TPTP Problem File: SLH0035^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Stalnaker_Logic/0000_Stalnaker_Logic/prob_00165_005109__6226604_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1638 ( 610 unt; 348 typ; 0 def)
% Number of atoms : 4032 (1432 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 12881 ( 400 ~; 61 |; 278 &;10255 @)
% ( 0 <=>;1887 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 1635 (1635 >; 0 *; 0 +; 0 <<)
% Number of symbols : 317 ( 314 usr; 22 con; 0-3 aty)
% Number of variables : 4535 ( 540 ^;3825 !; 170 ?;4535 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:47.619
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
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episte1905851639235817760t_unit: $tType ).
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,
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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,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
list_l3671048651222332332c_fm_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
set_li1092409891198325638c_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
set_li4204741992506657632c_fm_a: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
episte3561180207765753404c_fm_a: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
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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__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
list_s580375451141968640c_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__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__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,
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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,
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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__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (314)
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_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_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_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_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__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_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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte7127422172654481821c_fm_a: episte3561180207765753404c_fm_a > episte3561180207765753404c_fm_a > episte3561180207765753404c_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_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte8719284489194433015c_fm_a: episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a ).
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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte6664300826525126289c_fm_a: episte3561180207765753404c_fm_a > episte3561180207765753404c_fm_a > episte3561180207765753404c_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_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte9190102072588321003c_fm_a: episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a ).
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_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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte7828218607427978131c_fm_a: episte3561180207765753404c_fm_a > episte3561180207765753404c_fm_a > episte3561180207765753404c_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_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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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__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte9021198779603319828c_fm_a: list_Epistemic_fm_a > episte3561180207765753404c_fm_a > episte3561180207765753404c_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_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte3013695276341575662c_fm_a: set_Epistemic_fm_a > episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a ).
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_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,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001tf__a,type,
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episte7930623906115922400c_fm_a: ( list_Epistemic_fm_a > list_Epistemic_fm_a > $o ) > episte3561180207765753404c_fm_a > episte3561180207765753404c_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001tf__a,type,
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episte8246226516352692192c_fm_a: ( set_Epistemic_fm_a > set_Epistemic_fm_a > $o ) > episte5999154429349999510c_fm_a > episte5999154429349999510c_fm_a > $o ).
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,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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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_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
episte6227279731293865264c_fm_a: episte5999154429349999510c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__a,type,
epistemic_set_fm_a: epistemic_fm_a > set_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_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,
episte6390737319716712051t_unit: episte94448284482925344t_unit > set_Epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060W_062_001tf__a_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte3604461938431221107t_unit: episte1905851639235817760t_unit > set_li769143395467472256c_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 ).
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thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
the_el2173195877760541071c_fm_a: set_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
the_el8584007648813340437c_fm_a: set_li769143395467472256c_fm_a > list_Epistemic_fm_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
the_el1285646365920873327c_fm_a: set_se5208064806568342746c_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__a,type,
stalnaker_conjunct_a: list_Epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
accp_Epistemic_fm_a: ( epistemic_fm_a > epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
accp_l7472382826029824047c_fm_a: ( list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a > $o ) > list_l6083326122719238310c_fm_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
accp_l8971127668057225909c_fm_a: ( list_l3671048651222332332c_fm_a > list_l3671048651222332332c_fm_a > $o ) > list_l3671048651222332332c_fm_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_J,type,
accp_l1289336314643896847c_fm_a: ( list_l985973790606564998c_fm_a > list_l985973790606564998c_fm_a > $o ) > list_l985973790606564998c_fm_a > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
member6642669571620171971c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
member5906877432388582473c_fm_a: list_Epistemic_fm_a > set_li769143395467472256c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member7465314387053329231c_fm_a: list_l6083326122719238310c_fm_a > set_li1092409891198325638c_fm_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J,type,
member2328277852289759785c_fm_a: list_s580375451141968640c_fm_a > set_li4204741992506657632c_fm_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
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_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_A,type,
a2: epistemic_fm_a > $o ).
thf(sy_v_Ga____,type,
ga: list_Epistemic_fm_a ).
thf(sy_v_a____,type,
a3: epistemic_fm_a ).
thf(sy_v_i,type,
i: a ).
% Relevant facts (1270)
thf(fact_0__092_060open_062A_A_092_060turnstile_062_AK_Ai_A_Ia_A_092_060_094bold_062_092_060and_062_Aconjunct_AG_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_Aa_A_092_060_094bold_062_092_060and_062_AK_Ai_A_Iconjunct_AG_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( epistemic_Con_a @ a3 @ ( stalnaker_conjunct_a @ ga ) ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ i @ a3 ) @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ ga ) ) ) ) ).
% \<open>A \<turnstile> K i (a \<^bold>\<and> conjunct G) \<^bold>\<longrightarrow> K i a \<^bold>\<and> K i (conjunct G)\<close>
thf(fact_1__092_060open_062A_A_092_060turnstile_062_AK_Ai_A_Iconjunct_A_Ia_A_D_AG_J_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_Ia_A_092_060_094bold_062_092_060and_062_Aconjunct_AG_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ ( cons_Epistemic_fm_a @ a3 @ ga ) ) ) @ ( epistemic_K_a @ i @ ( epistemic_Con_a @ a3 @ ( stalnaker_conjunct_a @ ga ) ) ) ) ).
% \<open>A \<turnstile> K i (conjunct (a # G)) \<^bold>\<longrightarrow> K i (a \<^bold>\<and> conjunct G)\<close>
thf(fact_2_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_3_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_4_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_5_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_6_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_7_K__conjunction__in,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_Con_a @ P @ Q ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_8_K__conj__imply__factor,axiom,
! [A: epistemic_fm_a > $o,I: a,P: epistemic_fm_a,Q: epistemic_fm_a,R: epistemic_fm_a] : ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ Q ) ) @ R ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Con_a @ P @ Q ) ) @ R ) ) ) ).
% K_conj_imply_factor
thf(fact_9_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_10_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_11_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_12_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_13_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_14_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_15_list_Oinject,axiom,
! [X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a,Y21: list_Epistemic_fm_a,Y22: list_l6083326122719238310c_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X21 @ X22 )
= ( cons_l8134865115577406678c_fm_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_16_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_17_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_18_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_19_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_20_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_21_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_22_not__Cons__self2,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( cons_l8134865115577406678c_fm_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_23_not__Cons__self2,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_24_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_25_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_26_local_OCons,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ ga ) ) @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ i ) @ ga ) ) ) ).
% local.Cons
thf(fact_27_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ! [G: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G @ H @ 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_28_AK_Ocases,axiom,
! [A: epistemic_fm_a > $o,A2: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ A2 )
=> ( ~ ! [G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ 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_29_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_30_K__imply__conjunct,axiom,
! [A: epistemic_fm_a > $o,G3: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G3 ) @ P ) ) ) ).
% K_imply_conjunct
thf(fact_31_K__conjunct__imply,axiom,
! [A: epistemic_fm_a > $o,G3: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G3 ) @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ P ) ) ) ).
% K_conjunct_imply
thf(fact_32_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_33_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G3: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G3 ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_34_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G3: 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 ) @ G3 ) ) @ Q ) ) ).
% K_mp
thf(fact_35_member__rec_I1_J,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a] :
( ( member594239076476917947c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member594239076476917947c_fm_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_36_member__rec_I1_J,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( member6038508265109909045c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member6038508265109909045c_fm_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_37_map__ident,axiom,
( ( map_li3080774272885743684c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : X2 )
= ( ^ [Xs2: list_l6083326122719238310c_fm_a] : Xs2 ) ) ).
% map_ident
thf(fact_38_map__ident,axiom,
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X2: epistemic_fm_a] : X2 )
= ( ^ [Xs2: list_Epistemic_fm_a] : Xs2 ) ) ).
% map_ident
thf(fact_39_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_40_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_41_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_42_list_Omap__ident,axiom,
! [T: list_l6083326122719238310c_fm_a] :
( ( map_li3080774272885743684c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_43_list_Omap__ident,axiom,
! [T: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X2: epistemic_fm_a] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_44_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_45_K__distrib__K__imp,axiom,
! [A: epistemic_fm_a > $o,I: a,G3: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ ( epistemic_imply_a @ G3 @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G3 ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_46_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( map_Ep4939040189629079504c_fm_a @ F @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( cons_l2479501935080859996c_fm_a @ ( F @ X21 ) @ ( map_Ep4939040189629079504c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_47_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( cons_l8134865115577406678c_fm_a @ ( F @ X21 ) @ ( map_Ep1988204514693272906c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_48_list_Osimps_I9_J,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( ( map_li3858915945532548554c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) )
= ( cons_l2479501935080859996c_fm_a @ ( F @ X21 ) @ ( map_li3858915945532548554c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_49_list_Osimps_I9_J,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( ( map_li5516866612190465598c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) )
= ( cons_Epistemic_fm_a @ ( F @ X21 ) @ ( map_li5516866612190465598c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_50_list_Osimps_I9_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) )
= ( cons_l8134865115577406678c_fm_a @ ( F @ X21 ) @ ( map_li3080774272885743684c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_51_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( cons_Epistemic_fm_a @ ( F @ X21 ) @ ( map_Ep7084560364594560580c_fm_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_52_Cons__eq__map__D,axiom,
! [X: list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l2479501935080859996c_fm_a @ X @ Xs )
= ( map_Ep4939040189629079504c_fm_a @ F @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_Ep4939040189629079504c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_53_Cons__eq__map__D,axiom,
! [X: list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l2479501935080859996c_fm_a @ X @ Xs )
= ( map_li3858915945532548554c_fm_a @ F @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_li3858915945532548554c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_54_Cons__eq__map__D,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_Epistemic_fm_a @ X @ Xs )
= ( map_li5516866612190465598c_fm_a @ F @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_li5516866612190465598c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_55_Cons__eq__map__D,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ F @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_Ep1988204514693272906c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_56_Cons__eq__map__D,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X @ Xs )
= ( map_li3080774272885743684c_fm_a @ F @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_li3080774272885743684c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_57_Cons__eq__map__D,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_58_map__eq__Cons__D,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a,Y: list_l6083326122719238310c_fm_a,Ys: list_l3671048651222332332c_fm_a] :
( ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= ( cons_l2479501935080859996c_fm_a @ Y @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_Ep4939040189629079504c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_59_map__eq__Cons__D,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a,Ys: list_l3671048651222332332c_fm_a] :
( ( ( map_li3858915945532548554c_fm_a @ F @ Xs )
= ( cons_l2479501935080859996c_fm_a @ Y @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_li3858915945532548554c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_60_map__eq__Cons__D,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_li5516866612190465598c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_61_map__eq__Cons__D,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_Ep1988204514693272906c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_62_map__eq__Cons__D,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
=> ? [Z: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_li3080774272885743684c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_63_map__eq__Cons__D,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
=> ? [Z: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_64_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_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_66_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_67_mem__Collect__eq,axiom,
! [A2: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( member5906877432388582473c_fm_a @ A2 @ ( collec5191077796991884427c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_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_69_mem__Collect__eq,axiom,
! [A2: nat,P4: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_li769143395467472256c_fm_a] :
( ( collec5191077796991884427c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_77_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X3: epistemic_fm_a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_78_Collect__cong,axiom,
! [P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ! [X3: list_Epistemic_fm_a] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collec5191077796991884427c_fm_a @ P4 )
= ( collec5191077796991884427c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ! [X3: set_nat] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collect_set_nat @ P4 )
= ( collect_set_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_80_Collect__cong,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ! [X3: nat] :
( ( P4 @ X3 )
= ( Q4 @ X3 ) )
=> ( ( collect_nat @ P4 )
= ( collect_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_81_Cons__eq__map__conv,axiom,
! [X: list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l2479501935080859996c_fm_a @ X @ Xs )
= ( map_Ep4939040189629079504c_fm_a @ F @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_Ep4939040189629079504c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_82_Cons__eq__map__conv,axiom,
! [X: list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l2479501935080859996c_fm_a @ X @ Xs )
= ( map_li3858915945532548554c_fm_a @ F @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_li3858915945532548554c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_83_Cons__eq__map__conv,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_Epistemic_fm_a @ X @ Xs )
= ( map_li5516866612190465598c_fm_a @ F @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_li5516866612190465598c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_84_Cons__eq__map__conv,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ F @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_Ep1988204514693272906c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_85_Cons__eq__map__conv,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( cons_l8134865115577406678c_fm_a @ X @ Xs )
= ( map_li3080774272885743684c_fm_a @ F @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Ys
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_li3080774272885743684c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_86_Cons__eq__map__conv,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_87_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a,Y: list_l6083326122719238310c_fm_a,Ys: list_l3671048651222332332c_fm_a] :
( ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= ( cons_l2479501935080859996c_fm_a @ Y @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_Ep4939040189629079504c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_88_map__eq__Cons__conv,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a,Ys: list_l3671048651222332332c_fm_a] :
( ( ( map_li3858915945532548554c_fm_a @ F @ Xs )
= ( cons_l2479501935080859996c_fm_a @ Y @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_li3858915945532548554c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_89_map__eq__Cons__conv,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_li5516866612190465598c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_90_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_Ep1988204514693272906c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_91_map__eq__Cons__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) )
= ( ? [Z2: list_Epistemic_fm_a,Zs2: list_l6083326122719238310c_fm_a] :
( ( Xs
= ( cons_l8134865115577406678c_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_li3080774272885743684c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_92_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( ? [Z2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_93_duality__taut,axiom,
! [I: a,P: epistemic_fm_a,Q: epistemic_fm_a,G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ 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_94_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_95_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_96_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_97_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_98_conjunct__imply,axiom,
! [G3: list_Epistemic_fm_a,P: epistemic_fm_a,G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G3 ) @ P ) @ ( epistemic_imply_a @ G3 @ P ) ) ) ).
% conjunct_imply
thf(fact_99_imply__conjunct,axiom,
! [G3: list_Epistemic_fm_a,P: epistemic_fm_a,G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G3 @ P ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G3 ) @ P ) ) ) ).
% imply_conjunct
thf(fact_100_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_101_eval_Osimps_I5_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
=> ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_102_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H4: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H4 @ ( epistemic_K_a @ I @ P ) )
= ( H4 @ ( epistemic_K_a @ I @ P ) ) ) ).
% eval.simps(6)
thf(fact_103_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
thf(fact_104_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G3: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G3 ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ P ) ) ) ).
% K_Boole
thf(fact_105_eval_Osimps_I4_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
& ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_106_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_107_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_108_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G3: 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 @ G3 ) ) @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G3 ) ) @ R ) ) ) ).
% K_swap
thf(fact_109_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_110_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_111_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_112_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_113__092_060open_062A_A_092_060turnstile_062_AK_Ai_A_Iconjunct_A_091_093_J_A_092_060_094bold_062_092_060longrightarrow_062_Aconjunct_A_Imap_A_IK_Ai_J_A_091_093_J_092_060close_062,axiom,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ nil_Epistemic_fm_a ) ) @ ( stalnaker_conjunct_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ i ) @ nil_Epistemic_fm_a ) ) ) ).
% \<open>A \<turnstile> K i (conjunct []) \<^bold>\<longrightarrow> conjunct (map (K i) [])\<close>
thf(fact_114_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_115_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_116_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_117_map__eq__map__tailrec,axiom,
map_Ep1988204514693272906c_fm_a = map_ta2019361550626295956c_fm_a ).
% map_eq_map_tailrec
thf(fact_118_map__eq__map__tailrec,axiom,
map_li3080774272885743684c_fm_a = map_ta7821446597547650702c_fm_a ).
% map_eq_map_tailrec
thf(fact_119_map__eq__map__tailrec,axiom,
map_Ep4939040189629079504c_fm_a = map_ta456340477436210714c_fm_a ).
% map_eq_map_tailrec
thf(fact_120_map__eq__map__tailrec,axiom,
map_li3858915945532548554c_fm_a = map_ta6463483919923529492c_fm_a ).
% map_eq_map_tailrec
thf(fact_121_map__eq__map__tailrec,axiom,
map_li5516866612190465598c_fm_a = map_ta5548023648123488648c_fm_a ).
% map_eq_map_tailrec
thf(fact_122_map__eq__map__tailrec,axiom,
map_Ep7084560364594560580c_fm_a = map_ta875960198429730446c_fm_a ).
% map_eq_map_tailrec
thf(fact_123_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_124_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_125_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_126_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G3: list_Epistemic_fm_a,R: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G3 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G3 ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ R ) ) ) ) ) ).
% K_DisE
thf(fact_127_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_128_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_129_map__is__Nil__conv,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( ( map_se8248101584983471204c_fm_a @ F @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_130_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep6135353274472911908c_fm_a @ F @ Xs )
= nil_se102576565402781184c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_131_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_132_map__is__Nil__conv,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_133_map__is__Nil__conv,axiom,
! [F: set_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( ( map_se6909769656262425450c_fm_a @ F @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_134_map__is__Nil__conv,axiom,
! [F: list_Epistemic_fm_a > set_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( ( map_li6596013992111160350c_fm_a @ F @ Xs )
= nil_se102576565402781184c_fm_a )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_135_map__is__Nil__conv,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( ( map_se7333198834109155908c_fm_a @ F @ Xs )
= nil_se102576565402781184c_fm_a )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_136_map__is__Nil__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_137_map__is__Nil__conv,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= nil_li5178944226519631276c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% map_is_Nil_conv
thf(fact_138_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_139_Nil__is__map__conv,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( nil_Epistemic_fm_a
= ( map_se8248101584983471204c_fm_a @ F @ Xs ) )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_140_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_se102576565402781184c_fm_a
= ( map_Ep6135353274472911908c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_141_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_142_Nil__is__map__conv,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( nil_Epistemic_fm_a
= ( map_li5516866612190465598c_fm_a @ F @ Xs ) )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_143_Nil__is__map__conv,axiom,
! [F: set_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( map_se6909769656262425450c_fm_a @ F @ Xs ) )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_144_Nil__is__map__conv,axiom,
! [F: list_Epistemic_fm_a > set_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( nil_se102576565402781184c_fm_a
= ( map_li6596013992111160350c_fm_a @ F @ Xs ) )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_145_Nil__is__map__conv,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( nil_se102576565402781184c_fm_a
= ( map_se7333198834109155908c_fm_a @ F @ Xs ) )
= ( Xs = nil_se102576565402781184c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_146_Nil__is__map__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( map_li3080774272885743684c_fm_a @ F @ Xs ) )
= ( Xs = nil_li2451196919128234278c_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_147_Nil__is__map__conv,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a] :
( ( nil_li5178944226519631276c_fm_a
= ( map_Ep4939040189629079504c_fm_a @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_map_conv
thf(fact_148_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ A2 )
= nil_Epistemic_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_149_list_Omap__disc__iff,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a,A2: list_s580375451141968640c_fm_a] :
( ( ( map_se8248101584983471204c_fm_a @ F @ A2 )
= nil_Epistemic_fm_a )
= ( A2 = nil_se102576565402781184c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_150_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep6135353274472911908c_fm_a @ F @ A2 )
= nil_se102576565402781184c_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_151_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ A2 )
= nil_li2451196919128234278c_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_152_list_Omap__disc__iff,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,A2: list_l6083326122719238310c_fm_a] :
( ( ( map_li5516866612190465598c_fm_a @ F @ A2 )
= nil_Epistemic_fm_a )
= ( A2 = nil_li2451196919128234278c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_153_list_Omap__disc__iff,axiom,
! [F: set_Epistemic_fm_a > list_Epistemic_fm_a,A2: list_s580375451141968640c_fm_a] :
( ( ( map_se6909769656262425450c_fm_a @ F @ A2 )
= nil_li2451196919128234278c_fm_a )
= ( A2 = nil_se102576565402781184c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_154_list_Omap__disc__iff,axiom,
! [F: list_Epistemic_fm_a > set_Epistemic_fm_a,A2: list_l6083326122719238310c_fm_a] :
( ( ( map_li6596013992111160350c_fm_a @ F @ A2 )
= nil_se102576565402781184c_fm_a )
= ( A2 = nil_li2451196919128234278c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_155_list_Omap__disc__iff,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a,A2: list_s580375451141968640c_fm_a] :
( ( ( map_se7333198834109155908c_fm_a @ F @ A2 )
= nil_se102576565402781184c_fm_a )
= ( A2 = nil_se102576565402781184c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_156_list_Omap__disc__iff,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A2: list_l6083326122719238310c_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ A2 )
= nil_li2451196919128234278c_fm_a )
= ( A2 = nil_li2451196919128234278c_fm_a ) ) ).
% list.map_disc_iff
thf(fact_157_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,A2: list_Epistemic_fm_a] :
( ( ( map_Ep4939040189629079504c_fm_a @ F @ A2 )
= nil_li5178944226519631276c_fm_a )
= ( A2 = nil_Epistemic_fm_a ) ) ).
% list.map_disc_iff
thf(fact_158_transpose_Ocases,axiom,
! [X: list_l985973790606564998c_fm_a] :
( ( X != nil_li5433889876484989702c_fm_a )
=> ( ! [Xss: list_l985973790606564998c_fm_a] :
( X
!= ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss ) )
=> ~ ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Xss: list_l985973790606564998c_fm_a] :
( X
!= ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_159_transpose_Ocases,axiom,
! [X: list_l3671048651222332332c_fm_a] :
( ( X != nil_li5178944226519631276c_fm_a )
=> ( ! [Xss: list_l3671048651222332332c_fm_a] :
( X
!= ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss ) )
=> ~ ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Xss: list_l3671048651222332332c_fm_a] :
( X
!= ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_160_transpose_Ocases,axiom,
! [X: list_l6083326122719238310c_fm_a] :
( ( X != nil_li2451196919128234278c_fm_a )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( X
!= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ~ ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( X
!= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_161_list__nonempty__induct,axiom,
! [Xs: list_s580375451141968640c_fm_a,P4: list_s580375451141968640c_fm_a > $o] :
( ( Xs != nil_se102576565402781184c_fm_a )
=> ( ! [X3: set_Epistemic_fm_a] : ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ nil_se102576565402781184c_fm_a ) )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] :
( ( Xs3 != nil_se102576565402781184c_fm_a )
=> ( ( P4 @ Xs3 )
=> ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_162_list__nonempty__induct,axiom,
! [Xs: list_l6083326122719238310c_fm_a,P4: list_l6083326122719238310c_fm_a > $o] :
( ( Xs != nil_li2451196919128234278c_fm_a )
=> ( ! [X3: list_Epistemic_fm_a] : ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ nil_li2451196919128234278c_fm_a ) )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] :
( ( Xs3 != nil_li2451196919128234278c_fm_a )
=> ( ( P4 @ Xs3 )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_163_list__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ! [X3: epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] :
( ( Xs3 != nil_Epistemic_fm_a )
=> ( ( P4 @ Xs3 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_164_list__induct2_H,axiom,
! [P4: list_s580375451141968640c_fm_a > list_s580375451141968640c_fm_a > $o,Xs: list_s580375451141968640c_fm_a,Ys: list_s580375451141968640c_fm_a] :
( ( P4 @ nil_se102576565402781184c_fm_a @ nil_se102576565402781184c_fm_a )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] : ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ nil_se102576565402781184c_fm_a )
=> ( ! [Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] : ( P4 @ nil_se102576565402781184c_fm_a @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_165_list__induct2_H,axiom,
! [P4: list_s580375451141968640c_fm_a > list_Epistemic_fm_a > $o,Xs: list_s580375451141968640c_fm_a,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_se102576565402781184c_fm_a @ nil_Epistemic_fm_a )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] : ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_se102576565402781184c_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_166_list__induct2_H,axiom,
! [P4: list_s580375451141968640c_fm_a > list_l6083326122719238310c_fm_a > $o,Xs: list_s580375451141968640c_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( P4 @ nil_se102576565402781184c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] : ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ nil_li2451196919128234278c_fm_a )
=> ( ! [Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] : ( P4 @ nil_se102576565402781184c_fm_a @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_167_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_s580375451141968640c_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_s580375451141968640c_fm_a] :
( ( P4 @ nil_Epistemic_fm_a @ nil_se102576565402781184c_fm_a )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ nil_se102576565402781184c_fm_a )
=> ( ! [Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_168_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( P4 @ nil_Epistemic_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ nil_li2451196919128234278c_fm_a )
=> ( ! [Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_169_list__induct2_H,axiom,
! [P4: list_l6083326122719238310c_fm_a > list_s580375451141968640c_fm_a > $o,Xs: list_l6083326122719238310c_fm_a,Ys: list_s580375451141968640c_fm_a] :
( ( P4 @ nil_li2451196919128234278c_fm_a @ nil_se102576565402781184c_fm_a )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] : ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ nil_se102576565402781184c_fm_a )
=> ( ! [Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] : ( P4 @ nil_li2451196919128234278c_fm_a @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Y2: set_Epistemic_fm_a,Ys2: list_s580375451141968640c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_170_list__induct2_H,axiom,
! [P4: list_l6083326122719238310c_fm_a > list_Epistemic_fm_a > $o,Xs: list_l6083326122719238310c_fm_a,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_li2451196919128234278c_fm_a @ nil_Epistemic_fm_a )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] : ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_li2451196919128234278c_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_171_list__induct2_H,axiom,
! [P4: list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a > $o,Xs: list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a] :
( ( P4 @ nil_li2451196919128234278c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] : ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ nil_li2451196919128234278c_fm_a )
=> ( ! [Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] : ( P4 @ nil_li2451196919128234278c_fm_a @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Y2: list_Epistemic_fm_a,Ys2: list_l6083326122719238310c_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_172_list__induct2_H,axiom,
! [P4: list_Epistemic_fm_a > list_Epistemic_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( P4 @ nil_Epistemic_fm_a @ nil_Epistemic_fm_a )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs3 @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_173_neq__Nil__conv,axiom,
! [Xs: list_s580375451141968640c_fm_a] :
( ( Xs != nil_se102576565402781184c_fm_a )
= ( ? [Y3: set_Epistemic_fm_a,Ys3: list_s580375451141968640c_fm_a] :
( Xs
= ( cons_s4394448069393209776c_fm_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_174_neq__Nil__conv,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( Xs != nil_li2451196919128234278c_fm_a )
= ( ? [Y3: list_Epistemic_fm_a,Ys3: list_l6083326122719238310c_fm_a] :
( Xs
= ( cons_l8134865115577406678c_fm_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_175_neq__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
= ( ? [Y3: epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_176_remdups__adj_Ocases,axiom,
! [X: list_s580375451141968640c_fm_a] :
( ( X != nil_se102576565402781184c_fm_a )
=> ( ! [X3: set_Epistemic_fm_a] :
( X
!= ( cons_s4394448069393209776c_fm_a @ X3 @ nil_se102576565402781184c_fm_a ) )
=> ~ ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] :
( X
!= ( cons_s4394448069393209776c_fm_a @ X3 @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_177_remdups__adj_Ocases,axiom,
! [X: list_l6083326122719238310c_fm_a] :
( ( X != nil_li2451196919128234278c_fm_a )
=> ( ! [X3: list_Epistemic_fm_a] :
( X
!= ( cons_l8134865115577406678c_fm_a @ X3 @ nil_li2451196919128234278c_fm_a ) )
=> ~ ! [X3: list_Epistemic_fm_a,Y2: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a] :
( X
!= ( cons_l8134865115577406678c_fm_a @ X3 @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_178_remdups__adj_Ocases,axiom,
! [X: list_Epistemic_fm_a] :
( ( X != nil_Epistemic_fm_a )
=> ( ! [X3: epistemic_fm_a] :
( X
!= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
=> ~ ! [X3: epistemic_fm_a,Y2: epistemic_fm_a,Xs3: list_Epistemic_fm_a] :
( X
!= ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_179_min__list_Ocases,axiom,
! [X: list_s580375451141968640c_fm_a] :
( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a] :
( X
!= ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) )
=> ( X = nil_se102576565402781184c_fm_a ) ) ).
% min_list.cases
thf(fact_180_list_Oexhaust,axiom,
! [Y: list_s580375451141968640c_fm_a] :
( ( Y != nil_se102576565402781184c_fm_a )
=> ~ ! [X212: set_Epistemic_fm_a,X222: list_s580375451141968640c_fm_a] :
( Y
!= ( cons_s4394448069393209776c_fm_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_181_list_Oexhaust,axiom,
! [Y: list_l6083326122719238310c_fm_a] :
( ( Y != nil_li2451196919128234278c_fm_a )
=> ~ ! [X212: list_Epistemic_fm_a,X222: list_l6083326122719238310c_fm_a] :
( Y
!= ( cons_l8134865115577406678c_fm_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_182_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_183_list_OdiscI,axiom,
! [List: list_s580375451141968640c_fm_a,X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a] :
( ( List
= ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) )
=> ( List != nil_se102576565402781184c_fm_a ) ) ).
% list.discI
thf(fact_184_list_OdiscI,axiom,
! [List: list_l6083326122719238310c_fm_a,X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( ( List
= ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) )
=> ( List != nil_li2451196919128234278c_fm_a ) ) ).
% list.discI
thf(fact_185_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_186_list_Odistinct_I1_J,axiom,
! [X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a] :
( nil_se102576565402781184c_fm_a
!= ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_187_list_Odistinct_I1_J,axiom,
! [X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] :
( nil_li2451196919128234278c_fm_a
!= ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_188_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_189_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% list.simps(8)
thf(fact_190_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( map_Ep6135353274472911908c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% list.simps(8)
thf(fact_191_list_Osimps_I8_J,axiom,
! [F: set_Epistemic_fm_a > epistemic_fm_a] :
( ( map_se8248101584983471204c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_Epistemic_fm_a ) ).
% list.simps(8)
thf(fact_192_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% list.simps(8)
thf(fact_193_list_Osimps_I8_J,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a] :
( ( map_li5516866612190465598c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_Epistemic_fm_a ) ).
% list.simps(8)
thf(fact_194_list_Osimps_I8_J,axiom,
! [F: list_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( map_li6596013992111160350c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% list.simps(8)
thf(fact_195_list_Osimps_I8_J,axiom,
! [F: set_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( map_se6909769656262425450c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% list.simps(8)
thf(fact_196_list_Osimps_I8_J,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ( map_se7333198834109155908c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% list.simps(8)
thf(fact_197_list_Osimps_I8_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% list.simps(8)
thf(fact_198_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( map_Ep4939040189629079504c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_li5178944226519631276c_fm_a ) ).
% list.simps(8)
thf(fact_199_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_200_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_201_imply_Osimps_I1_J,axiom,
! [Q: epistemic_fm_a] :
( ( epistemic_imply_a @ nil_Epistemic_fm_a @ Q )
= Q ) ).
% imply.simps(1)
thf(fact_202_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_203_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_204_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_205_eval_Osimps_I3_J,axiom,
! [G5: list_char > $o,H4: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G5 @ H4 @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G5 @ H4 @ P )
| ( epistemic_eval_a @ G5 @ H4 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_206_member__rec_I2_J,axiom,
! [Y: list_Epistemic_fm_a] :
~ ( member594239076476917947c_fm_a @ nil_li2451196919128234278c_fm_a @ Y ) ).
% member_rec(2)
thf(fact_207_member__rec_I2_J,axiom,
! [Y: set_Epistemic_fm_a] :
~ ( member378061430901374997c_fm_a @ nil_se102576565402781184c_fm_a @ Y ) ).
% member_rec(2)
thf(fact_208_member__rec_I2_J,axiom,
! [Y: epistemic_fm_a] :
~ ( member6038508265109909045c_fm_a @ nil_Epistemic_fm_a @ Y ) ).
% member_rec(2)
thf(fact_209_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_210_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_211_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_212_map__rec,axiom,
( map_Ep4939040189629079504c_fm_a
= ( ^ [F2: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( rec_li3857189064575089956c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [X2: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_l2479501935080859996c_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_213_map__rec,axiom,
( map_li3858915945532548554c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( rec_li4767119888041796650c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [X2: list_Epistemic_fm_a,Uu2: list_l6083326122719238310c_fm_a] : ( cons_l2479501935080859996c_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_214_map__rec,axiom,
( map_li5516866612190465598c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > epistemic_fm_a] :
( rec_li2845402903074671286c_fm_a @ nil_Epistemic_fm_a
@ ^ [X2: list_Epistemic_fm_a,Uu2: list_l6083326122719238310c_fm_a] : ( cons_Epistemic_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_215_map__rec,axiom,
( map_Ep1988204514693272906c_fm_a
= ( ^ [F2: epistemic_fm_a > list_Epistemic_fm_a] :
( rec_li2622076421192168746c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [X2: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_216_map__rec,axiom,
( map_li3080774272885743684c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( rec_li5579148116651729200c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [X2: list_Epistemic_fm_a,Uu2: list_l6083326122719238310c_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_217_map__rec,axiom,
( map_Ep7084560364594560580c_fm_a
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a] :
( rec_li4360243526537435568c_fm_a @ nil_Epistemic_fm_a
@ ^ [X2: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_218_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_219_insert__Nil,axiom,
! [X: set_Epistemic_fm_a] :
( ( insert1868103382419917334c_fm_a @ X @ nil_se102576565402781184c_fm_a )
= ( cons_s4394448069393209776c_fm_a @ X @ nil_se102576565402781184c_fm_a ) ) ).
% insert_Nil
thf(fact_220_insert__Nil,axiom,
! [X: list_Epistemic_fm_a] :
( ( insert335162795743670844c_fm_a @ X @ nil_li2451196919128234278c_fm_a )
= ( cons_l8134865115577406678c_fm_a @ X @ nil_li2451196919128234278c_fm_a ) ) ).
% insert_Nil
thf(fact_221_insert__Nil,axiom,
! [X: epistemic_fm_a] :
( ( insert177310161492556854c_fm_a @ X @ nil_Epistemic_fm_a )
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ).
% insert_Nil
thf(fact_222_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_223_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_224_fm_Oset__cases,axiom,
! [E: list_Epistemic_fm_a,A2: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ A2 ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a] :
( ? [Z22: episte3561180207765753404c_fm_a] :
( A2
= ( episte6664300826525126289c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a,Z22: episte3561180207765753404c_fm_a] :
( ( A2
= ( episte6664300826525126289c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a] :
( ? [Z22: episte3561180207765753404c_fm_a] :
( A2
= ( episte7127422172654481821c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a,Z22: episte3561180207765753404c_fm_a] :
( ( A2
= ( episte7127422172654481821c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a] :
( ? [Z22: episte3561180207765753404c_fm_a] :
( A2
= ( episte7828218607427978131c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte3561180207765753404c_fm_a,Z22: episte3561180207765753404c_fm_a] :
( ( A2
= ( episte7828218607427978131c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte3561180207765753404c_fm_a] :
( A2
!= ( episte9021198779603319828c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: list_Epistemic_fm_a,Z22: episte3561180207765753404c_fm_a] :
( ( A2
= ( episte9021198779603319828c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( episte2454647923005522006c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_225_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A2: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ A2 ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte9190102072588321003c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte8719284489194433015c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a] :
( ? [Z22: episte5999154429349999510c_fm_a] :
( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z1 ) ) )
=> ( ! [Z1: episte5999154429349999510c_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte5200161025749276397c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) )
=> ( ! [Z22: episte5999154429349999510c_fm_a] :
( A2
!= ( episte3013695276341575662c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: episte5999154429349999510c_fm_a] :
( ( A2
= ( episte3013695276341575662c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( episte6227279731293865264c_fm_a @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_226_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_227_concat__map__singleton,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat5652308878773863783c_fm_a
@ ( map_Ep4245182426433019350c_fm_a
@ ^ [X2: epistemic_fm_a] : ( cons_l2479501935080859996c_fm_a @ ( F @ X2 ) @ nil_li5178944226519631276c_fm_a )
@ Xs ) )
= ( map_Ep4939040189629079504c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_228_concat__map__singleton,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat5652308878773863783c_fm_a
@ ( map_li5567427098789011152c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : ( cons_l2479501935080859996c_fm_a @ ( F @ X2 ) @ nil_li5178944226519631276c_fm_a )
@ Xs ) )
= ( map_li3858915945532548554c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_229_concat__map__singleton,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F @ X2 ) @ nil_Epistemic_fm_a )
@ Xs ) )
= ( map_li5516866612190465598c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_230_concat__map__singleton,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_Ep4939040189629079504c_fm_a
@ ^ [X2: epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F @ X2 ) @ nil_li2451196919128234278c_fm_a )
@ Xs ) )
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_231_concat__map__singleton,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ ( F @ X2 ) @ nil_li2451196919128234278c_fm_a )
@ Xs ) )
= ( map_li3080774272885743684c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_232_concat__map__singleton,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_Ep1988204514693272906c_fm_a
@ ^ [X2: epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F @ X2 ) @ nil_Epistemic_fm_a )
@ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_233_list__ex1__simps_I1_J,axiom,
! [P4: list_Epistemic_fm_a > $o] :
~ ( list_e1371837923195959913c_fm_a @ P4 @ nil_li2451196919128234278c_fm_a ) ).
% list_ex1_simps(1)
thf(fact_234_list__ex1__simps_I1_J,axiom,
! [P4: set_Epistemic_fm_a > $o] :
~ ( list_e3415556844334454979c_fm_a @ P4 @ nil_se102576565402781184c_fm_a ) ).
% list_ex1_simps(1)
thf(fact_235_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_236_arg__min__list_Osimps_I1_J,axiom,
! [F: epistemic_fm_a > nat,X: epistemic_fm_a] :
( ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) )
= X ) ).
% arg_min_list.simps(1)
thf(fact_237_map__tailrec__rev_Oelims,axiom,
! [X: set_Epistemic_fm_a > epistemic_fm_a,Xa: list_s580375451141968640c_fm_a,Xb: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( map_ta3870718651422774037c_fm_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_se102576565402781184c_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: set_Epistemic_fm_a,As: list_s580375451141968640c_fm_a] :
( ( Xa
= ( cons_s4394448069393209776c_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta3870718651422774037c_fm_a @ X @ As @ ( cons_Epistemic_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_238_map__tailrec__rev_Oelims,axiom,
! [X: set_Epistemic_fm_a > list_Epistemic_fm_a,Xa: list_s580375451141968640c_fm_a,Xb: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( map_ta7115970949835247003c_fm_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_se102576565402781184c_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: set_Epistemic_fm_a,As: list_s580375451141968640c_fm_a] :
( ( Xa
= ( cons_s4394448069393209776c_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta7115970949835247003c_fm_a @ X @ As @ ( cons_l8134865115577406678c_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_239_map__tailrec__rev_Oelims,axiom,
! [X: epistemic_fm_a > list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,Xb: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( map_ta3940434083879480187c_fm_a @ X @ 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_ta3940434083879480187c_fm_a @ X @ As @ ( cons_l8134865115577406678c_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_240_map__tailrec__rev_Oelims,axiom,
! [X: list_Epistemic_fm_a > epistemic_fm_a,Xa: list_l6083326122719238310c_fm_a,Xb: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( map_ta7469096181376672879c_fm_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_li2451196919128234278c_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: list_Epistemic_fm_a,As: list_l6083326122719238310c_fm_a] :
( ( Xa
= ( cons_l8134865115577406678c_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta7469096181376672879c_fm_a @ X @ As @ ( cons_Epistemic_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_241_map__tailrec__rev_Oelims,axiom,
! [X: list_Epistemic_fm_a > list_Epistemic_fm_a,Xa: list_l6083326122719238310c_fm_a,Xb: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( map_ta5197791105168353781c_fm_a @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_li2451196919128234278c_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: list_Epistemic_fm_a,As: list_l6083326122719238310c_fm_a] :
( ( Xa
= ( cons_l8134865115577406678c_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta5197791105168353781c_fm_a @ X @ As @ ( cons_l8134865115577406678c_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_242_map__tailrec__rev_Oelims,axiom,
! [X: 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 @ X @ 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 @ X @ As @ ( cons_Epistemic_fm_a @ ( X @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_243_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_244_arg__min__list_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_a > nat,X: list_Epistemic_fm_a,Y: list_Epistemic_fm_a,Zs3: list_l6083326122719238310c_fm_a] :
( ( arg_mi474574138286535628_a_nat @ F @ ( cons_l8134865115577406678c_fm_a @ X @ ( cons_l8134865115577406678c_fm_a @ Y @ Zs3 ) ) )
= ( if_lis2878681784746929638c_fm_a @ ( ord_less_eq_nat @ ( F @ X ) @ ( F @ ( arg_mi474574138286535628_a_nat @ F @ ( cons_l8134865115577406678c_fm_a @ Y @ Zs3 ) ) ) ) @ X @ ( arg_mi474574138286535628_a_nat @ F @ ( cons_l8134865115577406678c_fm_a @ Y @ Zs3 ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_245_arg__min__list_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > nat,X: epistemic_fm_a,Y: epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ X @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) )
= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( F @ X ) @ ( F @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) ) ) @ X @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_246_concat_Osimps_I1_J,axiom,
( ( concat6109960377493484129c_fm_a @ nil_li5178944226519631276c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% concat.simps(1)
thf(fact_247_concat_Osimps_I1_J,axiom,
( ( concat2941956077894552763c_fm_a @ nil_li5433889876484989702c_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% concat.simps(1)
thf(fact_248_concat_Osimps_I1_J,axiom,
( ( concat2780896542826320603c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_Epistemic_fm_a ) ).
% concat.simps(1)
thf(fact_249_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_250_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_251_fm_Oset__intros_I6_J,axiom,
! [Yf: list_Epistemic_fm_a,X52: episte3561180207765753404c_fm_a,X51: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Yf @ ( episte2454647923005522006c_fm_a @ X52 ) )
=> ( member5906877432388582473c_fm_a @ Yf @ ( episte2454647923005522006c_fm_a @ ( episte7828218607427978131c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_252_fm_Oset__intros_I6_J,axiom,
! [Yf: set_Epistemic_fm_a,X52: episte5999154429349999510c_fm_a,X51: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ X52 ) )
=> ( member536094252920883875c_fm_a @ Yf @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_253_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_254_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_255_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_256_fm_Oset__intros_I5_J,axiom,
! [Ye: list_Epistemic_fm_a,X51: episte3561180207765753404c_fm_a,X52: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Ye @ ( episte2454647923005522006c_fm_a @ X51 ) )
=> ( member5906877432388582473c_fm_a @ Ye @ ( episte2454647923005522006c_fm_a @ ( episte7828218607427978131c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_257_fm_Oset__intros_I5_J,axiom,
! [Ye: set_Epistemic_fm_a,X51: episte5999154429349999510c_fm_a,X52: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ X51 ) )
=> ( member536094252920883875c_fm_a @ Ye @ ( episte6227279731293865264c_fm_a @ ( episte5200161025749276397c_fm_a @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_258_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_259_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_260_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_261_fm_Oset__intros_I8_J,axiom,
! [Yg: list_Epistemic_fm_a,X62: episte3561180207765753404c_fm_a,X61: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Yg @ ( episte2454647923005522006c_fm_a @ X62 ) )
=> ( member5906877432388582473c_fm_a @ Yg @ ( episte2454647923005522006c_fm_a @ ( episte9021198779603319828c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_262_fm_Oset__intros_I8_J,axiom,
! [Yg: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a,X61: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ X62 ) )
=> ( member536094252920883875c_fm_a @ Yg @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_263_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_264_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_265_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_266_fm_Oset__intros_I7_J,axiom,
! [X61: list_Epistemic_fm_a,X62: episte3561180207765753404c_fm_a] : ( member5906877432388582473c_fm_a @ X61 @ ( episte2454647923005522006c_fm_a @ ( episte9021198779603319828c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_267_fm_Oset__intros_I7_J,axiom,
! [X61: set_Epistemic_fm_a,X62: episte5999154429349999510c_fm_a] : ( member536094252920883875c_fm_a @ X61 @ ( episte6227279731293865264c_fm_a @ ( episte3013695276341575662c_fm_a @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_268_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_269_map__concat,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_Ep1988204514693272906c_fm_a @ F @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat6109960377493484129c_fm_a @ ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_270_map__concat,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( map_li3080774272885743684c_fm_a @ F @ ( concat6109960377493484129c_fm_a @ Xs ) )
= ( concat6109960377493484129c_fm_a @ ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_271_map__concat,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_Ep4939040189629079504c_fm_a @ F @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat5652308878773863783c_fm_a @ ( map_li5567427098789011152c_fm_a @ ( map_Ep4939040189629079504c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_272_map__concat,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( map_li3858915945532548554c_fm_a @ F @ ( concat6109960377493484129c_fm_a @ Xs ) )
= ( concat5652308878773863783c_fm_a @ ( map_li3568483382057536586c_fm_a @ ( map_li3858915945532548554c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_273_map__concat,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( map_li5516866612190465598c_fm_a @ F @ ( concat6109960377493484129c_fm_a @ Xs ) )
= ( concat2780896542826320603c_fm_a @ ( map_li8743910666895923134c_fm_a @ ( map_li5516866612190465598c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_274_map__concat,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_275_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_276_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_277_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_278_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_279_fm_Oset__intros_I4_J,axiom,
! [Yd: list_Epistemic_fm_a,X42: episte3561180207765753404c_fm_a,X41: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Yd @ ( episte2454647923005522006c_fm_a @ X42 ) )
=> ( member5906877432388582473c_fm_a @ Yd @ ( episte2454647923005522006c_fm_a @ ( episte7127422172654481821c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_280_fm_Oset__intros_I4_J,axiom,
! [Yd: set_Epistemic_fm_a,X42: episte5999154429349999510c_fm_a,X41: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ X42 ) )
=> ( member536094252920883875c_fm_a @ Yd @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_281_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_282_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_283_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_284_fm_Oset__intros_I3_J,axiom,
! [Yc: list_Epistemic_fm_a,X41: episte3561180207765753404c_fm_a,X42: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Yc @ ( episte2454647923005522006c_fm_a @ X41 ) )
=> ( member5906877432388582473c_fm_a @ Yc @ ( episte2454647923005522006c_fm_a @ ( episte7127422172654481821c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_285_fm_Oset__intros_I3_J,axiom,
! [Yc: set_Epistemic_fm_a,X41: episte5999154429349999510c_fm_a,X42: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ X41 ) )
=> ( member536094252920883875c_fm_a @ Yc @ ( episte6227279731293865264c_fm_a @ ( episte8719284489194433015c_fm_a @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_286_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_287_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X24 ) ) ).
% fm.distinct(1)
thf(fact_288_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_289_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_290_fm_Oset__intros_I2_J,axiom,
! [Yb: list_Epistemic_fm_a,X32: episte3561180207765753404c_fm_a,X31: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Yb @ ( episte2454647923005522006c_fm_a @ X32 ) )
=> ( member5906877432388582473c_fm_a @ Yb @ ( episte2454647923005522006c_fm_a @ ( episte6664300826525126289c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_291_fm_Oset__intros_I2_J,axiom,
! [Yb: set_Epistemic_fm_a,X32: episte5999154429349999510c_fm_a,X31: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ X32 ) )
=> ( member536094252920883875c_fm_a @ Yb @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_292_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_293_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_294_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_295_fm_Oset__intros_I1_J,axiom,
! [Ya: list_Epistemic_fm_a,X31: episte3561180207765753404c_fm_a,X32: episte3561180207765753404c_fm_a] :
( ( member5906877432388582473c_fm_a @ Ya @ ( episte2454647923005522006c_fm_a @ X31 ) )
=> ( member5906877432388582473c_fm_a @ Ya @ ( episte2454647923005522006c_fm_a @ ( episte6664300826525126289c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_296_fm_Oset__intros_I1_J,axiom,
! [Ya: set_Epistemic_fm_a,X31: episte5999154429349999510c_fm_a,X32: episte5999154429349999510c_fm_a] :
( ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ X31 ) )
=> ( member536094252920883875c_fm_a @ Ya @ ( episte6227279731293865264c_fm_a @ ( episte9190102072588321003c_fm_a @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_297_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_298_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_299_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_300_eval_Osimps_I2_J,axiom,
! [G5: list_char > $o,Uw: epistemic_fm_a > $o,X: list_char] :
( ( epistemic_eval_a @ G5 @ Uw @ ( epistemic_Pro_a @ X ) )
= ( G5 @ X ) ) ).
% eval.simps(2)
thf(fact_301_list_Osimps_I7_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 @ ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ X22 ) ) ) ).
% list.simps(7)
thf(fact_302_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,A2: epistemic_fm_a,As2: list_Epistemic_fm_a,Bs: list_l6083326122719238310c_fm_a] :
( ( map_ta3940434083879480187c_fm_a @ F @ ( cons_Epistemic_fm_a @ A2 @ As2 ) @ Bs )
= ( map_ta3940434083879480187c_fm_a @ F @ As2 @ ( cons_l8134865115577406678c_fm_a @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_303_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,A2: list_Epistemic_fm_a,As2: list_l6083326122719238310c_fm_a,Bs: list_Epistemic_fm_a] :
( ( map_ta7469096181376672879c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ A2 @ As2 ) @ Bs )
= ( map_ta7469096181376672879c_fm_a @ F @ As2 @ ( cons_Epistemic_fm_a @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_304_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,A2: list_Epistemic_fm_a,As2: list_l6083326122719238310c_fm_a,Bs: list_l6083326122719238310c_fm_a] :
( ( map_ta5197791105168353781c_fm_a @ F @ ( cons_l8134865115577406678c_fm_a @ A2 @ As2 ) @ Bs )
= ( map_ta5197791105168353781c_fm_a @ F @ As2 @ ( cons_l8134865115577406678c_fm_a @ ( F @ A2 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_305_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_306_list_Osimps_I6_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( rec_li4360243526537435568c_fm_a @ F1 @ F22 @ nil_Epistemic_fm_a )
= F1 ) ).
% list.simps(6)
thf(fact_307_map__tailrec__rev_Osimps_I1_J,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Bs: list_Epistemic_fm_a] :
( ( map_ta6565535454553546997c_fm_a @ F @ nil_Epistemic_fm_a @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_308_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X3: epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_309_rec__list__Nil__imp,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( F
= ( rec_li4360243526537435568c_fm_a @ F1 @ F22 ) )
=> ( ( F @ nil_Epistemic_fm_a )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_310_rec__list__Cons__imp,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( F
= ( rec_li4360243526537435568c_fm_a @ F1 @ F22 ) )
=> ( ( F @ ( cons_Epistemic_fm_a @ X @ Xs ) )
= ( F22 @ X @ Xs @ ( F @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_311_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_312_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_313_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_314_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_315_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_316_dual__order_Orefl,axiom,
! [A2: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_317_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_318_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_319_order__refl,axiom,
! [X: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X @ X ) ).
% order_refl
thf(fact_320_order__refl,axiom,
! [X: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X @ X ) ).
% order_refl
thf(fact_321_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_322_order__refl,axiom,
! [X: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ X @ X ) ).
% order_refl
thf(fact_323_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_324_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_337_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] : ( Y4 = Z4 ) )
= ( ^ [X2: epistemic_fm_a > $o,Y3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y3 )
& ( ord_le4043730696559282883fm_a_o @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_338_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Epistemic_fm_a,Y3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y3 )
& ( ord_le3275665582123262618c_fm_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_339_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_340_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] : ( Y4 = Z4 ) )
= ( ^ [X2: set_li769143395467472256c_fm_a,Y3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X2 @ Y3 )
& ( ord_le2931979837777675552c_fm_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_341_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z4: set_set_nat] : ( Y4 = Z4 ) )
= ( ^ [X2: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y3 )
& ( ord_le6893508408891458716et_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_342_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_343_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_344_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_345_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_346_ord__eq__le__trans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( A2 = B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_347_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_348_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_349_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_350_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_351_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_352_ord__le__eq__trans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( B = C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_353_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_354_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_355_order__antisym,axiom,
! [X: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_356_order__antisym,axiom,
! [X: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_357_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_358_order__antisym,axiom,
! [X: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X @ Y )
=> ( ( ord_le2931979837777675552c_fm_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_359_order__antisym,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_360_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_361_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_362_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_363_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_364_order_Otrans,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_365_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_366_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_367_order__trans,axiom,
! [X: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_368_order__trans,axiom,
! [X: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z3 )
=> ( ord_le3275665582123262618c_fm_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_369_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_370_order__trans,axiom,
! [X: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a,Z3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X @ Y )
=> ( ( ord_le2931979837777675552c_fm_a @ Y @ Z3 )
=> ( ord_le2931979837777675552c_fm_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_371_order__trans,axiom,
! [X: set_set_nat,Y: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_372_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_373_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_374_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] : ( Y4 = Z4 ) )
= ( ^ [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_375_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] : ( Y4 = Z4 ) )
= ( ^ [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_376_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [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_377_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] : ( Y4 = Z4 ) )
= ( ^ [A4: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B3 @ A4 )
& ( ord_le2931979837777675552c_fm_a @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_378_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z4: set_set_nat] : ( Y4 = Z4 ) )
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_379_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_380_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_381_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_382_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_383_dual__order_Oantisym,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_384_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_385_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_386_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_387_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_388_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_389_dual__order_Otrans,axiom,
! [B: set_li769143395467472256c_fm_a,A2: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( ( ord_le2931979837777675552c_fm_a @ C @ B )
=> ( ord_le2931979837777675552c_fm_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_390_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_391_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_392_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_393_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_394_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_395_antisym,axiom,
! [A2: set_li769143395467472256c_fm_a,B: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_396_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_397_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_398_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] : ( Y4 = Z4 ) )
= ( ^ [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_399_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] : ( Y4 = Z4 ) )
= ( ^ [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_400_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [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_401_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] : ( Y4 = Z4 ) )
= ( ^ [A4: set_li769143395467472256c_fm_a,B3: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A4 @ B3 )
& ( ord_le2931979837777675552c_fm_a @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_402_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z4: set_set_nat] : ( Y4 = Z4 ) )
= ( ^ [A4: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_403_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_404_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_405_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_406_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_407_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 )
=> ( ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_408_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 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_409_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_410_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_411_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_412_order__subst1,axiom,
! [A2: nat,F: ( epistemic_fm_a > $o ) > nat,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X3: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_413_order__subst1,axiom,
! [A2: nat,F: set_li769143395467472256c_fm_a > nat,B: set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le2931979837777675552c_fm_a @ B @ C )
=> ( ! [X3: set_li769143395467472256c_fm_a,Y2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_414_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_415_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_416_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_417_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_418_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_419_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 )
=> ( ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_420_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_421_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 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_422_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le4043730696559282883fm_a_o @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_423_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le2931979837777675552c_fm_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_424_order__eq__refl,axiom,
! [X: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( X = Y )
=> ( ord_le4043730696559282883fm_a_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_425_order__eq__refl,axiom,
! [X: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( X = Y )
=> ( ord_le3275665582123262618c_fm_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_426_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_427_order__eq__refl,axiom,
! [X: set_li769143395467472256c_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( X = Y )
=> ( ord_le2931979837777675552c_fm_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_428_order__eq__refl,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( X = Y )
=> ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_429_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_430_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_431_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_434_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_435_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_436_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 )
=> ( ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_437_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_438_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 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_439_ord__eq__le__subst,axiom,
! [A2: epistemic_fm_a > $o,F: nat > epistemic_fm_a > $o,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_440_ord__eq__le__subst,axiom,
! [A2: set_li769143395467472256c_fm_a,F: nat > set_li769143395467472256c_fm_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_441_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_442_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_443_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_444_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_445_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 )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_446_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 )
=> ( ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_447_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 )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_448_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 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_449_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_450_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_li769143395467472256c_fm_a,C: set_li769143395467472256c_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le2931979837777675552c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_451_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_452_order__antisym__conv,axiom,
! [Y: epistemic_fm_a > $o,X: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_453_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X )
=> ( ( ord_le3275665582123262618c_fm_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_454_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_455_order__antisym__conv,axiom,
! [Y: set_li769143395467472256c_fm_a,X: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ Y @ X )
=> ( ( ord_le2931979837777675552c_fm_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_456_order__antisym__conv,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_457_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_458_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X )
=> ( Q4 @ X ) ) ) ).
% predicate1D
thf(fact_459_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X: epistemic_fm_a,Q4: epistemic_fm_a > $o] :
( ( P4 @ X )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X ) ) ) ).
% rev_predicate1D
thf(fact_460_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_461_le__funD,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G5 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G5 @ X ) ) ) ).
% le_funD
thf(fact_462_le__funE,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o,X: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G5 )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G5 @ X ) ) ) ).
% le_funE
thf(fact_463_le__funI,axiom,
! [F: epistemic_fm_a > $o,G5: epistemic_fm_a > $o] :
( ! [X3: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X3 ) @ ( G5 @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G5 ) ) ).
% le_funI
thf(fact_464_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G: epistemic_fm_a > $o] :
! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_465_S5__S5_H__assms,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_466_product__lists_Osimps_I2_J,axiom,
! [Xs: list_l6083326122719238310c_fm_a,Xss2: list_l3671048651222332332c_fm_a] :
( ( produc4433783857981728015c_fm_a @ ( cons_l2479501935080859996c_fm_a @ Xs @ Xss2 ) )
= ( concat5652308878773863783c_fm_a
@ ( map_li5567427098789011152c_fm_a
@ ^ [X2: list_Epistemic_fm_a] : ( map_li8347733892191492164c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X2 ) @ ( produc4433783857981728015c_fm_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_467_product__lists_Osimps_I2_J,axiom,
! [Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( produc1391074687832863881c_fm_a @ ( cons_l8134865115577406678c_fm_a @ Xs @ Xss2 ) )
= ( concat6109960377493484129c_fm_a
@ ( map_Ep4939040189629079504c_fm_a
@ ^ [X2: epistemic_fm_a] : ( map_li3080774272885743684c_fm_a @ ( cons_Epistemic_fm_a @ X2 ) @ ( produc1391074687832863881c_fm_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_468_product__lists_Osimps_I1_J,axiom,
( ( produc4433783857981728015c_fm_a @ nil_li5178944226519631276c_fm_a )
= ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ nil_li5178944226519631276c_fm_a ) ) ).
% product_lists.simps(1)
thf(fact_469_product__lists_Osimps_I1_J,axiom,
( ( produc3894803040673298025c_fm_a @ nil_li5433889876484989702c_fm_a )
= ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ nil_li5433889876484989702c_fm_a ) ) ).
% product_lists.simps(1)
thf(fact_470_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_471_fm_Orel__induct,axiom,
! [R2: a > a > $o,X: epistemic_fm_a,Y: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R2 @ X @ Y )
=> ( ( Q4 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q4 @ ( epistemic_Pro_a @ A22 ) @ ( 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] :
( ( R2 @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_472_fm_Orel__cases,axiom,
! [R2: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B != epistemic_FF_a ) )
=> ( ! [X3: list_char] :
( ( A2
= ( epistemic_Pro_a @ X3 ) )
=> ! [Y2: list_char] :
( ( B
= ( epistemic_Pro_a @ Y2 ) )
=> ( X3 != 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 @ R2 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 @ R2 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 @ R2 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ 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 ) )
=> ( ( R2 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R2 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_473_concat__map__maps,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat2780896542826320603c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F @ Xs ) )
= ( maps_E6072454603808916545c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_474_concat__map__maps,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ F @ Xs ) )
= ( maps_l2078986973461115323c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_475_concat__map__maps,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat6109960377493484129c_fm_a @ ( map_Ep4939040189629079504c_fm_a @ F @ Xs ) )
= ( maps_E7773696912818698439c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_476_concat__map__maps,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( concat6109960377493484129c_fm_a @ ( map_li3858915945532548554c_fm_a @ F @ Xs ) )
= ( maps_l3685687229838205249c_fm_a @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_477_maps__def,axiom,
( maps_E6072454603808916545c_fm_a
= ( ^ [F2: epistemic_fm_a > list_Epistemic_fm_a,Xs2: list_Epistemic_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_478_maps__def,axiom,
( maps_l2078986973461115323c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs2: list_l6083326122719238310c_fm_a] : ( concat2780896542826320603c_fm_a @ ( map_li3080774272885743684c_fm_a @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_479_maps__def,axiom,
( maps_E7773696912818698439c_fm_a
= ( ^ [F2: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs2: list_Epistemic_fm_a] : ( concat6109960377493484129c_fm_a @ ( map_Ep4939040189629079504c_fm_a @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_480_maps__def,axiom,
( maps_l3685687229838205249c_fm_a
= ( ^ [F2: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs2: list_l6083326122719238310c_fm_a] : ( concat6109960377493484129c_fm_a @ ( map_li3858915945532548554c_fm_a @ F2 @ Xs2 ) ) ) ) ).
% maps_def
thf(fact_481_map__eq__conv,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Xs ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_482_map__eq__conv,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= ( map_Ep4939040189629079504c_fm_a @ G5 @ Xs ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_483_map__eq__conv,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Xs ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_484_map__eq__conv,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( ( map_li3858915945532548554c_fm_a @ F @ Xs )
= ( map_li3858915945532548554c_fm_a @ G5 @ Xs ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_485_map__eq__conv,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > epistemic_fm_a] :
( ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= ( map_li5516866612190465598c_fm_a @ G5 @ Xs ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_486_map__eq__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Xs ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G5 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_487_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_488_in__set__insert,axiom,
! [X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_489_in__set__insert,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_490_in__set__insert,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( insert335162795743670844c_fm_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_491_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_492_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_493_not__in__set__insert,axiom,
! [X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ~ ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( insert1868103382419917334c_fm_a @ X @ Xs )
= ( cons_s4394448069393209776c_fm_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_494_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_495_not__in__set__insert,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ~ ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( insert335162795743670844c_fm_a @ X @ Xs )
= ( cons_l8134865115577406678c_fm_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_496_not__in__set__insert,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X @ Xs )
= ( cons_Epistemic_fm_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_497_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_498_subset__code_I1_J,axiom,
! [Xs: list_s580375451141968640c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ ( set_se8855941934996405007c_fm_a @ Xs ) @ B4 )
= ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( member536094252920883875c_fm_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_499_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ B4 )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_500_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_501_subset__code_I1_J,axiom,
! [Xs: list_l6083326122719238310c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ ( set_li8442223810127165109c_fm_a @ Xs ) @ B4 )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( member5906877432388582473c_fm_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_502_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_503_fm_Orel__mono,axiom,
! [R2: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R2 @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R2 ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_504_subset__CollectI,axiom,
! [B4: set_a,A: set_a,Q4: a > $o,P4: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a2 @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a2 @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_505_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 )
=> ( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_506_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 )
=> ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_507_subset__CollectI,axiom,
! [B4: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a,Q4: list_Epistemic_fm_a > $o,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ A )
=> ( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collec5191077796991884427c_fm_a
@ ^ [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_508_subset__CollectI,axiom,
! [B4: set_set_nat,A: set_set_nat,Q4: set_nat > $o,P4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B4 @ A )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_509_subset__CollectI,axiom,
! [B4: set_nat,A: set_nat,Q4: nat > $o,P4: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ B4 )
=> ( ( Q4 @ X3 )
=> ( P4 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ B4 )
& ( Q4 @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P4 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_510_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
@ ^ [X2: a] :
( ( member_a2 @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_511_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
@ ^ [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_512_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
@ ^ [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_513_subset__Collect__iff,axiom,
! [B4: set_li769143395467472256c_fm_a,A: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ B4 @ A )
=> ( ( ord_le2931979837777675552c_fm_a @ B4
@ ( collec5191077796991884427c_fm_a
@ ^ [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_514_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
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_515_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
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P4 @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B4 )
=> ( P4 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_516_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y4: a,Z4: a] : ( Y4 = Z4 ) )
= ( ^ [Y4: epistemic_fm_a,Z4: epistemic_fm_a] : ( Y4 = Z4 ) ) ) ).
% fm.rel_eq
thf(fact_517_fm_Orel__refl,axiom,
! [Ra: a > a > $o,X: epistemic_fm_a] :
( ! [X3: a] : ( Ra @ X3 @ X3 )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ X ) ) ).
% fm.rel_refl
thf(fact_518_list_Ocase__distrib,axiom,
! [H4: list_Epistemic_fm_a > list_Epistemic_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l8442927312758267104c_fm_a @ F1 @ F22 @ List ) )
= ( case_l8442927312758267104c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_519_list_Ocase__distrib,axiom,
! [H4: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l8442927312758267104c_fm_a @ F1 @ F22 @ List ) )
= ( case_l5707353781634818138c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_520_list_Ocase__distrib,axiom,
! [H4: list_Epistemic_fm_a > $o,F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l8442927312758267104c_fm_a @ F1 @ F22 @ List ) )
= ( case_l2021589626981272552c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_521_list_Ocase__distrib,axiom,
! [H4: list_l6083326122719238310c_fm_a > list_Epistemic_fm_a,F1: list_l6083326122719238310c_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l5707353781634818138c_fm_a @ F1 @ F22 @ List ) )
= ( case_l8442927312758267104c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_522_list_Ocase__distrib,axiom,
! [H4: list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a,F1: list_l6083326122719238310c_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l5707353781634818138c_fm_a @ F1 @ F22 @ List ) )
= ( case_l5707353781634818138c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_523_list_Ocase__distrib,axiom,
! [H4: list_l6083326122719238310c_fm_a > $o,F1: list_l6083326122719238310c_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l5707353781634818138c_fm_a @ F1 @ F22 @ List ) )
= ( case_l2021589626981272552c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_524_list_Ocase__distrib,axiom,
! [H4: $o > list_Epistemic_fm_a,F1: $o,F22: epistemic_fm_a > list_Epistemic_fm_a > $o,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l2021589626981272552c_fm_a @ F1 @ F22 @ List ) )
= ( case_l8442927312758267104c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_525_list_Ocase__distrib,axiom,
! [H4: $o > list_l6083326122719238310c_fm_a,F1: $o,F22: epistemic_fm_a > list_Epistemic_fm_a > $o,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l2021589626981272552c_fm_a @ F1 @ F22 @ List ) )
= ( case_l5707353781634818138c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_526_list_Ocase__distrib,axiom,
! [H4: $o > $o,F1: $o,F22: epistemic_fm_a > list_Epistemic_fm_a > $o,List: list_Epistemic_fm_a] :
( ( H4 @ ( case_l2021589626981272552c_fm_a @ F1 @ F22 @ List ) )
= ( case_l2021589626981272552c_fm_a @ ( H4 @ F1 )
@ ^ [X12: epistemic_fm_a,X25: list_Epistemic_fm_a] : ( H4 @ ( F22 @ X12 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_527_fm_Orel__inject_I5_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R2 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_528_fm_Orel__intros_I5_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_529_fm_Orel__inject_I6_J,axiom,
! [R2: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R2 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R2 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_530_fm_Orel__intros_I6_J,axiom,
! [R2: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R2 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_531_fm_Octr__transfer_I1_J,axiom,
! [R2: a > a > $o] : ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_532_fm_Orel__inject_I4_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R2 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_533_fm_Orel__intros_I4_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_534_fm_Orel__cong,axiom,
! [X: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R2: nat > nat > $o,Ra: nat > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: nat,Yb2: nat] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte3894023384580379906at_nat @ R2 @ X @ Y )
= ( episte3894023384580379906at_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_535_fm_Orel__cong,axiom,
! [X: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: nat > a > $o,Ra: nat > a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: nat,Yb2: a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte5492358210969815628_nat_a @ R2 @ X @ Y )
= ( episte5492358210969815628_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_536_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R2: a > nat > $o,Ra: a > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb2: nat] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte1460426709791529198_a_nat @ R2 @ X @ Y )
= ( episte1460426709791529198_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_537_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: a > a > $o,Ra: a > a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb2: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X @ Y )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_538_fm_Orel__cong,axiom,
! [X: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R2: nat > epistemic_fm_a > $o,Ra: nat > epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte9034525631817513832c_fm_a @ R2 @ X @ Y )
= ( episte9034525631817513832c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_539_fm_Orel__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R2: epistemic_fm_a > nat > $o,Ra: epistemic_fm_a > nat > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte8778020545599232650_a_nat @ R2 @ X @ Y )
= ( episte8778020545599232650_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_540_fm_Orel__cong,axiom,
! [X: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R2: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R2 @ X @ Y )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_541_fm_Orel__cong,axiom,
! [X: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R2: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R2 @ X @ Y )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_542_fm_Orel__cong,axiom,
! [X: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: episte3561180207765753404c_fm_a,Xa: episte3561180207765753404c_fm_a,R2: nat > list_Epistemic_fm_a > $o,Ra: nat > list_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: nat,Yb2: list_Epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member5906877432388582473c_fm_a @ Yb2 @ ( episte2454647923005522006c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte6321633799632875630c_fm_a @ R2 @ X @ Y )
= ( episte6321633799632875630c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_543_fm_Orel__cong,axiom,
! [X: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: episte5999154429349999510c_fm_a,Xa: episte5999154429349999510c_fm_a,R2: nat > set_Epistemic_fm_a > $o,Ra: nat > set_Epistemic_fm_a > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z: nat,Yb2: set_Epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Xa ) )
=> ( ( R2 @ Z @ Yb2 )
= ( Ra @ Z @ Yb2 ) ) ) )
=> ( ( episte6888114440546468168c_fm_a @ R2 @ X @ Y )
= ( episte6888114440546468168c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_544_fm_Orel__mono__strong,axiom,
! [R2: nat > nat > $o,X: epistemic_fm_nat,Y: epistemic_fm_nat,Ra: nat > nat > $o] :
( ( episte3894023384580379906at_nat @ R2 @ X @ Y )
=> ( ! [Z: nat,Yb2: nat] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte3894023384580379906at_nat @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_545_fm_Orel__mono__strong,axiom,
! [R2: nat > a > $o,X: epistemic_fm_nat,Y: epistemic_fm_a,Ra: nat > a > $o] :
( ( episte5492358210969815628_nat_a @ R2 @ X @ Y )
=> ( ! [Z: nat,Yb2: a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte5492358210969815628_nat_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_546_fm_Orel__mono__strong,axiom,
! [R2: a > nat > $o,X: epistemic_fm_a,Y: epistemic_fm_nat,Ra: a > nat > $o] :
( ( episte1460426709791529198_a_nat @ R2 @ X @ Y )
=> ( ! [Z: a,Yb2: nat] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte1460426709791529198_a_nat @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_547_fm_Orel__mono__strong,axiom,
! [R2: a > a > $o,X: epistemic_fm_a,Y: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R2 @ X @ Y )
=> ( ! [Z: a,Yb2: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_548_fm_Orel__mono__strong,axiom,
! [R2: nat > epistemic_fm_a > $o,X: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Ra: nat > epistemic_fm_a > $o] :
( ( episte9034525631817513832c_fm_a @ R2 @ X @ Y )
=> ( ! [Z: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte9034525631817513832c_fm_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_549_fm_Orel__mono__strong,axiom,
! [R2: epistemic_fm_a > nat > $o,X: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Ra: epistemic_fm_a > nat > $o] :
( ( episte8778020545599232650_a_nat @ R2 @ X @ Y )
=> ( ! [Z: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte8778020545599232650_a_nat @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_550_fm_Orel__mono__strong,axiom,
! [R2: epistemic_fm_a > a > $o,X: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R2 @ X @ Y )
=> ( ! [Z: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_551_fm_Orel__mono__strong,axiom,
! [R2: a > epistemic_fm_a > $o,X: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R2 @ X @ Y )
=> ( ! [Z: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_552_fm_Orel__mono__strong,axiom,
! [R2: nat > list_Epistemic_fm_a > $o,X: epistemic_fm_nat,Y: episte3561180207765753404c_fm_a,Ra: nat > list_Epistemic_fm_a > $o] :
( ( episte6321633799632875630c_fm_a @ R2 @ X @ Y )
=> ( ! [Z: nat,Yb2: list_Epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( ( member5906877432388582473c_fm_a @ Yb2 @ ( episte2454647923005522006c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte6321633799632875630c_fm_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_553_fm_Orel__mono__strong,axiom,
! [R2: nat > set_Epistemic_fm_a > $o,X: epistemic_fm_nat,Y: episte5999154429349999510c_fm_a,Ra: nat > set_Epistemic_fm_a > $o] :
( ( episte6888114440546468168c_fm_a @ R2 @ X @ Y )
=> ( ! [Z: nat,Yb2: set_Epistemic_fm_a] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( ( member536094252920883875c_fm_a @ Yb2 @ ( episte6227279731293865264c_fm_a @ Y ) )
=> ( ( R2 @ Z @ Yb2 )
=> ( Ra @ Z @ Yb2 ) ) ) )
=> ( episte6888114440546468168c_fm_a @ Ra @ X @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_554_fm_Orel__refl__strong,axiom,
! [X: epistemic_fm_nat,Ra: nat > nat > $o] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( epistemic_set_fm_nat @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( episte3894023384580379906at_nat @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_555_fm_Orel__refl__strong,axiom,
! [X: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( episte9089240958480457552c_fm_a @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_556_fm_Orel__refl__strong,axiom,
! [X: episte3561180207765753404c_fm_a,Ra: list_Epistemic_fm_a > list_Epistemic_fm_a > $o] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( episte2454647923005522006c_fm_a @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( episte7930623906115922400c_fm_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_557_fm_Orel__refl__strong,axiom,
! [X: episte5999154429349999510c_fm_a,Ra: set_Epistemic_fm_a > set_Epistemic_fm_a > $o] :
( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( episte6227279731293865264c_fm_a @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( episte8246226516352692192c_fm_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_558_fm_Orel__refl__strong,axiom,
! [X: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( epistemic_set_fm_a @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X @ X ) ) ).
% fm.rel_refl_strong
thf(fact_559_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_560_list_Oset__intros_I2_J,axiom,
! [Y: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a,X21: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ X22 ) )
=> ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_561_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_562_list_Oset__intros_I2_J,axiom,
! [Y: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a,X21: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Y @ ( set_li8442223810127165109c_fm_a @ X22 ) )
=> ( member5906877432388582473c_fm_a @ Y @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_563_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_564_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_565_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_a,X22: list_s580375451141968640c_fm_a] : ( member536094252920883875c_fm_a @ X21 @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_566_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_567_list_Oset__intros_I1_J,axiom,
! [X21: list_Epistemic_fm_a,X22: list_l6083326122719238310c_fm_a] : ( member5906877432388582473c_fm_a @ X21 @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_568_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_569_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_570_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_a,A2: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ A2 ) )
=> ( ! [Z22: list_s580375451141968640c_fm_a] :
( A2
!= ( cons_s4394448069393209776c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_a,Z22: list_s580375451141968640c_fm_a] :
( ( A2
= ( cons_s4394448069393209776c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member536094252920883875c_fm_a @ E @ ( set_se8855941934996405007c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_571_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_572_list_Oset__cases,axiom,
! [E: list_Epistemic_fm_a,A2: list_l6083326122719238310c_fm_a] :
( ( member5906877432388582473c_fm_a @ E @ ( set_li8442223810127165109c_fm_a @ A2 ) )
=> ( ! [Z22: list_l6083326122719238310c_fm_a] :
( A2
!= ( cons_l8134865115577406678c_fm_a @ E @ Z22 ) )
=> ~ ! [Z1: list_Epistemic_fm_a,Z22: list_l6083326122719238310c_fm_a] :
( ( A2
= ( cons_l8134865115577406678c_fm_a @ Z1 @ Z22 ) )
=> ~ ( member5906877432388582473c_fm_a @ E @ ( set_li8442223810127165109c_fm_a @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_573_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_574_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_575_set__ConsD,axiom,
! [Y: set_Epistemic_fm_a,X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member536094252920883875c_fm_a @ Y @ ( set_se8855941934996405007c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_576_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_577_set__ConsD,axiom,
! [Y: list_Epistemic_fm_a,X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( member5906877432388582473c_fm_a @ Y @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member5906877432388582473c_fm_a @ Y @ ( set_li8442223810127165109c_fm_a @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_578_set__ConsD,axiom,
! [Y: epistemic_fm_a,X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_579_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_580_set__subset__Cons,axiom,
! [Xs: list_l6083326122719238310c_fm_a,X: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( set_li8442223810127165109c_fm_a @ Xs ) @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_581_set__subset__Cons,axiom,
! [Xs: list_set_nat,X: set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ ( set_set_nat2 @ ( cons_set_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_582_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_a,X: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_583_fm_Orel__inject_I2_J,axiom,
! [R2: a > a > $o,X24: list_char,Y23: list_char] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.rel_inject(2)
thf(fact_584_fm_Orel__intros_I2_J,axiom,
! [X24: list_char,Y23: list_char,R2: a > a > $o] :
( ( X24 = Y23 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) ) ) ).
% fm.rel_intros(2)
thf(fact_585_fm_Orel__inject_I3_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R2 @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R2 @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_586_fm_Orel__intros_I3_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R2 @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R2 @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_587_list_Omap__cong,axiom,
! [X: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( X = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ X )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_588_list_Omap__cong,axiom,
! [X: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( X = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep4939040189629079504c_fm_a @ F @ X )
= ( map_Ep4939040189629079504c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_589_list_Omap__cong,axiom,
! [X: list_l6083326122719238310c_fm_a,Ya: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( X = Ya )
=> ( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ X )
= ( map_li3080774272885743684c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_590_list_Omap__cong,axiom,
! [X: list_l6083326122719238310c_fm_a,Ya: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( X = Ya )
=> ( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3858915945532548554c_fm_a @ F @ X )
= ( map_li3858915945532548554c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_591_list_Omap__cong,axiom,
! [X: list_l6083326122719238310c_fm_a,Ya: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,G5: list_Epistemic_fm_a > epistemic_fm_a] :
( ( X = Ya )
=> ( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li5516866612190465598c_fm_a @ F @ X )
= ( map_li5516866612190465598c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_592_list_Omap__cong,axiom,
! [X: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( X = Ya )
=> ( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_593_list_Omap__cong0,axiom,
! [X: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ X )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_594_list_Omap__cong0,axiom,
! [X: list_Epistemic_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep4939040189629079504c_fm_a @ F @ X )
= ( map_Ep4939040189629079504c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_595_list_Omap__cong0,axiom,
! [X: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ X )
= ( map_li3080774272885743684c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_596_list_Omap__cong0,axiom,
! [X: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li3858915945532548554c_fm_a @ F @ X )
= ( map_li3858915945532548554c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_597_list_Omap__cong0,axiom,
! [X: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,G5: list_Epistemic_fm_a > epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_li5516866612190465598c_fm_a @ F @ X )
= ( map_li5516866612190465598c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_598_list_Omap__cong0,axiom,
! [X: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( F @ Z )
= ( G5 @ Z ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ X ) ) ) ).
% list.map_cong0
thf(fact_599_list_Oinj__map__strong,axiom,
! [X: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,Fa: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Ep1988204514693272906c_fm_a @ F @ X )
= ( map_Ep1988204514693272906c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_600_list_Oinj__map__strong,axiom,
! [X: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Fa: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [Z: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Ep4939040189629079504c_fm_a @ F @ X )
= ( map_Ep4939040189629079504c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_601_list_Oinj__map__strong,axiom,
! [X: list_l6083326122719238310c_fm_a,Xa: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,Fa: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a,Za: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( member5906877432388582473c_fm_a @ Za @ ( set_li8442223810127165109c_fm_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li3080774272885743684c_fm_a @ F @ X )
= ( map_li3080774272885743684c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_602_list_Oinj__map__strong,axiom,
! [X: list_l6083326122719238310c_fm_a,Xa: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Fa: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [Z: list_Epistemic_fm_a,Za: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( member5906877432388582473c_fm_a @ Za @ ( set_li8442223810127165109c_fm_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li3858915945532548554c_fm_a @ F @ X )
= ( map_li3858915945532548554c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_603_list_Oinj__map__strong,axiom,
! [X: list_l6083326122719238310c_fm_a,Xa: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,Fa: list_Epistemic_fm_a > epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a,Za: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ X ) )
=> ( ( member5906877432388582473c_fm_a @ Za @ ( set_li8442223810127165109c_fm_a @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li5516866612190465598c_fm_a @ F @ X )
= ( map_li5516866612190465598c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_604_list_Oinj__map__strong,axiom,
! [X: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Fa: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ X ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Ep7084560364594560580c_fm_a @ F @ X )
= ( map_Ep7084560364594560580c_fm_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_605_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_606_list_Omap__ident__strong,axiom,
! [T: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [Z: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z @ ( set_se8855941934996405007c_fm_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_607_list_Omap__ident__strong,axiom,
! [T: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [Z: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Z @ ( set_li8442223810127165109c_fm_a @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_608_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_609_list_Omap__ident__strong,axiom,
! [T: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [Z: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z @ ( set_Epistemic_fm_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_610_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_611_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= ( map_Ep4939040189629079504c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_612_map__ext,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_613_map__ext,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li3858915945532548554c_fm_a @ F @ Xs )
= ( map_li3858915945532548554c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_614_map__ext,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,G5: list_Epistemic_fm_a > epistemic_fm_a] :
( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= ( map_li5516866612190465598c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_615_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Xs ) ) ) ).
% map_ext
thf(fact_616_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_617_map__idI,axiom,
! [Xs: list_s580375451141968640c_fm_a,F: set_Epistemic_fm_a > set_Epistemic_fm_a] :
( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( set_se8855941934996405007c_fm_a @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_se7333198834109155908c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_618_map__idI,axiom,
! [Xs: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_619_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_620_map__idI,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_621_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a,G5: epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep1988204514693272906c_fm_a @ F @ Xs )
= ( map_Ep1988204514693272906c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_622_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep4939040189629079504c_fm_a @ F @ Xs )
= ( map_Ep4939040189629079504c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_623_map__cong,axiom,
! [Xs: list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a,G5: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li3080774272885743684c_fm_a @ F @ Xs )
= ( map_li3080774272885743684c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_624_map__cong,axiom,
! [Xs: list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,G5: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li3858915945532548554c_fm_a @ F @ Xs )
= ( map_li3858915945532548554c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_625_map__cong,axiom,
! [Xs: list_l6083326122719238310c_fm_a,Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a,G5: list_Epistemic_fm_a > epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_li5516866612190465598c_fm_a @ F @ Xs )
= ( map_li5516866612190465598c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_626_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G5 @ X3 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ys ) ) ) ) ).
% map_cong
thf(fact_627_ex__map__conv,axiom,
! [Ys: list_l3671048651222332332c_fm_a,F: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( ? [Xs2: list_Epistemic_fm_a] :
( Ys
= ( map_Ep4939040189629079504c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ X2 @ ( set_li647609056391172027c_fm_a @ Ys ) )
=> ? [Y3: epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_628_ex__map__conv,axiom,
! [Ys: list_l3671048651222332332c_fm_a,F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( ? [Xs2: list_l6083326122719238310c_fm_a] :
( Ys
= ( map_li3858915945532548554c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ X2 @ ( set_li647609056391172027c_fm_a @ Ys ) )
=> ? [Y3: list_Epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_629_ex__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,F: list_Epistemic_fm_a > epistemic_fm_a] :
( ( ? [Xs2: list_l6083326122719238310c_fm_a] :
( Ys
= ( map_li5516866612190465598c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ? [Y3: list_Epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_630_ex__map__conv,axiom,
! [Ys: list_l6083326122719238310c_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( ? [Xs2: list_Epistemic_fm_a] :
( Ys
= ( map_Ep1988204514693272906c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ? [Y3: epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_631_ex__map__conv,axiom,
! [Ys: list_l6083326122719238310c_fm_a,F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( ? [Xs2: list_l6083326122719238310c_fm_a] :
( Ys
= ( map_li3080774272885743684c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Ys ) )
=> ? [Y3: list_Epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_632_ex__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ? [Xs2: list_Epistemic_fm_a] :
( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs2 ) ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ? [Y3: epistemic_fm_a] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_633_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P6: a > $o,Xs2: list_a] :
? [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
& ( P6 @ X2 )
& ! [Y3: a] :
( ( ( member_a2 @ Y3 @ ( set_a2 @ Xs2 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_634_list__ex1__iff,axiom,
( list_e3415556844334454979c_fm_a
= ( ^ [P6: set_Epistemic_fm_a > $o,Xs2: list_s580375451141968640c_fm_a] :
? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs2 ) )
& ( P6 @ X2 )
& ! [Y3: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ Y3 @ ( set_se8855941934996405007c_fm_a @ Xs2 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_635_list__ex1__iff,axiom,
( list_e1371837923195959913c_fm_a
= ( ^ [P6: list_Epistemic_fm_a > $o,Xs2: list_l6083326122719238310c_fm_a] :
? [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs2 ) )
& ( P6 @ X2 )
& ! [Y3: list_Epistemic_fm_a] :
( ( ( member5906877432388582473c_fm_a @ Y3 @ ( set_li8442223810127165109c_fm_a @ Xs2 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_636_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P6: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P6 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs2 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_637_list__ex1__iff,axiom,
( list_e2031426293596896995c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,Xs2: list_Epistemic_fm_a] :
? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
& ( P6 @ X2 )
& ! [Y3: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y3 @ ( set_Epistemic_fm_a2 @ Xs2 ) )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_638_list_Osimps_I5_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( case_l8442927312758267104c_fm_a @ F1 @ F22 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_639_list_Osimps_I5_J,axiom,
! [F1: list_l6083326122719238310c_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( case_l5707353781634818138c_fm_a @ F1 @ F22 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_640_list_Osimps_I5_J,axiom,
! [F1: $o,F22: epistemic_fm_a > list_Epistemic_fm_a > $o,X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( case_l2021589626981272552c_fm_a @ F1 @ F22 @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_641_list_Osimps_I4_J,axiom,
! [F1: list_Epistemic_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( case_l8442927312758267104c_fm_a @ F1 @ F22 @ nil_Epistemic_fm_a )
= F1 ) ).
% list.simps(4)
thf(fact_642_list_Osimps_I4_J,axiom,
! [F1: list_l6083326122719238310c_fm_a,F22: epistemic_fm_a > list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( case_l5707353781634818138c_fm_a @ F1 @ F22 @ nil_Epistemic_fm_a )
= F1 ) ).
% list.simps(4)
thf(fact_643_list_Osimps_I4_J,axiom,
! [F1: $o,F22: epistemic_fm_a > list_Epistemic_fm_a > $o] :
( ( case_l2021589626981272552c_fm_a @ F1 @ F22 @ nil_Epistemic_fm_a )
= F1 ) ).
% list.simps(4)
thf(fact_644_in__set__member,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_645_in__set__member,axiom,
! [X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
= ( member378061430901374997c_fm_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_646_in__set__member,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( member6038508265109909045c_fm_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_647_in__set__member,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
= ( member594239076476917947c_fm_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_648_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_649_fm_Orel__distinct_I30_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_650_fm_Orel__distinct_I29_J,axiom,
! [R2: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_651_fm_Orel__distinct_I7_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_652_fm_Orel__distinct_I8_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_653_fm_Orel__distinct_I9_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_654_fm_Orel__distinct_I10_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_655_fm_Orel__distinct_I26_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_656_fm_Orel__distinct_I25_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_657_fm_Orel__distinct_I28_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_658_fm_Orel__distinct_I27_J,axiom,
! [R2: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_659_fm_Orel__distinct_I16_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(16)
thf(fact_660_fm_Orel__distinct_I15_J,axiom,
! [R2: a > a > $o,X24: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_661_fm_Orel__distinct_I21_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_662_fm_Orel__distinct_I22_J,axiom,
! [R2: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_663_fm_Orel__distinct_I17_J,axiom,
! [R2: a > a > $o,X24: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_664_fm_Orel__distinct_I18_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(18)
thf(fact_665_fm_Orel__distinct_I23_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_666_fm_Orel__distinct_I24_J,axiom,
! [R2: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_667_fm_Orel__distinct_I5_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_668_fm_Orel__distinct_I6_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_669_fm_Orel__distinct_I2_J,axiom,
! [R2: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ Y23 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_670_fm_Orel__distinct_I1_J,axiom,
! [R2: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Pro_a @ Y23 ) ) ).
% fm.rel_distinct(1)
thf(fact_671_fm_Orel__distinct_I4_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_672_fm_Orel__distinct_I3_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_673_fm_Orel__distinct_I13_J,axiom,
! [R2: a > a > $o,X24: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_674_fm_Orel__distinct_I14_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(14)
thf(fact_675_fm_Orel__distinct_I19_J,axiom,
! [R2: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_676_fm_Orel__distinct_I20_J,axiom,
! [R2: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_677_fm_Orel__distinct_I12_J,axiom,
! [R2: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(12)
thf(fact_678_fm_Orel__distinct_I11_J,axiom,
! [R2: a > a > $o,X24: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R2 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_679_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_680_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X2: a,Xs2: list_a] : ( if_list_a @ ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_a @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_681_List_Oinsert__def,axiom,
( insert1868103382419917334c_fm_a
= ( ^ [X2: set_Epistemic_fm_a,Xs2: list_s580375451141968640c_fm_a] : ( if_lis3811715425919989702c_fm_a @ ( member536094252920883875c_fm_a @ X2 @ ( set_se8855941934996405007c_fm_a @ Xs2 ) ) @ Xs2 @ ( cons_s4394448069393209776c_fm_a @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_682_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_683_List_Oinsert__def,axiom,
( insert335162795743670844c_fm_a
= ( ^ [X2: list_Epistemic_fm_a,Xs2: list_l6083326122719238310c_fm_a] : ( if_lis342421823373228652c_fm_a @ ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs2 ) ) @ Xs2 @ ( cons_l8134865115577406678c_fm_a @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_684_List_Oinsert__def,axiom,
( insert177310161492556854c_fm_a
= ( ^ [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs2 ) ) @ Xs2 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_685_arg__min__list__in,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( Xs != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( arg_mi6265433823485604166_a_nat @ F @ Xs ) @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% arg_min_list_in
thf(fact_686_maps__simps_I2_J,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( maps_E7773696912818698439c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% maps_simps(2)
thf(fact_687_maps__simps_I2_J,axiom,
! [F: epistemic_fm_a > list_s580375451141968640c_fm_a] :
( ( maps_E2162433029970929185c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% maps_simps(2)
thf(fact_688_maps__simps_I2_J,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( maps_l2078986973461115323c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_Epistemic_fm_a ) ).
% maps_simps(2)
thf(fact_689_maps__simps_I2_J,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( maps_l3685687229838205249c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% maps_simps(2)
thf(fact_690_maps__simps_I2_J,axiom,
! [F: list_Epistemic_fm_a > list_s580375451141968640c_fm_a] :
( ( maps_l5051011768039730075c_fm_a @ F @ nil_li2451196919128234278c_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% maps_simps(2)
thf(fact_691_maps__simps_I2_J,axiom,
! [F: set_Epistemic_fm_a > list_Epistemic_fm_a] :
( ( maps_s4275181340481488481c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_Epistemic_fm_a ) ).
% maps_simps(2)
thf(fact_692_maps__simps_I2_J,axiom,
! [F: set_Epistemic_fm_a > list_l6083326122719238310c_fm_a] :
( ( maps_s5364767432190995175c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% maps_simps(2)
thf(fact_693_maps__simps_I2_J,axiom,
! [F: set_Epistemic_fm_a > list_s580375451141968640c_fm_a] :
( ( maps_s148483366175664193c_fm_a @ F @ nil_se102576565402781184c_fm_a )
= nil_se102576565402781184c_fm_a ) ).
% maps_simps(2)
thf(fact_694_maps__simps_I2_J,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( maps_E6072454603808916545c_fm_a @ F @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% maps_simps(2)
thf(fact_695_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs2: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs2 ) )
=> ! [G2: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs2 @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_696_transpose_Osimps_I3_J,axiom,
! [X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a,Xss2: list_l985973790606564998c_fm_a] :
( ( transp493887097527820938c_fm_a @ ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l1626872963913326262c_fm_a
@ ( cons_s4394448069393209776c_fm_a @ X
@ ( concat2941956077894552763c_fm_a
@ ( map_li1341364954853471300c_fm_a
@ ( case_l6572946392233364704c_fm_a @ nil_se102576565402781184c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_s4394448069393209776c_fm_a @ H @ nil_se102576565402781184c_fm_a ) )
@ Xss2 ) ) )
@ ( transp493887097527820938c_fm_a
@ ( cons_l1626872963913326262c_fm_a @ Xs
@ ( concat4130251116320034881c_fm_a
@ ( map_li7441020096712412618c_fm_a
@ ( case_l4953383812495590490c_fm_a @ nil_li5433889876484989702c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_l1626872963913326262c_fm_a @ T2 @ nil_li5433889876484989702c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_697_transpose_Osimps_I3_J,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Xss2: list_l3671048651222332332c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l2479501935080859996c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ X
@ ( concat6109960377493484129c_fm_a
@ ( map_li8347733892191492164c_fm_a
@ ( case_l116314299590685792c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l8134865115577406678c_fm_a @ H @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) )
@ ( transp7189280393105805488c_fm_a
@ ( cons_l2479501935080859996c_fm_a @ Xs
@ ( concat5652308878773863783c_fm_a
@ ( map_li3568483382057536586c_fm_a
@ ( case_l7601246044141707098c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l2479501935080859996c_fm_a @ T2 @ nil_li5178944226519631276c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_698_transpose_Osimps_I3_J,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_699_transpose_Oelims,axiom,
! [X: list_l985973790606564998c_fm_a,Y: list_l985973790606564998c_fm_a] :
( ( ( transp493887097527820938c_fm_a @ X )
= Y )
=> ( ( ( X = nil_li5433889876484989702c_fm_a )
=> ( Y != nil_li5433889876484989702c_fm_a ) )
=> ( ! [Xss: list_l985973790606564998c_fm_a] :
( ( X
= ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss ) )
=> ( Y
!= ( transp493887097527820938c_fm_a @ Xss ) ) )
=> ~ ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Xss: list_l985973790606564998c_fm_a] :
( ( X
= ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_l1626872963913326262c_fm_a
@ ( cons_s4394448069393209776c_fm_a @ X3
@ ( concat2941956077894552763c_fm_a
@ ( map_li1341364954853471300c_fm_a
@ ( case_l6572946392233364704c_fm_a @ nil_se102576565402781184c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_s4394448069393209776c_fm_a @ H @ nil_se102576565402781184c_fm_a ) )
@ Xss ) ) )
@ ( transp493887097527820938c_fm_a
@ ( cons_l1626872963913326262c_fm_a @ Xs3
@ ( concat4130251116320034881c_fm_a
@ ( map_li7441020096712412618c_fm_a
@ ( case_l4953383812495590490c_fm_a @ nil_li5433889876484989702c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_l1626872963913326262c_fm_a @ T2 @ nil_li5433889876484989702c_fm_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_700_transpose_Oelims,axiom,
! [X: list_l3671048651222332332c_fm_a,Y: list_l3671048651222332332c_fm_a] :
( ( ( transp7189280393105805488c_fm_a @ X )
= Y )
=> ( ( ( X = nil_li5178944226519631276c_fm_a )
=> ( Y != nil_li5178944226519631276c_fm_a ) )
=> ( ! [Xss: list_l3671048651222332332c_fm_a] :
( ( X
= ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss ) )
=> ( Y
!= ( transp7189280393105805488c_fm_a @ Xss ) ) )
=> ~ ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Xss: list_l3671048651222332332c_fm_a] :
( ( X
= ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_l2479501935080859996c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ X3
@ ( concat6109960377493484129c_fm_a
@ ( map_li8347733892191492164c_fm_a
@ ( case_l116314299590685792c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l8134865115577406678c_fm_a @ H @ nil_li2451196919128234278c_fm_a ) )
@ Xss ) ) )
@ ( transp7189280393105805488c_fm_a
@ ( cons_l2479501935080859996c_fm_a @ Xs3
@ ( concat5652308878773863783c_fm_a
@ ( map_li3568483382057536586c_fm_a
@ ( case_l7601246044141707098c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l2479501935080859996c_fm_a @ T2 @ nil_li5178944226519631276c_fm_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_701_transpose_Oelims,axiom,
! [X: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ X )
= Y )
=> ( ( ( X = nil_li2451196919128234278c_fm_a )
=> ( Y != nil_li2451196919128234278c_fm_a ) )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( ( X
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ( Y
!= ( transp2882070860200649898c_fm_a @ Xss ) ) )
=> ~ ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( X
= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X3
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs3
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_702_can__select__set__list__ex1,axiom,
! [P4: list_Epistemic_fm_a > $o,A: list_l6083326122719238310c_fm_a] :
( ( can_se7491264451046533021c_fm_a @ P4 @ ( set_li8442223810127165109c_fm_a @ A ) )
= ( list_e1371837923195959913c_fm_a @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_703_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_704_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_705_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a2 @ X2 @ A3 )
@ ^ [X2: a] : ( member_a2 @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_706_less__eq__set__def,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B5: set_se5208064806568342746c_fm_a] :
( ord_le1568123225352354019fm_a_o
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ A3 )
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_707_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ A3 )
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_708_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat2 @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_709_less__eq__set__def,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B5: set_li769143395467472256c_fm_a] :
( ord_le5289359451089373629fm_a_o
@ ^ [X2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X2 @ A3 )
@ ^ [X2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_710_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A3 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_711_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a2 @ X2 @ R2 )
@ ^ [X2: a] : ( member_a2 @ X2 @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_712_pred__subset__eq,axiom,
! [R2: set_se5208064806568342746c_fm_a,S: set_se5208064806568342746c_fm_a] :
( ( ord_le1568123225352354019fm_a_o
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ R2 )
@ ^ [X2: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X2 @ S ) )
= ( ord_le7112219575281605754c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_713_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat2 @ X2 @ R2 )
@ ^ [X2: nat] : ( member_nat2 @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_714_pred__subset__eq,axiom,
! [R2: set_li769143395467472256c_fm_a,S: set_li769143395467472256c_fm_a] :
( ( ord_le5289359451089373629fm_a_o
@ ^ [X2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X2 @ R2 )
@ ^ [X2: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ X2 @ S ) )
= ( ord_le2931979837777675552c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_715_pred__subset__eq,axiom,
! [R2: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ R2 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ S ) )
= ( ord_le6893508408891458716et_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_716_pred__subset__eq,axiom,
! [R2: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ R2 )
@ ^ [X2: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X2 @ S ) )
= ( ord_le3275665582123262618c_fm_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_717_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_718_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_719_subset__antisym,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B4 )
=> ( ( ord_le2931979837777675552c_fm_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_720_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_721_subsetI,axiom,
! [A: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ A )
=> ( member_a2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% subsetI
thf(fact_722_subsetI,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( member536094252920883875c_fm_a @ X3 @ B4 ) )
=> ( ord_le7112219575281605754c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_723_subsetI,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( member6642669571620171971c_fm_a @ X3 @ B4 ) )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_724_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( member_nat2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_725_subsetI,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ! [X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ A )
=> ( member5906877432388582473c_fm_a @ X3 @ B4 ) )
=> ( ord_le2931979837777675552c_fm_a @ A @ B4 ) ) ).
% subsetI
thf(fact_726_subsetI,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_727_the__elem__set,axiom,
! [X: set_Epistemic_fm_a] :
( ( the_el1285646365920873327c_fm_a @ ( set_se8855941934996405007c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X @ nil_se102576565402781184c_fm_a ) ) )
= X ) ).
% the_elem_set
thf(fact_728_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_729_the__elem__set,axiom,
! [X: list_Epistemic_fm_a] :
( ( the_el8584007648813340437c_fm_a @ ( set_li8442223810127165109c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ nil_li2451196919128234278c_fm_a ) ) )
= X ) ).
% the_elem_set
thf(fact_730_the__elem__set,axiom,
! [X: epistemic_fm_a] :
( ( the_el2173195877760541071c_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) )
= X ) ).
% the_elem_set
thf(fact_731_concat__eq__Nil__conv,axiom,
! [Xss2: list_l3671048651222332332c_fm_a] :
( ( ( concat6109960377493484129c_fm_a @ Xss2 )
= nil_li2451196919128234278c_fm_a )
= ( ! [X2: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ X2 @ ( set_li647609056391172027c_fm_a @ Xss2 ) )
=> ( X2 = nil_li2451196919128234278c_fm_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_732_concat__eq__Nil__conv,axiom,
! [Xss2: list_l985973790606564998c_fm_a] :
( ( ( concat2941956077894552763c_fm_a @ Xss2 )
= nil_se102576565402781184c_fm_a )
= ( ! [X2: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X2 @ ( set_li3422124509070804117c_fm_a @ Xss2 ) )
=> ( X2 = nil_se102576565402781184c_fm_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_733_concat__eq__Nil__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( ( concat2780896542826320603c_fm_a @ Xss2 )
= nil_Epistemic_fm_a )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X2 = nil_Epistemic_fm_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_734_Nil__eq__concat__conv,axiom,
! [Xss2: list_l3671048651222332332c_fm_a] :
( ( nil_li2451196919128234278c_fm_a
= ( concat6109960377493484129c_fm_a @ Xss2 ) )
= ( ! [X2: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ X2 @ ( set_li647609056391172027c_fm_a @ Xss2 ) )
=> ( X2 = nil_li2451196919128234278c_fm_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_735_Nil__eq__concat__conv,axiom,
! [Xss2: list_l985973790606564998c_fm_a] :
( ( nil_se102576565402781184c_fm_a
= ( concat2941956077894552763c_fm_a @ Xss2 ) )
= ( ! [X2: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X2 @ ( set_li3422124509070804117c_fm_a @ Xss2 ) )
=> ( X2 = nil_se102576565402781184c_fm_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_736_Nil__eq__concat__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( nil_Epistemic_fm_a
= ( concat2780896542826320603c_fm_a @ Xss2 ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X2 = nil_Epistemic_fm_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_737_can__select__def,axiom,
( can_select_nat
= ( ^ [P6: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
& ( P6 @ X2 )
& ! [Y3: nat] :
( ( ( member_nat2 @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_738_can__select__def,axiom,
( can_select_a
= ( ^ [P6: a > $o,A3: set_a] :
? [X2: a] :
( ( member_a2 @ X2 @ A3 )
& ( P6 @ X2 )
& ! [Y3: a] :
( ( ( member_a2 @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_739_can__select__def,axiom,
( can_se5173380710277125655c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,A3: set_Epistemic_fm_a] :
? [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
& ( P6 @ X2 )
& ! [Y3: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_740_can__select__def,axiom,
( can_se7491264451046533021c_fm_a
= ( ^ [P6: list_Epistemic_fm_a > $o,A3: set_li769143395467472256c_fm_a] :
? [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A3 )
& ( P6 @ X2 )
& ! [Y3: list_Epistemic_fm_a] :
( ( ( member5906877432388582473c_fm_a @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_741_can__select__def,axiom,
( can_se8336582597686632439c_fm_a
= ( ^ [P6: set_Epistemic_fm_a > $o,A3: set_se5208064806568342746c_fm_a] :
? [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
& ( P6 @ X2 )
& ! [Y3: set_Epistemic_fm_a] :
( ( ( member536094252920883875c_fm_a @ Y3 @ A3 )
& ( P6 @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_742_transpose__empty,axiom,
! [Xs: list_l3671048651222332332c_fm_a] :
( ( ( transp7189280393105805488c_fm_a @ Xs )
= nil_li5178944226519631276c_fm_a )
= ( ! [X2: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ X2 @ ( set_li647609056391172027c_fm_a @ Xs ) )
=> ( X2 = nil_li2451196919128234278c_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_743_transpose__empty,axiom,
! [Xs: list_l985973790606564998c_fm_a] :
( ( ( transp493887097527820938c_fm_a @ Xs )
= nil_li5433889876484989702c_fm_a )
= ( ! [X2: list_s580375451141968640c_fm_a] :
( ( member2328277852289759785c_fm_a @ X2 @ ( set_li3422124509070804117c_fm_a @ Xs ) )
=> ( X2 = nil_se102576565402781184c_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_744_transpose__empty,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( ! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( X2 = nil_Epistemic_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_745_transpose_Osimps_I1_J,axiom,
( ( transp2882070860200649898c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ).
% transpose.simps(1)
thf(fact_746_list_Odisc__eq__case_I2_J,axiom,
! [List: list_l6083326122719238310c_fm_a] :
( ( List != nil_li2451196919128234278c_fm_a )
= ( case_l3589819172248781038c_fm_a @ $false
@ ^ [Uu2: list_Epistemic_fm_a,Uv2: list_l6083326122719238310c_fm_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_747_list_Odisc__eq__case_I2_J,axiom,
! [List: list_s580375451141968640c_fm_a] :
( ( List != nil_se102576565402781184c_fm_a )
= ( case_l4166027460215902664c_fm_a @ $false
@ ^ [Uu2: set_Epistemic_fm_a,Uv2: list_s580375451141968640c_fm_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_748_list_Odisc__eq__case_I2_J,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
= ( case_l2021589626981272552c_fm_a @ $false
@ ^ [Uu2: epistemic_fm_a,Uv2: list_Epistemic_fm_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_749_list_Odisc__eq__case_I1_J,axiom,
! [List: list_l6083326122719238310c_fm_a] :
( ( List = nil_li2451196919128234278c_fm_a )
= ( case_l3589819172248781038c_fm_a @ $true
@ ^ [Uu2: list_Epistemic_fm_a,Uv2: list_l6083326122719238310c_fm_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_750_list_Odisc__eq__case_I1_J,axiom,
! [List: list_s580375451141968640c_fm_a] :
( ( List = nil_se102576565402781184c_fm_a )
= ( case_l4166027460215902664c_fm_a @ $true
@ ^ [Uu2: set_Epistemic_fm_a,Uv2: list_s580375451141968640c_fm_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_751_list_Odisc__eq__case_I1_J,axiom,
! [List: list_Epistemic_fm_a] :
( ( List = nil_Epistemic_fm_a )
= ( case_l2021589626981272552c_fm_a @ $true
@ ^ [Uu2: epistemic_fm_a,Uv2: list_Epistemic_fm_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_752_transpose_Osimps_I2_J,axiom,
! [Xss2: list_l3671048651222332332c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss2 ) )
= ( transp7189280393105805488c_fm_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_753_transpose_Osimps_I2_J,axiom,
! [Xss2: list_l985973790606564998c_fm_a] :
( ( transp493887097527820938c_fm_a @ ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss2 ) )
= ( transp493887097527820938c_fm_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_754_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_755_transpose__map__map,axiom,
! [F: list_Epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ Xs ) )
= ( map_li8347733892191492164c_fm_a @ ( map_li3080774272885743684c_fm_a @ F ) @ ( transp7189280393105805488c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_756_transpose__map__map,axiom,
! [F: list_Epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( transp9220707655006416950c_fm_a @ ( map_li3568483382057536586c_fm_a @ ( map_li3858915945532548554c_fm_a @ F ) @ Xs ) )
= ( map_li3568483382057536586c_fm_a @ ( map_li3858915945532548554c_fm_a @ F ) @ ( transp7189280393105805488c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_757_transpose__map__map,axiom,
! [F: epistemic_fm_a > list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( transp9220707655006416950c_fm_a @ ( map_li5567427098789011152c_fm_a @ ( map_Ep4939040189629079504c_fm_a @ F ) @ Xs ) )
= ( map_li5567427098789011152c_fm_a @ ( map_Ep4939040189629079504c_fm_a @ F ) @ ( transp2882070860200649898c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_758_transpose__map__map,axiom,
! [F: epistemic_fm_a > list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( transp7189280393105805488c_fm_a @ ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ Xs ) )
= ( map_li3858915945532548554c_fm_a @ ( map_Ep1988204514693272906c_fm_a @ F ) @ ( transp2882070860200649898c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_759_transpose__map__map,axiom,
! [F: list_Epistemic_fm_a > epistemic_fm_a,Xs: list_l3671048651222332332c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( map_li8743910666895923134c_fm_a @ ( map_li5516866612190465598c_fm_a @ F ) @ Xs ) )
= ( map_li8743910666895923134c_fm_a @ ( map_li5516866612190465598c_fm_a @ F ) @ ( transp7189280393105805488c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_760_transpose__map__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ Xs ) )
= ( map_li3080774272885743684c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( transp2882070860200649898c_fm_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_761_in__mono,axiom,
! [A: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( member_a2 @ X @ A )
=> ( member_a2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_762_in__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,X: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( member536094252920883875c_fm_a @ X @ A )
=> ( member536094252920883875c_fm_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_763_in__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,X: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ( member6642669571620171971c_fm_a @ X @ A )
=> ( member6642669571620171971c_fm_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_764_in__mono,axiom,
! [A: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat2 @ X @ A )
=> ( member_nat2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_765_in__mono,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a,X: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B4 )
=> ( ( member5906877432388582473c_fm_a @ X @ A )
=> ( member5906877432388582473c_fm_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_766_in__mono,axiom,
! [A: set_set_nat,B4: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_767_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_768_subsetD,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ( member536094252920883875c_fm_a @ C @ A )
=> ( member536094252920883875c_fm_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_769_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_770_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_771_subsetD,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a,C: list_Epistemic_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B4 )
=> ( ( member5906877432388582473c_fm_a @ C @ A )
=> ( member5906877432388582473c_fm_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_772_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_773_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_774_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_775_equalityE,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( A = B4 )
=> ~ ( ( ord_le2931979837777675552c_fm_a @ A @ B4 )
=> ~ ( ord_le2931979837777675552c_fm_a @ B4 @ A ) ) ) ).
% equalityE
thf(fact_776_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_777_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [X2: a] :
( ( member_a2 @ X2 @ A3 )
=> ( member_a2 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_778_subset__eq,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B5: set_se5208064806568342746c_fm_a] :
! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A3 )
=> ( member536094252920883875c_fm_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_779_subset__eq,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A3 )
=> ( member6642669571620171971c_fm_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_780_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
=> ( member_nat2 @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_781_subset__eq,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B5: set_li769143395467472256c_fm_a] :
! [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A3 )
=> ( member5906877432388582473c_fm_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_782_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( member_set_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_783_equalityD1,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ A @ B4 ) ) ).
% equalityD1
thf(fact_784_equalityD1,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_785_equalityD1,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( A = B4 )
=> ( ord_le2931979837777675552c_fm_a @ A @ B4 ) ) ).
% equalityD1
thf(fact_786_equalityD1,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A = B4 )
=> ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).
% equalityD1
thf(fact_787_equalityD2,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( A = B4 )
=> ( ord_le3275665582123262618c_fm_a @ B4 @ A ) ) ).
% equalityD2
thf(fact_788_equalityD2,axiom,
! [A: set_nat,B4: set_nat] :
( ( A = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% equalityD2
thf(fact_789_equalityD2,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a] :
( ( A = B4 )
=> ( ord_le2931979837777675552c_fm_a @ B4 @ A ) ) ).
% equalityD2
thf(fact_790_equalityD2,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A = B4 )
=> ( ord_le6893508408891458716et_nat @ B4 @ A ) ) ).
% equalityD2
thf(fact_791_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [T2: a] :
( ( member_a2 @ T2 @ A3 )
=> ( member_a2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_792_subset__iff,axiom,
( ord_le7112219575281605754c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a,B5: set_se5208064806568342746c_fm_a] :
! [T2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ T2 @ A3 )
=> ( member536094252920883875c_fm_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_793_subset__iff,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
! [T2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ T2 @ A3 )
=> ( member6642669571620171971c_fm_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_794_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat2 @ T2 @ A3 )
=> ( member_nat2 @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_795_subset__iff,axiom,
( ord_le2931979837777675552c_fm_a
= ( ^ [A3: set_li769143395467472256c_fm_a,B5: set_li769143395467472256c_fm_a] :
! [T2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ T2 @ A3 )
=> ( member5906877432388582473c_fm_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_796_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A3 )
=> ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_797_subset__refl,axiom,
! [A: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_798_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_799_subset__refl,axiom,
! [A: set_li769143395467472256c_fm_a] : ( ord_le2931979837777675552c_fm_a @ A @ A ) ).
% subset_refl
thf(fact_800_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_801_Collect__mono,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le7112219575281605754c_fm_a @ ( collec2519470961442302949c_fm_a @ P4 ) @ ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_802_Collect__mono,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X3: epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( collec4904205152690461189c_fm_a @ P4 ) @ ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_803_Collect__mono,axiom,
! [P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ! [X3: list_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q4 ) ) ) ).
% Collect_mono
thf(fact_804_Collect__mono,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ! [X3: set_nat] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P4 ) @ ( collect_set_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_805_Collect__mono,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ! [X3: nat] :
( ( P4 @ X3 )
=> ( Q4 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_806_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_807_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_808_subset__trans,axiom,
! [A: set_li769143395467472256c_fm_a,B4: set_li769143395467472256c_fm_a,C2: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A @ B4 )
=> ( ( ord_le2931979837777675552c_fm_a @ B4 @ C2 )
=> ( ord_le2931979837777675552c_fm_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_809_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_810_set__eq__subset,axiom,
( ( ^ [Y4: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] : ( Y4 = Z4 ) )
= ( ^ [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_811_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
= ( ^ [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_812_set__eq__subset,axiom,
( ( ^ [Y4: set_li769143395467472256c_fm_a,Z4: set_li769143395467472256c_fm_a] : ( Y4 = Z4 ) )
= ( ^ [A3: set_li769143395467472256c_fm_a,B5: set_li769143395467472256c_fm_a] :
( ( ord_le2931979837777675552c_fm_a @ A3 @ B5 )
& ( ord_le2931979837777675552c_fm_a @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_813_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z4: set_set_nat] : ( Y4 = Z4 ) )
= ( ^ [A3: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_814_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 ) )
= ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_815_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 ) )
= ( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_816_Collect__mono__iff,axiom,
! [P4: list_Epistemic_fm_a > $o,Q4: list_Epistemic_fm_a > $o] :
( ( ord_le2931979837777675552c_fm_a @ ( collec5191077796991884427c_fm_a @ P4 ) @ ( collec5191077796991884427c_fm_a @ Q4 ) )
= ( ! [X2: list_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_817_Collect__mono__iff,axiom,
! [P4: set_nat > $o,Q4: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P4 ) @ ( collect_set_nat @ Q4 ) )
= ( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_818_Collect__mono__iff,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) )
= ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_819_Collect__subset,axiom,
! [A: set_a,P4: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a2 @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_820_Collect__subset,axiom,
! [A: set_se5208064806568342746c_fm_a,P4: set_Epistemic_fm_a > $o] :
( ord_le7112219575281605754c_fm_a
@ ( collec2519470961442302949c_fm_a
@ ^ [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_821_Collect__subset,axiom,
! [A: set_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ord_le3275665582123262618c_fm_a
@ ( collec4904205152690461189c_fm_a
@ ^ [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_822_Collect__subset,axiom,
! [A: set_li769143395467472256c_fm_a,P4: list_Epistemic_fm_a > $o] :
( ord_le2931979837777675552c_fm_a
@ ( collec5191077796991884427c_fm_a
@ ^ [X2: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_823_Collect__subset,axiom,
! [A: set_set_nat,P4: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_824_Collect__subset,axiom,
! [A: set_nat,P4: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P4 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_825_transpose_Opsimps_I3_J,axiom,
! [X: set_Epistemic_fm_a,Xs: list_s580375451141968640c_fm_a,Xss2: list_l985973790606564998c_fm_a] :
( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X @ Xs ) @ Xss2 ) )
=> ( ( transp493887097527820938c_fm_a @ ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l1626872963913326262c_fm_a
@ ( cons_s4394448069393209776c_fm_a @ X
@ ( concat2941956077894552763c_fm_a
@ ( map_li1341364954853471300c_fm_a
@ ( case_l6572946392233364704c_fm_a @ nil_se102576565402781184c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_s4394448069393209776c_fm_a @ H @ nil_se102576565402781184c_fm_a ) )
@ Xss2 ) ) )
@ ( transp493887097527820938c_fm_a
@ ( cons_l1626872963913326262c_fm_a @ Xs
@ ( concat4130251116320034881c_fm_a
@ ( map_li7441020096712412618c_fm_a
@ ( case_l4953383812495590490c_fm_a @ nil_li5433889876484989702c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_l1626872963913326262c_fm_a @ T2 @ nil_li5433889876484989702c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_826_transpose_Opsimps_I3_J,axiom,
! [X: list_Epistemic_fm_a,Xs: list_l6083326122719238310c_fm_a,Xss2: list_l3671048651222332332c_fm_a] :
( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) @ Xss2 ) )
=> ( ( transp7189280393105805488c_fm_a @ ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l2479501935080859996c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ X
@ ( concat6109960377493484129c_fm_a
@ ( map_li8347733892191492164c_fm_a
@ ( case_l116314299590685792c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l8134865115577406678c_fm_a @ H @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) )
@ ( transp7189280393105805488c_fm_a
@ ( cons_l2479501935080859996c_fm_a @ Xs
@ ( concat5652308878773863783c_fm_a
@ ( map_li3568483382057536586c_fm_a
@ ( case_l7601246044141707098c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l2479501935080859996c_fm_a @ T2 @ nil_li5178944226519631276c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_827_transpose_Opsimps_I3_J,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Xss2 ) )
=> ( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Xss2 ) )
= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X
@ ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) ) )
@ ( transp2882070860200649898c_fm_a
@ ( cons_l8134865115577406678c_fm_a @ Xs
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_828_strong__soundness_092_060_094sub_062K,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_829_transpose_Opinduct,axiom,
! [A0: list_l985973790606564998c_fm_a,P4: list_l985973790606564998c_fm_a > $o] :
( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ A0 )
=> ( ( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ nil_li5433889876484989702c_fm_a )
=> ( P4 @ nil_li5433889876484989702c_fm_a ) )
=> ( ! [Xss: list_l985973790606564998c_fm_a] :
( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss ) )
=> ( ( P4 @ Xss )
=> ( P4 @ ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss ) ) ) )
=> ( ! [X3: set_Epistemic_fm_a,Xs3: list_s580375451141968640c_fm_a,Xss: list_l985973790606564998c_fm_a] :
( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( ( P4
@ ( cons_l1626872963913326262c_fm_a @ Xs3
@ ( concat4130251116320034881c_fm_a
@ ( map_li7441020096712412618c_fm_a
@ ( case_l4953383812495590490c_fm_a @ nil_li5433889876484989702c_fm_a
@ ^ [H: set_Epistemic_fm_a,T2: list_s580375451141968640c_fm_a] : ( cons_l1626872963913326262c_fm_a @ T2 @ nil_li5433889876484989702c_fm_a ) )
@ Xss ) ) ) )
=> ( P4 @ ( cons_l1626872963913326262c_fm_a @ ( cons_s4394448069393209776c_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) )
=> ( P4 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_830_transpose_Opinduct,axiom,
! [A0: list_l3671048651222332332c_fm_a,P4: list_l3671048651222332332c_fm_a > $o] :
( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ A0 )
=> ( ( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ nil_li5178944226519631276c_fm_a )
=> ( P4 @ nil_li5178944226519631276c_fm_a ) )
=> ( ! [Xss: list_l3671048651222332332c_fm_a] :
( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss ) )
=> ( ( P4 @ Xss )
=> ( P4 @ ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss ) ) ) )
=> ( ! [X3: list_Epistemic_fm_a,Xs3: list_l6083326122719238310c_fm_a,Xss: list_l3671048651222332332c_fm_a] :
( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( ( P4
@ ( cons_l2479501935080859996c_fm_a @ Xs3
@ ( concat5652308878773863783c_fm_a
@ ( map_li3568483382057536586c_fm_a
@ ( case_l7601246044141707098c_fm_a @ nil_li5178944226519631276c_fm_a
@ ^ [H: list_Epistemic_fm_a,T2: list_l6083326122719238310c_fm_a] : ( cons_l2479501935080859996c_fm_a @ T2 @ nil_li5178944226519631276c_fm_a ) )
@ Xss ) ) ) )
=> ( P4 @ ( cons_l2479501935080859996c_fm_a @ ( cons_l8134865115577406678c_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) )
=> ( P4 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_831_transpose_Opinduct,axiom,
! [A0: list_l6083326122719238310c_fm_a,P4: list_l6083326122719238310c_fm_a > $o] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ A0 )
=> ( ( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( P4 @ nil_li2451196919128234278c_fm_a ) )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ( ( P4 @ Xss )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) ) ) )
=> ( ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ Xss ) )
=> ( ( P4
@ ( cons_l8134865115577406678c_fm_a @ Xs3
@ ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss ) ) ) )
=> ( P4 @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ Xss ) ) ) )
=> ( P4 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_832_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ W2 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte8765170747386058258t_unit] :
( ( P4 @ M2 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte3911673118586344362_a_nat @ M2 @ X4 @ Xa2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_833_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte295617885132580260cs_a_a @ M2 @ X4 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_834_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte5333283044364550848c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_835_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1905851639235817760t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1905851639235817760t_unit,W2: list_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member5906877432388582473c_fm_a @ W2 @ ( episte3604461938431221107t_unit @ M3 ) )
=> ( episte3437425354345704390c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1905851639235817760t_unit] :
( ( P4 @ M2 )
=> ! [X4: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X4 @ ( episte3604461938431221107t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte3437425354345704390c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte3437425354345704390c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_836_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_837_subseqs_Osimps_I1_J,axiom,
( ( subseq3170398737635186805c_fm_a @ nil_li2451196919128234278c_fm_a )
= ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ nil_li5178944226519631276c_fm_a ) ) ).
% subseqs.simps(1)
thf(fact_838_subseqs_Osimps_I1_J,axiom,
( ( subseq7171631445414591695c_fm_a @ nil_se102576565402781184c_fm_a )
= ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ nil_li5433889876484989702c_fm_a ) ) ).
% subseqs.simps(1)
thf(fact_839_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_840_arg__min__list_Oelims,axiom,
! [X: set_Epistemic_fm_a > nat,Xa: list_s580375451141968640c_fm_a,Y: set_Epistemic_fm_a] :
( ( ( arg_mi7717146594641298214_a_nat @ X @ Xa )
= Y )
=> ( ! [X3: set_Epistemic_fm_a] :
( ( Xa
= ( cons_s4394448069393209776c_fm_a @ X3 @ nil_se102576565402781184c_fm_a ) )
=> ( Y != X3 ) )
=> ( ! [X3: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Zs: list_s580375451141968640c_fm_a] :
( ( Xa
= ( cons_s4394448069393209776c_fm_a @ X3 @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Zs ) ) )
=> ( Y
!= ( if_set2375185401397246656c_fm_a @ ( ord_less_eq_nat @ ( X @ X3 ) @ ( X @ ( arg_mi7717146594641298214_a_nat @ X @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Zs ) ) ) ) @ X3 @ ( arg_mi7717146594641298214_a_nat @ X @ ( cons_s4394448069393209776c_fm_a @ Y2 @ Zs ) ) ) ) )
=> ~ ( ( Xa = nil_se102576565402781184c_fm_a )
=> ( Y != undefi3292767070061752674c_fm_a ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_841_arg__min__list_Oelims,axiom,
! [X: list_Epistemic_fm_a > nat,Xa: list_l6083326122719238310c_fm_a,Y: list_Epistemic_fm_a] :
( ( ( arg_mi474574138286535628_a_nat @ X @ Xa )
= Y )
=> ( ! [X3: list_Epistemic_fm_a] :
( ( Xa
= ( cons_l8134865115577406678c_fm_a @ X3 @ nil_li2451196919128234278c_fm_a ) )
=> ( Y != X3 ) )
=> ( ! [X3: list_Epistemic_fm_a,Y2: list_Epistemic_fm_a,Zs: list_l6083326122719238310c_fm_a] :
( ( Xa
= ( cons_l8134865115577406678c_fm_a @ X3 @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Zs ) ) )
=> ( Y
!= ( if_lis2878681784746929638c_fm_a @ ( ord_less_eq_nat @ ( X @ X3 ) @ ( X @ ( arg_mi474574138286535628_a_nat @ X @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Zs ) ) ) ) @ X3 @ ( arg_mi474574138286535628_a_nat @ X @ ( cons_l8134865115577406678c_fm_a @ Y2 @ Zs ) ) ) ) )
=> ~ ( ( Xa = nil_li2451196919128234278c_fm_a )
=> ( Y != undefi2327196743408399240c_fm_a ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_842_arg__min__list_Oelims,axiom,
! [X: epistemic_fm_a > nat,Xa: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( ( arg_mi6265433823485604166_a_nat @ X @ Xa )
= Y )
=> ( ! [X3: epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
=> ( Y != X3 ) )
=> ( ! [X3: epistemic_fm_a,Y2: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Zs ) ) )
=> ( Y
!= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( X @ X3 ) @ ( X @ ( arg_mi6265433823485604166_a_nat @ X @ ( cons_Epistemic_fm_a @ Y2 @ Zs ) ) ) ) @ X3 @ ( arg_mi6265433823485604166_a_nat @ X @ ( cons_Epistemic_fm_a @ Y2 @ Zs ) ) ) ) )
=> ~ ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != undefi6158949259153642370c_fm_a ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_843_subseqs__refl,axiom,
! [Xs: list_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ Xs @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ).
% subseqs_refl
thf(fact_844_transpose_Opsimps_I1_J,axiom,
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ nil_li2451196919128234278c_fm_a )
=> ( ( transp2882070860200649898c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_li2451196919128234278c_fm_a ) ) ).
% transpose.psimps(1)
thf(fact_845_Cons__in__subseqsD,axiom,
! [Y: list_Epistemic_fm_a,Ys: list_l6083326122719238310c_fm_a,Xs: list_l6083326122719238310c_fm_a] :
( ( member7465314387053329231c_fm_a @ ( cons_l8134865115577406678c_fm_a @ Y @ Ys ) @ ( set_li647609056391172027c_fm_a @ ( subseq3170398737635186805c_fm_a @ Xs ) ) )
=> ( member7465314387053329231c_fm_a @ Ys @ ( set_li647609056391172027c_fm_a @ ( subseq3170398737635186805c_fm_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_846_Cons__in__subseqsD,axiom,
! [Y: epistemic_fm_a,Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Y @ Ys ) @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) )
=> ( member5906877432388582473c_fm_a @ Ys @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_847_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_l3671048651222332332c_fm_a] :
( ( accp_l8971127668057225909c_fm_a @ transp5805013372743109973c_fm_a @ ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss2 ) )
=> ( ( transp7189280393105805488c_fm_a @ ( cons_l2479501935080859996c_fm_a @ nil_li2451196919128234278c_fm_a @ Xss2 ) )
= ( transp7189280393105805488c_fm_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_848_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_l985973790606564998c_fm_a] :
( ( accp_l1289336314643896847c_fm_a @ transp578517656548729263c_fm_a @ ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss2 ) )
=> ( ( transp493887097527820938c_fm_a @ ( cons_l1626872963913326262c_fm_a @ nil_se102576565402781184c_fm_a @ Xss2 ) )
= ( transp493887097527820938c_fm_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_849_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
=> ( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss2 ) )
= ( transp2882070860200649898c_fm_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_850_generalization,axiom,
! [P: epistemic_fm_a,W: nat,M: episte8765170747386058258t_unit,I: a] :
( ! [M3: episte8765170747386058258t_unit,X3: nat] :
( ( member_nat2 @ X3 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X3 @ P ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_851_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I: a] :
( ! [M3: episte6182337868402532512t_unit,X3: a] :
( ( member_a2 @ X3 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X3 @ P ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_852_generalization,axiom,
! [P: epistemic_fm_a,W: epistemic_fm_a,M: episte94448284482925344t_unit,I: a] :
( ! [M3: episte94448284482925344t_unit,X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X3 @ P ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_853_generalization,axiom,
! [P: epistemic_fm_a,W: list_Epistemic_fm_a,M: episte1905851639235817760t_unit,I: a] :
( ! [M3: episte1905851639235817760t_unit,X3: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X3 @ ( episte3604461938431221107t_unit @ M3 ) )
=> ( episte3437425354345704390c_fm_a @ M3 @ X3 @ P ) )
=> ( ( member5906877432388582473c_fm_a @ W @ ( episte3604461938431221107t_unit @ M ) )
=> ( episte3437425354345704390c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_854_generalization,axiom,
! [P: epistemic_fm_a,W: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I: a] :
( ! [M3: episte1560738328020401952t_unit,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_855_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ! [M3: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_856_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_857_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ 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_858_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1905851639235817760t_unit > $o,P: epistemic_fm_a,M: episte1905851639235817760t_unit,W: list_Epistemic_fm_a] :
( ! [M3: episte1905851639235817760t_unit,W2: list_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member5906877432388582473c_fm_a @ W2 @ ( episte3604461938431221107t_unit @ M3 ) )
=> ( episte3437425354345704390c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member5906877432388582473c_fm_a @ W @ ( episte3604461938431221107t_unit @ M ) )
=> ( episte3437425354345704390c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_859_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ 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_860_main_092_060_094sub_062K,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_861_strong__completeness_092_060_094sub_062K,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_862_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte8765170747386058258t_unit] :
( ( P4 @ M2 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte3911673118586344362_a_nat @ M2 @ X4 @ Xa2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_863_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte6182337868402532512t_unit] :
( ( P4 @ M2 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X4 @ Xa2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_864_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte94448284482925344t_unit] :
( ( P4 @ M2 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_865_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1905851639235817760t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1905851639235817760t_unit,W2: list_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member5906877432388582473c_fm_a @ W2 @ ( episte3604461938431221107t_unit @ M3 ) )
=> ( episte3437425354345704390c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte1905851639235817760t_unit] :
( ( P4 @ M2 )
=> ! [X4: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X4 @ ( episte3604461938431221107t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte3437425354345704390c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte3437425354345704390c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_866_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M3 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_867_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_868_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_869_strong__soundness_092_060_094sub_062T,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_870_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) )
=> ! [M2: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M2 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_871_accp__subset,axiom,
! [R1: list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a > $o,R22: list_l6083326122719238310c_fm_a > list_l6083326122719238310c_fm_a > $o] :
( ( ord_le7821966774285193230fm_a_o @ R1 @ R22 )
=> ( ord_le2363486873338524983fm_a_o @ ( accp_l7472382826029824047c_fm_a @ R22 ) @ ( accp_l7472382826029824047c_fm_a @ R1 ) ) ) ).
% accp_subset
thf(fact_872_accp__subset,axiom,
! [R1: epistemic_fm_a > epistemic_fm_a > $o,R22: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3934200179093585166fm_a_o @ R1 @ R22 )
=> ( ord_le4043730696559282883fm_a_o @ ( accp_Epistemic_fm_a @ R22 ) @ ( accp_Epistemic_fm_a @ R1 ) ) ) ).
% accp_subset
thf(fact_873_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_874_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_875_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_876_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_877_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte1905851639235817760t_unit,W: list_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8696558436685192347t_unit @ M )
& ( episte5977966464841548710t_unit @ M )
& ( episte5775042212120887174t_unit @ M ) )
=> ( ( member5906877432388582473c_fm_a @ W @ ( episte3604461938431221107t_unit @ M ) )
=> ( episte3437425354345704390c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_878_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_879_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5'
thf(fact_880_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_881_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_882_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>5
thf(fact_883_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>S\<^sub>4
thf(fact_884_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_885_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_886_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_887_strong__completeness_092_060_094sub_062T,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_888_main_092_060_094sub_062T,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_889_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_890_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_891_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G3: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ Y3 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X2 @ P ) ) ) ) )
= ( ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ G3 )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_892_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_893_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs ) @ S2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_894_Collect__restrict,axiom,
! [X5: set_nat,P4: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
& ( P4 @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_895_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_896_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X5: set_nat,P4: nat > $o] :
( ( member_nat2 @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
& ( P4 @ X2 ) ) ) )
=> ( P4 @ X ) ) ) ).
% prop_restrict
thf(fact_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_conj__subset__def,axiom,
! [A: set_nat,P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( P4 @ X2 )
& ( Q4 @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P4 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q4 ) ) ) ) ).
% conj_subset_def
thf(fact_905_list__ex1__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_e2031426293596896995c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X @ Xs ) )
= ( ( ( P4 @ X )
=> ( list_a5841931967666341838c_fm_a
@ ^ [Y3: epistemic_fm_a] :
( ~ ( P4 @ Y3 )
| ( X = Y3 ) )
@ Xs ) )
& ( ~ ( P4 @ X )
=> ( list_e2031426293596896995c_fm_a @ P4 @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_906_transpose__aux__filter__tail,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( concat6109960377493484129c_fm_a
@ ( map_li3858915945532548554c_fm_a
@ ( case_l5707353781634818138c_fm_a @ nil_li2451196919128234278c_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_l8134865115577406678c_fm_a @ T2 @ nil_li2451196919128234278c_fm_a ) )
@ Xss2 ) )
= ( map_li3080774272885743684c_fm_a @ tl_Epistemic_fm_a
@ ( filter441479637125139101c_fm_a
@ ^ [Ys3: list_Epistemic_fm_a] : ( Ys3 != nil_Epistemic_fm_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_907_Nil__is__rev__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( rev_Epistemic_fm_a @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% Nil_is_rev_conv
thf(fact_908_rev__is__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( rev_Epistemic_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% rev_is_Nil_conv
thf(fact_909_list__all__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o,X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X @ Xs ) )
= ( ( P4 @ X )
& ( list_a5841931967666341838c_fm_a @ P4 @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_910_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_911_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_912_set__filter,axiom,
! [P4: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P4 @ Xs ) )
= ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( P4 @ X2 ) ) ) ) ).
% set_filter
thf(fact_913_list_Oset__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,V2: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( map_Ep7084560364594560580c_fm_a @ F @ V2 ) )
= ( image_4449434806354059013c_fm_a @ F @ ( set_Epistemic_fm_a2 @ V2 ) ) ) ).
% list.set_map
thf(fact_914_filter__False,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ! [X3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X3 ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a ) ) ).
% filter_False
thf(fact_915_singleton__rev__conv,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a )
= ( rev_Epistemic_fm_a @ Xs ) )
= ( ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_916_rev__singleton__conv,axiom,
! [Xs: list_Epistemic_fm_a,X: epistemic_fm_a] :
( ( ( rev_Epistemic_fm_a @ Xs )
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) )
= ( Xs
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ) ).
% rev_singleton_conv
thf(fact_917_rev_Osimps_I1_J,axiom,
( ( rev_Epistemic_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% rev.simps(1)
thf(fact_918_rev__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( rev_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ ( rev_Epistemic_fm_a @ Xs ) ) ) ).
% rev_map
thf(fact_919_filter_Osimps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( P4 @ X )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X @ Xs ) )
= ( cons_Epistemic_fm_a @ X @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) )
& ( ~ ( P4 @ X )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X @ Xs ) )
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_920_filter_Osimps_I1_J,axiom,
! [P4: epistemic_fm_a > $o] :
( ( filter7636273809395300631c_fm_a @ P4 @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% filter.simps(1)
thf(fact_921_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P4: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P4 @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P4 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_922_list_Osel_I3_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( tl_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_923_list_Osel_I2_J,axiom,
( ( tl_Epistemic_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% list.sel(2)
thf(fact_924_map__tl,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( tl_Epistemic_fm_a @ Xs ) )
= ( tl_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ) ).
% map_tl
thf(fact_925_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_926_list_Omap__cong__pred,axiom,
! [X: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G5: epistemic_fm_a > epistemic_fm_a] :
( ( X = Ya )
=> ( ( list_a5841931967666341838c_fm_a
@ ^ [Z2: epistemic_fm_a] :
( ( F @ Z2 )
= ( G5 @ Z2 ) )
@ Ya )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X )
= ( map_Ep7084560364594560580c_fm_a @ G5 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_927_empty__filter__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X2 ) ) ) ) ).
% empty_filter_conv
thf(fact_928_filter__empty__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a )
= ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X2 ) ) ) ) ).
% filter_empty_conv
thf(fact_929_image__set,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( image_4449434806354059013c_fm_a @ F @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( set_Epistemic_fm_a2 @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_930_tl__Nil,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( tl_Epistemic_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X2: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) ) ) ).
% tl_Nil
thf(fact_931_Nil__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( tl_Epistemic_fm_a @ Xs ) )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X2: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) ) ) ).
% Nil_tl
thf(fact_932_list_Oset__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,X: epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ ( tl_Epistemic_fm_a @ A2 ) ) )
=> ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_933_list_Omap__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ A2 ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ ( tl_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_934_tl__def,axiom,
( tl_Epistemic_fm_a
= ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [X213: epistemic_fm_a,X223: list_Epistemic_fm_a] : X223 ) ) ).
% tl_def
thf(fact_935_transpose__aux__filter__head,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_li3080774272885743684c_fm_a
@ ( case_l8442927312758267104c_fm_a @ nil_Epistemic_fm_a
@ ^ [H: epistemic_fm_a,T2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ H @ nil_Epistemic_fm_a ) )
@ Xss2 ) )
= ( map_li5516866612190465598c_fm_a @ hd_Epistemic_fm_a
@ ( filter441479637125139101c_fm_a
@ ^ [Ys3: list_Epistemic_fm_a] : ( Ys3 != nil_Epistemic_fm_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_936_list_Ocollapse,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
=> ( ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ List ) @ ( tl_Epistemic_fm_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_937_hd__Cons__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ Xs ) @ ( tl_Epistemic_fm_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_938_list_Osel_I1_J,axiom,
! [X21: epistemic_fm_a,X22: list_Epistemic_fm_a] :
( ( hd_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_939_hd__concat,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( Xs != nil_li2451196919128234278c_fm_a )
=> ( ( ( hd_lis6228752984540680715c_fm_a @ Xs )
!= nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( hd_Epistemic_fm_a @ ( hd_lis6228752984540680715c_fm_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_940_hd__in__set,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( hd_Epistemic_fm_a @ Xs ) @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_941_list_Oset__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( hd_Epistemic_fm_a @ A2 ) @ ( set_Epistemic_fm_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_942_hd__map,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( F @ ( hd_Epistemic_fm_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_943_list_Omap__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ A2 ) )
= ( F @ ( hd_Epistemic_fm_a @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_944_list_Oexpand,axiom,
! [List: list_Epistemic_fm_a,List2: list_Epistemic_fm_a] :
( ( ( List = nil_Epistemic_fm_a )
= ( List2 = nil_Epistemic_fm_a ) )
=> ( ( ( List != nil_Epistemic_fm_a )
=> ( ( List2 != nil_Epistemic_fm_a )
=> ( ( ( hd_Epistemic_fm_a @ List )
= ( hd_Epistemic_fm_a @ List2 ) )
& ( ( tl_Epistemic_fm_a @ List )
= ( tl_Epistemic_fm_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_945_list_Oexhaust__sel,axiom,
! [List: list_Epistemic_fm_a] :
( ( List != nil_Epistemic_fm_a )
=> ( List
= ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ List ) @ ( tl_Epistemic_fm_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_946_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat2 @ X2 ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_947_cSup__eq__maximum,axiom,
! [Z3: nat,X5: set_nat] :
( ( member_nat2 @ Z3 @ X5 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z3 ) ) ) ).
% cSup_eq_maximum
thf(fact_948_Cons__in__shuffles__iff,axiom,
! [Z3: epistemic_fm_a,Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z3 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Xs )
= Z3 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Ys )
= Z3 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ ( tl_Epistemic_fm_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_949_Nil__in__shuffles,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( ( Xs = nil_Epistemic_fm_a )
& ( Ys = nil_Epistemic_fm_a ) ) ) ).
% Nil_in_shuffles
thf(fact_950_Cons__shuffles__subset1,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ X ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_951_Cons__shuffles__subset2,axiom,
! [Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] : ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ Y ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_952_Union__eq,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X2: nat] :
? [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A3 )
& ( member_nat2 @ X2 @ Y3 ) ) ) ) ) ).
% Union_eq
thf(fact_953_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z3: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z3 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_954_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z3: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z3 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_955_Nil__in__shufflesI,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs = nil_Epistemic_fm_a )
=> ( ( Ys = nil_Epistemic_fm_a )
=> ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_956_shufflesE,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( ( ( Zs3 = Xs )
=> ( Ys != nil_Epistemic_fm_a ) )
=> ( ( ( Zs3 = Ys )
=> ( Xs != nil_Epistemic_fm_a ) )
=> ( ! [X3: epistemic_fm_a,Xs4: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ X3 @ Xs4 ) )
=> ! [Z: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z @ Zs4 ) )
=> ( ( X3 = Z )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y2: epistemic_fm_a,Ys4: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Y2 @ Ys4 ) )
=> ! [Z: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z @ Zs4 ) )
=> ( ( Y2 = Z )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_957_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_958_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G3: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G3 )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G3 )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_959_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
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) )
@ V
@ P ) ) ) ) ).
% truth_lemma
thf(fact_960_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_961_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
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_962_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
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_963_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
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_964_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
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_965_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W4: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W4 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) )
@ ^ [I2: a,V3: 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 ) @ V3 ) ) ) )
@ ( episte8239586592105053771t_unit
@ ^ [V3: set_Epistemic_fm_a,X2: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X2 ) @ V3 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_966_set__empty2,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ Xs ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% set_empty2
thf(fact_967_set__empty,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( set_Epistemic_fm_a2 @ Xs )
= bot_bo3626323581529592678c_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% set_empty
thf(fact_968_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_969_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_970_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_971_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_972_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ 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_973_cSup__least,axiom,
! [X5: set_nat,Z3: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_974_empty__set,axiom,
( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% empty_set
thf(fact_975_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M3: episte1560738328020401952t_unit,X3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X3 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_976_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_977_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_978_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_979_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_980_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def
thf(fact_981_min_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_min_nat @ A2 @ B )
= A2 ) ) ).
% min.absorb1
thf(fact_982_min_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_min_nat @ A2 @ B )
= B ) ) ).
% min.absorb2
thf(fact_983_min_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.bounded_iff
thf(fact_984_min_Omono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_985_min_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( ord_min_nat @ A2 @ B ) ) ) ).
% min.orderE
thf(fact_986_min_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% min.orderI
thf(fact_987_min_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.boundedE
thf(fact_988_min_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 @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_989_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( ord_min_nat @ A4 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_990_min_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ A2 ) ).
% min.cobounded1
thf(fact_991_min_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ B ) ).
% min.cobounded2
thf(fact_992_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_min_nat @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_993_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_994_min_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_995_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_996_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
| ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% min_le_iff_disj
thf(fact_997_min__def__raw,axiom,
( ord_min_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% min_def_raw
thf(fact_998_bdd__above_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_999_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M3: nat] :
~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_1000_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M4: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
=> ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1001_cSup__upper,axiom,
! [X: nat,X5: set_nat] :
( ( member_nat2 @ X @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1002_cSup__upper2,axiom,
! [X: nat,X5: set_nat,Y: nat] :
( ( member_nat2 @ X @ X5 )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1003_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_1004_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 )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ S )
=> ( ord_less_eq_nat @ X2 @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1005_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1006_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_1007_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_1008_le__inf__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_1009_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_1010_Int__Collect__mono,axiom,
! [A: set_nat,B4: set_nat,P4: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) )
=> ( 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_1011_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
& ( member_nat2 @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1012_Int__Collect,axiom,
! [X: nat,A: set_nat,P4: nat > $o] :
( ( member_nat2 @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P4 ) ) )
= ( ( member_nat2 @ X @ A )
& ( P4 @ X ) ) ) ).
% Int_Collect
thf(fact_1013_Collect__conj__eq,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P4 @ X2 )
& ( Q4 @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_conj_eq
thf(fact_1014_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_1015_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_1016_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_1017_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1018_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1019_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1020_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_1021_inf__greatest,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_1022_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_1023_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_1024_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1025_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1026_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1027_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1028_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_1029_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1030_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1031_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1032_le__infI2,axiom,
! [B: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI2
thf(fact_1033_le__infI1,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X ) ) ).
% le_infI1
thf(fact_1034_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_1035_le__infI,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1036_le__infE,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_1037_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1038_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1039_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1040_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1041_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat2 @ X2 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_1042_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_1043_le__sup__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_1044_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_1045_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1046_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1047_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1048_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1049_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1050_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1051_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1052_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1053_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1054_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1055_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1056_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_1057_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_1058_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1059_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1060_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1061_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1062_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1063_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1064_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1065_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1066_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1067_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1068_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1069_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1070_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1071_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1072_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1073_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1074_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1075_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1076_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1077_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1078_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1079_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1080_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1081_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_1082_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1083_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1084_complete__interval,axiom,
! [A2: nat,B: nat,P4: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P4 @ A2 )
=> ( ~ ( P4 @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P4 @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P4 @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1085_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1086_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1087_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_1088_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1089_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1090_less__induct,axiom,
! [P4: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X3 ) )
=> ( P4 @ A2 ) ) ).
% less_induct
thf(fact_1091_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1092_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1093_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_1094_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1095_exists__least__iff,axiom,
( ( ^ [P7: nat > $o] :
? [X6: nat] : ( P7 @ X6 ) )
= ( ^ [P6: nat > $o] :
? [N: nat] :
( ( P6 @ N )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ~ ( P6 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1096_linorder__less__wlog,axiom,
! [P4: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B2: nat] :
( ( ord_less_nat @ A5 @ B2 )
=> ( P4 @ A5 @ B2 ) )
=> ( ! [A5: nat] : ( P4 @ A5 @ A5 )
=> ( ! [A5: nat,B2: nat] :
( ( P4 @ B2 @ A5 )
=> ( P4 @ A5 @ B2 ) )
=> ( P4 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1097_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_1098_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1099_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1100_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1101_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1102_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1103_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1104_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1105_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_1106_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_1107_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1108_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1109_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1110_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1111_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1112_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1113_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P4: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P4 ) ) ).
% order_less_imp_triv
thf(fact_1114_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1115_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1116_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1117_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1118_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_1119_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_1120_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1121_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1122_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1123_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1124_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1125_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_1126_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_1127_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1128_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1129_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1130_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1131_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1132_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1133_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1134_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_1135_sup__least,axiom,
! [Y: nat,X: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_1136_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1137_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1138_le__supI2,axiom,
! [X: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1139_le__supI1,axiom,
! [X: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1140_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1141_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1142_le__supI,axiom,
! [A2: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1143_le__supE,axiom,
! [A2: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1144_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1145_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1146_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_1147_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_1148_shuffles_Osimps_I3_J,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( sup_su1755902861286427476c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ X ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ Y ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_1149_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_1150_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_1151_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1152_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat2 @ X2 @ A3 )
| ( member_nat2 @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1153_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat2 @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_1154_Collect__disj__eq,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P4 @ X2 )
| ( Q4 @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_disj_eq
thf(fact_1155_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_1156_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_1157_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1158_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1159_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_1160_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1161_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1162_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1163_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1164_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1165_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1166_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1167_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1168_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 ) )
=> ~ ! [V4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V4 ) @ V )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V4 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_1169_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ A2 ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1170_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1171_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_1172_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_1173_insert__def,axiom,
( insert_nat2
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X2: nat] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_1174_Collect__conv__if,axiom,
! [P4: nat > $o,A2: nat] :
( ( ( P4 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P4 @ X2 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P4 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1175_Collect__conv__if2,axiom,
! [P4: nat > $o,A2: nat] :
( ( ( P4 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P4 @ X2 ) ) )
= ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P4 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P4 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1176_insert__compr,axiom,
( insert_nat2
= ( ^ [A4: nat,B5: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A4 )
| ( member_nat2 @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_1177_insert__Collect,axiom,
! [A2: nat,P4: nat > $o] :
( ( insert_nat2 @ A2 @ ( collect_nat @ P4 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P4 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1178_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_1179_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 )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_a > $o] : ( epistemic_eval_a @ G4 @ H3 @ ( 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_1180_Pow__set_I2_J,axiom,
! [X: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X @ Xs ) ) )
= ( sup_su4305657963326700718c_fm_a @ ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) @ ( image_8582343580727714565c_fm_a @ ( insert7817948963269374698c_fm_a @ X ) @ ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1181_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G3: 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 @ G3 ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G3 @ P ) ) ) ).
% inconsistent_imply
thf(fact_1182_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 ) )
=> ~ ! [W5: 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 ) @ W5 ) @ ( 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 ) @ W5 )
=> ( ( finite3304564945125563331c_fm_a @ W5 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W5 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1183_finite__Collect__disjI,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P4 @ X2 )
| ( Q4 @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P4 ) )
& ( finite_finite_nat @ ( collect_nat @ Q4 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1184_finite__Collect__conjI,axiom,
! [P4: nat > $o,Q4: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P4 ) )
| ( finite_finite_nat @ ( collect_nat @ Q4 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P4 @ X2 )
& ( Q4 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1185_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1186_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_1187_finite__Collect__bex,axiom,
! [A: set_nat,Q4: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y3: nat] :
( ( member_nat2 @ Y3 @ A )
& ( Q4 @ X2 @ Y3 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Y3: nat] : ( Q4 @ Y3 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_1188_finite__UN__I,axiom,
! [A: set_nat,B4: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( finite_finite_nat @ ( B4 @ A5 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_1189_nat__seg__image__imp__finite,axiom,
! [A: set_nat,F: nat > nat,N2: nat] :
( ( A
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) )
=> ( finite_finite_nat @ A ) ) ).
% nat_seg_image_imp_finite
thf(fact_1190_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
? [N: nat,F2: nat > nat] :
( A3
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1191_shuffles_Osimps_I1_J,axiom,
! [Ys: list_Epistemic_fm_a] :
( ( shuffl511019022169630517c_fm_a @ nil_Epistemic_fm_a @ Ys )
= ( insert6567282714061225712c_fm_a @ Ys @ bot_bo8284763806281179116c_fm_a ) ) ).
% shuffles.simps(1)
thf(fact_1192_shuffles_Osimps_I2_J,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( shuffl511019022169630517c_fm_a @ Xs @ nil_Epistemic_fm_a )
= ( insert6567282714061225712c_fm_a @ Xs @ bot_bo8284763806281179116c_fm_a ) ) ).
% shuffles.simps(2)
thf(fact_1193_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1194_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B4: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ? [Xa3: nat] :
( ( member_nat2 @ Xa3 @ B4 )
& ( R2 @ X3 @ Xa3 ) ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat2 @ A4 @ A )
& ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1195_not__finite__existsD,axiom,
! [P4: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P4 ) )
=> ? [X_1: nat] : ( P4 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1196_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat2 @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1197_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_1198_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_1199_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_1200_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ Xa3 @ X3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1201_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1202_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_1203_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_1204_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 ) )
=> ? [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
& ( finite_finite_nat @ C5 )
& ( B4
= ( image_nat_nat @ F @ C5 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1205_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_1206_le__cSup__finite,axiom,
! [X5: set_nat,X: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat2 @ X @ X5 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1207_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1208_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ A )
=> ( ( ord_less_eq_nat @ Xa3 @ X3 )
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1209_finite__subset__Union,axiom,
! [A: set_nat,B7: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B7 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B7 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1210_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [X7: set_nat] :
( ( ord_less_eq_set_nat @ X7 @ A3 )
& ( finite_finite_nat @ X7 ) ) ) ) ) ).
% Fpow_def
thf(fact_1211_finite__subset__induct_H,axiom,
! [F4: set_nat,A: set_nat,P4: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [A5: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F5 @ A )
=> ( ~ ( member_nat2 @ A5 @ F5 )
=> ( ( P4 @ F5 )
=> ( P4 @ ( insert_nat2 @ A5 @ F5 ) ) ) ) ) ) )
=> ( P4 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1212_finite__subset__induct,axiom,
! [F4: set_nat,A: set_nat,P4: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [A5: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat2 @ A5 @ A )
=> ( ~ ( member_nat2 @ A5 @ F5 )
=> ( ( P4 @ F5 )
=> ( P4 @ ( insert_nat2 @ A5 @ F5 ) ) ) ) ) )
=> ( P4 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1213_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_1214_Fpow__Pow__finite,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A3 ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_1215_shuffles_Oelims,axiom,
! [X: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,Y: set_li769143395467472256c_fm_a] :
( ( ( shuffl511019022169630517c_fm_a @ X @ Xa )
= Y )
=> ( ( ( X = nil_Epistemic_fm_a )
=> ( Y
!= ( insert6567282714061225712c_fm_a @ Xa @ bot_bo8284763806281179116c_fm_a ) ) )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ( Y
!= ( insert6567282714061225712c_fm_a @ X @ bot_bo8284763806281179116c_fm_a ) ) )
=> ~ ! [X3: epistemic_fm_a,Xs3: list_Epistemic_fm_a] :
( ( X
= ( cons_Epistemic_fm_a @ X3 @ Xs3 ) )
=> ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_su1755902861286427476c_fm_a @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ X3 ) @ ( shuffl511019022169630517c_fm_a @ Xs3 @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) ) @ ( image_7387318756023638021c_fm_a @ ( cons_Epistemic_fm_a @ Y2 ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs3 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1216_Min_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1217_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_nat @ N @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1218_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_1219_Min__eqI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( member_nat2 @ X @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1220_Min__le,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X ) ) ) ).
% Min_le
thf(fact_1221_finite__ranking__induct,axiom,
! [S: set_nat,P4: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P4 @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat2 @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P4 @ S3 )
=> ( P4 @ ( insert_nat2 @ X3 @ S3 ) ) ) ) )
=> ( P4 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1222_Min__eq__iff,axiom,
! [A: set_nat,M6: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M6 )
= ( ( member_nat2 @ M6 @ A )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ M6 @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1223_Min__le__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
= ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1224_eq__Min__iff,axiom,
! [A: set_nat,M6: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M6
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat2 @ M6 @ A )
& ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ M6 @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1225_Min_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1226_Min_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X @ A5 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1227_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_1228_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_1229_Min__antimono,axiom,
! [M: set_nat,N3: set_nat] :
( ( ord_less_eq_set_nat @ M @ N3 )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N3 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1230_Min_Osubset,axiom,
! [A: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A ) )
= ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset
thf(fact_1231_Min__add__commute,axiom,
! [S: set_nat,F: nat > nat,K2: nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat
@ ( image_nat_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( F @ X2 ) @ K2 )
@ S ) )
= ( plus_plus_nat @ ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ S ) ) @ K2 ) ) ) ) ).
% Min_add_commute
thf(fact_1232_set__Cons__sing__Nil,axiom,
! [A: set_Epistemic_fm_a] :
( ( set_Co4868574198862222544c_fm_a @ A @ ( insert6567282714061225712c_fm_a @ nil_Epistemic_fm_a @ bot_bo8284763806281179116c_fm_a ) )
= ( image_4430763730358542731c_fm_a
@ ^ [X2: epistemic_fm_a] : ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a )
@ A ) ) ).
% set_Cons_sing_Nil
thf(fact_1233_Pow__fold,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( pow_nat @ A )
= ( finite4178521680790401110et_nat
@ ^ [X2: nat,A3: set_set_nat] : ( sup_sup_set_set_nat @ A3 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X2 ) @ A3 ) )
@ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
@ A ) ) ) ).
% Pow_fold
thf(fact_1234_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1235_nat__add__left__cancel__le,axiom,
! [K2: nat,M6: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M6 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M6 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1236_mono__nat__linear__lb,axiom,
! [F: nat > nat,M6: nat,K2: nat] :
( ! [M7: nat,N4: nat] :
( ( ord_less_nat @ M7 @ N4 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M6 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1237_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 ) )
=> ? [X3: nat] :
( ( P4 @ X3 )
& ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1238_nat__le__linear,axiom,
! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
| ( ord_less_eq_nat @ N2 @ M6 ) ) ).
% nat_le_linear
thf(fact_1239_le__antisym,axiom,
! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M6 )
=> ( M6 = N2 ) ) ) ).
% le_antisym
thf(fact_1240_eq__imp__le,axiom,
! [M6: nat,N2: nat] :
( ( M6 = N2 )
=> ( ord_less_eq_nat @ M6 @ N2 ) ) ).
% eq_imp_le
thf(fact_1241_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_1242_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1243_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1244_le__neq__implies__less,axiom,
! [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
=> ( ( M6 != N2 )
=> ( ord_less_nat @ M6 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1245_less__or__eq__imp__le,axiom,
! [M6: nat,N2: nat] :
( ( ( ord_less_nat @ M6 @ N2 )
| ( M6 = N2 ) )
=> ( ord_less_eq_nat @ M6 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1246_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1247_less__imp__le__nat,axiom,
! [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( ord_less_eq_nat @ M6 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1248_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( M5 != N ) ) ) ) ).
% nat_less_le
thf(fact_1249_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N: nat] :
? [K: nat] :
( N
= ( plus_plus_nat @ M5 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_1250_trans__le__add2,axiom,
! [I: nat,J: nat,M6: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M6 @ J ) ) ) ).
% trans_le_add2
thf(fact_1251_trans__le__add1,axiom,
! [I: nat,J: nat,M6: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M6 ) ) ) ).
% trans_le_add1
thf(fact_1252_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1253_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1254_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K2 @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1255_add__leD2,axiom,
! [M6: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M6 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1256_add__leD1,axiom,
! [M6: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M6 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M6 @ N2 ) ) ).
% add_leD1
thf(fact_1257_le__add2,axiom,
! [N2: nat,M6: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M6 @ N2 ) ) ).
% le_add2
thf(fact_1258_le__add1,axiom,
! [N2: nat,M6: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M6 ) ) ).
% le_add1
thf(fact_1259_add__leE,axiom,
! [M6: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M6 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M6 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1260_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_1261_finite__M__bounded__by__nat,axiom,
! [P4: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P4 @ K )
& ( ord_less_nat @ K @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1262_bounded__Max__nat,axiom,
! [P4: nat > $o,X: nat,M: nat] :
( ( P4 @ X )
=> ( ! [X3: nat] :
( ( P4 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M7: nat] :
( ( P4 @ M7 )
=> ~ ! [X4: nat] :
( ( P4 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1263_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat2 @ X2 @ N5 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1264_listset_Osimps_I1_J,axiom,
( ( listse5025355581707466111c_fm_a @ nil_se102576565402781184c_fm_a )
= ( insert6567282714061225712c_fm_a @ nil_Epistemic_fm_a @ bot_bo8284763806281179116c_fm_a ) ) ).
% listset.simps(1)
thf(fact_1265_Set__filter__fold,axiom,
! [A: set_nat,P4: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( filter_nat2 @ P4 @ A )
= ( finite5529483035118572448et_nat
@ ^ [X2: nat,A8: set_nat] : ( if_set_nat @ ( P4 @ X2 ) @ ( insert_nat2 @ X2 @ A8 ) @ A8 )
@ bot_bot_set_nat
@ A ) ) ) ).
% Set_filter_fold
thf(fact_1266_Set_Ofilter__def,axiom,
( filter_nat2
= ( ^ [P6: nat > $o,A3: set_nat] :
( collect_nat
@ ^ [A4: nat] :
( ( member_nat2 @ A4 @ A3 )
& ( P6 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1267_inter__Set__filter,axiom,
! [B4: set_nat,A: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( inf_inf_set_nat @ A @ B4 )
= ( filter_nat2
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A )
@ B4 ) ) ) ).
% inter_Set_filter
thf(fact_1268_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X2 ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_1269_Bex__fold,axiom,
! [A: set_nat,P4: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( P4 @ X2 ) ) )
= ( finite_fold_nat_o
@ ^ [K: nat,S4: $o] :
( S4
| ( P4 @ K ) )
@ $false
@ A ) ) ) ).
% Bex_fold
% Helper facts (19)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X: list_s580375451141968640c_fm_a,Y: list_s580375451141968640c_fm_a] :
( ( if_lis3811715425919989702c_fm_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( if_lis342421823373228652c_fm_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_J_T,axiom,
! [X: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( if_lis342421823373228652c_fm_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ ( cons_Epistemic_fm_a @ a3 @ ga ) ) ) @ ( epistemic_Con_a @ ( epistemic_K_a @ i @ a3 ) @ ( epistemic_K_a @ i @ ( stalnaker_conjunct_a @ ga ) ) ) ) ).
%------------------------------------------------------------------------------