TPTP Problem File: SLH0336^1.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_00145_004407__6166158_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1619 ( 494 unt; 327 typ; 0 def)
% Number of atoms : 4477 (1336 equ; 0 cnn)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 14649 ( 451 ~; 82 |; 303 &;11548 @)
% ( 0 <=>;2265 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 1702 (1702 >; 0 *; 0 +; 0 <<)
% Number of symbols : 301 ( 298 usr; 22 con; 0-4 aty)
% Number of variables : 4616 ( 306 ^;4087 !; 223 ?;4616 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:11:43.831
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
episte1560738328020401952t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Epistemic____Logic__Ofm_Itf__a_J_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
episte94448284482925344t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mt__Nat__Onat_Mt__Epistemic____Logic__Okripke__Okripke____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
episte8765170747386058258t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Oframe__Oframe____ext_Itf__a_Mtf__a_Mt__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
episte6182337868402532512t_unit: $tType ).
thf(ty_n_t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte1193835314949844379t_unit: $tType ).
thf(ty_n_t__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__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_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__Set__Oset_I_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_J,type,
set_Epistemic_fm_a_o: $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__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_It__Nat__Onat_J,type,
epistemic_fm_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Epistemic____Logic__Ofm_Itf__a_J,type,
epistemic_fm_a: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (298)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
bNF_Gr8437504134799245625c_fm_a: set_li769143395467472256c_fm_a > epistemic_fm_a > set_li769143395467472256c_fm_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
comple310706794042174646fm_a_o: set_Epistemic_fm_a_o > epistemic_fm_a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
condit2203544419062987614c_fm_a: set_se5208064806568342746c_fm_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(sy_c_Epistemic__Logic_OAK_001tf__a,type,
epistemic_AK_a: ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx4_001tf__a,type,
epistemic_Ax4_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAx5_001tf__a,type,
epistemic_Ax5_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxB_001tf__a,type,
epistemic_AxB_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OAxT_001tf__a,type,
epistemic_AxT_a: epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Epistemic____Logic__Ofm_Itf__a_J_Mt__Product____Type__Ounit_J,type,
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episte2449151000174023629t_unit: episte1560738328020401952t_unit > $o ).
thf(sy_c_Epistemic__Logic_OEuclidean_001tf__a_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
episte2339904321507024205t_unit: episte6182337868402532512t_unit > $o ).
thf(sy_c_Epistemic__Logic_Oconsistent_001tf__a,type,
episte2285483198712856226tent_a: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Oeval_001tf__a,type,
epistemic_eval_a: ( list_char > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte3685526487207141399c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001t__Nat__Onat,type,
epistemic_Con_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__a,type,
epistemic_Con_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte6088726764479022859c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Nat__Onat,type,
epistemic_Dis_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_ODis_001tf__a,type,
epistemic_Dis_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte5073044243917183961c_fm_a: episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001t__Nat__Onat,type,
epistemic_FF_nat: epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__a,type,
epistemic_FF_a: epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte260752218777527565c_fm_a: episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001t__Nat__Onat,type,
epistemic_Imp_nat: epistemic_fm_nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__a,type,
epistemic_Imp_a: epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte5657488632024175118c_fm_a: epistemic_fm_a > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Nat__Onat,type,
epistemic_K_nat: nat > epistemic_fm_nat > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__a,type,
epistemic_K_a: a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte3759128466173231372c_fm_a: list_char > episte740340785640729014c_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001t__Nat__Onat,type,
epistemic_Pro_nat: list_char > epistemic_fm_nat ).
thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__a,type,
epistemic_Pro_a: list_char > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
episte7774795710028497888c_fm_a: ( epistemic_fm_a > epistemic_fm_a > $o ) > episte740340785640729014c_fm_a > episte740340785640729014c_fm_a > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__a_J_001t__Nat__Onat,type,
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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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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__Nat__Onat,type,
episte3894023384580379906at_nat: ( nat > nat > $o ) > epistemic_fm_nat > epistemic_fm_nat > $o ).
thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__a_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
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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__Nat__Onat,type,
epistemic_set_fm_nat: epistemic_fm_nat > set_nat ).
thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__a,type,
epistemic_set_fm_a: epistemic_fm_a > set_a ).
thf(sy_c_Epistemic__Logic_Oframe_O_092_060K_062_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
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episte6926715892928323059t_unit: episte6182337868402532512t_unit > set_a ).
thf(sy_c_Epistemic__Logic_Oframe_Oframe__ext_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001tf__a_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
episte2888590659910966568t_unit: set_se5208064806568342746c_fm_a > ( a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a ) > episte1193835314949844379t_unit > episte1560738328020401952t_unit ).
thf(sy_c_Epistemic__Logic_Oframe_Omore_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Epistemic____Logic__Okripke__Okripke____ext_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Epistemic__Logic_Oimply_001tf__a,type,
epistemic_imply_a: list_Epistemic_fm_a > epistemic_fm_a > epistemic_fm_a ).
thf(sy_c_Epistemic__Logic_Okripke_O_092_060pi_062_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_001t__Product____Type__Ounit,type,
episte2398645135750866164t_unit: episte1560738328020401952t_unit > set_Epistemic_fm_a > list_char > $o ).
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image_971165786557580383c_fm_a: ( list_Epistemic_fm_a > set_Epistemic_fm_a ) > set_li769143395467472256c_fm_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_4193161545499529530fm_a_o: ( nat > epistemic_fm_a > $o ) > set_nat > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_2764307138536547683c_fm_a: ( nat > set_Epistemic_fm_a ) > set_nat > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
image_3809950421912819996fm_a_o: ( a > epistemic_fm_a > $o ) > set_a > set_Epistemic_fm_a_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J,type,
image_3480842985853347585c_fm_a: ( a > set_Epistemic_fm_a ) > set_a > set_se5208064806568342746c_fm_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__a_J,type,
insert7817948963269374698c_fm_a: epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
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_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__Epistemic____Logic__Ofm_Itf__a_J_J,type,
accp_l2570680006282209577c_fm_a: ( list_Epistemic_fm_a > list_Epistemic_fm_a > $o ) > list_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__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_member_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J,type,
member4486839677911940090fm_a_o: ( epistemic_fm_a > $o ) > set_Epistemic_fm_a_o > $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__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $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__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_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_p,type,
p: epistemic_fm_a ).
thf(sy_v_q,type,
q: epistemic_fm_a ).
% Relevant facts (1274)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_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_7_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_8_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_9_Ax,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a] :
( ( A @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% Ax
thf(fact_10_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_11_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_12_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_13_K__conjunct__imply,axiom,
! [A: epistemic_fm_a > $o,G: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% K_conjunct_imply
thf(fact_14_K__imply__conjunct,axiom,
! [A: epistemic_fm_a > $o,G: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) )
=> ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) ) ) ).
% K_imply_conjunct
thf(fact_15_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_16_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_17_fm_Odistinct_I9_J,axiom,
! [X61: a,X62: epistemic_fm_a] :
( epistemic_FF_a
!= ( epistemic_K_a @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_AK_Osimps,axiom,
( epistemic_AK_a
= ( ^ [A3: epistemic_fm_a > $o,A4: epistemic_fm_a] :
( ? [P2: epistemic_fm_a] :
( ( A4 = P2 )
& ! [G2: list_char > $o,H: epistemic_fm_a > $o] : ( epistemic_eval_a @ G2 @ 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 )
=> ( ~ ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ A2 )
=> ( ! [I3: a,P3: epistemic_fm_a,Q3: epistemic_fm_a] :
( A2
!= ( epistemic_Imp_a @ ( epistemic_Con_a @ ( epistemic_K_a @ I3 @ P3 ) @ ( epistemic_K_a @ I3 @ ( epistemic_Imp_a @ P3 @ Q3 ) ) ) @ ( epistemic_K_a @ I3 @ Q3 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P3: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ P3 )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_Imp_a @ P3 @ A2 ) ) )
=> ~ ! [P3: epistemic_fm_a] :
( ? [I3: a] :
( A2
= ( epistemic_K_a @ I3 @ P3 ) )
=> ~ ( epistemic_AK_a @ A @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_29_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_30_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_31_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_32_imply__conjunct,axiom,
! [G: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ G @ P ) @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) ) ) ).
% imply_conjunct
thf(fact_33_conjunct__imply,axiom,
! [G: list_Epistemic_fm_a,P: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( stalnaker_conjunct_a @ G ) @ P ) @ ( epistemic_imply_a @ G @ P ) ) ) ).
% conjunct_imply
thf(fact_34_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_35_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_36_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_37_eval_Osimps_I5_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Imp_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
=> ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_38_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H3: epistemic_fm_a > $o,I: a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ Ux @ H3 @ ( epistemic_K_a @ I @ P ) )
= ( H3 @ ( epistemic_K_a @ I @ P ) ) ) ).
% eval.simps(6)
thf(fact_39_A1,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [G5: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G5 @ H4 @ P )
=> ( epistemic_AK_a @ A @ P ) ) ).
% A1
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_eval_Osimps_I4_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Con_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
& ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_42_duality__taut,axiom,
! [I: a,P: epistemic_fm_a,Q: epistemic_fm_a,G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) ) @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ Q @ epistemic_FF_a ) ) @ epistemic_FF_a ) @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ).
% duality_taut
thf(fact_43_mem__Collect__eq,axiom,
! [A2: a,P4: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A2: nat,P4: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_a,P4: set_Epistemic_fm_a > $o] :
( ( member536094252920883875c_fm_a @ A2 @ ( collec2519470961442302949c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A2: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ A2 @ ( collec4904205152690461189c_fm_a @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] : ( member536094252920883875c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_a] :
( ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_51_Collect__cong,axiom,
! [P4: set_Epistemic_fm_a > $o,Q4: set_Epistemic_fm_a > $o] :
( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec2519470961442302949c_fm_a @ P4 )
= ( collec2519470961442302949c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_52_Collect__cong,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collec4904205152690461189c_fm_a @ P4 )
= ( collec4904205152690461189c_fm_a @ Q4 ) ) ) ).
% Collect_cong
thf(fact_53_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_54_K__Boole,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ G ) @ epistemic_FF_a ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% K_Boole
thf(fact_55_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_56_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_57_K__mp,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G: 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 ) @ G ) ) @ Q ) ) ).
% K_mp
thf(fact_58_K__ImpI,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G ) @ Q ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ ( epistemic_Imp_a @ P @ Q ) ) ) ) ).
% K_ImpI
thf(fact_59_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_60_K__distrib__K__imp,axiom,
! [A: epistemic_fm_a > $o,I: a,G: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_K_a @ I @ ( epistemic_imply_a @ G @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( map_Ep7084560364594560580c_fm_a @ ( epistemic_K_a @ I ) @ G ) @ ( epistemic_K_a @ I @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_61_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_62_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_63_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_64_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_65_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_66_imply_Osimps_I1_J,axiom,
! [Q: epistemic_fm_a] :
( ( epistemic_imply_a @ nil_Epistemic_fm_a @ Q )
= Q ) ).
% imply.simps(1)
thf(fact_67_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_68_tautology,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [G5: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G5 @ H4 @ P )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ).
% tautology
thf(fact_69_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_70_K__swap,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a,G: 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 @ G ) ) @ R ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ ( cons_Epistemic_fm_a @ P @ G ) ) @ R ) ) ) ).
% K_swap
thf(fact_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_predicate1I,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( P4 @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 ) ) ).
% predicate1I
thf(fact_80_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_81_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_82_order__refl,axiom,
! [X3: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ X3 @ X3 ) ).
% order_refl
thf(fact_83_order__refl,axiom,
! [X3: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ X3 @ X3 ) ).
% order_refl
thf(fact_84_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_85_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_86_dual__order_Orefl,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_87_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_88_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_89_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_90_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_91_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_92_le__funD,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G4 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funD
thf(fact_93_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_94_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_95_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_96_order__antisym__conv,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_97_order__antisym__conv,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_98_order__antisym__conv,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_99_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_100_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_101_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > nat,C: nat] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_nat,C: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Epistemic_fm_a > nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A2: nat,F: ( epistemic_fm_a > $o ) > nat,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_Epistemic_fm_a > set_nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_a,F: set_nat > set_Epistemic_fm_a,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_122_order__eq__refl,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( X3 = Y )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_123_order__eq__refl,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( X3 = Y )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_124_order__eq__refl,axiom,
! [X3: set_nat,Y: set_nat] :
( ( X3 = Y )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_125_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_126_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_128_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_129_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_132_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,F: ( epistemic_fm_a > $o ) > nat,C: nat] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,F: set_Epistemic_fm_a > set_nat,C: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_le3275665582123262618c_fm_a @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_137_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_138_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_139_order__subst1,axiom,
! [A2: nat,F: set_Epistemic_fm_a > nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_140_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a,B: nat,C: nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_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 )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A2: epistemic_fm_a > $o,F: nat > epistemic_fm_a > $o,B: nat,C: nat] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A2: set_Epistemic_fm_a,F: set_nat > set_Epistemic_fm_a,B: set_nat,C: set_nat] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A2: set_nat,F: set_Epistemic_fm_a > set_nat,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ C )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A4 @ B2 )
& ( ord_le4043730696559282883fm_a_o @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_147_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A4 @ B2 )
& ( ord_le3275665582123262618c_fm_a @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_148_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_149_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
& ( ord_less_eq_nat @ B2 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_150_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_151_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_152_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_153_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B2 @ A4 )
& ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_163_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B2 @ A4 )
& ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_164_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_166_linorder__wlog,axiom,
! [P4: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B3: nat] :
( ( ord_less_eq_nat @ A5 @ B3 )
=> ( P4 @ A5 @ B3 ) )
=> ( ! [A5: nat,B3: nat] :
( ( P4 @ B3 @ A5 )
=> ( P4 @ A5 @ B3 ) )
=> ( P4 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_167_order__trans,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ Z2 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_168_order__trans,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ Z2 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_169_order__trans,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_170_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_171_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_172_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_173_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_174_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_175_order__antisym,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_176_order__antisym,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_177_order__antisym,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_178_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: epistemic_fm_a > $o,Z: epistemic_fm_a > $o] : ( Y3 = Z ) )
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X @ Y4 )
& ( ord_le4043730696559282883fm_a_o @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_188_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Epistemic_fm_a,Z: set_Epistemic_fm_a] : ( Y3 = Z ) )
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X @ Y4 )
& ( ord_le3275665582123262618c_fm_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_189_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_190_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_191_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_192_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_193_not__Cons__self2,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_194_rev__predicate1D,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Q4: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( Q4 @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_195_predicate1D,axiom,
! [P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Q4 )
=> ( ( P4 @ X3 )
=> ( Q4 @ X3 ) ) ) ).
% predicate1D
thf(fact_196_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_197_list__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xs2 != nil_Epistemic_fm_a )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_198_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 )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ nil_Epistemic_fm_a )
=> ( ! [Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] : ( P4 @ nil_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Y2: epistemic_fm_a,Ys2: list_Epistemic_fm_a] :
( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_199_neq__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
= ( ? [Y4: epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_200_remdups__adj_Ocases,axiom,
! [X3: list_Epistemic_fm_a] :
( ( X3 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( X3
!= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_201_transpose_Ocases,axiom,
! [X3: list_l6083326122719238310c_fm_a] :
( ( X3 != nil_li2451196919128234278c_fm_a )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ~ ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( X3
!= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_202_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_203_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_204_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_205_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 ) )
= ( ? [Z3: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_206_Cons__eq__map__conv,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
= ( ? [Z3: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z3 @ Zs ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_207_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 ) )
=> ? [Z4: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_208_Cons__eq__map__D,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) )
=> ? [Z4: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Z4 @ Zs2 ) )
& ( X3
= ( F @ Z4 ) )
& ( Xs
= ( map_Ep7084560364594560580c_fm_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_209_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_210_le__fun__def,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [F2: epistemic_fm_a > $o,G2: epistemic_fm_a > $o] :
! [X: epistemic_fm_a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_fun_def
thf(fact_211_le__funI,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G4 @ X2 ) )
=> ( ord_le4043730696559282883fm_a_o @ F @ G4 ) ) ).
% le_funI
thf(fact_212_le__funE,axiom,
! [F: epistemic_fm_a > $o,G4: epistemic_fm_a > $o,X3: epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o @ F @ G4 )
=> ( ord_less_eq_o @ ( F @ X3 ) @ ( G4 @ X3 ) ) ) ).
% le_funE
thf(fact_213_insert__Nil,axiom,
! [X3: epistemic_fm_a] :
( ( insert177310161492556854c_fm_a @ X3 @ nil_Epistemic_fm_a )
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% insert_Nil
thf(fact_214_map__eq__map__tailrec,axiom,
map_Ep7084560364594560580c_fm_a = map_ta875960198429730446c_fm_a ).
% map_eq_map_tailrec
thf(fact_215_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_216_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_217_K__DisE,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: list_Epistemic_fm_a,R: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ G ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ Q @ G ) @ R ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ ( epistemic_Dis_a @ P @ Q ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ R ) ) ) ) ) ).
% K_DisE
thf(fact_218_map__tailrec__rev_Oelims,axiom,
! [X3: epistemic_fm_a > epistemic_fm_a,Xa: list_Epistemic_fm_a,Xb: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( map_ta6565535454553546997c_fm_a @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != Xb ) )
=> ~ ! [A5: epistemic_fm_a,As: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ A5 @ As ) )
=> ( Y
!= ( map_ta6565535454553546997c_fm_a @ X3 @ As @ ( cons_Epistemic_fm_a @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_219_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
~ ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ nil_Epistemic_fm_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_220_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_221_Greatest__equality,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ( order_253494837916242250fm_a_o @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_222_Greatest__equality,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ( order_4585748725732241747c_fm_a @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_223_Greatest__equality,axiom,
! [P4: set_nat > $o,X3: set_nat] :
( ( P4 @ X3 )
=> ( ! [Y2: set_nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X3 ) )
=> ( ( order_5724808138429204845et_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_224_Greatest__equality,axiom,
! [P4: nat > $o,X3: nat] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ( order_Greatest_nat @ P4 )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_225_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_226_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexord491902619044731238c_fm_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_227_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ Xs @ nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_228_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] : ( lexord491902619044731238c_fm_a @ Less @ nil_Epistemic_fm_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_229_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_230_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_231_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_232_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_233_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_234_eval_Osimps_I3_J,axiom,
! [G4: list_char > $o,H3: epistemic_fm_a > $o,P: epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_eval_a @ G4 @ H3 @ ( epistemic_Dis_a @ P @ Q ) )
= ( ( epistemic_eval_a @ G4 @ H3 @ P )
| ( epistemic_eval_a @ G4 @ H3 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_235_ord_Olexordp__eq_OCons,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Less @ X3 @ Y )
=> ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_236_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexord491902619044731238c_fm_a @ Less @ Xs @ Ys )
=> ( lexord491902619044731238c_fm_a @ Less @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_237_ord_Olexordp__eq_ONil,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] : ( lexord491902619044731238c_fm_a @ Less @ nil_Epistemic_fm_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_238_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_239_ord_Olexordp__eq_Ocases,axiom,
! [Less: epistemic_fm_a > epistemic_fm_a > $o,A1: list_Epistemic_fm_a,A22: list_Epistemic_fm_a] :
( ( lexord491902619044731238c_fm_a @ Less @ A1 @ A22 )
=> ( ( A1 != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ? [Xs2: list_Epistemic_fm_a] :
( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs2 ) )
=> ! [Y2: epistemic_fm_a] :
( ? [Ys2: list_Epistemic_fm_a] :
( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_a] :
( ( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexord491902619044731238c_fm_a @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_240_ord_Olexordp__eq_Osimps,axiom,
( lexord491902619044731238c_fm_a
= ( ^ [Less2: epistemic_fm_a > epistemic_fm_a > $o,A12: list_Epistemic_fm_a,A23: list_Epistemic_fm_a] :
( ? [Ys3: list_Epistemic_fm_a] :
( ( A12 = nil_Epistemic_fm_a )
& ( A23 = Ys3 ) )
| ? [X: epistemic_fm_a,Y4: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y4 @ Ys3 ) )
& ( Less2 @ X @ Y4 ) )
| ? [X: epistemic_fm_a,Y4: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X @ Y4 )
& ~ ( Less2 @ Y4 @ X )
& ( lexord491902619044731238c_fm_a @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_241_GreatestI2__order,axiom,
! [P4: ( epistemic_fm_a > $o ) > $o,X3: epistemic_fm_a > $o,Q4: ( epistemic_fm_a > $o ) > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ( P4 @ Y2 )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( P4 @ X2 )
=> ( ! [Y5: epistemic_fm_a > $o] :
( ( P4 @ Y5 )
=> ( ord_le4043730696559282883fm_a_o @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_253494837916242250fm_a_o @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_242_GreatestI2__order,axiom,
! [P4: set_Epistemic_fm_a > $o,X3: set_Epistemic_fm_a,Q4: set_Epistemic_fm_a > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ( P4 @ Y2 )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( P4 @ X2 )
=> ( ! [Y5: set_Epistemic_fm_a] :
( ( P4 @ Y5 )
=> ( ord_le3275665582123262618c_fm_a @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_4585748725732241747c_fm_a @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_243_GreatestI2__order,axiom,
! [P4: set_nat > $o,X3: set_nat,Q4: set_nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: set_nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X3 ) )
=> ( ! [X2: set_nat] :
( ( P4 @ X2 )
=> ( ! [Y5: set_nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_5724808138429204845et_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_244_GreatestI2__order,axiom,
! [P4: nat > $o,X3: nat,Q4: nat > $o] :
( ( P4 @ X3 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) )
=> ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q4 @ X2 ) ) )
=> ( Q4 @ ( order_Greatest_nat @ P4 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_245_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_246_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_247_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_248_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_249_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_250_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_251_listrelp_Ocases,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,A1: list_Epistemic_fm_a,A22: list_Epistemic_fm_a] :
( ( listre7830505053103709503c_fm_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_Epistemic_fm_a )
=> ( A22 != nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( A1
= ( cons_Epistemic_fm_a @ X2 @ Xs2 ) )
=> ! [Ys2: list_Epistemic_fm_a] :
( ( A22
= ( cons_Epistemic_fm_a @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listre7830505053103709503c_fm_a @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_252_listrelp_Osimps,axiom,
( listre7830505053103709503c_fm_a
= ( ^ [R2: epistemic_fm_a > epistemic_fm_a > $o,A12: list_Epistemic_fm_a,A23: list_Epistemic_fm_a] :
( ( ( A12 = nil_Epistemic_fm_a )
& ( A23 = nil_Epistemic_fm_a ) )
| ? [X: epistemic_fm_a,Y4: epistemic_fm_a,Xs3: list_Epistemic_fm_a,Ys3: list_Epistemic_fm_a] :
( ( A12
= ( cons_Epistemic_fm_a @ X @ Xs3 ) )
& ( A23
= ( cons_Epistemic_fm_a @ Y4 @ Ys3 ) )
& ( R2 @ X @ Y4 )
& ( listre7830505053103709503c_fm_a @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_253_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_254_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_255_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_256_singleton__rev__conv,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a )
= ( rev_Epistemic_fm_a @ Xs ) )
= ( ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_257_rev__singleton__conv,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ( rev_Epistemic_fm_a @ Xs )
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
= ( Xs
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ) ).
% rev_singleton_conv
thf(fact_258_rev_Osimps_I1_J,axiom,
( ( rev_Epistemic_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% rev.simps(1)
thf(fact_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_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_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_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_a
!= ( epistemic_Pro_a @ X24 ) ) ).
% fm.distinct(1)
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_eval_Osimps_I2_J,axiom,
! [G4: list_char > $o,Uw: epistemic_fm_a > $o,X3: list_char] :
( ( epistemic_eval_a @ G4 @ Uw @ ( epistemic_Pro_a @ X3 ) )
= ( G4 @ X3 ) ) ).
% eval.simps(2)
thf(fact_290_listrelp_OCons,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( R @ X3 @ Y )
=> ( ( listre7830505053103709503c_fm_a @ R @ Xs @ Ys )
=> ( listre7830505053103709503c_fm_a @ R @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_291_listrelp_ONil,axiom,
! [R: epistemic_fm_a > epistemic_fm_a > $o] : ( listre7830505053103709503c_fm_a @ R @ nil_Epistemic_fm_a @ nil_Epistemic_fm_a ) ).
% listrelp.Nil
thf(fact_292_fm_Orel__induct,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ( Q4 @ epistemic_FF_a @ epistemic_FF_a )
=> ( ! [A24: list_char,B22: list_char] :
( ( A24 = B22 )
=> ( Q4 @ ( epistemic_Pro_a @ A24 ) @ ( epistemic_Pro_a @ B22 ) ) )
=> ( ! [A31: epistemic_fm_a,A32: epistemic_fm_a,B31: epistemic_fm_a,B32: epistemic_fm_a] :
( ( Q4 @ A31 @ B31 )
=> ( ( Q4 @ A32 @ B32 )
=> ( Q4 @ ( epistemic_Dis_a @ A31 @ A32 ) @ ( epistemic_Dis_a @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_a,A42: epistemic_fm_a,B41: epistemic_fm_a,B42: epistemic_fm_a] :
( ( Q4 @ A41 @ B41 )
=> ( ( Q4 @ A42 @ B42 )
=> ( Q4 @ ( epistemic_Con_a @ A41 @ A42 ) @ ( epistemic_Con_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_a,A52: epistemic_fm_a,B51: epistemic_fm_a,B52: epistemic_fm_a] :
( ( Q4 @ A51 @ B51 )
=> ( ( Q4 @ A52 @ B52 )
=> ( Q4 @ ( epistemic_Imp_a @ A51 @ A52 ) @ ( epistemic_Imp_a @ B51 @ B52 ) ) ) )
=> ( ! [A61: a,A62: epistemic_fm_a,B61: a,B62: epistemic_fm_a] :
( ( R3 @ A61 @ B61 )
=> ( ( Q4 @ A62 @ B62 )
=> ( Q4 @ ( epistemic_K_a @ A61 @ A62 ) @ ( epistemic_K_a @ B61 @ B62 ) ) ) )
=> ( Q4 @ X3 @ Y ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_293_fm_Orel__cases,axiom,
! [R3: a > a > $o,A2: epistemic_fm_a,B: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ A2 @ B )
=> ( ( ( A2 = epistemic_FF_a )
=> ( B != epistemic_FF_a ) )
=> ( ! [X2: list_char] :
( ( A2
= ( epistemic_Pro_a @ X2 ) )
=> ! [Y2: list_char] :
( ( B
= ( epistemic_Pro_a @ Y2 ) )
=> ( X2 != Y2 ) ) )
=> ( ! [X1: epistemic_fm_a,X2a: epistemic_fm_a] :
( ( A2
= ( epistemic_Dis_a @ X1 @ X2a ) )
=> ! [Y1: epistemic_fm_a,Y2a: epistemic_fm_a] :
( ( B
= ( epistemic_Dis_a @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1 @ Y1 )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_a,X2b: epistemic_fm_a] :
( ( A2
= ( epistemic_Con_a @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_a,Y2b: epistemic_fm_a] :
( ( B
= ( epistemic_Con_a @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_a,X2c: epistemic_fm_a] :
( ( A2
= ( epistemic_Imp_a @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_a,Y2c: epistemic_fm_a] :
( ( B
= ( epistemic_Imp_a @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: a,X2d: epistemic_fm_a] :
( ( A2
= ( epistemic_K_a @ X1c @ X2d ) )
=> ! [Y1c: a,Y2d: epistemic_fm_a] :
( ( B
= ( epistemic_K_a @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_a_a @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_294_semantics_Osimps_I2_J,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,X3: list_char] :
( ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Pro_a @ X3 ) )
= ( episte2398645135750866164t_unit @ M @ W @ X3 ) ) ).
% semantics.simps(2)
thf(fact_295_rev__eq__Cons__iff,axiom,
! [Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( rev_Epistemic_fm_a @ Xs )
= ( cons_Epistemic_fm_a @ Y @ Ys ) )
= ( Xs
= ( append9179727413925872949c_fm_a @ ( rev_Epistemic_fm_a @ Ys ) @ ( cons_Epistemic_fm_a @ Y @ nil_Epistemic_fm_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_296_map__tailrec__rev,axiom,
( map_ta6565535454553546997c_fm_a
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a,As3: list_Epistemic_fm_a] : ( append9179727413925872949c_fm_a @ ( rev_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F2 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_297_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_a,Qs: list_Epistemic_fm_a,R: epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Ps ) @ ( set_Epistemic_fm_a2 @ Qs ) )
=> ! [G3: list_char > $o,H2: epistemic_fm_a > $o] : ( epistemic_eval_a @ G3 @ H2 @ ( epistemic_Imp_a @ ( epistemic_imply_a @ Ps @ R ) @ ( epistemic_imply_a @ Qs @ R ) ) ) ) ).
% tautology_imply_superset
thf(fact_298_member__rec_I2_J,axiom,
! [Y: epistemic_fm_a] :
~ ( member6038508265109909045c_fm_a @ nil_Epistemic_fm_a @ Y ) ).
% member_rec(2)
thf(fact_299_map__eq__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ Xs ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X )
= ( G4 @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_300_append_Oright__neutral,axiom,
! [A2: list_Epistemic_fm_a] :
( ( append9179727413925872949c_fm_a @ A2 @ nil_Epistemic_fm_a )
= A2 ) ).
% append.right_neutral
thf(fact_301_append__Nil2,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( append9179727413925872949c_fm_a @ Xs @ nil_Epistemic_fm_a )
= Xs ) ).
% append_Nil2
thf(fact_302_append__self__conv,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_Epistemic_fm_a ) ) ).
% append_self_conv
thf(fact_303_self__append__conv,axiom,
! [Y: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Y
= ( append9179727413925872949c_fm_a @ Y @ Ys ) )
= ( Ys = nil_Epistemic_fm_a ) ) ).
% self_append_conv
thf(fact_304_append__self__conv2,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% append_self_conv2
thf(fact_305_self__append__conv2,axiom,
! [Y: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( Y
= ( append9179727413925872949c_fm_a @ Xs @ Y ) )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% self_append_conv2
thf(fact_306_Nil__is__append__conv,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( ( Xs = nil_Epistemic_fm_a )
& ( Ys = nil_Epistemic_fm_a ) ) ) ).
% Nil_is_append_conv
thf(fact_307_append__is__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Xs @ Ys )
= nil_Epistemic_fm_a )
= ( ( Xs = nil_Epistemic_fm_a )
& ( Ys = nil_Epistemic_fm_a ) ) ) ).
% append_is_Nil_conv
thf(fact_308_map__append,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) @ ( map_Ep7084560364594560580c_fm_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_309_set__rev,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( rev_Epistemic_fm_a @ Xs ) )
= ( set_Epistemic_fm_a2 @ Xs ) ) ).
% set_rev
thf(fact_310_in__set__insert,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_311_in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_312_in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_313_append1__eq__conv,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a,Ys: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
= ( append9179727413925872949c_fm_a @ Ys @ ( cons_Epistemic_fm_a @ Y @ nil_Epistemic_fm_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_314_not__in__set__insert,axiom,
! [X3: a,Xs: list_a] :
( ~ ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X3 @ Xs )
= ( cons_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_315_not__in__set__insert,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X3 @ Xs )
= ( cons_nat @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_316_not__in__set__insert,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( insert177310161492556854c_fm_a @ X3 @ Xs )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_317_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_318_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 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_319_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_320_fm_Orel__mono,axiom,
! [R3: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R3 @ Ra )
=> ( ord_le3934200179093585166fm_a_o @ ( epistemic_rel_fm_a_a @ R3 ) @ ( epistemic_rel_fm_a_a @ Ra ) ) ) ).
% fm.rel_mono
thf(fact_321_in__set__member,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_322_in__set__member,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_323_in__set__member,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( member6038508265109909045c_fm_a @ Xs @ X3 ) ) ).
% in_set_member
thf(fact_324_fm_Orel__refl,axiom,
! [Ra: a > a > $o,X3: epistemic_fm_a] :
( ! [X2: a] : ( Ra @ X2 @ X2 )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl
thf(fact_325_fm_Orel__eq,axiom,
( ( epistemic_rel_fm_a_a
@ ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [Y3: epistemic_fm_a,Z: epistemic_fm_a] : ( Y3 = Z ) ) ) ).
% fm.rel_eq
thf(fact_326_split__list__first__prop__iff,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X ) ) )
= ( ? [Ys3: list_Epistemic_fm_a,X: epistemic_fm_a] :
( ? [Zs: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys3 @ ( cons_Epistemic_fm_a @ X @ Zs ) ) )
& ( P4 @ X )
& ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ ( set_Epistemic_fm_a2 @ Ys3 ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_327_split__list__last__prop__iff,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X ) ) )
= ( ? [Ys3: list_Epistemic_fm_a,X: epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys3 @ ( cons_Epistemic_fm_a @ X @ Zs ) ) )
& ( P4 @ X )
& ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ ( set_Epistemic_fm_a2 @ Zs ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_328_in__set__conv__decomp__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs ) ) )
& ~ ( member_a2 @ X3 @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_329_in__set__conv__decomp__first,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_330_in__set__conv__decomp__first,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys3 @ ( cons_Epistemic_fm_a @ X3 @ Zs ) ) )
& ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_331_in__set__conv__decomp__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs ) ) )
& ~ ( member_a2 @ X3 @ ( set_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_332_in__set__conv__decomp__last,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_333_in__set__conv__decomp__last,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys3 @ ( cons_Epistemic_fm_a @ X3 @ Zs ) ) )
& ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_334_split__list__first__propE,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a] :
( ? [Zs2: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
=> ( ( P4 @ X2 )
=> ~ ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_335_split__list__last__propE,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
=> ( ( P4 @ X2 )
=> ~ ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_336_split__list__first__prop,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a] :
( ? [Zs2: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
& ( P4 @ X2 )
& ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_337_split__list__last__prop,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
& ( P4 @ X2 )
& ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ ( set_Epistemic_fm_a2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_338_in__set__conv__decomp,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_339_in__set__conv__decomp,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_340_in__set__conv__decomp,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
= ( ? [Ys3: list_Epistemic_fm_a,Zs: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys3 @ ( cons_Epistemic_fm_a @ X3 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_341_append__Cons__eq__iff,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys4: list_a] :
( ~ ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_342_append__Cons__eq__iff,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys4: list_nat] :
( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_343_append__Cons__eq__iff,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Xs4: list_Epistemic_fm_a,Ys4: list_Epistemic_fm_a] :
( ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ Ys ) )
= ( append9179727413925872949c_fm_a @ Xs4 @ ( cons_Epistemic_fm_a @ X3 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_344_split__list__propE,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a] :
( ? [Zs2: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
=> ~ ( P4 @ X2 ) ) ) ).
% split_list_propE
thf(fact_345_split__list__first,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a2 @ X3 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_346_split__list__first,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_347_split__list__first,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X3 @ Zs2 ) ) )
& ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_348_split__list__prop,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_Epistemic_fm_a,X2: epistemic_fm_a] :
( ? [Zs2: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X2 @ Zs2 ) ) )
& ( P4 @ X2 ) ) ) ).
% split_list_prop
thf(fact_349_split__list__last,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) )
& ~ ( member_a2 @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_350_split__list__last,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ~ ( member_nat2 @ X3 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_351_split__list__last,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X3 @ Zs2 ) ) )
& ~ ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_352_split__list,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_353_split__list,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_354_split__list,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ? [Ys2: list_Epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( Xs
= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_355_fm_Orel__intros_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,Y51: epistemic_fm_a,X52: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_356_fm_Orel__inject_I5_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X51 @ Y51 )
& ( epistemic_rel_fm_a_a @ R3 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_357_fm_Orel__inject_I6_J,axiom,
! [R3: a > a > $o,X61: a,X62: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_358_fm_Orel__intros_I6_J,axiom,
! [R3: a > a > $o,X61: a,Y61: a,X62: epistemic_fm_a,Y62: epistemic_fm_a] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ X61 @ X62 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_359_fm_Octr__transfer_I1_J,axiom,
! [R3: a > a > $o] : ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ epistemic_FF_a ) ).
% fm.ctr_transfer(1)
thf(fact_360_append__Cons,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( append9179727413925872949c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Ys )
= ( cons_Epistemic_fm_a @ X3 @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_361_Cons__eq__appendI,axiom,
! [X3: epistemic_fm_a,Xs1: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append9179727413925872949c_fm_a @ Xs1 @ Zs3 ) )
=> ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( append9179727413925872949c_fm_a @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_362_append__Nil,axiom,
! [Ys: list_Epistemic_fm_a] :
( ( append9179727413925872949c_fm_a @ nil_Epistemic_fm_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_363_append_Oleft__neutral,axiom,
! [A2: list_Epistemic_fm_a] :
( ( append9179727413925872949c_fm_a @ nil_Epistemic_fm_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_364_eq__Nil__appendI,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs = Ys )
=> ( Xs
= ( append9179727413925872949c_fm_a @ nil_Epistemic_fm_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_365_fm_Orel__refl__strong,axiom,
! [X3: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_366_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_nat,Ra: nat > nat > $o] :
( ! [Z4: nat] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte3894023384580379906at_nat @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_367_fm_Orel__refl__strong,axiom,
! [X3: epistemic_fm_a,Ra: a > a > $o] :
( ! [Z4: a] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ X3 ) ) ).
% fm.rel_refl_strong
thf(fact_368_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > epistemic_fm_a > $o,X3: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte7774795710028497888c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_369_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > nat > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Ra: epistemic_fm_a > nat > $o] :
( ( episte8778020545599232650_a_nat @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte8778020545599232650_a_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_370_fm_Orel__mono__strong,axiom,
! [R3: nat > epistemic_fm_a > $o,X3: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Ra: nat > epistemic_fm_a > $o] :
( ( episte9034525631817513832c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte9034525631817513832c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_371_fm_Orel__mono__strong,axiom,
! [R3: nat > nat > $o,X3: epistemic_fm_nat,Y: epistemic_fm_nat,Ra: nat > nat > $o] :
( ( episte3894023384580379906at_nat @ R3 @ X3 @ Y )
=> ( ! [Z4: nat,Yb2: nat] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte3894023384580379906at_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_372_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_a > a > $o,X3: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Ra: epistemic_fm_a > a > $o] :
( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z4: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte4428145106359621316fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_373_fm_Orel__mono__strong,axiom,
! [R3: nat > a > $o,X3: epistemic_fm_nat,Y: epistemic_fm_a,Ra: nat > a > $o] :
( ( episte5492358210969815628_nat_a @ R3 @ X3 @ Y )
=> ( ! [Z4: nat,Yb2: a] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte5492358210969815628_nat_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_374_fm_Orel__mono__strong,axiom,
! [R3: a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Ra: a > epistemic_fm_a > $o] :
( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte8321036160184370300c_fm_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_375_fm_Orel__mono__strong,axiom,
! [R3: a > nat > $o,X3: epistemic_fm_a,Y: epistemic_fm_nat,Ra: a > nat > $o] :
( ( episte1460426709791529198_a_nat @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: nat] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte1460426709791529198_a_nat @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_376_fm_Orel__mono__strong,axiom,
! [R3: a > a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Ra: a > a > $o] :
( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
=> ( ! [Z4: a,Yb2: a] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ X3 ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Y ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( epistemic_rel_fm_a_a @ Ra @ X3 @ Y ) ) ) ).
% fm.rel_mono_strong
thf(fact_377_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: epistemic_fm_a > epistemic_fm_a > $o,Ra: epistemic_fm_a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: epistemic_fm_a,Yb2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte7774795710028497888c_fm_a @ R3 @ X3 @ Y )
= ( episte7774795710028497888c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_378_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: epistemic_fm_a > nat > $o,Ra: epistemic_fm_a > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: epistemic_fm_a,Yb2: nat] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte8778020545599232650_a_nat @ R3 @ X3 @ Y )
= ( episte8778020545599232650_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_379_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: nat > epistemic_fm_a > $o,Ra: nat > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: nat,Yb2: epistemic_fm_a] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte9034525631817513832c_fm_a @ R3 @ X3 @ Y )
= ( episte9034525631817513832c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_380_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: nat > nat > $o,Ra: nat > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: nat,Yb2: nat] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte3894023384580379906at_nat @ R3 @ X3 @ Y )
= ( episte3894023384580379906at_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_381_fm_Orel__cong,axiom,
! [X3: episte740340785640729014c_fm_a,Ya: episte740340785640729014c_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: epistemic_fm_a > a > $o,Ra: epistemic_fm_a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: epistemic_fm_a,Yb2: a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( episte9089240958480457552c_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte4428145106359621316fm_a_a @ R3 @ X3 @ Y )
= ( episte4428145106359621316fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_382_fm_Orel__cong,axiom,
! [X3: epistemic_fm_nat,Ya: epistemic_fm_nat,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: nat > a > $o,Ra: nat > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: nat,Yb2: a] :
( ( member_nat2 @ Z4 @ ( epistemic_set_fm_nat @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte5492358210969815628_nat_a @ R3 @ X3 @ Y )
= ( episte5492358210969815628_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_383_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: episte740340785640729014c_fm_a,Xa: episte740340785640729014c_fm_a,R3: a > epistemic_fm_a > $o,Ra: a > epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb2: epistemic_fm_a] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member6642669571620171971c_fm_a @ Yb2 @ ( episte9089240958480457552c_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte8321036160184370300c_fm_a @ R3 @ X3 @ Y )
= ( episte8321036160184370300c_fm_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_384_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_nat,Xa: epistemic_fm_nat,R3: a > nat > $o,Ra: a > nat > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb2: nat] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_nat2 @ Yb2 @ ( epistemic_set_fm_nat @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte1460426709791529198_a_nat @ R3 @ X3 @ Y )
= ( episte1460426709791529198_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_385_fm_Orel__cong,axiom,
! [X3: epistemic_fm_a,Ya: epistemic_fm_a,Y: epistemic_fm_a,Xa: epistemic_fm_a,R3: a > a > $o,Ra: a > a > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: a,Yb2: a] :
( ( member_a2 @ Z4 @ ( epistemic_set_fm_a @ Ya ) )
=> ( ( member_a2 @ Yb2 @ ( epistemic_set_fm_a @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X3 @ Y )
= ( epistemic_rel_fm_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_386_fm_Orel__intros_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,Y41: epistemic_fm_a,X42: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_387_fm_Orel__inject_I4_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X41 @ Y41 )
& ( epistemic_rel_fm_a_a @ R3 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_set__ConsD,axiom,
! [Y: a,X3: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_398_set__ConsD,axiom,
! [Y: nat,X3: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_399_set__ConsD,axiom,
! [Y: epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member6642669571620171971c_fm_a @ Y @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_400_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_401_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_402_append__eq__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Ys @ Zs3 )
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Us @ Vs ) )
& ( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Us ) )
& ( Zs3
= ( map_Ep7084560364594560580c_fm_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_403_map__eq__append__conv,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( append9179727413925872949c_fm_a @ Ys @ Zs3 ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Us @ Vs ) )
& ( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Us ) )
& ( Zs3
= ( map_Ep7084560364594560580c_fm_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_404_fm_Orel__intros_I2_J,axiom,
! [X24: list_char,Y23: list_char,R3: a > a > $o] :
( ( X24 = Y23 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) ) ) ).
% fm.rel_intros(2)
thf(fact_405_fm_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X24: list_char,Y23: list_char] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Pro_a @ Y23 ) )
= ( X24 = Y23 ) ) ).
% fm.rel_inject(2)
thf(fact_406_fm_Orel__intros_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,Y31: epistemic_fm_a,X32: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
=> ( ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 )
=> ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ) ) ).
% fm.rel_intros(3)
thf(fact_407_fm_Orel__inject_I3_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
( ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) )
= ( ( epistemic_rel_fm_a_a @ R3 @ X31 @ Y31 )
& ( epistemic_rel_fm_a_a @ R3 @ X32 @ Y32 ) ) ) ).
% fm.rel_inject(3)
thf(fact_408_list_Omap__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ( X3 = Ya )
=> ( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G4 @ Z4 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_409_list_Omap__cong0,axiom,
! [X3: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( F @ Z4 )
= ( G4 @ Z4 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ X3 ) ) ) ).
% list.map_cong0
thf(fact_410_list_Oinj__map__strong,axiom,
! [X3: list_Epistemic_fm_a,Xa: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,Fa: epistemic_fm_a > epistemic_fm_a] :
( ! [Z4: epistemic_fm_a,Za: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( member6642669571620171971c_fm_a @ Za @ ( set_Epistemic_fm_a2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_411_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z4: a] :
( ( member_a2 @ Z4 @ ( set_a2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_412_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z4: nat] :
( ( member_nat2 @ Z4 @ ( set_nat2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_413_list_Omap__ident__strong,axiom,
! [T: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_414_map__ext,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ Xs ) ) ) ).
% map_ext
thf(fact_415_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_416_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_417_map__idI,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_418_map__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G4 @ X2 ) ) )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ Xs )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ Ys ) ) ) ) ).
% map_cong
thf(fact_419_ex__map__conv,axiom,
! [Ys: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( ? [Xs3: list_Epistemic_fm_a] :
( Ys
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs3 ) ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ? [Y4: epistemic_fm_a] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_420_imply__append,axiom,
! [Ps: list_Epistemic_fm_a,Ps2: list_Epistemic_fm_a,Q: epistemic_fm_a] :
( ( epistemic_imply_a @ ( append9179727413925872949c_fm_a @ Ps @ Ps2 ) @ Q )
= ( epistemic_imply_a @ Ps @ ( epistemic_imply_a @ Ps2 @ Q ) ) ) ).
% imply_append
thf(fact_421_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P6: a > $o,Xs3: list_a] :
? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ ( set_a2 @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_422_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P6: nat > $o,Xs3: list_nat] :
? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: nat] :
( ( ( member_nat2 @ Y4 @ ( set_nat2 @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_423_list__ex1__iff,axiom,
( list_e2031426293596896995c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,Xs3: list_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ X )
& ! [Y4: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y4 @ ( set_Epistemic_fm_a2 @ Xs3 ) )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_424_fm_Orel__distinct_I30_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X51: epistemic_fm_a,X52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Imp_a @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_425_fm_Orel__distinct_I29_J,axiom,
! [R3: a > a > $o,X51: epistemic_fm_a,X52: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ X51 @ X52 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_426_fm_Orel__distinct_I8_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(8)
thf(fact_427_fm_Orel__distinct_I7_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_428_fm_Orel__distinct_I10_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(10)
thf(fact_429_fm_Orel__distinct_I9_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_430_fm_Orel__distinct_I25_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_431_fm_Orel__distinct_I26_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_432_fm_Orel__distinct_I28_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X41: epistemic_fm_a,X42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Con_a @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_433_fm_Orel__distinct_I27_J,axiom,
! [R3: a > a > $o,X41: epistemic_fm_a,X42: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ X41 @ X42 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_434_fm_Orel__distinct_I15_J,axiom,
! [R3: a > a > $o,X24: list_char,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(15)
thf(fact_435_fm_Orel__distinct_I16_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(16)
thf(fact_436_fm_Orel__distinct_I21_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y51: epistemic_fm_a,Y52: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Imp_a @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(21)
thf(fact_437_fm_Orel__distinct_I22_J,axiom,
! [R3: a > a > $o,Y51: epistemic_fm_a,Y52: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Imp_a @ Y51 @ Y52 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(22)
thf(fact_438_rev__induct,axiom,
! [P4: list_Epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( P4 @ nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( P4 @ Xs2 )
=> ( P4 @ ( append9179727413925872949c_fm_a @ Xs2 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) ) )
=> ( P4 @ Xs ) ) ) ).
% rev_induct
thf(fact_439_rev__exhaust,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ~ ! [Ys2: list_Epistemic_fm_a,Y2: epistemic_fm_a] :
( Xs
!= ( append9179727413925872949c_fm_a @ Ys2 @ ( cons_Epistemic_fm_a @ Y2 @ nil_Epistemic_fm_a ) ) ) ) ).
% rev_exhaust
thf(fact_440_Cons__eq__append__conv,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( append9179727413925872949c_fm_a @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_Epistemic_fm_a )
& ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= Zs3 ) )
| ? [Ys5: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append9179727413925872949c_fm_a @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_441_append__eq__Cons__conv,axiom,
! [Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Ys @ Zs3 )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( ( Ys = nil_Epistemic_fm_a )
& ( Zs3
= ( cons_Epistemic_fm_a @ X3 @ Xs ) ) )
| ? [Ys5: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ X3 @ Ys5 ) )
& ( ( append9179727413925872949c_fm_a @ Ys5 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_442_rev__nonempty__induct,axiom,
! [Xs: list_Epistemic_fm_a,P4: list_Epistemic_fm_a > $o] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] : ( P4 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xs2 != nil_Epistemic_fm_a )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( append9179727413925872949c_fm_a @ Xs2 @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_443_fm_Orel__distinct_I17_J,axiom,
! [R3: a > a > $o,X24: list_char,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(17)
thf(fact_444_fm_Orel__distinct_I18_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(18)
thf(fact_445_fm_Orel__distinct_I24_J,axiom,
! [R3: a > a > $o,Y61: a,Y62: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_K_a @ Y61 @ Y62 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(24)
thf(fact_446_fm_Orel__distinct_I23_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y61: a,Y62: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_K_a @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(23)
thf(fact_447_fm_Orel__distinct_I6_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(6)
thf(fact_448_fm_Orel__distinct_I5_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_449_fm_Orel__distinct_I1_J,axiom,
! [R3: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Pro_a @ Y23 ) ) ).
% fm.rel_distinct(1)
thf(fact_450_fm_Orel__distinct_I2_J,axiom,
! [R3: a > a > $o,Y23: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ Y23 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(2)
thf(fact_451_fm_Orel__distinct_I4_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ epistemic_FF_a ) ).
% fm.rel_distinct(4)
thf(fact_452_fm_Orel__distinct_I3_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ epistemic_FF_a @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(3)
thf(fact_453_fm_Orel__distinct_I13_J,axiom,
! [R3: a > a > $o,X24: list_char,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(13)
thf(fact_454_fm_Orel__distinct_I14_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(14)
thf(fact_455_fm_Orel__distinct_I20_J,axiom,
! [R3: a > a > $o,Y41: epistemic_fm_a,Y42: epistemic_fm_a,X31: epistemic_fm_a,X32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Con_a @ Y41 @ Y42 ) @ ( epistemic_Dis_a @ X31 @ X32 ) ) ).
% fm.rel_distinct(20)
thf(fact_456_fm_Orel__distinct_I19_J,axiom,
! [R3: a > a > $o,X31: epistemic_fm_a,X32: epistemic_fm_a,Y41: epistemic_fm_a,Y42: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ X31 @ X32 ) @ ( epistemic_Con_a @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(19)
thf(fact_457_fm_Orel__distinct_I12_J,axiom,
! [R3: a > a > $o,Y31: epistemic_fm_a,Y32: epistemic_fm_a,X24: list_char] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Dis_a @ Y31 @ Y32 ) @ ( epistemic_Pro_a @ X24 ) ) ).
% fm.rel_distinct(12)
thf(fact_458_fm_Orel__distinct_I11_J,axiom,
! [R3: a > a > $o,X24: list_char,Y31: epistemic_fm_a,Y32: epistemic_fm_a] :
~ ( epistemic_rel_fm_a_a @ R3 @ ( epistemic_Pro_a @ X24 ) @ ( epistemic_Dis_a @ Y31 @ Y32 ) ) ).
% fm.rel_distinct(11)
thf(fact_459_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_460_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X: a,Xs3: list_a] : ( if_list_a @ ( member_a2 @ X @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_461_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_462_List_Oinsert__def,axiom,
( insert177310161492556854c_fm_a
= ( ^ [X: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs3 ) ) @ Xs3 @ ( cons_Epistemic_fm_a @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_463_rev_Osimps_I2_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( rev_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( append9179727413925872949c_fm_a @ ( rev_Epistemic_fm_a @ Xs ) @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ) ).
% rev.simps(2)
thf(fact_464_member__rec_I1_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( member6038508265109909045c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member6038508265109909045c_fm_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_465_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_466_the__elem__set,axiom,
! [X3: epistemic_fm_a] :
( ( the_el2173195877760541071c_fm_a @ ( set_Epistemic_fm_a2 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) )
= X3 ) ).
% the_elem_set
thf(fact_467_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ Xa3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_468_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_469_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte8765170747386058258t_unit] :
( ( P4 @ M3 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ Xa3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_470_strong__soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% strong_soundness
thf(fact_471_strong__soundness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>5
thf(fact_472_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte6182337868402532512t_unit] :
( ( P4 @ M3 )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( episte6926715892928323059t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ Xa3 ) )
=> ( episte295617885132580260cs_a_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_473_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte94448284482925344t_unit] :
( ( P4 @ M3 )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( episte6390737319716712051t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte5333283044364550848c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_474_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte8765170747386058258t_unit] :
( ( P4 @ M3 )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( episte3616848269639615645t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ Xa3 ) )
=> ( episte3911673118586344362_a_nat @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_475_soundness__imply,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,Ps: list_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Ps @ P ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ ( set_Epistemic_fm_a2 @ Ps ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ) ).
% soundness_imply
thf(fact_476_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_477_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_478_can__select__def,axiom,
( can_select_a
= ( ^ [P6: a > $o,A3: set_a] :
? [X: a] :
( ( member_a2 @ X @ A3 )
& ( P6 @ X )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_479_can__select__def,axiom,
( can_se5173380710277125655c_fm_a
= ( ^ [P6: epistemic_fm_a > $o,A3: set_Epistemic_fm_a] :
? [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A3 )
& ( P6 @ X )
& ! [Y4: epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_480_can__select__def,axiom,
( can_select_nat
= ( ^ [P6: nat > $o,A3: set_nat] :
? [X: nat] :
( ( member_nat2 @ X @ A3 )
& ( P6 @ X )
& ! [Y4: nat] :
( ( ( member_nat2 @ Y4 @ A3 )
& ( P6 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_481_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2339904321507024205t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_482_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte4583239219080210381t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_483_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte3760347122651195639t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_484_soundness__Ax5,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax5_a @ P )
=> ( ( episte2449151000174023629t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax5
thf(fact_485_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_486_generalization,axiom,
! [P: epistemic_fm_a,W: a,M: episte6182337868402532512t_unit,I: a] :
( ! [M2: episte6182337868402532512t_unit,X2: a] :
( ( member_a2 @ X2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ X2 @ P ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_487_generalization,axiom,
! [P: epistemic_fm_a,W: epistemic_fm_a,M: episte94448284482925344t_unit,I: a] :
( ! [M2: episte94448284482925344t_unit,X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_488_generalization,axiom,
! [P: epistemic_fm_a,W: nat,M: episte8765170747386058258t_unit,I: a] :
( ! [M2: episte8765170747386058258t_unit,X2: nat] :
( ( member_nat2 @ X2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ X2 @ P ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_489_generalization,axiom,
! [P: epistemic_fm_a,W: set_Epistemic_fm_a,M: episte1560738328020401952t_unit,I: a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_K_a @ I @ P ) ) ) ) ).
% generalization
thf(fact_490_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte6182337868402532512t_unit > $o,P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ! [M2: episte6182337868402532512t_unit,W2: a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_a2 @ W2 @ ( episte6926715892928323059t_unit @ M2 ) )
=> ( episte295617885132580260cs_a_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_491_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte94448284482925344t_unit > $o,P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ! [M2: episte94448284482925344t_unit,W2: epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member6642669571620171971c_fm_a @ W2 @ ( episte6390737319716712051t_unit @ M2 ) )
=> ( episte5333283044364550848c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_492_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte8765170747386058258t_unit > $o,P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ! [M2: episte8765170747386058258t_unit,W2: nat,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member_nat2 @ W2 @ ( episte3616848269639615645t_unit @ M2 ) )
=> ( episte3911673118586344362_a_nat @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_493_soundness,axiom,
! [A: epistemic_fm_a > $o,P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,W2: set_Epistemic_fm_a,P3: epistemic_fm_a] :
( ( A @ P3 )
=> ( ( P4 @ M2 )
=> ( ( member536094252920883875c_fm_a @ W2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ W2 @ P3 ) ) ) )
=> ( ( epistemic_AK_a @ A @ P )
=> ( ( P4 @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ) ).
% soundness
thf(fact_494_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_495_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_496_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ ( coset_Epistemic_fm_a @ Ys ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ~ ( member6642669571620171971c_fm_a @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_497_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X @ A ) ) ) ) ).
% subset_code(2)
thf(fact_498_main_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>5
thf(fact_499_strong__completeness_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte2449151000174023629t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax5_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>5
thf(fact_500_strong__soundness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>B
thf(fact_501_strong__soundness_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>T
thf(fact_502_strong__soundness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K\<^sub>4
thf(fact_503_main_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>B
thf(fact_504_strong__completeness_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_AxB_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>B
thf(fact_505_strong__completeness_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>T
thf(fact_506_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_507_symm__trans__Euclid,axiom,
! [M: episte1560738328020401952t_unit] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( episte2449151000174023629t_unit @ M ) ) ) ).
% symm_trans_Euclid
thf(fact_508_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_509_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_510_soundness__AxT5,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax5_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte5734518988523130960t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT5
thf(fact_511_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_512_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte820475350133869606t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_513_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2438601301504999334t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_514_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte5734518988523130960t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_515_soundness__AxTB4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_AxB_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte5478016696552465318t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxTB4
thf(fact_516_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte8571156416534912283t_unit @ M )
& ( episte4939069199465351174t_unit @ M ) )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_517_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5633553332388074907t_unit @ M )
& ( episte2264091934940175238t_unit @ M ) )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_518_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte6990957894673201093t_unit @ M )
& ( episte2600384588920568880t_unit @ M ) )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_519_soundness__AxT4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( ( epistemic_AxT_a @ P )
| ( epistemic_Ax4_a @ P ) )
=> ( ( ( episte5648423998891577755t_unit @ M )
& ( episte8364071018013720454t_unit @ M ) )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT4
thf(fact_520_neg__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte820475350133869606t_unit @ M )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_521_neg__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2438601301504999334t_unit @ M )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_522_neg__introspection,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte5734518988523130960t_unit @ M )
=> ( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_523_neg__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5478016696552465318t_unit @ M )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) @ ( epistemic_K_a @ I @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ epistemic_FF_a ) ) ) ) ) ) ) ).
% neg_introspection
thf(fact_524_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_525_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_526_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_527_soundness__Ax4,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_Ax4_a @ P )
=> ( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_Ax4
thf(fact_528_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_529_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_530_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_AxT_a @ P )
=> ( ( episte6990957894673201093t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_531_soundness__AxT,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxT_a @ P )
=> ( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxT
thf(fact_532_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte6182337868402532512t_unit,W: a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte820475350133869606t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_533_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte94448284482925344t_unit,W: epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte2438601301504999334t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_534_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte8765170747386058258t_unit,W: nat] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5734518988523130960t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_535_soundness__AxB,axiom,
! [P: epistemic_fm_a,M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a] :
( ( epistemic_AxB_a @ P )
=> ( ( episte5478016696552465318t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ P ) ) ) ) ).
% soundness_AxB
thf(fact_536_pos__introspection,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte4939069199465351174t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_537_pos__introspection,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte2264091934940175238t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_538_pos__introspection,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte2600384588920568880t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_539_pos__introspection,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte8364071018013720454t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ ( epistemic_K_a @ I @ ( epistemic_K_a @ I @ P ) ) ) ) ) ) ).
% pos_introspection
thf(fact_540_truth,axiom,
! [M: episte6182337868402532512t_unit,W: a,I: a,P: epistemic_fm_a] :
( ( episte8571156416534912283t_unit @ M )
=> ( ( member_a2 @ W @ ( episte6926715892928323059t_unit @ M ) )
=> ( episte295617885132580260cs_a_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_541_truth,axiom,
! [M: episte94448284482925344t_unit,W: epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5633553332388074907t_unit @ M )
=> ( ( member6642669571620171971c_fm_a @ W @ ( episte6390737319716712051t_unit @ M ) )
=> ( episte5333283044364550848c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_542_truth,axiom,
! [M: episte8765170747386058258t_unit,W: nat,I: a,P: epistemic_fm_a] :
( ( episte6990957894673201093t_unit @ M )
=> ( ( member_nat2 @ W @ ( episte3616848269639615645t_unit @ M ) )
=> ( episte3911673118586344362_a_nat @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_543_truth,axiom,
! [M: episte1560738328020401952t_unit,W: set_Epistemic_fm_a,I: a,P: epistemic_fm_a] :
( ( episte5648423998891577755t_unit @ M )
=> ( ( member536094252920883875c_fm_a @ W @ ( episte8072386903178013299t_unit @ M ) )
=> ( episte7081087998767065248c_fm_a @ M @ W @ ( epistemic_Imp_a @ ( epistemic_K_a @ I @ P ) @ P ) ) ) ) ).
% truth
thf(fact_544_strong__completeness_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K\<^sub>4
thf(fact_545_main_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte8364071018013720454t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_Ax4_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K\<^sub>4
thf(fact_546_main_092_060_094sub_062T,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( episte5648423998891577755t_unit @ M4 )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a @ epistemic_AxT_a @ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>T
thf(fact_547_strong__soundness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5
thf(fact_548_strong__completeness_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( 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_549_main_092_060_094sub_062S_092_060_094sub_0625,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5
thf(fact_550_strong__soundness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte5478016696552465318t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>5'
thf(fact_551_map__ident,axiom,
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X: epistemic_fm_a] : X )
= ( ^ [Xs3: list_Epistemic_fm_a] : Xs3 ) ) ).
% map_ident
thf(fact_552_list_Omap__ident,axiom,
! [T: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a
@ ^ [X: epistemic_fm_a] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_553_subseqs_Osimps_I2_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( subseq859285839621985007c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( append1384085151619559355c_fm_a @ ( map_li3080774272885743684c_fm_a @ ( cons_Epistemic_fm_a @ X3 ) @ ( subseq859285839621985007c_fm_a @ Xs ) ) @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_554_arg__min__list_Osimps_I2_J,axiom,
! [F: epistemic_fm_a > nat,X3: epistemic_fm_a,Y: epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) )
= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) ) ) @ X3 @ ( arg_mi6265433823485604166_a_nat @ F @ ( cons_Epistemic_fm_a @ Y @ Zs3 ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_555_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_556_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_557_strong__soundness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a,P4: episte1560738328020401952t_unit > $o] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( P4 @ M3 )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>K
thf(fact_558_main_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>K
thf(fact_559_strong__completeness_092_060_094sub_062K,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a
@ ^ [Uu2: epistemic_fm_a] : $false
@ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ).
% strong_completeness\<^sub>K
thf(fact_560_S5__S5_H__assms,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_AxB_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_561_strong__completeness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( 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_562_main_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>4
thf(fact_563_strong__soundness_092_060_094sub_062S_092_060_094sub_0624,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ? [Qs2: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs2 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax4_a @ P2 ) )
@ ( epistemic_imply_a @ Qs2 @ P ) ) )
=> ! [M3: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M3 )
& ( episte8364071018013720454t_unit @ M3 ) )
=> ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ ( episte8072386903178013299t_unit @ M3 ) )
=> ( ! [Xa3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa3 @ G )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ Xa3 ) )
=> ( episte7081087998767065248c_fm_a @ M3 @ X4 @ P ) ) ) ) ) ).
% strong_soundness\<^sub>S\<^sub>4
thf(fact_564_strong__completeness_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ! [M2: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M2 )
& ( episte5478016696552465318t_unit @ M2 )
& ( episte8364071018013720454t_unit @ M2 ) )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( 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_565_main_092_060_094sub_062S_092_060_094sub_0625_H,axiom,
! [G: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( ! [M4: episte1560738328020401952t_unit] :
( ( ( episte5648423998891577755t_unit @ M4 )
& ( episte5478016696552465318t_unit @ M4 )
& ( episte8364071018013720454t_unit @ M4 ) )
=> ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ! [Y4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Y4 @ G )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ Y4 ) )
=> ( episte7081087998767065248c_fm_a @ M4 @ X @ P ) ) ) ) )
= ( ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ G )
& ( epistemic_AK_a
@ ^ [P2: epistemic_fm_a] :
( ( epistemic_AxT_a @ P2 )
| ( epistemic_Ax5_a @ P2 ) )
@ ( epistemic_imply_a @ Qs3 @ P ) ) ) ) ) ).
% main\<^sub>S\<^sub>5'
thf(fact_566_pred__subset__eq,axiom,
! [R3: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ R3 )
@ ^ [X: a] : ( member_a2 @ X @ S ) )
= ( ord_less_eq_set_a @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_567_pred__subset__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat2 @ X @ R3 )
@ ^ [X: nat] : ( member_nat2 @ X @ S ) )
= ( ord_less_eq_set_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_568_pred__subset__eq,axiom,
! [R3: set_Epistemic_fm_a,S: set_Epistemic_fm_a] :
( ( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ R3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ S ) )
= ( ord_le3275665582123262618c_fm_a @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_569_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a2 @ X @ A3 )
@ ^ [X: a] : ( member_a2 @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_570_less__eq__set__def,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [A3: set_Epistemic_fm_a,B5: set_Epistemic_fm_a] :
( ord_le4043730696559282883fm_a_o
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ A3 )
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_571_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat2 @ X @ A3 )
@ ^ [X: nat] : ( member_nat2 @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_572_map__rec,axiom,
( map_Ep7084560364594560580c_fm_a
= ( ^ [F2: epistemic_fm_a > epistemic_fm_a] :
( rec_li4360243526537435568c_fm_a @ nil_Epistemic_fm_a
@ ^ [X: epistemic_fm_a,Uu2: list_Epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_573_arg__min__list_Oelims,axiom,
! [X3: epistemic_fm_a > nat,Xa: list_Epistemic_fm_a,Y: epistemic_fm_a] :
( ( ( arg_mi6265433823485604166_a_nat @ X3 @ Xa )
= Y )
=> ( ! [X2: epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( Y != X2 ) )
=> ( ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Zs2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if_Epistemic_fm_a @ ( ord_less_eq_nat @ ( X3 @ X2 ) @ ( X3 @ ( arg_mi6265433823485604166_a_nat @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) ) ) @ X2 @ ( arg_mi6265433823485604166_a_nat @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa = nil_Epistemic_fm_a )
=> ( Y != undefi6158949259153642370c_fm_a ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_574_consistent__def,axiom,
( episte2285483198712856226tent_a
= ( ^ [A3: epistemic_fm_a > $o,S2: set_Epistemic_fm_a] :
~ ? [Qs3: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs3 ) @ S2 )
& ( epistemic_AK_a @ A3 @ ( epistemic_imply_a @ Qs3 @ epistemic_FF_a ) ) ) ) ) ).
% consistent_def
thf(fact_575_consistent__hereditary,axiom,
! [A: epistemic_fm_a > $o,S: set_Epistemic_fm_a,S3: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ S3 @ S )
=> ( episte2285483198712856226tent_a @ A @ S3 ) ) ) ).
% consistent_hereditary
thf(fact_576_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_577_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_578_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_579_ax__in__maximal,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( A @ P )
=> ( member6642669571620171971c_fm_a @ P @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_580_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_581_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_582_maximal__extension,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ V @ W4 )
=> ( ( episte2285483198712856226tent_a @ A @ W4 )
=> ~ ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W4 ) ) ) ) ).
% maximal_extension
thf(fact_583_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_584_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_585_Succ__def,axiom,
( bNF_Gr1807846264492387989c_fm_a
= ( ^ [Kl: set_li4204741992506657632c_fm_a,Kl2: list_s580375451141968640c_fm_a] :
( collec2519470961442302949c_fm_a
@ ^ [K: set_Epistemic_fm_a] : ( member2328277852289759785c_fm_a @ ( append5257919162946131733c_fm_a @ Kl2 @ ( cons_s4394448069393209776c_fm_a @ K @ nil_se102576565402781184c_fm_a ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_586_Succ__def,axiom,
( bNF_Gr1093487135560555701c_fm_a
= ( ^ [Kl: set_li769143395467472256c_fm_a,Kl2: list_Epistemic_fm_a] :
( collec4904205152690461189c_fm_a
@ ^ [K: epistemic_fm_a] : ( member5906877432388582473c_fm_a @ ( append9179727413925872949c_fm_a @ Kl2 @ ( cons_Epistemic_fm_a @ K @ nil_Epistemic_fm_a ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_587_SuccI,axiom,
! [Kl3: list_a,K2: a,Kl4: set_list_a] :
( ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K2 @ nil_a ) ) @ Kl4 )
=> ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_588_SuccI,axiom,
! [Kl3: list_nat,K2: nat,Kl4: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl4 )
=> ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_589_SuccI,axiom,
! [Kl3: list_Epistemic_fm_a,K2: epistemic_fm_a,Kl4: set_li769143395467472256c_fm_a] :
( ( member5906877432388582473c_fm_a @ ( append9179727413925872949c_fm_a @ Kl3 @ ( cons_Epistemic_fm_a @ K2 @ nil_Epistemic_fm_a ) ) @ Kl4 )
=> ( member6642669571620171971c_fm_a @ K2 @ ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_590_SuccD,axiom,
! [K2: a,Kl4: set_list_a,Kl3: list_a] :
( ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) )
=> ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K2 @ nil_a ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_591_SuccD,axiom,
! [K2: nat,Kl4: set_list_nat,Kl3: list_nat] :
( ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) )
=> ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_592_SuccD,axiom,
! [K2: epistemic_fm_a,Kl4: set_li769143395467472256c_fm_a,Kl3: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ K2 @ ( bNF_Gr1093487135560555701c_fm_a @ Kl4 @ Kl3 ) )
=> ( member5906877432388582473c_fm_a @ ( append9179727413925872949c_fm_a @ Kl3 @ ( cons_Epistemic_fm_a @ K2 @ nil_Epistemic_fm_a ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_593_concat__eq__append__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( concat2780896542826320603c_fm_a @ Xss2 )
= ( append9179727413925872949c_fm_a @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_li2451196919128234278c_fm_a )
=> ( ( Ys = nil_Epistemic_fm_a )
& ( Zs3 = nil_Epistemic_fm_a ) ) )
& ( ( Xss2 != nil_li2451196919128234278c_fm_a )
=> ? [Xss1: list_l6083326122719238310c_fm_a,Xs3: list_Epistemic_fm_a,Xs5: list_Epistemic_fm_a,Xss22: list_l6083326122719238310c_fm_a] :
( ( Xss2
= ( append1384085151619559355c_fm_a @ Xss1 @ ( cons_l8134865115577406678c_fm_a @ ( append9179727413925872949c_fm_a @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append9179727413925872949c_fm_a @ ( concat2780896542826320603c_fm_a @ Xss1 ) @ Xs3 ) )
& ( Zs3
= ( append9179727413925872949c_fm_a @ Xs5 @ ( concat2780896542826320603c_fm_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_594_concat__eq__Nil__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( ( concat2780896542826320603c_fm_a @ Xss2 )
= nil_Epistemic_fm_a )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_595_Nil__eq__concat__conv,axiom,
! [Xss2: list_l6083326122719238310c_fm_a] :
( ( nil_Epistemic_fm_a
= ( concat2780896542826320603c_fm_a @ Xss2 ) )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xss2 ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_596_concat__map__singleton,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( concat2780896542826320603c_fm_a
@ ( map_Ep1988204514693272906c_fm_a
@ ^ [X: epistemic_fm_a] : ( cons_Epistemic_fm_a @ ( F @ X ) @ nil_Epistemic_fm_a )
@ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_597_concat_Osimps_I1_J,axiom,
( ( concat2780896542826320603c_fm_a @ nil_li2451196919128234278c_fm_a )
= nil_Epistemic_fm_a ) ).
% concat.simps(1)
thf(fact_598_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_599_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
@ ^ [X: epistemic_fm_a] : ( map_li3080774272885743684c_fm_a @ ( cons_Epistemic_fm_a @ X ) @ ( produc1391074687832863881c_fm_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_600_empty__Shift,axiom,
! [Kl4: set_list_a,K2: a] :
( ( member_list_a @ nil_a @ Kl4 )
=> ( ( member_a2 @ K2 @ ( bNF_Greatest_Succ_a @ Kl4 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_601_empty__Shift,axiom,
! [Kl4: set_list_nat,K2: nat] :
( ( member_list_nat @ nil_nat @ Kl4 )
=> ( ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_602_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_603_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_604_butlast__snoc,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( butlas2200625790165101420c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_605_subset__subseqs,axiom,
! [X5: set_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X5 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member536094252920883875c_fm_a @ X5 @ ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_606_subset__subseqs,axiom,
! [X5: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X5 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_607_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_608_butlast_Osimps_I1_J,axiom,
( ( butlas2200625790165101420c_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% butlast.simps(1)
thf(fact_609_in__set__butlastD,axiom,
! [X3: a,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_610_in__set__butlastD,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_611_in__set__butlastD,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( butlas2200625790165101420c_fm_a @ Xs ) ) )
=> ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_612_map__butlast,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( map_Ep7084560364594560580c_fm_a @ F @ ( butlas2200625790165101420c_fm_a @ Xs ) )
= ( butlas2200625790165101420c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_613_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_614_butlast_Osimps_I2_J,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
=> ( ( butlas2200625790165101420c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= nil_Epistemic_fm_a ) )
& ( ( Xs != nil_Epistemic_fm_a )
=> ( ( butlas2200625790165101420c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( cons_Epistemic_fm_a @ X3 @ ( butlas2200625790165101420c_fm_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_615_butlast__append,axiom,
! [Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( Ys = nil_Epistemic_fm_a )
=> ( ( butlas2200625790165101420c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( butlas2200625790165101420c_fm_a @ Xs ) ) )
& ( ( Ys != nil_Epistemic_fm_a )
=> ( ( butlas2200625790165101420c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ Xs @ ( butlas2200625790165101420c_fm_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_616_in__set__butlast__appendI,axiom,
! [X3: a,Xs: list_a,Ys: list_a] :
( ( ( member_a2 @ X3 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a2 @ X3 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_617_in__set__butlast__appendI,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat2 @ X3 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat2 @ X3 @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat2 @ X3 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_618_in__set__butlast__appendI,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( butlas2200625790165101420c_fm_a @ Xs ) ) )
| ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( butlas2200625790165101420c_fm_a @ Ys ) ) ) )
=> ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( butlas2200625790165101420c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_619_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_620_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_621_transpose_Oelims,axiom,
! [X3: list_l6083326122719238310c_fm_a,Y: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ X3 )
= Y )
=> ( ( ( X3 = nil_li2451196919128234278c_fm_a )
=> ( Y != nil_li2451196919128234278c_fm_a ) )
=> ( ! [Xss: list_l6083326122719238310c_fm_a] :
( ( X3
= ( cons_l8134865115577406678c_fm_a @ nil_Epistemic_fm_a @ Xss ) )
=> ( Y
!= ( transp2882070860200649898c_fm_a @ Xss ) ) )
=> ~ ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( X3
= ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons_l8134865115577406678c_fm_a
@ ( cons_Epistemic_fm_a @ X2
@ ( 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 @ Xs2
@ ( 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_622_transpose_Osimps_I3_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,Xss2: list_l6083326122719238310c_fm_a] :
( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Xss2 ) )
= ( 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 ) )
@ 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_623_set__concat,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( set_Epistemic_fm_a2 @ ( concat2780896542826320603c_fm_a @ Xs ) )
= ( comple7868773486375872231c_fm_a @ ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ Xs ) ) ) ) ).
% set_concat
thf(fact_624_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 ) ) ) )
=> ( ! [X2: epistemic_fm_a,Xs2: list_Epistemic_fm_a,Xss: list_l6083326122719238310c_fm_a] :
( ( accp_l7472382826029824047c_fm_a @ transp597054356388329423c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) @ Xss ) )
=> ( ( P4
@ ( cons_l8134865115577406678c_fm_a @ Xs2
@ ( 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 @ X2 @ Xs2 ) @ Xss ) ) ) )
=> ( P4 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_625_append__butlast__last__id,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( append9179727413925872949c_fm_a @ ( butlas2200625790165101420c_fm_a @ Xs ) @ ( cons_Epistemic_fm_a @ ( last_Epistemic_fm_a @ Xs ) @ nil_Epistemic_fm_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_626_last__appendL,axiom,
! [Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( Ys = nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( last_Epistemic_fm_a @ Xs ) ) ) ).
% last_appendL
thf(fact_627_last__appendR,axiom,
! [Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( Ys != nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( last_Epistemic_fm_a @ Ys ) ) ) ).
% last_appendR
thf(fact_628_last__snoc,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( last_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) )
= X3 ) ).
% last_snoc
thf(fact_629_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_630_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_631_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_632_last_Osimps,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs != nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( last_Epistemic_fm_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_633_last__ConsL,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( Xs = nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_634_last__ConsR,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( last_Epistemic_fm_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_635_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a2 @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_636_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_637_last__in__set,axiom,
! [As2: list_Epistemic_fm_a] :
( ( As2 != nil_Epistemic_fm_a )
=> ( member6642669571620171971c_fm_a @ ( last_Epistemic_fm_a @ As2 ) @ ( set_Epistemic_fm_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_638_last__map,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( F @ ( last_Epistemic_fm_a @ Xs ) ) ) ) ).
% last_map
thf(fact_639_last__append,axiom,
! [Ys: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( Ys = nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( last_Epistemic_fm_a @ Xs ) ) )
& ( ( Ys != nil_Epistemic_fm_a )
=> ( ( last_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( last_Epistemic_fm_a @ Ys ) ) ) ) ).
% last_append
thf(fact_640_longest__common__suffix,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
? [Ss: list_Epistemic_fm_a,Xs6: list_Epistemic_fm_a,Ys6: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Xs6 @ Ss ) )
& ( Ys
= ( append9179727413925872949c_fm_a @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_Epistemic_fm_a )
| ( Ys6 = nil_Epistemic_fm_a )
| ( ( last_Epistemic_fm_a @ Xs6 )
!= ( last_Epistemic_fm_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_641_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_642_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_643_set__list__bind,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( bind_E8451893407412458119c_fm_a @ Xs @ F ) )
= ( comple7868773486375872231c_fm_a
@ ( image_1263732536703906021c_fm_a
@ ^ [X: epistemic_fm_a] : ( set_Epistemic_fm_a2 @ ( F @ X ) )
@ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_644_transpose__empty,axiom,
! [Xs: list_l6083326122719238310c_fm_a] :
( ( ( transp2882070860200649898c_fm_a @ Xs )
= nil_li2451196919128234278c_fm_a )
= ( ! [X: list_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ X @ ( set_li8442223810127165109c_fm_a @ Xs ) )
=> ( X = nil_Epistemic_fm_a ) ) ) ) ).
% transpose_empty
thf(fact_645_snoc__eq__iff__butlast,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
= Ys )
= ( ( Ys != nil_Epistemic_fm_a )
& ( ( butlas2200625790165101420c_fm_a @ Ys )
= Xs )
& ( ( last_Epistemic_fm_a @ Ys )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_646_transpose_Opsimps_I3_J,axiom,
! [X3: 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 @ X3 @ Xs ) @ Xss2 ) )
=> ( ( transp2882070860200649898c_fm_a @ ( cons_l8134865115577406678c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Xss2 ) )
= ( 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 ) )
@ 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_647_SUP__subset__mono,axiom,
! [A: set_a,B4: set_a,F: a > epistemic_fm_a > $o,G4: a > epistemic_fm_a > $o] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_648_SUP__subset__mono,axiom,
! [A: set_a,B4: set_a,F: a > set_Epistemic_fm_a,G4: a > set_Epistemic_fm_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_649_SUP__subset__mono,axiom,
! [A: set_a,B4: set_a,F: a > set_nat,G4: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_650_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,G4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_651_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,G4: epistemic_fm_a > set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_652_SUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,G4: epistemic_fm_a > set_nat] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_653_SUP__subset__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > epistemic_fm_a > $o,G4: nat > epistemic_fm_a > $o] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_654_SUP__subset__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_Epistemic_fm_a,G4: nat > set_Epistemic_fm_a] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_655_SUP__subset__mono,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_nat,G4: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_656_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_657_SUP__upper2,axiom,
! [I: a,A: set_a,U2: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_658_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_659_SUP__upper2,axiom,
! [I: nat,A: set_nat,U2: epistemic_fm_a > $o,F: nat > epistemic_fm_a > $o] :
( ( member_nat2 @ I @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ I ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_660_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_661_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_662_SUP__upper2,axiom,
! [I: nat,A: set_nat,U2: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a] :
( ( member_nat2 @ I @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ I ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_663_SUP__upper2,axiom,
! [I: a,A: set_a,U2: set_nat,F: a > set_nat] :
( ( member_a2 @ I @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_664_SUP__upper2,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,U2: set_nat,F: epistemic_fm_a > set_nat] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_665_SUP__upper2,axiom,
! [I: nat,A: set_nat,U2: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_666_Sup__eqI,axiom,
! [A: set_Epistemic_fm_a_o,X3: epistemic_fm_a > $o] :
( ! [Y2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Y2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Y2 @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [Z5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ Z5 @ A )
=> ( ord_le4043730696559282883fm_a_o @ Z5 @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_667_Sup__eqI,axiom,
! [A: set_se5208064806568342746c_fm_a,X3: set_Epistemic_fm_a] :
( ! [Y2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Y2 @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [Z5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z5 @ A )
=> ( ord_le3275665582123262618c_fm_a @ Z5 @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_668_Sup__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A )
=> ( ord_less_eq_set_nat @ Y2 @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [Z5: set_nat] :
( ( member_set_nat @ Z5 @ A )
=> ( ord_less_eq_set_nat @ Z5 @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_669_Sup__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ! [A5: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ A5 @ A )
=> ? [X4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X4 @ B4 )
& ( ord_le4043730696559282883fm_a_o @ A5 @ X4 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_mono
thf(fact_670_Sup__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ! [A5: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ A5 @ A )
=> ? [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ A5 @ X4 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_mono
thf(fact_671_Sup__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B4 )
& ( ord_less_eq_set_nat @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).
% Sup_mono
thf(fact_672_Sup__least,axiom,
! [A: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_673_Sup__least,axiom,
! [A: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_674_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_675_Sup__upper,axiom,
! [X3: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ A ) ) ) ).
% Sup_upper
thf(fact_676_Sup__upper,axiom,
! [X3: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ).
% Sup_upper
thf(fact_677_Sup__upper,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_678_Sup__le__iff,axiom,
! [A: set_Epistemic_fm_a_o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ B )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_679_Sup__le__iff,axiom,
! [A: set_se5208064806568342746c_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ B )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_680_Sup__le__iff,axiom,
! [A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_681_Sup__upper2,axiom,
! [U2: epistemic_fm_a > $o,A: set_Epistemic_fm_a_o,V2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ U2 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ V2 @ U2 )
=> ( ord_le4043730696559282883fm_a_o @ V2 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_682_Sup__upper2,axiom,
! [U2: set_Epistemic_fm_a,A: set_se5208064806568342746c_fm_a,V2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ U2 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ V2 @ U2 )
=> ( ord_le3275665582123262618c_fm_a @ V2 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_683_Sup__upper2,axiom,
! [U2: set_nat,A: set_set_nat,V2: set_nat] :
( ( member_set_nat @ U2 @ A )
=> ( ( ord_less_eq_set_nat @ V2 @ U2 )
=> ( ord_less_eq_set_nat @ V2 @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_684_accp__subset__induct,axiom,
! [D: epistemic_fm_a > $o,R3: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ D @ ( accp_Epistemic_fm_a @ R3 ) )
=> ( ! [X2: epistemic_fm_a,Z4: epistemic_fm_a] :
( ( D @ X2 )
=> ( ( R3 @ Z4 @ X2 )
=> ( D @ Z4 ) ) )
=> ( ( D @ X3 )
=> ( ! [X2: epistemic_fm_a] :
( ( D @ X2 )
=> ( ! [Z5: epistemic_fm_a] :
( ( R3 @ Z5 @ X2 )
=> ( P4 @ Z5 ) )
=> ( P4 @ X2 ) ) )
=> ( P4 @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_685_SUP__eq,axiom,
! [A: set_a,B4: set_a,F: a > set_nat,G4: a > set_nat] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X4: a] :
( ( member_a2 @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B4 )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_686_SUP__eq,axiom,
! [A: set_a,B4: set_nat,F: a > set_nat,G4: nat > set_nat] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat2 @ J @ B4 )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_687_SUP__eq,axiom,
! [A: set_nat,B4: set_a,F: nat > set_nat,G4: a > set_nat] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ? [X4: a] :
( ( member_a2 @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_688_SUP__eq,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_nat,G4: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat2 @ J @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_689_SUP__eq,axiom,
! [A: set_a,B4: set_a,F: a > set_Epistemic_fm_a,G4: a > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X4: a] :
( ( member_a2 @ X4 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B4 )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_690_SUP__eq,axiom,
! [A: set_a,B4: set_nat,F: a > set_Epistemic_fm_a,G4: nat > set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat2 @ J @ B4 )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_691_SUP__eq,axiom,
! [A: set_nat,B4: set_a,F: nat > set_Epistemic_fm_a,G4: a > set_Epistemic_fm_a] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ? [X4: a] :
( ( member_a2 @ X4 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_692_SUP__eq,axiom,
! [A: set_nat,B4: set_nat,F: nat > set_Epistemic_fm_a,G4: nat > set_Epistemic_fm_a] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ B4 )
& ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat2 @ J @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_le3275665582123262618c_fm_a @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
= ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_693_SUP__eq,axiom,
! [A: set_a,B4: set_Epistemic_fm_a,F: a > set_nat,G4: epistemic_fm_a > set_nat] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ J @ B4 )
=> ? [X4: a] :
( ( member_a2 @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_694_SUP__eq,axiom,
! [A: set_Epistemic_fm_a,B4: set_a,F: epistemic_fm_a > set_nat,G4: a > set_nat] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ? [X4: a] :
( ( member_a2 @ X4 @ B4 )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G4 @ X4 ) ) ) )
=> ( ! [J: a] :
( ( member_a2 @ J @ B4 )
=> ? [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G4 @ J ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G4 @ B4 ) ) ) ) ) ).
% SUP_eq
thf(fact_695_Sup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_696_Sup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_697_Sup__subset__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).
% Sup_subset_mono
thf(fact_698_SUP__eqI,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_699_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_700_SUP__eqI,axiom,
! [A: set_nat,F: nat > epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_701_SUP__eqI,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_702_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_703_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I4 ) @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_704_SUP__eqI,axiom,
! [A: set_a,F: a > set_nat,X3: set_nat] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [I4: a] :
( ( member_a2 @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_705_SUP__eqI,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,X3: set_nat] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [I4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_706_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_nat,X3: set_nat] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y2: set_nat] :
( ! [I4: nat] :
( ( member_nat2 @ I4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_707_SUP__least,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_708_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_709_SUP__least,axiom,
! [A: set_nat,F: nat > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I3 ) @ U2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_710_SUP__least,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_711_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_712_SUP__least,axiom,
! [A: set_nat,F: nat > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I3 ) @ U2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_713_SUP__least,axiom,
! [A: set_a,F: a > set_nat,U2: set_nat] :
( ! [I3: a] :
( ( member_a2 @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_714_SUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,U2: set_nat] :
( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_715_SUP__least,axiom,
! [A: set_nat,F: nat > set_nat,U2: set_nat] :
( ! [I3: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U2 ) ) ).
% SUP_least
thf(fact_716_SUP__upper,axiom,
! [I: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_717_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_718_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > epistemic_fm_a > $o] :
( ( member_nat2 @ I @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ I ) @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_719_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_720_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_721_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_Epistemic_fm_a] :
( ( member_nat2 @ I @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ I ) @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_722_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_nat] :
( ( member_a2 @ I @ A )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_723_SUP__upper,axiom,
! [I: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat] :
( ( member6642669571620171971c_fm_a @ I @ A )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_724_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat2 @ I @ A )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_725_cSup__eq__maximum,axiom,
! [Z2: epistemic_fm_a > $o,X5: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ Z2 @ X5 )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ( comple310706794042174646fm_a_o @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_726_cSup__eq__maximum,axiom,
! [Z2: set_Epistemic_fm_a,X5: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ Z2 @ X5 )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ( comple7868773486375872231c_fm_a @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_727_cSup__eq__maximum,axiom,
! [Z2: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z2 @ X5 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X5 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_728_cSup__eq__maximum,axiom,
! [Z2: nat,X5: set_nat] :
( ( member_nat2 @ Z2 @ X5 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_729_subseqs__powset,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( image_971165786557580383c_fm_a @ set_Epistemic_fm_a2 @ ( set_li8442223810127165109c_fm_a @ ( subseq859285839621985007c_fm_a @ Xs ) ) )
= ( pow_Epistemic_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_730_f__arg__min__list__f,axiom,
! [Xs: list_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( F @ ( arg_mi6265433823485604166_a_nat @ F @ Xs ) )
= ( lattic8721135487736765967in_nat @ ( image_3638449696541059749_a_nat @ F @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_731_min__list__Min,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( min_list_nat @ Xs )
= ( lattic8721135487736765967in_nat @ ( set_nat2 @ Xs ) ) ) ) ).
% min_list_Min
thf(fact_732_list__ex1__simps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_e2031426293596896995c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( ( P4 @ X3 )
=> ( list_a5841931967666341838c_fm_a
@ ^ [Y4: epistemic_fm_a] :
( ~ ( P4 @ Y4 )
| ( X3 = Y4 ) )
@ Xs ) )
& ( ~ ( P4 @ X3 )
=> ( list_e2031426293596896995c_fm_a @ P4 @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_733_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_734_rotate1_Osimps_I2_J,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( rotate3540958558250653293c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ) ).
% rotate1.simps(2)
thf(fact_735_filter__True,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X2 ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_736_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_737_list__all__simps_I1_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( list_a5841931967666341838c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( P4 @ X3 )
& ( list_a5841931967666341838c_fm_a @ P4 @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_738_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_739_rotate1__is__Nil__conv,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( rotate3540958558250653293c_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_740_set__rotate1,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( rotate3540958558250653293c_fm_a @ Xs ) )
= ( set_Epistemic_fm_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_741_set__filter,axiom,
! [P4: a > $o,Xs: list_a] :
( ( set_a2 @ ( filter_a @ P4 @ Xs ) )
= ( collect_a
@ ^ [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_742_set__filter,axiom,
! [P4: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P4 @ Xs ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_743_set__filter,axiom,
! [P4: set_Epistemic_fm_a > $o,Xs: list_s580375451141968640c_fm_a] :
( ( set_se8855941934996405007c_fm_a @ ( filter2620125754611450615c_fm_a @ P4 @ Xs ) )
= ( collec2519470961442302949c_fm_a
@ ^ [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( set_se8855941934996405007c_fm_a @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_744_set__filter,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) )
= ( collec4904205152690461189c_fm_a
@ ^ [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
& ( P4 @ X ) ) ) ) ).
% set_filter
thf(fact_745_filter__False,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X2 ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a ) ) ).
% filter_False
thf(fact_746_tl__append2,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_747_filter__cong,axiom,
! [Xs: list_a,Ys: list_a,P4: a > $o,Q4: a > $o] :
( ( Xs = Ys )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter_a @ P4 @ Xs )
= ( filter_a @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_748_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P4: nat > $o,Q4: nat > $o] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter_nat @ P4 @ Xs )
= ( filter_nat @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_749_filter__cong,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( Xs = Ys )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Ys ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= ( filter7636273809395300631c_fm_a @ Q4 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_750_filter__id__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= Xs )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X ) ) ) ) ).
% filter_id_conv
thf(fact_751_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_752_filter_Osimps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( P4 @ X3 )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( cons_Epistemic_fm_a @ X3 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) )
& ( ~ ( P4 @ X3 )
=> ( ( filter7636273809395300631c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_753_list_Osel_I2_J,axiom,
( ( tl_Epistemic_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% list.sel(2)
thf(fact_754_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_755_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_756_list__all__cong,axiom,
! [X3: list_a,Ya: list_a,P4: a > $o,Pa: a > $o] :
( ( X3 = Ya )
=> ( ! [Z4: a] :
( ( member_a2 @ Z4 @ ( set_a2 @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( list_all_a @ P4 @ X3 )
= ( list_all_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_757_list__all__cong,axiom,
! [X3: list_nat,Ya: list_nat,P4: nat > $o,Pa: nat > $o] :
( ( X3 = Ya )
=> ( ! [Z4: nat] :
( ( member_nat2 @ Z4 @ ( set_nat2 @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( list_all_nat @ P4 @ X3 )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_758_list__all__cong,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( X3 = Ya )
=> ( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ Ya ) )
=> ( ( P4 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
= ( list_a5841931967666341838c_fm_a @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_759_list_Opred__mono__strong,axiom,
! [P4: a > $o,X3: list_a,Pa: a > $o] :
( ( list_all_a @ P4 @ X3 )
=> ( ! [Z4: a] :
( ( member_a2 @ Z4 @ ( set_a2 @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( list_all_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_760_list_Opred__mono__strong,axiom,
! [P4: nat > $o,X3: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P4 @ X3 )
=> ( ! [Z4: nat] :
( ( member_nat2 @ Z4 @ ( set_nat2 @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( list_all_nat @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_761_list_Opred__mono__strong,axiom,
! [P4: epistemic_fm_a > $o,X3: list_Epistemic_fm_a,Pa: epistemic_fm_a > $o] :
( ( list_a5841931967666341838c_fm_a @ P4 @ X3 )
=> ( ! [Z4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Z4 @ ( set_Epistemic_fm_a2 @ X3 ) )
=> ( ( P4 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( list_a5841931967666341838c_fm_a @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_762_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_763_list_Opred__mono,axiom,
! [P4: epistemic_fm_a > $o,Pa: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ P4 @ Pa )
=> ( ord_le5289359451089373629fm_a_o @ ( list_a5841931967666341838c_fm_a @ P4 ) @ ( list_a5841931967666341838c_fm_a @ Pa ) ) ) ).
% list.pred_mono
thf(fact_764_rotate1_Osimps_I1_J,axiom,
( ( rotate3540958558250653293c_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% rotate1.simps(1)
thf(fact_765_rotate1__map,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( rotate3540958558250653293c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( map_Ep7084560364594560580c_fm_a @ F @ ( rotate3540958558250653293c_fm_a @ Xs ) ) ) ).
% rotate1_map
thf(fact_766_list_Omap__cong__pred,axiom,
! [X3: list_Epistemic_fm_a,Ya: list_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a,G4: epistemic_fm_a > epistemic_fm_a] :
( ( X3 = Ya )
=> ( ( list_a5841931967666341838c_fm_a
@ ^ [Z3: epistemic_fm_a] :
( ( F @ Z3 )
= ( G4 @ Z3 ) )
@ Ya )
=> ( ( map_Ep7084560364594560580c_fm_a @ F @ X3 )
= ( map_Ep7084560364594560580c_fm_a @ G4 @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_767_filter__is__subset,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ ( filter7636273809395300631c_fm_a @ P4 @ Xs ) ) @ ( set_Epistemic_fm_a2 @ Xs ) ) ).
% filter_is_subset
thf(fact_768_filter__is__subset,axiom,
! [P4: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P4 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_769_empty__filter__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( filter7636273809395300631c_fm_a @ P4 @ Xs ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_770_filter__empty__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ~ ( P4 @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_771_tl__Nil,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( tl_Epistemic_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ) ) ).
% tl_Nil
thf(fact_772_Nil__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( nil_Epistemic_fm_a
= ( tl_Epistemic_fm_a @ Xs ) )
= ( ( Xs = nil_Epistemic_fm_a )
| ? [X: epistemic_fm_a] :
( Xs
= ( cons_Epistemic_fm_a @ X @ nil_Epistemic_fm_a ) ) ) ) ).
% Nil_tl
thf(fact_773_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X3: a] :
( ( A2 != nil_a )
=> ( ( member_a2 @ X3 @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_774_list_Oset__sel_I2_J,axiom,
! [A2: list_nat,X3: nat] :
( ( A2 != nil_nat )
=> ( ( member_nat2 @ X3 @ ( set_nat2 @ ( tl_nat @ A2 ) ) )
=> ( member_nat2 @ X3 @ ( set_nat2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_775_list_Oset__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_a,X3: epistemic_fm_a] :
( ( A2 != nil_Epistemic_fm_a )
=> ( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( tl_Epistemic_fm_a @ A2 ) ) )
=> ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_776_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_777_tl__append__if,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( tl_Epistemic_fm_a @ Ys ) ) )
& ( ( Xs != nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_778_last__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
| ( ( tl_Epistemic_fm_a @ Xs )
!= nil_Epistemic_fm_a ) )
=> ( ( last_Epistemic_fm_a @ ( tl_Epistemic_fm_a @ Xs ) )
= ( last_Epistemic_fm_a @ Xs ) ) ) ).
% last_tl
thf(fact_779_Cons__eq__filterD,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( filter7636273809395300631c_fm_a @ P4 @ Ys ) )
=> ? [Us2: list_Epistemic_fm_a,Vs2: list_Epistemic_fm_a] :
( ( Ys
= ( append9179727413925872949c_fm_a @ Us2 @ ( cons_Epistemic_fm_a @ X3 @ Vs2 ) ) )
& ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Us2 ) )
=> ~ ( P4 @ X4 ) )
& ( P4 @ X3 )
& ( Xs
= ( filter7636273809395300631c_fm_a @ P4 @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_780_filter__eq__ConsD,axiom,
! [P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Ys )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) )
=> ? [Us2: list_Epistemic_fm_a,Vs2: list_Epistemic_fm_a] :
( ( Ys
= ( append9179727413925872949c_fm_a @ Us2 @ ( cons_Epistemic_fm_a @ X3 @ Vs2 ) ) )
& ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Us2 ) )
=> ~ ( P4 @ X4 ) )
& ( P4 @ X3 )
& ( Xs
= ( filter7636273809395300631c_fm_a @ P4 @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_781_Cons__eq__filter__iff,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] :
( ( ( cons_Epistemic_fm_a @ X3 @ Xs )
= ( filter7636273809395300631c_fm_a @ P4 @ Ys ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Ys
= ( append9179727413925872949c_fm_a @ Us @ ( cons_Epistemic_fm_a @ X3 @ Vs ) ) )
& ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Us ) )
=> ~ ( P4 @ X ) )
& ( P4 @ X3 )
& ( Xs
= ( filter7636273809395300631c_fm_a @ P4 @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_782_filter__eq__Cons__iff,axiom,
! [P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( filter7636273809395300631c_fm_a @ P4 @ Ys )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ? [Us: list_Epistemic_fm_a,Vs: list_Epistemic_fm_a] :
( ( Ys
= ( append9179727413925872949c_fm_a @ Us @ ( cons_Epistemic_fm_a @ X3 @ Vs ) ) )
& ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Us ) )
=> ~ ( P4 @ X ) )
& ( P4 @ X3 )
& ( Xs
= ( filter7636273809395300631c_fm_a @ P4 @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_783_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_784_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_785_min__list_Oelims,axiom,
! [X3: list_nat,Y: nat] :
( ( ( min_list_nat @ X3 )
= Y )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X3
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y
!= ( case_list_nat_nat @ X2
@ ^ [A4: nat,List2: list_nat] : ( ord_min_nat @ X2 @ ( min_list_nat @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X3 = nil_nat )
=> ( Y != undefined_nat ) ) ) ) ).
% min_list.elims
thf(fact_786_rotate1__hd__tl,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( rotate3540958558250653293c_fm_a @ Xs )
= ( append9179727413925872949c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ ( cons_Epistemic_fm_a @ ( hd_Epistemic_fm_a @ Xs ) @ nil_Epistemic_fm_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_787_hd__append2,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( hd_Epistemic_fm_a @ Xs ) ) ) ).
% hd_append2
thf(fact_788_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_789_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_790_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_791_min__absorb2,axiom,
! [Y: epistemic_fm_a > $o,X3: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( ord_mi8079229082941695292fm_a_o @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_792_min__absorb2,axiom,
! [Y: set_Epistemic_fm_a,X3: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( ord_mi5245583150369600225c_fm_a @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_793_min__absorb2,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_min_set_nat @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_794_min__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_min_nat @ X3 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_795_min__absorb1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
=> ( ( ord_mi8079229082941695292fm_a_o @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_796_min__absorb1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
=> ( ( ord_mi5245583150369600225c_fm_a @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_797_min__absorb1,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_min_set_nat @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_798_min__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_min_nat @ X3 @ Y )
= X3 ) ) ).
% min_absorb1
thf(fact_799_min__def,axiom,
( ord_mi8079229082941695292fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_800_min__def,axiom,
( ord_mi5245583150369600225c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_801_min__def,axiom,
( ord_min_set_nat
= ( ^ [A4: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_802_min__def,axiom,
( ord_min_nat
= ( ^ [A4: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def
thf(fact_803_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_804_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a2 @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_805_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_806_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_807_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a2 @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_808_list_Oset__sel_I1_J,axiom,
! [A2: list_nat] :
( ( A2 != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_809_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_810_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_811_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_812_longest__common__prefix,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
? [Ps3: list_Epistemic_fm_a,Xs6: list_Epistemic_fm_a,Ys6: list_Epistemic_fm_a] :
( ( Xs
= ( append9179727413925872949c_fm_a @ Ps3 @ Xs6 ) )
& ( Ys
= ( append9179727413925872949c_fm_a @ Ps3 @ Ys6 ) )
& ( ( Xs6 = nil_Epistemic_fm_a )
| ( Ys6 = nil_Epistemic_fm_a )
| ( ( hd_Epistemic_fm_a @ Xs6 )
!= ( hd_Epistemic_fm_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_813_hd__append,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( hd_Epistemic_fm_a @ Ys ) ) )
& ( ( Xs != nil_Epistemic_fm_a )
=> ( ( hd_Epistemic_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( hd_Epistemic_fm_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_814_list_Oexpand,axiom,
! [List: list_Epistemic_fm_a,List3: list_Epistemic_fm_a] :
( ( ( List = nil_Epistemic_fm_a )
= ( List3 = nil_Epistemic_fm_a ) )
=> ( ( ( List != nil_Epistemic_fm_a )
=> ( ( List3 != nil_Epistemic_fm_a )
=> ( ( ( hd_Epistemic_fm_a @ List )
= ( hd_Epistemic_fm_a @ List3 ) )
& ( ( tl_Epistemic_fm_a @ List )
= ( tl_Epistemic_fm_a @ List3 ) ) ) ) )
=> ( List = List3 ) ) ) ).
% list.expand
thf(fact_815_hd__Nil__eq__last,axiom,
( ( hd_Epistemic_fm_a @ nil_Epistemic_fm_a )
= ( last_Epistemic_fm_a @ nil_Epistemic_fm_a ) ) ).
% hd_Nil_eq_last
thf(fact_816_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_817_min__list_Osimps,axiom,
! [X3: nat,Xs: list_nat] :
( ( min_list_nat @ ( cons_nat @ X3 @ Xs ) )
= ( case_list_nat_nat @ X3
@ ^ [A4: nat,List2: list_nat] : ( ord_min_nat @ X3 @ ( min_list_nat @ Xs ) )
@ Xs ) ) ).
% min_list.simps
thf(fact_818_min__list_Opelims,axiom,
! [X3: list_nat,Y: nat] :
( ( ( min_list_nat @ X3 )
= Y )
=> ( ( accp_list_nat @ min_list_rel_nat @ X3 )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X3
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( Y
= ( case_list_nat_nat @ X2
@ ^ [A4: nat,List2: list_nat] : ( ord_min_nat @ X2 @ ( min_list_nat @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp_list_nat @ min_list_rel_nat @ ( cons_nat @ X2 @ Xs2 ) ) ) )
=> ~ ( ( X3 = nil_nat )
=> ( ( Y = undefined_nat )
=> ~ ( accp_list_nat @ min_list_rel_nat @ nil_nat ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_819_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_820_min_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_min_nat @ A2 @ B )
= B ) ) ).
% min.absorb2
thf(fact_821_min_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_min_nat @ A2 @ B )
= A2 ) ) ).
% min.absorb1
thf(fact_822_min_Omono,axiom,
! [A2: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_823_min_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( ord_min_nat @ A2 @ B ) ) ) ).
% min.orderE
thf(fact_824_min_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% min.orderI
thf(fact_825_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_826_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_827_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( A4
= ( ord_min_nat @ A4 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_828_min_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ A2 ) ).
% min.cobounded1
thf(fact_829_min_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B ) @ B ) ).
% min.cobounded2
thf(fact_830_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( ord_min_nat @ A4 @ B2 )
= A4 ) ) ) ).
% min.absorb_iff1
thf(fact_831_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( ord_min_nat @ A4 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_832_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_833_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_834_min__le__iff__disj,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
| ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_835_min__def__raw,axiom,
( ord_mi8079229082941695292fm_a_o
= ( ^ [A4: epistemic_fm_a > $o,B2: epistemic_fm_a > $o] : ( if_Epistemic_fm_a_o @ ( ord_le4043730696559282883fm_a_o @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_836_min__def__raw,axiom,
( ord_mi5245583150369600225c_fm_a
= ( ^ [A4: set_Epistemic_fm_a,B2: set_Epistemic_fm_a] : ( if_set2375185401397246656c_fm_a @ ( ord_le3275665582123262618c_fm_a @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_837_min__def__raw,axiom,
( ord_min_set_nat
= ( ^ [A4: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_838_min__def__raw,axiom,
( ord_min_nat
= ( ^ [A4: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B2 ) @ A4 @ B2 ) ) ) ).
% min_def_raw
thf(fact_839_distinct__adj__append__iff,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( distin8123941861610682872c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( ( distin8123941861610682872c_fm_a @ Xs )
& ( distin8123941861610682872c_fm_a @ Ys )
& ( ( Xs = nil_Epistemic_fm_a )
| ( Ys = nil_Epistemic_fm_a )
| ( ( last_Epistemic_fm_a @ Xs )
!= ( hd_Epistemic_fm_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_840_successively__append__iff,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( succes6989878846294031341c_fm_a @ P4 @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( ( succes6989878846294031341c_fm_a @ P4 @ Xs )
& ( succes6989878846294031341c_fm_a @ P4 @ Ys )
& ( ( Xs = nil_Epistemic_fm_a )
| ( Ys = nil_Epistemic_fm_a )
| ( P4 @ ( last_Epistemic_fm_a @ Xs ) @ ( hd_Epistemic_fm_a @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_841_distinct__adj__Cons__Cons,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( distin8123941861610682872c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) )
= ( ( X3 != Y )
& ( distin8123941861610682872c_fm_a @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_842_successively__map,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( succes6989878846294031341c_fm_a @ P4 @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
= ( succes6989878846294031341c_fm_a
@ ^ [X: epistemic_fm_a,Y4: epistemic_fm_a] : ( P4 @ ( F @ X ) @ ( F @ Y4 ) )
@ Xs ) ) ).
% successively_map
thf(fact_843_successively_Osimps_I3_J,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( succes6989878846294031341c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) )
= ( ( P4 @ X3 @ Y )
& ( succes6989878846294031341c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_844_successively_Oelims_I3_J,axiom,
! [X3: epistemic_fm_a > epistemic_fm_a > $o,Xa: list_Epistemic_fm_a] :
( ~ ( succes6989878846294031341c_fm_a @ X3 @ Xa )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) )
=> ( ( X3 @ X2 @ Y2 )
& ( succes6989878846294031341c_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_845_successively_Osimps_I1_J,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o] : ( succes6989878846294031341c_fm_a @ P4 @ nil_Epistemic_fm_a ) ).
% successively.simps(1)
thf(fact_846_successively__cong,axiom,
! [Xs: list_a,P4: a > a > $o,Q4: a > a > $o,Ys: list_a] :
( ! [X2: a,Y2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
= ( Q4 @ X2 @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_a @ P4 @ Xs )
= ( successively_a @ Q4 @ Ys ) ) ) ) ).
% successively_cong
thf(fact_847_successively__cong,axiom,
! [Xs: list_nat,P4: nat > nat > $o,Q4: nat > nat > $o,Ys: list_nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
= ( Q4 @ X2 @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_nat @ P4 @ Xs )
= ( successively_nat @ Q4 @ Ys ) ) ) ) ).
% successively_cong
thf(fact_848_successively__cong,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > epistemic_fm_a > $o,Q4: epistemic_fm_a > epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a,Y2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( member6642669571620171971c_fm_a @ Y2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
= ( Q4 @ X2 @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( succes6989878846294031341c_fm_a @ P4 @ Xs )
= ( succes6989878846294031341c_fm_a @ Q4 @ Ys ) ) ) ) ).
% successively_cong
thf(fact_849_successively__mono,axiom,
! [P4: a > a > $o,Xs: list_a,Q4: a > a > $o] :
( ( successively_a @ P4 @ Xs )
=> ( ! [X2: a,Y2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
=> ( Q4 @ X2 @ Y2 ) ) ) )
=> ( successively_a @ Q4 @ Xs ) ) ) ).
% successively_mono
thf(fact_850_successively__mono,axiom,
! [P4: nat > nat > $o,Xs: list_nat,Q4: nat > nat > $o] :
( ( successively_nat @ P4 @ Xs )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
=> ( Q4 @ X2 @ Y2 ) ) ) )
=> ( successively_nat @ Q4 @ Xs ) ) ) ).
% successively_mono
thf(fact_851_successively__mono,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,Xs: list_Epistemic_fm_a,Q4: epistemic_fm_a > epistemic_fm_a > $o] :
( ( succes6989878846294031341c_fm_a @ P4 @ Xs )
=> ( ! [X2: epistemic_fm_a,Y2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( member6642669571620171971c_fm_a @ Y2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ( P4 @ X2 @ Y2 )
=> ( Q4 @ X2 @ Y2 ) ) ) )
=> ( succes6989878846294031341c_fm_a @ Q4 @ Xs ) ) ) ).
% successively_mono
thf(fact_852_distinct__adj__mapD,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( distin8123941861610682872c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F @ Xs ) )
=> ( distin8123941861610682872c_fm_a @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_853_distinct__adj__Nil,axiom,
distin8123941861610682872c_fm_a @ nil_Epistemic_fm_a ).
% distinct_adj_Nil
thf(fact_854_distinct__adj__ConsD,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( distin8123941861610682872c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
=> ( distin8123941861610682872c_fm_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_855_successively_Oelims_I2_J,axiom,
! [X3: epistemic_fm_a > epistemic_fm_a > $o,Xa: list_Epistemic_fm_a] :
( ( succes6989878846294031341c_fm_a @ X3 @ Xa )
=> ( ( Xa != nil_Epistemic_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( Xa
!= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) )
=> ~ ( ( X3 @ X2 @ Y2 )
& ( succes6989878846294031341c_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_856_successively_Oelims_I1_J,axiom,
! [X3: epistemic_fm_a > epistemic_fm_a > $o,Xa: list_Epistemic_fm_a,Y: $o] :
( ( ( succes6989878846294031341c_fm_a @ X3 @ Xa )
= Y )
=> ( ( ( Xa = nil_Epistemic_fm_a )
=> ~ Y )
=> ( ( ? [X2: epistemic_fm_a] :
( Xa
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ Y )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( Xa
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X3 @ X2 @ Y2 )
& ( succes6989878846294031341c_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_857_successively_Osimps_I2_J,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a] : ( succes6989878846294031341c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% successively.simps(2)
thf(fact_858_distinct__adj__singleton,axiom,
! [X3: epistemic_fm_a] : ( distin8123941861610682872c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% distinct_adj_singleton
thf(fact_859_verit__comp__simplify1_I2_J,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_860_verit__comp__simplify1_I2_J,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_861_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_862_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_863_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_864_successively__Cons,axiom,
! [P4: epistemic_fm_a > epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( succes6989878846294031341c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( Xs = nil_Epistemic_fm_a )
| ( ( P4 @ X3 @ ( hd_Epistemic_fm_a @ Xs ) )
& ( succes6989878846294031341c_fm_a @ P4 @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_865_distinct__adj__Cons,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( distin8123941861610682872c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( ( Xs = nil_Epistemic_fm_a )
| ( ( X3
!= ( hd_Epistemic_fm_a @ Xs ) )
& ( distin8123941861610682872c_fm_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_866_Cons__in__shuffles__iff,axiom,
! [Z2: 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 @ Z2 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Xs )
= Z2 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ ( tl_Epistemic_fm_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_Epistemic_fm_a )
& ( ( hd_Epistemic_fm_a @ Ys )
= Z2 )
& ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ ( tl_Epistemic_fm_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_867_remdups__adj__append_H,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( ( Xs = nil_Epistemic_fm_a )
| ( Ys = nil_Epistemic_fm_a )
| ( ( last_Epistemic_fm_a @ Xs )
!= ( hd_Epistemic_fm_a @ Ys ) ) )
=> ( ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( remdup1727679340039403862c_fm_a @ Xs ) @ ( remdup1727679340039403862c_fm_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_868_Min_Oset__eq__fold,axiom,
! [X3: nat,Xs: list_nat] :
( ( lattic8721135487736765967in_nat @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
= ( fold_nat_nat @ ord_min_nat @ Xs @ X3 ) ) ).
% Min.set_eq_fold
thf(fact_869_remdups__adj__Nil__iff,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( ( remdup1727679340039403862c_fm_a @ Xs )
= nil_Epistemic_fm_a )
= ( Xs = nil_Epistemic_fm_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_870_remdups__adj__set,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( set_Epistemic_fm_a2 @ ( remdup1727679340039403862c_fm_a @ Xs ) )
= ( set_Epistemic_fm_a2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_871_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_872_remdups__adj__Cons__alt,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( cons_Epistemic_fm_a @ X3 @ ( tl_Epistemic_fm_a @ ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) )
= ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_873_Cons__shuffles__subset1,axiom,
! [X3: 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 @ X3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_874_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_875_remdups__adj_Osimps_I1_J,axiom,
( ( remdup1727679340039403862c_fm_a @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% remdups_adj.simps(1)
thf(fact_876_remdups__adj_Osimps_I3_J,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( X3 = Y )
=> ( ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) )
= ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) )
& ( ( X3 != Y )
=> ( ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) )
= ( cons_Epistemic_fm_a @ X3 @ ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_877_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_878_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z2: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z2 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ ( cons_Epistemic_fm_a @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_879_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_Epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Z2: epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ Z2 @ Zs3 ) @ ( shuffl511019022169630517c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_880_remdups__adj_Oelims,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( remdup1727679340039403862c_fm_a @ X3 )
= Y )
=> ( ( ( X3 = nil_Epistemic_fm_a )
=> ( Y != nil_Epistemic_fm_a ) )
=> ( ! [X2: epistemic_fm_a] :
( ( X3
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( Y
!= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( X3
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X2 = Y2 )
=> ( Y
= ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons_Epistemic_fm_a @ X2 @ ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_881_remdups__adj_Osimps_I2_J,axiom,
! [X3: epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) )
= ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ).
% remdups_adj.simps(2)
thf(fact_882_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 ) )
=> ( ! [X2: epistemic_fm_a,Xs6: list_Epistemic_fm_a] :
( ( Xs
= ( cons_Epistemic_fm_a @ X2 @ Xs6 ) )
=> ! [Z4: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z4 @ Zs4 ) )
=> ( ( X2 = Z4 )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y2: epistemic_fm_a,Ys6: list_Epistemic_fm_a] :
( ( Ys
= ( cons_Epistemic_fm_a @ Y2 @ Ys6 ) )
=> ! [Z4: epistemic_fm_a,Zs4: list_Epistemic_fm_a] :
( ( Zs3
= ( cons_Epistemic_fm_a @ Z4 @ Zs4 ) )
=> ( ( Y2 = Z4 )
=> ~ ( member5906877432388582473c_fm_a @ Zs4 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_883_successively__remdups__adj__iff,axiom,
! [Xs: list_a,P4: a > a > $o] :
( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( P4 @ X2 @ X2 ) )
=> ( ( successively_a @ P4 @ ( remdups_adj_a @ Xs ) )
= ( successively_a @ P4 @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_884_successively__remdups__adj__iff,axiom,
! [Xs: list_nat,P4: nat > nat > $o] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P4 @ X2 @ X2 ) )
=> ( ( successively_nat @ P4 @ ( remdups_adj_nat @ Xs ) )
= ( successively_nat @ P4 @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_885_successively__remdups__adj__iff,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X2 @ X2 ) )
=> ( ( succes6989878846294031341c_fm_a @ P4 @ ( remdup1727679340039403862c_fm_a @ Xs ) )
= ( succes6989878846294031341c_fm_a @ P4 @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_886_rev__conv__fold,axiom,
( rev_Epistemic_fm_a
= ( ^ [Xs3: list_Epistemic_fm_a] : ( fold_E7642651028368403345c_fm_a @ cons_Epistemic_fm_a @ Xs3 @ nil_Epistemic_fm_a ) ) ) ).
% rev_conv_fold
thf(fact_887_remdups__adj__append__two,axiom,
! [Xs: list_Epistemic_fm_a,X3: epistemic_fm_a,Y: epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y @ nil_Epistemic_fm_a ) ) ) )
= ( append9179727413925872949c_fm_a @ ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ) @ ( if_lis2878681784746929638c_fm_a @ ( X3 = Y ) @ nil_Epistemic_fm_a @ ( cons_Epistemic_fm_a @ Y @ nil_Epistemic_fm_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_888_fold__Cons__rev,axiom,
! [Xs: list_Epistemic_fm_a] :
( ( fold_E7642651028368403345c_fm_a @ cons_Epistemic_fm_a @ Xs )
= ( append9179727413925872949c_fm_a @ ( rev_Epistemic_fm_a @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_889_remdups__adj__Cons,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( case_l8442927312758267104c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a )
@ ^ [Y4: epistemic_fm_a,Xs3: list_Epistemic_fm_a] : ( if_lis2878681784746929638c_fm_a @ ( X3 = Y4 ) @ ( cons_Epistemic_fm_a @ Y4 @ Xs3 ) @ ( cons_Epistemic_fm_a @ X3 @ ( cons_Epistemic_fm_a @ Y4 @ Xs3 ) ) )
@ ( remdup1727679340039403862c_fm_a @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_890_remdups__adj__append,axiom,
! [Xs_1: list_Epistemic_fm_a,X3: epistemic_fm_a,Xs_2: list_Epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs_1 @ ( cons_Epistemic_fm_a @ X3 @ Xs_2 ) ) )
= ( append9179727413925872949c_fm_a @ ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs_1 @ ( cons_Epistemic_fm_a @ X3 @ nil_Epistemic_fm_a ) ) ) @ ( tl_Epistemic_fm_a @ ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_891_remdups__adj_Opelims,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( ( remdup1727679340039403862c_fm_a @ X3 )
= Y )
=> ( ( accp_l2570680006282209577c_fm_a @ remdup2768244663267211811c_fm_a @ X3 )
=> ( ( ( X3 = nil_Epistemic_fm_a )
=> ( ( Y = nil_Epistemic_fm_a )
=> ~ ( accp_l2570680006282209577c_fm_a @ remdup2768244663267211811c_fm_a @ nil_Epistemic_fm_a ) ) )
=> ( ! [X2: epistemic_fm_a] :
( ( X3
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ( ( Y
= ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) )
=> ~ ( accp_l2570680006282209577c_fm_a @ remdup2768244663267211811c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ nil_Epistemic_fm_a ) ) ) )
=> ~ ! [X2: epistemic_fm_a,Y2: epistemic_fm_a,Xs2: list_Epistemic_fm_a] :
( ( X3
= ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) )
=> ( ( ( ( X2 = Y2 )
=> ( Y
= ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons_Epistemic_fm_a @ X2 @ ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_l2570680006282209577c_fm_a @ remdup2768244663267211811c_fm_a @ ( cons_Epistemic_fm_a @ X2 @ ( cons_Epistemic_fm_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_892_tl__remdups__adj,axiom,
! [Ys: list_Epistemic_fm_a] :
( ( Ys != nil_Epistemic_fm_a )
=> ( ( tl_Epistemic_fm_a @ ( remdup1727679340039403862c_fm_a @ Ys ) )
= ( remdup1727679340039403862c_fm_a
@ ( dropWh6715642967443873729c_fm_a
@ ^ [X: epistemic_fm_a] :
( X
= ( hd_Epistemic_fm_a @ Ys ) )
@ ( tl_Epistemic_fm_a @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_893_dropWhile__eq__Nil__conv,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( dropWh6715642967443873729c_fm_a @ P4 @ Xs )
= nil_Epistemic_fm_a )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_894_dropWhile__append2,axiom,
! [Xs: list_a,P4: a > $o,Ys: list_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( P4 @ X2 ) )
=> ( ( dropWhile_a @ P4 @ ( append_a @ Xs @ Ys ) )
= ( dropWhile_a @ P4 @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_895_dropWhile__append2,axiom,
! [Xs: list_nat,P4: nat > $o,Ys: list_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P4 @ X2 ) )
=> ( ( dropWhile_nat @ P4 @ ( append_nat @ Xs @ Ys ) )
= ( dropWhile_nat @ P4 @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_896_dropWhile__append2,axiom,
! [Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( P4 @ X2 ) )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( dropWh6715642967443873729c_fm_a @ P4 @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_897_dropWhile__append1,axiom,
! [X3: a,Xs: list_a,P4: a > $o,Ys: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( dropWhile_a @ P4 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( dropWhile_a @ P4 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_898_dropWhile__append1,axiom,
! [X3: nat,Xs: list_nat,P4: nat > $o,Ys: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( dropWhile_nat @ P4 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( dropWhile_nat @ P4 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_899_dropWhile__append1,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Ys: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_900_dropWhile_Osimps_I2_J,axiom,
! [P4: epistemic_fm_a > $o,X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( ( P4 @ X3 )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) ) )
& ( ~ ( P4 @ X3 )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( cons_Epistemic_fm_a @ X3 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_901_dropWhile_Osimps_I1_J,axiom,
! [P4: epistemic_fm_a > $o] :
( ( dropWh6715642967443873729c_fm_a @ P4 @ nil_Epistemic_fm_a )
= nil_Epistemic_fm_a ) ).
% dropWhile.simps(1)
thf(fact_902_dropWhile__cong,axiom,
! [L: list_a,K2: list_a,P4: a > $o,Q4: a > $o] :
( ( L = K2 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ L ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( dropWhile_a @ P4 @ L )
= ( dropWhile_a @ Q4 @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_903_dropWhile__cong,axiom,
! [L: list_nat,K2: list_nat,P4: nat > $o,Q4: nat > $o] :
( ( L = K2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ L ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( dropWhile_nat @ P4 @ L )
= ( dropWhile_nat @ Q4 @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_904_dropWhile__cong,axiom,
! [L: list_Epistemic_fm_a,K2: list_Epistemic_fm_a,P4: epistemic_fm_a > $o,Q4: epistemic_fm_a > $o] :
( ( L = K2 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ L ) )
=> ( ( P4 @ X2 )
= ( Q4 @ X2 ) ) )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ L )
= ( dropWh6715642967443873729c_fm_a @ Q4 @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_905_set__dropWhileD,axiom,
! [X3: a,P4: a > $o,Xs: list_a] :
( ( member_a2 @ X3 @ ( set_a2 @ ( dropWhile_a @ P4 @ Xs ) ) )
=> ( member_a2 @ X3 @ ( set_a2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_906_set__dropWhileD,axiom,
! [X3: nat,P4: nat > $o,Xs: list_nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ ( dropWhile_nat @ P4 @ Xs ) ) )
=> ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_907_set__dropWhileD,axiom,
! [X3: epistemic_fm_a,P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) ) )
=> ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_908_dropWhile__append3,axiom,
! [P4: epistemic_fm_a > $o,Y: epistemic_fm_a,Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ~ ( P4 @ Y )
=> ( ( dropWh6715642967443873729c_fm_a @ P4 @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) )
= ( append9179727413925872949c_fm_a @ ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_909_dropWhile__eq__self__iff,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( dropWh6715642967443873729c_fm_a @ P4 @ Xs )
= Xs )
= ( ( Xs = nil_Epistemic_fm_a )
| ~ ( P4 @ ( hd_Epistemic_fm_a @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_910_hd__dropWhile,axiom,
! [P4: epistemic_fm_a > $o,Xs: list_Epistemic_fm_a] :
( ( ( dropWh6715642967443873729c_fm_a @ P4 @ Xs )
!= nil_Epistemic_fm_a )
=> ~ ( P4 @ ( hd_Epistemic_fm_a @ ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_911_dropWhile__last,axiom,
! [X3: a,Xs: list_a,P4: a > $o] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( last_a @ ( dropWhile_a @ P4 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_912_dropWhile__last,axiom,
! [X3: nat,Xs: list_nat,P4: nat > $o] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( last_nat @ ( dropWhile_nat @ P4 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_913_dropWhile__last,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,P4: epistemic_fm_a > $o] :
( ( member6642669571620171971c_fm_a @ X3 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( ~ ( P4 @ X3 )
=> ( ( last_Epistemic_fm_a @ ( dropWh6715642967443873729c_fm_a @ P4 @ Xs ) )
= ( last_Epistemic_fm_a @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_914_remdups__adj__Cons_H,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) )
= ( cons_Epistemic_fm_a @ X3
@ ( remdup1727679340039403862c_fm_a
@ ( dropWh6715642967443873729c_fm_a
@ ^ [Y4: epistemic_fm_a] : ( Y4 = X3 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_915_remdups__adj__append__dropWhile,axiom,
! [Xs: list_Epistemic_fm_a,Y: epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ Ys ) ) )
= ( append9179727413925872949c_fm_a @ ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ ( cons_Epistemic_fm_a @ Y @ nil_Epistemic_fm_a ) ) )
@ ( remdup1727679340039403862c_fm_a
@ ( dropWh6715642967443873729c_fm_a
@ ^ [X: epistemic_fm_a] : ( X = Y )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_916_remdups__adj__append_H_H,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a] :
( ( Xs != nil_Epistemic_fm_a )
=> ( ( remdup1727679340039403862c_fm_a @ ( append9179727413925872949c_fm_a @ Xs @ Ys ) )
= ( append9179727413925872949c_fm_a @ ( remdup1727679340039403862c_fm_a @ Xs )
@ ( remdup1727679340039403862c_fm_a
@ ( dropWh6715642967443873729c_fm_a
@ ^ [Y4: epistemic_fm_a] :
( Y4
= ( last_Epistemic_fm_a @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_917_map__lists__mono,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) @ B4 )
=> ( ord_le2931979837777675552c_fm_a @ ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) ) @ ( lists_Epistemic_fm_a @ B4 ) ) ) ).
% map_lists_mono
thf(fact_918_strong__completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,G: set_Epistemic_fm_a,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ G )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) )
=> ? [Qs4: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ Qs4 ) @ G )
& ( epistemic_AK_a @ A @ ( epistemic_imply_a @ Qs4 @ P ) ) ) ) ) ).
% strong_completeness
thf(fact_919_Euclidean__def,axiom,
( episte2449151000174023629t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) )
=> ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ) ).
% Euclidean_def
thf(fact_920_frame_Oext__inject,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit,W7: set_se5208064806568342746c_fm_a,K4: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More2: episte1193835314949844379t_unit] :
( ( ( episte2888590659910966568t_unit @ W6 @ K3 @ More )
= ( episte2888590659910966568t_unit @ W7 @ K4 @ More2 ) )
= ( ( W6 = W7 )
& ( K3 = K4 )
& ( More = More2 ) ) ) ).
% frame.ext_inject
thf(fact_921_kripke_Oext__inject,axiom,
! [Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit,Pi2: set_Epistemic_fm_a > list_char > $o,More2: product_unit] :
( ( ( episte8239586592105053771t_unit @ Pi @ More )
= ( episte8239586592105053771t_unit @ Pi2 @ More2 ) )
= ( ( Pi = Pi2 )
& ( More = More2 ) ) ) ).
% kripke.ext_inject
thf(fact_922_Cons__in__lists__iff,axiom,
! [X3: a,Xs: list_a,A: set_a] :
( ( member_list_a @ ( cons_a @ X3 @ Xs ) @ ( lists_a @ A ) )
= ( ( member_a2 @ X3 @ A )
& ( member_list_a @ Xs @ ( lists_a @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_923_Cons__in__lists__iff,axiom,
! [X3: nat,Xs: list_nat,A: set_nat] :
( ( member_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( lists_nat @ A ) )
= ( ( member_nat2 @ X3 @ A )
& ( member_list_nat @ Xs @ ( lists_nat @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_924_Cons__in__lists__iff,axiom,
! [X3: epistemic_fm_a,Xs: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ Xs ) @ ( lists_Epistemic_fm_a @ A ) )
= ( ( member6642669571620171971c_fm_a @ X3 @ A )
& ( member5906877432388582473c_fm_a @ Xs @ ( lists_Epistemic_fm_a @ A ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_925_in__listsI,axiom,
! [Xs: list_a,A: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X2 @ A ) )
=> ( member_list_a @ Xs @ ( lists_a @ A ) ) ) ).
% in_listsI
thf(fact_926_in__listsI,axiom,
! [Xs: list_nat,A: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X2 @ A ) )
=> ( member_list_nat @ Xs @ ( lists_nat @ A ) ) ) ).
% in_listsI
thf(fact_927_in__listsI,axiom,
! [Xs: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X2 @ A ) )
=> ( member5906877432388582473c_fm_a @ Xs @ ( lists_Epistemic_fm_a @ A ) ) ) ).
% in_listsI
thf(fact_928_frame_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte8072386903178013299t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= W6 ) ).
% frame.select_convs(1)
thf(fact_929_lists_OCons,axiom,
! [A2: a,A: set_a,L: list_a] :
( ( member_a2 @ A2 @ A )
=> ( ( member_list_a @ L @ ( lists_a @ A ) )
=> ( member_list_a @ ( cons_a @ A2 @ L ) @ ( lists_a @ A ) ) ) ) ).
% lists.Cons
thf(fact_930_lists_OCons,axiom,
! [A2: nat,A: set_nat,L: list_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( member_list_nat @ L @ ( lists_nat @ A ) )
=> ( member_list_nat @ ( cons_nat @ A2 @ L ) @ ( lists_nat @ A ) ) ) ) ).
% lists.Cons
thf(fact_931_lists_OCons,axiom,
! [A2: epistemic_fm_a,A: set_Epistemic_fm_a,L: list_Epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ A2 @ A )
=> ( ( member5906877432388582473c_fm_a @ L @ ( lists_Epistemic_fm_a @ A ) )
=> ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ A2 @ L ) @ ( lists_Epistemic_fm_a @ A ) ) ) ) ).
% lists.Cons
thf(fact_932_listsE,axiom,
! [X3: a,L: list_a,A: set_a] :
( ( member_list_a @ ( cons_a @ X3 @ L ) @ ( lists_a @ A ) )
=> ~ ( ( member_a2 @ X3 @ A )
=> ~ ( member_list_a @ L @ ( lists_a @ A ) ) ) ) ).
% listsE
thf(fact_933_listsE,axiom,
! [X3: nat,L: list_nat,A: set_nat] :
( ( member_list_nat @ ( cons_nat @ X3 @ L ) @ ( lists_nat @ A ) )
=> ~ ( ( member_nat2 @ X3 @ A )
=> ~ ( member_list_nat @ L @ ( lists_nat @ A ) ) ) ) ).
% listsE
thf(fact_934_listsE,axiom,
! [X3: epistemic_fm_a,L: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ ( cons_Epistemic_fm_a @ X3 @ L ) @ ( lists_Epistemic_fm_a @ A ) )
=> ~ ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ~ ( member5906877432388582473c_fm_a @ L @ ( lists_Epistemic_fm_a @ A ) ) ) ) ).
% listsE
thf(fact_935_in__lists__conv__set,axiom,
! [Xs: list_a,A: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_936_in__lists__conv__set,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat @ Xs @ ( lists_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_937_in__lists__conv__set,axiom,
! [Xs: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Xs @ ( lists_Epistemic_fm_a @ A ) )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X @ A ) ) ) ) ).
% in_lists_conv_set
thf(fact_938_in__listsD,axiom,
! [Xs: list_a,A: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A ) )
=> ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_939_in__listsD,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat @ Xs @ ( lists_nat @ A ) )
=> ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat2 @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_940_in__listsD,axiom,
! [Xs: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ Xs @ ( lists_Epistemic_fm_a @ A ) )
=> ! [X4: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X4 @ ( set_Epistemic_fm_a2 @ Xs ) )
=> ( member6642669571620171971c_fm_a @ X4 @ A ) ) ) ).
% in_listsD
thf(fact_941_kripke_Oselect__convs_I1_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte2398645135750866164t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= Pi ) ).
% kripke.select_convs(1)
thf(fact_942_Nil__in__lists,axiom,
! [A: set_Epistemic_fm_a] : ( member5906877432388582473c_fm_a @ nil_Epistemic_fm_a @ ( lists_Epistemic_fm_a @ A ) ) ).
% Nil_in_lists
thf(fact_943_frame_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte6250069432388174439t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= K3 ) ).
% frame.select_convs(2)
thf(fact_944_kripke_Ocases,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ ( episte8239586592105053771t_unit @ Pi3 @ product_Unity ) ) ) ).
% kripke.cases
thf(fact_945_kripke_Oext__induct,axiom,
! [P4: episte1193835314949844379t_unit > $o,R: episte1193835314949844379t_unit] :
( ! [Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] : ( P4 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) )
=> ( P4 @ R ) ) ).
% kripke.ext_induct
thf(fact_946_frame_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More3: episte1193835314949844379t_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ More3 ) ) ).
% frame.cases_scheme
thf(fact_947_kripke_Ocases__scheme,axiom,
! [R: episte1560738328020401952t_unit] :
~ ! [W8: set_se5208064806568342746c_fm_a,K5: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi3: set_Epistemic_fm_a > list_char > $o,More3: product_unit] :
( R
!= ( episte2888590659910966568t_unit @ W8 @ K5 @ ( episte8239586592105053771t_unit @ Pi3 @ More3 ) ) ) ).
% kripke.cases_scheme
thf(fact_948_eval__semantics,axiom,
! [Pi4: set_Epistemic_fm_a > list_char > $o,W: set_Epistemic_fm_a,W3: set_se5208064806568342746c_fm_a,R: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,P: epistemic_fm_a] :
( ( epistemic_eval_a @ ( Pi4 @ W ) @ ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W ) @ P )
= ( episte7081087998767065248c_fm_a @ ( episte2888590659910966568t_unit @ W3 @ R @ ( episte8239586592105053771t_unit @ Pi4 @ product_Unity ) ) @ W @ P ) ) ).
% eval_semantics
thf(fact_949_lists_Osimps,axiom,
! [A2: list_a,A: set_a] :
( ( member_list_a @ A2 @ ( lists_a @ A ) )
= ( ( A2 = nil_a )
| ? [A4: a,L2: list_a] :
( ( A2
= ( cons_a @ A4 @ L2 ) )
& ( member_a2 @ A4 @ A )
& ( member_list_a @ L2 @ ( lists_a @ A ) ) ) ) ) ).
% lists.simps
thf(fact_950_lists_Osimps,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat @ A2 @ ( lists_nat @ A ) )
= ( ( A2 = nil_nat )
| ? [A4: nat,L2: list_nat] :
( ( A2
= ( cons_nat @ A4 @ L2 ) )
& ( member_nat2 @ A4 @ A )
& ( member_list_nat @ L2 @ ( lists_nat @ A ) ) ) ) ) ).
% lists.simps
thf(fact_951_lists_Osimps,axiom,
! [A2: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ A2 @ ( lists_Epistemic_fm_a @ A ) )
= ( ( A2 = nil_Epistemic_fm_a )
| ? [A4: epistemic_fm_a,L2: list_Epistemic_fm_a] :
( ( A2
= ( cons_Epistemic_fm_a @ A4 @ L2 ) )
& ( member6642669571620171971c_fm_a @ A4 @ A )
& ( member5906877432388582473c_fm_a @ L2 @ ( lists_Epistemic_fm_a @ A ) ) ) ) ) ).
% lists.simps
thf(fact_952_lists_Ocases,axiom,
! [A2: list_a,A: set_a] :
( ( member_list_a @ A2 @ ( lists_a @ A ) )
=> ( ( A2 != nil_a )
=> ~ ! [A5: a,L3: list_a] :
( ( A2
= ( cons_a @ A5 @ L3 ) )
=> ( ( member_a2 @ A5 @ A )
=> ~ ( member_list_a @ L3 @ ( lists_a @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_953_lists_Ocases,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat @ A2 @ ( lists_nat @ A ) )
=> ( ( A2 != nil_nat )
=> ~ ! [A5: nat,L3: list_nat] :
( ( A2
= ( cons_nat @ A5 @ L3 ) )
=> ( ( member_nat2 @ A5 @ A )
=> ~ ( member_list_nat @ L3 @ ( lists_nat @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_954_lists_Ocases,axiom,
! [A2: list_Epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( member5906877432388582473c_fm_a @ A2 @ ( lists_Epistemic_fm_a @ A ) )
=> ( ( A2 != nil_Epistemic_fm_a )
=> ~ ! [A5: epistemic_fm_a,L3: list_Epistemic_fm_a] :
( ( A2
= ( cons_Epistemic_fm_a @ A5 @ L3 ) )
=> ( ( member6642669571620171971c_fm_a @ A5 @ A )
=> ~ ( member5906877432388582473c_fm_a @ L3 @ ( lists_Epistemic_fm_a @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_955_map__lists__surjective,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ( image_4449434806354059013c_fm_a @ F @ A )
= B4 )
=> ( ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) )
= ( lists_Epistemic_fm_a @ B4 ) ) ) ).
% map_lists_surjective
thf(fact_956_truth__lemma,axiom,
! [A: epistemic_fm_a > $o,V: set_Epistemic_fm_a,P: epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ V )
=> ( ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ V )
=> ( ( member6642669571620171971c_fm_a @ P @ V )
= ( episte7081087998767065248c_fm_a
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) )
@ V
@ P ) ) ) ) ).
% truth_lemma
thf(fact_957_lists__mono,axiom,
! [A: set_Epistemic_fm_a,B4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ord_le2931979837777675552c_fm_a @ ( lists_Epistemic_fm_a @ A ) @ ( lists_Epistemic_fm_a @ B4 ) ) ) ).
% lists_mono
thf(fact_958_lists__mono,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ord_le6045566169113846134st_nat @ ( lists_nat @ A ) @ ( lists_nat @ B4 ) ) ) ).
% lists_mono
thf(fact_959_lists__image,axiom,
! [F: epistemic_fm_a > epistemic_fm_a,A: set_Epistemic_fm_a] :
( ( lists_Epistemic_fm_a @ ( image_4449434806354059013c_fm_a @ F @ A ) )
= ( image_7387318756023638021c_fm_a @ ( map_Ep7084560364594560580c_fm_a @ F ) @ ( lists_Epistemic_fm_a @ A ) ) ) ).
% lists_image
thf(fact_960_reflexive_092_060_094sub_062T,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxT_a @ A )
=> ( episte5648423998891577755t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% reflexive\<^sub>T
thf(fact_961_transitive_092_060_094sub_062K_092_060_094sub_0624,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax4_a @ A )
=> ( episte8364071018013720454t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% transitive\<^sub>K\<^sub>4
thf(fact_962_symmetric_092_060_094sub_062K_092_060_094sub_062B,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_AxB_a @ A )
=> ( episte5478016696552465318t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% symmetric\<^sub>K\<^sub>B
thf(fact_963_Euclidean_092_060_094sub_062K_092_060_094sub_0625,axiom,
! [A: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ epistemic_Ax5_a @ A )
=> ( episte2449151000174023629t_unit
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) ) ) ).
% Euclidean\<^sub>K\<^sub>5
thf(fact_964_transitive__def,axiom,
( episte8364071018013720454t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Z3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Z3 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( ( member536094252920883875c_fm_a @ Z3 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
& ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Z3 ) ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) ) ) ) ) ) ) ) ).
% transitive_def
thf(fact_965_reflexive__def,axiom,
( episte5648423998891577755t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ).
% reflexive_def
thf(fact_966_symmetric__def,axiom,
( episte5478016696552465318t_unit
= ( ^ [M4: episte1560738328020401952t_unit] :
! [I2: a,X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ ( episte8072386903178013299t_unit @ M4 ) )
=> ! [Y4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ Y4 @ ( episte8072386903178013299t_unit @ M4 ) )
=> ( ( member536094252920883875c_fm_a @ X @ ( episte6250069432388174439t_unit @ M4 @ I2 @ Y4 ) )
= ( member536094252920883875c_fm_a @ Y4 @ ( episte6250069432388174439t_unit @ M4 @ I2 @ X ) ) ) ) ) ) ) ).
% symmetric_def
thf(fact_967_completeness,axiom,
! [P4: episte1560738328020401952t_unit > $o,P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit] :
( ( P4 @ M2 )
=> ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) ) )
=> ( ( P4
@ ( episte2888590659910966568t_unit
@ ( collec2519470961442302949c_fm_a
@ ^ [W5: set_Epistemic_fm_a] :
( ( episte2285483198712856226tent_a @ A @ W5 )
& ( maxima3264069584562520521c_fm_a @ ( episte2285483198712856226tent_a @ A ) @ W5 ) ) )
@ ^ [I2: a,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,X: list_char] : ( member6642669571620171971c_fm_a @ ( epistemic_Pro_a @ X ) @ V3 )
@ product_Unity ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ) ).
% completeness
thf(fact_968_kripke_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte8239586592105053771t_unit @ ( episte2398645135750866164t_unit @ R ) @ ( episte5479201149095757850t_unit @ R ) ) ) ) ).
% kripke.surjective
thf(fact_969_frame_Osurjective,axiom,
! [R: episte1560738328020401952t_unit] :
( R
= ( episte2888590659910966568t_unit @ ( episte8072386903178013299t_unit @ R ) @ ( episte6250069432388174439t_unit @ R ) @ ( episte3309513806868946049t_unit @ R ) ) ) ).
% frame.surjective
thf(fact_970_min__bot,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_mi5245583150369600225c_fm_a @ bot_bo3626323581529592678c_fm_a @ X3 )
= bot_bo3626323581529592678c_fm_a ) ).
% min_bot
thf(fact_971_min__bot,axiom,
! [X3: set_nat] :
( ( ord_min_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% min_bot
thf(fact_972_min__bot,axiom,
! [X3: nat] :
( ( ord_min_nat @ bot_bot_nat @ X3 )
= bot_bot_nat ) ).
% min_bot
thf(fact_973_min__bot2,axiom,
! [X3: set_Epistemic_fm_a] :
( ( ord_mi5245583150369600225c_fm_a @ X3 @ bot_bo3626323581529592678c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% min_bot2
thf(fact_974_min__bot2,axiom,
! [X3: set_nat] :
( ( ord_min_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% min_bot2
thf(fact_975_min__bot2,axiom,
! [X3: nat] :
( ( ord_min_nat @ X3 @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_976_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_977_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_978_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_979_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_980_bot_Oextremum__uniqueI,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
=> ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_981_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
=> ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_uniqueI
thf(fact_982_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_983_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_984_bot_Oextremum__unique,axiom,
! [A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ bot_bo6433428028861825271fm_a_o )
= ( A2 = bot_bo6433428028861825271fm_a_o ) ) ).
% bot.extremum_unique
thf(fact_985_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ bot_bo3626323581529592678c_fm_a )
= ( A2 = bot_bo3626323581529592678c_fm_a ) ) ).
% bot.extremum_unique
thf(fact_986_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_987_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_988_bot_Oextremum,axiom,
! [A2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ bot_bo6433428028861825271fm_a_o @ A2 ) ).
% bot.extremum
thf(fact_989_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ bot_bo3626323581529592678c_fm_a @ A2 ) ).
% bot.extremum
thf(fact_990_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_991_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_992_cSup__eq__non__empty,axiom,
! [X5: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( X5 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ A2 ) )
=> ( ! [Y2: epistemic_fm_a > $o] :
( ! [X4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X4 @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ X4 @ Y2 ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ Y2 ) )
=> ( ( comple310706794042174646fm_a_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_993_cSup__eq__non__empty,axiom,
! [X5: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( X5 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ A2 ) )
=> ( ! [Y2: set_Epistemic_fm_a] :
( ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ X4 @ Y2 ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ Y2 ) )
=> ( ( comple7868773486375872231c_fm_a @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_994_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X5 )
=> ( ord_less_eq_set_nat @ X2 @ A2 ) )
=> ( ! [Y2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_995_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_996_cSup__least,axiom,
! [X5: set_Epistemic_fm_a_o,Z2: epistemic_fm_a > $o] :
( ( X5 != bot_bo4908879468608929837fm_a_o )
=> ( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ Z2 ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_997_cSup__least,axiom,
! [X5: set_se5208064806568342746c_fm_a,Z2: set_Epistemic_fm_a] :
( ( X5 != bot_bo1868452413526482246c_fm_a )
=> ( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ Z2 ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_998_cSup__least,axiom,
! [X5: set_set_nat,Z2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X5 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_999_cSup__least,axiom,
! [X5: set_nat,Z2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1000_less__eq__Sup,axiom,
! [A: set_Epistemic_fm_a_o,U2: epistemic_fm_a > $o] :
( ! [V4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ V4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ V4 ) )
=> ( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1001_less__eq__Sup,axiom,
! [A: set_se5208064806568342746c_fm_a,U2: set_Epistemic_fm_a] :
( ! [V4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ V4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ V4 ) )
=> ( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1002_less__eq__Sup,axiom,
! [A: set_set_nat,U2: set_nat] :
( ! [V4: set_nat] :
( ( member_set_nat @ V4 @ A )
=> ( ord_less_eq_set_nat @ U2 @ V4 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1003_empty__set,axiom,
( bot_bo3626323581529592678c_fm_a
= ( set_Epistemic_fm_a2 @ nil_Epistemic_fm_a ) ) ).
% empty_set
thf(fact_1004_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1005_fm_Osimps_I90_J,axiom,
( ( episte9089240958480457552c_fm_a @ episte5073044243917183961c_fm_a )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(90)
thf(fact_1006_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_nat @ epistemic_FF_nat )
= bot_bot_set_nat ) ).
% fm.simps(90)
thf(fact_1007_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_a @ epistemic_FF_a )
= bot_bot_set_a ) ).
% fm.simps(90)
thf(fact_1008_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( episte9089240958480457552c_fm_a @ ( episte3759128466173231372c_fm_a @ X24 ) )
= bot_bo3626323581529592678c_fm_a ) ).
% fm.simps(91)
thf(fact_1009_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( epistemic_set_fm_nat @ ( epistemic_Pro_nat @ X24 ) )
= bot_bot_set_nat ) ).
% fm.simps(91)
thf(fact_1010_fm_Osimps_I91_J,axiom,
! [X24: list_char] :
( ( epistemic_set_fm_a @ ( epistemic_Pro_a @ X24 ) )
= bot_bot_set_a ) ).
% fm.simps(91)
thf(fact_1011_frame_Oselect__convs_I3_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,More: episte1193835314949844379t_unit] :
( ( episte3309513806868946049t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ More ) )
= More ) ).
% frame.select_convs(3)
thf(fact_1012_cSUP__least,axiom,
! [A: set_a,F: a > nat,M: nat] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1013_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1014_cSUP__least,axiom,
! [A: set_a,F: a > set_nat,M: set_nat] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1015_cSUP__least,axiom,
! [A: set_nat,F: nat > set_nat,M: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1016_cSUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > nat,M: nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1017_cSUP__least,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1018_cSUP__least,axiom,
! [A: set_nat,F: nat > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1019_cSUP__least,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,M: set_nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1020_cSUP__least,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1021_cSUP__least,axiom,
! [A: set_nat,F: nat > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1022_SUP__eq__iff,axiom,
! [I5: set_a,C: epistemic_fm_a > $o,F: a > epistemic_fm_a > $o] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1023_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: epistemic_fm_a > $o,F: epistemic_fm_a > epistemic_fm_a > $o] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1024_SUP__eq__iff,axiom,
! [I5: set_nat,C: epistemic_fm_a > $o,F: nat > epistemic_fm_a > $o] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ I5 )
=> ( ord_le4043730696559282883fm_a_o @ C @ ( F @ I3 ) ) )
=> ( ( ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1025_SUP__eq__iff,axiom,
! [I5: set_a,C: set_Epistemic_fm_a,F: a > set_Epistemic_fm_a] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1026_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1027_SUP__eq__iff,axiom,
! [I5: set_nat,C: set_Epistemic_fm_a,F: nat > set_Epistemic_fm_a] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ I5 )
=> ( ord_le3275665582123262618c_fm_a @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1028_SUP__eq__iff,axiom,
! [I5: set_a,C: set_nat,F: a > set_nat] :
( ( I5 != bot_bot_set_a )
=> ( ! [I3: a] :
( ( member_a2 @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ I5 ) )
= C )
= ( ! [X: a] :
( ( member_a2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1029_SUP__eq__iff,axiom,
! [I5: set_Epistemic_fm_a,C: set_nat,F: epistemic_fm_a > set_nat] :
( ( I5 != bot_bo3626323581529592678c_fm_a )
=> ( ! [I3: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ I5 ) )
= C )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1030_SUP__eq__iff,axiom,
! [I5: set_nat,C: set_nat,F: nat > set_nat] :
( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat2 @ I3 @ I5 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
= C )
= ( ! [X: nat] :
( ( member_nat2 @ X @ I5 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1031_completeness_092_060_094sub_062A,axiom,
! [P: epistemic_fm_a,A: epistemic_fm_a > $o] :
( ! [M2: episte1560738328020401952t_unit,X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ ( episte8072386903178013299t_unit @ M2 ) )
=> ( ! [Xa2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ Xa2 @ bot_bo3626323581529592678c_fm_a )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ Xa2 ) )
=> ( episte7081087998767065248c_fm_a @ M2 @ X2 @ P ) ) )
=> ( epistemic_AK_a @ A @ P ) ) ).
% completeness\<^sub>A
thf(fact_1032_frame_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R4: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R4 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R4 ) )
=> ( ( ( episte3309513806868946049t_unit @ R )
= ( episte3309513806868946049t_unit @ R4 ) )
=> ( R = R4 ) ) ) ) ).
% frame.equality
thf(fact_1033_kripke_Oselect__convs_I2_J,axiom,
! [W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte5479201149095757850t_unit @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= More ) ).
% kripke.select_convs(2)
thf(fact_1034_kripke_Oequality,axiom,
! [R: episte1560738328020401952t_unit,R4: episte1560738328020401952t_unit] :
( ( ( episte8072386903178013299t_unit @ R )
= ( episte8072386903178013299t_unit @ R4 ) )
=> ( ( ( episte6250069432388174439t_unit @ R )
= ( episte6250069432388174439t_unit @ R4 ) )
=> ( ( ( episte2398645135750866164t_unit @ R )
= ( episte2398645135750866164t_unit @ R4 ) )
=> ( ( ( episte5479201149095757850t_unit @ R )
= ( episte5479201149095757850t_unit @ R4 ) )
=> ( R = R4 ) ) ) ) ) ).
% kripke.equality
thf(fact_1035_kripke_Oupdate__convs_I2_J,axiom,
! [More2: product_unit > product_unit,W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte9120385895580347753c_fm_a @ More2 @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ ( More2 @ More ) ) ) ) ).
% kripke.update_convs(2)
thf(fact_1036_kripke_Oupdate__convs_I1_J,axiom,
! [Pi2: ( set_Epistemic_fm_a > list_char > $o ) > set_Epistemic_fm_a > list_char > $o,W6: set_se5208064806568342746c_fm_a,K3: a > set_Epistemic_fm_a > set_se5208064806568342746c_fm_a,Pi: set_Epistemic_fm_a > list_char > $o,More: product_unit] :
( ( episte1857908288096881731t_unit @ Pi2 @ ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ Pi @ More ) ) )
= ( episte2888590659910966568t_unit @ W6 @ K3 @ ( episte8239586592105053771t_unit @ ( Pi2 @ Pi ) @ More ) ) ) ).
% kripke.update_convs(1)
thf(fact_1037_cSUP__subset__mono,axiom,
! [A: set_a,G4: a > nat,B4: set_a,F: a > nat] :
( ( A != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1038_cSUP__subset__mono,axiom,
! [A: set_nat,G4: nat > nat,B4: set_nat,F: nat > nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1039_cSUP__subset__mono,axiom,
! [A: set_a,G4: a > set_nat,B4: set_a,F: a > set_nat] :
( ( A != bot_bot_set_a )
=> ( ( condit5477540289124974626et_nat @ ( image_a_set_nat @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1040_cSUP__subset__mono,axiom,
! [A: set_nat,G4: nat > set_nat,B4: set_nat,F: nat > set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1041_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G4: epistemic_fm_a > nat,B4: set_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_3638449696541059749_a_nat @ G4 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1042_cSUP__subset__mono,axiom,
! [A: set_a,G4: a > set_Epistemic_fm_a,B4: set_a,F: a > set_Epistemic_fm_a] :
( ( A != bot_bot_set_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1043_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G4: epistemic_fm_a > set_nat,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit5477540289124974626et_nat @ ( image_4248995238221578843et_nat @ G4 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1044_cSUP__subset__mono,axiom,
! [A: set_nat,G4: nat > set_Epistemic_fm_a,B4: set_nat,F: nat > set_Epistemic_fm_a] :
( ( A != bot_bot_set_nat )
=> ( ( condit2203544419062987614c_fm_a @ ( image_2764307138536547683c_fm_a @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1045_cSUP__subset__mono,axiom,
! [A: set_a,G4: a > epistemic_fm_a > $o,B4: set_a,F: a > epistemic_fm_a > $o] :
( ( A != bot_bot_set_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ G4 @ B4 ) )
=> ( ( ord_less_eq_set_a @ A @ B4 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1046_cSUP__subset__mono,axiom,
! [A: set_Epistemic_fm_a,G4: epistemic_fm_a > set_Epistemic_fm_a,B4: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ G4 @ B4 ) )
=> ( ( ord_le3275665582123262618c_fm_a @ A @ B4 )
=> ( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ ( G4 @ X2 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ G4 @ B4 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_1047_bdd__above_OI,axiom,
! [A: set_Epistemic_fm_a_o,M: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ M ) )
=> ( condit6663374420580310783fm_a_o @ A ) ) ).
% bdd_above.I
thf(fact_1048_bdd__above_OI,axiom,
! [A: set_se5208064806568342746c_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ M ) )
=> ( condit2203544419062987614c_fm_a @ A ) ) ).
% bdd_above.I
thf(fact_1049_bdd__above_OI,axiom,
! [A: set_set_nat,M: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ M ) )
=> ( condit5477540289124974626et_nat @ A ) ) ).
% bdd_above.I
thf(fact_1050_bdd__above_OI,axiom,
! [A: set_nat,M: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ( condit2214826472909112428ve_nat @ A ) ) ).
% bdd_above.I
thf(fact_1051_bdd__above_OI2,axiom,
! [A: set_a,F: a > nat,M: nat] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1052_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1053_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_nat,M: set_nat] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_a_set_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1054_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > set_nat,M: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1055_bdd__above_OI2,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > nat,M: nat] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1056_bdd__above_OI2,axiom,
! [A: set_a,F: a > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1057_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > set_Epistemic_fm_a,M: set_Epistemic_fm_a] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X2 ) @ M ) )
=> ( condit2203544419062987614c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1058_bdd__above_OI2,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,M: set_nat] :
( ! [X2: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M ) )
=> ( condit5477540289124974626et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1059_bdd__above_OI2,axiom,
! [A: set_a,F: a > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1060_bdd__above_OI2,axiom,
! [A: set_nat,F: nat > epistemic_fm_a > $o,M: epistemic_fm_a > $o] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X2 ) @ M ) )
=> ( condit6663374420580310783fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) ) ).
% bdd_above.I2
thf(fact_1061_cSup__upper,axiom,
! [X3: epistemic_fm_a > $o,X5: set_Epistemic_fm_a_o] :
( ( member4486839677911940090fm_a_o @ X3 @ X5 )
=> ( ( condit6663374420580310783fm_a_o @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( comple310706794042174646fm_a_o @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1062_cSup__upper,axiom,
! [X3: set_Epistemic_fm_a,X5: set_se5208064806568342746c_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X5 )
=> ( ( condit2203544419062987614c_fm_a @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( comple7868773486375872231c_fm_a @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1063_cSup__upper,axiom,
! [X3: set_nat,X5: set_set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ( condit5477540289124974626et_nat @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1064_cSup__upper,axiom,
! [X3: nat,X5: set_nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% cSup_upper
thf(fact_1065_cSup__upper2,axiom,
! [X3: epistemic_fm_a > $o,X5: set_Epistemic_fm_a_o,Y: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X3 @ X5 )
=> ( ( ord_le4043730696559282883fm_a_o @ Y @ X3 )
=> ( ( condit6663374420580310783fm_a_o @ X5 )
=> ( ord_le4043730696559282883fm_a_o @ Y @ ( comple310706794042174646fm_a_o @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1066_cSup__upper2,axiom,
! [X3: set_Epistemic_fm_a,X5: set_se5208064806568342746c_fm_a,Y: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X3 @ X5 )
=> ( ( ord_le3275665582123262618c_fm_a @ Y @ X3 )
=> ( ( condit2203544419062987614c_fm_a @ X5 )
=> ( ord_le3275665582123262618c_fm_a @ Y @ ( comple7868773486375872231c_fm_a @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1067_cSup__upper2,axiom,
! [X3: set_nat,X5: set_set_nat,Y: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( condit5477540289124974626et_nat @ X5 )
=> ( ord_less_eq_set_nat @ Y @ ( comple7399068483239264473et_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1068_cSup__upper2,axiom,
! [X3: nat,X5: set_nat,Y: nat] :
( ( member_nat2 @ X3 @ X5 )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( condit2214826472909112428ve_nat @ X5 )
=> ( ord_less_eq_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ) ).
% cSup_upper2
thf(fact_1069_bdd__above_Ounfold,axiom,
( condit6663374420580310783fm_a_o
= ( ^ [A3: set_Epistemic_fm_a_o] :
? [M4: epistemic_fm_a > $o] :
! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ A3 )
=> ( ord_le4043730696559282883fm_a_o @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1070_bdd__above_Ounfold,axiom,
( condit2203544419062987614c_fm_a
= ( ^ [A3: set_se5208064806568342746c_fm_a] :
? [M4: set_Epistemic_fm_a] :
! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ A3 )
=> ( ord_le3275665582123262618c_fm_a @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1071_bdd__above_Ounfold,axiom,
( condit5477540289124974626et_nat
= ( ^ [A3: set_set_nat] :
? [M4: set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1072_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat2 @ X @ A3 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_1073_bdd__above_OE,axiom,
! [A: set_Epistemic_fm_a_o] :
( ( condit6663374420580310783fm_a_o @ A )
=> ~ ! [M2: epistemic_fm_a > $o] :
~ ! [X4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X4 @ A )
=> ( ord_le4043730696559282883fm_a_o @ X4 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1074_bdd__above_OE,axiom,
! [A: set_se5208064806568342746c_fm_a] :
( ( condit2203544419062987614c_fm_a @ A )
=> ~ ! [M2: set_Epistemic_fm_a] :
~ ! [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ A )
=> ( ord_le3275665582123262618c_fm_a @ X4 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1075_bdd__above_OE,axiom,
! [A: set_set_nat] :
( ( condit5477540289124974626et_nat @ A )
=> ~ ! [M2: set_nat] :
~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1076_bdd__above_OE,axiom,
! [A: set_nat] :
( ( condit2214826472909112428ve_nat @ A )
=> ~ ! [M2: nat] :
~ ! [X4: nat] :
( ( member_nat2 @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_1077_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a2 @ X3 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1078_cSUP__upper,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1079_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > set_nat] :
( ( member_a2 @ X3 @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_a_set_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1080_cSUP__upper,axiom,
! [X3: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1081_cSUP__upper,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > nat] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( condit2214826472909112428ve_nat @ ( image_3638449696541059749_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1082_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > set_Epistemic_fm_a] :
( ( member_a2 @ X3 @ A )
=> ( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1083_cSUP__upper,axiom,
! [X3: nat,A: set_nat,F: nat > set_Epistemic_fm_a] :
( ( member_nat2 @ X3 @ A )
=> ( ( condit2203544419062987614c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X3 ) @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1084_cSUP__upper,axiom,
! [X3: epistemic_fm_a,A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat] :
( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( condit5477540289124974626et_nat @ ( image_4248995238221578843et_nat @ F @ A ) )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1085_cSUP__upper,axiom,
! [X3: a,A: set_a,F: a > epistemic_fm_a > $o] :
( ( member_a2 @ X3 @ A )
=> ( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1086_cSUP__upper,axiom,
! [X3: nat,A: set_nat,F: nat > epistemic_fm_a > $o] :
( ( member_nat2 @ X3 @ A )
=> ( ( condit6663374420580310783fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X3 ) @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) ) ) ) ).
% cSUP_upper
thf(fact_1087_cSUP__upper2,axiom,
! [F: a > nat,A: set_a,X3: a,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A ) )
=> ( ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1088_cSUP__upper2,axiom,
! [F: nat > nat,A: set_nat,X3: nat,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1089_cSUP__upper2,axiom,
! [F: a > set_nat,A: set_a,X3: a,U2: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_a_set_nat @ F @ A ) )
=> ( ( member_a2 @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1090_cSUP__upper2,axiom,
! [F: nat > set_nat,A: set_nat,X3: nat,U2: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1091_cSUP__upper2,axiom,
! [F: epistemic_fm_a > nat,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_3638449696541059749_a_nat @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1092_cSUP__upper2,axiom,
! [F: a > set_Epistemic_fm_a,A: set_a,X3: a,U2: set_Epistemic_fm_a] :
( ( condit2203544419062987614c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) )
=> ( ( member_a2 @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_3480842985853347585c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1093_cSUP__upper2,axiom,
! [F: nat > set_Epistemic_fm_a,A: set_nat,X3: nat,U2: set_Epistemic_fm_a] :
( ( condit2203544419062987614c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( ord_le3275665582123262618c_fm_a @ U2 @ ( F @ X3 ) )
=> ( ord_le3275665582123262618c_fm_a @ U2 @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1094_cSUP__upper2,axiom,
! [F: epistemic_fm_a > set_nat,A: set_Epistemic_fm_a,X3: epistemic_fm_a,U2: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_4248995238221578843et_nat @ F @ A ) )
=> ( ( member6642669571620171971c_fm_a @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ U2 @ ( F @ X3 ) )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1095_cSUP__upper2,axiom,
! [F: a > epistemic_fm_a > $o,A: set_a,X3: a,U2: epistemic_fm_a > $o] :
( ( condit6663374420580310783fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) )
=> ( ( member_a2 @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_3809950421912819996fm_a_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1096_cSUP__upper2,axiom,
! [F: nat > epistemic_fm_a > $o,A: set_nat,X3: nat,U2: epistemic_fm_a > $o] :
( ( condit6663374420580310783fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( ord_le4043730696559282883fm_a_o @ U2 @ ( F @ X3 ) )
=> ( ord_le4043730696559282883fm_a_o @ U2 @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_1097_cSup__mono,axiom,
! [B4: set_Epistemic_fm_a_o,A: set_Epistemic_fm_a_o] :
( ( B4 != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ A )
=> ( ! [B3: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ B3 @ B4 )
=> ? [X4: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X4 @ A )
& ( ord_le4043730696559282883fm_a_o @ B3 @ X4 ) ) )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ B4 ) @ ( comple310706794042174646fm_a_o @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1098_cSup__mono,axiom,
! [B4: set_se5208064806568342746c_fm_a,A: set_se5208064806568342746c_fm_a] :
( ( B4 != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ A )
=> ( ! [B3: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ B3 @ B4 )
=> ? [X4: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X4 @ A )
& ( ord_le3275665582123262618c_fm_a @ B3 @ X4 ) ) )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ B4 ) @ ( comple7868773486375872231c_fm_a @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1099_cSup__mono,axiom,
! [B4: set_set_nat,A: set_set_nat] :
( ( B4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A )
=> ( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ B4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ B3 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1100_cSup__mono,axiom,
! [B4: set_nat,A: set_nat] :
( ( B4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A )
=> ( ! [B3: nat] :
( ( member_nat2 @ B3 @ B4 )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A )
& ( ord_less_eq_nat @ B3 @ X4 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% cSup_mono
thf(fact_1101_cSup__le__iff,axiom,
! [S: set_Epistemic_fm_a_o,A2: epistemic_fm_a > $o] :
( ( S != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ S )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ S ) @ A2 )
= ( ! [X: epistemic_fm_a > $o] :
( ( member4486839677911940090fm_a_o @ X @ S )
=> ( ord_le4043730696559282883fm_a_o @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1102_cSup__le__iff,axiom,
! [S: set_se5208064806568342746c_fm_a,A2: set_Epistemic_fm_a] :
( ( S != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ S )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ S ) @ A2 )
= ( ! [X: set_Epistemic_fm_a] :
( ( member536094252920883875c_fm_a @ X @ S )
=> ( ord_le3275665582123262618c_fm_a @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1103_cSup__le__iff,axiom,
! [S: set_set_nat,A2: set_nat] :
( ( S != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ S )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S ) @ A2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ S )
=> ( ord_less_eq_set_nat @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1104_cSup__le__iff,axiom,
! [S: set_nat,A2: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ S )
=> ( ord_less_eq_nat @ X @ A2 ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_1105_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit6663374420580310783fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_2381060403230254136fm_a_o @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1106_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > epistemic_fm_a > $o,U2: epistemic_fm_a > $o] :
( ( A != bot_bot_set_nat )
=> ( ( condit6663374420580310783fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) )
=> ( ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( image_4193161545499529530fm_a_o @ F @ A ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_le4043730696559282883fm_a_o @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1107_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_1263732536703906021c_fm_a @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1108_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > set_Epistemic_fm_a,U2: set_Epistemic_fm_a] :
( ( A != bot_bot_set_nat )
=> ( ( condit2203544419062987614c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) )
=> ( ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( image_2764307138536547683c_fm_a @ F @ A ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_le3275665582123262618c_fm_a @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1109_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > set_nat,U2: set_nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit5477540289124974626et_nat @ ( image_4248995238221578843et_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4248995238221578843et_nat @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1110_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > set_nat,U2: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1111_cSUP__le__iff,axiom,
! [A: set_Epistemic_fm_a,F: epistemic_fm_a > nat,U2: nat] :
( ( A != bot_bo3626323581529592678c_fm_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_3638449696541059749_a_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_3638449696541059749_a_nat @ F @ A ) ) @ U2 )
= ( ! [X: epistemic_fm_a] :
( ( member6642669571620171971c_fm_a @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1112_cSUP__le__iff,axiom,
! [A: set_nat,F: nat > nat,U2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ ( F @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_1113_cSup__subset__mono,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] :
( ( A != bot_bo4908879468608929837fm_a_o )
=> ( ( condit6663374420580310783fm_a_o @ B4 )
=> ( ( ord_le4525777775358908921fm_a_o @ A @ B4 )
=> ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1114_cSup__subset__mono,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] :
( ( A != bot_bo1868452413526482246c_fm_a )
=> ( ( condit2203544419062987614c_fm_a @ B4 )
=> ( ( ord_le7112219575281605754c_fm_a @ A @ B4 )
=> ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1115_cSup__subset__mono,axiom,
! [A: set_set_nat,B4: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B4 )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_1116_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_1117_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_list_a @ Zs3 @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( filter_a
@ ^ [X: a] : ( member_a2 @ X @ ( set_a2 @ Ys ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1118_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( inf_in2882328776738922472c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ Ys ) )
= bot_bo3626323581529592678c_fm_a )
=> ( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( ( filter7636273809395300631c_fm_a
@ ^ [X: epistemic_fm_a] : ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1119_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat )
=> ( ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( filter_nat
@ ^ [X: nat] : ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1120_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_list_a @ Zs3 @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( filter_a
@ ^ [X: a] :
~ ( member_a2 @ X @ ( set_a2 @ Ys ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1121_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_Epistemic_fm_a,Ys: list_Epistemic_fm_a,Zs3: list_Epistemic_fm_a] :
( ( ( inf_in2882328776738922472c_fm_a @ ( set_Epistemic_fm_a2 @ Xs ) @ ( set_Epistemic_fm_a2 @ Ys ) )
= bot_bo3626323581529592678c_fm_a )
=> ( ( member5906877432388582473c_fm_a @ Zs3 @ ( shuffl511019022169630517c_fm_a @ Xs @ Ys ) )
=> ( ( filter7636273809395300631c_fm_a
@ ^ [X: epistemic_fm_a] :
~ ( member6642669571620171971c_fm_a @ X @ ( set_Epistemic_fm_a2 @ Ys ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1122_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat )
=> ( ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( filter_nat
@ ^ [X: nat] :
~ ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1123_le__inf__iff,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o,Z2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ Y @ Z2 ) )
= ( ( ord_le4043730696559282883fm_a_o @ X3 @ Y )
& ( ord_le4043730696559282883fm_a_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1124_le__inf__iff,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a,Z2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ Y @ Z2 ) )
= ( ( ord_le3275665582123262618c_fm_a @ X3 @ Y )
& ( ord_le3275665582123262618c_fm_a @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1125_le__inf__iff,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1126_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1127_inf_Obounded__iff,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o,C: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ ( inf_in3606484609122063093fm_a_o @ B @ C ) )
= ( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
& ( ord_le4043730696559282883fm_a_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1128_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a,C: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ ( inf_in2882328776738922472c_fm_a @ B @ C ) )
= ( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
& ( ord_le3275665582123262618c_fm_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1129_inf_Obounded__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1130_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_1131_lists__Int__eq,axiom,
! [A: set_nat,B4: set_nat] :
( ( lists_nat @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_inf_set_list_nat @ ( lists_nat @ A ) @ ( lists_nat @ B4 ) ) ) ).
% lists_Int_eq
thf(fact_1132_lists__IntI,axiom,
! [L: list_nat,A: set_nat,B4: set_nat] :
( ( member_list_nat @ L @ ( lists_nat @ A ) )
=> ( ( member_list_nat @ L @ ( lists_nat @ B4 ) )
=> ( member_list_nat @ L @ ( lists_nat @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ).
% lists_IntI
thf(fact_1133_Sup__inter__less__eq,axiom,
! [A: set_Epistemic_fm_a_o,B4: set_Epistemic_fm_a_o] : ( ord_le4043730696559282883fm_a_o @ ( comple310706794042174646fm_a_o @ ( inf_in8896176524268821035fm_a_o @ A @ B4 ) ) @ ( inf_in3606484609122063093fm_a_o @ ( comple310706794042174646fm_a_o @ A ) @ ( comple310706794042174646fm_a_o @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1134_Sup__inter__less__eq,axiom,
! [A: set_se5208064806568342746c_fm_a,B4: set_se5208064806568342746c_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( comple7868773486375872231c_fm_a @ ( inf_in1884693029477671368c_fm_a @ A @ B4 ) ) @ ( inf_in2882328776738922472c_fm_a @ ( comple7868773486375872231c_fm_a @ A ) @ ( comple7868773486375872231c_fm_a @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1135_Sup__inter__less__eq,axiom,
! [A: set_set_nat,B4: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1136_inf__sup__ord_I2_J,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1137_inf__sup__ord_I2_J,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1138_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1139_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1140_inf__sup__ord_I1_J,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1141_inf__sup__ord_I1_J,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1142_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1143_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1144_inf__le1,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1145_inf__le1,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1146_inf__le1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1147_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1148_inf__le2,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1149_inf__le2,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1150_inf__le2,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1151_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1152_le__infE,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) )
=> ~ ( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ~ ( ord_le4043730696559282883fm_a_o @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1153_le__infE,axiom,
! [X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) )
=> ~ ( ( ord_le3275665582123262618c_fm_a @ X3 @ A2 )
=> ~ ( ord_le3275665582123262618c_fm_a @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1154_le__infE,axiom,
! [X3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1155_le__infE,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_1156_le__infI,axiom,
! [X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X3 @ A2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X3 @ B )
=> ( ord_le4043730696559282883fm_a_o @ X3 @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1157_le__infI,axiom,
! [X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X3 @ A2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X3 @ B )
=> ( ord_le3275665582123262618c_fm_a @ X3 @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1158_le__infI,axiom,
! [X3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ B )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1159_le__infI,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_1160_inf__mono,axiom,
! [A2: epistemic_fm_a > $o,C: epistemic_fm_a > $o,B: epistemic_fm_a > $o,D2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ C )
=> ( ( ord_le4043730696559282883fm_a_o @ B @ D2 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ ( inf_in3606484609122063093fm_a_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1161_inf__mono,axiom,
! [A2: set_Epistemic_fm_a,C: set_Epistemic_fm_a,B: set_Epistemic_fm_a,D2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ C )
=> ( ( ord_le3275665582123262618c_fm_a @ B @ D2 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ ( inf_in2882328776738922472c_fm_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1162_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1163_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1164_le__infI1,axiom,
! [A2: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1165_le__infI1,axiom,
! [A2: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1166_le__infI1,axiom,
! [A2: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1167_le__infI1,axiom,
! [A2: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_1168_le__infI2,axiom,
! [B: epistemic_fm_a > $o,X3: epistemic_fm_a > $o,A2: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ B @ X3 )
=> ( ord_le4043730696559282883fm_a_o @ ( inf_in3606484609122063093fm_a_o @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1169_le__infI2,axiom,
! [B: set_Epistemic_fm_a,X3: set_Epistemic_fm_a,A2: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ B @ X3 )
=> ( ord_le3275665582123262618c_fm_a @ ( inf_in2882328776738922472c_fm_a @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1170_le__infI2,axiom,
! [B: set_nat,X3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1171_le__infI2,axiom,
! [B: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1172_inf_OorderE,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1173_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( A2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1174_inf_OorderE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1175_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_1176_inf_OorderI,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( A2
= ( inf_in3606484609122063093fm_a_o @ A2 @ B ) )
=> ( ord_le4043730696559282883fm_a_o @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1177_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( A2
= ( inf_in2882328776738922472c_fm_a @ A2 @ B ) )
=> ( ord_le3275665582123262618c_fm_a @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1178_inf_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1179_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_1180_inf__unique,axiom,
! [F: ( epistemic_fm_a > $o ) > ( epistemic_fm_a > $o ) > epistemic_fm_a > $o,X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o] : ( ord_le4043730696559282883fm_a_o @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: epistemic_fm_a > $o,Y2: epistemic_fm_a > $o,Z4: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ X2 @ Y2 )
=> ( ( ord_le4043730696559282883fm_a_o @ X2 @ Z4 )
=> ( ord_le4043730696559282883fm_a_o @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in3606484609122063093fm_a_o @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1181_inf__unique,axiom,
! [F: set_Epistemic_fm_a > set_Epistemic_fm_a > set_Epistemic_fm_a,X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a] : ( ord_le3275665582123262618c_fm_a @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_Epistemic_fm_a,Y2: set_Epistemic_fm_a,Z4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ X2 @ Y2 )
=> ( ( ord_le3275665582123262618c_fm_a @ X2 @ Z4 )
=> ( ord_le3275665582123262618c_fm_a @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in2882328776738922472c_fm_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1182_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_nat,Y2: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1183_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z4 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1184_le__iff__inf,axiom,
( ord_le4043730696559282883fm_a_o
= ( ^ [X: epistemic_fm_a > $o,Y4: epistemic_fm_a > $o] :
( ( inf_in3606484609122063093fm_a_o @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1185_le__iff__inf,axiom,
( ord_le3275665582123262618c_fm_a
= ( ^ [X: set_Epistemic_fm_a,Y4: set_Epistemic_fm_a] :
( ( inf_in2882328776738922472c_fm_a @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1186_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1187_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1188_inf_Oabsorb1,axiom,
! [A2: epistemic_fm_a > $o,B: epistemic_fm_a > $o] :
( ( ord_le4043730696559282883fm_a_o @ A2 @ B )
=> ( ( inf_in3606484609122063093fm_a_o @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1189_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_a,B: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ A2 @ B )
=> ( ( inf_in2882328776738922472c_fm_a @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1190_inf_Oabsorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1191_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_1192_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1193_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1194_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1195_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_1196_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_1197_inf__greatest,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1198_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( A4
= ( inf_inf_nat @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_1199_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_1200_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1201_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( inf_inf_nat @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1202_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_1203_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_1204_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_1205_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_1206_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_1207_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_1208_distrib__sup__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1209_distrib__inf__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1210_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_1211_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_1212_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_1213_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B2: nat] :
( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_1214_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1215_sup_Ocobounded2,axiom,
! [B: nat,A2: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded2
thf(fact_1216_sup_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B ) ) ).
% sup.cobounded1
thf(fact_1217_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_1218_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_1219_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_1220_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1221_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_1222_sup_Oabsorb2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( sup_sup_nat @ A2 @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1223_sup_Oabsorb1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( sup_sup_nat @ A2 @ B )
= A2 ) ) ).
% sup.absorb1
thf(fact_1224_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z4 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1225_sup_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% sup.orderI
thf(fact_1226_sup_OorderE,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B ) ) ) ).
% sup.orderE
thf(fact_1227_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_1228_sup__least,axiom,
! [Y: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_1229_sup__mono,axiom,
! [A2: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1230_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B ) ) ) ) ).
% sup.mono
thf(fact_1231_le__supI2,axiom,
! [X3: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI2
thf(fact_1232_le__supI1,axiom,
! [X3: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B ) ) ) ).
% le_supI1
thf(fact_1233_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_1234_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_1235_le__supI,axiom,
! [A2: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_1236_le__supE,axiom,
! [A2: nat,B: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_1237_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1238_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1239_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_1240_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 ) )
=> ~ ! [V5: list_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( set_Epistemic_fm_a2 @ V5 ) @ V )
=> ~ ( epistemic_AK_a @ A @ ( epistemic_imply_a @ ( cons_Epistemic_fm_a @ P @ V5 ) @ epistemic_FF_a ) ) ) ) ) ).
% inconsistent_subset
thf(fact_1241_inconsistent__imply,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,G: 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 @ G ) ) )
=> ( epistemic_AK_a @ A @ ( epistemic_imply_a @ G @ P ) ) ) ).
% inconsistent_imply
thf(fact_1242_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_1243_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 )
=> ( ! [G5: list_char > $o,H4: epistemic_fm_a > $o] : ( epistemic_eval_a @ G5 @ H4 @ ( 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_1244_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_a > $o,P: epistemic_fm_a,V: set_Epistemic_fm_a] :
( ~ ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ~ ! [W4: set_Epistemic_fm_a] :
( ( ord_le3275665582123262618c_fm_a @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ V ) )
=> ( ~ ( member6642669571620171971c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ W4 )
=> ( ( finite3304564945125563331c_fm_a @ W4 )
=> ( episte2285483198712856226tent_a @ A @ ( sup_su1367922730591523534c_fm_a @ ( insert7817948963269374698c_fm_a @ ( epistemic_Imp_a @ P @ epistemic_FF_a ) @ bot_bo3626323581529592678c_fm_a ) @ W4 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_1245_le__cSup__finite,axiom,
! [X5: set_nat,X3: nat] :
( ( finite_finite_nat @ X5 )
=> ( ( member_nat2 @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1246_Min_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1247_Inf__fin_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 @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1248_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_1249_Min__eqI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ord_less_eq_nat @ X3 @ Y2 ) )
=> ( ( member_nat2 @ X3 @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_1250_Min__le,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 ) ) ) ).
% Min_le
thf(fact_1251_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1252_Min__eq__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M5 )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ M5 @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1253_Min__le__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X3 )
= ( ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1254_eq__Min__iff,axiom,
! [A: set_nat,M5: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M5
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat2 @ M5 @ A )
& ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ M5 @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1255_Min_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A )
=> ( ord_less_eq_nat @ X3 @ A6 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1256_Min_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X3 @ A5 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1257_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B3: nat] :
( ( member_nat2 @ B3 @ A )
=> ( ord_less_eq_nat @ A2 @ B3 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_1258_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1259_Inf__fin_OboundedI,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat2 @ A5 @ A )
=> ( ord_less_eq_nat @ X3 @ A5 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1260_Inf__fin_OboundedE,axiom,
! [A: set_nat,X3: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A )
=> ( ord_less_eq_nat @ X3 @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1261_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_1262_Min__antimono,axiom,
! [M: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M @ N )
=> ( ( M != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% Min_antimono
thf(fact_1263_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1264_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1265_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1266_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1267_dual__Max,axiom,
( ( lattices_Max_nat
@ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X ) )
= lattic8721135487736765967in_nat ) ).
% dual_Max
thf(fact_1268_dual__max,axiom,
( ( max_nat
@ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X ) )
= ord_min_nat ) ).
% dual_max
thf(fact_1269_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1270_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1271_GreatestI__ex__nat,axiom,
! [P4: nat > $o,B: nat] :
( ? [X_1: nat] : ( P4 @ X_1 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1272_Greatest__le__nat,axiom,
! [P4: nat > $o,K2: nat,B: nat] :
( ( P4 @ K2 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% Greatest_le_nat
thf(fact_1273_GreatestI__nat,axiom,
! [P4: nat > $o,K2: nat,B: nat] :
( ( P4 @ K2 )
=> ( ! [Y2: nat] :
( ( P4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P4 @ ( order_Greatest_nat @ P4 ) ) ) ) ).
% GreatestI_nat
% Helper facts (17)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y: list_a] :
( ( if_list_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Epistemic____Logic__Ofm_Itf__a_J_T,axiom,
! [X3: epistemic_fm_a,Y: epistemic_fm_a] :
( ( if_Epistemic_fm_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_T,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( if_Epistemic_fm_a_o @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001_062_It__Epistemic____Logic__Ofm_Itf__a_J_M_Eo_J_T,axiom,
! [X3: epistemic_fm_a > $o,Y: epistemic_fm_a > $o] :
( ( if_Epistemic_fm_a_o @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: set_Epistemic_fm_a,Y: set_Epistemic_fm_a] :
( ( if_set2375185401397246656c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__a_J_J_T,axiom,
! [X3: list_Epistemic_fm_a,Y: list_Epistemic_fm_a] :
( ( if_lis2878681784746929638c_fm_a @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
epistemic_AK_a @ a2 @ ( epistemic_Imp_a @ ( epistemic_Con_a @ p @ q ) @ p ) ).
%------------------------------------------------------------------------------