,
sem_d_c_a_b: ( ( d > c ) > a > $o ) > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OStar_001tf__a_001tf__b_001tf__c_001tf__d,type,
star_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OWand_001tf__a_001tf__b_001tf__c_001tf__d,type,
wand_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d > assertion_a_b_c_d ).
thf(sy_c_UnboundedLogic_Oassertion_OWildcard_001tf__a_001tf__b_001tf__c_001tf__d,type,
wildcard_a_b_c_d: assertion_a_b_c_d > assertion_a_b_c_d ).
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__c_001tf__d,type,
entails_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oequal__outside_001tf__d_001tf__c,type,
equal_outside_d_c: ( d > c ) > ( d > c ) > set_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oequivalent_001tf__a_001tf__b_001tf__c_001tf__d,type,
equivalent_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Oframe__property_001tf__a_001tf__d_001tf__c,type,
frame_property_a_d_c: ( a > a > option_a ) > ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ointuitionistic_001tf__a_001tf__b_001tf__d_001tf__c,type,
intuit4720955538653295669_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > ( d > c ) > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Omodified_001tf__a_001tf__d_001tf__c,type,
modified_a_d_c: set_Pr7868159745199425715_a_d_c > set_d ).
thf(sy_c_UnboundedLogic_Ologic_Onot__in__fv_001tf__a_001tf__b_001tf__c_001tf__d,type,
not_in_fv_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > set_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Opure_001tf__a_001tf__b_001tf__c_001tf__d,type,
pure_a_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osafe_001tf__a_001tf__d_001tf__c,type,
safe_a_d_c: set_Pr7868159745199425715_a_d_c > product_prod_a_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osafety__monotonicity_001tf__a_001tf__d_001tf__c,type,
safety7280469885071620222_a_d_c: ( a > a > option_a ) > ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Osat_001tf__a_001tf__b_001tf__d_001tf__c,type,
sat_a_b_d_c: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > a > ( d > c ) > ( ( d > c ) > set_a ) > assertion_a_b_c_d > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ovalid__command_001tf__a_001tf__d_001tf__c,type,
valid_command_a_d_c: ( a > $o ) > set_Pr7868159745199425715_a_d_c > $o ).
thf(sy_c_UnboundedLogic_Ologic_Ovalid__hoare__triple_001tf__a_001tf__b_001tf__c_001tf__d,type,
valid_8824771084768397689_b_c_d: ( a > a > option_a ) > ( b > a > a ) > ( a > $o ) > assertion_a_b_c_d > set_Pr7868159745199425715_a_d_c > assertion_a_b_c_d > ( ( d > c ) > 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_Ofinite__psubset_001t__Option__Ooption_Itf__a_J,type,
finite4966134214920407047tion_a: set_Pr452613198074451719tion_a ).
thf(sy_c_Wellfounded_Ofinite__psubset_001tf__a,type,
finite_psubset_a: set_Pr5845495582615845127_set_a ).
thf(sy_c_Wellfounded_Omax__ext_001t__Option__Ooption_Itf__a_J,type,
max_ext_option_a: set_Pr7585778909603769095tion_a > set_Pr452613198074451719tion_a ).
thf(sy_c_Wellfounded_Omax__ext_001tf__a,type,
max_ext_a: set_Product_prod_a_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Wellfounded_Owf_001t__Option__Ooption_Itf__a_J,type,
wf_option_a: set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_Wellfounded_Owf_001tf__a,type,
wf_a: set_Product_prod_a_a > $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__Product____Type__Oprod_It__List__Olist_It__Option__Ooption_Itf__a_J_J_Mt__List__Olist_It__Option__Ooption_Itf__a_J_J_J,type,
member3116372912924030544tion_a: produc8176476770652931111tion_a > set_Pr2064693230030669831tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mt__Option__Ooption_Itf__a_J_J,type,
member5498148017924304208tion_a: produc3509355604313844263tion_a > set_Pr7585778909603769095tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_Itf__a_J_Mtf__a_J,type,
member6056235002698166154on_a_a: produc3083010940779526881on_a_a > set_Pr6308966090954093121on_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__a_M_062_Itf__d_Mtf__c_J_J_J_J,type,
member2052822272342364412_a_d_c: produc6150846815813599699_a_d_c > set_Pr7868159745199425715_a_d_c > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Option__Ooption_Itf__a_J_J_Mt__Set__Oset_It__Option__Ooption_Itf__a_J_J_J,type,
member5358692782348450128tion_a: produc8652252815484796455tion_a > set_Pr452613198074451719tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Option__Ooption_Itf__a_J_J,type,
member6937434987665551382tion_a: produc3964210925746912109tion_a > set_Pr3411724424142761165tion_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
member_set_option_a: set_option_a > set_set_option_a > $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_v_A,type,
a2: assertion_a_b_c_d ).
thf(sy_v_B,type,
b2: assertion_a_b_c_d ).
thf(sy_v__092_060Delta_062,type,
delta: ( d > c ) > 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 (1065)
thf(fact_0_commutative,axiom,
! [A: a,B: a] :
( ( plus @ A @ B )
= ( plus @ B @ A ) ) ).
% commutative
thf(fact_1_can__divide,axiom,
! [P: b,A: a,B: a] :
( ( ( mult @ P @ A )
= ( mult @ P @ B ) )
=> ( A = B ) ) ).
% can_divide
thf(fact_2_WildPos,axiom,
! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
=> ( entails_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ A2 ) @ Delta @ ( wildcard_a_b_c_d @ B2 ) ) ) ).
% WildPos
thf(fact_3_unique__inv,axiom,
! [A: a,P: b,B: a] :
( ( A
= ( mult @ P @ B ) )
= ( B
= ( mult @ ( sinv @ P ) @ A ) ) ) ).
% unique_inv
thf(fact_4_one__neutral,axiom,
! [A: a] :
( ( mult @ one @ A )
= A ) ).
% one_neutral
thf(fact_5_WildWild,axiom,
! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).
% WildWild
thf(fact_6_equivalent__def,axiom,
! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( equivalent_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
& ( entails_a_b_c_d @ plus @ mult @ valid @ B2 @ Delta @ A2 ) ) ) ).
% equivalent_def
thf(fact_7_WildStar1,axiom,
! [A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ).
% WildStar1
thf(fact_8_dot__and1,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% dot_and1
thf(fact_9_dot__and2,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) ) ).
% dot_and2
thf(fact_10_DotPos,axiom,
! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ).
% DotPos
thf(fact_11_valid__mono,axiom,
! [A: a,B: a] :
( ( ( valid @ A )
& ( pre_larger_a @ plus @ A @ B ) )
=> ( valid @ B ) ) ).
% valid_mono
thf(fact_12_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_13_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_14_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_15_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_16_sat_Osimps_I7_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).
% sat.simps(7)
thf(fact_17_larger__implies__compatible,axiom,
! [X: a,Y: a] :
( ( pre_larger_a @ plus @ X @ Y )
=> ( pre_compatible_a @ plus @ X @ Y ) ) ).
% larger_implies_compatible
thf(fact_18_compatible__smaller,axiom,
! [A: a,B: a,X: a] :
( ( pre_larger_a @ plus @ A @ B )
=> ( ( pre_compatible_a @ plus @ X @ A )
=> ( pre_compatible_a @ plus @ X @ B ) ) ) ).
% compatible_smaller
thf(fact_19_sat__mult,axiom,
! [Sigma: a,P: b,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ! [A3: a] :
( ( Sigma
= ( mult @ P @ A3 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ).
% sat_mult
thf(fact_20_sat_Osimps_I1_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
= ( ? [A4: a] :
( ( Sigma
= ( mult @ P @ A4 ) )
& ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 ) ) ) ) ).
% sat.simps(1)
thf(fact_21_entails__def,axiom,
! [A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 )
= ( ! [Sigma2: a,S2: d > c] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ B2 ) ) ) ) ).
% entails_def
thf(fact_22_entailsI,axiom,
! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ B2 ) )
=> ( entails_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ).
% entailsI
thf(fact_23_sat_Osimps_I12_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
= ( ? [A4: a,P2: b] :
( ( Sigma
= ( mult @ P2 @ A4 ) )
& ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 ) ) ) ) ).
% sat.simps(12)
thf(fact_24_equivalentI,axiom,
! [Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ B2 ) )
=> ( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ B2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ A2 ) )
=> ( equivalent_a_b_c_d @ plus @ mult @ valid @ A2 @ Delta @ B2 ) ) ) ).
% equivalentI
thf(fact_25_dot__star2,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ).
% dot_star2
thf(fact_26_dot__star1,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% dot_star1
thf(fact_27_WildDot,axiom,
! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).
% WildDot
thf(fact_28_DotWild,axiom,
! [P: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ).
% DotWild
thf(fact_29_DotStar,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% DotStar
thf(fact_30_DotAnd,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) ) ).
% DotAnd
thf(fact_31_pure__def,axiom,
! [A2: assertion_a_b_c_d] :
( ( pure_a_b_c_d @ plus @ mult @ valid @ A2 )
= ( ! [Sigma2: a,Sigma4: a,S2: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta2 @ A2 )
= ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S2 @ Delta3 @ A2 ) ) ) ) ).
% pure_def
thf(fact_32_intuitionistic__def,axiom,
! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S @ Delta @ A2 )
= ( ! [A4: a,B3: a] :
( ( ( pre_larger_a @ plus @ A4 @ B3 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ B3 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 ) ) ) ) ).
% intuitionistic_def
thf(fact_33_intuitionisticI,axiom,
! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ! [A3: a,B4: a] :
( ( ( pre_larger_a @ plus @ A3 @ B4 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ B4 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 ) )
=> ( intuit4720955538653295669_b_d_c @ plus @ mult @ valid @ S @ Delta @ A2 ) ) ).
% intuitionisticI
thf(fact_34_not__in__fv__def,axiom,
! [A2: assertion_a_b_c_d,S4: set_d] :
( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ S4 )
= ( ! [Sigma2: a,S2: d > c,Delta2: ( d > c ) > set_a,S5: d > c] :
( ( equal_outside_d_c @ S2 @ S5 @ S4 )
=> ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta2 @ A2 )
= ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S5 @ Delta2 @ A2 ) ) ) ) ) ).
% not_in_fv_def
thf(fact_35_sat_Osimps_I10_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ pred_a_b_c_d )
= ( member_a @ Sigma @ ( Delta @ S ) ) ) ).
% sat.simps(10)
thf(fact_36_WildOr,axiom,
! [A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ).
% WildOr
thf(fact_37_DotWand,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% DotWand
thf(fact_38_dot__wand2,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ).
% dot_wand2
thf(fact_39_dot__wand1,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% dot_wand1
thf(fact_40_DotOr,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% DotOr
thf(fact_41_dot__or2,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) ) ).
% dot_or2
thf(fact_42_dot__or1,axiom,
! [P: b,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ).
% dot_or1
thf(fact_43_DotDot,axiom,
! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( equivalent_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) ) ).
% DotDot
thf(fact_44_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: option_a,P3: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_option_a] :
( ( collect_option_a
@ ^ [X2: option_a] : ( member_option_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__cong,axiom,
! [P3: option_a > $o,Q2: option_a > $o] :
( ! [X3: option_a] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_option_a @ P3 )
= ( collect_option_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_49_dot__mult2,axiom,
! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) ) ).
% dot_mult2
thf(fact_50_dot__mult1,axiom,
! [P: b,Q: b,A2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] : ( entails_a_b_c_d @ plus @ mult @ valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( smult @ P @ Q ) @ A2 ) ) ).
% dot_mult1
thf(fact_51_sat__imp,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ).
% sat_imp
thf(fact_52_can__factorize,axiom,
! [Q: b,P: b] :
? [R: b] :
( Q
= ( smult @ R @ P ) ) ).
% can_factorize
thf(fact_53_smult__asso,axiom,
! [P: b,Q: b,R2: b] :
( ( smult @ ( smult @ P @ Q ) @ R2 )
= ( smult @ P @ ( smult @ Q @ R2 ) ) ) ).
% smult_asso
thf(fact_54_smult__comm,axiom,
! [P: b,Q: b] :
( ( smult @ P @ Q )
= ( smult @ Q @ P ) ) ).
% smult_comm
thf(fact_55_double__mult,axiom,
! [P: b,Q: b,A: a] :
( ( mult @ P @ ( mult @ Q @ A ) )
= ( mult @ ( smult @ P @ Q ) @ A ) ) ).
% double_mult
thf(fact_56_sone__neutral,axiom,
! [P: b] :
( ( smult @ one @ P )
= P ) ).
% sone_neutral
thf(fact_57_sinv__inverse,axiom,
! [P: b] :
( ( smult @ P @ ( sinv @ P ) )
= one ) ).
% sinv_inverse
thf(fact_58_sat_Osimps_I6_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
| ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).
% sat.simps(6)
thf(fact_59_sat_Osimps_I5_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ B2 ) ) ) ).
% sat.simps(5)
thf(fact_60_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_61_logic__axioms,axiom,
logic_a_b @ plus @ mult @ smult @ sadd @ sinv @ one @ valid ).
% logic_axioms
thf(fact_62_sat_Osimps_I3_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
= ( ! [A4: a,Sigma4: a] :
( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma4 )
= ( plus @ Sigma @ A4 ) ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S @ Delta @ B2 ) ) ) ) ).
% sat.simps(3)
thf(fact_63_sat__wand,axiom,
! [S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
( ! [A3: a,Sigma5: a] :
( ( ( sat_a_b_d_c @ plus @ mult @ valid @ A3 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma5 )
= ( plus @ Sigma @ A3 ) ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ).
% sat_wand
thf(fact_64_sat_Osimps_I2_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
= ( ? [A4: a,B3: a] :
( ( ( some_a @ Sigma )
= ( plus @ A4 @ B3 ) )
& ( sat_a_b_d_c @ plus @ mult @ valid @ A4 @ S @ Delta @ A2 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ B3 @ S @ Delta @ B2 ) ) ) ) ).
% sat.simps(2)
thf(fact_65_sat_Osimps_I11_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( bounded_a_b_c_d @ A2 ) )
= ( ( valid @ Sigma )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ A2 ) ) ) ).
% sat.simps(11)
thf(fact_66_sat_Osimps_I4_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( sem_d_c_a_b @ B ) )
= ( B @ S @ Sigma ) ) ).
% sat.simps(4)
thf(fact_67_assertion_Oinject_I7_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,Y71: assertion_a_b_c_d,Y72: assertion_a_b_c_d] :
( ( ( imp_a_b_c_d @ X71 @ X72 )
= ( imp_a_b_c_d @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% assertion.inject(7)
thf(fact_68_assertion_Oinject_I6_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,Y61: assertion_a_b_c_d,Y62: assertion_a_b_c_d] :
( ( ( and_a_b_c_d @ X61 @ X62 )
= ( and_a_b_c_d @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% assertion.inject(6)
thf(fact_69_assertion_Oinject_I4_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,Y41: assertion_a_b_c_d,Y42: assertion_a_b_c_d] :
( ( ( wand_a_b_c_d @ X41 @ X42 )
= ( wand_a_b_c_d @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% assertion.inject(4)
thf(fact_70_assertion_Oinject_I5_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,Y51: assertion_a_b_c_d,Y52: assertion_a_b_c_d] :
( ( ( or_a_b_c_d @ X51 @ X52 )
= ( or_a_b_c_d @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% assertion.inject(5)
thf(fact_71_sadd__comm,axiom,
! [P: b,Q: b] :
( ( sadd @ P @ Q )
= ( sadd @ Q @ P ) ) ).
% sadd_comm
thf(fact_72_move__sum,axiom,
! [A: a,A1: a,A22: a,B: a,B1: a,B22: a,X: a,X1: a,X22: a] :
( ( ( some_a @ A )
= ( plus @ A1 @ A22 ) )
=> ( ( ( some_a @ B )
= ( plus @ B1 @ B22 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X22 )
= ( plus @ A22 @ B22 ) )
=> ( ( some_a @ X )
= ( plus @ X1 @ X22 ) ) ) ) ) ) ) ).
% move_sum
thf(fact_73_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_74_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_75_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_76_sum__both__larger,axiom,
! [X4: a,A5: a,B5: a,X: a,A: a,B: a] :
( ( ( some_a @ X4 )
= ( plus @ A5 @ B5 ) )
=> ( ( ( some_a @ X )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ A5 @ A )
=> ( ( pre_larger_a @ plus @ B5 @ B )
=> ( pre_larger_a @ plus @ X4 @ X ) ) ) ) ) ).
% sum_both_larger
thf(fact_77_larger__first__sum,axiom,
! [Y: a,A: a,B: a,X: a] :
( ( ( some_a @ Y )
= ( plus @ A @ B ) )
=> ( ( pre_larger_a @ plus @ X @ Y )
=> ? [A6: a] :
( ( ( some_a @ X )
= ( plus @ A6 @ B ) )
& ( pre_larger_a @ plus @ A6 @ A ) ) ) ) ).
% larger_first_sum
thf(fact_78_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_79_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_80_assertion_Oinject_I2_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,Y21: b,Y22: assertion_a_b_c_d] :
( ( ( mult_b_a_c_d @ X21 @ X222 )
= ( mult_b_a_c_d @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% assertion.inject(2)
thf(fact_81_assertion_Oinject_I11_J,axiom,
! [X12: assertion_a_b_c_d,Y12: assertion_a_b_c_d] :
( ( ( wildcard_a_b_c_d @ X12 )
= ( wildcard_a_b_c_d @ Y12 ) )
= ( X12 = Y12 ) ) ).
% assertion.inject(11)
thf(fact_82_assertion_Oinject_I3_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,Y31: assertion_a_b_c_d,Y32: assertion_a_b_c_d] :
( ( ( star_a_b_c_d @ X31 @ X32 )
= ( star_a_b_c_d @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% assertion.inject(3)
thf(fact_83_assertion_Oinject_I10_J,axiom,
! [X11: assertion_a_b_c_d,Y11: assertion_a_b_c_d] :
( ( ( bounded_a_b_c_d @ X11 )
= ( bounded_a_b_c_d @ Y11 ) )
= ( X11 = Y11 ) ) ).
% assertion.inject(10)
thf(fact_84_assertion_Oinject_I1_J,axiom,
! [X1: ( d > c ) > a > $o,Y1: ( d > c ) > a > $o] :
( ( ( sem_d_c_a_b @ X1 )
= ( sem_d_c_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% assertion.inject(1)
thf(fact_85_distrib__mult,axiom,
! [P: b,Q: b,X: a] :
( ( some_a @ ( mult @ ( sadd @ P @ Q ) @ X ) )
= ( plus @ ( mult @ P @ X ) @ ( mult @ Q @ X ) ) ) ).
% distrib_mult
thf(fact_86_assertion_Odistinct_I19_J,axiom,
! [X1: ( d > c ) > a > $o,X11: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(19)
thf(fact_87_logic_Oasso1,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus @ A @ B )
= ( some_a @ Ab ) )
& ( ( Plus @ B @ C )
= ( some_a @ Bc ) ) )
=> ( ( Plus @ Ab @ C )
= ( Plus @ A @ Bc ) ) ) ) ).
% logic.asso1
thf(fact_88_logic_Omove__sum,axiom,
! [Plus: 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,X: a,X1: a,X22: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus @ A1 @ A22 ) )
=> ( ( ( some_a @ B )
= ( Plus @ B1 @ B22 ) )
=> ( ( ( some_a @ X )
= ( Plus @ A @ B ) )
=> ( ( ( some_a @ X1 )
= ( Plus @ A1 @ B1 ) )
=> ( ( ( some_a @ X22 )
= ( Plus @ A22 @ B22 ) )
=> ( ( some_a @ X )
= ( Plus @ X1 @ X22 ) ) ) ) ) ) ) ) ).
% logic.move_sum
thf(fact_89_logic_Oplus__mult,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ A )
= ( Plus @ B @ C ) )
=> ( ( some_a @ ( Mult @ P @ A ) )
= ( Plus @ ( Mult @ P @ B ) @ ( Mult @ P @ C ) ) ) ) ) ).
% logic.plus_mult
thf(fact_90_logic_Osadd__comm,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Sadd @ P @ Q )
= ( Sadd @ Q @ P ) ) ) ).
% logic.sadd_comm
thf(fact_91_logic_Ocan__divide,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Mult @ P @ A )
= ( Mult @ P @ B ) )
=> ( A = B ) ) ) ).
% logic.can_divide
thf(fact_92_logic_Osmult__asso,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ ( Smult @ P @ Q ) @ R2 )
= ( Smult @ P @ ( Smult @ Q @ R2 ) ) ) ) ).
% logic.smult_asso
thf(fact_93_logic_Osmult__comm,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ Q )
= ( Smult @ Q @ P ) ) ) ).
% logic.smult_comm
thf(fact_94_logic_Ounique__inv,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( A
= ( Mult @ P @ B ) )
= ( B
= ( Mult @ ( Sinv @ P ) @ A ) ) ) ) ).
% logic.unique_inv
thf(fact_95_logic_Ocommutative,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Plus @ A @ B )
= ( Plus @ B @ A ) ) ) ).
% logic.commutative
thf(fact_96_logic_Odouble__mult,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ P @ ( Mult @ Q @ A ) )
= ( Mult @ ( Smult @ P @ Q ) @ A ) ) ) ).
% logic.double_mult
thf(fact_97_logic_Oone__neutral,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Mult @ One @ A )
= A ) ) ).
% logic.one_neutral
thf(fact_98_logic_Odistrib__mult,axiom,
! [Plus: 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,X: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( some_a @ ( Mult @ ( Sadd @ P @ Q ) @ X ) )
= ( Plus @ ( Mult @ P @ X ) @ ( Mult @ Q @ X ) ) ) ) ).
% logic.distrib_mult
thf(fact_99_logic_Osinv__inverse,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sinv @ P ) )
= One ) ) ).
% logic.sinv_inverse
thf(fact_100_logic_Osone__neutral,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ One @ P )
= P ) ) ).
% logic.sone_neutral
thf(fact_101_logic_Ocan__factorize,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ? [R: b] :
( Q
= ( Smult @ R @ P ) ) ) ).
% logic.can_factorize
thf(fact_102_logic_Osmult__distrib,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( Smult @ P @ ( Sadd @ Q @ R2 ) )
= ( Sadd @ ( Smult @ P @ Q ) @ ( Smult @ P @ R2 ) ) ) ) ).
% logic.smult_distrib
thf(fact_103_logic_Olarger__first__sum,axiom,
! [Plus: 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,X: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ Y )
= ( Plus @ A @ B ) )
=> ( ( pre_larger_a @ Plus @ X @ Y )
=> ? [A6: a] :
( ( ( some_a @ X )
= ( Plus @ A6 @ B ) )
& ( pre_larger_a @ Plus @ A6 @ A ) ) ) ) ) ).
% logic.larger_first_sum
thf(fact_104_logic_Osum__both__larger,axiom,
! [Plus: 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,A5: a,B5: a,X: a,A: a,B: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( some_a @ X4 )
= ( Plus @ A5 @ B5 ) )
=> ( ( ( some_a @ X )
= ( Plus @ A @ B ) )
=> ( ( pre_larger_a @ Plus @ A5 @ A )
=> ( ( pre_larger_a @ Plus @ B5 @ B )
=> ( pre_larger_a @ Plus @ X4 @ X ) ) ) ) ) ) ).
% logic.sum_both_larger
thf(fact_105_logic_Oasso3,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ~ ( pre_compatible_a @ Plus @ A @ B )
=> ( ( ( Plus @ B @ C )
= ( some_a @ Bc ) )
=> ~ ( pre_compatible_a @ Plus @ A @ Bc ) ) ) ) ).
% logic.asso3
thf(fact_106_logic_Oasso2,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( ( Plus @ A @ B )
= ( some_a @ Ab ) )
& ~ ( pre_compatible_a @ Plus @ B @ C ) )
=> ~ ( pre_compatible_a @ Plus @ Ab @ C ) ) ) ).
% logic.asso2
thf(fact_107_logic_Osat_Osimps_I4_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,B: ( d > c ) > a > $o] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( sem_d_c_a_b @ B ) )
= ( B @ S @ Sigma ) ) ) ).
% logic.sat.simps(4)
thf(fact_108_logic_Osat_Osimps_I11_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( bounded_a_b_c_d @ A2 ) )
= ( ( Valid @ Sigma )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(11)
thf(fact_109_assertion_Odistinct_I1_J,axiom,
! [X1: ( d > c ) > a > $o,X21: b,X222: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( mult_b_a_c_d @ X21 @ X222 ) ) ).
% assertion.distinct(1)
thf(fact_110_assertion_Odistinct_I21_J,axiom,
! [X1: ( d > c ) > a > $o,X12: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(21)
thf(fact_111_assertion_Odistinct_I3_J,axiom,
! [X1: ( d > c ) > a > $o,X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( star_a_b_c_d @ X31 @ X32 ) ) ).
% assertion.distinct(3)
thf(fact_112_assertion_Odistinct_I7_J,axiom,
! [X1: ( d > c ) > a > $o,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( or_a_b_c_d @ X51 @ X52 ) ) ).
% assertion.distinct(7)
thf(fact_113_assertion_Odistinct_I5_J,axiom,
! [X1: ( d > c ) > a > $o,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( wand_a_b_c_d @ X41 @ X42 ) ) ).
% assertion.distinct(5)
thf(fact_114_assertion_Odistinct_I9_J,axiom,
! [X1: ( d > c ) > a > $o,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( and_a_b_c_d @ X61 @ X62 ) ) ).
% assertion.distinct(9)
thf(fact_115_logic_Ocompatible__iff,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus @ A @ B )
= ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_iff
thf(fact_116_logic_Ocompatible__imp,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus @ A @ B )
=> ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.compatible_imp
thf(fact_117_logic_Ocompatible__multiples,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_compatible_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ Q @ B ) )
=> ( pre_compatible_a @ Plus @ A @ B ) ) ) ).
% logic.compatible_multiples
thf(fact_118_assertion_Odistinct_I39_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(39)
thf(fact_119_assertion_Odistinct_I11_J,axiom,
! [X1: ( d > c ) > a > $o,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(11)
thf(fact_120_logic_Ovalid__mono,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( Valid @ A )
& ( pre_larger_a @ Plus @ A @ B ) )
=> ( Valid @ B ) ) ) ).
% logic.valid_mono
thf(fact_121_logic_Olarger__same,axiom,
! [Plus: 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 @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus @ A @ B )
= ( pre_larger_a @ Plus @ ( Mult @ P @ A ) @ ( Mult @ P @ B ) ) ) ) ).
% logic.larger_same
thf(fact_122_assertion_Odistinct_I17_J,axiom,
! [X1: ( d > c ) > a > $o] :
( ( sem_d_c_a_b @ X1 )
!= pred_a_b_c_d ) ).
% assertion.distinct(17)
thf(fact_123_assertion_Odistinct_I131_J,axiom,
! [X11: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( bounded_a_b_c_d @ X11 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(131)
thf(fact_124_logic_Osat_Osimps_I2_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( star_a_b_c_d @ A2 @ B2 ) )
= ( ? [A4: a,B3: a] :
( ( ( some_a @ Sigma )
= ( Plus @ A4 @ B3 ) )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B3 @ S @ Delta @ B2 ) ) ) ) ) ).
% logic.sat.simps(2)
thf(fact_125_logic_Osat__wand,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,Sigma: a,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a,Sigma5: a] :
( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma5 )
= ( Plus @ Sigma @ A3 ) ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S @ Delta @ B2 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.sat_wand
thf(fact_126_logic_Osat_Osimps_I3_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wand_a_b_c_d @ A2 @ B2 ) )
= ( ! [A4: a,Sigma4: a] :
( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 )
& ( ( some_a @ Sigma4 )
= ( Plus @ Sigma @ A4 ) ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S @ Delta @ B2 ) ) ) ) ) ).
% logic.sat.simps(3)
thf(fact_127_assertion_Odistinct_I57_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(57)
thf(fact_128_pre__logic_Olarger__def,axiom,
( pre_larger_a
= ( ^ [Plus2: a > a > option_a,A4: a,B3: a] :
? [C2: a] :
( ( some_a @ A4 )
= ( Plus2 @ B3 @ C2 ) ) ) ) ).
% pre_logic.larger_def
thf(fact_129_assertion_Odistinct_I87_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(87)
thf(fact_130_assertion_Odistinct_I73_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(73)
thf(fact_131_assertion_Odistinct_I99_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(99)
thf(fact_132_assertion_Odistinct_I109_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( imp_a_b_c_d @ X71 @ X72 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(109)
thf(fact_133_assertion_Odistinct_I127_J,axiom,
! [X11: assertion_a_b_c_d] :
( pred_a_b_c_d
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(127)
thf(fact_134_logic__def,axiom,
( logic_a_b
= ( ^ [Plus2: a > a > option_a,Mult2: b > a > a,Smult2: b > b > b,Sadd2: b > b > b,Sinv2: b > b,One2: b,Valid2: a > $o] :
( ! [A4: a,B3: a] :
( ( Plus2 @ A4 @ B3 )
= ( Plus2 @ B3 @ A4 ) )
& ! [A4: a,B3: a,Ab2: a,C2: a,Bc2: a] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_a @ Ab2 ) )
& ( ( Plus2 @ B3 @ C2 )
= ( some_a @ Bc2 ) ) )
=> ( ( Plus2 @ Ab2 @ C2 )
= ( Plus2 @ A4 @ Bc2 ) ) )
& ! [A4: a,B3: a,Ab2: a,C2: a] :
( ( ( ( Plus2 @ A4 @ B3 )
= ( some_a @ Ab2 ) )
& ~ ( pre_compatible_a @ Plus2 @ B3 @ C2 ) )
=> ~ ( pre_compatible_a @ Plus2 @ Ab2 @ C2 ) )
& ! [P2: b] :
( ( Smult2 @ P2 @ ( Sinv2 @ P2 ) )
= One2 )
& ! [P2: b] :
( ( Smult2 @ One2 @ P2 )
= P2 )
& ! [P2: b,Q3: b] :
( ( Sadd2 @ P2 @ Q3 )
= ( Sadd2 @ Q3 @ P2 ) )
& ! [P2: b,Q3: b] :
( ( Smult2 @ P2 @ Q3 )
= ( Smult2 @ Q3 @ P2 ) )
& ! [P2: b,Q3: b,R3: b] :
( ( Smult2 @ P2 @ ( Sadd2 @ Q3 @ R3 ) )
= ( Sadd2 @ ( Smult2 @ P2 @ Q3 ) @ ( Smult2 @ P2 @ R3 ) ) )
& ! [P2: b,Q3: b,R3: b] :
( ( Smult2 @ ( Smult2 @ P2 @ Q3 ) @ R3 )
= ( Smult2 @ P2 @ ( Smult2 @ Q3 @ R3 ) ) )
& ! [P2: b,Q3: b,A4: a] :
( ( Mult2 @ P2 @ ( Mult2 @ Q3 @ A4 ) )
= ( Mult2 @ ( Smult2 @ P2 @ Q3 ) @ A4 ) )
& ! [A4: a,B3: a,C2: a,P2: b] :
( ( ( some_a @ A4 )
= ( Plus2 @ B3 @ C2 ) )
=> ( ( some_a @ ( Mult2 @ P2 @ A4 ) )
= ( Plus2 @ ( Mult2 @ P2 @ B3 ) @ ( Mult2 @ P2 @ C2 ) ) ) )
& ! [P2: b,Q3: b,X2: a] :
( ( some_a @ ( Mult2 @ ( Sadd2 @ P2 @ Q3 ) @ X2 ) )
= ( Plus2 @ ( Mult2 @ P2 @ X2 ) @ ( Mult2 @ Q3 @ X2 ) ) )
& ! [A4: a] :
( ( Mult2 @ One2 @ A4 )
= A4 )
& ! [A4: a,B3: a] :
( ( ( Valid2 @ A4 )
& ( pre_larger_a @ Plus2 @ A4 @ B3 ) )
=> ( Valid2 @ B3 ) ) ) ) ) ).
% logic_def
thf(fact_135_logic_Osat_Osimps_I1_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,P: b,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) )
= ( ? [A4: a] :
( ( Sigma
= ( Mult @ P @ A4 ) )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 ) ) ) ) ) ).
% logic.sat.simps(1)
thf(fact_136_logic_Osat__mult,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a] :
( ( Sigma
= ( Mult @ P @ A3 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( mult_b_a_c_d @ P @ A2 ) ) ) ) ).
% logic.sat_mult
thf(fact_137_logic_Osat_Osimps_I12_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( wildcard_a_b_c_d @ A2 ) )
= ( ? [A4: a,P2: b] :
( ( Sigma
= ( Mult @ P2 @ A4 ) )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 ) ) ) ) ) ).
% logic.sat.simps(12)
thf(fact_138_logic_Osat_Osimps_I6_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( or_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
| ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(6)
thf(fact_139_logic_Osat_Osimps_I7_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( and_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(7)
thf(fact_140_logic_Osat__imp,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.sat_imp
thf(fact_141_logic_Osat_Osimps_I5_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( imp_a_b_c_d @ A2 @ B2 ) )
= ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ B2 ) ) ) ) ).
% logic.sat.simps(5)
thf(fact_142_logic_Osat_Osimps_I10_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ pred_a_b_c_d )
= ( member_a @ Sigma @ ( Delta @ S ) ) ) ) ).
% logic.sat.simps(10)
thf(fact_143_logic_OentailsI,axiom,
! [Plus: 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: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ B2 ) )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ).
% logic.entailsI
thf(fact_144_logic_Oentails__def,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( ! [Sigma2: a,S2: d > c] :
( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ B2 ) ) ) ) ) ).
% logic.entails_def
thf(fact_145_logic_OequivalentI,axiom,
! [Plus: 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: ( d > c ) > set_a,A2: assertion_a_b_c_d,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ B2 ) )
=> ( ! [Sigma3: a,S3: d > c] :
( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ B2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ A2 ) )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 ) ) ) ) ).
% logic.equivalentI
thf(fact_146_logic_Ocompatible__smaller,axiom,
! [Plus: 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,X: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus @ A @ B )
=> ( ( pre_compatible_a @ Plus @ X @ A )
=> ( pre_compatible_a @ Plus @ X @ B ) ) ) ) ).
% logic.compatible_smaller
thf(fact_147_logic_Olarger__implies__compatible,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,X: a,Y: a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pre_larger_a @ Plus @ X @ Y )
=> ( pre_compatible_a @ Plus @ X @ Y ) ) ) ).
% logic.larger_implies_compatible
thf(fact_148_logic_Oequivalent__def,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
& ( entails_a_b_c_d @ Plus @ Mult @ Valid @ B2 @ Delta @ A2 ) ) ) ) ).
% logic.equivalent_def
thf(fact_149_logic_Opure__def,axiom,
! [Plus: 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_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( pure_a_b_c_d @ Plus @ Mult @ Valid @ A2 )
= ( ! [Sigma2: a,Sigma4: a,S2: d > c,Delta2: ( d > c ) > set_a,Delta3: ( d > c ) > set_a] :
( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta2 @ A2 )
= ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S2 @ Delta3 @ A2 ) ) ) ) ) ).
% logic.pure_def
thf(fact_150_logic_Odot__mult2,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) @ Delta @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) ) ) ).
% logic.dot_mult2
thf(fact_151_logic_Odot__mult1,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) ) ) ).
% logic.dot_mult1
thf(fact_152_logic_ODotPos,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d,Pi: b] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
= ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ Pi @ A2 ) @ Delta @ ( mult_b_a_c_d @ Pi @ B2 ) ) ) ) ).
% logic.DotPos
thf(fact_153_logic_OWildPos,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a,B2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( entails_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ Delta @ B2 )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ A2 ) @ Delta @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).
% logic.WildPos
thf(fact_154_logic_ODotDot,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( mult_b_a_c_d @ Q @ A2 ) ) @ Delta @ ( mult_b_a_c_d @ ( Smult @ P @ Q ) @ A2 ) ) ) ).
% logic.DotDot
thf(fact_155_logic_OWildWild,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).
% logic.WildWild
thf(fact_156_logic_OintuitionisticI,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [A3: a,B4: a] :
( ( ( pre_larger_a @ Plus @ A3 @ B4 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B4 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A3 @ S @ Delta @ A2 ) )
=> ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S @ Delta @ A2 ) ) ) ).
% logic.intuitionisticI
thf(fact_157_logic_Ointuitionistic__def,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,S: d > c,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( intuit4720955538653295669_b_d_c @ Plus @ Mult @ Valid @ S @ Delta @ A2 )
= ( ! [A4: a,B3: a] :
( ( ( pre_larger_a @ Plus @ A4 @ B3 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ B3 @ S @ Delta @ A2 ) )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ A4 @ S @ Delta @ A2 ) ) ) ) ) ).
% logic.intuitionistic_def
thf(fact_158_logic_Onot__in__fv__def,axiom,
! [Plus: 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_c_d,S4: set_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ S4 )
= ( ! [Sigma2: a,S2: d > c,Delta2: ( d > c ) > set_a,S5: d > c] :
( ( equal_outside_d_c @ S2 @ S5 @ S4 )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta2 @ A2 )
= ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S5 @ Delta2 @ A2 ) ) ) ) ) ) ).
% logic.not_in_fv_def
thf(fact_159_logic_Odot__star1,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.dot_star1
thf(fact_160_logic_Odot__star2,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.dot_star2
thf(fact_161_logic_Odot__or1,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.dot_or1
thf(fact_162_logic_Odot__or2,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.dot_or2
thf(fact_163_logic_Odot__wand1,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.dot_wand1
thf(fact_164_logic_Odot__wand2,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.dot_wand2
thf(fact_165_logic_Odot__and2,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.dot_and2
thf(fact_166_logic_Odot__and1,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.dot_and1
thf(fact_167_logic_OWildStar1,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( entails_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).
% logic.WildStar1
thf(fact_168_logic_ODotWild,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wildcard_a_b_c_d @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).
% logic.DotWild
thf(fact_169_logic_OWildDot,axiom,
! [Plus: 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_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) ) @ Delta @ ( wildcard_a_b_c_d @ A2 ) ) ) ).
% logic.WildDot
thf(fact_170_logic_ODotStar,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( star_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( star_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.DotStar
thf(fact_171_logic_ODotOr,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.DotOr
thf(fact_172_logic_ODotWand,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( mult_b_a_c_d @ P @ ( wand_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( wand_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) ) ) ).
% logic.DotWand
thf(fact_173_logic_ODotAnd,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( and_a_b_c_d @ ( mult_b_a_c_d @ P @ A2 ) @ ( mult_b_a_c_d @ P @ B2 ) ) @ Delta @ ( mult_b_a_c_d @ P @ ( and_a_b_c_d @ A2 @ B2 ) ) ) ) ).
% logic.DotAnd
thf(fact_174_logic_OWildOr,axiom,
! [Plus: 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_c_d,B2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( equivalent_a_b_c_d @ Plus @ Mult @ Valid @ ( wildcard_a_b_c_d @ ( or_a_b_c_d @ A2 @ B2 ) ) @ Delta @ ( or_a_b_c_d @ ( wildcard_a_b_c_d @ A2 ) @ ( wildcard_a_b_c_d @ B2 ) ) ) ) ).
% logic.WildOr
thf(fact_175_logic_Osat_Ocong,axiom,
sat_a_b_d_c = sat_a_b_d_c ).
% logic.sat.cong
thf(fact_176_logic_Oentails_Ocong,axiom,
entails_a_b_c_d = entails_a_b_c_d ).
% logic.entails.cong
thf(fact_177_pre__logic_Ocompatible_Ocong,axiom,
pre_compatible_a = pre_compatible_a ).
% pre_logic.compatible.cong
thf(fact_178_pre__logic_Olarger_Ocong,axiom,
pre_larger_a = pre_larger_a ).
% pre_logic.larger.cong
thf(fact_179_logic_Oequivalent_Ocong,axiom,
equivalent_a_b_c_d = equivalent_a_b_c_d ).
% logic.equivalent.cong
thf(fact_180_logic_Onot__in__fv_Ocong,axiom,
not_in_fv_a_b_c_d = not_in_fv_a_b_c_d ).
% logic.not_in_fv.cong
thf(fact_181_assertion_Odistinct_I41_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(41)
thf(fact_182_assertion_Odistinct_I23_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( star_a_b_c_d @ X31 @ X32 ) ) ).
% assertion.distinct(23)
thf(fact_183_assertion_Odistinct_I27_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( or_a_b_c_d @ X51 @ X52 ) ) ).
% assertion.distinct(27)
thf(fact_184_assertion_Odistinct_I25_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( wand_a_b_c_d @ X41 @ X42 ) ) ).
% assertion.distinct(25)
thf(fact_185_assertion_Odistinct_I29_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( and_a_b_c_d @ X61 @ X62 ) ) ).
% assertion.distinct(29)
thf(fact_186_assertion_Odistinct_I31_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(31)
thf(fact_187_assertion_Odistinct_I59_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(59)
thf(fact_188_assertion_Odistinct_I37_J,axiom,
! [X21: b,X222: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= pred_a_b_c_d ) ).
% assertion.distinct(37)
thf(fact_189_assertion_Odistinct_I89_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(89)
thf(fact_190_assertion_Odistinct_I75_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(75)
thf(fact_191_assertion_Odistinct_I101_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(101)
thf(fact_192_assertion_Odistinct_I45_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( or_a_b_c_d @ X51 @ X52 ) ) ).
% assertion.distinct(45)
thf(fact_193_assertion_Odistinct_I43_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( wand_a_b_c_d @ X41 @ X42 ) ) ).
% assertion.distinct(43)
thf(fact_194_assertion_Odistinct_I47_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( and_a_b_c_d @ X61 @ X62 ) ) ).
% assertion.distinct(47)
thf(fact_195_assertion_Odistinct_I61_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( or_a_b_c_d @ X51 @ X52 ) ) ).
% assertion.distinct(61)
thf(fact_196_assertion_Odistinct_I77_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( and_a_b_c_d @ X61 @ X62 ) ) ).
% assertion.distinct(77)
thf(fact_197_assertion_Odistinct_I111_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( imp_a_b_c_d @ X71 @ X72 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(111)
thf(fact_198_assertion_Odistinct_I63_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( and_a_b_c_d @ X61 @ X62 ) ) ).
% assertion.distinct(63)
thf(fact_199_assertion_Odistinct_I49_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(49)
thf(fact_200_assertion_Odistinct_I79_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(79)
thf(fact_201_assertion_Odistinct_I65_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(65)
thf(fact_202_assertion_Odistinct_I91_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= ( imp_a_b_c_d @ X71 @ X72 ) ) ).
% assertion.distinct(91)
thf(fact_203_assertion_Odistinct_I129_J,axiom,
! [X12: assertion_a_b_c_d] :
( pred_a_b_c_d
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(129)
thf(fact_204_assertion_Odistinct_I55_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= pred_a_b_c_d ) ).
% assertion.distinct(55)
thf(fact_205_assertion_Odistinct_I85_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= pred_a_b_c_d ) ).
% assertion.distinct(85)
thf(fact_206_assertion_Odistinct_I71_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= pred_a_b_c_d ) ).
% assertion.distinct(71)
thf(fact_207_assertion_Odistinct_I97_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= pred_a_b_c_d ) ).
% assertion.distinct(97)
thf(fact_208_assertion_Odistinct_I107_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d] :
( ( imp_a_b_c_d @ X71 @ X72 )
!= pred_a_b_c_d ) ).
% assertion.distinct(107)
thf(fact_209_hoare__triple__input,axiom,
! [P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ Q2 @ Delta )
= ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ ( bounded_a_b_c_d @ P3 ) @ C @ Q2 @ Delta ) ) ).
% hoare_triple_input
thf(fact_210_hoare__triple__output,axiom,
! [C: set_Pr7868159745199425715_a_d_c,P3: assertion_a_b_c_d,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( valid_command_a_d_c @ valid @ C )
=> ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ Q2 @ Delta )
= ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ ( bounded_a_b_c_d @ Q2 ) @ Delta ) ) ) ).
% hoare_triple_output
thf(fact_211_option_Oinject,axiom,
! [X22: a,Y2: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_212_compatible__def,axiom,
! [A: a,B: a] :
( ( pre_compatible_a @ plus @ A @ B )
= ( ( plus @ A @ B )
!= none_a ) ) ).
% compatible_def
thf(fact_213_assertion_Oexhaust,axiom,
! [Y: assertion_a_b_c_d] :
( ! [X13: ( d > c ) > a > $o] :
( Y
!= ( sem_d_c_a_b @ X13 ) )
=> ( ! [X212: b,X223: assertion_a_b_c_d] :
( Y
!= ( mult_b_a_c_d @ X212 @ X223 ) )
=> ( ! [X312: assertion_a_b_c_d,X322: assertion_a_b_c_d] :
( Y
!= ( star_a_b_c_d @ X312 @ X322 ) )
=> ( ! [X412: assertion_a_b_c_d,X422: assertion_a_b_c_d] :
( Y
!= ( wand_a_b_c_d @ X412 @ X422 ) )
=> ( ! [X512: assertion_a_b_c_d,X522: assertion_a_b_c_d] :
( Y
!= ( or_a_b_c_d @ X512 @ X522 ) )
=> ( ! [X612: assertion_a_b_c_d,X622: assertion_a_b_c_d] :
( Y
!= ( and_a_b_c_d @ X612 @ X622 ) )
=> ( ! [X712: assertion_a_b_c_d,X722: assertion_a_b_c_d] :
( Y
!= ( imp_a_b_c_d @ X712 @ X722 ) )
=> ( ! [X81: d,X82: assertion_a_b_c_d] :
( Y
!= ( exists_d_a_b_c @ X81 @ X82 ) )
=> ( ! [X91: d,X92: assertion_a_b_c_d] :
( Y
!= ( forall_d_a_b_c @ X91 @ X92 ) )
=> ( ( Y != pred_a_b_c_d )
=> ( ! [X112: assertion_a_b_c_d] :
( Y
!= ( bounded_a_b_c_d @ X112 ) )
=> ~ ! [X122: assertion_a_b_c_d] :
( Y
!= ( wildcard_a_b_c_d @ X122 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% assertion.exhaust
thf(fact_214_sat_Osimps_I8_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,X: d,A2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( exists_d_a_b_c @ X @ A2 ) )
= ( ? [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(8)
thf(fact_215_sat__forall,axiom,
! [Sigma: a,S: d > c,X: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ! [V2: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V2 ) @ Delta @ A2 )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X @ A2 ) ) ) ).
% sat_forall
thf(fact_216_sat_Osimps_I9_J,axiom,
! [Sigma: a,S: d > c,Delta: ( d > c ) > set_a,X: d,A2: assertion_a_b_c_d] :
( ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X @ A2 ) )
= ( ! [V: c] : ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V ) @ Delta @ A2 ) ) ) ).
% sat.simps(9)
thf(fact_217_logic_Ohoare__triple__input,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ Q2 @ Delta )
= ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ ( bounded_a_b_c_d @ P3 ) @ C @ Q2 @ Delta ) ) ) ).
% logic.hoare_triple_input
thf(fact_218_assertion_Oinject_I9_J,axiom,
! [X912: d,X922: assertion_a_b_c_d,Y91: d,Y92: assertion_a_b_c_d] :
( ( ( forall_d_a_b_c @ X912 @ X922 )
= ( forall_d_a_b_c @ Y91 @ Y92 ) )
= ( ( X912 = Y91 )
& ( X922 = Y92 ) ) ) ).
% assertion.inject(9)
thf(fact_219_assertion_Oinject_I8_J,axiom,
! [X812: d,X822: assertion_a_b_c_d,Y81: d,Y82: assertion_a_b_c_d] :
( ( ( exists_d_a_b_c @ X812 @ X822 )
= ( exists_d_a_b_c @ Y81 @ Y82 ) )
= ( ( X812 = Y81 )
& ( X822 = Y82 ) ) ) ).
% assertion.inject(8)
thf(fact_220_not__Some__eq,axiom,
! [X: option_a] :
( ( ! [Y3: a] :
( X
!= ( some_a @ Y3 ) ) )
= ( X = none_a ) ) ).
% not_Some_eq
thf(fact_221_not__None__eq,axiom,
! [X: option_a] :
( ( X != none_a )
= ( ? [Y3: a] :
( X
= ( some_a @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_222_assertion_Odistinct_I113_J,axiom,
! [X812: d,X822: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( exists_d_a_b_c @ X812 @ X822 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(113)
thf(fact_223_logic_Ovalid__hoare__triple_Ocong,axiom,
valid_8824771084768397689_b_c_d = valid_8824771084768397689_b_c_d ).
% logic.valid_hoare_triple.cong
thf(fact_224_combine__options__cases,axiom,
! [X: option_a,P3: option_a > option_a > $o,Y: option_a] :
( ( ( X = none_a )
=> ( P3 @ X @ Y ) )
=> ( ( ( Y = none_a )
=> ( P3 @ X @ Y ) )
=> ( ! [A3: a,B4: a] :
( ( X
= ( some_a @ A3 ) )
=> ( ( Y
= ( some_a @ B4 ) )
=> ( P3 @ X @ Y ) ) )
=> ( P3 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_225_split__option__all,axiom,
( ( ^ [P4: option_a > $o] :
! [X5: option_a] : ( P4 @ X5 ) )
= ( ^ [P5: option_a > $o] :
( ( P5 @ none_a )
& ! [X2: a] : ( P5 @ ( some_a @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_226_split__option__ex,axiom,
( ( ^ [P4: option_a > $o] :
? [X5: option_a] : ( P4 @ X5 ) )
= ( ^ [P5: option_a > $o] :
( ( P5 @ none_a )
| ? [X2: a] : ( P5 @ ( some_a @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_227_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X23: a] :
( Y
!= ( some_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_228_option_OdiscI,axiom,
! [Option: option_a,X22: a] :
( ( Option
= ( some_a @ X22 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_229_option_Odistinct_I1_J,axiom,
! [X22: a] :
( none_a
!= ( some_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_230_logic_Ovalid__command_Ocong,axiom,
valid_command_a_d_c = valid_command_a_d_c ).
% logic.valid_command.cong
thf(fact_231_logic_Osat_Osimps_I9_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,X: d,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X @ A2 ) )
= ( ! [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(9)
thf(fact_232_logic_Osat__forall,axiom,
! [Plus: 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,S: d > c,X: d,Delta: ( d > c ) > set_a,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [V2: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V2 ) @ Delta @ A2 )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( forall_d_a_b_c @ X @ A2 ) ) ) ) ).
% logic.sat_forall
thf(fact_233_logic_Osat_Osimps_I8_J,axiom,
! [Plus: 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,S: d > c,Delta: ( d > c ) > set_a,X: d,A2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ S @ Delta @ ( exists_d_a_b_c @ X @ A2 ) )
= ( ? [V: c] : ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma @ ( fun_upd_d_c @ S @ X @ V ) @ Delta @ A2 ) ) ) ) ).
% logic.sat.simps(8)
thf(fact_234_assertion_Odistinct_I35_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(35)
thf(fact_235_assertion_Odistinct_I33_J,axiom,
! [X21: b,X222: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( mult_b_a_c_d @ X21 @ X222 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(33)
thf(fact_236_assertion_Odistinct_I125_J,axiom,
! [X912: d,X922: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( forall_d_a_b_c @ X912 @ X922 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(125)
thf(fact_237_assertion_Odistinct_I119_J,axiom,
! [X812: d,X822: assertion_a_b_c_d,X12: assertion_a_b_c_d] :
( ( exists_d_a_b_c @ X812 @ X822 )
!= ( wildcard_a_b_c_d @ X12 ) ) ).
% assertion.distinct(119)
thf(fact_238_assertion_Odistinct_I53_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(53)
thf(fact_239_assertion_Odistinct_I83_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(83)
thf(fact_240_assertion_Odistinct_I69_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(69)
thf(fact_241_assertion_Odistinct_I51_J,axiom,
! [X31: assertion_a_b_c_d,X32: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( star_a_b_c_d @ X31 @ X32 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(51)
thf(fact_242_assertion_Odistinct_I95_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(95)
thf(fact_243_pre__logic_Ocompatible__def,axiom,
( pre_compatible_a
= ( ^ [Plus2: a > a > option_a,A4: a,B3: a] :
( ( Plus2 @ A4 @ B3 )
!= none_a ) ) ) ).
% pre_logic.compatible_def
thf(fact_244_assertion_Odistinct_I81_J,axiom,
! [X51: assertion_a_b_c_d,X52: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( or_a_b_c_d @ X51 @ X52 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(81)
thf(fact_245_assertion_Odistinct_I67_J,axiom,
! [X41: assertion_a_b_c_d,X42: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( wand_a_b_c_d @ X41 @ X42 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(67)
thf(fact_246_assertion_Odistinct_I93_J,axiom,
! [X61: assertion_a_b_c_d,X62: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( and_a_b_c_d @ X61 @ X62 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(93)
thf(fact_247_assertion_Odistinct_I123_J,axiom,
! [X912: d,X922: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( forall_d_a_b_c @ X912 @ X922 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(123)
thf(fact_248_assertion_Odistinct_I105_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X912: d,X922: assertion_a_b_c_d] :
( ( imp_a_b_c_d @ X71 @ X72 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(105)
thf(fact_249_assertion_Odistinct_I117_J,axiom,
! [X812: d,X822: assertion_a_b_c_d,X11: assertion_a_b_c_d] :
( ( exists_d_a_b_c @ X812 @ X822 )
!= ( bounded_a_b_c_d @ X11 ) ) ).
% assertion.distinct(117)
thf(fact_250_assertion_Odistinct_I15_J,axiom,
! [X1: ( d > c ) > a > $o,X912: d,X922: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( forall_d_a_b_c @ X912 @ X922 ) ) ).
% assertion.distinct(15)
thf(fact_251_assertion_Odistinct_I103_J,axiom,
! [X71: assertion_a_b_c_d,X72: assertion_a_b_c_d,X812: d,X822: assertion_a_b_c_d] :
( ( imp_a_b_c_d @ X71 @ X72 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(103)
thf(fact_252_assertion_Odistinct_I13_J,axiom,
! [X1: ( d > c ) > a > $o,X812: d,X822: assertion_a_b_c_d] :
( ( sem_d_c_a_b @ X1 )
!= ( exists_d_a_b_c @ X812 @ X822 ) ) ).
% assertion.distinct(13)
thf(fact_253_assertion_Odistinct_I121_J,axiom,
! [X912: d,X922: assertion_a_b_c_d] :
( ( forall_d_a_b_c @ X912 @ X922 )
!= pred_a_b_c_d ) ).
% assertion.distinct(121)
thf(fact_254_assertion_Odistinct_I115_J,axiom,
! [X812: d,X822: assertion_a_b_c_d] :
( ( exists_d_a_b_c @ X812 @ X822 )
!= pred_a_b_c_d ) ).
% assertion.distinct(115)
thf(fact_255_logic_Ohoare__triple__output,axiom,
! [Plus: 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_Pr7868159745199425715_a_d_c,P3: assertion_a_b_c_d,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_d_c @ Valid @ C )
=> ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ Q2 @ Delta )
= ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ ( bounded_a_b_c_d @ Q2 ) @ Delta ) ) ) ) ).
% logic.hoare_triple_output
thf(fact_256_frame__rule,axiom,
! [C: set_Pr7868159745199425715_a_d_c,P3: assertion_a_b_c_d,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,R4: assertion_a_b_c_d] :
( ( valid_command_a_d_c @ valid @ C )
=> ( ( safety7280469885071620222_a_d_c @ plus @ valid @ C )
=> ( ( frame_property_a_d_c @ plus @ valid @ C )
=> ( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ Q2 @ Delta )
=> ( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ R4 @ ( modified_a_d_c @ C ) )
=> ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ ( star_a_b_c_d @ P3 @ R4 ) @ C @ ( star_a_b_c_d @ Q2 @ R4 ) @ Delta ) ) ) ) ) ) ).
% frame_rule
thf(fact_257_logic_Oframe__rule,axiom,
! [Plus: 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_Pr7868159745199425715_a_d_c,P3: assertion_a_b_c_d,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a,R4: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_d_c @ Valid @ C )
=> ( ( safety7280469885071620222_a_d_c @ Plus @ Valid @ C )
=> ( ( frame_property_a_d_c @ Plus @ Valid @ C )
=> ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ Q2 @ Delta )
=> ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ R4 @ ( modified_a_d_c @ C ) )
=> ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ ( star_a_b_c_d @ P3 @ R4 ) @ C @ ( star_a_b_c_d @ Q2 @ R4 ) @ Delta ) ) ) ) ) ) ) ).
% logic.frame_rule
thf(fact_258_not__in__fv__mod,axiom,
! [A2: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S: d > c,Sigma6: a,S6: d > c,X: a,Delta: ( d > c ) > set_a] :
( ( not_in_fv_a_b_c_d @ plus @ mult @ valid @ A2 @ ( modified_a_d_c @ C ) )
=> ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_d_c @ plus @ mult @ valid @ X @ S @ Delta @ A2 )
= ( sat_a_b_d_c @ plus @ mult @ valid @ X @ S6 @ Delta @ A2 ) ) ) ) ).
% not_in_fv_mod
thf(fact_259_valid__command__def,axiom,
! [C: set_Pr7868159745199425715_a_d_c] :
( ( valid_command_a_d_c @ valid @ C )
= ( ! [A4: a,B3: a,Sa: d > c,Sb: d > c] :
( ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ A4 @ Sa ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ B3 @ Sb ) ) ) @ C )
& ( valid @ A4 ) )
=> ( valid @ B3 ) ) ) ) ).
% valid_command_def
thf(fact_260_logic_Osafety__monotonicity_Ocong,axiom,
safety7280469885071620222_a_d_c = safety7280469885071620222_a_d_c ).
% logic.safety_monotonicity.cong
thf(fact_261_logic_Oframe__property_Ocong,axiom,
frame_property_a_d_c = frame_property_a_d_c ).
% logic.frame_property.cong
thf(fact_262_logic_Ovalid__command__def,axiom,
! [Plus: 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_Pr7868159745199425715_a_d_c] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_command_a_d_c @ Valid @ C )
= ( ! [A4: a,B3: a,Sa: d > c,Sb: d > c] :
( ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ A4 @ Sa ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ B3 @ Sb ) ) ) @ C )
& ( Valid @ A4 ) )
=> ( Valid @ B3 ) ) ) ) ) ).
% logic.valid_command_def
thf(fact_263_logic_Onot__in__fv__mod,axiom,
! [Plus: 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_c_d,C: set_Pr7868159745199425715_a_d_c,Sigma: a,S: d > c,Sigma6: a,S6: d > c,X: a,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( not_in_fv_a_b_c_d @ Plus @ Mult @ Valid @ A2 @ ( modified_a_d_c @ C ) )
=> ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma @ S ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma6 @ S6 ) ) ) @ C )
=> ( ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X @ S @ Delta @ A2 )
= ( sat_a_b_d_c @ Plus @ Mult @ Valid @ X @ S6 @ Delta @ A2 ) ) ) ) ) ).
% logic.not_in_fv_mod
thf(fact_264_valid__hoare__triple__def,axiom,
! [P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ Q2 @ Delta )
= ( ! [Sigma2: a,S2: d > c] :
( ( ( valid @ Sigma2 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma2 @ S2 @ Delta @ P3 ) )
=> ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) )
& ! [Sigma4: a,S5: d > c] :
( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) ) @ C )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma4 @ S5 @ Delta @ Q2 ) ) ) ) ) ) ).
% valid_hoare_triple_def
thf(fact_265_valid__hoare__tripleI,axiom,
! [Delta: ( d > c ) > set_a,P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d] :
( ! [Sigma3: a,S3: d > c] :
( ( ( valid @ Sigma3 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ P3 ) )
=> ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma3 @ S3 ) ) )
=> ( ! [Sigma3: a,S3: d > c,Sigma5: a,S7: d > c] :
( ( ( valid @ Sigma3 )
& ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma3 @ S3 @ Delta @ P3 ) )
=> ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S3 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S7 ) ) ) @ C )
=> ( sat_a_b_d_c @ plus @ mult @ valid @ Sigma5 @ S7 @ Delta @ Q2 ) ) )
=> ( valid_8824771084768397689_b_c_d @ plus @ mult @ valid @ P3 @ C @ Q2 @ Delta ) ) ) ).
% valid_hoare_tripleI
thf(fact_266_frame__property__def,axiom,
! [C: set_Pr7868159745199425715_a_d_c] :
( ( frame_property_a_d_c @ plus @ valid @ C )
= ( ! [Sigma2: a,Sigma_0: a,R3: a,Sigma4: a,S2: d > c,S5: d > c] :
( ( ( valid @ Sigma2 )
& ( valid @ Sigma4 )
& ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma_0 @ S2 ) )
& ( ( some_a @ Sigma2 )
= ( plus @ Sigma_0 @ R3 ) )
& ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) ) @ C ) )
=> ? [Sigma_02: a] :
( ( ( some_a @ Sigma4 )
= ( plus @ Sigma_02 @ R3 ) )
& ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma_0 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma_02 @ S5 ) ) ) @ C ) ) ) ) ) ).
% frame_property_def
thf(fact_267_safety__monotonicity__def,axiom,
! [C: set_Pr7868159745199425715_a_d_c] :
( ( safety7280469885071620222_a_d_c @ plus @ valid @ C )
= ( ! [Sigma2: a,Sigma4: a,S2: d > c] :
( ( ( valid @ Sigma4 )
& ( pre_larger_a @ plus @ Sigma4 @ Sigma2 )
& ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) ) )
=> ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S2 ) ) ) ) ) ).
% safety_monotonicity_def
thf(fact_268_logic_Oframe__property__def,axiom,
! [Plus: 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_Pr7868159745199425715_a_d_c] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( frame_property_a_d_c @ Plus @ Valid @ C )
= ( ! [Sigma2: a,Sigma_0: a,R3: a,Sigma4: a,S2: d > c,S5: d > c] :
( ( ( Valid @ Sigma2 )
& ( Valid @ Sigma4 )
& ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma_0 @ S2 ) )
& ( ( some_a @ Sigma2 )
= ( Plus @ Sigma_0 @ R3 ) )
& ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) ) @ C ) )
=> ? [Sigma_02: a] :
( ( ( some_a @ Sigma4 )
= ( Plus @ Sigma_02 @ R3 ) )
& ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma_0 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma_02 @ S5 ) ) ) @ C ) ) ) ) ) ) ).
% logic.frame_property_def
thf(fact_269_logic_Ovalid__hoare__tripleI,axiom,
! [Plus: 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: ( d > c ) > set_a,P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ! [Sigma3: a,S3: d > c] :
( ( ( Valid @ Sigma3 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ P3 ) )
=> ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma3 @ S3 ) ) )
=> ( ! [Sigma3: a,S3: d > c,Sigma5: a,S7: d > c] :
( ( ( Valid @ Sigma3 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma3 @ S3 @ Delta @ P3 ) )
=> ( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma3 @ S3 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma5 @ S7 ) ) ) @ C )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma5 @ S7 @ Delta @ Q2 ) ) )
=> ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ Q2 @ Delta ) ) ) ) ).
% logic.valid_hoare_tripleI
thf(fact_270_logic_Osafety__monotonicity__def,axiom,
! [Plus: 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_Pr7868159745199425715_a_d_c] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( safety7280469885071620222_a_d_c @ Plus @ Valid @ C )
= ( ! [Sigma2: a,Sigma4: a,S2: d > c] :
( ( ( Valid @ Sigma4 )
& ( pre_larger_a @ Plus @ Sigma4 @ Sigma2 )
& ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) ) )
=> ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma4 @ S2 ) ) ) ) ) ) ).
% logic.safety_monotonicity_def
thf(fact_271_logic_Ovalid__hoare__triple__def,axiom,
! [Plus: a > a > option_a,Mult: b > a > a,Smult: b > b > b,Sadd: b > b > b,Sinv: b > b,One: b,Valid: a > $o,P3: assertion_a_b_c_d,C: set_Pr7868159745199425715_a_d_c,Q2: assertion_a_b_c_d,Delta: ( d > c ) > set_a] :
( ( logic_a_b @ Plus @ Mult @ Smult @ Sadd @ Sinv @ One @ Valid )
=> ( ( valid_8824771084768397689_b_c_d @ Plus @ Mult @ Valid @ P3 @ C @ Q2 @ Delta )
= ( ! [Sigma2: a,S2: d > c] :
( ( ( Valid @ Sigma2 )
& ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma2 @ S2 @ Delta @ P3 ) )
=> ( ( safe_a_d_c @ C @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) )
& ! [Sigma4: a,S5: d > c] :
( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) ) @ C )
=> ( sat_a_b_d_c @ Plus @ Mult @ Valid @ Sigma4 @ S5 @ Delta @ Q2 ) ) ) ) ) ) ) ).
% logic.valid_hoare_triple_def
thf(fact_272_option_Ocollapse,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( ( some_a @ ( the_a @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_273_option_Osel,axiom,
! [X22: a] :
( ( the_a @ ( some_a @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_274_option_Oexpand,axiom,
! [Option: option_a,Option2: option_a] :
( ( ( Option = none_a )
= ( Option2 = none_a ) )
=> ( ( ( Option != none_a )
=> ( ( Option2 != none_a )
=> ( ( the_a @ Option )
= ( the_a @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_275_option_Oexhaust__sel,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( Option
= ( some_a @ ( the_a @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_276_option_Osplit__sel,axiom,
! [P3: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( P3 @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( ( ( Option = none_a )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_a @ ( the_a @ Option ) ) )
=> ( P3 @ ( F2 @ ( the_a @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_277_option_Osplit__sel__asm,axiom,
! [P3: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( P3 @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( ~ ( ( ( Option = none_a )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_a @ ( the_a @ Option ) ) )
& ~ ( P3 @ ( F2 @ ( the_a @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_278_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_279_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_280_option_Ocase__eq__if,axiom,
( case_option_o_a
= ( ^ [F12: $o,F22: a > $o,Option3: option_a] :
( ( ( Option3 = none_a )
=> F12 )
& ( ( Option3 != none_a )
=> ( F22 @ ( the_a @ Option3 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_281_option_Oset__sel,axiom,
! [A: option_option_a] :
( ( A != none_option_a )
=> ( member_option_a @ ( the_option_a @ A ) @ ( set_option_option_a2 @ A ) ) ) ).
% option.set_sel
thf(fact_282_option_Oset__sel,axiom,
! [A: option_a] :
( ( A != none_a )
=> ( member_a @ ( the_a @ A ) @ ( set_option_a3 @ A ) ) ) ).
% option.set_sel
thf(fact_283_bind__eq__None__conv,axiom,
! [A: option_a,F: a > option_a] :
( ( ( bind_a_a @ A @ F )
= none_a )
= ( ( A = none_a )
| ( ( F @ ( the_a @ A ) )
= none_a ) ) ) ).
% bind_eq_None_conv
thf(fact_284_elem__set,axiom,
! [X: option_a,Xo: option_option_a] :
( ( member_option_a @ X @ ( set_option_option_a2 @ Xo ) )
= ( Xo
= ( some_option_a @ X ) ) ) ).
% elem_set
thf(fact_285_elem__set,axiom,
! [X: a,Xo: option_a] :
( ( member_a @ X @ ( set_option_a3 @ Xo ) )
= ( Xo
= ( some_a @ X ) ) ) ).
% elem_set
thf(fact_286_bind__runit,axiom,
! [X: option_a] :
( ( bind_a_a @ X @ some_a )
= X ) ).
% bind_runit
thf(fact_287_fun__upd__None__if__notin__dom,axiom,
! [K: a,M: a > option_a] :
( ~ ( member_a @ K @ ( dom_a_a @ M ) )
=> ( ( fun_upd_a_option_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_288_fun__upd__None__if__notin__dom,axiom,
! [K: option_a,M: option_a > option_a] :
( ~ ( member_option_a @ K @ ( dom_option_a_a @ M ) )
=> ( ( fun_up1079276522633388797tion_a @ M @ K @ none_a )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_289_bind__option__cong__code,axiom,
! [X: option_a,Y: option_a,F: a > option_a] :
( ( X = Y )
=> ( ( bind_a_a @ X @ F )
= ( bind_a_a @ Y @ F ) ) ) ).
% bind_option_cong_code
thf(fact_290_bind__option__cong,axiom,
! [X: option_a,Y: option_a,F: a > option_a,G: a > option_a] :
( ( X = Y )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_option_a3 @ Y ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( bind_a_a @ X @ F )
= ( bind_a_a @ Y @ G ) ) ) ) ).
% bind_option_cong
thf(fact_291_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_292_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_293_domD,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
=> ? [B4: a] :
( ( M @ A )
= ( some_a @ B4 ) ) ) ).
% domD
thf(fact_294_domD,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
=> ? [B4: a] :
( ( M @ A )
= ( some_a @ B4 ) ) ) ).
% domD
thf(fact_295_domIff,axiom,
! [A: a,M: a > option_a] :
( ( member_a @ A @ ( dom_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_296_domIff,axiom,
! [A: option_a,M: option_a > option_a] :
( ( member_option_a @ A @ ( dom_option_a_a @ M ) )
= ( ( M @ A )
!= none_a ) ) ).
% domIff
thf(fact_297_bind_Obind__lunit,axiom,
! [X: a,F: a > option_a] :
( ( bind_a_a @ ( some_a @ X ) @ F )
= ( F @ X ) ) ).
% bind.bind_lunit
thf(fact_298_Option_Obind__cong,axiom,
! [X: option_a,Y: option_a,F: a > option_a,G: a > option_a] :
( ( X = Y )
=> ( ! [A3: a] :
( ( Y
= ( some_a @ A3 ) )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( bind_a_a @ X @ F )
= ( bind_a_a @ Y @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_299_bind__eq__Some__conv,axiom,
! [F: option_a,G: a > option_a,X: a] :
( ( ( bind_a_a @ F @ G )
= ( some_a @ X ) )
= ( ? [Y3: a] :
( ( F
= ( some_a @ Y3 ) )
& ( ( G @ Y3 )
= ( some_a @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_300_bind_Obind__lzero,axiom,
! [F: a > option_a] :
( ( bind_a_a @ none_a @ F )
= none_a ) ).
% bind.bind_lzero
thf(fact_301_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_302_option_Oset__cases,axiom,
! [E: a,A: option_a] :
( ( member_a @ E @ ( set_option_a3 @ A ) )
=> ( A
= ( some_a @ E ) ) ) ).
% option.set_cases
thf(fact_303_option_Oset__intros,axiom,
! [X22: option_a] : ( member_option_a @ X22 @ ( set_option_option_a2 @ ( some_option_a @ X22 ) ) ) ).
% option.set_intros
thf(fact_304_option_Oset__intros,axiom,
! [X22: a] : ( member_a @ X22 @ ( set_option_a3 @ ( some_a @ X22 ) ) ) ).
% option.set_intros
thf(fact_305_ospec,axiom,
! [A2: option_a,P3: a > $o,X: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_option_a3 @ A2 ) )
=> ( P3 @ X3 ) )
=> ( ( A2
= ( some_a @ X ) )
=> ( P3 @ X ) ) ) ).
% ospec
thf(fact_306_option_Osimps_I5_J,axiom,
! [F1: $o,F2: a > $o,X22: a] :
( ( case_option_o_a @ F1 @ F2 @ ( some_a @ X22 ) )
= ( F2 @ X22 ) ) ).
% option.simps(5)
thf(fact_307_option_Osimps_I4_J,axiom,
! [F1: $o,F2: a > $o] :
( ( case_option_o_a @ F1 @ F2 @ none_a )
= F1 ) ).
% option.simps(4)
thf(fact_308_bind__split__asm,axiom,
! [P3: option_a > $o,M: option_a,F: a > option_a] :
( ( P3 @ ( bind_a_a @ M @ F ) )
= ( ~ ( ( ( M = none_a )
& ~ ( P3 @ none_a ) )
| ? [X2: a] :
( ( M
= ( some_a @ X2 ) )
& ~ ( P3 @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_309_bind__split,axiom,
! [P3: option_a > $o,M: option_a,F: a > option_a] :
( ( P3 @ ( bind_a_a @ M @ F ) )
= ( ( ( M = none_a )
=> ( P3 @ none_a ) )
& ! [V: a] :
( ( M
= ( some_a @ V ) )
=> ( P3 @ ( F @ V ) ) ) ) ) ).
% bind_split
thf(fact_310_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_a @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_311_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_312_set__empty__eq,axiom,
! [Xo: option_a] :
( ( ( set_option_a3 @ Xo )
= bot_bot_set_a )
= ( Xo = none_a ) ) ).
% set_empty_eq
thf(fact_313_dom__eq__empty__conv,axiom,
! [F: a > option_a] :
( ( ( dom_a_a @ F )
= bot_bot_set_a )
= ( F
= ( ^ [X2: a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_314_dom__eq__empty__conv,axiom,
! [F: option_a > option_a] :
( ( ( dom_option_a_a @ F )
= bot_bot_set_option_a )
= ( F
= ( ^ [X2: option_a] : none_a ) ) ) ).
% dom_eq_empty_conv
thf(fact_315_is__none__code_I2_J,axiom,
! [X: a] :
~ ( is_none_a @ ( some_a @ X ) ) ).
% is_none_code(2)
thf(fact_316_is__none__code_I1_J,axiom,
is_none_a @ none_a ).
% is_none_code(1)
thf(fact_317_is__none__simps_I2_J,axiom,
! [X: a] :
~ ( is_none_a @ ( some_a @ X ) ) ).
% is_none_simps(2)
thf(fact_318_Option_Ois__none__def,axiom,
( is_none_a
= ( ^ [X2: option_a] : ( X2 = none_a ) ) ) ).
% Option.is_none_def
thf(fact_319_is__none__simps_I1_J,axiom,
is_none_a @ none_a ).
% is_none_simps(1)
thf(fact_320_option_Osimps_I14_J,axiom,
( ( set_option_option_a2 @ none_option_a )
= bot_bot_set_option_a ) ).
% option.simps(14)
thf(fact_321_option_Osimps_I14_J,axiom,
( ( set_option_a3 @ none_a )
= bot_bot_set_a ) ).
% option.simps(14)
thf(fact_322_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_323_option_Osimps_I15_J,axiom,
! [X22: a] :
( ( set_option_a3 @ ( some_a @ X22 ) )
= ( insert_a @ X22 @ bot_bot_set_a ) ) ).
% option.simps(15)
thf(fact_324_restrict__out,axiom,
! [X: a,A2: set_a,M: a > option_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( restrict_map_a_a @ M @ A2 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_325_restrict__out,axiom,
! [X: option_a,A2: set_option_a,M: option_a > option_a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( restri3984065703976872170on_a_a @ M @ A2 @ X )
= none_a ) ) ).
% restrict_out
thf(fact_326_restrict__map__to__empty,axiom,
! [M: a > option_a] :
( ( restrict_map_a_a @ M @ bot_bot_set_a )
= ( ^ [X2: a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_327_restrict__map__to__empty,axiom,
! [M: option_a > option_a] :
( ( restri3984065703976872170on_a_a @ M @ bot_bot_set_option_a )
= ( ^ [X2: option_a] : none_a ) ) ).
% restrict_map_to_empty
thf(fact_328_restrict__map__def,axiom,
( restrict_map_a_a
= ( ^ [M2: a > option_a,A7: set_a,X2: a] : ( if_option_a @ ( member_a @ X2 @ A7 ) @ ( M2 @ X2 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_329_restrict__map__def,axiom,
( restri3984065703976872170on_a_a
= ( ^ [M2: option_a > option_a,A7: set_option_a,X2: option_a] : ( if_option_a @ ( member_option_a @ X2 @ A7 ) @ ( M2 @ X2 ) @ none_a ) ) ) ).
% restrict_map_def
thf(fact_330_insert__dom,axiom,
! [F: a > option_a,X: a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_a @ X @ ( dom_a_a @ F ) )
= ( dom_a_a @ F ) ) ) ).
% insert_dom
thf(fact_331_insert__dom,axiom,
! [F: option_a > option_a,X: option_a,Y: a] :
( ( ( F @ X )
= ( some_a @ Y ) )
=> ( ( insert_option_a @ X @ ( dom_option_a_a @ F ) )
= ( dom_option_a_a @ F ) ) ) ).
% insert_dom
thf(fact_332_restrict__upd__same,axiom,
! [M: a > option_a,X: a,Y: a] :
( ( restrict_map_a_a @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( restrict_map_a_a @ M @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% restrict_upd_same
thf(fact_333_restrict__upd__same,axiom,
! [M: option_a > option_a,X: option_a,Y: a] :
( ( restri3984065703976872170on_a_a @ ( fun_up1079276522633388797tion_a @ M @ X @ ( some_a @ Y ) ) @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( restri3984065703976872170on_a_a @ M @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ).
% restrict_upd_same
thf(fact_334_fun__upd__None__restrict,axiom,
! [X: a,D: set_a,M: a > option_a] :
( ( ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
& ( ~ ( member_a @ X @ D )
=> ( ( fun_upd_a_option_a @ ( restrict_map_a_a @ M @ D ) @ X @ none_a )
= ( restrict_map_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_335_fun__upd__None__restrict,axiom,
! [X: option_a,D: set_option_a,M: option_a > option_a] :
( ( ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ ( minus_1574173051537231627tion_a @ D @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) )
& ( ~ ( member_option_a @ X @ D )
=> ( ( fun_up1079276522633388797tion_a @ ( restri3984065703976872170on_a_a @ M @ D ) @ X @ none_a )
= ( restri3984065703976872170on_a_a @ M @ D ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_336_dom__fun__upd,axiom,
! [Y: option_a,F: a > option_a,X: a] :
( ( ( Y = none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_a_a @ ( fun_upd_a_option_a @ F @ X @ Y ) )
= ( insert_a @ X @ ( dom_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_337_dom__fun__upd,axiom,
! [Y: option_a,F: option_a > option_a,X: option_a] :
( ( ( Y = none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) )
& ( ( Y != none_a )
=> ( ( dom_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) )
= ( insert_option_a @ X @ ( dom_option_a_a @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_338_dom__minus,axiom,
! [F: a > option_a,X: a,A2: set_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_minus_set_a @ ( dom_a_a @ F ) @ ( insert_a @ X @ A2 ) )
= ( minus_minus_set_a @ ( dom_a_a @ F ) @ A2 ) ) ) ).
% dom_minus
thf(fact_339_dom__minus,axiom,
! [F: option_a > option_a,X: option_a,A2: set_option_a] :
( ( ( F @ X )
= none_a )
=> ( ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ ( insert_option_a @ X @ A2 ) )
= ( minus_1574173051537231627tion_a @ ( dom_option_a_a @ F ) @ A2 ) ) ) ).
% dom_minus
thf(fact_340_restrict__complement__singleton__eq,axiom,
! [F: a > option_a,X: a] :
( ( restrict_map_a_a @ F @ ( uminus_uminus_set_a @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( fun_upd_a_option_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_341_restrict__complement__singleton__eq,axiom,
! [F: option_a > option_a,X: option_a] :
( ( restri3984065703976872170on_a_a @ F @ ( uminus6205308855922866075tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) )
= ( fun_up1079276522633388797tion_a @ F @ X @ none_a ) ) ).
% restrict_complement_singleton_eq
thf(fact_342_ran__map__upd__Some,axiom,
! [M: a > option_a,X: a,Y: a,Z2: a] :
( ( ( M @ X )
= ( 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 @ X @ ( 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_343_these__insert__Some,axiom,
! [X: option_a,A2: set_option_option_a] :
( ( these_option_a @ ( insert605063979879581146tion_a @ ( some_option_a @ X ) @ A2 ) )
= ( insert_option_a @ X @ ( these_option_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_344_these__insert__Some,axiom,
! [X: a,A2: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X ) @ A2 ) )
= ( insert_a @ X @ ( these_a @ A2 ) ) ) ).
% these_insert_Some
thf(fact_345_Field__insert,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) )
= ( sup_sup_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) ) ) ).
% Field_insert
thf(fact_346_Field__insert,axiom,
! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a] :
( ( field_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) )
= ( sup_sup_set_option_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) ) ) ).
% Field_insert
thf(fact_347_inj__on__empty,axiom,
! [F: a > option_a] : ( inj_on_a_option_a @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_348_these__empty,axiom,
( ( these_option_a @ bot_bo4163488203964334806tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_349_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_350_inj__on__map__add__dom,axiom,
! [M: a > option_a,M3: a > option_a] :
( ( inj_on_a_option_a @ ( map_add_a_a @ M @ M3 ) @ ( dom_a_a @ M3 ) )
= ( inj_on_a_option_a @ M3 @ ( dom_a_a @ M3 ) ) ) ).
% inj_on_map_add_dom
thf(fact_351_these__insert__None,axiom,
! [A2: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A2 ) )
= ( these_a @ A2 ) ) ).
% these_insert_None
thf(fact_352_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_353_inj__Some,axiom,
! [A2: set_a] : ( inj_on_a_option_a @ some_a @ A2 ) ).
% inj_Some
thf(fact_354_FieldI1,axiom,
! [I: a,J: a,R4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
=> ( member_a @ I @ ( field_a @ R4 ) ) ) ).
% FieldI1
thf(fact_355_FieldI1,axiom,
! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
=> ( member_option_a @ I @ ( field_option_a @ R4 ) ) ) ).
% FieldI1
thf(fact_356_FieldI2,axiom,
! [I: a,J: a,R4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
=> ( member_a @ J @ ( field_a @ R4 ) ) ) ).
% FieldI2
thf(fact_357_FieldI2,axiom,
! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
=> ( member_option_a @ J @ ( field_option_a @ R4 ) ) ) ).
% FieldI2
thf(fact_358_inj__on__inverseI,axiom,
! [A2: set_a,G: option_a > a,F: a > option_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_359_inj__on__contraD,axiom,
! [F: a > option_a,A2: set_a,X: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_360_inj__on__eq__iff,axiom,
! [F: a > option_a,A2: set_a,X: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_361_inj__on__cong,axiom,
! [A2: set_a,F: a > option_a,G: a > option_a] :
( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_a_option_a @ F @ A2 )
= ( inj_on_a_option_a @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_362_inj__on__def,axiom,
( inj_on_a_option_a
= ( ^ [F3: a > option_a,A7: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A7 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_363_inj__onI,axiom,
! [A2: set_a,F: a > option_a] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_a_option_a @ F @ A2 ) ) ).
% inj_onI
thf(fact_364_inj__onD,axiom,
! [F: a > option_a,A2: set_a,X: a,Y: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_365_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_366_in__these__eq,axiom,
! [X: option_a,A2: set_option_option_a] :
( ( member_option_a @ X @ ( these_option_a @ A2 ) )
= ( member5113800082084363315tion_a @ ( some_option_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_367_in__these__eq,axiom,
! [X: a,A2: set_option_a] :
( ( member_a @ X @ ( these_a @ A2 ) )
= ( member_option_a @ ( some_a @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_368_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_369_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_370_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_371_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_372_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_a2 @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) ) @ ( image_option_a_a2 @ F @ ( minus_1574173051537231627tion_a @ B2 @ A2 ) ) )
= bot_bot_set_a ) ) ) ).
% inj_on_Un
thf(fact_373_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_a2 @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ B2 @ A2 ) ) )
= bot_bot_set_option_a ) ) ) ).
% inj_on_Un
thf(fact_374_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_a2 @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_375_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_a2 @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_376_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_a2 @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_377_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_378_refl__on__singleton,axiom,
! [X: a] : ( refl_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_379_refl__on__singleton,axiom,
! [X: option_a] : ( refl_on_option_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ bot_bo235252021745139059tion_a ) ) ).
% refl_on_singleton
thf(fact_380_image__map__upd,axiom,
! [X: a,A2: set_a,M: a > option_a,Y: a] :
( ~ ( member_a @ X @ A2 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ M @ X @ ( some_a @ Y ) ) @ A2 )
= ( image_a_option_a2 @ M @ A2 ) ) ) ).
% image_map_upd
thf(fact_381_image__map__upd,axiom,
! [X: option_a,A2: set_option_a,M: option_a > option_a,Y: a] :
( ~ ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ M @ X @ ( some_a @ Y ) ) @ A2 )
= ( image_7439109396645324421tion_a @ M @ A2 ) ) ) ).
% image_map_upd
thf(fact_382_these__image__Some__eq,axiom,
! [A2: set_a] :
( ( these_a @ ( image_a_option_a2 @ some_a @ A2 ) )
= A2 ) ).
% these_image_Some_eq
thf(fact_383_inj__on__image__iff,axiom,
! [A2: set_a,G: a > option_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Xa: a] :
( ( member_a @ Xa @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A2 )
=> ( ( inj_on_a_option_a @ G @ ( image_a_a2 @ F @ A2 ) )
= ( inj_on_a_option_a @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_384_refl__onD2,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_385_refl__onD2,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( refl_on_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( member_option_a @ Y @ A2 ) ) ) ).
% refl_onD2
thf(fact_386_refl__onD1,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( refl_on_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_387_refl__onD1,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( refl_on_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( member_option_a @ X @ A2 ) ) ) ).
% refl_onD1
thf(fact_388_refl__onD,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A2 @ R2 )
=> ( ( member_a @ A @ A2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_389_refl__onD,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( refl_on_option_a @ A2 @ R2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ) ).
% refl_onD
thf(fact_390_inj__img__insertE,axiom,
! [F: a > a,A2: set_a,X: a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ B2 )
= ( image_a_a2 @ F @ A2 ) )
=> ~ ! [X6: a,A8: set_a] :
( ~ ( member_a @ X6 @ A8 )
=> ( ( A2
= ( insert_a @ X6 @ A8 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_a_a2 @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_391_inj__img__insertE,axiom,
! [F: option_a > a,A2: set_option_a,X: a,B2: set_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ B2 )
= ( image_option_a_a2 @ F @ A2 ) )
=> ~ ! [X6: option_a,A8: set_option_a] :
( ~ ( member_option_a @ X6 @ A8 )
=> ( ( A2
= ( insert_option_a @ X6 @ A8 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_option_a_a2 @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_392_inj__img__insertE,axiom,
! [F: a > option_a,A2: set_a,X: option_a,B2: set_option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ X @ B2 )
=> ( ( ( insert_option_a @ X @ B2 )
= ( image_a_option_a2 @ F @ A2 ) )
=> ~ ! [X6: a,A8: set_a] :
( ~ ( member_a @ X6 @ A8 )
=> ( ( A2
= ( insert_a @ X6 @ A8 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_a_option_a2 @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_393_inj__img__insertE,axiom,
! [F: option_a > option_a,A2: set_option_a,X: option_a,B2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ~ ( member_option_a @ X @ B2 )
=> ( ( ( insert_option_a @ X @ B2 )
= ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ~ ! [X6: option_a,A8: set_option_a] :
( ~ ( member_option_a @ X6 @ A8 )
=> ( ( A2
= ( insert_option_a @ X6 @ A8 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_7439109396645324421tion_a @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_394_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_a2 @ F @ A2 )
= ( image_option_a_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_395_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_a2 @ F @ A2 )
= ( image_a_option_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_396_None__notin__image__Some,axiom,
! [A2: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a2 @ some_a @ A2 ) ) ).
% None_notin_image_Some
thf(fact_397_inj__on__fun__updI,axiom,
! [F: option_a > a,A2: set_option_a,Y: a,X: option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ~ ( member_a @ Y @ ( image_option_a_a2 @ F @ A2 ) )
=> ( inj_on_option_a_a @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_398_inj__on__fun__updI,axiom,
! [F: a > option_a,A2: set_a,Y: option_a,X: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ~ ( member_option_a @ Y @ ( image_a_option_a2 @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_399_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > a,Y: a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_a2 @ ( fun_upd_a_a @ F @ X @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_a2 @ ( fun_upd_a_a @ F @ X @ Y ) @ A2 )
= ( image_a_a2 @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_400_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > option_a,Y: option_a] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_a_option_a2 @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_option_a2 @ ( fun_upd_a_option_a @ F @ X @ Y ) @ A2 )
= ( image_a_option_a2 @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_401_fun__upd__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > a,Y: a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 )
= ( insert_a @ Y @ ( image_option_a_a2 @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( image_option_a_a2 @ ( fun_upd_option_a_a @ F @ X @ Y ) @ A2 )
= ( image_option_a_a2 @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_402_fun__upd__image,axiom,
! [X: option_a,A2: set_option_a,F: option_a > option_a,Y: option_a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A2 )
= ( insert_option_a @ Y @ ( image_7439109396645324421tion_a @ F @ ( minus_1574173051537231627tion_a @ A2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) ) ) ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( image_7439109396645324421tion_a @ ( fun_up1079276522633388797tion_a @ F @ X @ Y ) @ A2 )
= ( image_7439109396645324421tion_a @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_403_refl__on__domain,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( refl_on_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A2 )
& ( member_a @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_404_refl__on__domain,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( refl_on_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ( member_option_a @ A @ A2 )
& ( member_option_a @ B @ A2 ) ) ) ) ).
% refl_on_domain
thf(fact_405_linear__order__on__singleton,axiom,
! [X: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_406_linear__order__on__singleton,axiom,
! [X: option_a] : ( order_7850372301378808569tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ bot_bo235252021745139059tion_a ) ) ).
% linear_order_on_singleton
thf(fact_407_the__inv__into__onto,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( image_a_option_a2 @ ( the_in1757154643552616557on_a_a @ A2 @ F ) @ ( image_option_a_a2 @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_408_the__inv__into__onto,axiom,
! [F: a > option_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( image_option_a_a2 @ ( the_in8758012798868597241tion_a @ A2 @ F ) @ ( image_a_option_a2 @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_409_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_a2 @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_option_a_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_410_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_a2 @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_411_the__inv__f__f,axiom,
! [F: a > option_a,X: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( the_in8758012798868597241tion_a @ top_top_set_a @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_412_surjD,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ? [X3: option_a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_413_surjD,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ? [X3: option_a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_414_surjD,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
=> ? [X3: a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_415_surjD,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X3: a] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_416_surjE,axiom,
! [F: option_a > option_a,Y: option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
=> ~ ! [X3: option_a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_417_surjE,axiom,
! [F: option_a > a,Y: a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ~ ! [X3: option_a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_418_surjE,axiom,
! [F: a > option_a,Y: option_a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
=> ~ ! [X3: a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_419_surjE,axiom,
! [F: a > a,Y: a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X3: a] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_420_surjI,axiom,
! [G: option_a > option_a,F: option_a > option_a] :
( ! [X3: option_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_7439109396645324421tion_a @ G @ top_top_set_option_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_421_surjI,axiom,
! [G: option_a > a,F: a > option_a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_option_a_a2 @ G @ top_top_set_option_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_422_surjI,axiom,
! [G: a > option_a,F: option_a > a] :
( ! [X3: option_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_option_a2 @ G @ top_top_set_a )
= top_top_set_option_a ) ) ).
% surjI
thf(fact_423_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_a2 @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_424_surj__def,axiom,
! [F: option_a > option_a] :
( ( ( image_7439109396645324421tion_a @ F @ top_top_set_option_a )
= top_top_set_option_a )
= ( ! [Y3: option_a] :
? [X2: option_a] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_425_surj__def,axiom,
! [F: option_a > a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X2: option_a] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_426_surj__def,axiom,
! [F: a > option_a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
= ( ! [Y3: option_a] :
? [X2: a] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_427_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y3: a] :
? [X2: a] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_428_injD,axiom,
! [F: a > option_a,X: a,Y: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_429_injI,axiom,
! [F: a > option_a] :
( ! [X3: a,Y4: a] :
( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) )
=> ( inj_on_a_option_a @ F @ top_top_set_a ) ) ).
% injI
thf(fact_430_inj__eq,axiom,
! [F: a > option_a,X: a,Y: a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_431_inj__def,axiom,
! [F: a > option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
= ( ! [X2: a,Y3: a] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_432_the__inv__into__f__eq,axiom,
! [F: a > option_a,A2: set_a,X: a,Y: option_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( ( F @ X )
= Y )
=> ( ( member_a @ X @ A2 )
=> ( ( the_in8758012798868597241tion_a @ A2 @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_433_the__inv__into__f__f,axiom,
! [F: a > option_a,A2: set_a,X: a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( the_in8758012798868597241tion_a @ A2 @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_434_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_435_UNIV__option__conv,axiom,
( top_top_set_option_a
= ( insert_option_a @ none_a @ ( image_a_option_a2 @ some_a @ top_top_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_436_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_a2 @ F @ top_top_set_option_a ) )
= ( ? [X2: option_a] :
( ( B
= ( F @ X2 ) )
& ! [Y3: option_a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_437_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 ) )
= ( ? [X2: option_a] :
( ( B
= ( F @ X2 ) )
& ! [Y3: option_a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_438_range__ex1__eq,axiom,
! [F: a > a,B: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ B @ ( image_a_a2 @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B
= ( F @ X2 ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_439_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_a2 @ F @ top_top_set_a ) )
= ( ? [X2: a] :
( ( B
= ( F @ X2 ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_440_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_a2 @ F @ A2 )
= ( image_option_a_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_441_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_a2 @ F @ A2 )
= ( image_a_option_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_442_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_a2 @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_443_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_444_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_a2 @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_445_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_a2 @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_446_notin__range__Some,axiom,
! [X: option_option_a] :
( ( ~ ( member5113800082084363315tion_a @ X @ ( image_2132136900116418507tion_a @ some_option_a @ top_top_set_option_a ) ) )
= ( X = none_option_a ) ) ).
% notin_range_Some
thf(fact_447_notin__range__Some,axiom,
! [X: option_a] :
( ( ~ ( member_option_a @ X @ ( image_a_option_a2 @ some_a @ top_top_set_a ) ) )
= ( X = none_a ) ) ).
% notin_range_Some
thf(fact_448_reflI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ X3 ) @ R2 )
=> ( refl_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% reflI
thf(fact_449_reflI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 )
=> ( refl_on_a @ top_top_set_a @ R2 ) ) ).
% reflI
thf(fact_450_reflD,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( refl_on_option_a @ top_top_set_option_a @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ).
% reflD
thf(fact_451_reflD,axiom,
! [R2: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ top_top_set_a @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ).
% reflD
thf(fact_452_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_a2 @ F @ A2 ) )
=> ( ( F @ ( the_in1757154643552616557on_a_a @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_453_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_a2 @ F @ A2 ) )
=> ( ( F @ ( the_in8758012798868597241tion_a @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_454_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_a2 @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_455_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_a2 @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_456_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_a2 @ F @ ( inf_inf_set_option_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_option_a_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) ) ) ) ).
% image_Int
thf(fact_457_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_a2 @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_option_a @ ( image_a_option_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) ) ) ) ).
% image_Int
thf(fact_458_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_459_finite__range__updI,axiom,
! [F: a > option_a,A: a,B: a] :
( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ F @ top_top_set_a ) )
=> ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( fun_upd_a_option_a @ F @ A @ ( some_a @ B ) ) @ top_top_set_a ) ) ) ).
% finite_range_updI
thf(fact_460_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_a2 @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) @ ( uminus_uminus_set_a @ ( image_option_a_a2 @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_461_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_a2 @ F @ ( uminus_uminus_set_a @ A2 ) ) @ ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F @ A2 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_462_trans__singleton,axiom,
! [A: option_a] : ( trans_on_option_a @ top_top_set_option_a @ ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ bot_bo235252021745139059tion_a ) ) ).
% trans_singleton
thf(fact_463_trans__singleton,axiom,
! [A: a] : ( trans_on_a @ top_top_set_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ A ) @ bot_bo3357376287454694259od_a_a ) ) ).
% trans_singleton
thf(fact_464_Linear__order__in__diff__Id,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
= ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_465_Linear__order__in__diff__Id,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
= ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_466_finite__option__UNIV,axiom,
( ( finite8114217219359860531tion_a @ top_to1659475022456381882tion_a )
= ( finite1674126218327898605tion_a @ top_top_set_option_a ) ) ).
% finite_option_UNIV
thf(fact_467_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_468_trans__onD,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a,Z2: a] :
( ( trans_on_a @ A2 @ R2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( member_a @ Z2 @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_469_trans__onD,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a,Z2: option_a] :
( ( trans_on_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( member_option_a @ Y @ A2 )
=> ( ( member_option_a @ Z2 @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).
% trans_onD
thf(fact_470_trans__onI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a,Z: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( ( member_a @ Z @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z ) @ R2 ) ) ) ) ) )
=> ( trans_on_a @ A2 @ R2 ) ) ).
% trans_onI
thf(fact_471_trans__onI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a,Z: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( ( member_option_a @ Y4 @ A2 )
=> ( ( member_option_a @ Z @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z ) @ R2 ) ) ) ) ) )
=> ( trans_on_option_a @ A2 @ R2 ) ) ).
% trans_onI
thf(fact_472_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_473_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_474_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_475_finite__range__Some,axiom,
( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ some_a @ top_top_set_a ) )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_range_Some
thf(fact_476_transD,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a,Z2: option_a] :
( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Z2 ) @ R2 ) ) ) ) ).
% transD
thf(fact_477_transD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z2: a] :
( ( trans_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ R2 ) ) ) ) ).
% transD
thf(fact_478_transE,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a,Z2: option_a] :
( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ Z2 ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Z2 ) @ R2 ) ) ) ) ).
% transE
thf(fact_479_transE,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a,Z2: a] :
( ( trans_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z2 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z2 ) @ R2 ) ) ) ) ).
% transE
thf(fact_480_transI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a,Z: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z ) @ R2 ) ) )
=> ( trans_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% transI
thf(fact_481_transI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z ) @ R2 ) ) )
=> ( trans_on_a @ top_top_set_a @ R2 ) ) ).
% transI
thf(fact_482_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_a2 @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_483_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_a2 @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_484_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_a2 @ F @ A2 ) )
= ( member_option_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_485_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_486_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_a2 @ F @ A2 )
= ( image_option_a_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_487_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_a2 @ F @ A2 )
= ( image_a_option_a2 @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_488_finite__ran,axiom,
! [P: option_a > option_option_a] :
( ( finite1674126218327898605tion_a @ ( dom_op4724496951392727122tion_a @ P ) )
=> ( finite1674126218327898605tion_a @ ( ran_op6317565877353657455tion_a @ P ) ) ) ).
% finite_ran
thf(fact_489_finite__ran,axiom,
! [P: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ P ) )
=> ( finite_finite_a @ ( ran_option_a_a @ P ) ) ) ).
% finite_ran
thf(fact_490_finite__ran,axiom,
! [P: a > option_option_a] :
( ( finite_finite_a @ ( dom_a_option_a @ P ) )
=> ( finite1674126218327898605tion_a @ ( ran_a_option_a @ P ) ) ) ).
% finite_ran
thf(fact_491_finite__ran,axiom,
! [P: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ P ) )
=> ( finite_finite_a @ ( ran_a_a @ P ) ) ) ).
% finite_ran
thf(fact_492_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_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) )
= ( ord_le1955136853071979460tion_a @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_493_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_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_494_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_a2 @ F @ ( inf_inf_set_option_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_option_a_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_495_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_a2 @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_option_a @ ( image_a_option_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_496_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_a2 @ F @ ( minus_1574173051537231627tion_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_option_a_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_497_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_a2 @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_1574173051537231627tion_a @ ( image_a_option_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_498_finite__map__freshness,axiom,
! [F: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ F ) )
=> ( ~ ( finite1674126218327898605tion_a @ top_top_set_option_a )
=> ? [X3: option_a] :
( ( F @ X3 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_499_finite__map__freshness,axiom,
! [F: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ F ) )
=> ( ~ ( finite_finite_a @ top_top_set_a )
=> ? [X3: a] :
( ( F @ X3 )
= none_a ) ) ) ).
% finite_map_freshness
thf(fact_500_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_501_surj__Compl__image__subset,axiom,
! [F: option_a > a,A2: set_option_a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_option_a_a2 @ F @ A2 ) ) @ ( image_option_a_a2 @ F @ ( uminus6205308855922866075tion_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_502_surj__Compl__image__subset,axiom,
! [F: a > option_a,A2: set_a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ord_le1955136853071979460tion_a @ ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F @ A2 ) ) @ ( image_a_option_a2 @ F @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_503_surj__Compl__image__subset,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a2 @ F @ A2 ) ) @ ( image_a_a2 @ F @ ( uminus_uminus_set_a @ A2 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_504_the__inv__into__into,axiom,
! [F: a > a,A2: set_a,X: a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( member_a @ X @ ( image_a_a2 @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a @ ( the_inv_into_a_a @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_505_the__inv__into__into,axiom,
! [F: option_a > a,A2: set_option_a,X: a,B2: set_option_a] :
( ( inj_on_option_a_a @ F @ A2 )
=> ( ( member_a @ X @ ( image_option_a_a2 @ F @ A2 ) )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( member_option_a @ ( the_in1757154643552616557on_a_a @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_506_the__inv__into__into,axiom,
! [F: a > option_a,A2: set_a,X: option_a,B2: set_a] :
( ( inj_on_a_option_a @ F @ A2 )
=> ( ( member_option_a @ X @ ( image_a_option_a2 @ F @ A2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a @ ( the_in8758012798868597241tion_a @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_507_the__inv__into__into,axiom,
! [F: option_a > option_a,A2: set_option_a,X: option_a,B2: set_option_a] :
( ( inj_on8559383841115902449tion_a @ F @ A2 )
=> ( ( member_option_a @ X @ ( image_7439109396645324421tion_a @ F @ A2 ) )
=> ( ( ord_le1955136853071979460tion_a @ A2 @ B2 )
=> ( member_option_a @ ( the_in2538339130118444083tion_a @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_508_Linear__order__wf__diff__Id,axiom,
! [R2: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( wf_a @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) )
= ( ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( field_a @ R2 ) )
=> ( ( A7 != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A7 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_509_Linear__order__wf__diff__Id,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( wf_option_a @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) )
= ( ! [A7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R2 ) )
=> ( ( A7 != bot_bot_set_option_a )
=> ? [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
& ! [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_510_total__onI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_a @ A2 @ R2 ) ) ).
% total_onI
thf(fact_511_total__onI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( ( member_option_a @ Y4 @ A2 )
=> ( ( X3 != Y4 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
| ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on_option_a @ A2 @ R2 ) ) ).
% total_onI
thf(fact_512_totalI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( X3 != Y4 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
| ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 ) ) )
=> ( total_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% totalI
thf(fact_513_totalI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( X3 != Y4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 ) ) )
=> ( total_on_a @ top_top_set_a @ R2 ) ) ).
% totalI
thf(fact_514_wf__eq__minimal2,axiom,
( wf_a
= ( ^ [R3: set_Product_prod_a_a] :
! [A7: set_a] :
( ( ( ord_less_eq_set_a @ A7 @ ( field_a @ R3 ) )
& ( A7 != bot_bot_set_a ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A7 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A7 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_515_wf__eq__minimal2,axiom,
( wf_option_a
= ( ^ [R3: set_Pr7585778909603769095tion_a] :
! [A7: set_option_a] :
( ( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R3 ) )
& ( A7 != bot_bot_set_option_a ) )
=> ? [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
& ! [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X2 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_516_wfE__min_H,axiom,
! [R4: set_Product_prod_a_a,Q2: set_a] :
( ( wf_a @ R4 )
=> ( ( Q2 != bot_bot_set_a )
=> ~ ! [Z: a] :
( ( member_a @ Z @ Q2 )
=> ~ ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ R4 )
=> ~ ( member_a @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_517_wfE__min_H,axiom,
! [R4: set_Pr7585778909603769095tion_a,Q2: set_option_a] :
( ( wf_option_a @ R4 )
=> ( ( Q2 != bot_bot_set_option_a )
=> ~ ! [Z: option_a] :
( ( member_option_a @ Z @ Q2 )
=> ~ ! [Y5: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z ) @ R4 )
=> ~ ( member_option_a @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_518_wfE__min,axiom,
! [R4: set_Product_prod_a_a,X: a,Q2: set_a] :
( ( wf_a @ R4 )
=> ( ( member_a @ X @ Q2 )
=> ~ ! [Z: a] :
( ( member_a @ Z @ Q2 )
=> ~ ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Z ) @ R4 )
=> ~ ( member_a @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_519_wfE__min,axiom,
! [R4: set_Pr7585778909603769095tion_a,X: option_a,Q2: set_option_a] :
( ( wf_option_a @ R4 )
=> ( ( member_option_a @ X @ Q2 )
=> ~ ! [Z: option_a] :
( ( member_option_a @ Z @ Q2 )
=> ~ ! [Y5: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ Z ) @ R4 )
=> ~ ( member_option_a @ Y5 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_520_wfI__min,axiom,
! [R4: set_Product_prod_a_a] :
( ! [X3: a,Q4: set_a] :
( ( member_a @ X3 @ Q4 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ Q4 )
& ! [Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Xa2 ) @ R4 )
=> ~ ( member_a @ Y4 @ Q4 ) ) ) )
=> ( wf_a @ R4 ) ) ).
% wfI_min
thf(fact_521_wfI__min,axiom,
! [R4: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Q4: set_option_a] :
( ( member_option_a @ X3 @ Q4 )
=> ? [Xa2: option_a] :
( ( member_option_a @ Xa2 @ Q4 )
& ! [Y4: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Xa2 ) @ R4 )
=> ~ ( member_option_a @ Y4 @ Q4 ) ) ) )
=> ( wf_option_a @ R4 ) ) ).
% wfI_min
thf(fact_522_wf__eq__minimal,axiom,
( wf_a
= ( ^ [R3: set_Product_prod_a_a] :
! [Q5: set_a] :
( ? [X2: a] : ( member_a @ X2 @ Q5 )
=> ? [X2: a] :
( ( member_a @ X2 @ Q5 )
& ! [Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ R3 )
=> ~ ( member_a @ Y3 @ Q5 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_523_wf__eq__minimal,axiom,
( wf_option_a
= ( ^ [R3: set_Pr7585778909603769095tion_a] :
! [Q5: set_option_a] :
( ? [X2: option_a] : ( member_option_a @ X2 @ Q5 )
=> ? [X2: option_a] :
( ( member_option_a @ X2 @ Q5 )
& ! [Y3: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X2 ) @ R3 )
=> ~ ( member_option_a @ Y3 @ Q5 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_524_wo__rel_Ocases__Total3,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) )
=> ( ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ ( minus_6817036919807184750od_a_a @ R2 @ id_a2 ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_525_wo__rel_Ocases__Total3,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) )
=> ( ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) )
| ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ ( minus_6512073291116468334tion_a @ R2 @ id_option_a2 ) ) )
=> ( Phi @ A @ B ) )
=> ( ( ( A = B )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_526_finite__range__map__of__map__add,axiom,
! [F: option_a > option_a,L: list_P6260409590414597735on_a_a] :
( ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ F @ top_top_set_option_a ) )
=> ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( map_add_option_a_a @ F @ ( map_of_option_a_a @ L ) ) @ top_top_set_option_a ) ) ) ).
% finite_range_map_of_map_add
thf(fact_527_finite__range__map__of__map__add,axiom,
! [F: a > option_a,L: list_P1396940483166286381od_a_a] :
( ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ F @ top_top_set_a ) )
=> ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( map_add_a_a @ F @ ( map_of_a_a @ L ) ) @ top_top_set_a ) ) ) ).
% finite_range_map_of_map_add
thf(fact_528_under__incr,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ord_le1955136853071979460tion_a @ ( order_under_option_a @ R2 @ A ) @ ( order_under_option_a @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_529_under__incr,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( trans_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ord_less_eq_set_a @ ( order_under_a @ R2 @ A ) @ ( order_under_a @ R2 @ B ) ) ) ) ).
% under_incr
thf(fact_530_wo__rel_Oin__notinI,axiom,
! [R2: set_Product_prod_a_a,J: a,I: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ J @ I ) @ R2 )
| ( J = I ) )
=> ( ( member_a @ I @ ( field_a @ R2 ) )
=> ( ( member_a @ J @ ( field_a @ R2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_531_wo__rel_Oin__notinI,axiom,
! [R2: set_Pr7585778909603769095tion_a,J: option_a,I: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ J @ I ) @ R2 )
| ( J = I ) )
=> ( ( member_option_a @ I @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ J @ ( field_option_a @ R2 ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R2 ) ) ) ) ) ).
% wo_rel.in_notinI
thf(fact_532_well__order__induct__imp,axiom,
! [R2: set_Product_prod_a_a,P3: a > $o,A: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ! [X3: a] :
( ! [Y5: a] :
( ( ( Y5 != X3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_a @ Y5 @ ( field_a @ R2 ) )
=> ( P3 @ Y5 ) ) )
=> ( ( member_a @ X3 @ ( field_a @ R2 ) )
=> ( P3 @ X3 ) ) )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( P3 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_533_well__order__induct__imp,axiom,
! [R2: set_Pr7585778909603769095tion_a,P3: option_a > $o,A: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ! [X3: option_a] :
( ! [Y5: option_a] :
( ( ( Y5 != X3 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y5 @ X3 ) @ R2 ) )
=> ( ( member_option_a @ Y5 @ ( field_option_a @ R2 ) )
=> ( P3 @ Y5 ) ) )
=> ( ( member_option_a @ X3 @ ( field_option_a @ R2 ) )
=> ( P3 @ X3 ) ) )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( P3 @ A ) ) ) ) ).
% well_order_induct_imp
thf(fact_534_wo__rel_Ocases__Total,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a,Phi: a > a > $o] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_535_wo__rel_Ocases__Total,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,Phi: option_a > option_a > $o] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( field_option_a @ R2 ) )
=> ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 )
=> ( Phi @ A @ B ) )
=> ( Phi @ A @ B ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_536_finite__range__map__of,axiom,
! [Xys: list_P6260409590414597735on_a_a] : ( finite1674126218327898605tion_a @ ( image_7439109396645324421tion_a @ ( map_of_option_a_a @ Xys ) @ top_top_set_option_a ) ) ).
% finite_range_map_of
thf(fact_537_finite__range__map__of,axiom,
! [Xys: list_P1396940483166286381od_a_a] : ( finite1674126218327898605tion_a @ ( image_a_option_a2 @ ( map_of_a_a @ Xys ) @ top_top_set_a ) ) ).
% finite_range_map_of
thf(fact_538_wo__rel_Oequals__minim,axiom,
! [R2: set_Product_prod_a_a,B2: set_a,A: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
=> ( ( member_a @ A @ B2 )
=> ( ! [B4: a] :
( ( member_a @ B4 @ B2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B4 ) @ R2 ) )
=> ( A
= ( bNF_We5615626441682584778inim_a @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_539_wo__rel_Oequals__minim,axiom,
! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,A: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ A @ B2 )
=> ( ! [B4: option_a] :
( ( member_option_a @ B4 @ B2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B4 ) @ R2 ) )
=> ( A
= ( bNF_We6579146059749918992tion_a @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_540_wo__rel_Ominim__least,axiom,
! [R2: set_Product_prod_a_a,B2: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ B2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We5615626441682584778inim_a @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_541_wo__rel_Ominim__least,axiom,
! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ B2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We6579146059749918992tion_a @ R2 @ B2 ) @ B ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_542_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
& ( member_a @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) @ ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_543_wo__rel_Omax2__greater__among,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
& ( member_option_a @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) @ ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_544_Linear__order__Well__order__iff,axiom,
! [R2: set_Product_prod_a_a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( order_6972113574731384220r_on_a @ ( field_a @ R2 ) @ R2 )
= ( ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( field_a @ R2 ) )
=> ( ( A7 != bot_bot_set_a )
=> ? [X2: a] :
( ( member_a @ X2 @ A7 )
& ! [Y3: a] :
( ( member_a @ Y3 @ A7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_545_Linear__order__Well__order__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( order_4821795997958563554tion_a @ ( field_option_a @ R2 ) @ R2 )
= ( ! [A7: set_option_a] :
( ( ord_le1955136853071979460tion_a @ A7 @ ( field_option_a @ R2 ) )
=> ( ( A7 != bot_bot_set_option_a )
=> ? [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
& ! [Y3: option_a] :
( ( member_option_a @ Y3 @ A7 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_546_well__order__on__domain,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_6972113574731384220r_on_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A2 )
& ( member_a @ B @ A2 ) ) ) ) ).
% well_order_on_domain
thf(fact_547_well__order__on__domain,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_4821795997958563554tion_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ( member_option_a @ A @ A2 )
& ( member_option_a @ B @ A2 ) ) ) ) ).
% well_order_on_domain
thf(fact_548_wo__rel_Omax2__equals1,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ( bNF_We3763454674811381836max2_a @ R2 @ A @ B )
= A )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_549_wo__rel_Omax2__equals1,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ( bNF_We4567742444881707410tion_a @ R2 @ A @ B )
= A )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_550_wo__rel_Omax2__equals2,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ( bNF_We3763454674811381836max2_a @ R2 @ A @ B )
= B )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_551_wo__rel_Omax2__equals2,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ( bNF_We4567742444881707410tion_a @ R2 @ A @ B )
= B )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_552_wo__rel_Omax2__greater,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We3763454674811381836max2_a @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_553_wo__rel_Omax2__greater,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We4567742444881707410tion_a @ R2 @ A @ B ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_554_underS__incl__iff,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A ) @ ( order_underS_a @ R2 @ B ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_555_underS__incl__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_7850372301378808569tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R2 @ A ) @ ( order_8525669848891258378tion_a @ R2 @ B ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_556_max__ext_Omax__extI,axiom,
! [X7: set_a,Y6: set_a,R4: set_Product_prod_a_a] :
( ( finite_finite_a @ X7 )
=> ( ( finite_finite_a @ Y6 )
=> ( ( Y6 != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ X7 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ Y6 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Xa2 ) @ R4 ) ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X7 @ Y6 ) @ ( max_ext_a @ R4 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_557_max__ext_Omax__extI,axiom,
! [X7: set_option_a,Y6: set_option_a,R4: set_Pr7585778909603769095tion_a] :
( ( finite1674126218327898605tion_a @ X7 )
=> ( ( finite1674126218327898605tion_a @ Y6 )
=> ( ( Y6 != bot_bot_set_option_a )
=> ( ! [X3: option_a] :
( ( member_option_a @ X3 @ X7 )
=> ? [Xa2: option_a] :
( ( member_option_a @ Xa2 @ Y6 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Xa2 ) @ R4 ) ) )
=> ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ X7 @ Y6 ) @ ( max_ext_option_a @ R4 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_558_max__ext_Osimps,axiom,
! [A1: set_a,A22: set_a,R4: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R4 ) )
= ( ( finite_finite_a @ A1 )
& ( finite_finite_a @ A22 )
& ( A22 != bot_bot_set_a )
& ! [X2: a] :
( ( member_a @ X2 @ A1 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ A22 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R4 ) ) ) ) ) ).
% max_ext.simps
thf(fact_559_max__ext_Osimps,axiom,
! [A1: set_option_a,A22: set_option_a,R4: set_Pr7585778909603769095tion_a] :
( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A1 @ A22 ) @ ( max_ext_option_a @ R4 ) )
= ( ( finite1674126218327898605tion_a @ A1 )
& ( finite1674126218327898605tion_a @ A22 )
& ( A22 != bot_bot_set_option_a )
& ! [X2: option_a] :
( ( member_option_a @ X2 @ A1 )
=> ? [Y3: option_a] :
( ( member_option_a @ Y3 @ A22 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y3 ) @ R4 ) ) ) ) ) ).
% max_ext.simps
thf(fact_560_underS__E,axiom,
! [I: a,R4: set_Product_prod_a_a,J: a] :
( ( member_a @ I @ ( order_underS_a @ R4 @ J ) )
=> ( ( I != J )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 ) ) ) ).
% underS_E
thf(fact_561_underS__E,axiom,
! [I: option_a,R4: set_Pr7585778909603769095tion_a,J: option_a] :
( ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R4 @ J ) )
=> ( ( I != J )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 ) ) ) ).
% underS_E
thf(fact_562_underS__I,axiom,
! [I: a,J: a,R4: set_Product_prod_a_a] :
( ( I != J )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R4 )
=> ( member_a @ I @ ( order_underS_a @ R4 @ J ) ) ) ) ).
% underS_I
thf(fact_563_underS__I,axiom,
! [I: option_a,J: option_a,R4: set_Pr7585778909603769095tion_a] :
( ( I != J )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ I @ J ) @ R4 )
=> ( member_option_a @ I @ ( order_8525669848891258378tion_a @ R4 @ J ) ) ) ) ).
% underS_I
thf(fact_564_max__ext_Ocases,axiom,
! [A1: set_a,A22: set_a,R4: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A1 @ A22 ) @ ( max_ext_a @ R4 ) )
=> ~ ( ( finite_finite_a @ A1 )
=> ( ( finite_finite_a @ A22 )
=> ( ( A22 != bot_bot_set_a )
=> ~ ! [X8: a] :
( ( member_a @ X8 @ A1 )
=> ? [Xa: a] :
( ( member_a @ Xa @ A22 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X8 @ Xa ) @ R4 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_565_max__ext_Ocases,axiom,
! [A1: set_option_a,A22: set_option_a,R4: set_Pr7585778909603769095tion_a] :
( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A1 @ A22 ) @ ( max_ext_option_a @ R4 ) )
=> ~ ( ( finite1674126218327898605tion_a @ A1 )
=> ( ( finite1674126218327898605tion_a @ A22 )
=> ( ( A22 != bot_bot_set_option_a )
=> ~ ! [X8: option_a] :
( ( member_option_a @ X8 @ A1 )
=> ? [Xa: option_a] :
( ( member_option_a @ Xa @ A22 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X8 @ Xa ) @ R4 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_566_underS__incr,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( antisym_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ord_le1955136853071979460tion_a @ ( order_8525669848891258378tion_a @ R2 @ A ) @ ( order_8525669848891258378tion_a @ R2 @ B ) ) ) ) ) ).
% underS_incr
thf(fact_567_underS__incr,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( trans_on_a @ top_top_set_a @ R2 )
=> ( ( antisym_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A ) @ ( order_underS_a @ R2 @ B ) ) ) ) ) ).
% underS_incr
thf(fact_568_wo__rel_OisMinim__def,axiom,
! [R2: set_Product_prod_a_a,A2: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( bNF_We6697304935525757620inim_a @ R2 @ A2 @ B )
= ( ( member_a @ B @ A2 )
& ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_569_wo__rel_OisMinim__def,axiom,
! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( bNF_We2467337426749329402tion_a @ R2 @ A2 @ B )
= ( ( member_option_a @ B @ A2 )
& ! [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ X2 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_570_antisym__onI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisym_on_a @ A2 @ R2 ) ) ).
% antisym_onI
thf(fact_571_antisym__onI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( ( member_option_a @ Y4 @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 )
=> ( X3 = Y4 ) ) ) ) )
=> ( antisym_on_option_a @ A2 @ R2 ) ) ).
% antisym_onI
thf(fact_572_antisym__onD,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( antisym_on_a @ A2 @ R2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_573_antisym__onD,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( antisym_on_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( member_option_a @ Y @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ) ) ).
% antisym_onD
thf(fact_574_image__strict__mono,axiom,
! [F: option_a > a,B2: set_option_a,A2: set_option_a] :
( ( inj_on_option_a_a @ F @ B2 )
=> ( ( ord_le5631237216984945872tion_a @ A2 @ B2 )
=> ( ord_less_set_a @ ( image_option_a_a2 @ F @ A2 ) @ ( image_option_a_a2 @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_575_image__strict__mono,axiom,
! [F: a > option_a,B2: set_a,A2: set_a] :
( ( inj_on_a_option_a @ F @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_le5631237216984945872tion_a @ ( image_a_option_a2 @ F @ A2 ) @ ( image_a_option_a2 @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_576_antisymD,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( antisym_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ).
% antisymD
thf(fact_577_antisymD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a] :
( ( antisym_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R2 )
=> ( X = Y ) ) ) ) ).
% antisymD
thf(fact_578_antisymI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 )
=> ( X3 = Y4 ) ) )
=> ( antisym_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% antisymI
thf(fact_579_antisymI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 )
=> ( X3 = Y4 ) ) )
=> ( antisym_on_a @ top_top_set_a @ R2 ) ) ).
% antisymI
thf(fact_580_in__finite__psubset,axiom,
! [A2: set_option_a,B2: set_option_a] :
( ( member5358692782348450128tion_a @ ( produc8179951581375851543tion_a @ A2 @ B2 ) @ finite4966134214920407047tion_a )
= ( ( ord_le5631237216984945872tion_a @ A2 @ B2 )
& ( finite1674126218327898605tion_a @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_581_in__finite__psubset,axiom,
! [A2: set_a,B2: set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A2 @ B2 ) @ finite_psubset_a )
= ( ( ord_less_set_a @ A2 @ B2 )
& ( finite_finite_a @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_582_wo__rel_Osuc__greater,axiom,
! [R2: set_Product_prod_a_a,B2: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
=> ( ( ( order_AboveS_a @ R2 @ B2 )
!= bot_bot_set_a )
=> ( ( member_a @ B @ B2 )
=> ( ( ( bNF_We6154283375207884895_suc_a @ R2 @ B2 )
!= B )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_583_wo__rel_Osuc__greater,axiom,
! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
=> ( ( ( order_6500638856667293583tion_a @ R2 @ B2 )
!= bot_bot_set_option_a )
=> ( ( member_option_a @ B @ B2 )
=> ( ( ( bNF_We5356091070762920229tion_a @ R2 @ B2 )
!= B )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_584_wo__rel_Osuc__least__AboveS,axiom,
! [R2: set_Product_prod_a_a,A: a,B2: set_a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( member_a @ A @ ( order_AboveS_a @ R2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_585_wo__rel_Osuc__least__AboveS,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B2: set_option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) @ A ) @ R2 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_586_wo__rel_Oequals__suc__AboveS,axiom,
! [R2: set_Product_prod_a_a,B2: set_a,A: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( field_a @ R2 ) )
=> ( ( member_a @ A @ ( order_AboveS_a @ R2 @ B2 ) )
=> ( ! [A6: a] :
( ( member_a @ A6 @ ( order_AboveS_a @ R2 @ B2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A6 ) @ R2 ) )
=> ( A
= ( bNF_We6154283375207884895_suc_a @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_587_wo__rel_Oequals__suc__AboveS,axiom,
! [R2: set_Pr7585778909603769095tion_a,B2: set_option_a,A: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ A @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
=> ( ! [A6: option_a] :
( ( member_option_a @ A6 @ ( order_6500638856667293583tion_a @ R2 @ B2 ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A6 ) @ R2 ) )
=> ( A
= ( bNF_We5356091070762920229tion_a @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_588_Not__Domain__rtrancl,axiom,
! [X: a,R4: set_Product_prod_a_a,Y: a] :
( ~ ( member_a @ X @ ( domain_a_a @ R4 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R4 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_589_Not__Domain__rtrancl,axiom,
! [X: option_a,R4: set_Pr7585778909603769095tion_a,Y: option_a] :
( ~ ( member_option_a @ X @ ( domain5649462347324568460tion_a @ R4 ) )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ ( transi330218190764880583tion_a @ R4 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_590_wo__rel_Osuc__ofilter__in,axiom,
! [R2: set_Product_prod_a_a,A2: set_a,B: a] :
( ( bNF_We1162827675446709994_rel_a @ R2 )
=> ( ( order_ofilter_a @ R2 @ A2 )
=> ( ( ( order_AboveS_a @ R2 @ A2 )
!= bot_bot_set_a )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ ( bNF_We6154283375207884895_suc_a @ R2 @ A2 ) ) @ R2 )
=> ( ( B
!= ( bNF_We6154283375207884895_suc_a @ R2 @ A2 ) )
=> ( member_a @ B @ A2 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_591_wo__rel_Osuc__ofilter__in,axiom,
! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a,B: option_a] :
( ( bNF_We8432232079604507440tion_a @ R2 )
=> ( ( order_6420974439381506266tion_a @ R2 @ A2 )
=> ( ( ( order_6500638856667293583tion_a @ R2 @ A2 )
!= bot_bot_set_option_a )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ ( bNF_We5356091070762920229tion_a @ R2 @ A2 ) ) @ R2 )
=> ( ( B
!= ( bNF_We5356091070762920229tion_a @ R2 @ A2 ) )
=> ( member_option_a @ B @ A2 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_592_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_a2 @ F @ A2 ) )
=> ( ord_le1955136853071979460tion_a @ ( vimage_option_a_a @ F @ B2 ) @ A2 ) ) ) ).
% vimage_subsetI
thf(fact_593_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_a2 @ F @ A2 ) )
=> ( ord_less_eq_set_a @ ( vimage_a_option_a @ F @ B2 ) @ A2 ) ) ) ).
% vimage_subsetI
thf(fact_594_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_595_option_Oinj__map,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( inj_on8559383841115902449tion_a @ ( map_option_a_a2 @ F ) @ top_top_set_option_a ) ) ).
% option.inj_map
thf(fact_596_map__option__eq__Some,axiom,
! [F: a > a,Xo: option_a,Y: a] :
( ( ( map_option_a_a2 @ F @ Xo )
= ( some_a @ Y ) )
= ( ? [Z3: a] :
( ( Xo
= ( some_a @ Z3 ) )
& ( ( F @ Z3 )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_597_None__eq__map__option__iff,axiom,
! [F: a > a,X: option_a] :
( ( none_a
= ( map_option_a_a2 @ F @ X ) )
= ( X = none_a ) ) ).
% None_eq_map_option_iff
thf(fact_598_map__option__is__None,axiom,
! [F: a > a,Opt: option_a] :
( ( ( map_option_a_a2 @ F @ Opt )
= none_a )
= ( Opt = none_a ) ) ).
% map_option_is_None
thf(fact_599_option_Omap__disc__iff,axiom,
! [F: a > a,A: option_a] :
( ( ( map_option_a_a2 @ F @ A )
= none_a )
= ( A = none_a ) ) ).
% option.map_disc_iff
thf(fact_600_is__none__map__option,axiom,
! [F: a > a,X: option_a] :
( ( is_none_a @ ( map_option_a_a2 @ F @ X ) )
= ( is_none_a @ X ) ) ).
% is_none_map_option
thf(fact_601_map__option__cong,axiom,
! [X: option_a,Y: option_a,F: a > a,G: a > a] :
( ( X = Y )
=> ( ! [A3: a] :
( ( Y
= ( some_a @ A3 ) )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( map_option_a_a2 @ F @ X )
= ( map_option_a_a2 @ G @ Y ) ) ) ) ).
% map_option_cong
thf(fact_602_option_Osimps_I9_J,axiom,
! [F: a > a,X22: a] :
( ( map_option_a_a2 @ F @ ( some_a @ X22 ) )
= ( some_a @ ( F @ X22 ) ) ) ).
% option.simps(9)
thf(fact_603_option_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_option_a_a2 @ F @ none_a )
= none_a ) ).
% option.simps(8)
thf(fact_604_option_Omap__cong,axiom,
! [X: option_a,Ya: option_a,F: a > a,G: a > a] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_option_a3 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_option_a_a2 @ F @ X )
= ( map_option_a_a2 @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_605_option_Omap__cong0,axiom,
! [X: option_a,F: a > a,G: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_option_a3 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_option_a_a2 @ F @ X )
= ( map_option_a_a2 @ G @ X ) ) ) ).
% option.map_cong0
thf(fact_606_option_Oinj__map__strong,axiom,
! [X: option_a,Xa3: option_a,F: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_option_a3 @ X ) )
=> ( ( member_a @ Za @ ( set_option_a3 @ Xa3 ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_option_a_a2 @ F @ X )
= ( map_option_a_a2 @ Fa @ Xa3 ) )
=> ( X = Xa3 ) ) ) ).
% option.inj_map_strong
thf(fact_607_option_Omap__ident__strong,axiom,
! [T: option_option_a,F: option_a > option_a] :
( ! [Z: option_a] :
( ( member_option_a @ Z @ ( set_option_option_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_op788413144570152203tion_a @ F @ T )
= T ) ) ).
% option.map_ident_strong
thf(fact_608_option_Omap__ident__strong,axiom,
! [T: option_a,F: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_option_a3 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_option_a_a2 @ F @ T )
= T ) ) ).
% option.map_ident_strong
thf(fact_609_map__option__idI,axiom,
! [X: option_option_a,F: option_a > option_a] :
( ! [Y4: option_a] :
( ( member_option_a @ Y4 @ ( set_option_option_a2 @ X ) )
=> ( ( F @ Y4 )
= Y4 ) )
=> ( ( map_op788413144570152203tion_a @ F @ X )
= X ) ) ).
% map_option_idI
thf(fact_610_map__option__idI,axiom,
! [X: option_a,F: a > a] :
( ! [Y4: a] :
( ( member_a @ Y4 @ ( set_option_a3 @ X ) )
=> ( ( F @ Y4 )
= Y4 ) )
=> ( ( map_option_a_a2 @ F @ X )
= X ) ) ).
% map_option_idI
thf(fact_611_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_612_surj__image__vimage__eq,axiom,
! [F: option_a > a,A2: set_a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_option_a_a2 @ F @ ( vimage_option_a_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_613_surj__image__vimage__eq,axiom,
! [F: a > option_a,A2: set_option_a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_a_option_a2 @ F @ ( vimage_a_option_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_614_surj__image__vimage__eq,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a2 @ F @ ( vimage_a_a @ F @ A2 ) )
= A2 ) ) ).
% surj_image_vimage_eq
thf(fact_615_option_Oset__map,axiom,
! [F: a > option_a,V3: option_a] :
( ( set_option_option_a2 @ ( map_op2340691886215429841tion_a @ F @ V3 ) )
= ( image_a_option_a2 @ F @ ( set_option_a3 @ V3 ) ) ) ).
% option.set_map
thf(fact_616_option_Oset__map,axiom,
! [F: option_a > a,V3: option_option_a] :
( ( set_option_a3 @ ( map_op4563205767754224965on_a_a @ F @ V3 ) )
= ( image_option_a_a2 @ F @ ( set_option_option_a2 @ V3 ) ) ) ).
% option.set_map
thf(fact_617_option_Oset__map,axiom,
! [F: a > a,V3: option_a] :
( ( set_option_a3 @ ( map_option_a_a2 @ F @ V3 ) )
= ( image_a_a2 @ F @ ( set_option_a3 @ V3 ) ) ) ).
% option.set_map
thf(fact_618_option_Omap__sel,axiom,
! [A: option_a,F: a > a] :
( ( A != none_a )
=> ( ( the_a @ ( map_option_a_a2 @ F @ A ) )
= ( F @ ( the_a @ A ) ) ) ) ).
% option.map_sel
thf(fact_619_the__map__option,axiom,
! [X: option_a,F: a > a] :
( ~ ( is_none_a @ X )
=> ( ( the_a @ ( map_option_a_a2 @ F @ X ) )
= ( F @ ( the_a @ X ) ) ) ) ).
% the_map_option
thf(fact_620_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_621_surj__vimage__empty,axiom,
! [F: option_a > a,A2: set_a] :
( ( ( image_option_a_a2 @ 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_622_surj__vimage__empty,axiom,
! [F: a > option_a,A2: set_option_a] :
( ( ( image_a_option_a2 @ 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_623_surj__vimage__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a2 @ 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_624_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_625_vimage__subsetD,axiom,
! [F: option_a > a,B2: set_a,A2: set_option_a] :
( ( ( image_option_a_a2 @ 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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_626_vimage__subsetD,axiom,
! [F: a > option_a,B2: set_option_a,A2: set_a] :
( ( ( image_a_option_a2 @ 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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_627_vimage__subsetD,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( ( image_a_a2 @ 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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subsetD
thf(fact_628_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_a2 @ F @ A2 ) )
= A2 ) ) ).
% inj_vimage_image_eq
thf(fact_629_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_a2 @ F @ A2 ) )
= A2 ) ) ).
% inj_vimage_image_eq
thf(fact_630_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_631_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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_632_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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_633_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_a2 @ F @ A2 ) ) ) ) ).
% vimage_subset_eq
thf(fact_634_asymI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 ) )
=> ( asym_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% asymI
thf(fact_635_asymI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 ) )
=> ( asym_on_a @ top_top_set_a @ R2 ) ) ).
% asymI
thf(fact_636_comp__apply,axiom,
( comp_option_a_a_a
= ( ^ [F3: option_a > a,G2: a > option_a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_637_asym__onI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 ) ) ) )
=> ( asym_on_a @ A2 @ R2 ) ) ).
% asym_onI
thf(fact_638_asym__onI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
( ! [X3: option_a,Y4: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( ( member_option_a @ Y4 @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ X3 ) @ R2 ) ) ) )
=> ( asym_on_option_a @ A2 @ R2 ) ) ).
% asym_onI
thf(fact_639_case__map__option,axiom,
! [G: a,H: option_a > a,F: a > option_a,X: option_a] :
( ( case_o926465512965637841tion_a @ G @ H @ ( map_op2340691886215429841tion_a @ F @ X ) )
= ( case_option_a_a @ G @ ( comp_option_a_a_a @ H @ F ) @ X ) ) ).
% case_map_option
thf(fact_640_case__map__option,axiom,
! [G: $o,H: a > $o,F: a > a,X: option_a] :
( ( case_option_o_a @ G @ H @ ( map_option_a_a2 @ F @ X ) )
= ( case_option_o_a @ G @ ( comp_a_o_a @ H @ F ) @ X ) ) ).
% case_map_option
thf(fact_641_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_642_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_643_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_644_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_a2 @ ( comp_option_a_a_a @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_645_map__option_Ocompositionality,axiom,
! [F: a > a,G: a > a,Option: option_a] :
( ( map_option_a_a2 @ F @ ( map_option_a_a2 @ G @ Option ) )
= ( map_option_a_a2 @ ( comp_a_a_a @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_646_option_Omap__comp,axiom,
! [G: option_a > a,F: a > option_a,V3: option_a] :
( ( map_op4563205767754224965on_a_a @ G @ ( map_op2340691886215429841tion_a @ F @ V3 ) )
= ( map_option_a_a2 @ ( comp_option_a_a_a @ G @ F ) @ V3 ) ) ).
% option.map_comp
thf(fact_647_option_Omap__comp,axiom,
! [G: a > a,F: a > a,V3: option_a] :
( ( map_option_a_a2 @ G @ ( map_option_a_a2 @ F @ V3 ) )
= ( map_option_a_a2 @ ( comp_a_a_a @ G @ F ) @ V3 ) ) ).
% option.map_comp
thf(fact_648_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_a2 @ ( comp_option_a_a_a @ F @ G ) ) ) ).
% map_option.comp
thf(fact_649_map__option_Ocomp,axiom,
! [F: a > a,G: a > a] :
( ( comp_o3154387707078715297tion_a @ ( map_option_a_a2 @ F ) @ ( map_option_a_a2 @ G ) )
= ( map_option_a_a2 @ ( comp_a_a_a @ F @ G ) ) ) ).
% map_option.comp
thf(fact_650_vimage__comp,axiom,
! [F: a > option_a,G: option_a > a,X: set_a] :
( ( vimage_a_option_a @ F @ ( vimage_option_a_a @ G @ X ) )
= ( vimage_a_a @ ( comp_option_a_a_a @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_651_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_652_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_653_image__comp,axiom,
! [F: option_a > option_a,G: a > option_a,R2: set_a] :
( ( image_7439109396645324421tion_a @ F @ ( image_a_option_a2 @ G @ R2 ) )
= ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_654_image__comp,axiom,
! [F: a > a,G: option_a > a,R2: set_option_a] :
( ( image_a_a2 @ F @ ( image_option_a_a2 @ G @ R2 ) )
= ( image_option_a_a2 @ ( comp_a_a_option_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_655_image__comp,axiom,
! [F: a > option_a,G: a > a,R2: set_a] :
( ( image_a_option_a2 @ F @ ( image_a_a2 @ G @ R2 ) )
= ( image_a_option_a2 @ ( comp_a_option_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_656_image__comp,axiom,
! [F: a > option_a,G: option_a > a,R2: set_option_a] :
( ( image_a_option_a2 @ F @ ( image_option_a_a2 @ G @ R2 ) )
= ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_657_image__comp,axiom,
! [F: option_a > a,G: option_a > option_a,R2: set_option_a] :
( ( image_option_a_a2 @ F @ ( image_7439109396645324421tion_a @ G @ R2 ) )
= ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_658_image__comp,axiom,
! [F: option_a > a,G: a > option_a,R2: set_a] :
( ( image_option_a_a2 @ F @ ( image_a_option_a2 @ G @ R2 ) )
= ( image_a_a2 @ ( comp_option_a_a_a @ F @ G ) @ R2 ) ) ).
% image_comp
thf(fact_659_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_a2 @ G @ B2 ) )
=> ( ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ F ) @ A2 )
= ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_660_image__eq__imp__comp,axiom,
! [F: a > a,A2: set_a,G: option_a > a,B2: set_option_a,H: a > option_a] :
( ( ( image_a_a2 @ F @ A2 )
= ( image_option_a_a2 @ G @ B2 ) )
=> ( ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ F ) @ A2 )
= ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_661_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_a2 @ F @ A2 )
= ( image_7439109396645324421tion_a @ G @ B2 ) )
=> ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F ) @ A2 )
= ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_662_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_a2 @ F @ A2 )
= ( image_a_option_a2 @ G @ B2 ) )
=> ( ( image_a_a2 @ ( comp_option_a_a_a @ H @ F ) @ A2 )
= ( image_a_a2 @ ( comp_option_a_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_663_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_a2 @ F @ A2 )
= ( image_a_option_a2 @ G @ B2 ) )
=> ( ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ F ) @ A2 )
= ( image_a_option_a2 @ ( comp_o6087033147929006299on_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_664_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_a2 @ F @ A2 )
= ( image_a_a2 @ G @ B2 ) )
=> ( ( image_7439109396645324421tion_a @ ( comp_a6249931511552232923tion_a @ H @ F ) @ A2 )
= ( image_a_option_a2 @ ( comp_a_option_a_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_665_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_a2 @ F @ A2 )
= ( image_option_a_a2 @ G @ B2 ) )
=> ( ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ F ) @ A2 )
= ( image_option_a_a2 @ ( comp_a_a_option_a @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_666_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_a2 @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_667_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_a2 @ F @ A2 )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_668_comp__def,axiom,
( comp_option_a_a_a
= ( ^ [F3: option_a > a,G2: a > option_a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_669_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_670_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_671_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_672_comp__eq__dest,axiom,
! [A: option_a > a,B: a > option_a,C: option_a > a,D2: a > option_a,V3: a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_option_a_a_a @ C @ D2 ) )
=> ( ( A @ ( B @ V3 ) )
= ( C @ ( D2 @ V3 ) ) ) ) ).
% comp_eq_dest
thf(fact_673_comp__eq__elim,axiom,
! [A: option_a > a,B: a > option_a,C: option_a > a,D2: a > option_a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_option_a_a_a @ C @ D2 ) )
=> ! [V4: a] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_674_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_675_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_676_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_677_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_678_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_679_comp__eq__dest__lhs,axiom,
! [A: option_a > a,B: a > option_a,C: a > a,V3: a] :
( ( ( comp_option_a_a_a @ A @ B )
= C )
=> ( ( A @ ( B @ V3 ) )
= ( C @ V3 ) ) ) ).
% comp_eq_dest_lhs
thf(fact_680_bij__betw__comp__iff,axiom,
! [F: a > option_a,A2: set_a,A9: set_option_a,F4: option_a > a,A10: set_a] :
( ( bij_betw_a_option_a @ F @ A2 @ A9 )
=> ( ( bij_betw_option_a_a @ F4 @ A9 @ A10 )
= ( bij_betw_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 @ A10 ) ) ) ).
% bij_betw_comp_iff
thf(fact_681_bij__betw__iff__bijections,axiom,
( bij_betw_a_a
= ( ^ [F3: a > a,A7: set_a,B6: set_a] :
? [G2: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ( ( member_a @ ( F3 @ X2 ) @ B6 )
& ( ( G2 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B6 )
=> ( ( member_a @ ( G2 @ X2 ) @ A7 )
& ( ( F3 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_682_bij__betw__iff__bijections,axiom,
( bij_betw_option_a_a
= ( ^ [F3: option_a > a,A7: set_option_a,B6: set_a] :
? [G2: a > option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
=> ( ( member_a @ ( F3 @ X2 ) @ B6 )
& ( ( G2 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B6 )
=> ( ( member_option_a @ ( G2 @ X2 ) @ A7 )
& ( ( F3 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_683_bij__betw__iff__bijections,axiom,
( bij_betw_a_option_a
= ( ^ [F3: a > option_a,A7: set_a,B6: set_option_a] :
? [G2: option_a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ( ( member_option_a @ ( F3 @ X2 ) @ B6 )
& ( ( G2 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: option_a] :
( ( member_option_a @ X2 @ B6 )
=> ( ( member_a @ ( G2 @ X2 ) @ A7 )
& ( ( F3 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_684_bij__betw__iff__bijections,axiom,
( bij_be5431266891817924854tion_a
= ( ^ [F3: option_a > option_a,A7: set_option_a,B6: set_option_a] :
? [G2: option_a > option_a] :
( ! [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
=> ( ( member_option_a @ ( F3 @ X2 ) @ B6 )
& ( ( G2 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: option_a] :
( ( member_option_a @ X2 @ B6 )
=> ( ( member_option_a @ ( G2 @ X2 ) @ A7 )
& ( ( F3 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_685_fun__upd__comp,axiom,
! [F: option_a > a,G: a > option_a,X: a,Y: option_a] :
( ( comp_option_a_a_a @ F @ ( fun_upd_a_option_a @ G @ X @ Y ) )
= ( fun_upd_a_a @ ( comp_option_a_a_a @ F @ G ) @ X @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_bij__iff,axiom,
! [F: option_a > option_a] :
( ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a )
= ( ! [X2: option_a] :
? [Y3: option_a] :
( ( ( F @ Y3 )
= X2 )
& ! [Z3: option_a] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_695_bij__iff,axiom,
! [F: option_a > a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
= ( ! [X2: a] :
? [Y3: option_a] :
( ( ( F @ Y3 )
= X2 )
& ! [Z3: option_a] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_696_bij__iff,axiom,
! [F: a > option_a] :
( ( bij_betw_a_option_a @ F @ top_top_set_a @ top_top_set_option_a )
= ( ! [X2: option_a] :
? [Y3: a] :
( ( ( F @ Y3 )
= X2 )
& ! [Z3: a] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_697_bij__iff,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
= ( ! [X2: a] :
? [Y3: a] :
( ( ( F @ Y3 )
= X2 )
& ! [Z3: a] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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 )
=> ~ ! [X3: option_a] :
( ( Y
= ( F @ X3 ) )
=> ~ ! [X9: option_a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X3 ) ) ) ) ).
% bij_pointE
thf(fact_707_bij__pointE,axiom,
! [F: option_a > a,Y: a] :
( ( bij_betw_option_a_a @ F @ top_top_set_option_a @ top_top_set_a )
=> ~ ! [X3: option_a] :
( ( Y
= ( F @ X3 ) )
=> ~ ! [X9: option_a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X3 ) ) ) ) ).
% bij_pointE
thf(fact_708_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 )
=> ~ ! [X3: a] :
( ( Y
= ( F @ X3 ) )
=> ~ ! [X9: a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X3 ) ) ) ) ).
% bij_pointE
thf(fact_709_bij__pointE,axiom,
! [F: a > a,Y: a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ~ ! [X3: a] :
( ( Y
= ( F @ X3 ) )
=> ~ ! [X9: a] :
( ( Y
= ( F @ X9 ) )
=> ( X9 = X3 ) ) ) ) ).
% bij_pointE
thf(fact_710_involuntory__imp__bij,axiom,
! [F: option_a > option_a] :
( ! [X3: option_a] :
( ( F @ ( F @ X3 ) )
= X3 )
=> ( bij_be5431266891817924854tion_a @ F @ top_top_set_option_a @ top_top_set_option_a ) ) ).
% involuntory_imp_bij
thf(fact_711_involuntory__imp__bij,axiom,
! [F: a > a] :
( ! [X3: a] :
( ( F @ ( F @ X3 ) )
= X3 )
=> ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a ) ) ).
% involuntory_imp_bij
thf(fact_712_asym__onD,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( asym_on_a @ A2 @ R2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R2 ) ) ) ) ) ).
% asym_onD
thf(fact_713_asym__onD,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( asym_on_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( member_option_a @ Y @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X ) @ R2 ) ) ) ) ) ).
% asym_onD
thf(fact_714_bij__betw__comp__iff2,axiom,
! [F4: option_a > a,A9: set_option_a,A10: set_a,F: a > option_a,A2: set_a] :
( ( bij_betw_option_a_a @ F4 @ A9 @ A10 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A2 ) @ A9 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A9 )
= ( bij_betw_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 @ A10 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_715_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_716_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_a2 @ F @ top_top_set_a )
= top_top_set_option_a ) ) ).
% bij_betw_imp_surj
thf(fact_717_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_a2 @ F @ top_top_set_option_a )
= top_top_set_a ) ) ).
% bij_betw_imp_surj
thf(fact_718_bij__betw__imp__surj,axiom,
! [F: a > a,A2: set_a] :
( ( bij_betw_a_a @ F @ A2 @ top_top_set_a )
=> ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a ) ) ).
% bij_betw_imp_surj
thf(fact_719_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_720_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_a2 @ F @ top_top_set_option_a )
= top_top_set_a ) ) ).
% bij_is_surj
thf(fact_721_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_a2 @ F @ top_top_set_a )
= top_top_set_option_a ) ) ).
% bij_is_surj
thf(fact_722_bij__is__surj,axiom,
! [F: a > a] :
( ( bij_betw_a_a @ F @ top_top_set_a @ top_top_set_a )
=> ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a ) ) ).
% bij_is_surj
thf(fact_723_bij__betw__subset,axiom,
! [F: a > option_a,A2: set_a,A9: set_option_a,B2: set_a,B7: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ A9 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ( image_a_option_a2 @ F @ B2 )
= B7 )
=> ( bij_betw_a_option_a @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_724_bij__betw__subset,axiom,
! [F: option_a > a,A2: set_option_a,A9: set_a,B2: set_option_a,B7: set_a] :
( ( bij_betw_option_a_a @ F @ A2 @ A9 )
=> ( ( ord_le1955136853071979460tion_a @ B2 @ A2 )
=> ( ( ( image_option_a_a2 @ F @ B2 )
= B7 )
=> ( bij_betw_option_a_a @ F @ B2 @ B7 ) ) ) ) ).
% bij_betw_subset
thf(fact_725_bij__betw__byWitness,axiom,
! [A2: set_a,F4: option_a > a,F: a > option_a,A9: set_option_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F4 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A9 )
=> ( ( F @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F @ A2 ) @ A9 )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F4 @ A9 ) @ A2 )
=> ( bij_betw_a_option_a @ F @ A2 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_726_bij__betw__byWitness,axiom,
! [A2: set_option_a,F4: a > option_a,F: option_a > a,A9: set_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( ( F4 @ ( F @ X3 ) )
= X3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A9 )
=> ( ( F @ ( F4 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq_set_a @ ( image_option_a_a2 @ F @ A2 ) @ A9 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_a_option_a2 @ F4 @ A9 ) @ A2 )
=> ( bij_betw_option_a_a @ F @ A2 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_727_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_728_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_729_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_a2 @ F @ A2 )
= B2 )
=> ( bij_betw_option_a_a @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_730_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_a2 @ F @ A2 )
= B2 )
=> ( bij_betw_a_option_a @ F @ A2 @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_731_bij__betw__def,axiom,
( bij_betw_option_a_a
= ( ^ [F3: option_a > a,A7: set_option_a,B6: set_a] :
( ( inj_on_option_a_a @ F3 @ A7 )
& ( ( image_option_a_a2 @ F3 @ A7 )
= B6 ) ) ) ) ).
% bij_betw_def
thf(fact_732_bij__betw__def,axiom,
( bij_betw_a_option_a
= ( ^ [F3: a > option_a,A7: set_a,B6: set_option_a] :
( ( inj_on_a_option_a @ F3 @ A7 )
& ( ( image_a_option_a2 @ F3 @ A7 )
= B6 ) ) ) ) ).
% bij_betw_def
thf(fact_733_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_734_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_735_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_736_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_737_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_738_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_a2 @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_option_a_a2 @ ( comp_o3864519266390211175tion_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_739_comp__surj,axiom,
! [F: option_a > a,G: a > option_a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_option_a2 @ 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_740_comp__surj,axiom,
! [F: option_a > a,G: a > a] :
( ( ( image_option_a_a2 @ F @ top_top_set_option_a )
= top_top_set_a )
=> ( ( ( image_a_a2 @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_option_a_a2 @ ( comp_a_a_option_a @ G @ F ) @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_741_comp__surj,axiom,
! [F: a > option_a,G: option_a > option_a] :
( ( ( image_a_option_a2 @ 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_a2 @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_742_comp__surj,axiom,
! [F: a > option_a,G: option_a > a] :
( ( ( image_a_option_a2 @ F @ top_top_set_a )
= top_top_set_option_a )
=> ( ( ( image_option_a_a2 @ G @ top_top_set_option_a )
= top_top_set_a )
=> ( ( image_a_a2 @ ( comp_option_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_743_comp__surj,axiom,
! [F: a > a,G: a > option_a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_option_a2 @ G @ top_top_set_a )
= top_top_set_option_a )
=> ( ( image_a_option_a2 @ ( comp_a_option_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% comp_surj
thf(fact_744_comp__surj,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( image_a_a2 @ G @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a2 @ ( comp_a_a_a @ G @ F ) @ top_top_set_a )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_745_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_a2 @ F @ A2 ) )
= ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_746_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_a2 @ F @ A2 ) )
= ( inj_on_a_option_a @ ( comp_a_option_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_747_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_a2 @ F @ A2 ) )
= ( inj_on_a_a @ ( comp_option_a_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_748_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_a2 @ F @ A2 ) )
= ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ F4 @ F ) @ A2 ) ) ) ).
% comp_inj_on_iff
thf(fact_749_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_750_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_751_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_752_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_a2 @ F @ A2 ) ) ) ).
% inj_on_imageI
thf(fact_753_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_a2 @ F @ A2 ) )
=> ( inj_on8559383841115902449tion_a @ ( comp_a6249931511552232923tion_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_754_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_a2 @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( comp_a_option_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_755_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_a2 @ F @ A2 ) )
=> ( inj_on_a_a @ ( comp_option_a_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_756_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_a2 @ F @ A2 ) )
=> ( inj_on_a_option_a @ ( comp_o6087033147929006299on_a_a @ G @ F ) @ A2 ) ) ) ).
% comp_inj_on
thf(fact_757_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_758_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_759_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_760_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_761_bind__map__option,axiom,
! [F: a > a,X: option_a,G: a > option_a] :
( ( bind_a_a @ ( map_option_a_a2 @ F @ X ) @ G )
= ( bind_a_a @ X @ ( comp_a_option_a_a @ G @ F ) ) ) ).
% bind_map_option
thf(fact_762_map__option__bind,axiom,
! [F: a > a,X: option_a,G: a > option_a] :
( ( map_option_a_a2 @ F @ ( bind_a_a @ X @ G ) )
= ( bind_a_a @ X @ ( comp_o6087033147929006299on_a_a @ ( map_option_a_a2 @ F ) @ G ) ) ) ).
% map_option_bind
thf(fact_763_asymD,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( asym_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y @ X ) @ R2 ) ) ) ).
% asymD
thf(fact_764_asymD,axiom,
! [R2: set_Product_prod_a_a,X: a,Y: a] :
( ( asym_on_a @ top_top_set_a @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R2 ) ) ) ).
% asymD
thf(fact_765_asym__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ( asym_on_option_a @ top_top_set_option_a @ R2 )
= ( ! [X2: option_a,Y3: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X2 @ Y3 ) @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X2 ) @ R2 ) ) ) ) ).
% asym_iff
thf(fact_766_asym__iff,axiom,
! [R2: set_Product_prod_a_a] :
( ( asym_on_a @ top_top_set_a @ R2 )
= ( ! [X2: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ R2 ) ) ) ) ).
% asym_iff
thf(fact_767_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_768_bijI,axiom,
! [F: option_a > a] :
( ( inj_on_option_a_a @ F @ top_top_set_option_a )
=> ( ( ( image_option_a_a2 @ 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_769_bijI,axiom,
! [F: a > option_a] :
( ( inj_on_a_option_a @ F @ top_top_set_a )
=> ( ( ( image_a_option_a2 @ 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_770_bijI,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( ( image_a_a2 @ 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_771_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_772_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_a2 @ F @ top_top_set_option_a )
= top_top_set_a ) ) ) ).
% bij_def
thf(fact_773_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_a2 @ F @ top_top_set_a )
= top_top_set_option_a ) ) ) ).
% bij_def
thf(fact_774_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_a2 @ F @ top_top_set_a )
= top_top_set_a ) ) ) ).
% bij_def
thf(fact_775_notIn__Un__bij__betw3,axiom,
! [B: a,A2: set_a,F: a > a,A9: set_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_a_a @ F @ A2 @ A9 )
= ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_776_notIn__Un__bij__betw3,axiom,
! [B: a,A2: set_a,F: a > option_a,A9: set_option_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A9 )
= ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_777_notIn__Un__bij__betw3,axiom,
! [B: option_a,A2: set_option_a,F: option_a > a,A9: set_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_option_a_a @ F @ A2 @ A9 )
= ( 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 @ A9 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_778_notIn__Un__bij__betw3,axiom,
! [B: option_a,A2: set_option_a,F: option_a > option_a,A9: set_option_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A9 )
=> ( ( bij_be5431266891817924854tion_a @ F @ A2 @ A9 )
= ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_779_notIn__Un__bij__betw,axiom,
! [B: a,A2: set_a,F: a > a,A9: set_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_a_a @ F @ A2 @ A9 )
=> ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_a @ A9 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_780_notIn__Un__bij__betw,axiom,
! [B: a,A2: set_a,F: a > option_a,A9: set_option_a] :
( ~ ( member_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_a_option_a @ F @ A2 @ A9 )
=> ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_781_notIn__Un__bij__betw,axiom,
! [B: option_a,A2: set_option_a,F: option_a > a,A9: set_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_a @ ( F @ B ) @ A9 )
=> ( ( bij_betw_option_a_a @ F @ A2 @ A9 )
=> ( 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 @ A9 @ ( insert_a @ ( F @ B ) @ bot_bot_set_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_782_notIn__Un__bij__betw,axiom,
! [B: option_a,A2: set_option_a,F: option_a > option_a,A9: set_option_a] :
( ~ ( member_option_a @ B @ A2 )
=> ( ~ ( member_option_a @ ( F @ B ) @ A9 )
=> ( ( bij_be5431266891817924854tion_a @ F @ A2 @ A9 )
=> ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( sup_sup_set_option_a @ A9 @ ( insert_option_a @ ( F @ B ) @ bot_bot_set_option_a ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_783_bij__betw__partition,axiom,
! [F: a > a,A2: set_a,C3: set_a,B2: set_a,D: set_a] :
( ( bij_betw_a_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D ) )
=> ( ( bij_betw_a_a @ F @ C3 @ D )
=> ( ( ( inf_inf_set_a @ A2 @ C3 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ B2 @ D )
= bot_bot_set_a )
=> ( bij_betw_a_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_784_bij__betw__partition,axiom,
! [F: a > option_a,A2: set_a,C3: set_a,B2: set_option_a,D: set_option_a] :
( ( bij_betw_a_option_a @ F @ ( sup_sup_set_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D ) )
=> ( ( bij_betw_a_option_a @ F @ C3 @ D )
=> ( ( ( inf_inf_set_a @ A2 @ C3 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_option_a @ B2 @ D )
= bot_bot_set_option_a )
=> ( bij_betw_a_option_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_785_bij__betw__partition,axiom,
! [F: option_a > a,A2: set_option_a,C3: set_option_a,B2: set_a,D: set_a] :
( ( bij_betw_option_a_a @ F @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_a @ B2 @ D ) )
=> ( ( bij_betw_option_a_a @ F @ C3 @ D )
=> ( ( ( inf_inf_set_option_a @ A2 @ C3 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_a @ B2 @ D )
= bot_bot_set_a )
=> ( bij_betw_option_a_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_786_bij__betw__partition,axiom,
! [F: option_a > option_a,A2: set_option_a,C3: set_option_a,B2: set_option_a,D: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ ( sup_sup_set_option_a @ A2 @ C3 ) @ ( sup_sup_set_option_a @ B2 @ D ) )
=> ( ( bij_be5431266891817924854tion_a @ F @ C3 @ D )
=> ( ( ( inf_inf_set_option_a @ A2 @ C3 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_option_a @ B2 @ D )
= bot_bot_set_option_a )
=> ( bij_be5431266891817924854tion_a @ F @ A2 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_787_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_788_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_a2 @ F @ ( uminus6205308855922866075tion_a @ A2 ) )
= ( uminus_uminus_set_a @ ( image_option_a_a2 @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_789_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_a2 @ F @ ( uminus_uminus_set_a @ A2 ) )
= ( uminus6205308855922866075tion_a @ ( image_a_option_a2 @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_790_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_a2 @ F @ ( uminus_uminus_set_a @ A2 ) )
= ( uminus_uminus_set_a @ ( image_a_a2 @ F @ A2 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_791_the__inv__into__comp,axiom,
! [F: a > a,G: option_a > a,A2: set_option_a,X: a] :
( ( inj_on_a_a @ F @ ( image_option_a_a2 @ G @ A2 ) )
=> ( ( inj_on_option_a_a @ G @ A2 )
=> ( ( member_a @ X @ ( image_a_a2 @ F @ ( image_option_a_a2 @ G @ A2 ) ) )
=> ( ( the_in1757154643552616557on_a_a @ A2 @ ( comp_a_a_option_a @ F @ G ) @ X )
= ( comp_a_option_a_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_inv_into_a_a @ ( image_option_a_a2 @ G @ A2 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_792_the__inv__into__comp,axiom,
! [F: option_a > a,G: a > option_a,A2: set_a,X: a] :
( ( inj_on_option_a_a @ F @ ( image_a_option_a2 @ G @ A2 ) )
=> ( ( inj_on_a_option_a @ G @ A2 )
=> ( ( member_a @ X @ ( image_option_a_a2 @ F @ ( image_a_option_a2 @ G @ A2 ) ) )
=> ( ( the_inv_into_a_a @ A2 @ ( comp_option_a_a_a @ F @ G ) @ X )
= ( comp_option_a_a_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in1757154643552616557on_a_a @ ( image_a_option_a2 @ G @ A2 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_793_the__inv__into__comp,axiom,
! [F: option_a > option_a,G: a > option_a,A2: set_a,X: option_a] :
( ( inj_on8559383841115902449tion_a @ F @ ( image_a_option_a2 @ G @ A2 ) )
=> ( ( inj_on_a_option_a @ G @ A2 )
=> ( ( member_option_a @ X @ ( image_7439109396645324421tion_a @ F @ ( image_a_option_a2 @ G @ A2 ) ) )
=> ( ( the_in8758012798868597241tion_a @ A2 @ ( comp_o6087033147929006299on_a_a @ F @ G ) @ X )
= ( comp_o3864519266390211175tion_a @ ( the_in8758012798868597241tion_a @ A2 @ G ) @ ( the_in2538339130118444083tion_a @ ( image_a_option_a2 @ G @ A2 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_794_the__inv__into__comp,axiom,
! [F: a > option_a,G: option_a > a,A2: set_option_a,X: option_a] :
( ( inj_on_a_option_a @ F @ ( image_option_a_a2 @ G @ A2 ) )
=> ( ( inj_on_option_a_a @ G @ A2 )
=> ( ( member_option_a @ X @ ( image_a_option_a2 @ F @ ( image_option_a_a2 @ G @ A2 ) ) )
=> ( ( the_in2538339130118444083tion_a @ A2 @ ( comp_a6249931511552232923tion_a @ F @ G ) @ X )
= ( comp_a6249931511552232923tion_a @ ( the_in1757154643552616557on_a_a @ A2 @ G ) @ ( the_in8758012798868597241tion_a @ ( image_option_a_a2 @ G @ A2 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_795_map__conv__bind__option,axiom,
( map_option_a_a2
= ( ^ [F3: a > a,X2: option_a] : ( bind_a_a @ X2 @ ( comp_a_option_a_a @ some_a @ F3 ) ) ) ) ).
% map_conv_bind_option
thf(fact_796_comp__the__Some,axiom,
( ( comp_option_a_a_a @ the_a @ some_a )
= id_a ) ).
% comp_the_Some
thf(fact_797_Image__singleton__iff,axiom,
! [B: a,R2: set_Product_prod_a_a,A: a] :
( ( member_a @ B @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_798_Image__singleton__iff,axiom,
! [B: option_a,R2: set_Pr3411724424142761165tion_a,A: a] :
( ( member_option_a @ B @ ( image_a_option_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_799_Image__singleton__iff,axiom,
! [B: a,R2: set_Pr6308966090954093121on_a_a,A: option_a] :
( ( member_a @ B @ ( image_option_a_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_800_Image__singleton__iff,axiom,
! [B: option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_801_id__apply,axiom,
( id_a
= ( ^ [X2: a] : X2 ) ) ).
% id_apply
thf(fact_802_image__id,axiom,
( ( image_a_a2 @ id_a )
= id_set_a ) ).
% image_id
thf(fact_803_ImageI,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,A2: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A2 )
=> ( member_a @ B @ ( image_a_a @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_804_ImageI,axiom,
! [A: a,B: option_a,R2: set_Pr3411724424142761165tion_a,A2: set_a] :
( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 )
=> ( ( member_a @ A @ A2 )
=> ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_805_ImageI,axiom,
! [A: option_a,B: a,R2: set_Pr6308966090954093121on_a_a,A2: set_option_a] :
( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_806_ImageI,axiom,
! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ( member_option_a @ A @ A2 )
=> ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) ) ) ) ).
% ImageI
thf(fact_807_bij__betw__id,axiom,
! [A2: set_a] : ( bij_betw_a_a @ id_a @ A2 @ A2 ) ).
% bij_betw_id
thf(fact_808_vimage__id,axiom,
( ( vimage_a_a @ id_a )
= id_set_a ) ).
% vimage_id
thf(fact_809_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_810_comp__eq__id__dest,axiom,
! [A: option_a > a,B: a > option_a,C: a > a,V3: a] :
( ( ( comp_option_a_a_a @ A @ B )
= ( comp_a_a_a @ id_a @ C ) )
=> ( ( A @ ( B @ V3 ) )
= ( C @ V3 ) ) ) ).
% comp_eq_id_dest
thf(fact_811_rev__ImageI,axiom,
! [A: a,A2: set_a,B: a,R2: set_Product_prod_a_a] :
( ( member_a @ A @ A2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( member_a @ B @ ( image_a_a @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_812_rev__ImageI,axiom,
! [A: a,A2: set_a,B: option_a,R2: set_Pr3411724424142761165tion_a] :
( ( member_a @ A @ A2 )
=> ( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ R2 )
=> ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_813_rev__ImageI,axiom,
! [A: option_a,A2: set_option_a,B: a,R2: set_Pr6308966090954093121on_a_a] :
( ( member_option_a @ A @ A2 )
=> ( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ R2 )
=> ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_814_rev__ImageI,axiom,
! [A: option_a,A2: set_option_a,B: option_a,R2: set_Pr7585778909603769095tion_a] :
( ( member_option_a @ A @ A2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) ) ) ) ).
% rev_ImageI
thf(fact_815_ImageE,axiom,
! [B: a,R2: set_Product_prod_a_a,A2: set_a] :
( ( member_a @ B @ ( image_a_a @ R2 @ A2 ) )
=> ~ ! [X3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ B ) @ R2 )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_816_ImageE,axiom,
! [B: a,R2: set_Pr6308966090954093121on_a_a,A2: set_option_a] :
( ( member_a @ B @ ( image_option_a_a @ R2 @ A2 ) )
=> ~ ! [X3: option_a] :
( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X3 @ B ) @ R2 )
=> ~ ( member_option_a @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_817_ImageE,axiom,
! [B: option_a,R2: set_Pr3411724424142761165tion_a,A2: set_a] :
( ( member_option_a @ B @ ( image_a_option_a @ R2 @ A2 ) )
=> ~ ! [X3: a] :
( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X3 @ B ) @ R2 )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_818_ImageE,axiom,
! [B: option_a,R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
( ( member_option_a @ B @ ( image_4442594622209975379tion_a @ R2 @ A2 ) )
=> ~ ! [X3: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ B ) @ R2 )
=> ~ ( member_option_a @ X3 @ A2 ) ) ) ).
% ImageE
thf(fact_819_eq__id__iff,axiom,
! [F: a > a] :
( ( ! [X2: a] :
( ( F @ X2 )
= X2 ) )
= ( F = id_a ) ) ).
% eq_id_iff
thf(fact_820_id__def,axiom,
( id_a
= ( ^ [X2: a] : X2 ) ) ).
% id_def
thf(fact_821_option_Omap__id,axiom,
! [T: option_a] :
( ( map_option_a_a2 @ id_a @ T )
= T ) ).
% option.map_id
thf(fact_822_option_Omap__id0,axiom,
( ( map_option_a_a2 @ id_a )
= id_option_a ) ).
% option.map_id0
thf(fact_823_inj__on__id,axiom,
! [A2: set_a] : ( inj_on_a_a @ id_a @ A2 ) ).
% inj_on_id
thf(fact_824_surj__id,axiom,
( ( image_7439109396645324421tion_a @ id_option_a @ top_top_set_option_a )
= top_top_set_option_a ) ).
% surj_id
thf(fact_825_surj__id,axiom,
( ( image_a_a2 @ id_a @ top_top_set_a )
= top_top_set_a ) ).
% surj_id
thf(fact_826_bij__id,axiom,
bij_be5431266891817924854tion_a @ id_option_a @ top_top_set_option_a @ top_top_set_option_a ).
% bij_id
thf(fact_827_bij__id,axiom,
bij_betw_a_a @ id_a @ top_top_set_a @ top_top_set_a ).
% bij_id
thf(fact_828_subset__Image1__Image1__iff,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a] :
( ( order_preorder_on_a @ ( field_a @ R2 ) @ R2 )
=> ( ( member_a @ A @ ( field_a @ R2 ) )
=> ( ( member_a @ B @ ( field_a @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_829_subset__Image1__Image1__iff,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( order_4134995541221112539tion_a @ ( field_option_a @ R2 ) @ R2 )
=> ( ( member_option_a @ A @ ( field_option_a @ R2 ) )
=> ( ( member_option_a @ B @ ( field_option_a @ R2 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B @ A ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_830_equiv__class__nondisjoint,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,A: a,B: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_a @ X @ ( inf_inf_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_831_equiv__class__nondisjoint,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,A: option_a,B: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ ( inf_inf_set_option_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_832_equiv__class__eq__iff,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
= ( ( ( image_a_a @ R2 @ ( insert_a @ X @ bot_bot_set_a ) )
= ( image_a_a @ R2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
& ( member_a @ X @ A2 )
& ( member_a @ Y @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_833_equiv__class__eq__iff,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
= ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
& ( member_option_a @ X @ A2 )
& ( member_option_a @ Y @ A2 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_834_eq__equiv__class__iff,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( image_a_a @ R2 @ ( insert_a @ X @ bot_bot_set_a ) )
= ( image_a_a @ R2 @ ( insert_a @ Y @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_835_eq__equiv__class__iff,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( member_option_a @ Y @ A2 )
=> ( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_836_equiv__class__eq,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_eq
thf(fact_837_equiv__class__eq,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).
% equiv_class_eq
thf(fact_838_eq__equiv__class,axiom,
! [R2: set_Product_prod_a_a,A: a,B: a,A2: set_a] :
( ( ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
=> ( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_a @ B @ A2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_839_eq__equiv__class,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a,A2: set_option_a] :
( ( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
=> ( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_option_a @ B @ A2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_840_subset__equiv__class,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,B: a,A: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ A2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_841_subset__equiv__class,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,B: option_a,A: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
=> ( ( member_option_a @ B @ A2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_842_equiv__class__subset,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ).
% equiv_class_subset
thf(fact_843_equiv__class__subset,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 )
=> ( ord_le1955136853071979460tion_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) ) ) ) ).
% equiv_class_subset
thf(fact_844_proj__iff,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( ord_less_eq_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A2 )
=> ( ( ( equiv_proj_a_a @ R2 @ X )
= ( equiv_proj_a_a @ R2 @ Y ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_845_proj__iff,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ A2 )
=> ( ( ( equiv_6865337221296424970tion_a @ R2 @ X )
= ( equiv_6865337221296424970tion_a @ R2 @ Y ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_846_disjnt__equiv__class,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( disjnt_a @ ( image_a_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_847_disjnt__equiv__class,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a,B: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( disjnt_option_a @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) @ ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ B @ bot_bot_set_option_a ) ) )
= ( ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_848_in__quotient__imp__in__rel,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X7: set_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_set_a @ X7 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ X7 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_849_in__quotient__imp__in__rel,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X7: set_option_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_set_option_a @ X7 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( ord_le1955136853071979460tion_a @ ( insert_option_a @ X @ ( insert_option_a @ Y @ bot_bot_set_option_a ) ) @ X7 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_850_quotient__eqI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X7: set_a,Y6: set_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_set_a @ X7 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( member_a @ X @ X7 )
=> ( ( member_a @ Y @ Y6 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( X7 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_851_quotient__eqI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X7: set_option_a,Y6: set_option_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_set_option_a @ X7 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( member_option_a @ X @ X7 )
=> ( ( member_option_a @ Y @ Y6 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( X7 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_852_quotient__eq__iff,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X7: set_a,Y6: set_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_set_a @ X7 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( member_set_a @ Y6 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( member_a @ X @ X7 )
=> ( ( member_a @ Y @ Y6 )
=> ( ( X7 = Y6 )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_853_quotient__eq__iff,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X7: set_option_a,Y6: set_option_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_set_option_a @ X7 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( member_set_option_a @ Y6 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( member_option_a @ X @ X7 )
=> ( ( member_option_a @ Y @ Y6 )
=> ( ( X7 = Y6 )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_854_in__quotient__imp__closed,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X7: set_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_set_a @ X7 @ ( equiv_quotient_a @ A2 @ R2 ) )
=> ( ( member_a @ X @ X7 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member_a @ Y @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_855_in__quotient__imp__closed,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X7: set_option_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_set_option_a @ X7 @ ( equiv_2859340374733651339tion_a @ A2 @ R2 ) )
=> ( ( member_option_a @ X @ X7 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 )
=> ( member_option_a @ Y @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_856_eq__equiv__class__iff2,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
( ( equiv_equiv_a @ A2 @ R2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ( ( equiv_quotient_a @ ( insert_a @ X @ bot_bot_set_a ) @ R2 )
= ( equiv_quotient_a @ ( insert_a @ Y @ bot_bot_set_a ) @ R2 ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_857_eq__equiv__class__iff2,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,X: option_a,Y: option_a] :
( ( equiv_equiv_option_a @ A2 @ R2 )
=> ( ( member_option_a @ X @ A2 )
=> ( ( member_option_a @ Y @ A2 )
=> ( ( ( equiv_2859340374733651339tion_a @ ( insert_option_a @ X @ bot_bot_set_option_a ) @ R2 )
= ( equiv_2859340374733651339tion_a @ ( insert_option_a @ Y @ bot_bot_set_option_a ) @ R2 ) )
= ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_858_option_Osize__gen__o__map,axiom,
! [F: a > nat,G: a > a] :
( ( comp_o8583038678572498833tion_a @ ( size_option_a @ F ) @ ( map_option_a_a2 @ G ) )
= ( size_option_a @ ( comp_a_nat_a @ F @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_859_map__upds__Cons,axiom,
! [M: a > option_a,A: a,As: list_a,B: a,Bs: list_a] :
( ( map_upds_a_a @ M @ ( cons_a @ A @ As ) @ ( cons_a @ B @ Bs ) )
= ( map_upds_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs ) ) ).
% map_upds_Cons
thf(fact_860_map__upds__twist,axiom,
! [A: option_a,As: list_option_a,M: option_a > option_a,B: a,Bs: list_a] :
( ~ ( member_option_a @ A @ ( set_option_a2 @ As ) )
=> ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
= ( fun_up1079276522633388797tion_a @ ( map_upds_option_a_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).
% map_upds_twist
thf(fact_861_map__upds__twist,axiom,
! [A: a,As: list_a,M: a > option_a,B: a,Bs: list_a] :
( ~ ( member_a @ A @ ( set_a2 @ As ) )
=> ( ( map_upds_a_a @ ( fun_upd_a_option_a @ M @ A @ ( some_a @ B ) ) @ As @ Bs )
= ( fun_upd_a_option_a @ ( map_upds_a_a @ M @ As @ Bs ) @ A @ ( some_a @ B ) ) ) ) ).
% map_upds_twist
thf(fact_862_restrict__map__upds,axiom,
! [Xs: list_a,Ys: list_a,D: set_a,M: a > option_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ D )
=> ( ( restrict_map_a_a @ ( map_upds_a_a @ M @ Xs @ Ys ) @ D )
= ( map_upds_a_a @ ( restrict_map_a_a @ M @ ( minus_minus_set_a @ D @ ( set_a2 @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_863_map__of__eq__None__iff,axiom,
! [Xys: list_P1396940483166286381od_a_a,X: a] :
( ( ( map_of_a_a @ Xys @ X )
= none_a )
= ( ~ ( member_a @ X @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_864_map__of__eq__None__iff,axiom,
! [Xys: list_P6260409590414597735on_a_a,X: option_a] :
( ( ( map_of_option_a_a @ Xys @ X )
= none_a )
= ( ~ ( member_option_a @ X @ ( image_3098826861768462248tion_a @ produc8941638570267940413on_a_a @ ( set_Pr1233600038994746358on_a_a @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_865_dom__map__upds,axiom,
! [M: a > option_a,Xs: list_a,Ys: list_a] :
( ( dom_a_a @ ( map_upds_a_a @ M @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) @ ( dom_a_a @ M ) ) ) ).
% dom_map_upds
thf(fact_866_map__upd__upds__conv__if,axiom,
! [X: option_a,Ys: list_a,Xs: list_option_a,F: option_a > option_a,Y: a] :
( ( ( member_option_a @ X @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
=> ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ ( some_a @ Y ) ) @ Xs @ Ys )
= ( map_upds_option_a_a @ F @ Xs @ Ys ) ) )
& ( ~ ( member_option_a @ X @ ( set_option_a2 @ ( take_option_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
=> ( ( map_upds_option_a_a @ ( fun_up1079276522633388797tion_a @ F @ X @ ( some_a @ Y ) ) @ Xs @ Ys )
= ( fun_up1079276522633388797tion_a @ ( map_upds_option_a_a @ F @ Xs @ Ys ) @ X @ ( some_a @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_867_map__upd__upds__conv__if,axiom,
! [X: a,Ys: list_a,Xs: list_a,F: a > option_a,Y: a] :
( ( ( member_a @ X @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
=> ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F @ X @ ( some_a @ Y ) ) @ Xs @ Ys )
= ( map_upds_a_a @ F @ Xs @ Ys ) ) )
& ( ~ ( member_a @ X @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Ys ) @ Xs ) ) )
=> ( ( map_upds_a_a @ ( fun_upd_a_option_a @ F @ X @ ( some_a @ Y ) ) @ Xs @ Ys )
= ( fun_upd_a_option_a @ ( map_upds_a_a @ F @ Xs @ Ys ) @ X @ ( some_a @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_868_map__of__zip__upd,axiom,
! [Ys: list_a,Xs: list_option_a,Zs: list_a,X: option_a,Y: a,Z2: a] :
( ( ( size_size_list_a @ Ys )
= ( size_s3078493964004954806tion_a @ Xs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s3078493964004954806tion_a @ Xs ) )
=> ( ~ ( member_option_a @ X @ ( set_option_a2 @ Xs ) )
=> ( ( ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) @ X @ ( some_a @ Y ) )
= ( fun_up1079276522633388797tion_a @ ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Zs ) ) @ X @ ( some_a @ Z2 ) ) )
=> ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) )
= ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_869_map__of__zip__upd,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a,X: a,Y: a,Z2: a] :
( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Xs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) @ X @ ( some_a @ Y ) )
= ( fun_upd_a_option_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) @ X @ ( some_a @ Z2 ) ) )
=> ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_870_map__of__zip__is__None,axiom,
! [Xs: list_option_a,Ys: list_a,X: option_a] :
( ( ( size_s3078493964004954806tion_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X )
= none_a )
= ( ~ ( member_option_a @ X @ ( set_option_a2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_871_map__of__zip__is__None,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X )
= none_a )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_872_dom__map__of__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( dom_a_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) )
= ( set_a2 @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_873_map__of__zip__is__Some,axiom,
! [Xs: list_option_a,Ys: list_a,X: option_a] :
( ( ( size_s3078493964004954806tion_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_option_a @ X @ ( set_option_a2 @ Xs ) )
= ( ? [Y3: a] :
( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ Ys ) @ X )
= ( some_a @ Y3 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_874_map__of__zip__is__Some,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Y3: a] :
( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ X )
= ( some_a @ Y3 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_875_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( zip_a_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_876_in__set__impl__in__set__zip1,axiom,
! [Xs: list_option_a,Ys: list_a,X: option_a] :
( ( ( size_s3078493964004954806tion_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_option_a @ X @ ( set_option_a2 @ Xs ) )
=> ~ ! [Y4: a] :
~ ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X @ Y4 ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_877_in__set__impl__in__set__zip1,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ~ ! [Y4: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_878_zip__same,axiom,
! [A: option_a,B: option_a,Xs: list_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( set_Pr948796958549772220tion_a @ ( zip_op6411647709037274935tion_a @ Xs @ Xs ) ) )
= ( ( member_option_a @ A @ ( set_option_a2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_879_zip__same,axiom,
! [A: a,B: a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
= ( ( member_a @ A @ ( set_a2 @ Xs ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_880_in__set__zipE,axiom,
! [X: option_a,Y: option_a,Xs: list_option_a,Ys: list_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ Y ) @ ( set_Pr948796958549772220tion_a @ ( zip_op6411647709037274935tion_a @ Xs @ Ys ) ) )
=> ~ ( ( member_option_a @ X @ ( set_option_a2 @ Xs ) )
=> ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_881_in__set__zipE,axiom,
! [X: option_a,Y: a,Xs: list_option_a,Ys: list_a] :
( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ X @ Y ) @ ( set_Pr1233600038994746358on_a_a @ ( zip_option_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_option_a @ X @ ( set_option_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_882_in__set__zipE,axiom,
! [X: a,Y: option_a,Xs: list_a,Ys: list_option_a] :
( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ~ ( member_option_a @ Y @ ( set_option_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_883_in__set__zipE,axiom,
! [X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
=> ~ ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_884_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
( ( ( zip_a_a @ Xs @ Ys )
= ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
=> ~ ! [X3: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Y4: a,Ys2: list_a] :
( ( Ys
= ( cons_a @ Y4 @ Ys2 ) )
=> ( ( Xy
= ( product_Pair_a_a @ X3 @ Y4 ) )
=> ( Xys
!= ( zip_a_a @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_885_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys: list_option_a,Y: option_a] :
( ( ( size_size_list_a @ Xs )
= ( size_s3078493964004954806tion_a @ Ys ) )
=> ( ( member_option_a @ Y @ ( set_option_a2 @ Ys ) )
=> ~ ! [X3: a] :
~ ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ X3 @ Y ) @ ( set_Pr2114800023962131586tion_a @ ( zip_a_option_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_886_in__set__impl__in__set__zip2,axiom,
! [Xs: list_a,Ys: list_a,Y: a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ~ ! [X3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_887_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R2 ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R2 ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_888_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R2 ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_889_lenlex__trans,axiom,
! [X: list_option_a,Y: list_option_a,R2: set_Pr7585778909603769095tion_a,Z2: list_option_a] :
( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X @ Y ) @ ( lenlex_option_a @ R2 ) )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Y @ Z2 ) @ ( lenlex_option_a @ R2 ) )
=> ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X @ Z2 ) @ ( lenlex_option_a @ R2 ) ) ) ) ) ).
% lenlex_trans
thf(fact_890_lenlex__trans,axiom,
! [X: list_a,Y: list_a,R2: set_Product_prod_a_a,Z2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lenlex_a @ R2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Z2 ) @ ( lenlex_a @ R2 ) )
=> ( ( trans_on_a @ top_top_set_a @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Z2 ) @ ( lenlex_a @ R2 ) ) ) ) ) ).
% lenlex_trans
thf(fact_891_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_892_partition__set,axiom,
! [P3: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
( ( ( partition_a @ P3 @ Xs )
= ( produc6837034575241423639list_a @ Yes @ No ) )
=> ( ( sup_sup_set_a @ ( set_a2 @ Yes ) @ ( set_a2 @ No ) )
= ( set_a2 @ Xs ) ) ) ).
% partition_set
thf(fact_893_partition__P,axiom,
! [P3: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
( ( ( partition_a @ P3 @ Xs )
= ( produc6837034575241423639list_a @ Yes @ No ) )
=> ( ! [X8: a] :
( ( member_a @ X8 @ ( set_a2 @ Yes ) )
=> ( P3 @ X8 ) )
& ! [X8: a] :
( ( member_a @ X8 @ ( set_a2 @ No ) )
=> ~ ( P3 @ X8 ) ) ) ) ).
% partition_P
thf(fact_894_lex__take__index,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I2 @ Xs )
= ( take_a @ I2 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_895_ran__map__of__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( distinct_a @ Xs )
=> ( ( ran_a_a @ ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) ) )
= ( set_a2 @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_896_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I )
= ( product_Pair_a_a @ ( nth_a @ Xs @ I ) @ ( nth_a @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_897_map__of__zip__nth,axiom,
! [Xs: list_a,Ys: list_a,I: nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) @ ( nth_a @ Xs @ I ) )
= ( some_a @ ( nth_a @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_898_map__of__zip__inject,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Xs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Xs ) )
=> ( ( distinct_a @ Xs )
=> ( ( ( map_of_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( map_of_a_a @ ( zip_a_a @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_899_map__of__zip__map,axiom,
! [Xs: list_option_a,F: option_a > a] :
( ( map_of_option_a_a @ ( zip_option_a_a @ Xs @ ( map_option_a_a @ F @ Xs ) ) )
= ( ^ [X2: option_a] : ( if_option_a @ ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) ) @ ( some_a @ ( F @ X2 ) ) @ none_a ) ) ) ).
% map_of_zip_map
thf(fact_900_map__of__zip__map,axiom,
! [Xs: list_a,F: a > a] :
( ( map_of_a_a @ ( zip_a_a @ Xs @ ( map_a_a @ F @ Xs ) ) )
= ( ^ [X2: a] : ( if_option_a @ ( member_a @ X2 @ ( set_a2 @ Xs ) ) @ ( some_a @ ( F @ X2 ) ) @ none_a ) ) ) ).
% map_of_zip_map
thf(fact_901_find__Some__iff2,axiom,
! [X: a,P3: a > $o,Xs: list_a] :
( ( ( some_a @ X )
= ( find_a @ P3 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( P3 @ ( nth_a @ Xs @ I3 ) )
& ( X
= ( nth_a @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ I3 )
=> ~ ( P3 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_902_find__Some__iff,axiom,
! [P3: a > $o,Xs: list_a,X: a] :
( ( ( find_a @ P3 @ Xs )
= ( some_a @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( P3 @ ( nth_a @ Xs @ I3 ) )
& ( X
= ( nth_a @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ I3 )
=> ~ ( P3 @ ( nth_a @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_903_find_Osimps_I2_J,axiom,
! [P3: a > $o,X: a,Xs: list_a] :
( ( ( P3 @ X )
=> ( ( find_a @ P3 @ ( cons_a @ X @ Xs ) )
= ( some_a @ X ) ) )
& ( ~ ( P3 @ X )
=> ( ( find_a @ P3 @ ( cons_a @ X @ Xs ) )
= ( find_a @ P3 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_904_find__None__iff2,axiom,
! [P3: option_a > $o,Xs: list_option_a] :
( ( none_option_a
= ( find_option_a @ P3 @ Xs ) )
= ( ~ ? [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
& ( P3 @ X2 ) ) ) ) ).
% find_None_iff2
thf(fact_905_find__None__iff2,axiom,
! [P3: a > $o,Xs: list_a] :
( ( none_a
= ( find_a @ P3 @ Xs ) )
= ( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P3 @ X2 ) ) ) ) ).
% find_None_iff2
thf(fact_906_find__None__iff,axiom,
! [P3: option_a > $o,Xs: list_option_a] :
( ( ( find_option_a @ P3 @ Xs )
= none_option_a )
= ( ~ ? [X2: option_a] :
( ( member_option_a @ X2 @ ( set_option_a2 @ Xs ) )
& ( P3 @ X2 ) ) ) ) ).
% find_None_iff
thf(fact_907_find__None__iff,axiom,
! [P3: a > $o,Xs: list_a] :
( ( ( find_a @ P3 @ Xs )
= none_a )
= ( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P3 @ X2 ) ) ) ) ).
% find_None_iff
thf(fact_908_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ ( nth_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_909_rtrancl__listrel1__ConsI2,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R2 ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_910_Cons__listrel1__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_911_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_912_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_913_rtrancl__listrel1__eq__len,axiom,
! [X: list_a,Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R2 ) ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_914_rtrancl__listrel1__ConsI1,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R2 ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_915_listrel1I2,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I2
thf(fact_916_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R2 ) )
=> ( ! [X3: a] :
( ( Xs
= ( cons_a @ X3 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_917_Cons__listrel1E1,axiom,
! [X: a,Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys ) @ ( listrel1_a @ R2 ) )
=> ( ! [Y4: a] :
( ( Ys
= ( cons_a @ Y4 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ R2 ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_918_listrel1I1,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R2 ) ) ) ).
% listrel1I1
thf(fact_919_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [X3: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_920_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_a @ R2 ) )
=> ~ ! [Y4: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y4 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys3 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_921_listrel_OCons,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_922_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
= ( ? [Y3: a,N2: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ Y3 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_923_lexord__cons__cons,axiom,
! [A: a,X: list_a,B: a,Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R2 ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
| ( ( A = B )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_924_lexord__partial__trans,axiom,
! [Xs: list_option_a,R2: set_Pr7585778909603769095tion_a,Ys: list_option_a,Zs: list_option_a] :
( ! [X3: option_a,Y4: option_a,Z: option_a] :
( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Y4 ) @ R2 )
=> ( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Z ) @ R2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ Z ) @ R2 ) ) ) )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Xs @ Ys ) @ ( lexord_option_a @ R2 ) )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Ys @ Zs ) @ ( lexord_option_a @ R2 ) )
=> ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Xs @ Zs ) @ ( lexord_option_a @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_925_lexord__partial__trans,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a,Ys: list_a,Zs: list_a] :
( ! [X3: a,Y4: a,Z: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z ) @ R2 ) ) ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lexord_a @ R2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs ) @ ( lexord_a @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_926_lexord__trans,axiom,
! [X: list_option_a,Y: list_option_a,R2: set_Pr7585778909603769095tion_a,Z2: list_option_a] :
( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X @ Y ) @ ( lexord_option_a @ R2 ) )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Y @ Z2 ) @ ( lexord_option_a @ R2 ) )
=> ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X @ Z2 ) @ ( lexord_option_a @ R2 ) ) ) ) ) ).
% lexord_trans
thf(fact_927_lexord__trans,axiom,
! [X: list_a,Y: list_a,R2: set_Product_prod_a_a,Z2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Z2 ) @ ( lexord_a @ R2 ) )
=> ( ( trans_on_a @ top_top_set_a @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Z2 ) @ ( lexord_a @ R2 ) ) ) ) ) ).
% lexord_trans
thf(fact_928_lexord__asymmetric,axiom,
! [R4: set_Pr7585778909603769095tion_a,A: list_option_a,B: list_option_a] :
( ( asym_on_option_a @ top_top_set_option_a @ R4 )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ A @ B ) @ ( lexord_option_a @ R4 ) )
=> ~ ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ B @ A ) @ ( lexord_option_a @ R4 ) ) ) ) ).
% lexord_asymmetric
thf(fact_929_lexord__asymmetric,axiom,
! [R4: set_Product_prod_a_a,A: list_a,B: list_a] :
( ( asym_on_a @ top_top_set_a @ R4 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ ( lexord_a @ R4 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ B @ A ) @ ( lexord_a @ R4 ) ) ) ) ).
% lexord_asymmetric
thf(fact_930_lexord__lex,axiom,
! [X: list_a,Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lex_a @ R2 ) )
= ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R2 ) )
& ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y ) ) ) ) ).
% lexord_lex
thf(fact_931_lexord__take__index__conv,axiom,
! [X: list_a,Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R2 ) )
= ( ( ( ord_less_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) )
& ( ( take_a @ ( size_size_list_a @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_a @ X ) @ ( size_size_list_a @ Y ) ) )
& ( ( take_a @ I3 @ X )
= ( take_a @ I3 @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ X @ I3 ) @ ( nth_a @ Y @ I3 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_932_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R2 ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_933_lexord__Nil__left,axiom,
! [Y: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R2 ) )
= ( ? [A4: a,X2: list_a] :
( Y
= ( cons_a @ A4 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_934_lexord__Nil__right,axiom,
! [X: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ nil_a ) @ ( lexord_a @ R2 ) ) ).
% lexord_Nil_right
thf(fact_935_partition_Osimps_I1_J,axiom,
! [P3: a > $o] :
( ( partition_a @ P3 @ nil_a )
= ( produc6837034575241423639list_a @ nil_a @ nil_a ) ) ).
% partition.simps(1)
thf(fact_936_find_Osimps_I1_J,axiom,
! [Uu: a > $o] :
( ( find_a @ Uu @ nil_a )
= none_a ) ).
% find.simps(1)
thf(fact_937_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ~ ! [X3: a,Xs3: list_a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_938_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( ! [Xs3: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs3 @ nil_a ) )
=> ~ ! [X3: a,Xs3: list_a,Y4: a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_939_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ~ ! [P6: a > a > $o,X3: a,Ys3: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_940_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ( ! [P6: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P6: a > a > $o,X3: a,Y4: a,Xs3: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X3 @ ( cons_a @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_941_Nil2__notin__lex,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_942_Nil__notin__lex,axiom,
! [Ys: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R2 ) ) ).
% Nil_notin_lex
thf(fact_943_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_944_not__listrel1__Nil,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_945_not__Nil__listrel1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_946_listrel__Nil2,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R2 ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_947_listrel__Nil1,axiom,
! [Xs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R2 ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_948_listrel_ONil,axiom,
! [R2: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R2 ) ) ).
% listrel.Nil
thf(fact_949_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R2 ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X3: a,Y4: a,Xs3: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_950_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R2 ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X2: a,Y3: a,Xs4: list_a,Ys4: list_a] :
( ( A1
= ( cons_a @ X2 @ Xs4 ) )
& ( A22
= ( cons_a @ Y3 @ Ys4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R2 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs4 @ Ys4 ) @ ( listrel_a_a @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_951_map__upds__append1,axiom,
! [Xs: list_a,Ys: list_a,M: a > option_a,X: a] :
( ( ord_less_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( ( map_upds_a_a @ M @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ Ys )
= ( fun_upd_a_option_a @ ( map_upds_a_a @ M @ Xs @ Ys ) @ X @ ( some_a @ ( nth_a @ Ys @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_952_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R2 ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_953_fun__upds__append__drop,axiom,
! [Xs: list_a,Ys: list_a,M: a > option_a,Zs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( map_upds_a_a @ M @ ( append_a @ Xs @ Zs ) @ Ys )
= ( map_upds_a_a @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_954_fun__upds__append2__drop,axiom,
! [Xs: list_a,Ys: list_a,M: a > option_a,Zs: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( map_upds_a_a @ M @ Xs @ ( append_a @ Ys @ Zs ) )
= ( map_upds_a_a @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_955_lexord__append__rightI,axiom,
! [Y: list_a,X: list_a,R2: set_Product_prod_a_a] :
( ? [B8: a,Z4: list_a] :
( Y
= ( cons_a @ B8 @ Z4 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ ( append_a @ X @ Y ) ) @ ( lexord_a @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_956_lexord__sufE,axiom,
! [Xs: list_a,Zs: list_a,Ys: list_a,Qs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Zs ) @ ( append_a @ Ys @ Qs ) ) @ ( lexord_a @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Qs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lexord_a @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_957_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_958_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R4: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R4 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys ) ) @ ( lenlex_a @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_959_listrel1I,axiom,
! [X: a,Y: a,R2: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_960_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R2 ) )
=> ~ ! [X3: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
=> ! [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ ( cons_a @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us2 @ ( cons_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_961_lexord__append__left__rightI,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_962_lexord__same__pref__iff,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lexord_a @ R2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X2 ) @ R2 ) )
| ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_963_lexord__sufI,axiom,
! [U: list_a,W: list_a,R2: set_Product_prod_a_a,V3: list_a,Z2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ W ) @ ( lexord_a @ R2 ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ W ) @ ( size_size_list_a @ U ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ V3 ) @ ( append_a @ W @ Z2 ) ) @ ( lexord_a @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_964_extract__SomeE,axiom,
! [P3: a > $o,Xs: list_a,Ys: list_a,Y: a,Zs: list_a] :
( ( ( extract_a @ P3 @ Xs )
= ( some_P5354654743593010357list_a @ ( produc3204708664006668352list_a @ Ys @ ( produc6670463072477821725list_a @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y @ Zs ) ) )
& ( P3 @ Y )
& ~ ? [X8: a] :
( ( member_a @ X8 @ ( set_a2 @ Ys ) )
& ( P3 @ X8 ) ) ) ) ).
% extract_SomeE
thf(fact_965_extract__Some__iff,axiom,
! [P3: a > $o,Xs: list_a,Ys: list_a,Y: a,Zs: list_a] :
( ( ( extract_a @ P3 @ Xs )
= ( some_P5354654743593010357list_a @ ( produc3204708664006668352list_a @ Ys @ ( produc6670463072477821725list_a @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y @ Zs ) ) )
& ( P3 @ Y )
& ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( P3 @ X2 ) ) ) ) ).
% extract_Some_iff
thf(fact_966_extract__Nil__code,axiom,
! [P3: a > $o] :
( ( extract_a @ P3 @ nil_a )
= none_P5893993846586699057list_a ) ).
% extract_Nil_code
thf(fact_967_extract__None__iff,axiom,
! [P3: a > $o,Xs: list_a] :
( ( ( extract_a @ P3 @ Xs )
= none_P5893993846586699057list_a )
= ( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P3 @ X2 ) ) ) ) ).
% extract_None_iff
thf(fact_968_lenlex__append2,axiom,
! [R4: set_Pr7585778909603769095tion_a,Us: list_option_a,Xs: list_option_a,Ys: list_option_a] :
( ( irrefl_on_option_a @ top_top_set_option_a @ R4 )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ ( append_option_a @ Us @ Xs ) @ ( append_option_a @ Us @ Ys ) ) @ ( lenlex_option_a @ R4 ) )
= ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Xs @ Ys ) @ ( lenlex_option_a @ R4 ) ) ) ) ).
% lenlex_append2
thf(fact_969_lenlex__append2,axiom,
! [R4: set_Product_prod_a_a,Us: list_a,Xs: list_a,Ys: list_a] :
( ( irrefl_on_a @ top_top_set_a @ R4 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Xs ) @ ( append_a @ Us @ Ys ) ) @ ( lenlex_a @ R4 ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lenlex_a @ R4 ) ) ) ) ).
% lenlex_append2
thf(fact_970_lexord__same__pref__if__irrefl,axiom,
! [R2: set_Pr7585778909603769095tion_a,Xs: list_option_a,Ys: list_option_a,Zs: list_option_a] :
( ( irrefl_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ ( append_option_a @ Xs @ Ys ) @ ( append_option_a @ Xs @ Zs ) ) @ ( lexord_option_a @ R2 ) )
= ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ Ys @ Zs ) @ ( lexord_option_a @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_971_lexord__same__pref__if__irrefl,axiom,
! [R2: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( irrefl_on_a @ top_top_set_a @ R2 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lexord_a @ R2 ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_972_irrefl__onI,axiom,
! [A2: set_a,R2: set_Product_prod_a_a] :
( ! [A3: a] :
( ( member_a @ A3 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R2 ) )
=> ( irrefl_on_a @ A2 @ R2 ) ) ).
% irrefl_onI
thf(fact_973_irrefl__onI,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a] :
( ! [A3: option_a] :
( ( member_option_a @ A3 @ A2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A3 @ A3 ) @ R2 ) )
=> ( irrefl_on_option_a @ A2 @ R2 ) ) ).
% irrefl_onI
thf(fact_974_irrefl__onD,axiom,
! [A2: set_a,R2: set_Product_prod_a_a,A: a] :
( ( irrefl_on_a @ A2 @ R2 )
=> ( ( member_a @ A @ A2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% irrefl_onD
thf(fact_975_irrefl__onD,axiom,
! [A2: set_option_a,R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( irrefl_on_option_a @ A2 @ R2 )
=> ( ( member_option_a @ A @ A2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ A ) @ R2 ) ) ) ).
% irrefl_onD
thf(fact_976_irreflI,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ! [A3: option_a] :
~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A3 @ A3 ) @ R2 )
=> ( irrefl_on_option_a @ top_top_set_option_a @ R2 ) ) ).
% irreflI
thf(fact_977_irreflI,axiom,
! [R2: set_Product_prod_a_a] :
( ! [A3: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R2 )
=> ( irrefl_on_a @ top_top_set_a @ R2 ) ) ).
% irreflI
thf(fact_978_irreflD,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: option_a] :
( ( irrefl_on_option_a @ top_top_set_option_a @ R2 )
=> ~ ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X @ X ) @ R2 ) ) ).
% irreflD
thf(fact_979_irreflD,axiom,
! [R2: set_Product_prod_a_a,X: a] :
( ( irrefl_on_a @ top_top_set_a @ R2 )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R2 ) ) ).
% irreflD
thf(fact_980_lexl__not__refl,axiom,
! [R2: set_Pr7585778909603769095tion_a,X: list_option_a] :
( ( irrefl_on_option_a @ top_top_set_option_a @ R2 )
=> ~ ( member3116372912924030544tion_a @ ( produc6071813090954082327tion_a @ X @ X ) @ ( lex_option_a @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_981_lexl__not__refl,axiom,
! [R2: set_Product_prod_a_a,X: list_a] :
( ( irrefl_on_a @ top_top_set_a @ R2 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ X ) @ ( lex_a @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_982_lexn__length,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,N: nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lexn_a @ R2 @ N ) )
=> ( ( ( size_size_list_a @ Xs )
= N )
& ( ( size_size_list_a @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_983_inv__image__partition,axiom,
! [Xs: list_option_a,P3: option_a > $o,Ys: list_option_a] :
( ! [X3: option_a] :
( ( member_option_a @ X3 @ ( set_option_a2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( ! [Y4: option_a] :
( ( member_option_a @ Y4 @ ( set_option_a2 @ Ys ) )
=> ~ ( P3 @ Y4 ) )
=> ( ( vimage3571068517928849688tion_a @ ( partition_option_a @ P3 ) @ ( insert3212945957549752823tion_a @ ( produc6071813090954082327tion_a @ Xs @ Ys ) @ bot_bo6007253611978100339tion_a ) )
= ( shuffles_option_a @ Xs @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_984_inv__image__partition,axiom,
! [Xs: list_a,P3: a > $o,Ys: list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys ) )
=> ~ ( P3 @ Y4 ) )
=> ( ( vimage4558233055442567774list_a @ ( partition_a @ P3 ) @ ( insert1856800524785285367list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ bot_bo2955605580254355571list_a ) )
= ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_985_Pair__vimage__Sigma,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( vimage5143925195038468708od_a_a @ ( product_Pair_a_a @ X ) @ ( product_Sigma_a_a @ A2 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( vimage5143925195038468708od_a_a @ ( product_Pair_a_a @ X ) @ ( product_Sigma_a_a @ A2 @ F ) )
= bot_bot_set_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_986_Pair__vimage__Sigma,axiom,
! [X: option_a,A2: set_option_a,F: option_a > set_a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( vimage7079528361448986782on_a_a @ ( produc3446707977624461905on_a_a @ X ) @ ( produc3962846827955709570on_a_a @ A2 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( vimage7079528361448986782on_a_a @ ( produc3446707977624461905on_a_a @ X ) @ ( produc3962846827955709570on_a_a @ A2 @ F ) )
= bot_bot_set_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_987_Pair__vimage__Sigma,axiom,
! [X: a,A2: set_a,F: a > set_option_a] :
( ( ( member_a @ X @ A2 )
=> ( ( vimage1039097674533424484tion_a @ ( produc1224194096085666781tion_a @ X ) @ ( produc1740332946416914446tion_a @ A2 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( vimage1039097674533424484tion_a @ ( produc1224194096085666781tion_a @ X ) @ ( produc1740332946416914446tion_a @ A2 @ F ) )
= bot_bot_set_option_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_988_Pair__vimage__Sigma,axiom,
! [X: option_a,A2: set_option_a,F: option_a > set_option_a] :
( ( ( member_option_a @ X @ A2 )
=> ( ( vimage6204385730559098270tion_a @ ( produc9011544418120257559tion_a @ X ) @ ( produc269287337874323144tion_a @ A2 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member_option_a @ X @ A2 )
=> ( ( vimage6204385730559098270tion_a @ ( produc9011544418120257559tion_a @ X ) @ ( produc269287337874323144tion_a @ A2 @ F ) )
= bot_bot_set_option_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_989_mem__Sigma__iff,axiom,
! [A: a,B: a,A2: set_a,B2: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B2 ) )
= ( ( member_a @ A @ A2 )
& ( member_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_990_mem__Sigma__iff,axiom,
! [A: a,B: option_a,A2: set_a,B2: a > set_option_a] :
( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ ( produc1740332946416914446tion_a @ A2 @ B2 ) )
= ( ( member_a @ A @ A2 )
& ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_991_mem__Sigma__iff,axiom,
! [A: option_a,B: a,A2: set_option_a,B2: option_a > set_a] :
( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ ( produc3962846827955709570on_a_a @ A2 @ B2 ) )
= ( ( member_option_a @ A @ A2 )
& ( member_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_992_mem__Sigma__iff,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
= ( ( member_option_a @ A @ A2 )
& ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_993_SigmaI,axiom,
! [A: a,A2: set_a,B: a,B2: a > set_a] :
( ( member_a @ A @ A2 )
=> ( ( member_a @ B @ ( B2 @ A ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_994_SigmaI,axiom,
! [A: a,A2: set_a,B: option_a,B2: a > set_option_a] :
( ( member_a @ A @ A2 )
=> ( ( member_option_a @ B @ ( B2 @ A ) )
=> ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ ( produc1740332946416914446tion_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_995_SigmaI,axiom,
! [A: option_a,A2: set_option_a,B: a,B2: option_a > set_a] :
( ( member_option_a @ A @ A2 )
=> ( ( member_a @ B @ ( B2 @ A ) )
=> ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ ( produc3962846827955709570on_a_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_996_SigmaI,axiom,
! [A: option_a,A2: set_option_a,B: option_a,B2: option_a > set_option_a] :
( ( member_option_a @ A @ A2 )
=> ( ( member_option_a @ B @ ( B2 @ A ) )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_997_SigmaE2,axiom,
! [A: a,B: a,A2: set_a,B2: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Sigma_a_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ A @ A2 )
=> ~ ( member_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_998_SigmaE2,axiom,
! [A: a,B: option_a,A2: set_a,B2: a > set_option_a] :
( ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B ) @ ( produc1740332946416914446tion_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ A @ A2 )
=> ~ ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_999_SigmaE2,axiom,
! [A: option_a,B: a,A2: set_option_a,B2: option_a > set_a] :
( ( member6056235002698166154on_a_a @ ( produc3446707977624461905on_a_a @ A @ B ) @ ( produc3962846827955709570on_a_a @ A2 @ B2 ) )
=> ~ ( ( member_option_a @ A @ A2 )
=> ~ ( member_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1000_SigmaE2,axiom,
! [A: option_a,B: option_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ~ ( ( member_option_a @ A @ A2 )
=> ~ ( member_option_a @ B @ ( B2 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1001_SigmaE,axiom,
! [C: product_prod_a_a,A2: set_a,B2: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A2 @ B2 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B2 @ X3 ) )
=> ( C
!= ( product_Pair_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_1002_SigmaE,axiom,
! [C: produc3964210925746912109tion_a,A2: set_a,B2: a > set_option_a] :
( ( member6937434987665551382tion_a @ C @ ( produc1740332946416914446tion_a @ A2 @ B2 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y4: option_a] :
( ( member_option_a @ Y4 @ ( B2 @ X3 ) )
=> ( C
!= ( produc1224194096085666781tion_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_1003_SigmaE,axiom,
! [C: produc3083010940779526881on_a_a,A2: set_option_a,B2: option_a > set_a] :
( ( member6056235002698166154on_a_a @ C @ ( produc3962846827955709570on_a_a @ A2 @ B2 ) )
=> ~ ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B2 @ X3 ) )
=> ( C
!= ( produc3446707977624461905on_a_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_1004_SigmaE,axiom,
! [C: produc3509355604313844263tion_a,A2: set_option_a,B2: option_a > set_option_a] :
( ( member5498148017924304208tion_a @ C @ ( produc269287337874323144tion_a @ A2 @ B2 ) )
=> ~ ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ! [Y4: option_a] :
( ( member_option_a @ Y4 @ ( B2 @ X3 ) )
=> ( C
!= ( produc9011544418120257559tion_a @ X3 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_1005_image__split__eq__Sigma,axiom,
! [F: a > a,G: a > option_a,A2: set_a] :
( ( image_1318920958782419572tion_a
@ ^ [X2: a] : ( produc1224194096085666781tion_a @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( produc1740332946416914446tion_a @ ( image_a_a2 @ F @ A2 )
@ ^ [X2: a] : ( image_a_option_a2 @ G @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ A2 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_1006_image__split__eq__Sigma,axiom,
! [F: option_a > a,G: option_a > a,A2: set_option_a] :
( ( image_7456799861883459304od_a_a
@ ^ [X2: option_a] : ( product_Pair_a_a @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( product_Sigma_a_a @ ( image_option_a_a2 @ F @ A2 )
@ ^ [X2: a] : ( image_option_a_a2 @ G @ ( inf_inf_set_option_a @ ( vimage_option_a_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ A2 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_1007_image__split__eq__Sigma,axiom,
! [F: option_a > option_a,G: option_a > a,A2: set_option_a] :
( ( image_2176871512121339682on_a_a
@ ^ [X2: option_a] : ( produc3446707977624461905on_a_a @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( produc3962846827955709570on_a_a @ ( image_7439109396645324421tion_a @ F @ A2 )
@ ^ [X2: option_a] : ( image_option_a_a2 @ G @ ( inf_inf_set_option_a @ ( vimage1562710927270423099tion_a @ F @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) @ A2 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_1008_image__split__eq__Sigma,axiom,
! [F: a > option_a,G: a > option_a,A2: set_a] :
( ( image_7468189554597481134tion_a
@ ^ [X2: a] : ( produc9011544418120257559tion_a @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( produc269287337874323144tion_a @ ( image_a_option_a2 @ F @ A2 )
@ ^ [X2: option_a] : ( image_a_option_a2 @ G @ ( inf_inf_set_a @ ( vimage_a_option_a @ F @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) @ A2 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_1009_map__of__map__restrict,axiom,
! [F: a > a,Ks: list_a] :
( ( map_of_a_a
@ ( map_a_7860052162900579309od_a_a
@ ^ [K2: a] : ( product_Pair_a_a @ K2 @ ( F @ K2 ) )
@ Ks ) )
= ( restrict_map_a_a @ ( comp_a_option_a_a @ some_a @ F ) @ ( set_a2 @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_1010_bind__assoc,axiom,
! [X: option_a,F: a > option_a,G: a > option_a] :
( ( bind_a_a @ ( bind_a_a @ X @ F ) @ G )
= ( bind_a_a @ X
@ ^ [Y3: a] : ( bind_a_a @ ( F @ Y3 ) @ G ) ) ) ).
% bind_assoc
thf(fact_1011_bind__rzero,axiom,
! [X: option_a] :
( ( bind_a_a @ X
@ ^ [X2: a] : none_a )
= none_a ) ).
% bind_rzero
thf(fact_1012_dom__const,axiom,
! [F: option_a > a] :
( ( dom_option_a_a
@ ^ [X2: option_a] : ( some_a @ ( F @ X2 ) ) )
= top_top_set_option_a ) ).
% dom_const
thf(fact_1013_dom__const,axiom,
! [F: a > a] :
( ( dom_a_a
@ ^ [X2: a] : ( some_a @ ( F @ X2 ) ) )
= top_top_set_a ) ).
% dom_const
thf(fact_1014_dom__empty,axiom,
( ( dom_a_a
@ ^ [X2: a] : none_a )
= bot_bot_set_a ) ).
% dom_empty
thf(fact_1015_dom__empty,axiom,
( ( dom_option_a_a
@ ^ [X2: option_a] : none_a )
= bot_bot_set_option_a ) ).
% dom_empty
thf(fact_1016_insert__Times__insert,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ( product_Sigma_a_a @ ( insert_a @ A @ A2 )
@ ^ [Uu2: a] : ( insert_a @ B @ B2 ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B )
@ ( sup_su3048258781599657691od_a_a
@ ( product_Sigma_a_a @ A2
@ ^ [Uu2: a] : ( insert_a @ B @ B2 ) )
@ ( product_Sigma_a_a @ ( insert_a @ A @ A2 )
@ ^ [Uu2: a] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1017_insert__Times__insert,axiom,
! [A: a,A2: set_a,B: option_a,B2: set_option_a] :
( ( produc1740332946416914446tion_a @ ( insert_a @ A @ A2 )
@ ^ [Uu2: a] : ( insert_option_a @ B @ B2 ) )
= ( insert6939662406020284733tion_a @ ( produc1224194096085666781tion_a @ A @ B )
@ ( sup_su167856392462661793tion_a
@ ( produc1740332946416914446tion_a @ A2
@ ^ [Uu2: a] : ( insert_option_a @ B @ B2 ) )
@ ( produc1740332946416914446tion_a @ ( insert_a @ A @ A2 )
@ ^ [Uu2: a] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1018_insert__Times__insert,axiom,
! [A: option_a,A2: set_option_a,B: a,B2: set_a] :
( ( produc3962846827955709570on_a_a @ ( insert_option_a @ A @ A2 )
@ ^ [Uu2: option_a] : ( insert_a @ B @ B2 ) )
= ( insert6058462421052899505on_a_a @ ( produc3446707977624461905on_a_a @ A @ B )
@ ( sup_su3065098059273993749on_a_a
@ ( produc3962846827955709570on_a_a @ A2
@ ^ [Uu2: option_a] : ( insert_a @ B @ B2 ) )
@ ( produc3962846827955709570on_a_a @ ( insert_option_a @ A @ A2 )
@ ^ [Uu2: option_a] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1019_insert__Times__insert,axiom,
! [A: option_a,A2: set_option_a,B: option_a,B2: set_option_a] :
( ( produc269287337874323144tion_a @ ( insert_option_a @ A @ A2 )
@ ^ [Uu2: option_a] : ( insert_option_a @ B @ B2 ) )
= ( insert1246254401036548087tion_a @ ( produc9011544418120257559tion_a @ A @ B )
@ ( sup_su1214438497309894875tion_a
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu2: option_a] : ( insert_option_a @ B @ B2 ) )
@ ( produc269287337874323144tion_a @ ( insert_option_a @ A @ A2 )
@ ^ [Uu2: option_a] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_1020_trancl__subset__Sigma__aux,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,A2: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( transitive_rtrancl_a @ R2 ) )
=> ( ( ord_le746702958409616551od_a_a @ R2
@ ( product_Sigma_a_a @ A2
@ ^ [Uu2: a] : A2 ) )
=> ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_1021_trancl__subset__Sigma__aux,axiom,
! [A: option_a,B: option_a,R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B ) @ ( transi330218190764880583tion_a @ R2 ) )
=> ( ( ord_le4471550158292877991tion_a @ R2
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu2: option_a] : A2 ) )
=> ( ( A = B )
| ( member_option_a @ A @ A2 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_1022_refl__onI,axiom,
! [R2: set_Product_prod_a_a,A2: set_a] :
( ( ord_le746702958409616551od_a_a @ R2
@ ( product_Sigma_a_a @ A2
@ ^ [Uu2: a] : A2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R2 ) )
=> ( refl_on_a @ A2 @ R2 ) ) ) ).
% refl_onI
thf(fact_1023_refl__onI,axiom,
! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a] :
( ( ord_le4471550158292877991tion_a @ R2
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu2: option_a] : A2 ) )
=> ( ! [X3: option_a] :
( ( member_option_a @ X3 @ A2 )
=> ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ X3 @ X3 ) @ R2 ) )
=> ( refl_on_option_a @ A2 @ R2 ) ) ) ).
% refl_onI
thf(fact_1024_wfI,axiom,
! [R2: set_Product_prod_a_a,A2: set_a,B2: set_a] :
( ( ord_le746702958409616551od_a_a @ R2
@ ( product_Sigma_a_a @ A2
@ ^ [Uu2: a] : B2 ) )
=> ( ! [X3: a,P6: a > $o] :
( ! [Xa2: a] :
( ! [Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Xa2 ) @ R2 )
=> ( P6 @ Y4 ) )
=> ( P6 @ Xa2 ) )
=> ( ( member_a @ X3 @ A2 )
=> ( ( member_a @ X3 @ B2 )
=> ( P6 @ X3 ) ) ) )
=> ( wf_a @ R2 ) ) ) ).
% wfI
thf(fact_1025_wfI,axiom,
! [R2: set_Pr7585778909603769095tion_a,A2: set_option_a,B2: set_option_a] :
( ( ord_le4471550158292877991tion_a @ R2
@ ( produc269287337874323144tion_a @ A2
@ ^ [Uu2: option_a] : B2 ) )
=> ( ! [X3: option_a,P6: option_a > $o] :
( ! [Xa2: option_a] :
( ! [Y4: option_a] :
( ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y4 @ Xa2 ) @ R2 )
=> ( P6 @ Y4 ) )
=> ( P6 @ Xa2 ) )
=> ( ( member_option_a @ X3 @ A2 )
=> ( ( member_option_a @ X3 @ B2 )
=> ( P6 @ X3 ) ) ) )
=> ( wf_option_a @ R2 ) ) ) ).
% wfI
thf(fact_1026_wf__finite__segments,axiom,
! [R2: set_Pr7585778909603769095tion_a] :
( ( irrefl_on_option_a @ top_top_set_option_a @ R2 )
=> ( ( trans_on_option_a @ top_top_set_option_a @ R2 )
=> ( ! [X3: option_a] :
( finite1674126218327898605tion_a
@ ( collect_option_a
@ ^ [Y3: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ Y3 @ X3 ) @ R2 ) ) )
=> ( wf_option_a @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_1027_wf__finite__segments,axiom,
! [R2: set_Product_prod_a_a] :
( ( irrefl_on_a @ top_top_set_a @ R2 )
=> ( ( trans_on_a @ top_top_set_a @ R2 )
=> ( ! [X3: a] :
( finite_finite_a
@ ( collect_a
@ ^ [Y3: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R2 ) ) )
=> ( wf_a @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_1028_finite__Map__induct,axiom,
! [M: option_a > option_a,P3: ( option_a > option_a ) > $o] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M ) )
=> ( ( P3
@ ^ [X2: option_a] : none_a )
=> ( ! [K3: option_a,V2: a,M4: option_a > option_a] :
( ( finite1674126218327898605tion_a @ ( dom_option_a_a @ M4 ) )
=> ( ~ ( member_option_a @ K3 @ ( dom_option_a_a @ M4 ) )
=> ( ( P3 @ M4 )
=> ( P3 @ ( fun_up1079276522633388797tion_a @ M4 @ K3 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P3 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1029_finite__Map__induct,axiom,
! [M: a > option_a,P3: ( a > option_a ) > $o] :
( ( finite_finite_a @ ( dom_a_a @ M ) )
=> ( ( P3
@ ^ [X2: a] : none_a )
=> ( ! [K3: a,V2: a,M4: a > option_a] :
( ( finite_finite_a @ ( dom_a_a @ M4 ) )
=> ( ~ ( member_a @ K3 @ ( dom_a_a @ M4 ) )
=> ( ( P3 @ M4 )
=> ( P3 @ ( fun_upd_a_option_a @ M4 @ K3 @ ( some_a @ V2 ) ) ) ) ) )
=> ( P3 @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_1030_case__optionE,axiom,
! [P3: $o,Q2: a > $o,X: option_a] :
( ( case_option_o_a @ P3 @ Q2 @ X )
=> ( ( ( X = none_a )
=> ~ P3 )
=> ~ ! [Y4: a] :
( ( X
= ( some_a @ Y4 ) )
=> ~ ( Q2 @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_1031_map__option__case,axiom,
( map_option_a_a2
= ( ^ [F3: a > a] :
( case_o3148979394504432965on_a_a @ none_a
@ ^ [X2: a] : ( some_a @ ( F3 @ X2 ) ) ) ) ) ).
% map_option_case
thf(fact_1032_Some__image__these__eq,axiom,
! [A2: set_option_a] :
( ( image_a_option_a2 @ some_a @ ( these_a @ A2 ) )
= ( collect_option_a
@ ^ [X2: option_a] :
( ( member_option_a @ X2 @ A2 )
& ( X2 != none_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_1033_Option_Othese__def,axiom,
( these_a
= ( ^ [A7: set_option_a] :
( image_option_a_a2 @ the_a
@ ( collect_option_a
@ ^ [X2: option_a] :
( ( member_option_a @ X2 @ A7 )
& ( X2 != none_a ) ) ) ) ) ) ).
% Option.these_def
thf(fact_1034_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_a] :
( ( Option != none_a )
= ( case_option_o_a @ $false
@ ^ [Uu2: a] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_1035_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_a] :
( ( Option = none_a )
= ( case_option_o_a @ $true
@ ^ [Uu2: a] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_1036_dom__def,axiom,
( dom_option_a_a
= ( ^ [M2: option_a > option_a] :
( collect_option_a
@ ^ [A4: option_a] :
( ( M2 @ A4 )
!= none_a ) ) ) ) ).
% dom_def
thf(fact_1037_map__option_Oidentity,axiom,
( ( map_option_a_a2
@ ^ [X2: a] : X2 )
= id_option_a ) ).
% map_option.identity
thf(fact_1038_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( H @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( case_option_o_a @ ( H @ F1 )
@ ^ [X2: a] : ( H @ ( F2 @ X2 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_1039_option_Omap__ident,axiom,
! [T: option_a] :
( ( map_option_a_a2
@ ^ [X2: a] : X2
@ T )
= T ) ).
% option.map_ident
thf(fact_1040_finite__set__of__finite__maps,axiom,
! [A2: set_option_a,B2: set_option_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ B2 )
=> ( finite3089212293327181890tion_a
@ ( collec2458836999851688832tion_a
@ ^ [M2: option_a > option_option_a] :
( ( ( dom_op4724496951392727122tion_a @ M2 )
= A2 )
& ( ord_le1955136853071979460tion_a @ ( ran_op6317565877353657455tion_a @ M2 ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1041_finite__set__of__finite__maps,axiom,
! [A2: set_option_a,B2: set_a] :
( ( finite1674126218327898605tion_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite8942580144290239484tion_a
@ ( collec5803847638788715578tion_a
@ ^ [M2: option_a > option_a] :
( ( ( dom_option_a_a @ M2 )
= A2 )
& ( ord_less_eq_set_a @ ( ran_option_a_a @ M2 ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1042_finite__set__of__finite__maps,axiom,
! [A2: set_a,B2: set_option_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite1674126218327898605tion_a @ B2 )
=> ( finite595534299322690568tion_a
@ ( collec6680173830675942470tion_a
@ ^ [M2: a > option_option_a] :
( ( ( dom_a_option_a @ M2 )
= A2 )
& ( ord_le1955136853071979460tion_a @ ( ran_a_option_a @ M2 ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1043_finite__set__of__finite__maps,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite5998080203967203522tion_a
@ ( collect_a_option_a
@ ^ [M2: a > option_a] :
( ( ( dom_a_a @ M2 )
= A2 )
& ( ord_less_eq_set_a @ ( ran_a_a @ M2 ) @ B2 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_1044_under__def,axiom,
( order_under_option_a
= ( ^ [R3: set_Pr7585778909603769095tion_a,A4: option_a] :
( collect_option_a
@ ^ [B3: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B3 @ A4 ) @ R3 ) ) ) ) ).
% under_def
thf(fact_1045_underS__def,axiom,
( order_8525669848891258378tion_a
= ( ^ [R3: set_Pr7585778909603769095tion_a,A4: option_a] :
( collect_option_a
@ ^ [B3: option_a] :
( ( B3 != A4 )
& ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ B3 @ A4 ) @ R3 ) ) ) ) ) ).
% underS_def
thf(fact_1046_fst__diag__id,axiom,
! [Z2: a] :
( ( comp_P5977721380588955012_a_a_a @ product_fst_a_a
@ ^ [X2: a] : ( product_Pair_a_a @ X2 @ X2 )
@ Z2 )
= ( id_a @ Z2 ) ) ).
% fst_diag_id
thf(fact_1047_snd__diag__id,axiom,
! [Z2: a] :
( ( comp_P5977721380588955012_a_a_a @ product_snd_a_a
@ ^ [X2: a] : ( product_Pair_a_a @ X2 @ X2 )
@ Z2 )
= ( id_a @ Z2 ) ) ).
% snd_diag_id
thf(fact_1048_map__of__map__keys,axiom,
! [Xs: list_a,M: a > option_a] :
( ( ( set_a2 @ Xs )
= ( dom_a_a @ M ) )
=> ( ( map_of_a_a
@ ( map_a_7860052162900579309od_a_a
@ ^ [K2: a] : ( product_Pair_a_a @ K2 @ ( the_a @ ( M @ K2 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_1049_Image__singleton,axiom,
! [R2: set_Pr3411724424142761165tion_a,A: a] :
( ( image_a_option_a @ R2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( collect_option_a
@ ^ [B3: option_a] : ( member6937434987665551382tion_a @ ( produc1224194096085666781tion_a @ A @ B3 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_1050_Image__singleton,axiom,
! [R2: set_Pr7585778909603769095tion_a,A: option_a] :
( ( image_4442594622209975379tion_a @ R2 @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( collect_option_a
@ ^ [B3: option_a] : ( member5498148017924304208tion_a @ ( produc9011544418120257559tion_a @ A @ B3 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_1051_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_a2 @ F @ A2 ) @ ( image_a_a2 @ G @ B2 ) )
= bot_bot_set_a )
=> ( inj_on_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1052_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_a2 @ F @ A2 ) @ ( image_option_a_a2 @ G @ B2 ) )
= bot_bot_set_a )
=> ( inj_on_option_a_a
@ ^ [X2: option_a] : ( if_a @ ( member_option_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1053_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_a2 @ F @ A2 ) @ ( image_a_option_a2 @ G @ B2 ) )
= bot_bot_set_option_a )
=> ( inj_on_a_option_a
@ ^ [X2: a] : ( if_option_a @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1054_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
@ ^ [X2: option_a] : ( if_option_a @ ( member_option_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1055_inj__singleton,axiom,
! [A2: set_a] :
( inj_on_a_set_a
@ ^ [X2: a] : ( insert_a @ X2 @ bot_bot_set_a )
@ A2 ) ).
% inj_singleton
thf(fact_1056_inj__singleton,axiom,
! [A2: set_option_a] :
( inj_on7881382345526841553tion_a
@ ^ [X2: option_a] : ( insert_option_a @ X2 @ bot_bot_set_option_a )
@ A2 ) ).
% inj_singleton
thf(fact_1057_bij__betw__disjoint__Un,axiom,
! [F: a > a,A2: set_a,C3: set_a,G: a > a,B2: set_a,D: set_a] :
( ( bij_betw_a_a @ F @ A2 @ C3 )
=> ( ( bij_betw_a_a @ G @ B2 @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ C3 @ D )
= bot_bot_set_a )
=> ( bij_betw_a_a
@ ^ [X2: a] : ( if_a @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B2 )
@ ( sup_sup_set_a @ C3 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1058_bij__betw__disjoint__Un,axiom,
! [F: a > option_a,A2: set_a,C3: set_option_a,G: a > option_a,B2: set_a,D: set_option_a] :
( ( bij_betw_a_option_a @ F @ A2 @ C3 )
=> ( ( bij_betw_a_option_a @ G @ B2 @ D )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_option_a @ C3 @ D )
= bot_bot_set_option_a )
=> ( bij_betw_a_option_a
@ ^ [X2: a] : ( if_option_a @ ( member_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_a @ A2 @ B2 )
@ ( sup_sup_set_option_a @ C3 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1059_bij__betw__disjoint__Un,axiom,
! [F: option_a > a,A2: set_option_a,C3: set_a,G: option_a > a,B2: set_option_a,D: set_a] :
( ( bij_betw_option_a_a @ F @ A2 @ C3 )
=> ( ( bij_betw_option_a_a @ G @ B2 @ D )
=> ( ( ( inf_inf_set_option_a @ A2 @ B2 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_a @ C3 @ D )
= bot_bot_set_a )
=> ( bij_betw_option_a_a
@ ^ [X2: option_a] : ( if_a @ ( member_option_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 )
@ ( sup_sup_set_a @ C3 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1060_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,D: set_option_a] :
( ( bij_be5431266891817924854tion_a @ F @ A2 @ C3 )
=> ( ( bij_be5431266891817924854tion_a @ G @ B2 @ D )
=> ( ( ( inf_inf_set_option_a @ A2 @ B2 )
= bot_bot_set_option_a )
=> ( ( ( inf_inf_set_option_a @ C3 @ D )
= bot_bot_set_option_a )
=> ( bij_be5431266891817924854tion_a
@ ^ [X2: option_a] : ( if_option_a @ ( member_option_a @ X2 @ A2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup_set_option_a @ A2 @ B2 )
@ ( sup_sup_set_option_a @ C3 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_1061_dom__eq__singleton__conv,axiom,
! [F: a > option_a,X: a] :
( ( ( dom_a_a @ F )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ? [V: a] :
( F
= ( fun_upd_a_option_a
@ ^ [X2: a] : none_a
@ X
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1062_dom__eq__singleton__conv,axiom,
! [F: option_a > option_a,X: option_a] :
( ( ( dom_option_a_a @ F )
= ( insert_option_a @ X @ bot_bot_set_option_a ) )
= ( ? [V: a] :
( F
= ( fun_up1079276522633388797tion_a
@ ^ [X2: option_a] : none_a
@ X
@ ( some_a @ V ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_1063_set__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) )
= ( collec3336397797384452498od_a_a
@ ^ [Uu2: product_prod_a_a] :
? [I3: nat] :
( ( Uu2
= ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) )
& ( ord_less_nat @ I3 @ ( ord_min_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_1064_modified__def,axiom,
( modified_a_d_c
= ( ^ [C2: set_Pr7868159745199425715_a_d_c] :
( collect_d
@ ^ [Uu2: d] :
? [X2: d] :
( ( Uu2 = X2 )
& ? [Sigma2: a,S2: d > c,Sigma4: a,S5: d > c] :
( ( member2052822272342364412_a_d_c @ ( produc5208860900648697099_a_d_c @ ( product_Pair_a_d_c @ Sigma2 @ S2 ) @ ( some_P377817780860425132_a_d_c @ ( product_Pair_a_d_c @ Sigma4 @ S5 ) ) ) @ C2 )
& ( ( S2 @ X2 )
!= ( S5 @ X2 ) ) ) ) ) ) ) ).
% modified_def
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
! [X: option_a,Y: option_a] :
( ( if_option_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
entails_a_b_c_d @ plus @ mult @ valid @ ( wildcard_a_b_c_d @ ( and_a_b_c_d @ a2 @ b2 ) ) @ delta @ ( and_a_b_c_d @ ( wildcard_a_b_c_d @ a2 ) @ ( wildcard_a_b_c_d @ b2 ) ) ).
%------------------------------------------------------------------------------