TPTP Problem File: SLH0461^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Stalnaker_Logic/0000_Stalnaker_Logic/prob_00104_002795__6069596_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1607 ( 524 unt; 324 typ; 0 def)
% Number of atoms : 4226 (1041 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 13493 ( 354 ~; 70 |; 304 &;10638 @)
% ( 0 <=>;2127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 1614 (1614 >; 0 *; 0 +; 0 <<)
% Number of symbols : 295 ( 292 usr; 26 con; 0-4 aty)
% Number of variables : 4428 ( 376 ^;3910 !; 142 ?;4428 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:31.034
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
episte1560738328020401952t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
episte94448284482925344t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Nat__Onat_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
episte8765170747386058258t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mtf__a_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
episte6182337868402532512t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte1193835314949844379t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
set_Pr8948481099588399239c_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
list_l6083326122719238310c_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
set_li769143395467472256c_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
set_se5208064806568342746c_fm_a: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte740340785640729014c_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
list_Epistemic_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
set_Epistemic_fm_a: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Set__Oset_It__Nat__Onat_J_J,type,
filter_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__Nat__Onat_J,type,
epistemic_fm_nat: $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__Epistemic____Logic__Ofm_Itf__a_J,type,
epistemic_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (292)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
bNF_Ca1305897159876240246c_fm_a: set_Epistemic_fm_a > set_Pr8948481099588399239c_fm_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
bNF_Gr8437504134799245625c_fm_a: set_li769143395467472256c_fm_a > epistemic_fm_a > set_li769143395467472256c_fm_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
bNF_Gr1093487135560555701c_fm_a: set_li769143395467472256c_fm_a > list_Epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Epistemic__Logic_OAK_001tf__a,type,
epistemic_AK_a: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__a,type,
epistemic_Ax4_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__a,type,
epistemic_Ax5_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__a,type,
epistemic_AxB_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__a,type,
epistemic_AxT_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Product____Type__Ounit_J,type,
episte4583239219080210381t_unit: episte94448284482925344t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte3760347122651195639t_unit: episte8765170747386058258t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte2449151000174023629t_unit: episte1560738328020401952t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
episte2339904321507024205t_unit: episte6182337868402532512t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001tf__a,type,
episte2285483198712856226tent_a: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__a,type,
epistemic_eval_a: ( list_char > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte3685526487207141399c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001t__Nat__Onat,type,
epistemic_Con_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__a,type,
epistemic_Con_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte6088726764479022859c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Nat__Onat,type,
epistemic_Dis_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__a,type,
epistemic_Dis_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte5073044243917183961c_fm_a: episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001t__Nat__Onat,type,
epistemic_FF_nat: epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__a,type,
epistemic_FF_a: epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte260752218777527565c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001t__Nat__Onat,type,
epistemic_Imp_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__a,type,
epistemic_Imp_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte5657488632024175118c_fm_a: epistemic_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Nat__Onat,type,
epistemic_K_nat: nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__a,type,
epistemic_K_a: a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte3759128466173231372c_fm_a: list_char > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001t__Nat__Onat,type,
epistemic_Pro_nat: list_char > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__a,type,
epistemic_Pro_a: list_char > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte7774795710028497888c_fm_a: ( epistemic_fm_a > epistemic_fm_a > $o ) > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Nat__Onat,type,
episte8778020545599232650_a_nat: ( epistemic_fm_a > nat > $o ) > episte740340785640729014c_fm_a > epistemic_fm_nat > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001tf__a,type,
episte4428145106359621316fm_a_a: ( epistemic_fm_a > a > $o ) > episte740340785640729014c_fm_a > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte9034525631817513832c_fm_a: ( nat > epistemic_fm_a > $o ) > epistemic_fm_nat > episte740340785640729014c_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001t__Nat__Onat,type,
episte3894023384580379906at_nat: ( nat > nat > $o ) > epistemic_fm_nat > epistemic_fm_nat > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte8321036160184370300c_fm_a: ( a > epistemic_fm_a > $o ) > epistemic_fm_a > episte740340785640729014c_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001t__Nat__Onat,type,
episte1460426709791529198_a_nat: ( a > nat > $o ) > epistemic_fm_a > epistemic_fm_nat > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001tf__a,type,
epistemic_rel_fm_a_a: ( a > a > $o ) > epistemic_fm_a > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte9089240958480457552c_fm_a: episte740340785640729014c_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__Nat__Onat,type,
epistemic_set_fm_nat: epistemic_fm_nat > set_nat ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__a,type,
epistemic_set_fm_a: epistemic_fm_a > set_a ).
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episte6390737319716712051t_unit: episte94448284482925344t_unit > set_Epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__a_001t__Nat__Onat_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
episte3616848269639615645t_unit: episte8765170747386058258t_unit > set_nat ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte8072386903178013299t_unit: episte1560738328020401952t_unit > set_se5208064806568342746c_fm_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__a_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
episte6926715892928323059t_unit: episte6182337868402532512t_unit > set_a ).
thf(sy_c_Epistemic__Logic_Oframe_Oframe__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte2888590659910966568t_unit: set_se5208064806568342746c_fm_a > ( a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ) > episte1193835314949844379t_unit > episte1560738328020401952t_unit ).
thf(sy_c_Epistemic__Logic_Oframe_Omore_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte3309513806868946049t_unit: episte1560738328020401952t_unit > episte1193835314949844379t_unit ).
thf(sy_c_Epistemic__Logic_Oimply_001tf__a,type,
epistemic_imply_a: list_Epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Product____Type__Ounit,type,
episte2398645135750866164t_unit: episte1560738328020401952t_unit > set_Epistemic_fm_a > list_char > $o ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062__update_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001tf__a_001t__Product____Type__Ounit,type,
episte1857908288096881731t_unit: ( ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o ) > episte1560738328020401952t_unit > episte1560738328020401952t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Okripke__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Product____Type__Ounit,type,
episte8239586592105053771t_unit: ( set_Epistemic_fm_a > list_char > $o ) > product_unit > episte1193835314949844379t_unit ).
thf(sy_c_Epistemic__Logic_Okripke_Omore_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member536094252920883875c_fm_a: set_Epistemic_fm_a > set_se5208064806568342746c_fm_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_A,type,
a2: epistemic_fm_a > $o ).
thf(sy_v_G,type,
g: list_Epistemic_fm_a ).
thf(sy_v_p,type,
p: epistemic_fm_a ).
% Relevant facts (1271)
thf(fact_0_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_1_R1,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ Q ) ) ) ).
% R1
thf(fact_2_imply__conjunct,axiom,
! [G2: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G2 @ P ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G2 ) @ P ) ) ) ).
% imply_conjunct
thf(fact_3_assms,axiom,
epistemic_AK_a @ a2 @ ( epistemic_imply_a @ g @ p ) ).
% assms
thf(fact_4_K__imp__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans
thf(fact_5_K__imp__trans_H,axiom,
! [A: epistemic_fm_a > $o,Q: epistemic_fm_a,R: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ Q @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_imp_trans'
thf(fact_6_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( ( epistemic_Imp_a @ X51 @ X52 )
= ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_7_K__trans,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ Q ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ Q @ R ) @ ( epistemic_Imp_a @ P @ R ) ) ) ) ).
% K_trans
thf(fact_8_conjunct__imply,axiom,
! [G2: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G2 ) @ P ) @ ( epistemic_imply_a @ G2 @ P ) ) ) ).
% conjunct_imply
thf(fact_9_K__right__mp,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ ( epistemic_Imp_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_10_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_11_K__imply__multi,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ B ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ C ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Con_a @ B @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_12_K__multi__imply,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a,B: epistemic_fm_a,C: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ A2 @ ( epistemic_Imp_a @ B @ C ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ A2 @ B ) @ C ) ) ) ).
% K_multi_imply
thf(fact_13_eval_Osimps_I5_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_14_K__A2_H,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_A2'
thf(fact_15_K__map,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_map
thf(fact_16_conjunct_Osimps_I1_J,axiom,
( ( stalnaker_conjunct_a @ nil_Epistemic_fm_a )
= ( epistemic_Imp_a @ epistemic_FF_a @ epistemic_FF_a ) ) ).
% conjunct.simps(1)
thf(fact_17_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G2: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ G2 ) ) @ Q ) ) ).
% K_mp
thf(fact_18_fm_Oinject_I5_J,axiom,
! [X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( ( epistemic_K_a @ X61 @ X62 )
= ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_19_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( ( epistemic_Con_a @ X41 @ X42 )
= ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_20_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H3: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H3 @ ( epistemic_K_a @ I @ P ) )
= ( H3 @ ( epistemic_K_a @ I @ P ) ) ) ).
% eval.simps(6)
thf(fact_21_eval_Osimps_I4_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
& ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_22_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_a > $o] :
~ ( epistemic_eval_a @ Uu @ Uv @ epistemic_FF_a ) ).
% eval.simps(1)
thf(fact_23_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_24_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_25_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_26_imply_Osimps_I1_J,axiom,
! [Q: epistemic_fm_a] :
( ( epistemic_imply_a @ nil_Epistemic_fm_a @ Q )
= Q ) ).
% imply.simps(1)
thf(fact_27_duality__taut,axiom,
! [I: a,P: epistemic_fm_a,Q: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_28_imply_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q )
= ( epistemic_Imp_a @ P @ ( epistemic_imply_a @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_29_K__imply__head,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ P ) ) ).
% K_imply_head
thf(fact_30_K__imply__Cons,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_31_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G2: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ ( cons_Epistemic_fm_a @ Q @ G2 ) ) @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G2 ) ) @ R ) ) ) ).
% K_swap
thf(fact_32_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Imp_a @ X51 @ X52 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_33_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_34_R2,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( epistemic_AK_a @ A @ P )
=> ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ P ) ) ) ).
% R2
thf(fact_35_K__thm,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Con_a @ P @ Q ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ).
% K_thm
thf(fact_36_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G2: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G2 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G2 @ P ) ) ) ).
% K_Boole
thf(fact_37_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ! [G5: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G5 @ H4 @ P2 ) )
| ? [I2: a,P2: epistemic_fm_a,Q2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I2 @ P2 ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ Q2 ) ) ) @ ( epistemic_K_a @ I2 @ Q2 ) ) )
| ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ( A3 @ P2 ) )
| ? [P2: epistemic_fm_a,Q2: epistemic_fm_a] :
( ( A4 = Q2 )
& ( epistemic_AK_a @ A3 @ P2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_Imp_a @ P2 @ Q2 ) ) )
| ? [P2: epistemic_fm_a,I2: a] :
( ( A4
= ( epistemic_K_a @ I2 @ P2 ) )
& ( epistemic_AK_a @ A3 @ P2 ) ) ) ) ) ).
% AK.simps
thf(fact_38_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ A2 )
=> ( ! [I3: a,P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P3 ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) ) @ ( epistemic_K_a @ I3 @ Q3 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P3: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P3 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P3 @ A2 ) ) )
=> ~ ! [P3: epistemic_fm_a] :
( ? [I3: a] :
( A2
= ( epistemic_K_a @ I3 @ P3 ) )
=> ~ ( epistemic_AK_a @ A @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_39_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Con_a @ X41 @ X42 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_40_K__L__dual,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P ) ) ) ).
% K_L_dual
thf(fact_41_K__LK,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ).
% K_LK
thf(fact_42_A2,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% A2
thf(fact_43_conjunct_Osimps_I2_J,axiom,
! [P: epistemic_fm_a,Ps: list_Epistemic_fm_a] :
( ( stalnaker_conjunct_a @ ( cons_Epistemic_fm_a @ P @ Ps ) )
= ( epistemic_Con_a @ P @ ( stalnaker_conjunct_a @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_44_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A2: set_nat,P4: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A2: list_nat,P4: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A2: nat,P4: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_56_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_57_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ! [X2: set_nat] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collect_set_nat @ P4 )
= ( collect_set_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_59_Collect__cong,axiom,
! [P4: list_nat > $o,Q4: list_nat > $o] :
( ! [X2: list_nat] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collect_list_nat @ P4 )
= ( collect_list_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ! [X2: nat] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collect_nat @ P4 )
= ( collect_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_61_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G2: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G2 ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G2 @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_62_list_Oinject,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a,Y21: epistemic_fm_a,Y22: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X21 @ X22 )
= ( cons_Epistemic_fm_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_63_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_64_list_Odistinct_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( nil_Epistemic_fm_a
!= ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_65_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_66_list_OdiscI,axiom,
! [List: list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( List
= ( cons_Epistemic_fm_a @ X21 @ X22 ) )
=> ( List != nil_Epistemic_fm_a ) ) ).
% list.discI
thf(fact_67_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_68_list_Oexhaust,axiom,
! [Y: list_Epistemic_fm_a] :
( ( Y != nil_Epistemic_fm_a )
=> ~ ! [X212: epistemic_fm_a,X222: list_Epistemic_fm_a] :
( Y
!= ( cons_Epistemic_fm_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_69_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_70_min__list_Ocases,axiom,
! [X3: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X3
!= ( cons_nat @ X2 @ Xs ) )
=> ( X3 = nil_nat ) ) ).
% min_list.cases
thf(fact_71_transpose_Ocases,axiom,
! [X3: list_l6083326122719238310c_fm_a] :
( ( X3 != nil_li2451196919128234278c_fm_a )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ~ ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_72_transpose_Ocases,axiom,
! [X3: list_list_nat] :
( ( X3 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X3
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X3
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_73_remdups__adj_Ocases,axiom,
! [X3: list_Epistemic_fm_a] :
( ( X3 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_74_remdups__adj_Ocases,axiom,
! [X3: list_nat] :
( ( X3 != nil_nat )
=> ( ! [X2: nat] :
( X3
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( X3
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_75_not__Cons__self2,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_76_not__Cons__self2,axiom,
! [X3: nat,Xs2: list_nat] :
( ( cons_nat @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_77_list__nonempty__induct,axiom,
! [Xs2: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( Xs2 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( P4 @ Xs )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) ) ) )
=> ( P4 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_78_list__nonempty__induct,axiom,
! [Xs2: list_nat,P4: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P4 @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P4 @ Xs )
=> ( P4 @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P4 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_79_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_Epistemic_fm_a > $o,Xs2: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_Epistemic_fm_a @ nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_80_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_nat > $o,Xs2: list_Epistemic_fm_a,Ys: list_nat] :
( ( P4 @ nil_Epistemic_fm_a @ nil_nat )
=> ( ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P4 @ nil_Epistemic_fm_a @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y2: nat,Ys2: list_nat] :
( ( P4 @ Xs @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_81_list__induct2_H,axiom,
! [P4: list_nat > list_Epistemic_fm_a > $o,Xs2: list_nat,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_nat @ nil_Epistemic_fm_a )
=> ( ! [X2: nat,Xs: list_nat] : ( P4 @ ( cons_nat @ X2 @ Xs ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_nat @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs @ Ys2 )
=> ( P4 @ ( cons_nat @ X2 @ Xs ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_82_list__induct2_H,axiom,
! [P4: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P4 @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P4 @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P4 @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys2: list_nat] :
( ( P4 @ Xs @ Ys2 )
=> ( P4 @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_83_neq__Nil__conv,axiom,
! [Xs2: list_Epistemic_fm_a] :
( ( Xs2 != nil_Epistemic_fm_a )
= ( ? [Y3: epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( Xs2
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_84_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_85_product__lists_Osimps_I1_J,axiom,
( ( produc1391074687832863881c_fm_a @ nil_li2451196919128234278c_fm_a )
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ nil_li2451196919128234278c_fm_a ) ) ).
% product_lists.simps(1)
thf(fact_86_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_87_subseqs_Osimps_I1_J,axiom,
( ( subseq859285839621985007c_fm_a @ nil_Epistemic_fm_a )
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ nil_li2451196919128234278c_fm_a ) ) ).
% subseqs.simps(1)
thf(fact_88_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_89_insert__Nil,axiom,
! [X3: epistemic_fm_a] :
( ( insert177310161492556854c_fm_a @ X3 @ nil_Epistemic_fm_a )
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% insert_Nil
thf(fact_90_insert__Nil,axiom,
! [X3: nat] :
( ( insert_nat @ X3 @ nil_nat )
= ( cons_nat @ X3 @ nil_nat ) ) ).
% insert_Nil
thf(fact_91_AxB_Ointros,axiom,
! [P: epistemic_fm_a,I: a] : ( epistemic_AxB_a @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% AxB.intros
thf(fact_92_AxB_Osimps,axiom,
( epistemic_AxB_a
= ( ^ [A4: epistemic_fm_a] :
? [P2: epistemic_fm_a,I2: a] :
( A4
= ( epistemic_Imp_a @ P2 @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% AxB.simps
thf(fact_93_AxB_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxB_a @ A2 )
=> ~ ! [P3: epistemic_fm_a,I3: a] :
( A2
!= ( epistemic_Imp_a @ P3 @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% AxB.cases
thf(fact_94_Ax5_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_Ax5_a @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% Ax5.intros
thf(fact_95_Ax5_Osimps,axiom,
( epistemic_Ax5_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ ( epistemic_Imp_a @ P2 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_96_Ax5_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax5_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ).
% Ax5.cases
thf(fact_97_K5__L,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% K5_L
thf(fact_98_list__ex1__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o] :
~ ( list_e2031426293596896995c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list_ex1_simps(1)
thf(fact_99_list__ex1__simps_I1_J,axiom,
! [P4: nat > $o] :
~ ( list_ex1_nat @ P4 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_100_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G2: list_Epistemic_fm_a,R: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G2 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G2 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G2 @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G2 @ R ) ) ) ) ) ).
% K_DisE
thf(fact_101_K__DisL,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,P5: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ Ps ) @ Q ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P5 @ Ps ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Dis_a @ P @ P5 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_102_map__tailrec__rev_Oelims,axiom,
! [X3: epistemic_fm_a > epistemic_fm_a,Xa: list_Epistemic_fm_a,Xb: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( map_ta6565535454553546997c_fm_a @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: epistemic_fm_a,As: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta6565535454553546997c_fm_a @ X3 @ As @ ( cons_Epistemic_fm_a @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_103_map__tailrec__rev_Oelims,axiom,
! [X3: epistemic_fm_a > nat,Xa: list_Epistemic_fm_a,Xb: list_nat,Y: list_nat] :
( ( ( map_ta3766586382360151221_a_nat @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: epistemic_fm_a,As: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta3766586382360151221_a_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_104_map__tailrec__rev_Oelims,axiom,
! [X3: nat > epistemic_fm_a,Xa: list_nat,Xb: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( map_ta4023091468578432403c_fm_a @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A5: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A5 @ As ) )
=> ( Y
!= ( map_ta4023091468578432403c_fm_a @ X3 @ As @ ( cons_Epistemic_fm_a @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_105_map__tailrec__rev_Oelims,axiom,
! [X3: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A5: nat,As: list_nat] :
( ( Xa
= ( cons_nat @ A5 @ As ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X3 @ As @ ( cons_nat @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_106_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
~ ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ nil_Epistemic_fm_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_107_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X3: nat,Xs2: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_108_bind__simps_I1_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( bind_E8451893407412458119c_fm_a @ nil_Epistemic_fm_a @ F )
= nil_Epistemic_fm_a ) ).
% bind_simps(1)
thf(fact_109_bind__simps_I1_J,axiom,
! [F: epistemic_fm_a > list_nat] :
( ( bind_E29171742398319523_a_nat @ nil_Epistemic_fm_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_110_bind__simps_I1_J,axiom,
! [F: nat > list_Epistemic_fm_a] :
( ( bind_n285676828616600705c_fm_a @ nil_nat @ F )
= nil_Epistemic_fm_a ) ).
% bind_simps(1)
thf(fact_111_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_112_listrelp_Ocases,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,A1: list_Epistemic_fm_a,A22: list_Epistemic_fm_a] :
( ( listre7830505053103709503c_fm_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_a )
=> ( A22 != nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs ) )
=> ! [Ys2: list_Epistemic_fm_a] :
( ( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listre7830505053103709503c_fm_a @ R @ Xs @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_113_listrelp_Ocases,axiom,
! [R: epistemic_fm_a > nat > $o,A1: list_Epistemic_fm_a,A22: list_nat] :
( ( listre8997185868192620267_a_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: epistemic_fm_a,Y2: nat,Xs: list_Epistemic_fm_a] :
( ( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listre8997185868192620267_a_nat @ R @ Xs @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_114_listrelp_Ocases,axiom,
! [R: nat > epistemic_fm_a > $o,A1: list_nat,A22: list_Epistemic_fm_a] :
( ( listre30318917556125641c_fm_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_Epistemic_fm_a ) )
=> ~ ! [X2: nat,Y2: epistemic_fm_a,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys2: list_Epistemic_fm_a] :
( ( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listre30318917556125641c_fm_a @ R @ Xs @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_115_listrelp_Ocases,axiom,
! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( listrelp_nat_nat @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listrelp_nat_nat @ R @ Xs @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_116_listrelp_Osimps,axiom,
( listre7830505053103709503c_fm_a
= ( ^ [R2: epistemic_fm_a > epistemic_fm_a > $o,A12: list_Epistemic_fm_a,A23: list_Epistemic_fm_a] :
( ( ( A12 = nil_Epistemic_fm_a )
& ( A23 = nil_Epistemic_fm_a ) )
| ? [X: epistemic_fm_a,Y3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) )
& ( R2 @ X @ Y3 )
& ( listre7830505053103709503c_fm_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_117_listrelp_Osimps,axiom,
( listre8997185868192620267_a_nat
= ( ^ [R2: epistemic_fm_a > nat > $o,A12: list_Epistemic_fm_a,A23: list_nat] :
( ( ( A12 = nil_Epistemic_fm_a )
& ( A23 = nil_nat ) )
| ? [X: epistemic_fm_a,Y3: nat,Xs3: list_Epistemic_fm_a,Ys3: list_nat] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( R2 @ X @ Y3 )
& ( listre8997185868192620267_a_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_118_listrelp_Osimps,axiom,
( listre30318917556125641c_fm_a
= ( ^ [R2: nat > epistemic_fm_a > $o,A12: list_nat,A23: list_Epistemic_fm_a] :
( ( ( A12 = nil_nat )
& ( A23 = nil_Epistemic_fm_a ) )
| ? [X: nat,Y3: epistemic_fm_a,Xs3: list_nat,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_nat @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) )
& ( R2 @ X @ Y3 )
& ( listre30318917556125641c_fm_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_119_listrelp_Osimps,axiom,
( listrelp_nat_nat
= ( ^ [R2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( ( A12 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( R2 @ X @ Y3 )
& ( listrelp_nat_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_120_fm_Oinject_I2_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( ( epistemic_Dis_a @ X31 @ X32 )
= ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(2)
thf(fact_121_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexord491902619044731238c_fm_a @ Less @ Xs2 @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_122_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: nat > nat > $o,X3: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( lexordp_eq_nat @ Less @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq_nat @ Less @ Xs2 @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_123_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] : ( lexord491902619044731238c_fm_a @ Less @ nil_Epistemic_fm_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_124_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_125_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Xs2: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ Xs2 @ nil_Epistemic_fm_a )
= ( Xs2 = nil_Epistemic_fm_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_126_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs2: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_127_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_128_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_129_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_130_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Dis_a @ X31 @ X32 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_131_eval_Osimps_I3_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
| ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_132_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexord491902619044731238c_fm_a @ Less @ Xs2 @ Ys )
=> ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_133_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: nat > nat > $o,X3: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq_nat @ Less @ Xs2 @ Ys )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_134_ord_Olexordp__eq_OCons,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Less @ X3 @ Y )
=> ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_135_ord_Olexordp__eq_OCons,axiom,
! [Less: nat > nat > $o,X3: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_136_ord_Olexordp__eq_ONil,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] : ( lexord491902619044731238c_fm_a @ Less @ nil_Epistemic_fm_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_137_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_138_listrelp_OCons,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( R @ X3 @ Y )
=> ( ( listre7830505053103709503c_fm_a @ R @ Xs2 @ Ys )
=> ( listre7830505053103709503c_fm_a @ R @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_139_listrelp_OCons,axiom,
! [R: epistemic_fm_a > nat > $o,X3: epistemic_fm_a,Y: nat,Xs2: list_Epistemic_fm_a,Ys: list_nat] :
( ( R @ X3 @ Y )
=> ( ( listre8997185868192620267_a_nat @ R @ Xs2 @ Ys )
=> ( listre8997185868192620267_a_nat @ R @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_140_listrelp_OCons,axiom,
! [R: nat > epistemic_fm_a > $o,X3: nat,Y: epistemic_fm_a,Xs2: list_nat,Ys: list_Epistemic_fm_a] :
( ( R @ X3 @ Y )
=> ( ( listre30318917556125641c_fm_a @ R @ Xs2 @ Ys )
=> ( listre30318917556125641c_fm_a @ R @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_141_listrelp_OCons,axiom,
! [R: nat > nat > $o,X3: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
( ( R @ X3 @ Y )
=> ( ( listrelp_nat_nat @ R @ Xs2 @ Ys )
=> ( listrelp_nat_nat @ R @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_142_listrelp_ONil,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o] : ( listre7830505053103709503c_fm_a @ R @ nil_Epistemic_fm_a @ nil_Epistemic_fm_a ) ).
% listrelp.Nil
thf(fact_143_listrelp_ONil,axiom,
! [R: epistemic_fm_a > nat > $o] : ( listre8997185868192620267_a_nat @ R @ nil_Epistemic_fm_a @ nil_nat ) ).
% listrelp.Nil
thf(fact_144_listrelp_ONil,axiom,
! [R: nat > epistemic_fm_a > $o] : ( listre30318917556125641c_fm_a @ R @ nil_nat @ nil_Epistemic_fm_a ) ).
% listrelp.Nil
thf(fact_145_listrelp_ONil,axiom,
! [R: nat > nat > $o] : ( listrelp_nat_nat @ R @ nil_nat @ nil_nat ) ).
% listrelp.Nil
thf(fact_146_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A2: epistemic_fm_a,As2: list_Epistemic_fm_a,Bs: list_Epistemic_fm_a] :
( ( map_ta6565535454553546997c_fm_a @ F @ ( cons_Epistemic_fm_a @ A2 @ As2 ) @ Bs )
= ( map_ta6565535454553546997c_fm_a @ F @ As2 @ ( cons_Epistemic_fm_a @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_147_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > nat,A2: epistemic_fm_a,As2: list_Epistemic_fm_a,Bs: list_nat] :
( ( map_ta3766586382360151221_a_nat @ F @ ( cons_Epistemic_fm_a @ A2 @ As2 ) @ Bs )
= ( map_ta3766586382360151221_a_nat @ F @ As2 @ ( cons_nat @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_148_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > epistemic_fm_a,A2: nat,As2: list_nat,Bs: list_Epistemic_fm_a] :
( ( map_ta4023091468578432403c_fm_a @ F @ ( cons_nat @ A2 @ As2 ) @ Bs )
= ( map_ta4023091468578432403c_fm_a @ F @ As2 @ ( cons_Epistemic_fm_a @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_149_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A2: nat,As2: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A2 @ As2 ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_150_ord_Olexordp__eq_Osimps,axiom,
( lexord491902619044731238c_fm_a
= ( ^ [Less2: epistemic_fm_a > epistemic_fm_a > $o,A12: list_Epistemic_fm_a,A23: list_Epistemic_fm_a] :
( ? [Ys3: list_Epistemic_fm_a] :
( ( A12 = nil_Epistemic_fm_a )
& ( A23 = Ys3 ) )
| ? [X: epistemic_fm_a,Y3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) )
& ( Less2 @ X @ Y3 ) )
| ? [X: epistemic_fm_a,Y3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X @ Y3 )
& ~ ( Less2 @ Y3 @ X )
& ( lexord491902619044731238c_fm_a @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_151_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Ys3: list_nat] :
( ( A12 = nil_nat )
& ( A23 = Ys3 ) )
| ? [X: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ( Less2 @ X @ Y3 ) )
| ? [X: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs3 ) )
& ( A23
= ( cons_nat @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X @ Y3 )
& ~ ( Less2 @ Y3 @ X )
& ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_152_ord_Olexordp__eq_Ocases,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,A1: list_Epistemic_fm_a,A22: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ A1 @ A22 )
=> ( ( A1 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ? [Xs: list_Epistemic_fm_a] :
( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs ) )
=> ! [Y2: epistemic_fm_a] :
( ? [Ys2: list_Epistemic_fm_a] :
( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs ) )
=> ! [Ys2: list_Epistemic_fm_a] :
( ( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexord491902619044731238c_fm_a @ Less @ Xs @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_153_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( lexordp_eq_nat @ Less @ A1 @ A22 )
=> ( ( A1 != nil_nat )
=> ( ! [X2: nat] :
( ? [Xs: list_nat] :
( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys2: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq_nat @ Less @ Xs @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_154_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_155_KB4__5,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% KB4_5
thf(fact_156_S5_H__B,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ) ) ).
% S5'_B
thf(fact_157_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_158_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_159_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_160_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_161_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_162_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_163_order__refl,axiom,
! [X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ X3 ) ).
% order_refl
thf(fact_164_order__refl,axiom,
! [X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ X3 ) ).
% order_refl
thf(fact_165_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_166_order__refl,axiom,
! [X3: set_set_nat] : ( ord_le6893508408891458716et_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_167_order__refl,axiom,
! [X3: $o > nat] : ( ord_less_eq_o_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_168_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_169_K4__L,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% K4_L
thf(fact_170_S5_H__4,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% S5'_4
thf(fact_171_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G5: epistemic_fm_a > $o] :
! [X: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G5 @ X ) ) ) ) ).
% le_fun_def
thf(fact_172_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G5: $o > nat] :
! [X: $o] : ( ord_less_eq_nat @ ( F2 @ X ) @ ( G5 @ X ) ) ) ) ).
% le_fun_def
thf(fact_173_le__funI,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G4 ) ) ).
% le_funI
thf(fact_174_le__funI,axiom,
! [F: $o > nat,G4: $o > nat] :
( ! [X2: $o] : ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_less_eq_o_nat @ F @ G4 ) ) ).
% le_funI
thf(fact_175_le__funE,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G4 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funE
thf(fact_176_le__funE,axiom,
! [F: $o > nat,G4: $o > nat,X3: $o] :
( ( ord_less_eq_o_nat @ F @ G4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funE
thf(fact_177_AxT_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_AxT_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_178_AxT_Osimps,axiom,
( epistemic_AxT_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P2 ) @ P2 ) ) ) ) ).
% AxT.simps
thf(fact_179_AxT_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_AxT_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ).
% AxT.intros
thf(fact_180_Ax4_Ocases,axiom,
! [A2: epistemic_fm_a] :
( ( epistemic_Ax4_a @ A2 )
=> ~ ! [I3: a,P3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_K_a @ I3 @ P3 ) @ ( epistemic_K_a @ I3 @ ( epistemic_K_a @ I3 @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_181_Ax4_Osimps,axiom,
( epistemic_Ax4_a
= ( ^ [A4: epistemic_fm_a] :
? [I2: a,P2: epistemic_fm_a] :
( A4
= ( epistemic_Imp_a @ ( epistemic_K_a @ I2 @ P2 ) @ ( epistemic_K_a @ I2 @ ( epistemic_K_a @ I2 @ P2 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_182_Ax4_Ointros,axiom,
! [I: a,P: epistemic_fm_a] : ( epistemic_Ax4_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% Ax4.intros
thf(fact_183_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_184_le__cases3,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_185_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y4 = Z2 ) )
= ( ^ [X: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y3 )
& ( ord_le4043730696559282883fm_a_o @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_186_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y4 = Z2 ) )
= ( ^ [X: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y3 )
& ( ord_le3275665582123262618c_fm_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_187_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_188_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [X: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y3 )
& ( ord_le6893508408891458716et_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_189_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [X: $o > nat,Y3: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y3 )
& ( ord_less_eq_o_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_190_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_191_ord__eq__le__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2 = B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2 = B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_194_ord__eq__le__trans,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( A2 = B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_195_ord__eq__le__trans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( A2 = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_196_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_197_ord__le__eq__trans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( B = C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_198_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_199_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_200_ord__le__eq__trans,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_201_ord__le__eq__trans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_202_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_203_order__antisym,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_204_order__antisym,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_205_order__antisym,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_206_order__antisym,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_207_order__antisym,axiom,
! [X3: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_208_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_209_order_Otrans,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_210_order_Otrans,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_211_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_212_order_Otrans,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_213_order_Otrans,axiom,
! [A2: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_214_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_215_order__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_216_order__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_217_order__trans,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_218_order__trans,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z )
=> ( ord_le6893508408891458716et_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_219_order__trans,axiom,
! [X3: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_eq_o_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_220_order__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_221_linorder__wlog,axiom,
! [P4: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
=> ( P4 @ A5 @ B2 ) )
=> ( ! [A5: nat,B2: nat] :
( ( P4 @ B2 @ A5 )
=> ( P4 @ A5 @ B2 ) )
=> ( P4 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_222_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: epistemic_fm_a > $o,B3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B3 @ A4 )
& ( ord_le4043730696559282883fm_a_o @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_223_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B3 @ A4 )
& ( ord_le3275665582123262618c_fm_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_224_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_225_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_226_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A4 )
& ( ord_less_eq_o_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_227_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_228_dual__order_Oantisym,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_229_dual__order_Oantisym,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_230_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_231_dual__order_Oantisym,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_232_dual__order_Oantisym,axiom,
! [B: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_233_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_234_dual__order_Otrans,axiom,
! [B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_235_dual__order_Otrans,axiom,
! [B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_236_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_237_dual__order_Otrans,axiom,
! [B: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_238_dual__order_Otrans,axiom,
! [B: $o > nat,A2: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A2 )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_239_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_240_antisym,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_241_antisym,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_242_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_243_antisym,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_244_antisym,axiom,
! [A2: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( ord_less_eq_o_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_245_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_246_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: epistemic_fm_a > $o,B3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B3 )
& ( ord_le4043730696559282883fm_a_o @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_247_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_a,B3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B3 )
& ( ord_le3275665582123262618c_fm_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_248_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_249_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_250_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B3 )
& ( ord_less_eq_o_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_251_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_252_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_253_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_254_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_255_order__subst1,axiom,
! [A2: nat,F: set_Epistemic_fm_a > nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_256_order__subst1,axiom,
! [A2: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_257_order__subst1,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ! [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_258_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a,B: nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_259_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_260_order__subst1,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_261_order__subst1,axiom,
! [A2: $o > nat,F: nat > $o > nat,B: nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_262_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_263_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_264_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_265_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_266_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_267_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_o_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_268_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_269_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_270_order__subst2,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_271_order__subst2,axiom,
! [A2: $o > nat,B: $o > nat,F: ( $o > nat ) > nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_272_order__eq__refl,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( X3 = Y )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_273_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( X3 = Y )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_274_order__eq__refl,axiom,
! [X3: set_nat,Y: set_nat] :
( ( X3 = Y )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_275_order__eq__refl,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( X3 = Y )
=> ( ord_le6893508408891458716et_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_276_order__eq__refl,axiom,
! [X3: $o > nat,Y: $o > nat] :
( ( X3 = Y )
=> ( ord_less_eq_o_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_277_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_278_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_279_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_280_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_281_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_282_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_283_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_284_ord__eq__le__subst,axiom,
! [A2: $o > nat,F: nat > $o > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_285_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Epistemic_fm_a > nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_286_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_287_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_nat > nat,B: set_set_nat,C: set_set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_288_ord__eq__le__subst,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B: $o > nat,C: $o > nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ! [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_289_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_290_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_291_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_294_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > $o > nat,C: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_295_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_296_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_297_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_298_ord__le__eq__subst,axiom,
! [A2: $o > nat,B: $o > nat,F: ( $o > nat ) > nat,C: nat] :
( ( ord_less_eq_o_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_299_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_300_order__antisym__conv,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_301_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_302_order__antisym__conv,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_303_order__antisym__conv,axiom,
! [Y: set_set_nat,X3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X3 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_304_order__antisym__conv,axiom,
! [Y: $o > nat,X3: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X3 )
=> ( ( ord_less_eq_o_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_305_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_306_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) ) ).
% predicate1D
thf(fact_307_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Q4: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_308_T__L,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) ) @ epistemic_FF_a ) ) ) ) ).
% T_L
thf(fact_309_le__funD,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G4 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funD
thf(fact_310_le__funD,axiom,
! [F: $o > nat,G4: $o > nat,X3: $o] :
( ( ord_less_eq_o_nat @ F @ G4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funD
thf(fact_311_fm_Oexhaust,axiom,
! [Y: epistemic_fm_a] :
( ( Y != epistemic_FF_a )
=> ( ! [X23: list_char] :
( Y
!= ( epistemic_Pro_a @ X23 ) )
=> ( ! [X312: epistemic_fm_a,X322: epistemic_fm_a] :
( Y
!= ( epistemic_Dis_a @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_a,X422: epistemic_fm_a] :
( Y
!= ( epistemic_Con_a @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_a,X522: epistemic_fm_a] :
( Y
!= ( epistemic_Imp_a @ X512 @ X522 ) )
=> ~ ! [X612: a,X622: epistemic_fm_a] :
( Y
!= ( epistemic_K_a @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_312_fm_Oset__cases,axiom,
! [E: nat,A2: epistemic_fm_nat] :
( ( member_nat2 @ E @ ( epistemic_set_fm_nat @ A2 ) )
=> ( ! [Z1: epistemic_fm_nat] :
( ? [Z22: epistemic_fm_nat] :
( A2
= ( epistemic_Dis_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_nat,Z22: epistemic_fm_nat] :
( ( A2
= ( epistemic_Dis_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_nat] :
( ? [Z22: epistemic_fm_nat] :
( A2
= ( epistemic_Con_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_nat,Z22: epistemic_fm_nat] :
( ( A2
= ( epistemic_Con_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_nat] :
( ? [Z22: epistemic_fm_nat] :
( A2
= ( epistemic_Imp_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_nat,Z22: epistemic_fm_nat] :
( ( A2
= ( epistemic_Imp_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_nat] :
( A2
!= ( epistemic_K_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: epistemic_fm_nat] :
( ( A2
= ( epistemic_K_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( epistemic_set_fm_nat @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_313_fm_Oset__cases,axiom,
! [E: epistemic_fm_a,A2: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ A2 ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte6088726764479022859c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte3685526487207141399c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a] :
( ? [Z22: episte740340785640729014c_fm_a] :
( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte740340785640729014c_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte260752218777527565c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte740340785640729014c_fm_a] :
( A2
!= ( episte5657488632024175118c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: episte740340785640729014c_fm_a] :
( ( A2
= ( episte5657488632024175118c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( episte9089240958480457552c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_314_fm_Oset__cases,axiom,
! [E: a,A2: epistemic_fm_a] :
( ( member_a2 @ E @ ( epistemic_set_fm_a @ A2 ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_a] :
( ? [Z22: epistemic_fm_a] :
( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_a] :
( A2
!= ( epistemic_K_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( epistemic_set_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_315_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_253494837916242250fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_316_GreatestI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Q4: set_Epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_4585748725732241747c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_317_GreatestI2__order,axiom,
! [P4: set_nat > $o,X3: set_nat,Q4: set_nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X3 ) )
=> ( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( ! [Y5: set_nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_5724808138429204845et_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_318_GreatestI2__order,axiom,
! [P4: set_set_nat > $o,X3: set_set_nat,Q4: set_set_nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_set_nat] :
( ( P4 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X3 ) )
=> ( ! [X2: set_set_nat] :
( ( P4 @ X2 )
=> ( ! [Y5: set_set_nat] :
( ( P4 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_1279421399067128355et_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_319_GreatestI2__order,axiom,
! [P4: ( $o > nat ) > $o,X3: $o > nat,Q4: ( $o > nat ) > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: $o > nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X3 ) )
=> ( ! [X2: $o > nat] :
( ( P4 @ X2 )
=> ( ! [Y5: $o > nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_Greatest_o_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_320_GreatestI2__order,axiom,
! [P4: nat > $o,X3: nat,Q4: nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_Greatest_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_321_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( order_253494837916242250fm_a_o @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_322_Greatest__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( order_4585748725732241747c_fm_a @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_323_Greatest__equality,axiom,
! [P4: set_nat > $o,X3: set_nat] :
( ( P4 @ X3 )
=> ( ! [Y2: set_nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X3 ) )
=> ( ( order_5724808138429204845et_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_324_Greatest__equality,axiom,
! [P4: set_set_nat > $o,X3: set_set_nat] :
( ( P4 @ X3 )
=> ( ! [Y2: set_set_nat] :
( ( P4 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ Y2 @ X3 ) )
=> ( ( order_1279421399067128355et_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_325_Greatest__equality,axiom,
! [P4: ( $o > nat ) > $o,X3: $o > nat] :
( ( P4 @ X3 )
=> ( ! [Y2: $o > nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X3 ) )
=> ( ( order_Greatest_o_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_326_Greatest__equality,axiom,
! [P4: nat > $o,X3: nat] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( order_Greatest_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_327_distribution,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a,Q: epistemic_fm_a] : ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ P @ Q ) ) ) @ ( epistemic_K_a @ I @ Q ) ) ) ).
% distribution
thf(fact_328_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs ) )
=> ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_329_lexordp__eq__simps_I3_J,axiom,
! [X3: nat,Xs2: list_nat] :
~ ( ord_lexordp_eq_nat @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat ) ).
% lexordp_eq_simps(3)
thf(fact_330_member__rec_I2_J,axiom,
! [Y: epistemic_fm_a] :
~ ( member6038508265109909045c_fm_a @ nil_Epistemic_fm_a @ Y ) ).
% member_rec(2)
thf(fact_331_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_332_fm_Oinject_I1_J,axiom,
! [X24: list_char,Y23: list_char] :
( ( ( epistemic_Pro_a @ X24 )
= ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.inject(1)
thf(fact_333_Imp__intro,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) ) ) ).
% Imp_intro
thf(fact_334_lexordp__eq__simps_I1_J,axiom,
! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).
% lexordp_eq_simps(1)
thf(fact_335_lexordp__eq__simps_I2_J,axiom,
! [Xs2: list_nat] :
( ( ord_lexordp_eq_nat @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% lexordp_eq_simps(2)
thf(fact_336_in__set__insert,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_337_in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_338_in__set__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_339_not__in__set__insert,axiom,
! [X3: a,Xs2: list_a] :
( ~ ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X3 @ Xs2 )
= ( cons_a @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_340_not__in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs2 )
= ( cons_Epistemic_fm_a @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_341_not__in__set__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X3 @ Xs2 )
= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_342_subset__code_I1_J,axiom,
! [Xs2: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B4 )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( member_a2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_343_subset__code_I1_J,axiom,
! [Xs2: list_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) @ B4 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs2 ) )
=> ( member6642669571620171971c_fm_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_344_subset__code_I1_J,axiom,
! [Xs2: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( member_nat2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_345_subset__code_I1_J,axiom,
! [Xs2: list_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B4 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_346_in__set__member,axiom,
! [X3: a,Xs2: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
= ( member_a @ Xs2 @ X3 ) ) ).
% in_set_member
thf(fact_347_in__set__member,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
= ( member6038508265109909045c_fm_a @ Xs2 @ X3 ) ) ).
% in_set_member
thf(fact_348_in__set__member,axiom,
! [X3: nat,Xs2: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
= ( member_nat @ Xs2 @ X3 ) ) ).
% in_set_member
thf(fact_349_set__ConsD,axiom,
! [Y: a,X3: a,Xs2: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member_a2 @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_350_set__ConsD,axiom,
! [Y: epistemic_fm_a,X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_351_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs2: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) )
=> ( ( Y = X3 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_352_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a2 @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z22: list_a] :
( A2
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_353_list_Oset__cases,axiom,
! [E: epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ A2 ) )
=> ( ! [Z22: list_Epistemic_fm_a] :
( A2
!= ( cons_Epistemic_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_a,Z22: list_Epistemic_fm_a] :
( ( A2
= ( cons_Epistemic_fm_a @ Z1 @ Z22 ) )
=> ~ ( member6642669571620171971c_fm_a @ E @ ( set_Epistemic_fm_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_354_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_355_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_356_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X21 @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_357_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_358_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a2 @ Y @ ( set_a2 @ X22 ) )
=> ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_359_list_Oset__intros_I2_J,axiom,
! [Y: epistemic_fm_a,X22: list_Epistemic_fm_a,X21: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ X22 ) )
=> ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_360_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_361_set__subset__Cons,axiom,
! [Xs2: list_Epistemic_fm_a,X3: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_362_set__subset__Cons,axiom,
! [Xs2: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_363_set__subset__Cons,axiom,
! [Xs2: list_set_nat,X3: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ ( set_set_nat2 @ ( cons_set_nat @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_364_semantics_Osimps_I5_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(5)
thf(fact_365_fm_Oset__intros_I6_J,axiom,
! [Yf: nat,X52: epistemic_fm_nat,X51: epistemic_fm_nat] :
( ( member_nat2 @ Yf @ ( epistemic_set_fm_nat @ X52 ) )
=> ( member_nat2 @ Yf @ ( epistemic_set_fm_nat @ ( epistemic_Imp_nat @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_366_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_a,X52: episte740340785640729014c_fm_a,X51: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ X52 ) )
=> ( member6642669571620171971c_fm_a @ Yf @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_367_fm_Oset__intros_I6_J,axiom,
! [Yf: a,X52: epistemic_fm_a,X51: epistemic_fm_a] :
( ( member_a2 @ Yf @ ( epistemic_set_fm_a @ X52 ) )
=> ( member_a2 @ Yf @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_368_fm_Oset__intros_I5_J,axiom,
! [Ye: nat,X51: epistemic_fm_nat,X52: epistemic_fm_nat] :
( ( member_nat2 @ Ye @ ( epistemic_set_fm_nat @ X51 ) )
=> ( member_nat2 @ Ye @ ( epistemic_set_fm_nat @ ( epistemic_Imp_nat @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_369_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_a,X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ X51 ) )
=> ( member6642669571620171971c_fm_a @ Ye @ ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_370_fm_Oset__intros_I5_J,axiom,
! [Ye: a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( member_a2 @ Ye @ ( epistemic_set_fm_a @ X51 ) )
=> ( member_a2 @ Ye @ ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_371_fm_Oset__intros_I7_J,axiom,
! [X61: nat,X62: epistemic_fm_nat] : ( member_nat2 @ X61 @ ( epistemic_set_fm_nat @ ( epistemic_K_nat @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_372_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] : ( member6642669571620171971c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_373_fm_Oset__intros_I7_J,axiom,
! [X61: a,X62: epistemic_fm_a] : ( member_a2 @ X61 @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_374_fm_Oset__intros_I8_J,axiom,
! [Yg: nat,X62: epistemic_fm_nat,X61: nat] :
( ( member_nat2 @ Yg @ ( epistemic_set_fm_nat @ X62 ) )
=> ( member_nat2 @ Yg @ ( epistemic_set_fm_nat @ ( epistemic_K_nat @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_375_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_a,X62: episte740340785640729014c_fm_a,X61: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ X62 ) )
=> ( member6642669571620171971c_fm_a @ Yg @ ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_376_fm_Oset__intros_I8_J,axiom,
! [Yg: a,X62: epistemic_fm_a,X61: a] :
( ( member_a2 @ Yg @ ( epistemic_set_fm_a @ X62 ) )
=> ( member_a2 @ Yg @ ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_377_semantics_Osimps_I1_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
~ ( episte7081087998767065248c_fm_a @ M @ W @ epistemic_FF_a ) ).
% semantics.simps(1)
thf(fact_378_fm_Odistinct_I15_J,axiom,
! [X24: list_char,X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_379_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: a,X62: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_380_semantics_Osimps_I4_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Con_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
& ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(4)
thf(fact_381_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X24 ) ) ).
% fm.distinct(1)
thf(fact_382_fm_Oset__intros_I3_J,axiom,
! [Yc: nat,X41: epistemic_fm_nat,X42: epistemic_fm_nat] :
( ( member_nat2 @ Yc @ ( epistemic_set_fm_nat @ X41 ) )
=> ( member_nat2 @ Yc @ ( epistemic_set_fm_nat @ ( epistemic_Con_nat @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_383_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_a,X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ X41 ) )
=> ( member6642669571620171971c_fm_a @ Yc @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_384_fm_Oset__intros_I3_J,axiom,
! [Yc: a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( member_a2 @ Yc @ ( epistemic_set_fm_a @ X41 ) )
=> ( member_a2 @ Yc @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_385_fm_Oset__intros_I4_J,axiom,
! [Yd: nat,X42: epistemic_fm_nat,X41: epistemic_fm_nat] :
( ( member_nat2 @ Yd @ ( epistemic_set_fm_nat @ X42 ) )
=> ( member_nat2 @ Yd @ ( epistemic_set_fm_nat @ ( epistemic_Con_nat @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_386_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_a,X42: episte740340785640729014c_fm_a,X41: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ X42 ) )
=> ( member6642669571620171971c_fm_a @ Yd @ ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_387_fm_Oset__intros_I4_J,axiom,
! [Yd: a,X42: epistemic_fm_a,X41: epistemic_fm_a] :
( ( member_a2 @ Yd @ ( epistemic_set_fm_a @ X42 ) )
=> ( member_a2 @ Yd @ ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_388_semantics_Osimps_I3_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( episte7081087998767065248c_fm_a @ M @ W @ P )
| ( episte7081087998767065248c_fm_a @ M @ W @ Q ) ) ) ).
% semantics.simps(3)
thf(fact_389_fm_Oset__intros_I2_J,axiom,
! [Yb: nat,X32: epistemic_fm_nat,X31: epistemic_fm_nat] :
( ( member_nat2 @ Yb @ ( epistemic_set_fm_nat @ X32 ) )
=> ( member_nat2 @ Yb @ ( epistemic_set_fm_nat @ ( epistemic_Dis_nat @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_390_fm_Oset__intros_I2_J,axiom,
! [Yb: epistemic_fm_a,X32: episte740340785640729014c_fm_a,X31: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ X32 ) )
=> ( member6642669571620171971c_fm_a @ Yb @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_391_fm_Oset__intros_I2_J,axiom,
! [Yb: a,X32: epistemic_fm_a,X31: epistemic_fm_a] :
( ( member_a2 @ Yb @ ( epistemic_set_fm_a @ X32 ) )
=> ( member_a2 @ Yb @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_392_fm_Oset__intros_I1_J,axiom,
! [Ya: nat,X31: epistemic_fm_nat,X32: epistemic_fm_nat] :
( ( member_nat2 @ Ya @ ( epistemic_set_fm_nat @ X31 ) )
=> ( member_nat2 @ Ya @ ( epistemic_set_fm_nat @ ( epistemic_Dis_nat @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_393_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_a,X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ X31 ) )
=> ( member6642669571620171971c_fm_a @ Ya @ ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_394_fm_Oset__intros_I1_J,axiom,
! [Ya: a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( member_a2 @ Ya @ ( epistemic_set_fm_a @ X31 ) )
=> ( member_a2 @ Ya @ ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_395_fm_Odistinct_I13_J,axiom,
! [X24: list_char,X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_396_fm_Odistinct_I11_J,axiom,
! [X24: list_char,X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_Pro_a @ X24 )
!= ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.distinct(11)
thf(fact_397_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_398_lexordp__eq_ONil,axiom,
! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).
% lexordp_eq.Nil
thf(fact_399_eval_Osimps_I2_J,axiom,
! [G4: list_char > $o,Uw: epistemic_fm_a > $o,X3: list_char] :
( ( epistemic_eval_a @ G4 @ Uw @ ( epistemic_Pro_a @ X3 ) )
= ( G4 @ X3 ) ) ).
% eval.simps(2)
thf(fact_400_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P6: a > $o,Xs3: list_a] :
? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y3: a] :
( ( ( member_a2 @ Y3 @ ( set_a2 @ Xs3 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_401_list__ex1__iff,axiom,
( list_e2031426293596896995c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,Xs3: list_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y3: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y3 @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_402_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P6: nat > $o,Xs3: list_nat] :
? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs3 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_403_K__imply__weaken,axiom,
! [A: epistemic_fm_a > $o,Ps: list_Epistemic_fm_a,Q: epistemic_fm_a,Ps2: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ Q ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Ps2 ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_404_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X: a,Xs3: list_a] : ( if_list_a @ ( member_a2 @ X @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_405_List_Oinsert__def,axiom,
( insert177310161492556854c_fm_a
= ( ^ [X: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_406_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_407_member__rec_I1_J,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( member6038508265109909045c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) @ Y )
= ( ( X3 = Y )
| ( member6038508265109909045c_fm_a @ Xs2 @ Y ) ) ) ).
% member_rec(1)
thf(fact_408_member__rec_I1_J,axiom,
! [X3: nat,Xs2: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X3 @ Xs2 ) @ Y )
= ( ( X3 = Y )
| ( member_nat @ Xs2 @ Y ) ) ) ).
% member_rec(1)
thf(fact_409_can__select__set__list__ex1,axiom,
! [P4: epistemic_fm_a > $o,A: list_Epistemic_fm_a] :
( ( can_se5173380710277125655c_fm_a @ P4 @ ( set_Epistemic_fm_a2 @ A ) )
= ( list_e2031426293596896995c_fm_a @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_410_can__select__set__list__ex1,axiom,
! [P4: nat > $o,A: list_nat] :
( ( can_select_nat @ P4 @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_411_subset__antisym,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_412_subset__antisym,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_413_subset__antisym,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_414_subsetI,axiom,
! [A: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_a2 @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% subsetI
thf(fact_415_subsetI,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ X2 @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_416_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ X2 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_417_subsetI,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_418_the__elem__set,axiom,
! [X3: epistemic_fm_a] :
( ( the_el2173195877760541071c_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_419_the__elem__set,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X3 @ nil_nat ) ) )
= X3 ) ).
% the_elem_set
thf(fact_420_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte8765170747386058258t_unit] :
( ( P4 @ M3 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ Xa2 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_421_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_422_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_423_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_424_fm_Orel__induct,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ( Q4 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A24: list_char,B22: list_char] :
( ( A24 = B22 )
=> ( Q4 @ ( epistemic_Pro_a @ A24 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q4 @ A31 @ B31 )
=> ( ( Q4 @ A32 @ B32 )
=> ( Q4 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q4 @ A41 @ B41 )
=> ( ( Q4 @ A42 @ B42 )
=> ( Q4 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q4 @ A51 @ B51 )
=> ( ( Q4 @ A52 @ B52 )
=> ( Q4 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R3 @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_425_fm_Orel__cases,axiom,
! [R3: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B != epistemic_FF_a ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_a @ X2 ) )
=> ! [Y2: list_char] :
( ( B
= ( epistemic_Pro_a @ Y2 ) )
=> ( X2 != Y2 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_426_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_427_fm_Orel__mono,axiom,
! [R3: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R3 @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R3 ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_428_fm_Orel__refl,axiom,
! [Ra: a > a > $o,X3: epistemic_fm_a] :
( ! [X2: a] : ( Ra @ X2 @ X2 )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl
thf(fact_429_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [Y4: epistemic_fm_a,Z2: epistemic_fm_a] : ( Y4 = Z2 ) ) ) ).
% fm.rel_eq
thf(fact_430_fm_Orel__intros_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_431_fm_Orel__inject_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_432_fm_Orel__inject_I6_J,axiom,
! [R3: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_433_fm_Orel__intros_I6_J,axiom,
! [R3: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_434_fm_Octr__transfer_I1_J,axiom,
! [R3: a > a > $o] : ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_435_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte3894023384580379906at_nat @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_436_fm_Orel__refl__strong,axiom,
! [X3: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_437_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ X3 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_438_fm_Orel__mono__strong,axiom,
! [R3: nat > nat > $o,X3: epistemic_fm_nat,Y: epistemic_fm_nat,Ra: nat > nat > $o] :
( ( episte3894023384580379906at_nat @ R3 @ X3 @ Y )
=> ( ! [Z3: nat,Yb2: nat] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte3894023384580379906at_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_439_fm_Orel__mono__strong,axiom,
! [R3: nat > epistemic_fm_a > $o,X3: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Ra: nat > epistemic_fm_a > $o] :
( ( episte9034525631817513832c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z3: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte9034525631817513832c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_440_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > nat > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Ra: epistemic_fm_a > nat > $o] :
( ( episte8778020545599232650_a_nat @ R3 @ X3 @ Y )
=> ( ! [Z3: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte8778020545599232650_a_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_441_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > epistemic_fm_a > $o,X3: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z3: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_442_fm_Orel__mono__strong,axiom,
! [R3: nat > a > $o,X3: epistemic_fm_nat,Y: epistemic_fm_a,Ra: nat > a > $o] :
( ( episte5492358210969815628_nat_a @ R3 @ X3 @ Y )
=> ( ! [Z3: nat,Yb2: a] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte5492358210969815628_nat_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_443_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > a > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z3: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_444_fm_Orel__mono__strong,axiom,
! [R3: a > nat > $o,X3: epistemic_fm_a,Y: epistemic_fm_nat,Ra: a > nat > $o] :
( ( episte1460426709791529198_a_nat @ R3 @ X3 @ Y )
=> ( ! [Z3: a,Yb2: nat] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte1460426709791529198_a_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_445_fm_Orel__mono__strong,axiom,
! [R3: a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z3: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_446_fm_Orel__mono__strong,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z3: a,Yb2: a] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_447_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: nat > nat > $o,Ra: nat > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb2: nat] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte3894023384580379906at_nat @ R3 @ X3 @ Y )
= ( episte3894023384580379906at_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_448_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: nat > epistemic_fm_a > $o,Ra: nat > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte9034525631817513832c_fm_a @ R3 @ X3 @ Y )
= ( episte9034525631817513832c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_449_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: epistemic_fm_a > nat > $o,Ra: epistemic_fm_a > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte8778020545599232650_a_nat @ R3 @ X3 @ Y )
= ( episte8778020545599232650_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_450_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_451_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: nat > a > $o,Ra: nat > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb2: a] :
( ( member_nat2 @ Z3 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte5492358210969815628_nat_a @ R3 @ X3 @ Y )
= ( episte5492358210969815628_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_452_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_453_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: a > nat > $o,Ra: a > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb2: nat] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte1460426709791529198_a_nat @ R3 @ X3 @ Y )
= ( episte1460426709791529198_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_454_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_455_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb2: a] :
( ( member_a2 @ Z3 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_456_fm_Orel__inject_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_457_fm_Orel__intros_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_458_fm_Orel__intros_I2_J,axiom,
! [X24: list_char,Y23: list_char,R3: a > a > $o] :
( ( X24 = Y23 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) ) ) ).
% fm.rel_intros(2)
thf(fact_459_fm_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X24: list_char,Y23: list_char] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.rel_inject(2)
thf(fact_460_fm_Orel__intros_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_461_fm_Orel__inject_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_462_can__select__def,axiom,
( can_select_nat
= ( ^ [P6: nat > $o,A3: set_nat] :
? [X: nat] :
( ( member_nat2 @ X @ A3 )
& ( P6 @ X )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_463_can__select__def,axiom,
( can_select_a
= ( ^ [P6: a > $o,A3: set_a] :
? [X: a] :
( ( member_a2 @ X @ A3 )
& ( P6 @ X )
& ! [Y3: a] :
( ( ( member_a2 @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_464_can__select__def,axiom,
( can_se5173380710277125655c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,A3: set_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
& ( P6 @ X )
& ! [Y3: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% can_select_def
thf(fact_465_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte3760347122651195639t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_466_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_467_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte4583239219080210381t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_468_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_469_fm_Orel__distinct_I30_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_470_fm_Orel__distinct_I29_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_471_fm_Orel__distinct_I8_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_472_fm_Orel__distinct_I7_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_473_fm_Orel__distinct_I10_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_474_fm_Orel__distinct_I9_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_475_fm_Orel__distinct_I26_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_476_fm_Orel__distinct_I25_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_477_fm_Orel__distinct_I28_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_478_fm_Orel__distinct_I27_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_479_fm_Orel__distinct_I15_J,axiom,
! [R3: a > a > $o,X24: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_480_fm_Orel__distinct_I16_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(16)
thf(fact_481_fm_Orel__distinct_I22_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_482_fm_Orel__distinct_I21_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_483_fm_Orel__distinct_I17_J,axiom,
! [R3: a > a > $o,X24: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_484_fm_Orel__distinct_I18_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(18)
thf(fact_485_fm_Orel__distinct_I24_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_486_fm_Orel__distinct_I23_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_487_fm_Orel__distinct_I6_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_488_fm_Orel__distinct_I5_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_489_fm_Orel__distinct_I1_J,axiom,
! [R3: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Pro_a @ Y23 ) ) ).
% fm.rel_distinct(1)
thf(fact_490_fm_Orel__distinct_I2_J,axiom,
! [R3: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ Y23 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_491_fm_Orel__distinct_I4_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_492_fm_Orel__distinct_I3_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_493_fm_Orel__distinct_I13_J,axiom,
! [R3: a > a > $o,X24: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_494_fm_Orel__distinct_I14_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(14)
thf(fact_495_fm_Orel__distinct_I20_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_496_fm_Orel__distinct_I19_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_497_fm_Orel__distinct_I12_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(12)
thf(fact_498_fm_Orel__distinct_I11_J,axiom,
! [R3: a > a > $o,X24: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_499_Cons__in__subseqsD,axiom,
! [Y: epistemic_fm_a,Ys: list_Epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Y @ Ys ) @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs2 ) ) )
=> ( member5906877432388582473c_fm_a @ Ys @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_500_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_501_generalization,axiom,
! [P: epistemic_fm_a,W: nat,M: episte8765170747386058258t_unit,I: a] :
( ! [M2: episte8765170747386058258t_unit,X2: nat] :
( ( member_nat2 @ X2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ X2 @ P ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_502_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I: a] :
( ! [M2: episte6182337868402532512t_unit,X2: a] :
( ( member_a2 @ X2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X2 @ P ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_503_generalization,axiom,
! [P: epistemic_fm_a,W: epistemic_fm_a,M: episte94448284482925344t_unit,I: a] :
( ! [M2: episte94448284482925344t_unit,X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_504_generalization,axiom,
! [P: epistemic_fm_a,W: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I: a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_505_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_506_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_507_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_508_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_509_in__mono,axiom,
! [A: set_a,B4: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( member_a2 @ X3 @ A )
=> ( member_a2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_510_in__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( member6642669571620171971c_fm_a @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_511_in__mono,axiom,
! [A: set_nat,B4: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ X3 @ A )
=> ( member_nat2 @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_512_in__mono,axiom,
! [A: set_set_nat,B4: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_513_subsetD,axiom,
! [A: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_514_subsetD,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( member6642669571620171971c_fm_a @ C @ A )
=> ( member6642669571620171971c_fm_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_515_subsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ C @ A )
=> ( member_nat2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_516_subsetD,axiom,
! [A: set_set_nat,B4: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_517_equalityE,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ~ ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ) ).
% equalityE
thf(fact_518_equalityE,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% equalityE
thf(fact_519_equalityE,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A = B4 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ~ ( ord_le6893508408891458716et_nat @ B4 @ A ) ) ) ).
% equalityE
thf(fact_520_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [X: a] :
( ( member_a2 @ X @ A3 )
=> ( member_a2 @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_521_subset__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
=> ( member6642669571620171971c_fm_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_522_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( member_nat2 @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_523_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( member_set_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_524_equalityD1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% equalityD1
thf(fact_525_equalityD1,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_526_equalityD1,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A = B4 )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_527_equalityD2,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ).
% equalityD2
thf(fact_528_equalityD2,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% equalityD2
thf(fact_529_equalityD2,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A = B4 )
=> ( ord_le6893508408891458716et_nat @ B4 @ A ) ) ).
% equalityD2
thf(fact_530_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [T: a] :
( ( member_a2 @ T @ A3 )
=> ( member_a2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_531_subset__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
! [T: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ T @ A3 )
=> ( member6642669571620171971c_fm_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_532_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat2 @ T @ A3 )
=> ( member_nat2 @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_533_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_534_subset__refl,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_535_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_536_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_537_Collect__mono,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_538_Collect__mono,axiom,
! [P4: list_nat > $o,Q4: list_nat > $o] :
( ! [X2: list_nat] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P4 ) @ ( collect_list_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_539_Collect__mono,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_540_Collect__mono,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P4 ) @ ( collect_set_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_541_Collect__mono,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ! [X2: nat] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_542_subset__trans,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_543_subset__trans,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_544_subset__trans,axiom,
! [A: set_set_nat,B4: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_545_set__eq__subset,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A3 @ B5 )
& ( ord_le3275665582123262618c_fm_a @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_546_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_547_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_548_Collect__mono__iff,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_549_Collect__mono__iff,axiom,
! [P4: list_nat > $o,Q4: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P4 ) @ ( collect_list_nat @ Q4 ) )
= ( ! [X: list_nat] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_550_Collect__mono__iff,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) )
= ( ! [X: epistemic_fm_a] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_551_Collect__mono__iff,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P4 ) @ ( collect_set_nat @ Q4 ) )
= ( ! [X: set_nat] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_552_Collect__mono__iff,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) )
= ( ! [X: nat] :
( ( P4 @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_553_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_554_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_555_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte8765170747386058258t_unit] :
( ( P4 @ M3 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ Xa2 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_556_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_557_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_558_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_559_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_560_strong__soundness_092_060_094sub_062T,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_561_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_562_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_563_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_564_main_092_060_094sub_062T,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_565_strong__completeness_092_060_094sub_062T,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_566_refl__Euclid__equiv,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) ) ) ) ).
% refl_Euclid_equiv
thf(fact_567_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_568_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte5734518988523130960t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_569_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_570_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_571_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_572_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte5734518988523130960t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_573_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_574_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_575_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_576_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_577_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_578_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_579_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_580_neg__introspection,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte5734518988523130960t_unit @ M )
=> ( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_581_neg__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_582_neg__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2438601301504999334t_unit @ M )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_583_neg__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_584_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_585_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_586_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_587_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_588_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_AxT_a @ P )
=> ( ( episte6990957894673201093t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_589_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_590_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_591_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_592_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5734518988523130960t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_593_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_594_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte2438601301504999334t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_595_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5478016696552465318t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_596_pos__introspection,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_597_pos__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_598_pos__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_599_pos__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_600_truth,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte6990957894673201093t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_601_truth,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_602_truth,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_603_truth,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_604_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_605_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_606_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_607_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_608_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_609_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_610_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_611_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_612_Collect__subset,axiom,
! [A: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_613_Collect__subset,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_614_Collect__subset,axiom,
! [A: set_list_nat,P4: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_615_Collect__subset,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_616_Collect__subset,axiom,
! [A: set_set_nat,P4: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_617_Collect__subset,axiom,
! [A: set_nat,P4: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ A )
& ( P4 @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_618_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ A3 )
@ ^ [X: a] : ( member_a2 @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_619_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_620_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat2 @ X @ A3 )
@ ^ [X: nat] : ( member_nat2 @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_621_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_622_arg__min__list_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > nat,X3: epistemic_fm_a,Y: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Zs ) ) )
= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs ) ) ) ) @ X3 @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_623_arg__min__list_Osimps_I2_J,axiom,
! [F: nat > nat,X3: nat,Y: nat,Zs: list_nat] :
( ( arg_min_list_nat_nat @ F @ ( cons_nat @ X3 @ ( cons_nat @ Y @ Zs ) ) )
= ( if_nat @ ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ ( arg_min_list_nat_nat @ F @ ( cons_nat @ Y @ Zs ) ) ) ) @ X3 @ ( arg_min_list_nat_nat @ F @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_624_S5__S5_H,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ P ) ) ).
% S5_S5'
thf(fact_625_S5_H__S5,axiom,
! [P: epistemic_fm_a] :
( ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ P )
=> ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ P ) ) ).
% S5'_S5
thf(fact_626_strong__soundness_092_060_094sub_062K,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_627_strong__completeness_092_060_094sub_062K,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_628_main_092_060_094sub_062K,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_629_S5__S5_H__assms,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_630_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_631_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_632_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G2: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G2 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_633_pred__subset__eq,axiom,
! [R3: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ R3 )
@ ^ [X: a] : ( member_a2 @ X @ S ) )
= ( ord_less_eq_set_a @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_634_pred__subset__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat2 @ X @ R3 )
@ ^ [X: nat] : ( member_nat2 @ X @ S ) )
= ( ord_less_eq_set_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_635_pred__subset__eq,axiom,
! [R3: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ R3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
= ( ord_le6893508408891458716et_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_636_pred__subset__eq,axiom,
! [R3: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ R3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ S ) )
= ( ord_le3275665582123262618c_fm_a @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_637_arg__min__list_Oelims,axiom,
! [X3: epistemic_fm_a > nat,Xa: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( ( arg_mi6265433823485604166_a_nat @ X3 @ Xa )
= Y )
=> ( ! [X2: epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( Y != X2 ) )
=> ( ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( X3 @ X2 ) @ ( X3 @ ( arg_mi6265433823485604166_a_nat @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) ) ) @ X2 @ ( arg_mi6265433823485604166_a_nat @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != undefi6158949259153642370c_fm_a ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_638_arg__min__list_Oelims,axiom,
! [X3: nat > nat,Xa: list_nat,Y: nat] :
( ( ( arg_min_list_nat_nat @ X3 @ Xa )
= Y )
=> ( ! [X2: nat] :
( ( Xa
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Y != X2 ) )
=> ( ! [X2: nat,Y2: nat,Zs2: list_nat] :
( ( Xa
= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if_nat @ ( ord_less_eq_nat @ ( X3 @ X2 ) @ ( X3 @ ( arg_min_list_nat_nat @ X3 @ ( cons_nat @ Y2 @ Zs2 ) ) ) ) @ X2 @ ( arg_min_list_nat_nat @ X3 @ ( cons_nat @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa = nil_nat )
=> ( Y != undefined_nat ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_639_subset__code_I3_J,axiom,
~ ( ord_le3275665582123262618c_fm_a @ ( coset_Epistemic_fm_a @ nil_Epistemic_fm_a ) @ ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% subset_code(3)
thf(fact_640_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_641_subset__code_I3_J,axiom,
~ ( ord_le6893508408891458716et_nat @ ( coset_set_nat @ nil_set_nat ) @ ( set_set_nat2 @ nil_set_nat ) ) ).
% subset_code(3)
thf(fact_642_semantics_Osimps_I2_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,X3: list_char] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Pro_a @ X3 ) )
= ( episte2398645135750866164t_unit @ M @ W @ X3 ) ) ).
% semantics.simps(2)
thf(fact_643_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ S2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs4 @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_644_consistent__hereditary,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S )
=> ( episte2285483198712856226tent_a @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_645_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_646_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( coset_Epistemic_fm_a @ Ys ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ~ ( member6642669571620171971c_fm_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_647_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_648_subset__code_I2_J,axiom,
! [A: set_set_nat,Ys: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( coset_set_nat @ Ys ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Ys ) )
=> ~ ( member_set_nat @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_649_AxB__symmetric_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ W3 ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_650_Ax5__Euclidean,axiom,
! [A: epistemic_fm_a > $o,U: set_Epistemic_fm_a,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ U )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ U )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ W3 )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W3 )
=> ( ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ U ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ U ) ) ) ) )
=> ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_651_Ax4__transitive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,W3: set_Epistemic_fm_a,I: a,U: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member536094252920883875c_fm_a @ W3
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) )
=> ( ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ W3 ) ) ) ) )
=> ( member536094252920883875c_fm_a @ U
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_652_AxT__reflexive,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,I: a] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( member536094252920883875c_fm_a @ V
@ ( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I @ P2 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_653_conj__subset__def,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ A
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ P4 ) )
& ( ord_le7112219575281605754c_fm_a @ A @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_654_conj__subset__def,axiom,
! [A: set_list_nat,P4: list_nat > $o,Q4: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ P4 ) )
& ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_655_conj__subset__def,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ P4 ) )
& ( ord_le3275665582123262618c_fm_a @ A @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_656_conj__subset__def,axiom,
! [A: set_set_nat,P4: set_nat > $o,Q4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P4 ) )
& ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_657_conj__subset__def,axiom,
! [A: set_nat,P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P4 @ X )
& ( Q4 @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P4 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_658_ax__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_659_consequent__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ Q ) @ V )
=> ( member6642669571620171971c_fm_a @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_660_deriv__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( epistemic_AK_a @ A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_661_maximal__extension,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V @ W4 )
=> ( ( episte2285483198712856226tent_a @ A @ W4 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) ) ) ).
% maximal_extension
thf(fact_662_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_663_subset__Collect__iff,axiom,
! [B4: set_a,A: set_a,P4: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: a] :
( ( member_a2 @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_664_subset__Collect__iff,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ( ord_le7112219575281605754c_fm_a @ B4
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_665_subset__Collect__iff,axiom,
! [B4: set_list_nat,A: set_list_nat,P4: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A )
=> ( ( ord_le6045566169113846134st_nat @ B4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_666_subset__Collect__iff,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ B4
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_667_subset__Collect__iff,axiom,
! [B4: set_set_nat,A: set_set_nat,P4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ( ord_le6893508408891458716et_nat @ B4
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_668_subset__Collect__iff,axiom,
! [B4: set_nat,A: set_nat,P4: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ A )
& ( P4 @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ B4 )
=> ( P4 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_669_subset__CollectI,axiom,
! [B4: set_a,A: set_a,Q4: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_670_subset__CollectI,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a,Q4: set_Epistemic_fm_a > $o,P4: set_Epistemic_fm_a > $o] :
( ( ord_le7112219575281605754c_fm_a @ B4 @ A )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_671_subset__CollectI,axiom,
! [B4: set_list_nat,A: set_list_nat,Q4: list_nat > $o,P4: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_672_subset__CollectI,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a,Q4: epistemic_fm_a > $o,P4: epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_673_subset__CollectI,axiom,
! [B4: set_set_nat,A: set_set_nat,Q4: set_nat > $o,P4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_674_subset__CollectI,axiom,
! [B4: set_nat,A: set_nat,Q4: nat > $o,P4: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B4 )
=> ( ( Q4 @ X2 )
=> ( P4 @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ B4 )
& ( Q4 @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ A )
& ( P4 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_675_Collect__restrict,axiom,
! [X5: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_676_Collect__restrict,axiom,
! [X5: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_677_Collect__restrict,axiom,
! [X5: set_list_nat,P4: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_678_Collect__restrict,axiom,
! [X5: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_679_Collect__restrict,axiom,
! [X5: set_set_nat,P4: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_680_Collect__restrict,axiom,
! [X5: set_nat,P4: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ X5 )
& ( P4 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_681_prop__restrict,axiom,
! [X3: a,Z4: set_a,X5: set_a,P4: a > $o] :
( ( member_a2 @ X3 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_682_prop__restrict,axiom,
! [X3: set_Epistemic_fm_a,Z4: set_se5208064806568342746c_fm_a,X5: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ X3 @ Z4 )
=> ( ( ord_le7112219575281605754c_fm_a @ Z4
@ ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_683_prop__restrict,axiom,
! [X3: list_nat,Z4: set_list_nat,X5: set_list_nat,P4: list_nat > $o] :
( ( member_list_nat @ X3 @ Z4 )
=> ( ( ord_le6045566169113846134st_nat @ Z4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_684_prop__restrict,axiom,
! [X3: epistemic_fm_a,Z4: set_Epistemic_fm_a,X5: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X3 @ Z4 )
=> ( ( ord_le3275665582123262618c_fm_a @ Z4
@ ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_685_prop__restrict,axiom,
! [X3: set_nat,Z4: set_set_nat,X5: set_set_nat,P4: set_nat > $o] :
( ( member_set_nat @ X3 @ Z4 )
=> ( ( ord_le6893508408891458716et_nat @ Z4
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_686_prop__restrict,axiom,
! [X3: nat,Z4: set_nat,X5: set_nat,P4: nat > $o] :
( ( member_nat2 @ X3 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ X5 )
& ( P4 @ X ) ) ) )
=> ( P4 @ X3 ) ) ) ).
% prop_restrict
thf(fact_687_subset__subseqs,axiom,
! [X5: set_Epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X5 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
=> ( member536094252920883875c_fm_a @ X5 @ ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_688_subset__subseqs,axiom,
! [X5: set_nat,Xs2: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_689_subset__subseqs,axiom,
! [X5: set_set_nat,Xs2: list_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ ( set_set_nat2 @ Xs2 ) )
=> ( member_set_set_nat @ X5 @ ( image_8726355809080528601et_nat @ set_set_nat2 @ ( set_list_set_nat2 @ ( subseqs_set_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_690_transpose__empty,axiom,
! [Xs2: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ Xs2 )
= nil_li2451196919128234278c_fm_a )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs2 ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_691_transpose__empty,axiom,
! [Xs2: list_list_nat] :
( ( ( transpose_nat @ Xs2 )
= nil_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( X = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_692_image__Collect__subsetI,axiom,
! [P4: nat > $o,F: nat > a,B4: set_a] :
( ! [X2: nat] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_693_image__Collect__subsetI,axiom,
! [P4: nat > $o,F: nat > nat,B4: set_nat] :
( ! [X2: nat] :
( ( P4 @ X2 )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_694_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > a,B4: set_a] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_695_image__Collect__subsetI,axiom,
! [P4: set_nat > $o,F: set_nat > a,B4: set_a] :
( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_set_nat_a @ F @ ( collect_set_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_696_image__Collect__subsetI,axiom,
! [P4: list_nat > $o,F: list_nat > a,B4: set_a] :
( ! [X2: list_nat] :
( ( P4 @ X2 )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_list_nat_a @ F @ ( collect_list_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_697_image__Collect__subsetI,axiom,
! [P4: nat > $o,F: nat > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: nat] :
( ( P4 @ X2 )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F @ ( collect_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_698_image__Collect__subsetI,axiom,
! [P4: epistemic_fm_a > $o,F: epistemic_fm_a > nat,B4: set_nat] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_3638449696541059749_a_nat @ F @ ( collec4904205152690461189c_fm_a @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_699_image__Collect__subsetI,axiom,
! [P4: set_nat > $o,F: set_nat > nat,B4: set_nat] :
( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ ( collect_set_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_700_image__Collect__subsetI,axiom,
! [P4: list_nat > $o,F: list_nat > nat,B4: set_nat] :
( ! [X2: list_nat] :
( ( P4 @ X2 )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_list_nat_nat @ F @ ( collect_list_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_701_image__Collect__subsetI,axiom,
! [P4: nat > $o,F: nat > set_nat,B4: set_set_nat] :
( ! [X2: nat] :
( ( P4 @ X2 )
=> ( member_set_nat @ ( F @ X2 ) @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P4 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_702_image__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ ( image_4449434806354059013c_fm_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_703_image__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( image_3638449696541059749_a_nat @ F @ A ) @ ( image_3638449696541059749_a_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_704_image__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_4248995238221578843et_nat @ F @ A ) @ ( image_4248995238221578843et_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_705_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > epistemic_fm_a] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F @ A ) @ ( image_3894954782759340931c_fm_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_706_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_707_image__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_708_image__mono,axiom,
! [A: set_set_nat,B4: set_set_nat,F: set_nat > epistemic_fm_a] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3663150724827809357c_fm_a @ F @ A ) @ ( image_3663150724827809357c_fm_a @ F @ B4 ) ) ) ).
% image_mono
thf(fact_709_image__mono,axiom,
! [A: set_set_nat,B4: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ ( image_set_nat_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_710_image__mono,axiom,
! [A: set_set_nat,B4: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ ( image_7916887816326733075et_nat @ F @ B4 ) ) ) ).
% image_mono
thf(fact_711_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B4: set_a] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_712_image__subsetI,axiom,
! [A: set_a,F: a > a,B4: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_713_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_714_image__subsetI,axiom,
! [A: set_a,F: a > nat,B4: set_nat] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_715_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > a,B4: set_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_a @ ( image_7228943142519265129fm_a_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_716_image__subsetI,axiom,
! [A: set_nat,F: nat > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_717_image__subsetI,axiom,
! [A: set_a,F: a > epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member6642669571620171971c_fm_a @ ( F @ X2 ) @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( image_1898462159489238305c_fm_a @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_718_image__subsetI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > nat,B4: set_nat] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_3638449696541059749_a_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_719_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B4: set_set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_720_image__subsetI,axiom,
! [A: set_a,F: a > set_nat,B4: set_set_nat] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_721_subset__imageE,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B4
!= ( image_4449434806354059013c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_722_subset__imageE,axiom,
! [B4: set_Epistemic_fm_a,F: nat > epistemic_fm_a,A: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_3894954782759340931c_fm_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B4
!= ( image_3894954782759340931c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_723_subset__imageE,axiom,
! [B4: set_Epistemic_fm_a,F: set_nat > epistemic_fm_a,A: set_set_nat] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_3663150724827809357c_fm_a @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B4
!= ( image_3663150724827809357c_fm_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_724_subset__imageE,axiom,
! [B4: set_nat,F: epistemic_fm_a > nat,A: set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ B4 @ ( image_3638449696541059749_a_nat @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B4
!= ( image_3638449696541059749_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_725_subset__imageE,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B4
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_726_subset__imageE,axiom,
! [B4: set_nat,F: set_nat > nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_set_nat_nat @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B4
!= ( image_set_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_727_subset__imageE,axiom,
! [B4: set_set_nat,F: epistemic_fm_a > set_nat,A: set_Epistemic_fm_a] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_4248995238221578843et_nat @ F @ A ) )
=> ~ ! [C3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C3 @ A )
=> ( B4
!= ( image_4248995238221578843et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_728_subset__imageE,axiom,
! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B4
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_729_subset__imageE,axiom,
! [B4: set_set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B4
!= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_730_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B4 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ ( F @ X ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_731_subset__image__iff,axiom,
! [B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B4
= ( image_4449434806354059013c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_732_subset__image__iff,axiom,
! [B4: set_Epistemic_fm_a,F: nat > epistemic_fm_a,A: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_3894954782759340931c_fm_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_3894954782759340931c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_733_subset__image__iff,axiom,
! [B4: set_Epistemic_fm_a,F: set_nat > epistemic_fm_a,A: set_set_nat] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ ( image_3663150724827809357c_fm_a @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B4
= ( image_3663150724827809357c_fm_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_734_subset__image__iff,axiom,
! [B4: set_nat,F: epistemic_fm_a > nat,A: set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ B4 @ ( image_3638449696541059749_a_nat @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B4
= ( image_3638449696541059749_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_735_subset__image__iff,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_736_subset__image__iff,axiom,
! [B4: set_nat,F: set_nat > nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_set_nat_nat @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B4
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_737_subset__image__iff,axiom,
! [B4: set_set_nat,F: epistemic_fm_a > set_nat,A: set_Epistemic_fm_a] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_4248995238221578843et_nat @ F @ A ) )
= ( ? [AA: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ AA @ A )
& ( B4
= ( image_4248995238221578843et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_738_subset__image__iff,axiom,
! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B4
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_739_subset__image__iff,axiom,
! [B4: set_set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_7916887816326733075et_nat @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B4
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_740_transpose_Osimps_I2_J,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
= ( transp2882070860200649898c_fm_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_741_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_742_all__subset__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ ( image_4449434806354059013c_fm_a @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ A )
=> ( P4 @ ( image_4449434806354059013c_fm_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_743_all__subset__image,axiom,
! [F: nat > epistemic_fm_a,A: set_nat,P4: set_Epistemic_fm_a > $o] :
( ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ ( image_3894954782759340931c_fm_a @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P4 @ ( image_3894954782759340931c_fm_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_744_all__subset__image,axiom,
! [F: set_nat > epistemic_fm_a,A: set_set_nat,P4: set_Epistemic_fm_a > $o] :
( ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ ( image_3663150724827809357c_fm_a @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A )
=> ( P4 @ ( image_3663150724827809357c_fm_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_745_all__subset__image,axiom,
! [F: epistemic_fm_a > nat,A: set_Epistemic_fm_a,P4: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_3638449696541059749_a_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ A )
=> ( P4 @ ( image_3638449696541059749_a_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_746_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P4: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P4 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_747_all__subset__image,axiom,
! [F: set_nat > nat,A: set_set_nat,P4: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_set_nat_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A )
=> ( P4 @ ( image_set_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_748_all__subset__image,axiom,
! [F: epistemic_fm_a > set_nat,A: set_Epistemic_fm_a,P4: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_4248995238221578843et_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B5 @ A )
=> ( P4 @ ( image_4248995238221578843et_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_749_all__subset__image,axiom,
! [F: nat > set_nat,A: set_nat,P4: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_nat_set_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A )
=> ( P4 @ ( image_nat_set_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_750_all__subset__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P4: set_set_nat > $o] :
( ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ A )
=> ( P4 @ ( image_7916887816326733075et_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_751_subseqs__powset,axiom,
! [Xs2: list_Epistemic_fm_a] :
( ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs2 ) ) )
= ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_752_subseqs__powset,axiom,
! [Xs2: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
= ( pow_nat @ ( set_nat2 @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_753_f__arg__min__list__f,axiom,
! [Xs2: list_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( Xs2 != nil_Epistemic_fm_a )
=> ( ( F @ ( arg_mi6265433823485604166_a_nat @ F @ Xs2 ) )
= ( lattic8721135487736765967in_nat @ ( image_3638449696541059749_a_nat @ F @ ( set_Epistemic_fm_a2 @ Xs2 ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_754_f__arg__min__list__f,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( F @ ( arg_min_list_nat_nat @ F @ Xs2 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_755_list__ex1__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( list_e2031426293596896995c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) )
= ( ( ( P4 @ X3 )
=> ( list_a5841931967666341838c_fm_a
@ ^ [Y3: epistemic_fm_a] :
( ~ ( P4 @ Y3 )
| ( X3 = Y3 ) )
@ Xs2 ) )
& ( ~ ( P4 @ X3 )
=> ( list_e2031426293596896995c_fm_a @ P4 @ Xs2 ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_756_list__ex1__simps_I2_J,axiom,
! [P4: nat > $o,X3: nat,Xs2: list_nat] :
( ( list_ex1_nat @ P4 @ ( cons_nat @ X3 @ Xs2 ) )
= ( ( ( P4 @ X3 )
=> ( list_all_nat
@ ^ [Y3: nat] :
( ~ ( P4 @ Y3 )
| ( X3 = Y3 ) )
@ Xs2 ) )
& ( ~ ( P4 @ X3 )
=> ( list_ex1_nat @ P4 @ Xs2 ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_757_Shift__def,axiom,
( bNF_Gr8437504134799245625c_fm_a
= ( ^ [Kl: set_li769143395467472256c_fm_a,K: epistemic_fm_a] :
( collec5191077796991884427c_fm_a
@ ^ [Kl2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_758_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl: set_list_nat,K: nat] :
( collect_list_nat
@ ^ [Kl2: list_nat] : ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_759_PowI,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B4 ) ) ) ).
% PowI
thf(fact_760_PowI,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( member_set_nat @ A @ ( pow_nat @ B4 ) ) ) ).
% PowI
thf(fact_761_PowI,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( member_set_set_nat @ A @ ( pow_set_nat @ B4 ) ) ) ).
% PowI
thf(fact_762_Pow__iff,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B4 ) )
= ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% Pow_iff
thf(fact_763_Pow__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% Pow_iff
thf(fact_764_Pow__iff,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B4 ) )
= ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% Pow_iff
thf(fact_765_list__all__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) )
= ( ( P4 @ X3 )
& ( list_a5841931967666341838c_fm_a @ P4 @ Xs2 ) ) ) ).
% list_all_simps(1)
thf(fact_766_list__all__simps_I1_J,axiom,
! [P4: nat > $o,X3: nat,Xs2: list_nat] :
( ( list_all_nat @ P4 @ ( cons_nat @ X3 @ Xs2 ) )
= ( ( P4 @ X3 )
& ( list_all_nat @ P4 @ Xs2 ) ) ) ).
% list_all_simps(1)
thf(fact_767_list_Opred__inject_I2_J,axiom,
! [P4: epistemic_fm_a > $o,A2: epistemic_fm_a,Aa: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ A2 @ Aa ) )
= ( ( P4 @ A2 )
& ( list_a5841931967666341838c_fm_a @ P4 @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_768_list_Opred__inject_I2_J,axiom,
! [P4: nat > $o,A2: nat,Aa: list_nat] :
( ( list_all_nat @ P4 @ ( cons_nat @ A2 @ Aa ) )
= ( ( P4 @ A2 )
& ( list_all_nat @ P4 @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_769_list__all__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o] : ( list_a5841931967666341838c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list_all_simps(2)
thf(fact_770_list__all__simps_I2_J,axiom,
! [P4: nat > $o] : ( list_all_nat @ P4 @ nil_nat ) ).
% list_all_simps(2)
thf(fact_771_Pow__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le7112219575281605754c_fm_a @ ( pow_Epistemic_fm_a @ A ) @ ( pow_Epistemic_fm_a @ B4 ) ) ) ).
% Pow_mono
thf(fact_772_Pow__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% Pow_mono
thf(fact_773_Pow__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_le9131159989063066194et_nat @ ( pow_set_nat @ A ) @ ( pow_set_nat @ B4 ) ) ) ).
% Pow_mono
thf(fact_774_PowD,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A @ ( pow_Epistemic_fm_a @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% PowD
thf(fact_775_PowD,axiom,
! [A: set_nat,B4: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% PowD
thf(fact_776_PowD,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% PowD
thf(fact_777_list_Opred__inject_I1_J,axiom,
! [P4: epistemic_fm_a > $o] : ( list_a5841931967666341838c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% list.pred_inject(1)
thf(fact_778_list_Opred__inject_I1_J,axiom,
! [P4: nat > $o] : ( list_all_nat @ P4 @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_779_list__all__cong,axiom,
! [X3: list_a,Ya: list_a,P4: a > $o,Pa: a > $o] :
( ( X3 = Ya )
=> ( ! [Z3: a] :
( ( member_a2 @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( list_all_a @ P4 @ X3 )
= ( list_all_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_780_list__all__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
= ( list_a5841931967666341838c_fm_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_781_list__all__cong,axiom,
! [X3: list_nat,Ya: list_nat,P4: nat > $o,Pa: nat > $o] :
( ( X3 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( P4 @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( list_all_nat @ P4 @ X3 )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_782_list_Opred__mono__strong,axiom,
! [P4: a > $o,X3: list_a,Pa: a > $o] :
( ( list_all_a @ P4 @ X3 )
=> ( ! [Z3: a] :
( ( member_a2 @ Z3 @ ( set_a2 @ X3 ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( list_all_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_783_list_Opred__mono__strong,axiom,
! [P4: epistemic_fm_a > $o,X3: list_Epistemic_fm_a,Pa: epistemic_fm_a > $o] :
( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
=> ( ! [Z3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z3 @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( list_a5841931967666341838c_fm_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_784_list_Opred__mono__strong,axiom,
! [P4: nat > $o,X3: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P4 @ X3 )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ ( set_nat2 @ X3 ) )
=> ( ( P4 @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( list_all_nat @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_785_list_Opred__mono,axiom,
! [P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Pa )
=> ( ord_le5289359451089373629fm_a_o @ ( list_a5841931967666341838c_fm_a @ P4 ) @ ( list_a5841931967666341838c_fm_a @ Pa ) ) ) ).
% list.pred_mono
thf(fact_786_Pow__def,axiom,
( pow_Epistemic_fm_a
= ( ^ [A3: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [B5: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B5 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_787_Pow__def,axiom,
( pow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_788_Pow__def,axiom,
( pow_set_nat
= ( ^ [A3: set_set_nat] :
( collect_set_set_nat
@ ^ [B5: set_set_nat] : ( ord_le6893508408891458716et_nat @ B5 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_789_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_790_ShiftD,axiom,
! [Kl3: list_Epistemic_fm_a,Kl4: set_li769143395467472256c_fm_a,K2: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Kl3 @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K2 ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ K2 @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_791_ShiftD,axiom,
! [Kl3: list_nat,Kl4: set_list_nat,K2: nat] :
( ( member_list_nat @ Kl3 @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K2 ) )
=> ( member_list_nat @ ( cons_nat @ K2 @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_792_min__list__Min,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( min_list_nat @ Xs2 )
= ( lattic8721135487736765967in_nat @ ( set_nat2 @ Xs2 ) ) ) ) ).
% min_list_Min
thf(fact_793_empty__Shift,axiom,
! [Kl4: set_list_a,K2: a] :
( ( member_list_a @ nil_a @ Kl4 )
=> ( ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_794_empty__Shift,axiom,
! [Kl4: set_li769143395467472256c_fm_a,K2: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ Kl4 )
=> ( ( member6642669571620171971c_fm_a @ K2 @ ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ nil_Epistemic_fm_a ) )
=> ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_795_empty__Shift,axiom,
! [Kl4: set_list_nat,K2: nat] :
( ( member_list_nat @ nil_nat @ Kl4 )
=> ( ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_796_Succ__Shift,axiom,
! [Kl4: set_li769143395467472256c_fm_a,K2: epistemic_fm_a,Kl3: list_Epistemic_fm_a] :
( ( bNF_Gr1093487135560555701c_fm_a @ ( bNF_Gr8437504134799245625c_fm_a @ Kl4 @ K2 ) @ Kl3 )
= ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ ( cons_Epistemic_fm_a @ K2 @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_797_Succ__Shift,axiom,
! [Kl4: set_list_nat,K2: nat,Kl3: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K2 ) @ Kl3 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ ( cons_nat @ K2 @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_798_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G2: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G2 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) )
=> ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G2 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_799_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ Y3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y3 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_800_frame_Oext__inject,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W7: set_se5208064806568342746c_fm_a,K4: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W6 @ K3 @ More )
= ( episte2888590659910966568t_unit @ W7 @ K4 @ More2 ) )
= ( ( W6 = W7 )
& ( K3 = K4 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_801_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_a > list_char > $o,More2: product_unit] :
( ( ( episte8239586592105053771t_unit @ Pi @ More )
= ( episte8239586592105053771t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_802_kripke_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_803_frame_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= K3 ) ).
% frame.select_convs(2)
thf(fact_804_kripke_Ocases,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ ( episte8239586592105053771t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_805_kripke_Oext__induct,axiom,
! [P4: episte1193835314949844379t_unit > $o,R: episte1193835314949844379t_unit] :
( ! [Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] : ( P4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R ) ) ).
% kripke.ext_induct
thf(fact_806_frame_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_807_kripke_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_808_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_a > list_char > $o,W: set_Epistemic_fm_a,W3: set_se5208064806568342746c_fm_a,R: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ ( Pi4 @ W ) @ ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_809_frame_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= W6 ) ).
% frame.select_convs(1)
thf(fact_810_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) )
@ V
@ P ) ) ) ) ).
% truth_lemma
thf(fact_811_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_812_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_813_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_814_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_815_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( ( member536094252920883875c_fm_a @ Z5 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y3 ) )
& ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Z5 ) ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y3 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_816_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_817_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y3 ) )
= ( member536094252920883875c_fm_a @ Y3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_818_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_819_kripke_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte8239586592105053771t_unit @ ( episte2398645135750866164t_unit @ R ) @ ( episte5479201149095757850t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_820_frame_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte3309513806868946049t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_821_kripke_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R4: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R4 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R4 ) )
=> ( ( ( episte2398645135750866164t_unit @ R )
= ( episte2398645135750866164t_unit @ R4 ) )
=> ( ( ( episte5479201149095757850t_unit @ R )
= ( episte5479201149095757850t_unit @ R4 ) )
=> ( R = R4 ) ) ) ) ) ).
% kripke.equality
thf(fact_822_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte9120385895580347753c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_823_subset__empty,axiom,
! [A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a )
= ( A = bot_bo3626323581529592678c_fm_a ) ) ).
% subset_empty
thf(fact_824_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_825_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_826_empty__subsetI,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A ) ).
% empty_subsetI
thf(fact_827_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_828_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_829_set__empty,axiom,
! [Xs2: list_Epistemic_fm_a] :
( ( ( set_Epistemic_fm_a2 @ Xs2 )
= bot_bo3626323581529592678c_fm_a )
= ( Xs2 = nil_Epistemic_fm_a ) ) ).
% set_empty
thf(fact_830_set__empty,axiom,
! [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= bot_bot_set_nat )
= ( Xs2 = nil_nat ) ) ).
% set_empty
thf(fact_831_set__empty2,axiom,
! [Xs2: list_Epistemic_fm_a] :
( ( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ Xs2 ) )
= ( Xs2 = nil_Epistemic_fm_a ) ) ).
% set_empty2
thf(fact_832_set__empty2,axiom,
! [Xs2: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% set_empty2
thf(fact_833_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ bot_bo6433428028861825271fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_834_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_835_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_836_bot_Oextremum,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% bot.extremum
thf(fact_837_bot_Oextremum,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ bot_bot_o_nat @ A2 ) ).
% bot.extremum
thf(fact_838_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_839_bot_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
= ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_unique
thf(fact_840_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
= ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_841_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_842_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_843_bot_Oextremum__unique,axiom,
! [A2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ bot_bot_o_nat )
= ( A2 = bot_bot_o_nat ) ) ).
% bot.extremum_unique
thf(fact_844_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_845_bot_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
=> ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_846_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
=> ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_847_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_848_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_849_bot_Oextremum__uniqueI,axiom,
! [A2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ bot_bot_o_nat )
=> ( A2 = bot_bot_o_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_850_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_851_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_852_subset__emptyI,axiom,
! [A: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ A @ bot_bo3626323581529592678c_fm_a ) ) ).
% subset_emptyI
thf(fact_853_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_854_subset__emptyI,axiom,
! [A: set_set_nat] :
( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_855_empty__set,axiom,
( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% empty_set
thf(fact_856_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_857_fm_Osimps_I90_J,axiom,
( ( episte9089240958480457552c_fm_a @ episte5073044243917183961c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(90)
thf(fact_858_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_nat @ epistemic_FF_nat )
= bot_bot_set_nat ) ).
% fm.simps(90)
thf(fact_859_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_860_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( episte9089240958480457552c_fm_a @ ( episte3759128466173231372c_fm_a @ X24 ) )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(91)
thf(fact_861_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( epistemic_set_fm_nat @ ( epistemic_Pro_nat @ X24 ) )
= bot_bot_set_nat ) ).
% fm.simps(91)
thf(fact_862_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( epistemic_set_fm_a @ ( epistemic_Pro_a @ X24 ) )
= bot_bot_set_a ) ).
% fm.simps(91)
thf(fact_863_frame_Oselect__convs_I3_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_864_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_865_frame_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R4: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R4 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R4 ) )
=> ( ( ( episte3309513806868946049t_unit @ R )
= ( episte3309513806868946049t_unit @ R4 ) )
=> ( R = R4 ) ) ) ) ).
% frame.equality
thf(fact_866_kripke_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_867_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte1857908288096881731t_unit @ Pi2 @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_868_image__Fpow__mono,axiom,
! [F: nat > epistemic_fm_a,A: set_nat,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F @ A ) @ B4 )
=> ( ord_le7112219575281605754c_fm_a @ ( image_1008147270575274029c_fm_a @ ( image_3894954782759340931c_fm_a @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite7994277033128312448c_fm_a @ B4 ) ) ) ).
% image_Fpow_mono
thf(fact_869_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B4 ) ) ) ).
% image_Fpow_mono
thf(fact_870_image__Fpow__mono,axiom,
! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_set_nat @ B4 ) ) ) ).
% image_Fpow_mono
thf(fact_871_inconsistent__subset,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [V3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V3 ) @ V )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V3 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_872_Pow__set_I1_J,axiom,
( ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) )
= ( insert7129737658439086282c_fm_a @ bot_bo3626323581529592678c_fm_a @ bot_bo1868452413526482246c_fm_a ) ) ).
% Pow_set(1)
thf(fact_873_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_874_canonical__model_I1_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,V2: set_Epistemic_fm_a] :
( collec2519470961442302949c_fm_a
@ ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [P2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ ( epistemic_K_a @ I2 @ P2 ) @ V2 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V2: set_Epistemic_fm_a,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V2 )
@ product_Unity ) )
@ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S )
@ P ) ) ) ).
% canonical_model(1)
thf(fact_875_insert__subset,axiom,
! [X3: a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X3 @ A ) @ B4 )
= ( ( member_a2 @ X3 @ B4 )
& ( ord_less_eq_set_a @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_876_insert__subset,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ A ) @ B4 )
= ( ( member6642669571620171971c_fm_a @ X3 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_877_insert__subset,axiom,
! [X3: nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X3 @ A ) @ B4 )
= ( ( member_nat2 @ X3 @ B4 )
& ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_878_insert__subset,axiom,
! [X3: set_nat,A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A ) @ B4 )
= ( ( member_set_nat @ X3 @ B4 )
& ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_879_Un__subset__iff,axiom,
! [A: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_880_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,C2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ C2 )
= ( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
& ( ord_le3275665582123262618c_fm_a @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_881_Un__subset__iff,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_882_Un__subset__iff,axiom,
! [A: set_set_nat,B4: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_883_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ A2 @ A )
= ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_884_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_885_singleton__insert__inj__eq_H,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A2 = B )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_886_singleton__insert__inj__eq,axiom,
! [B: epistemic_fm_a,A2: epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a )
= ( insert7817948963269374698c_fm_a @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_887_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_888_singleton__insert__inj__eq,axiom,
! [B: set_nat,A2: set_nat,A: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_889_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( insert7817948963269374698c_fm_a @ X21 @ ( set_Epistemic_fm_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_890_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_891_fm_Osimps_I95_J,axiom,
! [X61: epistemic_fm_a,X62: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte5657488632024175118c_fm_a @ X61 @ X62 ) )
= ( insert7817948963269374698c_fm_a @ X61 @ ( episte9089240958480457552c_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_892_fm_Osimps_I95_J,axiom,
! [X61: nat,X62: epistemic_fm_nat] :
( ( epistemic_set_fm_nat @ ( epistemic_K_nat @ X61 @ X62 ) )
= ( insert_nat2 @ X61 @ ( epistemic_set_fm_nat @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_893_fm_Osimps_I95_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_K_a @ X61 @ X62 ) )
= ( insert_a2 @ X61 @ ( epistemic_set_fm_a @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_894_List_Oset__insert,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( insert177310161492556854c_fm_a @ X3 @ Xs2 ) )
= ( insert7817948963269374698c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_895_List_Oset__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ( set_nat2 @ ( insert_nat @ X3 @ Xs2 ) )
= ( insert_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_896_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ( sup_sup_set_a @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_897_subset__Un__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
( ( sup_su1367922730591523534c_fm_a @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_898_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_899_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_900_subset__UnE,axiom,
! [C2: set_a,A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B4 ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B4 )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_901_subset__UnE,axiom,
! [C2: set_Epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) )
=> ~ ! [A6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A6 @ A )
=> ! [B6: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B6 @ B4 )
=> ( C2
!= ( sup_su1367922730591523534c_fm_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_902_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B4 ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B4 )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_903_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B4 ) )
=> ~ ! [A6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A6 @ A )
=> ! [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B6 @ B4 )
=> ( C2
!= ( sup_sup_set_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_904_Un__absorb2,axiom,
! [B4: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ( sup_sup_set_a @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_905_Un__absorb2,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B4 @ A )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_906_Un__absorb2,axiom,
! [B4: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( sup_sup_set_nat @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_907_Un__absorb2,axiom,
! [B4: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ( sup_sup_set_set_nat @ A @ B4 )
= A ) ) ).
% Un_absorb2
thf(fact_908_Un__absorb1,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( sup_sup_set_a @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_909_Un__absorb1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( sup_su1367922730591523534c_fm_a @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_910_Un__absorb1,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( sup_sup_set_nat @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_911_Un__absorb1,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( sup_sup_set_set_nat @ A @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_912_Un__upper2,axiom,
! [B4: set_a,A: set_a] : ( ord_less_eq_set_a @ B4 @ ( sup_sup_set_a @ A @ B4 ) ) ).
% Un_upper2
thf(fact_913_Un__upper2,axiom,
! [B4: set_Epistemic_fm_a,A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ).
% Un_upper2
thf(fact_914_Un__upper2,axiom,
! [B4: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_upper2
thf(fact_915_Un__upper2,axiom,
! [B4: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B4 @ ( sup_sup_set_set_nat @ A @ B4 ) ) ).
% Un_upper2
thf(fact_916_Un__upper1,axiom,
! [A: set_a,B4: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B4 ) ) ).
% Un_upper1
thf(fact_917_Un__upper1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ).
% Un_upper1
thf(fact_918_Un__upper1,axiom,
! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_upper1
thf(fact_919_Un__upper1,axiom,
! [A: set_set_nat,B4: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B4 ) ) ).
% Un_upper1
thf(fact_920_Un__least,axiom,
! [A: set_a,C2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_921_Un__least,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ C2 )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_922_Un__least,axiom,
! [A: set_nat,C2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_923_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_924_Un__mono,axiom,
! [A: set_a,C2: set_a,B4: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B4 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B4 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_925_Un__mono,axiom,
! [A: set_Epistemic_fm_a,C2: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,D: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ C2 )
=> ( ( ord_le3275665582123262618c_fm_a @ B4 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) @ ( sup_su1367922730591523534c_fm_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_926_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B4: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_927_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B4: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B4 @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_928_subset__insertI2,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,B: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_929_subset__insertI2,axiom,
! [A: set_nat,B4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_930_subset__insertI2,axiom,
! [A: set_set_nat,B4: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_931_subset__insertI,axiom,
! [B4: set_Epistemic_fm_a,A2: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ B4 @ ( insert7817948963269374698c_fm_a @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_932_subset__insertI,axiom,
! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat2 @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_933_subset__insertI,axiom,
! [B4: set_set_nat,A2: set_nat] : ( ord_le6893508408891458716et_nat @ B4 @ ( insert_set_nat @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_934_subset__insert,axiom,
! [X3: a,A: set_a,B4: set_a] :
( ~ ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X3 @ B4 ) )
= ( ord_less_eq_set_a @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_935_subset__insert,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X3 @ B4 ) )
= ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_936_subset__insert,axiom,
! [X3: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat2 @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X3 @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_937_subset__insert,axiom,
! [X3: set_nat,A: set_set_nat,B4: set_set_nat] :
( ~ ( member_set_nat @ X3 @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X3 @ B4 ) )
= ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_938_insert__mono,axiom,
! [C2: set_Epistemic_fm_a,D: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C2 @ D )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ A2 @ C2 ) @ ( insert7817948963269374698c_fm_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_939_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_940_insert__mono,axiom,
! [C2: set_set_nat,D: set_set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A2 @ C2 ) @ ( insert_set_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_941_insert__subsetI,axiom,
! [X3: a,A: set_a,X5: set_a] :
( ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_942_insert__subsetI,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,X5: set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ X5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_943_insert__subsetI,axiom,
! [X3: nat,A: set_nat,X5: set_nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_944_insert__subsetI,axiom,
! [X3: set_nat,A: set_set_nat,X5: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( ord_le6893508408891458716et_nat @ X5 @ A )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_945_canonical__model_I3_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ) ).
% canonical_model(3)
thf(fact_946_maximal__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% maximal_Extend
thf(fact_947_consistent__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ).
% consistent_Extend
thf(fact_948_canonical__model_I2_J,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( member6642669571620171971c_fm_a @ P @ S )
=> ( episte2285483198712856226tent_a @ A @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ) ) ).
% canonical_model(2)
thf(fact_949_Extend__subset,axiom,
! [S: set_Epistemic_fm_a,A: epistemic_fm_a > $o] : ( ord_le3275665582123262618c_fm_a @ S @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% Extend_subset
thf(fact_950_top_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
=> ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_uniqueI
thf(fact_951_top_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
=> ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_uniqueI
thf(fact_952_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_953_top_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
=> ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_954_top_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ top_to6710954824118802707fm_a_o @ A2 )
= ( A2 = top_to6710954824118802707fm_a_o ) ) ).
% top.extremum_unique
thf(fact_955_top_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ top_to7796028867103199306c_fm_a @ A2 )
= ( A2 = top_to7796028867103199306c_fm_a ) ) ).
% top.extremum_unique
thf(fact_956_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_957_top_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ top_top_set_set_nat @ A2 )
= ( A2 = top_top_set_set_nat ) ) ).
% top.extremum_unique
thf(fact_958_top__greatest,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ top_to6710954824118802707fm_a_o ) ).
% top_greatest
thf(fact_959_top__greatest,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ top_to7796028867103199306c_fm_a ) ).
% top_greatest
thf(fact_960_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_961_top__greatest,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ top_top_set_set_nat ) ).
% top_greatest
thf(fact_962_Un__Pow__subset,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] : ( ord_le7112219575281605754c_fm_a @ ( sup_su4305657963326700718c_fm_a @ ( pow_Epistemic_fm_a @ A ) @ ( pow_Epistemic_fm_a @ B4 ) ) @ ( pow_Epistemic_fm_a @ ( sup_su1367922730591523534c_fm_a @ A @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_963_Un__Pow__subset,axiom,
! [A: set_a,B4: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B4 ) ) @ ( pow_a @ ( sup_sup_set_a @ A @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_964_Un__Pow__subset,axiom,
! [A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_965_subset__UNIV,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ top_to7796028867103199306c_fm_a ) ).
% subset_UNIV
thf(fact_966_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_967_subset__UNIV,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ top_top_set_set_nat ) ).
% subset_UNIV
thf(fact_968_consistent__disjuncts,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ ( epistemic_Dis_a @ P @ Q ) @ V )
=> ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P @ bot_bo3626323581529592678c_fm_a ) @ V ) )
| ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_969_maximal__def,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] :
( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S )
= ( ! [P2: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ P2 @ bot_bo3626323581529592678c_fm_a ) @ S ) )
=> ( member6642669571620171971c_fm_a @ P2 @ S ) ) ) ) ).
% maximal_def
thf(fact_970_subset__singleton__iff,axiom,
! [X5: set_Epistemic_fm_a,A2: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X5 @ ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) )
= ( ( X5 = bot_bo3626323581529592678c_fm_a )
| ( X5
= ( insert7817948963269374698c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_971_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_972_subset__singleton__iff,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_set_nat )
| ( X5
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_973_subset__singletonD,axiom,
! [A: set_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) )
=> ( ( A = bot_bo3626323581529592678c_fm_a )
| ( A
= ( insert7817948963269374698c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a ) ) ) ) ).
% subset_singletonD
thf(fact_974_subset__singletonD,axiom,
! [A: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_975_subset__singletonD,axiom,
! [A: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
=> ( ( A = bot_bot_set_set_nat )
| ( A
= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_976_consistent__consequent,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_977_range__subsetD,axiom,
! [F: nat > a,B4: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B4 )
=> ( member_a2 @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_978_range__subsetD,axiom,
! [F: set_nat > a,B4: set_a,I: set_nat] :
( ( ord_less_eq_set_a @ ( image_set_nat_a @ F @ top_top_set_set_nat ) @ B4 )
=> ( member_a2 @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_979_range__subsetD,axiom,
! [F: nat > epistemic_fm_a,B4: set_Epistemic_fm_a,I: nat] :
( ( ord_le3275665582123262618c_fm_a @ ( image_3894954782759340931c_fm_a @ F @ top_top_set_nat ) @ B4 )
=> ( member6642669571620171971c_fm_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_980_range__subsetD,axiom,
! [F: set_nat > epistemic_fm_a,B4: set_Epistemic_fm_a,I: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ ( image_3663150724827809357c_fm_a @ F @ top_top_set_set_nat ) @ B4 )
=> ( member6642669571620171971c_fm_a @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_981_range__subsetD,axiom,
! [F: nat > nat,B4: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B4 )
=> ( member_nat2 @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_982_range__subsetD,axiom,
! [F: set_nat > nat,B4: set_nat,I: set_nat] :
( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ top_top_set_set_nat ) @ B4 )
=> ( member_nat2 @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_983_range__subsetD,axiom,
! [F: nat > set_nat,B4: set_set_nat,I: nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ top_top_set_nat ) @ B4 )
=> ( member_set_nat @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_984_range__subsetD,axiom,
! [F: set_nat > set_nat,B4: set_set_nat,I: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ top_top_set_set_nat ) @ B4 )
=> ( member_set_nat @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_985_consistent__consequent_H,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
=> ( ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ ( epistemic_Imp_a @ P @ Q ) )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ Q @ bot_bo3626323581529592678c_fm_a ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_986_fm_Osimps_I94_J,axiom,
! [X51: episte740340785640729014c_fm_a,X52: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte260752218777527565c_fm_a @ X51 @ X52 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X51 ) @ ( episte9089240958480457552c_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_987_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_nat,X52: epistemic_fm_nat] :
( ( epistemic_set_fm_nat @ ( epistemic_Imp_nat @ X51 @ X52 ) )
= ( sup_sup_set_nat @ ( epistemic_set_fm_nat @ X51 ) @ ( epistemic_set_fm_nat @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_988_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_a,X52: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Imp_a @ X51 @ X52 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X51 ) @ ( epistemic_set_fm_a @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_989_fm_Osimps_I93_J,axiom,
! [X41: episte740340785640729014c_fm_a,X42: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte3685526487207141399c_fm_a @ X41 @ X42 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X41 ) @ ( episte9089240958480457552c_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_990_fm_Osimps_I93_J,axiom,
! [X41: epistemic_fm_nat,X42: epistemic_fm_nat] :
( ( epistemic_set_fm_nat @ ( epistemic_Con_nat @ X41 @ X42 ) )
= ( sup_sup_set_nat @ ( epistemic_set_fm_nat @ X41 ) @ ( epistemic_set_fm_nat @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_991_fm_Osimps_I93_J,axiom,
! [X41: epistemic_fm_a,X42: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Con_a @ X41 @ X42 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X41 ) @ ( epistemic_set_fm_a @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_992_fm_Osimps_I92_J,axiom,
! [X31: episte740340785640729014c_fm_a,X32: episte740340785640729014c_fm_a] :
( ( episte9089240958480457552c_fm_a @ ( episte6088726764479022859c_fm_a @ X31 @ X32 ) )
= ( sup_su1367922730591523534c_fm_a @ ( episte9089240958480457552c_fm_a @ X31 ) @ ( episte9089240958480457552c_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_993_fm_Osimps_I92_J,axiom,
! [X31: epistemic_fm_nat,X32: epistemic_fm_nat] :
( ( epistemic_set_fm_nat @ ( epistemic_Dis_nat @ X31 @ X32 ) )
= ( sup_sup_set_nat @ ( epistemic_set_fm_nat @ X31 ) @ ( epistemic_set_fm_nat @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_994_fm_Osimps_I92_J,axiom,
! [X31: epistemic_fm_a,X32: epistemic_fm_a] :
( ( epistemic_set_fm_a @ ( epistemic_Dis_a @ X31 @ X32 ) )
= ( sup_sup_set_a @ ( epistemic_set_fm_a @ X31 ) @ ( epistemic_set_fm_a @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_995_UNIV__coset,axiom,
( top_to7796028867103199306c_fm_a
= ( coset_Epistemic_fm_a @ nil_Epistemic_fm_a ) ) ).
% UNIV_coset
thf(fact_996_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_997_UNIV__coset,axiom,
( top_top_set_set_nat
= ( coset_set_nat @ nil_set_nat ) ) ).
% UNIV_coset
thf(fact_998_Fpow__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le7112219575281605754c_fm_a @ ( finite7994277033128312448c_fm_a @ A ) @ ( finite7994277033128312448c_fm_a @ B4 ) ) ) ).
% Fpow_mono
thf(fact_999_Fpow__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B4 ) ) ) ).
% Fpow_mono
thf(fact_1000_Fpow__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_le9131159989063066194et_nat @ ( finite_Fpow_set_nat @ A ) @ ( finite_Fpow_set_nat @ B4 ) ) ) ).
% Fpow_mono
thf(fact_1001_Fpow__subset__Pow,axiom,
! [A: set_nat] : ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( pow_nat @ A ) ) ).
% Fpow_subset_Pow
thf(fact_1002_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G2: list_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ ( set_Epistemic_fm_a2 @ G2 ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G2 @ P ) ) ) ).
% inconsistent_imply
thf(fact_1003_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1004_sup_Obounded__iff,axiom,
! [B: epistemic_fm_a > $o,C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ B @ C ) @ A2 )
= ( ( ord_le4043730696559282883fm_a_o @ B @ A2 )
& ( ord_le4043730696559282883fm_a_o @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1005_sup_Obounded__iff,axiom,
! [B: set_Epistemic_fm_a,C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ B @ C ) @ A2 )
= ( ( ord_le3275665582123262618c_fm_a @ B @ A2 )
& ( ord_le3275665582123262618c_fm_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1006_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1007_sup_Obounded__iff,axiom,
! [B: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1008_sup_Obounded__iff,axiom,
! [B: $o > nat,C: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_o_nat @ B @ A2 )
& ( ord_less_eq_o_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1009_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_1010_le__sup__iff,axiom,
! [X3: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X3 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1011_le__sup__iff,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( sup_su6697315146409554831fm_a_o @ X3 @ Y ) @ Z )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Z )
& ( ord_le4043730696559282883fm_a_o @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1012_le__sup__iff,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ X3 @ Y ) @ Z )
= ( ( ord_le3275665582123262618c_fm_a @ X3 @ Z )
& ( ord_le3275665582123262618c_fm_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1013_le__sup__iff,axiom,
! [X3: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X3 @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1014_le__sup__iff,axiom,
! [X3: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X3 @ Y ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X3 @ Z )
& ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1015_le__sup__iff,axiom,
! [X3: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_o_nat @ X3 @ Z )
& ( ord_less_eq_o_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1016_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1017_set__union,axiom,
! [Xs2: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( union_Epistemic_fm_a @ Xs2 @ Ys ) )
= ( sup_su1367922730591523534c_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) @ ( set_Epistemic_fm_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1018_set__union,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs2 @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1019_set__union,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_1020_extendS__def,axiom,
! [A: epistemic_fm_a > $o,N: epistemic_fm_a,Prev: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S @ N @ Prev )
= ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) ) )
& ( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ N @ bot_bo3626323581529592678c_fm_a ) @ Prev ) )
=> ( ( maxima5458213586468375476c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_1021_maximal_H__Extend,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a] : ( maxima2621630783216038545c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ ( maxima2580775624958445067c_fm_a @ ( bNF_Ca1305897159876240246c_fm_a @ top_to7796028867103199306c_fm_a ) @ ( episte2285483198712856226tent_a @ A ) @ S ) ) ).
% maximal'_Extend
thf(fact_1022_Pow__set_I2_J,axiom,
! [X3: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs2 ) ) )
= ( sup_su4305657963326700718c_fm_a @ ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) ) @ ( image_8582343580727714565c_fm_a @ ( insert7817948963269374698c_fm_a @ X3 ) @ ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs2 ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1023_Pow__set_I2_J,axiom,
! [X3: nat,Xs2: list_nat] :
( ( pow_nat @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ ( set_nat2 @ Xs2 ) ) @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X3 ) @ ( pow_nat @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1024_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1025_sup_OcoboundedI2,axiom,
! [C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ B )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1026_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ B )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1027_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1028_sup_OcoboundedI2,axiom,
! [C: set_set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1029_sup_OcoboundedI2,axiom,
! [C: $o > nat,B: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ ( sup_sup_o_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1030_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1031_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1032_sup_OcoboundedI1,axiom,
! [C: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ C @ A2 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( sup_su6697315146409554831fm_a_o @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1033_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ C @ A2 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( sup_su1367922730591523534c_fm_a @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1034_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1035_sup_OcoboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1036_sup_OcoboundedI1,axiom,
! [C: $o > nat,A2: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ C @ A2 )
=> ( ord_less_eq_o_nat @ C @ ( sup_sup_o_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1037_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1038_sup_Oabsorb__iff2,axiom,
( ord_less_eq_o_nat
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( sup_sup_o_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1039_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1040_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1041_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1042_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1043_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1044_sup_OboundedI,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_1045_sup_OboundedE,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_1046_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1047_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_1048_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1049_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1050_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1051_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1052_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1053_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( sup_sup_nat @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1054_sup__least,axiom,
! [Y: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_1055_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1056_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1057_le__supI2,axiom,
! [X3: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1058_le__supI1,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1059_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_1060_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_1061_le__supI,axiom,
! [A2: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_1062_le__supE,axiom,
! [A2: nat,B: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_1063_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1064_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1065_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W4 )
=> ( ( finite3304564945125563331c_fm_a @ W4 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1066_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_1067_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1068_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1069_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1070_rev__finite__subset,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_1071_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1072_finite__subset,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_1073_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs: list_nat] :
( ( set_nat2 @ Xs )
= A ) ) ).
% finite_list
thf(fact_1074_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1075_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1076_finite__surj,axiom,
! [A: set_nat,B4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B4 ) ) ) ).
% finite_surj
thf(fact_1077_finite__subset__image,axiom,
! [B4: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B4
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1078_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P4: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
& ( P4 @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A )
& ( P4 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1079_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P4: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
=> ( P4 @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A ) )
=> ( P4 @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1080_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [X6: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ A3 )
& ( finite_finite_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_1081_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P4: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat2 @ A5 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_nat2 @ A5 @ F4 ) ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1082_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P4: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ~ ( member_nat2 @ A5 @ F4 )
=> ( ( P4 @ F4 )
=> ( P4 @ ( insert_nat2 @ A5 @ F4 ) ) ) ) ) )
=> ( P4 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1083_Min_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1084_Min__antimono,axiom,
! [M: set_nat,N2: set_nat] :
( ( ord_less_eq_set_nat @ M @ N2 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N2 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1085_Min_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1086_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ A )
=> ( ord_less_eq_nat @ A2 @ B2 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_1087_Min_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_1088_Min__eqI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ord_less_eq_nat @ X3 @ Y2 ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_1089_Min__le,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 ) ) ) ).
% Min_le
thf(fact_1090_finite__ranking__induct,axiom,
! [S: set_nat,P4: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [X2: nat,S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ! [Y5: nat] :
( ( member_nat2 @ Y5 @ S4 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P4 @ S4 )
=> ( P4 @ ( insert_nat2 @ X2 @ S4 ) ) ) ) )
=> ( P4 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1091_Min_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X3 @ A5 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1092_Min_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X3 @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1093_eq__Min__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M5
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ M5 @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1094_Min__le__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 )
= ( ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1095_Min__eq__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M5 )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ M5 @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1096_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_1097_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1098_finite__subset__Union,axiom,
! [A: set_nat,B7: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B7 ) )
=> ~ ! [F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ B7 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1099_le__cSup__finite,axiom,
! [X5: set_nat,X3: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1100_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat2 @ Z @ X5 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1101_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1102_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1103_mono__Min__commute,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_1104_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1105_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1106_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1107_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B4 ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( A = B4 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1108_mono__sup,axiom,
! [F: nat > nat,A: nat,B4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F @ A ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup_nat @ A @ B4 ) ) ) ) ).
% mono_sup
thf(fact_1109_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X6: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_1110_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R5: nat,S5: nat] :
( ( member_nat2 @ R5 @ A )
=> ( ( member_nat2 @ S5 @ A )
=> ( ( ord_less_eq_nat @ R5 @ S5 )
=> ( ord_less_eq_nat @ ( F @ R5 ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_1111_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R: nat,S6: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat2 @ R @ A )
=> ( ( member_nat2 @ S6 @ A )
=> ( ( ord_less_eq_nat @ R @ S6 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S6 ) ) ) ) ) ) ).
% mono_onD
thf(fact_1112_mono__imp__mono__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_1113_monoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_1114_monoE,axiom,
! [F: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_1115_monoD,axiom,
! [F: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_1116_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B4: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( monotone_on_nat_nat @ B4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_1117_antimonoI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ F ) ) ).
% antimonoI
thf(fact_1118_antimonoE,axiom,
! [F: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ F )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoE
thf(fact_1119_antimonoD,axiom,
! [F: nat > nat,X3: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ F )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ).
% antimonoD
thf(fact_1120_sorted__list__of__set__nonempty,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1121_mono__cSup,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( F @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ) ).
% mono_cSup
thf(fact_1122_bdd__above_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1123_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M2: nat] :
~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1124_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1125_cSup__upper2,axiom,
! [X3: nat,X5: set_nat,Y: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1126_cSup__upper,axiom,
! [X3: nat,X5: set_nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1127_cSup__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_nat @ B2 @ X4 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1128_cSup__le__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ S )
=> ( ord_less_eq_nat @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1129_cSup__subset__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1130_bdd__above__image__mono,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_mono
thf(fact_1131_remove__induct,axiom,
! [P4: set_nat > $o,B4: set_nat] :
( ( P4 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P4 @ B4 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B4 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A8 )
=> ( P4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B4 ) ) ) ) ).
% remove_induct
thf(fact_1132_finite__remove__induct,axiom,
! [B4: set_nat,P4: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B4 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A8 )
=> ( P4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P4 @ A8 ) ) ) ) )
=> ( P4 @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_1133_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat2 @ X3 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X3
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_1134_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat2 @ X3 @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat2 @ X3 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X3
@ ( linord2614967742042102400et_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_1135_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X3
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_1136_mono__Max__commute,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_1137_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X3 @ ( linord2614967742042102400et_nat @ A ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1138_Max_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_1139_Max_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).
% Max.coboundedI
thf(fact_1140_Max__eq__if,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ? [Xa3: nat] :
( ( member_nat2 @ Xa3 @ B4 )
& ( ord_less_eq_nat @ X2 @ Xa3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B4 )
=> ? [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
& ( ord_less_eq_nat @ X2 @ Xa3 ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A )
= ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_1141_Max__eqI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( lattic8265883725875713057ax_nat @ A )
= X3 ) ) ) ) ).
% Max_eqI
thf(fact_1142_Max__ge,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).
% Max_ge
thf(fact_1143_Max_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ A5 @ X3 ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 ) ) ) ) ).
% Max.boundedI
thf(fact_1144_Max_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X3 ) ) ) ) ) ).
% Max.boundedE
thf(fact_1145_eq__Max__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M5
= ( lattic8265883725875713057ax_nat @ A ) )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1146_Max__ge__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) )
= ( ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_1147_Max__eq__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A )
= M5 )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1148_Max__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A2 ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Max_insert2
thf(fact_1149_Max_Osubset__imp,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_1150_Max__mono,axiom,
! [M: set_nat,N2: set_nat] :
( ( ord_less_eq_set_nat @ M @ N2 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N2 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N2 ) ) ) ) ) ).
% Max_mono
thf(fact_1151_dual__Min,axiom,
( ( lattices_Min_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X ) )
= lattic8265883725875713057ax_nat ) ).
% dual_Min
thf(fact_1152_Least__mono,axiom,
! [F: nat > nat,S: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ? [X4: nat] :
( ( member_nat2 @ X4 @ S )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ S )
=> ( ord_less_eq_nat @ X4 @ Xa2 ) ) )
=> ( ( ord_Least_nat
@ ^ [Y3: nat] : ( member_nat2 @ Y3 @ ( image_nat_nat @ F @ S ) ) )
= ( F
@ ( ord_Least_nat
@ ^ [X: nat] : ( member_nat2 @ X @ S ) ) ) ) ) ) ).
% Least_mono
thf(fact_1153_Least1I,axiom,
! [P4: nat > $o] :
( ? [X4: nat] :
( ( P4 @ X4 )
& ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P4 @ Y2 )
& ! [Ya2: nat] :
( ( P4 @ Ya2 )
=> ( ord_less_eq_nat @ Y2 @ Ya2 ) ) )
=> ( Y2 = X4 ) ) )
=> ( P4 @ ( ord_Least_nat @ P4 ) ) ) ).
% Least1I
thf(fact_1154_Least1__le,axiom,
! [P4: nat > $o,Z: nat] :
( ? [X4: nat] :
( ( P4 @ X4 )
& ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P4 @ Y2 )
& ! [Ya2: nat] :
( ( P4 @ Ya2 )
=> ( ord_less_eq_nat @ Y2 @ Ya2 ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P4 @ Z )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P4 ) @ Z ) ) ) ).
% Least1_le
thf(fact_1155_LeastI2__order,axiom,
! [P4: nat > $o,X3: nat,Q4: nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) )
=> ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ X2 @ Y5 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( ord_Least_nat @ P4 ) ) ) ) ) ).
% LeastI2_order
thf(fact_1156_Least__equality,axiom,
! [P4: nat > $o,X3: nat] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ X3 @ Y2 ) )
=> ( ( ord_Least_nat @ P4 )
= X3 ) ) ) ).
% Least_equality
thf(fact_1157_LeastI2__wellorder,axiom,
! [P4: nat > $o,A2: nat,Q4: nat > $o] :
( ( P4 @ A2 )
=> ( ! [A5: nat] :
( ( P4 @ A5 )
=> ( ! [B8: nat] :
( ( P4 @ B8 )
=> ( ord_less_eq_nat @ A5 @ B8 ) )
=> ( Q4 @ A5 ) ) )
=> ( Q4 @ ( ord_Least_nat @ P4 ) ) ) ) ).
% LeastI2_wellorder
thf(fact_1158_LeastI2__wellorder__ex,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ? [X_1: nat] : ( P4 @ X_1 )
=> ( ! [A5: nat] :
( ( P4 @ A5 )
=> ( ! [B8: nat] :
( ( P4 @ B8 )
=> ( ord_less_eq_nat @ A5 @ B8 ) )
=> ( Q4 @ A5 ) ) )
=> ( Q4 @ ( ord_Least_nat @ P4 ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_1159_Least__le,axiom,
! [P4: nat > $o,K2: nat] :
( ( P4 @ K2 )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P4 ) @ K2 ) ) ).
% Least_le
thf(fact_1160_bdd__above__image__antimono,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ F )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_1161_bdd__belowI,axiom,
! [A: set_nat,M5: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ M5 @ X2 ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_belowI
thf(fact_1162_bdd__below_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ M @ X2 ) )
=> ( condit1738341127787009408ow_nat @ A ) ) ).
% bdd_below.I
thf(fact_1163_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= A )
& ( distinct_nat @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1164_bdd__below_Ounfold,axiom,
( condit1738341127787009408ow_nat
= ( ^ [A3: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( ord_less_eq_nat @ M4 @ X ) ) ) ) ).
% bdd_below.unfold
thf(fact_1165_bdd__below_OE,axiom,
! [A: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ~ ! [M2: nat] :
~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( ord_less_eq_nat @ M2 @ X4 ) ) ) ).
% bdd_below.E
thf(fact_1166_bdd__below__image__mono,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bdd_below_image_mono
thf(fact_1167_bdd__below__image__antimono,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
@ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
@ F )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_1168_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ( distinct_nat @ Xs3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1169_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1170_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1171_Int__Collect__mono,axiom,
! [A: set_nat,B4: set_nat,P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P4 ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q4 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1172_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1173_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1174_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1175_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1176_le__infE,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1177_le__infI,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1178_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1179_le__infI1,axiom,
! [A2: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1180_le__infI2,axiom,
! [B: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1181_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1182_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1183_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1184_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( inf_inf_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1185_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1186_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1187_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1188_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1189_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1190_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1191_inf__greatest,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1192_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1193_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1194_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1195_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1196_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1197_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1198_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1199_distrib__sup__le,axiom,
! [X3: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1200_distrib__inf__le,axiom,
! [X3: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1201_mono__inf,axiom,
! [F: nat > nat,A: nat,B4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A @ B4 ) ) @ ( inf_inf_nat @ ( F @ A ) @ ( F @ B4 ) ) ) ) ).
% mono_inf
thf(fact_1202_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1203_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1204_cSup__inter__less__eq,axiom,
! [A: set_nat,B4: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit2214826472909112428ve_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_1205_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1206_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1207_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1208_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1209_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1210_diff__is__0__eq,axiom,
! [M5: nat,N: nat] :
( ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% diff_is_0_eq
thf(fact_1211_diff__is__0__eq_H,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1212_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1213_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1214_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1215_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1216_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1217_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1218_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1219_eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( M5 = N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% eq_imp_le
thf(fact_1220_le__antisym,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( ord_less_eq_nat @ N @ M5 )
=> ( M5 = N ) ) ) ).
% le_antisym
thf(fact_1221_GreatestI__nat,axiom,
! [P4: nat > $o,K2: nat,B: nat] :
( ( P4 @ K2 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% GreatestI_nat
thf(fact_1222_nat__le__linear,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
| ( ord_less_eq_nat @ N @ M5 ) ) ).
% nat_le_linear
thf(fact_1223_Greatest__le__nat,axiom,
! [P4: nat > $o,K2: nat,B: nat] :
( ( P4 @ K2 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% Greatest_le_nat
thf(fact_1224_GreatestI__ex__nat,axiom,
! [P4: nat > $o,B: nat] :
( ? [X_1: nat] : ( P4 @ X_1 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1225_Nat_Oex__has__greatest__nat,axiom,
! [P4: nat > $o,K2: nat,B: nat] :
( ( P4 @ K2 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P4 @ X2 )
& ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1226_diff__le__mono2,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_1227_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1228_diff__le__self,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N ) @ M5 ) ).
% diff_le_self
thf(fact_1229_diff__le__mono,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1230_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M5 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1231_le__diff__iff,axiom,
! [K2: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1232_eq__diff__iff,axiom,
! [K2: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M5 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M5 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M5 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1233_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_1234_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X: nat] :
( ( member_nat2 @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1235_bounded__Max__nat,axiom,
! [P4: nat > $o,X3: nat,M: nat] :
( ( P4 @ X3 )
=> ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M7: nat] :
( ( P4 @ M7 )
=> ~ ! [X4: nat] :
( ( P4 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1236_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1237_cInf__greatest,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X5 ) ) ) ) ).
% cInf_greatest
thf(fact_1238_cInf__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ A2 @ X2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ord_less_eq_nat @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A2 ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1239_cInf__eq__minimum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat2 @ Z @ X5 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_1240_cInf__eq,axiom,
! [X5: set_nat,A2: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ A2 @ X2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ X5 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ord_less_eq_nat @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf_nat @ X5 )
= A2 ) ) ) ).
% cInf_eq
thf(fact_1241_wellorder__Inf__le1,axiom,
! [K2: nat,A: set_nat] :
( ( member_nat2 @ K2 @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ K2 ) ) ).
% wellorder_Inf_le1
thf(fact_1242_cInf__lower2,axiom,
! [X3: nat,X5: set_nat,Y: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( condit1738341127787009408ow_nat @ X5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X5 ) @ Y ) ) ) ) ).
% cInf_lower2
thf(fact_1243_cInf__lower,axiom,
! [X3: nat,X5: set_nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( condit1738341127787009408ow_nat @ X5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X5 ) @ X3 ) ) ) ).
% cInf_lower
thf(fact_1244_cInf__le__finite,axiom,
! [X5: set_nat,X3: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X5 ) @ X3 ) ) ) ).
% cInf_le_finite
thf(fact_1245_cInf__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ B2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ).
% cInf_mono
thf(fact_1246_le__cInf__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ S )
=> ( ( ord_less_eq_nat @ A2 @ ( complete_Inf_Inf_nat @ S ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ S )
=> ( ord_less_eq_nat @ A2 @ X ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_1247_cInf__superset__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B4 ) @ ( complete_Inf_Inf_nat @ A ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_1248_less__eq__cInf__inter,axiom,
! [A: set_nat,B4: set_nat] :
( ( condit1738341127787009408ow_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A @ B4 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_1249_cInf__le__cSup,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_1250_mono__cInf,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_nat @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% mono_cInf
thf(fact_1251_lfp__induct__set,axiom,
! [A2: nat,F: set_nat > set_nat,P4: nat > $o] :
( ( member_nat2 @ A2 @ ( comple7975543026063415949et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( F @ ( inf_inf_set_nat @ ( comple7975543026063415949et_nat @ F ) @ ( collect_nat @ P4 ) ) ) )
=> ( P4 @ X2 ) )
=> ( P4 @ A2 ) ) ) ) ).
% lfp_induct_set
thf(fact_1252_def__lfp__induct__set,axiom,
! [A: set_nat,F: set_nat > set_nat,A2: nat,P4: nat > $o] :
( ( A
= ( comple7975543026063415949et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat2 @ A2 @ A )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( F @ ( inf_inf_set_nat @ A @ ( collect_nat @ P4 ) ) ) )
=> ( P4 @ X2 ) )
=> ( P4 @ A2 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_1253_def__Collect__coinduct,axiom,
! [A: set_nat,P4: set_nat > nat > $o,A2: nat,X5: set_nat] :
( ( A
= ( comple1596078789208929544et_nat
@ ^ [W9: set_nat] : ( collect_nat @ ( P4 @ W9 ) ) ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat
@ ^ [W9: set_nat] : ( collect_nat @ ( P4 @ W9 ) ) )
=> ( ( member_nat2 @ A2 @ X5 )
=> ( ! [Z3: nat] :
( ( member_nat2 @ Z3 @ X5 )
=> ( P4 @ ( sup_sup_set_nat @ X5 @ A ) @ Z3 ) )
=> ( member_nat2 @ A2 @ A ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_1254_member__le__sum,axiom,
! [I: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ I @ A )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ) ) ).
% member_le_sum
thf(fact_1255_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1256_sum__le__included,axiom,
! [S6: set_nat,T3: set_nat,G4: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S6 )
=> ( ( finite_finite_nat @ T3 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ T3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G4 @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ S6 )
=> ? [Xa3: nat] :
( ( member_nat2 @ Xa3 @ T3 )
& ( ( I @ Xa3 )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ Xa3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S6 ) @ ( groups3542108847815614940at_nat @ G4 @ T3 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1257_sum__nonneg__0,axiom,
! [S6: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ S6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S6 )
= zero_zero_nat )
=> ( ( member_nat2 @ I @ S6 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1258_sum__nonneg__leq__bound,axiom,
! [S6: set_nat,F: nat > nat,B4: nat,I: nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ S6 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S6 )
= B4 )
=> ( ( member_nat2 @ I @ S6 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B4 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1259_sum__mono__inv,axiom,
! [F: nat > nat,I4: set_nat,G4: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I4 )
= ( groups3542108847815614940at_nat @ G4 @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G4 @ I3 ) ) )
=> ( ( member_nat2 @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G4 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1260_sum_Osame__carrier,axiom,
! [C2: set_nat,A: set_nat,B4: set_nat,G4: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G4 @ A5 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ ( minus_minus_set_nat @ C2 @ B4 ) )
=> ( ( H3 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G4 @ A )
= ( groups3542108847815614940at_nat @ H3 @ B4 ) )
= ( ( groups3542108847815614940at_nat @ G4 @ C2 )
= ( groups3542108847815614940at_nat @ H3 @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_1261_sum_Osame__carrierI,axiom,
! [C2: set_nat,A: set_nat,B4: set_nat,G4: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ ( minus_minus_set_nat @ C2 @ A ) )
=> ( ( G4 @ A5 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ ( minus_minus_set_nat @ C2 @ B4 ) )
=> ( ( H3 @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G4 @ C2 )
= ( groups3542108847815614940at_nat @ H3 @ C2 ) )
=> ( ( groups3542108847815614940at_nat @ G4 @ A )
= ( groups3542108847815614940at_nat @ H3 @ B4 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_1262_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S: set_nat,G4: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G4 @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G4 @ S )
= ( groups3542108847815614940at_nat @ G4 @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_1263_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S: set_nat,G4: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G4 @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G4 @ T2 )
= ( groups3542108847815614940at_nat @ G4 @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_1264_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S: set_nat,H3: nat > nat,G4: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( H3 @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ S )
=> ( ( G4 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G4 @ S )
= ( groups3542108847815614940at_nat @ H3 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_1265_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S: set_nat,G4: nat > nat,H3: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( minus_minus_set_nat @ T2 @ S ) )
=> ( ( G4 @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ S )
=> ( ( G4 @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G4 @ T2 )
= ( groups3542108847815614940at_nat @ H3 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_1266_sum__diff__nat,axiom,
! [B4: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1267_sum__mono2,axiom,
! [B4: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [B2: nat] :
( ( member_nat2 @ B2 @ ( minus_minus_set_nat @ B4 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).
% sum_mono2
thf(fact_1268_sum__strict__mono2,axiom,
! [B4: set_nat,A: set_nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ B @ ( minus_minus_set_nat @ B4 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1269_eventually__finite__subsets__at__top__weakI,axiom,
! [A: set_nat,P4: set_nat > $o] :
( ! [X7: set_nat] :
( ( finite_finite_nat @ X7 )
=> ( ( ord_less_eq_set_nat @ X7 @ A )
=> ( P4 @ X7 ) ) )
=> ( eventually_set_nat @ P4 @ ( finite3254316476582989077op_nat @ A ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_1270_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > nat,G4: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G4 @ X4 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G4 @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
% Helper facts (11)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ g ) @ p ) ).
%------------------------------------------------------------------------------