TPTP Problem File: SLH0480^1.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_00397_015051__6814402_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1536 ( 585 unt; 249 typ; 0 def)
% Number of atoms : 3835 (1242 equ; 0 cnn)
% Maximal formula atoms : 21 ( 2 avg)
% Number of connectives : 13440 ( 463 ~; 55 |; 266 &;11020 @)
% ( 0 <=>;1636 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 987 ( 987 >; 0 *; 0 +; 0 <<)
% Number of symbols : 227 ( 224 usr; 21 con; 0-4 aty)
% Number of variables : 3939 ( 535 ^;3305 !; 99 ?;3939 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 16:12:47.469
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
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set_Pr4658907567593863815c_fm_i: $tType ).
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thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
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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,
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thf(ty_n_t__Set__Oset_Itf__i_J,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__i,type,
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% Explicit typings (224)
thf(sy_c_Epistemic__Logic_OAK_001tf__i,type,
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thf(sy_c_Epistemic__Logic_OAx4_001tf__i,type,
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thf(sy_c_Epistemic__Logic_OAx5_001tf__i,type,
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thf(sy_c_Epistemic__Logic_OAxB_001tf__i,type,
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thf(sy_c_Epistemic__Logic_OAxT_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Oconsistent_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Oconsistent_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Oeval_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_OCon_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_ODis_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_OFF_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_OFF_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_OImp_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_OK_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_OK_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_OPro_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001t__Epistemic____Logic__Ofm_Itf__i_J_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_Orel__fm_001tf__i_001tf__i,type,
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thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Epistemic__Logic_Ofm_Oset__fm_001tf__i,type,
epistemic_set_fm_i: epistemic_fm_i > set_i ).
thf(sy_c_Epistemic__Logic_Oimply_001tf__i,type,
epistemic_imply_i: list_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Finite__Set_Ofinite_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Otranspose_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
transp2882070894626480306c_fm_i: list_l4360594050116504750c_fm_i > list_l4360594050116504750c_fm_i ).
thf(sy_c_List_Ounion_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
union_Epistemic_fm_i: list_Epistemic_fm_i > list_Epistemic_fm_i > list_Epistemic_fm_i ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima1924290493700099598c_fm_i: ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima5540402109999912942c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima2128808098672023021c_fm_i: ( set_Ep6763084006362001438c_fm_i > $o ) > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
maxima7510630710825918413c_fm_i: ( set_Ep7768176982351688190c_fm_i > $o ) > set_Ep7768176982351688190c_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima3264069618988350929c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS_Omaximal_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
maxima7741914846562630577c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima8116622286610175645c_fm_i: set_Pr4658907567593863815c_fm_i > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_001t__Nat__Onat,type,
maxima4501515429357695765rd_nat: set_Pr1261947904930325089at_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Maximal__Consistent__Sets_OMCS__Lim__Ord_OextendS_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
maxima5458213620894205884c_fm_i: ( set_Epistemic_fm_i > $o ) > set_Epistemic_fm_i > epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
size_s3357407047445144980c_fm_i: list_Epistemic_fm_i > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
bot_bo2580527446789968623fm_i_o: epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
bot_bo6089404950617257231fm_i_o: set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
bot_bo4194595901900360558c_fm_i: set_Epistemic_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
bot_bo145720340923748686c_fm_i: set_se3485332733965609186c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
bot_bo4781621276559889198c_fm_i: set_se7339729205154126530c_fm_i ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__i_J,type,
bot_bot_set_i: set_i ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Epistemic____Logic__Ofm_Itf__i_J_M_Eo_J,type,
ord_le190830114487426235fm_i_o: ( epistemic_fm_i > $o ) > ( epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_M_Eo_J,type,
ord_le5706303257367236315fm_i_o: ( set_Epistemic_fm_i > $o ) > ( set_Epistemic_fm_i > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
ord_le3843937902494030498c_fm_i: set_Epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
ord_le5389487502678872194c_fm_i: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_J,type,
ord_le2772500130360476258c_fm_i: set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_Set_OCollect_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
collec4904205187116291597c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec5759350117362652307c_fm_i: ( list_Epistemic_fm_i > $o ) > set_li8269783359719514504c_fm_i ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2200325868635342451c_fm_i: ( list_s8081015415394010888c_fm_i > $o ) > set_li2934481837907835752c_fm_i ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
collec3087743281813070829c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
collec2761885132556217805c_fm_i: ( set_se3485332733965609186c_fm_i > $o ) > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_OCollect_001tf__i,type,
collect_i: ( i > $o ) > set_i ).
thf(sy_c_Set_Ofilter_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
filter6053540608743173075c_fm_i: ( epistemic_fm_i > $o ) > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat2: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
filter8450032351001387443c_fm_i: ( set_Epistemic_fm_i > $o ) > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
insert7817948997695205106c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
insert7698009978809854162c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
insert7111703059596746418c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > set_se7339729205154126530c_fm_i ).
thf(sy_c_Set_Oinsert_001tf__i,type,
insert_i: i > set_i > set_i ).
thf(sy_c_Set_Ois__empty_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_emp7643256682846039796c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Ois__singleton_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
is_sin4923117429174228502c_fm_i: set_Epistemic_fm_i > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
is_sin2358732406619996150c_fm_i: set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_Set_Oremove_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
remove5552923401211475591c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
remove1392734035364174951c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i ).
thf(sy_c_Set_Othe__elem_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
the_el2173195912186371479c_fm_i: set_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_Stalnaker__Logic_OAx__2_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
stalna8619858462013048343c_fm_i: episte1308613106011496894c_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_OAx__2_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
stalna2151822877101976567c_fm_i: episte4276422356747265950c_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_OAx__2_001tf__i,type,
stalnaker_Ax_2_i: epistemic_fm_i > $o ).
thf(sy_c_Stalnaker__Logic_Oconjunct_001tf__i,type,
stalnaker_conjunct_i: list_Epistemic_fm_i > epistemic_fm_i ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1686840210572743143c_fm_i: episte1308613106011496894c_fm_i > set_Ep6763084006362001438c_fm_i > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member3637625068036862919c_fm_i: episte4276422356747265950c_fm_i > set_Ep7768176982351688190c_fm_i > $o ).
thf(sy_c_member_001t__Epistemic____Logic__Ofm_Itf__i_J,type,
member6642669606046002379c_fm_i: epistemic_fm_i > set_Epistemic_fm_i > $o ).
thf(sy_c_member_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member6475149752759350353c_fm_i: list_Epistemic_fm_i > set_li8269783359719514504c_fm_i > $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__i_J_J_J,type,
member605545779687026225c_fm_i: list_s8081015415394010888c_fm_i > set_li2934481837907835752c_fm_i > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J,type,
member1104366573291651755c_fm_i: set_Epistemic_fm_i > set_se3485332733965609186c_fm_i > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J,type,
member1461078328125707403c_fm_i: set_se3485332733965609186c_fm_i > set_se7339729205154126530c_fm_i > $o ).
thf(sy_c_member_001tf__i,type,
member_i: i > set_i > $o ).
thf(sy_v_A,type,
a: epistemic_fm_i > $o ).
thf(sy_v_S_H____,type,
s: list_Epistemic_fm_i ).
thf(sy_v_U,type,
u: set_Epistemic_fm_i ).
thf(sy_v_V,type,
v: set_Epistemic_fm_i ).
thf(sy_v_W,type,
w: set_Epistemic_fm_i ).
thf(sy_v_X1____,type,
x1: set_Epistemic_fm_i ).
thf(sy_v_Z_H____,type,
z: set_Epistemic_fm_i ).
thf(sy_v_i,type,
i2: i ).
% Relevant facts (1277)
thf(fact_0_assms_I1_J,axiom,
! [P: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ P )
=> ( a @ P ) ) ).
% assms(1)
thf(fact_1_assms_I6_J,axiom,
episte2285483198712856234tent_i @ a @ u ).
% assms(6)
thf(fact_2_assms_I4_J,axiom,
episte2285483198712856234tent_i @ a @ w ).
% assms(4)
thf(fact_3_assms_I7_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ u ).
% assms(7)
thf(fact_4_assms_I5_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ w ).
% assms(5)
thf(fact_5_z1,axiom,
( ( episte2285483198712856234tent_i @ a @ z )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ z ) ) ).
% z1
thf(fact_6_assms_I2_J,axiom,
episte2285483198712856234tent_i @ a @ v ).
% assms(2)
thf(fact_7_assms_I3_J,axiom,
maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ v ).
% assms(3)
thf(fact_8_exactly__one__in__maximal,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,P2: episte1308613106011496894c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( member1686840210572743143c_fm_i @ P2 @ V )
= ( ~ ( member1686840210572743143c_fm_i @ ( episte260752253203357973c_fm_i @ P2 @ episte5073044278343014369c_fm_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_9_exactly__one__in__maximal,axiom,
! [A: episte4276422356747265950c_fm_i > $o,V: set_Ep7768176982351688190c_fm_i,P2: episte4276422356747265950c_fm_i] :
( ( episte7789141329995100326c_fm_i @ A @ V )
=> ( ( maxima7510630710825918413c_fm_i @ ( episte7789141329995100326c_fm_i @ A ) @ V )
=> ( ( member3637625068036862919c_fm_i @ P2 @ V )
= ( ~ ( member3637625068036862919c_fm_i @ ( episte5768433346120044277c_fm_i @ P2 @ episte5042310466404390337c_fm_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_10_exactly__one__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V )
= ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ V ) ) ) ) ) ).
% exactly_one_in_maximal
thf(fact_11_x1,axiom,
( ( episte2285483198712856234tent_i @ a @ x1 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ x1 ) ) ).
% x1
thf(fact_12_mcs__conjunction,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,P2: episte1308613106011496894c_fm_i,Q: episte1308613106011496894c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( ( member1686840210572743143c_fm_i @ P2 @ V )
& ( member1686840210572743143c_fm_i @ Q @ V ) )
=> ( member1686840210572743143c_fm_i @ ( episte3685526521632971807c_fm_i @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_13_mcs__conjunction,axiom,
! [A: episte4276422356747265950c_fm_i > $o,V: set_Ep7768176982351688190c_fm_i,P2: episte4276422356747265950c_fm_i,Q: episte4276422356747265950c_fm_i] :
( ( episte7789141329995100326c_fm_i @ A @ V )
=> ( ( maxima7510630710825918413c_fm_i @ ( episte7789141329995100326c_fm_i @ A ) @ V )
=> ( ( ( member3637625068036862919c_fm_i @ P2 @ V )
& ( member3637625068036862919c_fm_i @ Q @ V ) )
=> ( member3637625068036862919c_fm_i @ ( episte64184772710425087c_fm_i @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_14_mcs__conjunction,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( member6642669606046002379c_fm_i @ P2 @ V )
& ( member6642669606046002379c_fm_i @ Q @ V ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Con_i @ P2 @ Q ) @ V ) ) ) ) ).
% mcs_conjunction
thf(fact_15_ax__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( A @ P2 )
=> ( member6642669606046002379c_fm_i @ P2 @ V ) ) ) ) ).
% ax_in_maximal
thf(fact_16_MCS_Omaximal_Ocong,axiom,
maxima7741914846562630577c_fm_i = maxima7741914846562630577c_fm_i ).
% MCS.maximal.cong
thf(fact_17_MCS_Omaximal_Ocong,axiom,
maxima3264069618988350929c_fm_i = maxima3264069618988350929c_fm_i ).
% MCS.maximal.cong
thf(fact_18_consequent__in__maximal,axiom,
! [A: episte1308613106011496894c_fm_i > $o,V: set_Ep6763084006362001438c_fm_i,P2: episte1308613106011496894c_fm_i,Q: episte1308613106011496894c_fm_i] :
( ( episte704884158563092166c_fm_i @ A @ V )
=> ( ( maxima2128808098672023021c_fm_i @ ( episte704884158563092166c_fm_i @ A ) @ V )
=> ( ( member1686840210572743143c_fm_i @ P2 @ V )
=> ( ( member1686840210572743143c_fm_i @ ( episte260752253203357973c_fm_i @ P2 @ Q ) @ V )
=> ( member1686840210572743143c_fm_i @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_19_consequent__in__maximal,axiom,
! [A: episte4276422356747265950c_fm_i > $o,V: set_Ep7768176982351688190c_fm_i,P2: episte4276422356747265950c_fm_i,Q: episte4276422356747265950c_fm_i] :
( ( episte7789141329995100326c_fm_i @ A @ V )
=> ( ( maxima7510630710825918413c_fm_i @ ( episte7789141329995100326c_fm_i @ A ) @ V )
=> ( ( member3637625068036862919c_fm_i @ P2 @ V )
=> ( ( member3637625068036862919c_fm_i @ ( episte5768433346120044277c_fm_i @ P2 @ Q ) @ V )
=> ( member3637625068036862919c_fm_i @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_20_consequent__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ Q ) @ V )
=> ( member6642669606046002379c_fm_i @ Q @ V ) ) ) ) ) ).
% consequent_in_maximal
thf(fact_21_Ax__2_Ocases,axiom,
! [A2: episte1308613106011496894c_fm_i] :
( ( stalna8619858462013048343c_fm_i @ A2 )
=> ~ ! [I: epistemic_fm_i,P3: episte1308613106011496894c_fm_i] :
( A2
!= ( episte260752253203357973c_fm_i @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ P3 ) @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) @ ( episte5657488666450005526c_fm_i @ I @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I @ ( episte260752253203357973c_fm_i @ P3 @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_22_Ax__2_Ocases,axiom,
! [A2: episte4276422356747265950c_fm_i] :
( ( stalna2151822877101976567c_fm_i @ A2 )
=> ~ ! [I: set_Epistemic_fm_i,P3: episte4276422356747265950c_fm_i] :
( A2
!= ( episte5768433346120044277c_fm_i @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I @ P3 ) @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) @ ( episte3581967596712343542c_fm_i @ I @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I @ ( episte5768433346120044277c_fm_i @ P3 @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_23_Ax__2_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( stalnaker_Ax_2_i @ A2 )
=> ~ ! [I: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P3 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax_2.cases
thf(fact_24_Ax__2_Osimps,axiom,
( stalna8619858462013048343c_fm_i
= ( ^ [A3: episte1308613106011496894c_fm_i] :
? [I2: epistemic_fm_i,P4: episte1308613106011496894c_fm_i] :
( A3
= ( episte260752253203357973c_fm_i @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ P4 ) @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I2 @ ( episte260752253203357973c_fm_i @ P4 @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_25_Ax__2_Osimps,axiom,
( stalna2151822877101976567c_fm_i
= ( ^ [A3: episte4276422356747265950c_fm_i] :
? [I2: set_Epistemic_fm_i,P4: episte4276422356747265950c_fm_i] :
( A3
= ( episte5768433346120044277c_fm_i @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I2 @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I2 @ P4 ) @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) @ ( episte3581967596712343542c_fm_i @ I2 @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I2 @ ( episte5768433346120044277c_fm_i @ P4 @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_26_Ax__2_Osimps,axiom,
( stalnaker_Ax_2_i
= ( ^ [A3: epistemic_fm_i] :
? [I2: i,P4: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P4 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax_2.simps
thf(fact_27_Ax__2_Ointros,axiom,
! [I3: epistemic_fm_i,P2: episte1308613106011496894c_fm_i] : ( stalna8619858462013048343c_fm_i @ ( episte260752253203357973c_fm_i @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I3 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I3 @ P2 ) @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) @ ( episte5657488666450005526c_fm_i @ I3 @ ( episte260752253203357973c_fm_i @ ( episte5657488666450005526c_fm_i @ I3 @ ( episte260752253203357973c_fm_i @ P2 @ episte5073044278343014369c_fm_i ) ) @ episte5073044278343014369c_fm_i ) ) ) ) ).
% Ax_2.intros
thf(fact_28_Ax__2_Ointros,axiom,
! [I3: set_Epistemic_fm_i,P2: episte4276422356747265950c_fm_i] : ( stalna2151822877101976567c_fm_i @ ( episte5768433346120044277c_fm_i @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I3 @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I3 @ P2 ) @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) @ ( episte3581967596712343542c_fm_i @ I3 @ ( episte5768433346120044277c_fm_i @ ( episte3581967596712343542c_fm_i @ I3 @ ( episte5768433346120044277c_fm_i @ P2 @ episte5042310466404390337c_fm_i ) ) @ episte5042310466404390337c_fm_i ) ) ) ) ).
% Ax_2.intros
thf(fact_29_Ax__2_Ointros,axiom,
! [I3: i,P2: epistemic_fm_i] : ( stalnaker_Ax_2_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax_2.intros
thf(fact_30__092_060open_062_092_060nexists_062X_O_Aconsistent_AA_AX_A_092_060and_062_Amaximal_AA_AX_A_092_060and_062_AX_A_092_060in_062_A_123Wa_O_Aknown_AW_Ai_A_092_060subseteq_062_AWa_125_A_092_060inter_062_A_123W_O_Aknown_AU_Ai_A_092_060subseteq_062_AW_125_092_060close_062,axiom,
~ ? [X: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ a @ X )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ X )
& ( member1104366573291651755c_fm_i @ X
@ ( inf_in161960956874937808c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ w ) ) ) )
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ u ) ) ) ) ) ) ) ).
% \<open>\<nexists>X. consistent A X \<and> maximal A X \<and> X \<in> {Wa. known W i \<subseteq> Wa} \<inter> {W. known U i \<subseteq> W}\<close>
thf(fact_31__092_060open_062_092_060not_062_Aconsistent_AA_A_Iknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_J_092_060close_062,axiom,
~ ( episte2285483198712856234tent_i @ a
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ u ) ) ) ) ).
% \<open>\<not> consistent A (known W i \<union> known U i)\<close>
thf(fact_32__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062X1_O_A_092_060lbrakk_062consistent_AA_AX1_A_092_060and_062_Amaximal_AA_AX1_059_AX1_A_092_060in_062_A_123W_O_Aknown_AV_Ai_A_092_060subseteq_062_AW_125_059_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AX1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X1: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ a @ X1 )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ X1 ) )
=> ( ( member1104366573291651755c_fm_i @ X1
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) )
=> ~ ( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ X1 ) ) ) ).
% \<open>\<And>thesis. (\<And>X1. \<lbrakk>consistent A X1 \<and> maximal A X1; X1 \<in> {W. known V i \<subseteq> W}; L i (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> X1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_33__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Z_H_O_A_092_060lbrakk_062consistent_AA_AZ_H_A_092_060and_062_Amaximal_AA_AZ_H_059_AZ_H_A_092_060in_062_A_123W_O_Aknown_AV_Ai_A_092_060subseteq_062_AW_125_059_AK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060and_062_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AZ_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Z: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ a @ Z )
& ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ a ) @ Z ) )
=> ( ( member1104366573291651755c_fm_i @ Z
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) )
=> ~ ( member6642669606046002379c_fm_i
@ ( epistemic_Con_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ Z ) ) ) ).
% \<open>\<And>thesis. (\<And>Z'. \<lbrakk>consistent A Z' \<and> maximal A Z'; Z' \<in> {W. known V i \<subseteq> W}; K i (conjunct (filter (\<lambda>p. p \<in> known W i) S')) \<^bold>\<and> L i (conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> Z'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_34_assms_I8_J,axiom,
( member1104366573291651755c_fm_i @ w
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) ) ).
% assms(8)
thf(fact_35_assms_I9_J,axiom,
( member1104366573291651755c_fm_i @ u
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) ) ).
% assms(9)
thf(fact_36_x2,axiom,
( member1104366573291651755c_fm_i @ x1
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) ) ).
% x2
thf(fact_37_z2,axiom,
( member1104366573291651755c_fm_i @ z
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ v ) ) ) ) ) ).
% z2
thf(fact_38_fm_Oinject_I5_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,Y61: epistemic_fm_i,Y62: episte1308613106011496894c_fm_i] :
( ( ( episte5657488666450005526c_fm_i @ X61 @ X62 )
= ( episte5657488666450005526c_fm_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_39_fm_Oinject_I5_J,axiom,
! [X61: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i,Y61: set_Epistemic_fm_i,Y62: episte4276422356747265950c_fm_i] :
( ( ( episte3581967596712343542c_fm_i @ X61 @ X62 )
= ( episte3581967596712343542c_fm_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_40_fm_Oinject_I5_J,axiom,
! [X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( ( epistemic_K_i @ X61 @ X62 )
= ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_41_fm_Oinject_I4_J,axiom,
! [X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i,Y51: episte1308613106011496894c_fm_i,Y52: episte1308613106011496894c_fm_i] :
( ( ( episte260752253203357973c_fm_i @ X51 @ X52 )
= ( episte260752253203357973c_fm_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_42_fm_Oinject_I4_J,axiom,
! [X51: episte4276422356747265950c_fm_i,X52: episte4276422356747265950c_fm_i,Y51: episte4276422356747265950c_fm_i,Y52: episte4276422356747265950c_fm_i] :
( ( ( episte5768433346120044277c_fm_i @ X51 @ X52 )
= ( episte5768433346120044277c_fm_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_43_fm_Oinject_I4_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( ( epistemic_Imp_i @ X51 @ X52 )
= ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_44_fm_Oinject_I3_J,axiom,
! [X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i,Y41: episte1308613106011496894c_fm_i,Y42: episte1308613106011496894c_fm_i] :
( ( ( episte3685526521632971807c_fm_i @ X41 @ X42 )
= ( episte3685526521632971807c_fm_i @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_45_fm_Oinject_I3_J,axiom,
! [X41: episte4276422356747265950c_fm_i,X42: episte4276422356747265950c_fm_i,Y41: episte4276422356747265950c_fm_i,Y42: episte4276422356747265950c_fm_i] :
( ( ( episte64184772710425087c_fm_i @ X41 @ X42 )
= ( episte64184772710425087c_fm_i @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_46_fm_Oinject_I3_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( ( epistemic_Con_i @ X41 @ X42 )
= ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_47__092_060open_062_092_060_094bold_062_092_060not_062_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AX1_A_092_060and_062_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AX1_092_060close_062,axiom,
( ( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ epistemic_FF_i )
@ x1 )
& ( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ x1 ) ) ).
% \<open>\<^bold>\<not> L i (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> X1 \<and> L i (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> X1\<close>
thf(fact_48_x3,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ x1 ) ).
% x3
thf(fact_49_z3,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Con_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ z ) ).
% z3
thf(fact_50_c4_H,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ w ) ).
% c4'
thf(fact_51_c4,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ u ) ).
% c4
thf(fact_52_c5,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ v ) ).
% c5
thf(fact_53__092_060open_062K_Ai_A_IL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060in_062_AV_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ v ) ).
% \<open>K i (L i (conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<in> V\<close>
thf(fact_54__092_060open_062L_Ai_A_IK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060in_062_AV_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ v ) ).
% \<open>L i (K i (conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<in> V\<close>
thf(fact_55_o3,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) )
@ v ) ).
% o3
thf(fact_56_o2,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ v ) ).
% o2
thf(fact_57__092_060open_062K_Ai_A_IL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060_094bold_062_092_060and_062_AL_Ai_A_IK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_J_A_092_060in_062_AV_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Con_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ v ) ).
% \<open>K i (L i (conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<^bold>\<and> L i (K i (conjunct (filter (\<lambda>p. p \<in> known W i) S'))) \<in> V\<close>
thf(fact_58__092_060open_062L_Ai_A_IL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060and_062_AK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_J_A_092_060in_062_AV_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ v ) ).
% \<open>L i (L i (conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<^bold>\<and> K i (conjunct (filter (\<lambda>p. p \<in> known W i) S'))) \<in> V\<close>
thf(fact_59__092_060open_062L_Ai_A_IK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060and_062_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060in_062_AV_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ v ) ).
% \<open>L i (K i (conjunct (filter (\<lambda>p. p \<in> known W i) S')) \<^bold>\<and> L i (conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<in> V\<close>
thf(fact_60_x4,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ x1 ) ).
% x4
thf(fact_61__092_060open_062K_Ai_A_I_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_AL_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AX1_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ ( epistemic_Imp_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ epistemic_FF_i ) )
@ x1 ) ).
% \<open>K i (\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known W i) S'))) \<^bold>\<longrightarrow> \<^bold>\<not> L i (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> X1\<close>
thf(fact_62__092_060open_062K_Ai_A_I_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060in_062_AX1_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) )
@ x1 ) ).
% \<open>K i (\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known W i) S'))) \<^bold>\<longrightarrow> K i (\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<in> X1\<close>
thf(fact_63_mem__Collect__eq,axiom,
! [A2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( member1461078328125707403c_fm_i @ A2 @ ( collec2761885132556217805c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A2: list_s8081015415394010888c_fm_i,P5: list_s8081015415394010888c_fm_i > $o] :
( ( member605545779687026225c_fm_i @ A2 @ ( collec2200325868635342451c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A2: i,P5: i > $o] :
( ( member_i @ A2 @ ( collect_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A2: set_nat,P5: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A2: list_Epistemic_fm_i,P5: list_Epistemic_fm_i > $o] :
( ( member6475149752759350353c_fm_i @ A2 @ ( collec5759350117362652307c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A2: list_nat,P5: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A2: epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A2: nat,P5: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_li2934481837907835752c_fm_i] :
( ( collec2200325868635342451c_fm_i
@ ^ [X2: list_s8081015415394010888c_fm_i] : ( member605545779687026225c_fm_i @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_i] :
( ( collect_i
@ ^ [X2: i] : ( member_i @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_li8269783359719514504c_fm_i] :
( ( collec5759350117362652307c_fm_i
@ ^ [X2: list_Epistemic_fm_i] : ( member6475149752759350353c_fm_i @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A: set_Epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_80_Collect__cong,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ! [X3: set_Epistemic_fm_i] :
( ( P5 @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec3087743281813070829c_fm_i @ P5 )
= ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ).
% Collect_cong
thf(fact_81_Collect__cong,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P5 @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collec4904205187116291597c_fm_i @ P5 )
= ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ).
% Collect_cong
thf(fact_82_Collect__cong,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P5 @ X3 )
= ( Q3 @ X3 ) )
=> ( ( collect_nat @ P5 )
= ( collect_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_83__092_060open_062K_Ai_A_I_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_J_A_092_060in_062_AX1_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) )
@ x1 ) ).
% \<open>K i (\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known W i) S')) \<^bold>\<longrightarrow> \<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S'))) \<in> X1\<close>
thf(fact_84__092_060open_062_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AX1_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ x1 ) ).
% \<open>\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known W i) S')) \<^bold>\<longrightarrow> \<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known W i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> X1\<close>
thf(fact_85_z4,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ epistemic_FF_i ) )
@ epistemic_FF_i )
@ z ) ).
% z4
thf(fact_86__C_K_C_I2_J,axiom,
( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ s )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ u ) ) ) ) ).
% "*"(2)
thf(fact_87_fm_Odistinct_I5_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_88_fm_Odistinct_I7_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_89_fm_Odistinct_I9_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_90_fm_Odistinct_I25_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_91_fm_Odistinct_I27_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Con_i @ X41 @ X42 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_92_fm_Odistinct_I29_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Imp_i @ X51 @ X52 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_93_reach__dualK,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W: set_Epistemic_fm_i,I3: i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W )
=> ( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) )
=> ! [P: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P @ W )
=> ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V ) ) ) ) ) ) ) ).
% reach_dualK
thf(fact_94_dual__reach,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I3: i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ V )
=> ? [W2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ W2
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) )
& ( member6642669606046002379c_fm_i @ P2 @ W2 ) ) ) ) ) ).
% dual_reach
thf(fact_95_consistent__hereditary,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( episte2285483198712856234tent_i @ A @ S2 ) ) ) ).
% consistent_hereditary
thf(fact_96_maximal__extension,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ V @ W2 )
=> ( ( episte2285483198712856234tent_i @ A @ W2 )
=> ~ ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W2 ) ) ) ) ).
% maximal_extension
thf(fact_97_o1,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) ).
% o1
thf(fact_98__092_060open_062A_A_092_060turnstile_062_AK_Ai_A_IK_Ai_A_I_092_060_094bold_062_092_060not_062_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060and_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_J_J_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_K_i @ i2
@ ( epistemic_K_i @ i2
@ ( epistemic_Imp_i
@ ( epistemic_Con_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ epistemic_FF_i ) ) ) ) ).
% \<open>A \<turnstile> K i (K i (\<^bold>\<not> (conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<and> conjunct (filter (\<lambda>p. p \<in> known W i) S'))))\<close>
thf(fact_99__092_060open_062set_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060union_062_Aset_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_061_Aset_AS_H_092_060close_062,axiom,
( ( sup_su1936195050962291414c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
= ( set_Epistemic_fm_i2 @ s ) ) ).
% \<open>set (filter (\<lambda>p. p \<in> known U i) S') \<union> set (filter (\<lambda>p. p \<in> known W i) S') = set S'\<close>
thf(fact_100__092_060open_062A_A_092_060turnstile_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S')\<close>
thf(fact_101__092_060open_062set_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060subseteq_062_Aknown_AU_Ai_092_060close_062,axiom,
( ord_le3843937902494030498c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ u ) ) ) ).
% \<open>set (filter (\<lambda>p. p \<in> known U i) S') \<subseteq> known U i\<close>
thf(fact_102__092_060open_062set_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060subseteq_062_Aknown_AW_Ai_092_060close_062,axiom,
( ord_le3843937902494030498c_fm_i
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ w ) ) ) ).
% \<open>set (filter (\<lambda>p. p \<in> known W i) S') \<subseteq> known W i\<close>
thf(fact_103__092_060open_062conjunct_A_Imap_A_IK_Ai_J_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060_094bold_062_092_060longrightarrow_062_AK_Ai_A_Iconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060in_062_AU_092_060close_062,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) ) )
@ u ) ).
% \<open>conjunct (map (K i) (filter (\<lambda>p. p \<in> known U i) S')) \<^bold>\<longrightarrow> K i (conjunct (filter (\<lambda>p. p \<in> known U i) S')) \<in> U\<close>
thf(fact_104_c3,axiom,
( member6642669606046002379c_fm_i
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ ( epistemic_K_i @ i2
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) ) )
@ w ) ).
% c3
thf(fact_105__092_060open_062_092_060forall_062p_O_Ap_A_092_060in_062_Aset_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060longrightarrow_062_AK_Ai_Ap_A_092_060in_062_AU_092_060close_062,axiom,
! [P: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [Q2: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ Q2
@ ( collec4904205187116291597c_fm_i
@ ^ [R: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ R ) @ u ) ) )
@ s ) ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ u ) ) ).
% \<open>\<forall>p. p \<in> set (filter (\<lambda>p. p \<in> known U i) S') \<longrightarrow> K i p \<in> U\<close>
thf(fact_106__092_060open_062_092_060forall_062p_O_Ap_A_092_060in_062_Aset_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_A_092_060longrightarrow_062_AK_Ai_Ap_A_092_060in_062_AW_092_060close_062,axiom,
! [P: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ P
@ ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [Q2: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ Q2
@ ( collec4904205187116291597c_fm_i
@ ^ [R: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ R ) @ w ) ) )
@ s ) ) )
=> ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P ) @ w ) ) ).
% \<open>\<forall>p. p \<in> set (filter (\<lambda>p. p \<in> known W i) S') \<longrightarrow> K i p \<in> W\<close>
thf(fact_107_Un__Int__eq_I1_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_108_Un__Int__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_109_subsetI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( member1104366573291651755c_fm_i @ X3 @ B ) )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_110_subsetI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( member6642669606046002379c_fm_i @ X3 @ B ) )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% subsetI
thf(fact_111_subset__antisym,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_112_IntI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_113_IntI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% IntI
thf(fact_114_Int__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_115_Int__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Int_iff
thf(fact_116_UnCI,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ A ) )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_117_UnCI,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ A ) )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnCI
thf(fact_118_Un__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
| ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_119_Un__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
| ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Un_iff
thf(fact_120_Int__subset__iff,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
& ( ord_le5389487502678872194c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_121_Int__subset__iff,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= ( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
& ( ord_le3843937902494030498c_fm_i @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_122_Un__subset__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
& ( ord_le3843937902494030498c_fm_i @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_123_Int__Un__eq_I4_J,axiom,
! [T: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ T @ ( inf_in161960956874937808c_fm_i @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_124_Int__Un__eq_I4_J,axiom,
! [T: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ T @ ( inf_in3450601097109690352c_fm_i @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_125_Int__Un__eq_I3_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ S @ ( inf_in161960956874937808c_fm_i @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_126_Int__Un__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ S @ ( inf_in3450601097109690352c_fm_i @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_127_Int__Un__eq_I2_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_128_Int__Un__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_129_Int__Un__eq_I1_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_130_Int__Un__eq_I1_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_131_Un__Int__eq_I4_J,axiom,
! [T: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ T @ ( sup_su2582925890723967158c_fm_i @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_132_Un__Int__eq_I4_J,axiom,
! [T: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ T @ ( sup_su1936195050962291414c_fm_i @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_133_Un__Int__eq_I3_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ S @ ( sup_su2582925890723967158c_fm_i @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_134_Un__Int__eq_I3_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ S @ ( sup_su1936195050962291414c_fm_i @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_135_Un__Int__eq_I2_J,axiom,
! [S: set_se3485332733965609186c_fm_i,T: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_136_Un__Int__eq_I2_J,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_137_c2_H,axiom,
( member6642669606046002379c_fm_i
@ ( stalnaker_conjunct_i
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ w ) ).
% c2'
thf(fact_138_c2,axiom,
( member6642669606046002379c_fm_i
@ ( stalnaker_conjunct_i
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ u ) ).
% c2
thf(fact_139_c1,axiom,
( ord_le3843937902494030498c_fm_i
@ ( set_Epistemic_fm_i2
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) ) )
@ w ) ).
% c1
thf(fact_140__092_060open_062set_A_Imap_A_IK_Ai_J_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_J_A_092_060subseteq_062_AU_092_060close_062,axiom,
( ord_le3843937902494030498c_fm_i
@ ( set_Epistemic_fm_i2
@ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ i2 )
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) ) )
@ u ) ).
% \<open>set (map (K i) (filter (\<lambda>p. p \<in> known U i) S')) \<subseteq> U\<close>
thf(fact_141__C_K_C_I3_J,axiom,
finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ s ) ).
% "*"(3)
thf(fact_142__C_K_C_I1_J,axiom,
epistemic_AK_i @ a @ ( epistemic_imply_i @ s @ epistemic_FF_i ) ).
% "*"(1)
thf(fact_143_Ax,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i] :
( ( A @ P2 )
=> ( epistemic_AK_i @ A @ P2 ) ) ).
% Ax
thf(fact_144_R2,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I3: i] :
( ( epistemic_AK_i @ A @ P2 )
=> ( epistemic_AK_i @ A @ ( epistemic_K_i @ I3 @ P2 ) ) ) ).
% R2
thf(fact_145_R1,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P2 )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( epistemic_AK_i @ A @ Q ) ) ) ).
% R1
thf(fact_146_K__trans,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ Q ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ Q @ R2 ) @ ( epistemic_Imp_i @ P2 @ R2 ) ) ) ) ).
% K_trans
thf(fact_147_K__imp__trans,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ R2 ) ) ) ) ).
% K_imp_trans
thf(fact_148_K__imp__trans_H,axiom,
! [A: epistemic_fm_i > $o,Q: epistemic_fm_i,R2: epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ Q @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ Q ) @ ( epistemic_Imp_i @ P2 @ R2 ) ) ) ) ).
% K_imp_trans'
thf(fact_149_K__conjunction__in__mult,axiom,
! [A: epistemic_fm_i > $o,I3: i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( stalnaker_conjunct_i @ G ) ) @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I3 ) @ G ) ) ) ) ).
% K_conjunction_in_mult
thf(fact_150_K__conjunction__out__mult,axiom,
! [A: epistemic_fm_i > $o,I3: i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I3 ) @ G ) ) @ ( epistemic_K_i @ I3 @ ( stalnaker_conjunct_i @ G ) ) ) ) ).
% K_conjunction_out_mult
thf(fact_151_K__A2_H,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ Q ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_A2'
thf(fact_152_K__map,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,I3: i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_map
thf(fact_153_K__imply__multi,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ B2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ C ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Con_i @ B2 @ C ) ) ) ) ) ).
% K_imply_multi
thf(fact_154_K__multi__imply,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i,C: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ A2 @ ( epistemic_Imp_i @ B2 @ C ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ A2 @ B2 ) @ C ) ) ) ).
% K_multi_imply
thf(fact_155_deriv__in__maximal,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( epistemic_AK_i @ A @ P2 )
=> ( member6642669606046002379c_fm_i @ P2 @ V ) ) ) ) ).
% deriv_in_maximal
thf(fact_156_K__LK,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ epistemic_FF_i ) ) ) ).
% K_LK
thf(fact_157_K__L__dual,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ P2 ) ) ) ).
% K_L_dual
thf(fact_158_A2,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ Q ) ) ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ).
% A2
thf(fact_159_K__conjunction__in,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P2 @ Q ) ) @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_conjunction_in
thf(fact_160_K__conjunction__out,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ Q ) ) @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P2 @ Q ) ) ) ) ).
% K_conjunction_out
thf(fact_161_K__conj__imply__factor,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i,R2: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ Q ) ) @ R2 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Con_i @ P2 @ Q ) ) @ R2 ) ) ) ).
% K_conj_imply_factor
thf(fact_162_in__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( member1104366573291651755c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_163_in__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( member6642669606046002379c_fm_i @ X4 @ B ) ) ) ).
% in_mono
thf(fact_164_subsetD,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_165_subsetD,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% subsetD
thf(fact_166_equalityE,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ~ ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ) ).
% equalityE
thf(fact_167_subset__eq,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A4 )
=> ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_168_subset__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A4 )
=> ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_169_equalityD1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% equalityD1
thf(fact_170_equalityD2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( A = B )
=> ( ord_le3843937902494030498c_fm_i @ B @ A ) ) ).
% equalityD2
thf(fact_171_subset__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
! [T2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ T2 @ A4 )
=> ( member1104366573291651755c_fm_i @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_172_subset__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
! [T2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ T2 @ A4 )
=> ( member6642669606046002379c_fm_i @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_173_subset__refl,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ A ) ).
% subset_refl
thf(fact_174_Collect__mono,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ! [X3: set_Epistemic_fm_i] :
( ( P5 @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ).
% Collect_mono
thf(fact_175_Collect__mono,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P5 @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P5 ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_176_Collect__mono,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( P5 @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ).
% Collect_mono
thf(fact_177_subset__trans,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_178_set__eq__subset,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y = Z2 ) )
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A4 @ B3 )
& ( ord_le3843937902494030498c_fm_i @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_179_Collect__mono__iff,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q3 ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
=> ( Q3 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_180_Collect__mono__iff,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P5 ) @ ( collect_nat @ Q3 ) )
= ( ! [X2: nat] :
( ( P5 @ X2 )
=> ( Q3 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_181_Collect__mono__iff,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q3 ) )
= ( ! [X2: epistemic_fm_i] :
( ( P5 @ X2 )
=> ( Q3 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_IntE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_183_IntE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% IntE
thf(fact_184_IntD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_185_IntD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% IntD1
thf(fact_186_IntD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_187_IntD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% IntD2
thf(fact_188_Int__assoc,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_189_Int__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ A )
= A ) ).
% Int_absorb
thf(fact_190_Int__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_191_Int__left__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Int_left_absorb
thf(fact_192_Int__left__commute,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ B @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_193_UnE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_194_UnE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% UnE
thf(fact_195_UnI1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_196_UnI1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI1
thf(fact_197_UnI2,axiom,
! [C: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( sup_su2582925890723967158c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_198_UnI2,axiom,
! [C: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% UnI2
thf(fact_199_bex__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
& ( P5 @ X2 ) ) )
= ( ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
& ( P5 @ X2 ) )
| ? [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ( P5 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_200_ball__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ( P5 @ X2 ) ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ( P5 @ X2 ) )
& ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
=> ( P5 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_201_Un__assoc,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_202_Un__absorb,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ A )
= A ) ).
% Un_absorb
thf(fact_203_Un__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_204_Un__left__absorb,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_left_absorb
thf(fact_205_Un__left__commute,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_206_K__thm,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Con_i @ P2 @ Q ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ).
% K_thm
thf(fact_207_Collect__subset,axiom,
! [A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_208_Collect__subset,axiom,
! [A: set_nat,P5: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P5 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_209_Collect__subset,axiom,
! [A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_210_Int__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A4 )
& ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_211_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_212_Int__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A4 )
& ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_213_Int__Collect,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P5 ) ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( P5 @ X4 ) ) ) ).
% Int_Collect
thf(fact_214_Int__Collect,axiom,
! [X4: nat,A: set_nat,P5: nat > $o] :
( ( member_nat @ X4 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P5 ) ) )
= ( ( member_nat @ X4 @ A )
& ( P5 @ X4 ) ) ) ).
% Int_Collect
thf(fact_215_Int__Collect,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P5 ) ) )
= ( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( P5 @ X4 ) ) ) ).
% Int_Collect
thf(fact_216_Collect__conj__eq,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) )
= ( inf_in3450601097109690352c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ).
% Collect_conj_eq
thf(fact_217_Collect__conj__eq,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P5 ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_conj_eq
thf(fact_218_Collect__conj__eq,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) )
= ( inf_in161960956874937808c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ).
% Collect_conj_eq
thf(fact_219_Un__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A4 )
| ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_220_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_221_Un__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A4 )
| ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_222_Collect__disj__eq,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) )
= ( sup_su2582925890723967158c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ).
% Collect_disj_eq
thf(fact_223_Collect__disj__eq,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P5 ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_disj_eq
thf(fact_224_Collect__disj__eq,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) )
= ( sup_su1936195050962291414c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ).
% Collect_disj_eq
thf(fact_225_Int__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,D: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ C2 )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ D )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_226_Int__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_227_Int__lower1,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_228_Int__lower1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_229_Int__lower2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_230_Int__lower2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_231_Int__absorb1,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_232_Int__absorb1,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_233_Int__absorb2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_234_Int__absorb2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_235_Int__greatest,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ C2 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ C2 @ B )
=> ( ord_le5389487502678872194c_fm_i @ C2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_236_Int__greatest,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ C2 @ B )
=> ( ord_le3843937902494030498c_fm_i @ C2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_237_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P5: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P5 @ X3 )
=> ( Q3 @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P5 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_238_Int__Collect__mono,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( P5 @ X3 )
=> ( Q3 @ X3 ) ) )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P5 ) ) @ ( inf_in161960956874937808c_fm_i @ B @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_239_Int__Collect__mono,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( P5 @ X3 )
=> ( Q3 @ X3 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P5 ) ) @ ( inf_in3450601097109690352c_fm_i @ B @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_240_Un__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i,D: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_241_Un__least,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_242_Un__upper1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper1
thf(fact_243_Un__upper2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_upper2
thf(fact_244_Un__absorb1,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_245_Un__absorb2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_246_subset__UnE,axiom,
! [C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) )
=> ~ ! [A5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A5 @ A )
=> ! [B4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B4 @ B )
=> ( C2
!= ( sup_su1936195050962291414c_fm_i @ A5 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_247_subset__Un__eq,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_248_Un__Int__crazy,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) )
= ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_249_Un__Int__crazy,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_250_Int__Un__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_251_Int__Un__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_252_Un__Int__distrib,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) @ ( sup_su2582925890723967158c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_253_Un__Int__distrib,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ ( sup_su1936195050962291414c_fm_i @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_254_Int__Un__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) @ A )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_255_Int__Un__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) @ A )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ A ) @ ( inf_in3450601097109690352c_fm_i @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_256_Un__Int__distrib2,axiom,
! [B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) @ A )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ B @ A ) @ ( sup_su2582925890723967158c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_257_Un__Int__distrib2,axiom,
! [B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) @ A )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B @ A ) @ ( sup_su1936195050962291414c_fm_i @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_258_Un__Int__assoc__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) ) )
= ( ord_le5389487502678872194c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_259_Un__Int__assoc__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) )
= ( ord_le3843937902494030498c_fm_i @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_260__092_060open_062A_A_092_060turnstile_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_J_A_092_060_094bold_062_092_060longrightarrow_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s ) )
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> conjunct (filter (\<lambda>p. p \<in> known U i) S') \<^bold>\<longrightarrow> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
thf(fact_261__092_060open_062A_A_092_060turnstile_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_Imp_i
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S')\<close>
thf(fact_262__092_060open_062A_A_092_060turnstile_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AU_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_Afilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_092_060close_062,axiom,
( epistemic_AK_i @ a
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ u ) ) )
@ s )
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i ) ) ) ).
% \<open>A \<turnstile> filter (\<lambda>p. p \<in> known U i) S' \<^bold>\<leadsto> filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
thf(fact_263_set__filter,axiom,
! [P5: set_Epistemic_fm_i > $o,Xs: list_s8081015415394010888c_fm_i] :
( ( set_se200842218512397079c_fm_i @ ( filter3188398074982218495c_fm_i @ P5 @ Xs ) )
= ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
& ( P5 @ X2 ) ) ) ) ).
% set_filter
thf(fact_264_set__filter,axiom,
! [P5: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P5 @ Xs ) )
= ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( P5 @ X2 ) ) ) ) ).
% set_filter
thf(fact_265_set__filter,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) )
= ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( P5 @ X2 ) ) ) ) ).
% set_filter
thf(fact_266_filter__True,axiom,
! [Xs: list_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P5 @ X3 ) )
=> ( ( filter7636273843821131039c_fm_i @ P5 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_267_map__eq__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ Xs ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_268_inf__sup__absorb,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( sup_su2582925890723967158c_fm_i @ X4 @ Y2 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_269_inf__sup__absorb,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_270_sup__inf__absorb,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_271_sup__inf__absorb,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_272_t,axiom,
! [X5: epistemic_fm_i,Y3: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_Con_i @ X5 @ Y3 ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_Con_i @ Y3 @ X5 ) @ epistemic_FF_i ) ) ) ).
% t
thf(fact_273_le__sup__iff,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ Z3 )
= ( ( ord_le3843937902494030498c_fm_i @ X4 @ Z3 )
& ( ord_le3843937902494030498c_fm_i @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_274_le__sup__iff,axiom,
! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y2 ) @ Z3 )
= ( ( ord_less_eq_nat @ X4 @ Z3 )
& ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_275_inf__right__idem,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ Y2 )
= ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) ) ).
% inf_right_idem
thf(fact_276_inf_Oright__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ B2 )
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_277_inf__left__idem,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) )
= ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) ) ).
% inf_left_idem
thf(fact_278_inf_Oleft__idem,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_279_inf__idem,axiom,
! [X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_280_inf_Oidem,axiom,
! [A2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_281_sup_Oright__idem,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ B2 )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_282_sup__left__idem,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) )
= ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_283_sup_Oleft__idem,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_284_sup__idem,axiom,
! [X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_285_sup_Oidem,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_286_map__ident,axiom,
( ( map_Ep2755178516647988292c_fm_i
@ ^ [X2: epistemic_fm_i] : X2 )
= ( ^ [Xs2: list_Epistemic_fm_i] : Xs2 ) ) ).
% map_ident
thf(fact_287_filter__filter,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter7636273843821131039c_fm_i @ P5 @ ( filter7636273843821131039c_fm_i @ Q3 @ Xs ) )
= ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( Q3 @ X2 )
& ( P5 @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_288_inf_Obounded__iff,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) )
= ( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
& ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_289_inf_Obounded__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
= ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
& ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_290_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_291_le__inf__iff,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( ( ord_le5389487502678872194c_fm_i @ X4 @ Y2 )
& ( ord_le5389487502678872194c_fm_i @ X4 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_292_le__inf__iff,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) )
= ( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
& ( ord_le3843937902494030498c_fm_i @ X4 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_293_le__inf__iff,axiom,
! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z3 ) )
= ( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_294_sup_Obounded__iff,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 )
= ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
& ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_295_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_296_List_Ofinite__set,axiom,
! [Xs: list_Epistemic_fm_i] : ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% List.finite_set
thf(fact_297_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_298__092_060open_062tautology_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_A_092_060_094bold_062_092_060longrightarrow_062_A_092_060_094bold_062_092_060not_062_Aconjunct_A_Ifilter_A_I_092_060lambda_062p_O_Ap_A_092_060in_062_Aknown_AW_Ai_J_AS_H_J_J_092_060close_062,axiom,
! [G3: list_char > $o,H: epistemic_fm_i > $o] :
( epistemic_eval_i @ G3 @ H
@ ( epistemic_Imp_i
@ ( epistemic_imply_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s )
@ epistemic_FF_i )
@ ( epistemic_Imp_i
@ ( stalnaker_conjunct_i
@ ( filter7636273843821131039c_fm_i
@ ^ [P4: epistemic_fm_i] :
( member6642669606046002379c_fm_i @ P4
@ ( collec4904205187116291597c_fm_i
@ ^ [Q2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ Q2 ) @ w ) ) )
@ s ) )
@ epistemic_FF_i ) ) ) ).
% \<open>tautology (filter (\<lambda>p. p \<in> known W i) S' \<^bold>\<leadsto> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<not> conjunct (filter (\<lambda>p. p \<in> known W i) S'))\<close>
thf(fact_299__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_H_O_A_092_060lbrakk_062A_A_092_060turnstile_062_AS_H_A_092_060_094bold_062_092_060leadsto_062_A_092_060_094bold_062_092_060bottom_062_059_Aset_AS_H_A_092_060subseteq_062_Aknown_AW_Ai_A_092_060union_062_Aknown_AU_Ai_059_Afinite_A_Iset_AS_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S3: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ a @ ( epistemic_imply_i @ S3 @ epistemic_FF_i ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S3 )
@ ( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ w ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ i2 @ P4 ) @ u ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>S'. \<lbrakk>A \<turnstile> S' \<^bold>\<leadsto> \<^bold>\<bottom>; set S' \<subseteq> known W i \<union> known U i; finite (set S')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_300_finite__list,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ? [Xs3: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_301_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_302_imply__conjunct,axiom,
! [G: list_Epistemic_fm_i,P2: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ G @ P2 ) @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P2 ) ) ) ).
% imply_conjunct
thf(fact_303_conjunct__imply,axiom,
! [G: list_Epistemic_fm_i,P2: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P2 ) @ ( epistemic_imply_i @ G @ P2 ) ) ) ).
% conjunct_imply
thf(fact_304_inf__set__def,axiom,
( inf_in3450601097109690352c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( inf_in8976956063904982253fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A4 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_305_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_306_inf__set__def,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( inf_in853329969422372621fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A4 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_307_sup__set__def,axiom,
( sup_su2582925890723967158c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( sup_su6723579903606884263fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A4 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_308_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_309_sup__set__def,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( sup_su2844414564337698183fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A4 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_310_less__eq__set__def,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( ord_le5706303257367236315fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A4 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_311_less__eq__set__def,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( ord_le190830114487426235fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A4 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_312_eval_Osimps_I6_J,axiom,
! [Ux: list_char > $o,H2: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] :
( ( epistemic_eval_i @ Ux @ H2 @ ( epistemic_K_i @ I3 @ P2 ) )
= ( H2 @ ( epistemic_K_i @ I3 @ P2 ) ) ) ).
% eval.simps(6)
thf(fact_313_eval_Osimps_I5_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G2 @ H2 @ ( epistemic_Imp_i @ P2 @ Q ) )
= ( ( epistemic_eval_i @ G2 @ H2 @ P2 )
=> ( epistemic_eval_i @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(5)
thf(fact_314_eval_Osimps_I1_J,axiom,
! [Uu: list_char > $o,Uv: epistemic_fm_i > $o] :
~ ( epistemic_eval_i @ Uu @ Uv @ epistemic_FF_i ) ).
% eval.simps(1)
thf(fact_315_A1,axiom,
! [P2: epistemic_fm_i,A: epistemic_fm_i > $o] :
( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ P2 )
=> ( epistemic_AK_i @ A @ P2 ) ) ).
% A1
thf(fact_316_eval_Osimps_I4_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G2 @ H2 @ ( epistemic_Con_i @ P2 @ Q ) )
= ( ( epistemic_eval_i @ G2 @ H2 @ P2 )
& ( epistemic_eval_i @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(4)
thf(fact_317_tautology__imply__superset,axiom,
! [Ps: list_Epistemic_fm_i,Qs: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Qs ) )
=> ! [G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_imply_i @ Ps @ R2 ) @ ( epistemic_imply_i @ Qs @ R2 ) ) ) ) ).
% tautology_imply_superset
thf(fact_318_K__right__mp,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ P2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ ( epistemic_Imp_i @ P2 @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) ) ) ) ).
% K_right_mp
thf(fact_319_inconsistent__finite,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ S )
=> ? [S3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
& ( finite3304564979551393739c_fm_i @ S3 )
& ~ ( episte2285483198712856234tent_i @ A @ S3 ) ) ) ).
% inconsistent_finite
thf(fact_320_K__imply__conjunct,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P2 ) ) ) ).
% K_imply_conjunct
thf(fact_321_K__conjunct__imply,axiom,
! [A: epistemic_fm_i > $o,G: list_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( stalnaker_conjunct_i @ G ) @ P2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P2 ) ) ) ).
% K_conjunct_imply
thf(fact_322_inf__left__commute,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ Y2 @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) ) ).
% inf_left_commute
thf(fact_323_inf_Oleft__commute,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ B2 @ ( inf_in161960956874937808c_fm_i @ A2 @ C ) )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_324_inf__commute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X2: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_325_inf_Ocommute,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_326_inf__assoc,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ Z3 )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) ) ) ).
% inf_assoc
thf(fact_327_inf_Oassoc,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C )
= ( inf_in161960956874937808c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_328_inf__sup__aci_I1_J,axiom,
( inf_in161960956874937808c_fm_i
= ( ^ [X2: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] : ( inf_in161960956874937808c_fm_i @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_329_inf__sup__aci_I2_J,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ Z3 )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) ) ) ).
% inf_sup_aci(2)
thf(fact_330_inf__sup__aci_I3_J,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ Y2 @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) ) ).
% inf_sup_aci(3)
thf(fact_331_inf__sup__aci_I4_J,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) )
= ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_332_sup__left__commute,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ Y2 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_333_sup_Oleft__commute,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ C ) )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_334_sup__commute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_335_sup_Ocommute,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ B5 @ A3 ) ) ) ).
% sup.commute
thf(fact_336_sup__assoc,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ Z3 )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) ) ) ).
% sup_assoc
thf(fact_337_sup_Oassoc,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ C )
= ( sup_su1936195050962291414c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_338_inf__sup__aci_I5_J,axiom,
( sup_su1936195050962291414c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_339_inf__sup__aci_I6_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ Z3 )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_340_inf__sup__aci_I7_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ Y2 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_341_inf__sup__aci_I8_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) )
= ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_342_K__imply__weaken,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,Ps2: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ps ) @ ( set_Epistemic_fm_i2 @ Ps2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps2 @ Q ) ) ) ) ).
% K_imply_weaken
thf(fact_343_K__distrib__K__imp,axiom,
! [A: epistemic_fm_i > $o,I3: i,G: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_K_i @ I3 @ ( epistemic_imply_i @ G @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( map_Ep2755178516647988292c_fm_i @ ( epistemic_K_i @ I3 ) @ G ) @ ( epistemic_K_i @ I3 @ Q ) ) ) ) ).
% K_distrib_K_imp
thf(fact_344_duality__taut,axiom,
! [I3: i,P2: epistemic_fm_i,Q: epistemic_fm_i,G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) ) @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ Q @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ epistemic_FF_i ) ) ) ) ).
% duality_taut
thf(fact_345_list_Omap__ident,axiom,
! [T3: list_Epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i
@ ^ [X2: epistemic_fm_i] : X2
@ T3 )
= T3 ) ).
% list.map_ident
thf(fact_346_AK_Osimps,axiom,
( epistemic_AK_i
= ( ^ [A4: epistemic_fm_i > $o,A3: epistemic_fm_i] :
( ? [P4: epistemic_fm_i] :
( ( A3 = P4 )
& ! [G5: list_char > $o,H4: epistemic_fm_i > $o] : ( epistemic_eval_i @ G5 @ H4 @ P4 ) )
| ? [I2: i,P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I2 @ P4 ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P4 @ Q2 ) ) ) @ ( epistemic_K_i @ I2 @ Q2 ) ) )
| ? [P4: epistemic_fm_i] :
( ( A3 = P4 )
& ( A4 @ P4 ) )
| ? [P4: epistemic_fm_i,Q2: epistemic_fm_i] :
( ( A3 = Q2 )
& ( epistemic_AK_i @ A4 @ P4 )
& ( epistemic_AK_i @ A4 @ ( epistemic_Imp_i @ P4 @ Q2 ) ) )
| ? [P4: epistemic_fm_i,I2: i] :
( ( A3
= ( epistemic_K_i @ I2 @ P4 ) )
& ( epistemic_AK_i @ A4 @ P4 ) ) ) ) ) ).
% AK.simps
thf(fact_347_AK_Ocases,axiom,
! [A: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ A2 )
=> ( ~ ! [G3: list_char > $o,H: epistemic_fm_i > $o] : ( epistemic_eval_i @ G3 @ H @ A2 )
=> ( ! [I: i,P3: epistemic_fm_i,Q4: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Con_i @ ( epistemic_K_i @ I @ P3 ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ Q4 ) ) ) @ ( epistemic_K_i @ I @ Q4 ) ) )
=> ( ~ ( A @ A2 )
=> ( ! [P3: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ P3 )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P3 @ A2 ) ) )
=> ~ ! [P3: epistemic_fm_i] :
( ? [I: i] :
( A2
= ( epistemic_K_i @ I @ P3 ) )
=> ~ ( epistemic_AK_i @ A @ P3 ) ) ) ) ) ) ) ).
% AK.cases
thf(fact_348_consistent__def,axiom,
( episte2285483198712856234tent_i
= ( ^ [A4: epistemic_fm_i > $o,S4: set_Epistemic_fm_i] :
~ ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ S4 )
& ( epistemic_AK_i @ A4 @ ( epistemic_imply_i @ Qs2 @ epistemic_FF_i ) ) ) ) ) ).
% consistent_def
thf(fact_349_mcs__conjunction__mult,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,S: list_Epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ S ) @ V )
& ( finite3304564979551393739c_fm_i @ ( set_Epistemic_fm_i2 @ S ) ) )
=> ( member6642669606046002379c_fm_i @ ( stalnaker_conjunct_i @ S ) @ V ) ) ) ) ).
% mcs_conjunction_mult
thf(fact_350_inf_OcoboundedI2,axiom,
! [B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_351_inf_OcoboundedI2,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_352_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_353_inf_OcoboundedI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_354_inf_OcoboundedI1,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_355_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_356_inf_Oabsorb__iff2,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [B5: set_se3485332733965609186c_fm_i,A3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_357_inf_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_358_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_359_inf_Oabsorb__iff1,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_360_inf_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_361_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( inf_inf_nat @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_362_inf_Ocobounded2,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_363_inf_Ocobounded2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_364_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_365_inf_Ocobounded1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_366_inf_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_367_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_368_inf_Oorder__iff,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [A3: set_se3485332733965609186c_fm_i,B5: set_se3485332733965609186c_fm_i] :
( A3
= ( inf_in161960956874937808c_fm_i @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_369_inf_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( A3
= ( inf_in3450601097109690352c_fm_i @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_370_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( A3
= ( inf_inf_nat @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_371_inf__greatest,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ Y2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ Z3 )
=> ( ord_le5389487502678872194c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_372_inf__greatest,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_373_inf__greatest,axiom,
! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_374_inf_OboundedI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_375_inf_OboundedI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_376_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_377_inf_OboundedE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B2 @ C ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_378_inf_OboundedE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B2 @ C ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_379_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_380_inf__absorb2,axiom,
! [Y2: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ Y2 @ X4 )
=> ( ( inf_in161960956874937808c_fm_i @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_381_inf__absorb2,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X4 )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_382_inf__absorb2,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_383_inf__absorb1,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ Y2 )
=> ( ( inf_in161960956874937808c_fm_i @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_384_inf__absorb1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_385_inf__absorb1,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= X4 ) ) ).
% inf_absorb1
thf(fact_386_inf_Oabsorb2,axiom,
! [B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ A2 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_387_inf_Oabsorb2,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_388_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_389_inf_Oabsorb1,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_390_inf_Oabsorb1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_391_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_392_le__iff__inf,axiom,
( ord_le5389487502678872194c_fm_i
= ( ^ [X2: set_se3485332733965609186c_fm_i,Y4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_393_le__iff__inf,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_394_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_395_inf__unique,axiom,
! [F: set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i > set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i,Z4: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X3 @ Y5 )
=> ( ( ord_le5389487502678872194c_fm_i @ X3 @ Z4 )
=> ( ord_le5389487502678872194c_fm_i @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_396_inf__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Z4 )
=> ( ord_le3843937902494030498c_fm_i @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_397_inf__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y2: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_398_inf_OorderI,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
=> ( ord_le5389487502678872194c_fm_i @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_399_inf_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_400_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_401_inf_OorderE,axiom,
! [A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ B2 )
=> ( A2
= ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_402_inf_OorderE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( A2
= ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_403_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_404_le__infI2,axiom,
! [B2: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ B2 @ X4 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_405_le__infI2,axiom,
! [B2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_406_le__infI2,axiom,
! [B2: nat,X4: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI2
thf(fact_407_le__infI1,axiom,
! [A2: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ X4 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_408_le__infI1,axiom,
! [A2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_409_le__infI1,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X4 ) ) ).
% le_infI1
thf(fact_410_inf__mono,axiom,
! [A2: set_se3485332733965609186c_fm_i,C: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,D2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A2 @ C )
=> ( ( ord_le5389487502678872194c_fm_i @ B2 @ D2 )
=> ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) @ ( inf_in161960956874937808c_fm_i @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_411_inf__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) @ ( inf_in3450601097109690352c_fm_i @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_412_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_413_le__infI,axiom,
! [X4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ A2 )
=> ( ( ord_le5389487502678872194c_fm_i @ X4 @ B2 )
=> ( ord_le5389487502678872194c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_414_le__infI,axiom,
! [X4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_415_le__infI,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_416_le__infE,axiom,
! [X4: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) )
=> ~ ( ( ord_le5389487502678872194c_fm_i @ X4 @ A2 )
=> ~ ( ord_le5389487502678872194c_fm_i @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_417_le__infE,axiom,
! [X4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ A2 @ B2 ) )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ X4 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_418_le__infE,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X4 @ A2 )
=> ~ ( ord_less_eq_nat @ X4 @ B2 ) ) ) ).
% le_infE
thf(fact_419_inf__le2,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_420_inf__le2,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_421_inf__le2,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_422_inf__le1,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_423_inf__le1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_424_inf__le1,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ X4 ) ).
% inf_le1
thf(fact_425_inf__sup__ord_I1_J,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_426_inf__sup__ord_I1_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_427_inf__sup__ord_I1_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_428_inf__sup__ord_I2_J,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_429_inf__sup__ord_I2_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_430_inf__sup__ord_I2_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_431_sup_OcoboundedI2,axiom,
! [C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_432_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_433_sup_OcoboundedI1,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_434_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_435_sup_Oabsorb__iff2,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_436_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B5: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_437_sup_Oabsorb__iff1,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_438_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_439_sup_Ocobounded2,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_440_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_441_sup_Ocobounded1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_442_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_443_sup_Oorder__iff,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [B5: set_Epistemic_fm_i,A3: set_Epistemic_fm_i] :
( A3
= ( sup_su1936195050962291414c_fm_i @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_444_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_445_sup_OboundedI,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_446_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_447_sup_OboundedE,axiom,
! [B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ~ ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_448_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_449_sup__absorb2,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_450_sup__absorb2,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_451_sup__absorb1,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X4 )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 )
= X4 ) ) ).
% sup_absorb1
thf(fact_452_sup__absorb1,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= X4 ) ) ).
% sup_absorb1
thf(fact_453_sup_Oabsorb2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_454_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_455_sup_Oabsorb1,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_456_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_457_sup__unique,axiom,
! [F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y5 @ X3 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z4 @ X3 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_458_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y2: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y2 )
= ( F @ X4 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_459_sup_OorderI,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
=> ( ord_le3843937902494030498c_fm_i @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_460_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_461_sup_OorderE,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( A2
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_462_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_463_le__iff__sup,axiom,
( ord_le3843937902494030498c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_464_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( sup_sup_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_465_sup__least,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z3 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) @ X4 ) ) ) ).
% sup_least
thf(fact_466_sup__least,axiom,
! [Y2: nat,X4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z3 ) @ X4 ) ) ) ).
% sup_least
thf(fact_467_sup__mono,axiom,
! [A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ C )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ D2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_468_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_469_sup_Omono,axiom,
! [C: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,D2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D2 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ C @ D2 ) @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_470_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_471_le__supI2,axiom,
! [X4: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_472_le__supI2,axiom,
! [X4: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_473_le__supI1,axiom,
! [X4: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ A2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_474_le__supI1,axiom,
! [X4: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_475_sup__ge2,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y2 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% sup_ge2
thf(fact_476_sup__ge2,axiom,
! [Y2: nat,X4: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% sup_ge2
thf(fact_477_sup__ge1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% sup_ge1
thf(fact_478_sup__ge1,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% sup_ge1
thf(fact_479_le__supI,axiom,
! [A2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ X4 )
=> ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_480_le__supI,axiom,
! [A2: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X4 )
=> ( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_481_le__supE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_le3843937902494030498c_fm_i @ A2 @ X4 )
=> ~ ( ord_le3843937902494030498c_fm_i @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_482_le__supE,axiom,
! [A2: nat,B2: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X4 )
=> ~ ( ord_less_eq_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_483_inf__sup__ord_I3_J,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_484_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_485_inf__sup__ord_I4_J,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ Y2 @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_486_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X4: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X4 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_487_subset__code_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ B )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( member1104366573291651755c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_488_subset__code_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ B )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( member6642669606046002379c_fm_i @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_489_sup__inf__distrib2,axiom,
! [Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y2 @ X4 ) @ ( sup_su2582925890723967158c_fm_i @ Z3 @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_490_sup__inf__distrib2,axiom,
! [Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y2 @ X4 ) @ ( sup_su1936195050962291414c_fm_i @ Z3 @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_491_sup__inf__distrib1,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X4 @ Y2 ) @ ( sup_su2582925890723967158c_fm_i @ X4 @ Z3 ) ) ) ).
% sup_inf_distrib1
thf(fact_492_sup__inf__distrib1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ).
% sup_inf_distrib1
thf(fact_493_inf__sup__distrib2,axiom,
! [Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y2 @ X4 ) @ ( inf_in161960956874937808c_fm_i @ Z3 @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_494_inf__sup__distrib2,axiom,
! [Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y2 @ X4 ) @ ( inf_in3450601097109690352c_fm_i @ Z3 @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_495_inf__sup__distrib1,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) ) ).
% inf_sup_distrib1
thf(fact_496_inf__sup__distrib1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z3 ) ) ) ).
% inf_sup_distrib1
thf(fact_497_distrib__imp2,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i,Z4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X3 @ ( inf_in161960956874937808c_fm_i @ Y5 @ Z4 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X3 @ Y5 ) @ ( sup_su2582925890723967158c_fm_i @ X3 @ Z4 ) ) )
=> ( ( inf_in161960956874937808c_fm_i @ X4 @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) ) ) ).
% distrib_imp2
thf(fact_498_distrib__imp2,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X3 @ ( inf_in3450601097109690352c_fm_i @ Y5 @ Z4 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y5 ) @ ( sup_su1936195050962291414c_fm_i @ X3 @ Z4 ) ) )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z3 ) ) ) ) ).
% distrib_imp2
thf(fact_499_distrib__imp1,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i,Z4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X3 @ ( sup_su2582925890723967158c_fm_i @ Y5 @ Z4 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y5 ) @ ( inf_in161960956874937808c_fm_i @ X3 @ Z4 ) ) )
=> ( ( sup_su2582925890723967158c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X4 @ Y2 ) @ ( sup_su2582925890723967158c_fm_i @ X4 @ Z3 ) ) ) ) ).
% distrib_imp1
thf(fact_500_distrib__imp1,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i,Z4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X3 @ ( sup_su1936195050962291414c_fm_i @ Y5 @ Z4 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y5 ) @ ( inf_in3450601097109690352c_fm_i @ X3 @ Z4 ) ) )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ) ).
% distrib_imp1
thf(fact_501_ex__map__conv,axiom,
! [Ys: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( ? [Xs2: list_Epistemic_fm_i] :
( Ys
= ( map_Ep2755178516647988292c_fm_i @ F @ Xs2 ) ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ? [Y4: epistemic_fm_i] :
( X2
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_502_map__cong,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_503_map__idI,axiom,
! [Xs: list_s8081015415394010888c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_se1449444348732536900c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_504_map__idI,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_505_map__ext,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_506_list_Omap__ident__strong,axiom,
! [T3: list_s8081015415394010888c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ! [Z4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( set_se200842218512397079c_fm_i @ T3 ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_se1449444348732536900c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_507_list_Omap__ident__strong,axiom,
! [T3: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( set_Epistemic_fm_i2 @ T3 ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_508_list_Oinj__map__strong,axiom,
! [X4: list_Epistemic_fm_i,Xa: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,Fa: epistemic_fm_i > epistemic_fm_i] :
( ! [Z4: epistemic_fm_i,Za: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( set_Epistemic_fm_i2 @ X4 ) )
=> ( ( member6642669606046002379c_fm_i @ Za @ ( set_Epistemic_fm_i2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_Ep2755178516647988292c_fm_i @ F @ X4 )
= ( map_Ep2755178516647988292c_fm_i @ Fa @ Xa ) )
=> ( X4 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_509_list_Omap__cong0,axiom,
! [X4: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i] :
( ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( set_Epistemic_fm_i2 @ X4 ) )
=> ( ( F @ Z4 )
= ( G2 @ Z4 ) ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ X4 )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ X4 ) ) ) ).
% list.map_cong0
thf(fact_510_list_Omap__cong,axiom,
! [X4: list_Epistemic_fm_i,Ya: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i] :
( ( X4 = Ya )
=> ( ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( set_Epistemic_fm_i2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G2 @ Z4 ) ) )
=> ( ( map_Ep2755178516647988292c_fm_i @ F @ X4 )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_511_filter__id__conv,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( ( filter7636273843821131039c_fm_i @ P5 @ Xs )
= Xs )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( P5 @ X2 ) ) ) ) ).
% filter_id_conv
thf(fact_512_filter__cong,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ( ( P5 @ X3 )
= ( Q3 @ X3 ) ) )
=> ( ( filter3188398074982218495c_fm_i @ P5 @ Xs )
= ( filter3188398074982218495c_fm_i @ Q3 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_513_filter__cong,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( Xs = Ys )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ( ( P5 @ X3 )
= ( Q3 @ X3 ) ) )
=> ( ( filter7636273843821131039c_fm_i @ P5 @ Xs )
= ( filter7636273843821131039c_fm_i @ Q3 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_514_distrib__sup__le,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) ) @ ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X4 @ Y2 ) @ ( sup_su2582925890723967158c_fm_i @ X4 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_515_distrib__sup__le,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) ) @ ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_516_distrib__sup__le,axiom,
! [X4: nat,Y2: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ ( inf_inf_nat @ Y2 @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X4 @ Y2 ) @ ( sup_sup_nat @ X4 @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_517_distrib__inf__le,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] : ( ord_le5389487502678872194c_fm_i @ ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) @ ( inf_in161960956874937808c_fm_i @ X4 @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_518_distrib__inf__le,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z3 ) ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_519_distrib__inf__le,axiom,
! [X4: nat,Y2: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X4 @ Y2 ) @ ( inf_inf_nat @ X4 @ Z3 ) ) @ ( inf_inf_nat @ X4 @ ( sup_sup_nat @ Y2 @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_520_filter__is__subset,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% filter_is_subset
thf(fact_521_inter__set__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_522_inter__set__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( set_se200842218512397079c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_523_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_524_finite__Collect__subsets,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [B3: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B3 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_525_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_526_finite__Un,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) )
= ( ( finite3304564979551393739c_fm_i @ F2 )
& ( finite3304564979551393739c_fm_i @ G ) ) ) ).
% finite_Un
thf(fact_527_finite__Int,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( ( finite3304564979551393739c_fm_i @ F2 )
| ( finite3304564979551393739c_fm_i @ G ) )
=> ( finite3304564979551393739c_fm_i @ ( inf_in3450601097109690352c_fm_i @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_528_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_529_finite__Int,axiom,
! [F2: set_se3485332733965609186c_fm_i,G: set_se3485332733965609186c_fm_i] :
( ( ( finite7933139204641697195c_fm_i @ F2 )
| ( finite7933139204641697195c_fm_i @ G ) )
=> ( finite7933139204641697195c_fm_i @ ( inf_in161960956874937808c_fm_i @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_530_set__union,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( union_Epistemic_fm_i @ Xs @ Ys ) )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ).
% set_union
thf(fact_531_finite__Collect__conjI,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
| ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q3 ) ) )
=> ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_532_finite__Collect__conjI,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
| ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q3 ) ) )
=> ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_533_finite__Collect__conjI,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P5 ) )
| ( finite_finite_nat @ ( collect_nat @ Q3 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_534_finite__Collect__disjI,axiom,
! [P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) ) )
= ( ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
& ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_535_finite__Collect__disjI,axiom,
! [P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) ) )
= ( ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
& ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_536_finite__Collect__disjI,axiom,
! [P5: nat > $o,Q3: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P5 @ X2 )
| ( Q3 @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P5 ) )
& ( finite_finite_nat @ ( collect_nat @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_537_dual__order_Orefl,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_538_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_539_order__refl,axiom,
! [X4: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ X4 @ X4 ) ).
% order_refl
thf(fact_540_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_541_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_542_le__cases3,axiom,
! [X4: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_543_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y = Z2 ) )
= ( ^ [X2: set_Epistemic_fm_i,Y4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X2 @ Y4 )
& ( ord_le3843937902494030498c_fm_i @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_544_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_545_ord__eq__le__trans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2 = B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_546_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_547_ord__le__eq__trans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_548_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_549_order__antisym,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_550_order__antisym,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_551_order_Otrans,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ C ) ) ) ).
% order.trans
thf(fact_552_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_553_order__trans,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
=> ( ( ord_le3843937902494030498c_fm_i @ Y2 @ Z3 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ Z3 ) ) ) ).
% order_trans
thf(fact_554_order__trans,axiom,
! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% order_trans
thf(fact_555_linorder__wlog,axiom,
! [P5: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P5 @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P5 @ B6 @ A6 )
=> ( P5 @ A6 @ B6 ) )
=> ( P5 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_556_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y = Z2 ) )
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B5 @ A3 )
& ( ord_le3843937902494030498c_fm_i @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_557_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_558_dual__order_Oantisym,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_559_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_560_dual__order_Otrans,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( ( ord_le3843937902494030498c_fm_i @ C @ B2 )
=> ( ord_le3843937902494030498c_fm_i @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_561_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_562_antisym,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_563_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_564_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y = Z2 ) )
= ( ^ [A3: set_Epistemic_fm_i,B5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A3 @ B5 )
& ( ord_le3843937902494030498c_fm_i @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_565_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z2: nat] : ( Y = Z2 ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_566_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_567_order__subst1,axiom,
! [A2: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B2: nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_568_order__subst1,axiom,
! [A2: nat,F: set_Epistemic_fm_i > nat,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_569_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_570_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_571_order__subst2,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_572_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le3843937902494030498c_fm_i @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_573_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_574_order__eq__refl,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( X4 = Y2 )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_575_order__eq__refl,axiom,
! [X4: nat,Y2: nat] :
( ( X4 = Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_576_linorder__linear,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_577_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_578_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Epistemic_fm_i > nat,B2: set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3843937902494030498c_fm_i @ B2 @ C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_579_ord__eq__le__subst,axiom,
! [A2: set_Epistemic_fm_i,F: nat > set_Epistemic_fm_i,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_580_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_581_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_582_ord__le__eq__subst,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat,C: nat] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_583_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Epistemic_fm_i,C: set_Epistemic_fm_i] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_le3843937902494030498c_fm_i @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_584_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_585_linorder__le__cases,axiom,
! [X4: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_586_order__antisym__conv,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ Y2 @ X4 )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_587_order__antisym__conv,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_588_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,R3: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ? [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_589_pigeonhole__infinite__rel,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_nat,R3: set_Epistemic_fm_i > nat > $o] :
( ~ ( finite7933139204641697195c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite7933139204641697195c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [A3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_590_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,R3: epistemic_fm_i > epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ? [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_591_pigeonhole__infinite__rel,axiom,
! [A: set_Epistemic_fm_i,B: set_nat,R3: epistemic_fm_i > nat > $o] :
( ~ ( finite3304564979551393739c_fm_i @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite3304564979551393739c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [A3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_592_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_Epistemic_fm_i,R3: nat > epistemic_fm_i > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_593_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_594_not__finite__existsD,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ~ ( finite7933139204641697195c_fm_i @ ( collec3087743281813070829c_fm_i @ P5 ) )
=> ? [X_1: set_Epistemic_fm_i] : ( P5 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_595_not__finite__existsD,axiom,
! [P5: epistemic_fm_i > $o] :
( ~ ( finite3304564979551393739c_fm_i @ ( collec4904205187116291597c_fm_i @ P5 ) )
=> ? [X_1: epistemic_fm_i] : ( P5 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_596_not__finite__existsD,axiom,
! [P5: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P5 ) )
=> ? [X_1: nat] : ( P5 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_597_finite__has__maximal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( ord_le3843937902494030498c_fm_i @ A2 @ X3 )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_598_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_599_finite__has__minimal2,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ( ord_le3843937902494030498c_fm_i @ X3 @ A2 )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_600_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_601_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_602_finite__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( finite3304564979551393739c_fm_i @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_subset
thf(fact_603_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_604_infinite__super,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S @ T )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ T ) ) ) ).
% infinite_super
thf(fact_605_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_606_rev__finite__subset,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% rev_finite_subset
thf(fact_607_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_608_finite__UnI,axiom,
! [F2: set_Epistemic_fm_i,G: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( finite3304564979551393739c_fm_i @ G )
=> ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_609_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_610_Un__infinite,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T ) ) ) ).
% Un_infinite
thf(fact_611_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_612_infinite__Un,axiom,
! [S: set_Epistemic_fm_i,T: set_Epistemic_fm_i] :
( ( ~ ( finite3304564979551393739c_fm_i @ ( sup_su1936195050962291414c_fm_i @ S @ T ) ) )
= ( ~ ( finite3304564979551393739c_fm_i @ S )
| ~ ( finite3304564979551393739c_fm_i @ T ) ) ) ).
% infinite_Un
thf(fact_613_sup__Un__eq,axiom,
! [R3: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( sup_su6723579903606884263fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ R3 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ S ) )
= ( ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ ( sup_su2582925890723967158c_fm_i @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_614_sup__Un__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( sup_su2844414564337698183fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ R3 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ S ) )
= ( ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ ( sup_su1936195050962291414c_fm_i @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_615_inf__Int__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( inf_in8976956063904982253fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ R3 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ S ) )
= ( ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ ( inf_in3450601097109690352c_fm_i @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_616_inf__Int__eq,axiom,
! [R3: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( inf_in853329969422372621fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ R3 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ S ) )
= ( ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ ( inf_in161960956874937808c_fm_i @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_617_pred__subset__eq,axiom,
! [R3: set_se3485332733965609186c_fm_i,S: set_se3485332733965609186c_fm_i] :
( ( ord_le5706303257367236315fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ R3 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ S ) )
= ( ord_le5389487502678872194c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_618_pred__subset__eq,axiom,
! [R3: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ R3 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ S ) )
= ( ord_le3843937902494030498c_fm_i @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_619_AxB__symmetric_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W )
=> ( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ W ) ) ) ) ) ) ) ) ) ) ) ).
% AxB_symmetric'
thf(fact_620_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( B
= ( inf_in161960956874937808c_fm_i @ K @ B2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ B )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_621_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_se3485332733965609186c_fm_i,K: set_se3485332733965609186c_fm_i,A2: set_se3485332733965609186c_fm_i,B2: set_se3485332733965609186c_fm_i] :
( ( A
= ( inf_in161960956874937808c_fm_i @ K @ A2 ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B2 )
= ( inf_in161960956874937808c_fm_i @ K @ ( inf_in161960956874937808c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_622_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Epistemic_fm_i,K: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( B
= ( sup_su1936195050962291414c_fm_i @ K @ B2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A2 @ B )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_623_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Epistemic_fm_i,K: set_Epistemic_fm_i,A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( A
= ( sup_su1936195050962291414c_fm_i @ K @ A2 ) )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B2 )
= ( sup_su1936195050962291414c_fm_i @ K @ ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_624_AxB_Ointros,axiom,
! [P2: epistemic_fm_i,I3: i] : ( epistemic_AxB_i @ ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% AxB.intros
thf(fact_625_AxB_Osimps,axiom,
( epistemic_AxB_i
= ( ^ [A3: epistemic_fm_i] :
? [P4: epistemic_fm_i,I2: i] :
( A3
= ( epistemic_Imp_i @ P4 @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% AxB.simps
thf(fact_626_AxB_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxB_i @ A2 )
=> ~ ! [P3: epistemic_fm_i,I: i] :
( A2
!= ( epistemic_Imp_i @ P3 @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% AxB.cases
thf(fact_627_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y2 @ X4 ) @ ( sup_su2582925890723967158c_fm_i @ Z3 @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_628_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y2 @ X4 ) @ ( sup_su1936195050962291414c_fm_i @ Z3 @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_629_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ Y2 @ X4 ) @ ( inf_in161960956874937808c_fm_i @ Z3 @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_630_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,X4: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) @ X4 )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ Y2 @ X4 ) @ ( inf_in3450601097109690352c_fm_i @ Z3 @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_631_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ X4 @ ( inf_in161960956874937808c_fm_i @ Y2 @ Z3 ) )
= ( inf_in161960956874937808c_fm_i @ ( sup_su2582925890723967158c_fm_i @ X4 @ Y2 ) @ ( sup_su2582925890723967158c_fm_i @ X4 @ Z3 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_632_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ ( inf_in3450601097109690352c_fm_i @ Y2 @ Z3 ) )
= ( inf_in3450601097109690352c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) @ ( sup_su1936195050962291414c_fm_i @ X4 @ Z3 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_633_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_se3485332733965609186c_fm_i,Y2: set_se3485332733965609186c_fm_i,Z3: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ ( sup_su2582925890723967158c_fm_i @ Y2 @ Z3 ) )
= ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ X4 @ Y2 ) @ ( inf_in161960956874937808c_fm_i @ X4 @ Z3 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_634_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ( inf_in3450601097109690352c_fm_i @ X4 @ ( sup_su1936195050962291414c_fm_i @ Y2 @ Z3 ) )
= ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X4 @ Y2 ) @ ( inf_in3450601097109690352c_fm_i @ X4 @ Z3 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_635_KB4__5,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxB_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% KB4_5
thf(fact_636_Ax5__Euclidean,axiom,
! [A: epistemic_fm_i > $o,U: set_Epistemic_fm_i,V: set_Epistemic_fm_i,W: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ U )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ U )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ W )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ W )
=> ( ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ U ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ U ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Ax5_Euclidean
thf(fact_637_Ax4__transitive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,W: set_Epistemic_fm_i,I3: i,U: set_Epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( ( member1104366573291651755c_fm_i @ W
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) )
=> ( ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ W ) ) ) ) )
=> ( member1104366573291651755c_fm_i @ U
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) ) ) ) ) ) ) ).
% Ax4_transitive
thf(fact_638_K5__L,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ P2 ) ) ) ) ).
% K5_L
thf(fact_639_AxT__reflexive,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ V )
=> ( member1104366573291651755c_fm_i @ V
@ ( collec3087743281813070829c_fm_i
@ ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [P4: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ ( epistemic_K_i @ I3 @ P4 ) @ V ) ) ) ) ) ) ) ) ).
% AxT_reflexive
thf(fact_640_S5__S5_H,axiom,
! [P2: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ P2 )
=> ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ P2 ) ) ).
% S5_S5'
thf(fact_641_S5_H__S5,axiom,
! [P2: epistemic_fm_i] :
( ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ P2 )
=> ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ P2 ) ) ).
% S5'_S5
thf(fact_642_S5__S5_H__assms,axiom,
! [G: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_AxB_i @ P4 )
| ( epistemic_Ax4_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) )
= ( ? [Qs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Qs2 ) @ G )
& ( epistemic_AK_i
@ ^ [P4: epistemic_fm_i] :
( ( epistemic_AxT_i @ P4 )
| ( epistemic_Ax5_i @ P4 ) )
@ ( epistemic_imply_i @ Qs2 @ P2 ) ) ) ) ) ).
% S5_S5'_assms
thf(fact_643_AxT_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_AxT_i @ A2 )
=> ~ ! [I: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P3 ) @ P3 ) ) ) ).
% AxT.cases
thf(fact_644_AxT_Osimps,axiom,
( epistemic_AxT_i
= ( ^ [A3: epistemic_fm_i] :
? [I2: i,P4: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P4 ) @ P4 ) ) ) ) ).
% AxT.simps
thf(fact_645_AxT_Ointros,axiom,
! [I3: i,P2: epistemic_fm_i] : ( epistemic_AxT_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ P2 ) ) ).
% AxT.intros
thf(fact_646_S5_H__4,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P2 ) ) ) ) ) ) ).
% S5'_4
thf(fact_647_Ax4_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax4_i @ A2 )
=> ~ ! [I: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_K_i @ I @ P3 ) @ ( epistemic_K_i @ I @ ( epistemic_K_i @ I @ P3 ) ) ) ) ) ).
% Ax4.cases
thf(fact_648_Ax4_Osimps,axiom,
( epistemic_Ax4_i
= ( ^ [A3: epistemic_fm_i] :
? [I2: i,P4: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ P4 ) @ ( epistemic_K_i @ I2 @ ( epistemic_K_i @ I2 @ P4 ) ) ) ) ) ) ).
% Ax4.simps
thf(fact_649_Ax4_Ointros,axiom,
! [I3: i,P2: epistemic_fm_i] : ( epistemic_Ax4_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ P2 ) @ ( epistemic_K_i @ I3 @ ( epistemic_K_i @ I3 @ P2 ) ) ) ) ).
% Ax4.intros
thf(fact_650_S5_H__B,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( ( ord_le190830114487426235fm_i_o @ epistemic_Ax5_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% S5'_B
thf(fact_651_Ax5_Ocases,axiom,
! [A2: epistemic_fm_i] :
( ( epistemic_Ax5_i @ A2 )
=> ~ ! [I: i,P3: epistemic_fm_i] :
( A2
!= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ ( epistemic_K_i @ I @ ( epistemic_Imp_i @ P3 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ).
% Ax5.cases
thf(fact_652_Ax5_Osimps,axiom,
( epistemic_Ax5_i
= ( ^ [A3: epistemic_fm_i] :
? [I2: i,P4: epistemic_fm_i] :
( A3
= ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I2 @ ( epistemic_Imp_i @ P4 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ) ) ).
% Ax5.simps
thf(fact_653_Ax5_Ointros,axiom,
! [I3: i,P2: epistemic_fm_i] : ( epistemic_Ax5_i @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% Ax5.intros
thf(fact_654_T__L,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,I3: i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_AxT_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% T_L
thf(fact_655_K4__L,axiom,
! [A: epistemic_fm_i > $o,I3: i,P2: epistemic_fm_i] :
( ( ord_le190830114487426235fm_i_o @ epistemic_Ax4_i @ A )
=> ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ epistemic_FF_i ) ) @ epistemic_FF_i ) @ ( epistemic_Imp_i @ ( epistemic_K_i @ I3 @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) ) @ epistemic_FF_i ) ) ) ) ).
% K4_L
thf(fact_656_map__eq__map__tailrec,axiom,
map_Ep2755178516647988292c_fm_i = map_ta5769950387337933966c_fm_i ).
% map_eq_map_tailrec
thf(fact_657_inter__Set__filter,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= ( filter6053540608743173075c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_658_inter__Set__filter,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= ( filter_nat2
@ ^ [X2: nat] : ( member_nat @ X2 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_659_inter__Set__filter,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= ( filter8450032351001387443c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A )
@ B ) ) ) ).
% inter_Set_filter
thf(fact_660_conj__subset__def,axiom,
! [A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o,Q3: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ A
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ P5 ) )
& ( ord_le5389487502678872194c_fm_i @ A @ ( collec3087743281813070829c_fm_i @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_661_conj__subset__def,axiom,
! [A: set_nat,P5: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X2: nat] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P5 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_662_conj__subset__def,axiom,
! [A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o,Q3: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ A
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( P5 @ X2 )
& ( Q3 @ X2 ) ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ P5 ) )
& ( ord_le3843937902494030498c_fm_i @ A @ ( collec4904205187116291597c_fm_i @ Q3 ) ) ) ) ).
% conj_subset_def
thf(fact_663_prop__restrict,axiom,
! [X4: set_Epistemic_fm_i,Z5: set_se3485332733965609186c_fm_i,X6: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( member1104366573291651755c_fm_i @ X4 @ Z5 )
=> ( ( ord_le5389487502678872194c_fm_i @ Z5
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ X6 )
& ( P5 @ X2 ) ) ) )
=> ( P5 @ X4 ) ) ) ).
% prop_restrict
thf(fact_664_prop__restrict,axiom,
! [X4: nat,Z5: set_nat,X6: set_nat,P5: nat > $o] :
( ( member_nat @ X4 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P5 @ X2 ) ) ) )
=> ( P5 @ X4 ) ) ) ).
% prop_restrict
thf(fact_665_prop__restrict,axiom,
! [X4: epistemic_fm_i,Z5: set_Epistemic_fm_i,X6: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( member6642669606046002379c_fm_i @ X4 @ Z5 )
=> ( ( ord_le3843937902494030498c_fm_i @ Z5
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ X6 )
& ( P5 @ X2 ) ) ) )
=> ( P5 @ X4 ) ) ) ).
% prop_restrict
thf(fact_666_member__filter,axiom,
! [X4: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( filter8450032351001387443c_fm_i @ P5 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( P5 @ X4 ) ) ) ).
% member_filter
thf(fact_667_member__filter,axiom,
! [X4: epistemic_fm_i,P5: epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( filter6053540608743173075c_fm_i @ P5 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( P5 @ X4 ) ) ) ).
% member_filter
thf(fact_668_Set_Ofilter__def,axiom,
( filter8450032351001387443c_fm_i
= ( ^ [P6: set_Epistemic_fm_i > $o,A4: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [A3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A3 @ A4 )
& ( P6 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_669_Set_Ofilter__def,axiom,
( filter6053540608743173075c_fm_i
= ( ^ [P6: epistemic_fm_i > $o,A4: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [A3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A3 @ A4 )
& ( P6 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_670_Set_Ofilter__def,axiom,
( filter_nat2
= ( ^ [P6: nat > $o,A4: set_nat] :
( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A4 )
& ( P6 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_671_finite__filter,axiom,
! [S: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( finite3304564979551393739c_fm_i @ ( filter6053540608743173075c_fm_i @ P5 @ S ) ) ) ).
% finite_filter
thf(fact_672_finite__filter,axiom,
! [S: set_nat,P5: nat > $o] :
( ( finite_finite_nat @ S )
=> ( finite_finite_nat @ ( filter_nat2 @ P5 @ S ) ) ) ).
% finite_filter
thf(fact_673_filter__set,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( filter6053540608743173075c_fm_i @ P5 @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ).
% filter_set
thf(fact_674_Collect__restrict,axiom,
! [X6: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ X6 )
& ( P5 @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_675_Collect__restrict,axiom,
! [X6: set_nat,P5: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P5 @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_676_Collect__restrict,axiom,
! [X6: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ X6 )
& ( P5 @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_677_subset__Collect__iff,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ B
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ B )
=> ( P5 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_678_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P5: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P5 @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P5 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_679_subset__Collect__iff,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ B
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
=> ( P5 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_680_subset__CollectI,axiom,
! [B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,Q3: set_Epistemic_fm_i > $o,P5: set_Epistemic_fm_i > $o] :
( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P5 @ X3 ) ) )
=> ( ord_le5389487502678872194c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ B )
& ( Q3 @ X2 ) ) )
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_681_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q3: nat > $o,P5: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P5 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B )
& ( Q3 @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P5 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_682_subset__CollectI,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i,Q3: epistemic_fm_i > $o,P5: epistemic_fm_i > $o] :
( ( ord_le3843937902494030498c_fm_i @ B @ A )
=> ( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ B )
=> ( ( Q3 @ X3 )
=> ( P5 @ X3 ) ) )
=> ( ord_le3843937902494030498c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ B )
& ( Q3 @ X2 ) ) )
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
& ( P5 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_683_K__Boole,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ G ) @ epistemic_FF_i ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P2 ) ) ) ).
% K_Boole
thf(fact_684_union__coset__filter,axiom,
! [Xs: list_s8081015415394010888c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( coset_1342939534705115381c_fm_i @ Xs ) @ A )
= ( coset_1342939534705115381c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_685_union__coset__filter,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( coset_Epistemic_fm_i @ Xs ) @ A )
= ( coset_Epistemic_fm_i
@ ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_686_inconsistent__imply,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G: list_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ ( set_Epistemic_fm_i2 @ G ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ P2 ) ) ) ).
% inconsistent_imply
thf(fact_687_empty__Collect__eq,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ P5 ) )
= ( ! [X2: set_Epistemic_fm_i] :
~ ( P5 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_688_empty__Collect__eq,axiom,
! [P5: epistemic_fm_i > $o] :
( ( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ P5 ) )
= ( ! [X2: epistemic_fm_i] :
~ ( P5 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_689_empty__Collect__eq,axiom,
! [P5: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P5 ) )
= ( ! [X2: nat] :
~ ( P5 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_690_Collect__empty__eq,axiom,
! [P5: set_Epistemic_fm_i > $o] :
( ( ( collec3087743281813070829c_fm_i @ P5 )
= bot_bo145720340923748686c_fm_i )
= ( ! [X2: set_Epistemic_fm_i] :
~ ( P5 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_691_Collect__empty__eq,axiom,
! [P5: epistemic_fm_i > $o] :
( ( ( collec4904205187116291597c_fm_i @ P5 )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X2: epistemic_fm_i] :
~ ( P5 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_692_Collect__empty__eq,axiom,
! [P5: nat > $o] :
( ( ( collect_nat @ P5 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P5 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_693_all__not__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ! [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ A ) )
= ( A = bot_bo145720340923748686c_fm_i ) ) ).
% all_not_in_conv
thf(fact_694_all__not__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ! [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ A ) )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% all_not_in_conv
thf(fact_695_empty__iff,axiom,
! [C: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ C @ bot_bo145720340923748686c_fm_i ) ).
% empty_iff
thf(fact_696_empty__iff,axiom,
! [C: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ C @ bot_bo4194595901900360558c_fm_i ) ).
% empty_iff
thf(fact_697_insert__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_698_insert__iff,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_699_insertCI,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i] :
( ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) ) ).
% insertCI
thf(fact_700_insertCI,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertCI
thf(fact_701_subset__empty,axiom,
! [A: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% subset_empty
thf(fact_702_empty__subsetI,axiom,
! [A: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A ) ).
% empty_subsetI
thf(fact_703_inf__bot__left,axiom,
! [X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X4 )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_left
thf(fact_704_inf__bot__right,axiom,
! [X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% inf_bot_right
thf(fact_705_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ X4 )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_left
thf(fact_706_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% boolean_algebra.conj_zero_right
thf(fact_707_sup__bot__left,axiom,
! [X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_708_sup__bot__right,axiom,
! [X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i )
= X4 ) ).
% sup_bot_right
thf(fact_709_bot__eq__sup__iff,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 ) )
= ( ( X4 = bot_bo4194595901900360558c_fm_i )
& ( Y2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% bot_eq_sup_iff
thf(fact_710_sup__eq__bot__iff,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ X4 @ Y2 )
= bot_bo4194595901900360558c_fm_i )
= ( ( X4 = bot_bo4194595901900360558c_fm_i )
& ( Y2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_eq_bot_iff
thf(fact_711_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A2 @ B2 )
= bot_bo4194595901900360558c_fm_i )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_712_sup__bot_Oleft__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_713_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( sup_su1936195050962291414c_fm_i @ A2 @ B2 ) )
= ( ( A2 = bot_bo4194595901900360558c_fm_i )
& ( B2 = bot_bo4194595901900360558c_fm_i ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_714_sup__bot_Oright__neutral,axiom,
! [A2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_715_singletonI,axiom,
! [A2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singletonI
thf(fact_716_singletonI,axiom,
! [A2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singletonI
thf(fact_717_finite__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ).
% finite_insert
thf(fact_718_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_719_insert__subset,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) @ B )
= ( ( member1104366573291651755c_fm_i @ X4 @ B )
& ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_720_insert__subset,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( ( member6642669606046002379c_fm_i @ X4 @ B )
& ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% insert_subset
thf(fact_721_Un__empty,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B = bot_bo4194595901900360558c_fm_i ) ) ) ).
% Un_empty
thf(fact_722_Int__insert__left__if0,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_723_Int__insert__left__if0,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_724_Int__insert__left__if1,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_725_Int__insert__left__if1,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_726_insert__inter__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_727_Int__insert__right__if0,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_728_Int__insert__right__if0,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_729_Int__insert__right__if1,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_730_Int__insert__right__if1,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_731_Un__insert__left,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_732_Un__insert__right,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_733_singleton__conv,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( X2 = A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv
thf(fact_734_singleton__conv,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] : ( X2 = A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv
thf(fact_735_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_736_singleton__conv2,axiom,
! [A2: set_Epistemic_fm_i] :
( ( collec3087743281813070829c_fm_i
@ ( ^ [Y: set_Epistemic_fm_i,Z2: set_Epistemic_fm_i] : ( Y = Z2 )
@ A2 ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) ).
% singleton_conv2
thf(fact_737_singleton__conv2,axiom,
! [A2: epistemic_fm_i] :
( ( collec4904205187116291597c_fm_i
@ ( ^ [Y: epistemic_fm_i,Z2: epistemic_fm_i] : ( Y = Z2 )
@ A2 ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ).
% singleton_conv2
thf(fact_738_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y: nat,Z2: nat] : ( Y = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_739_singleton__insert__inj__eq,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i )
= ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_740_singleton__insert__inj__eq_H,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( A2 = B2 )
& ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_741_list_Osimps_I15_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( insert7817948997695205106c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_742_insert__disjoint_I1_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_743_insert__disjoint_I1_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% insert_disjoint(1)
thf(fact_744_insert__disjoint_I2_J,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_745_insert__disjoint_I2_J,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ A ) @ B ) )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_746_disjoint__insert_I1_J,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ A ) )
= bot_bo4194595901900360558c_fm_i )
= ( ~ ( member6642669606046002379c_fm_i @ A2 @ B )
& ( ( inf_in3450601097109690352c_fm_i @ B @ A )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_747_disjoint__insert_I1_J,axiom,
! [B: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ A2 @ A ) )
= bot_bo145720340923748686c_fm_i )
= ( ~ ( member1104366573291651755c_fm_i @ A2 @ B )
& ( ( inf_in161960956874937808c_fm_i @ B @ A )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% disjoint_insert(1)
thf(fact_748_disjoint__insert_I2_J,axiom,
! [A: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) )
= ( ~ ( member6642669606046002379c_fm_i @ B2 @ A )
& ( bot_bo4194595901900360558c_fm_i
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_749_disjoint__insert_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) )
= ( ~ ( member1104366573291651755c_fm_i @ B2 @ A )
& ( bot_bo145720340923748686c_fm_i
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_750_insert__subsetI,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X6: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ X6 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_751_insert__subsetI,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,X6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_752_Int__insert__right,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) )
= ( inf_in3450601097109690352c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_753_Int__insert__right,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_754_Int__insert__left,axiom,
! [A2: epistemic_fm_i,C2: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( insert7817948997695205106c_fm_i @ A2 @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ A2 @ C2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in3450601097109690352c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_755_Int__insert__left,axiom,
! [A2: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( insert7698009978809854162c_fm_i @ A2 @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ A2 @ C2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( insert7698009978809854162c_fm_i @ A2 @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_756_map__eq__Cons__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,Y2: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( cons_Epistemic_fm_i @ Y2 @ Ys ) )
= ( ? [Z6: epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( Xs
= ( cons_Epistemic_fm_i @ Z6 @ Zs ) )
& ( ( F @ Z6 )
= Y2 )
& ( ( map_Ep2755178516647988292c_fm_i @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_757_Cons__eq__map__conv,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X4 @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ F @ Ys ) )
= ( ? [Z6: epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( Ys
= ( cons_Epistemic_fm_i @ Z6 @ Zs ) )
& ( X4
= ( F @ Z6 ) )
& ( Xs
= ( map_Ep2755178516647988292c_fm_i @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_758_map__eq__Cons__D,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,Y2: epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( cons_Epistemic_fm_i @ Y2 @ Ys ) )
=> ? [Z4: epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Xs
= ( cons_Epistemic_fm_i @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_Ep2755178516647988292c_fm_i @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_759_Cons__eq__map__D,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( cons_Epistemic_fm_i @ X4 @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ F @ Ys ) )
=> ? [Z4: epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( Ys
= ( cons_Epistemic_fm_i @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_Ep2755178516647988292c_fm_i @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_760_list_Osimps_I9_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,X21: epistemic_fm_i,X22: list_Epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ ( cons_Epistemic_fm_i @ X21 @ X22 ) )
= ( cons_Epistemic_fm_i @ ( F @ X21 ) @ ( map_Ep2755178516647988292c_fm_i @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_761_filter_Osimps_I2_J,axiom,
! [P5: epistemic_fm_i > $o,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( P5 @ X4 )
=> ( ( filter7636273843821131039c_fm_i @ P5 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( cons_Epistemic_fm_i @ X4 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) )
& ( ~ ( P5 @ X4 )
=> ( ( filter7636273843821131039c_fm_i @ P5 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_762_subset__singletonD,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ( A = bot_bo4194595901900360558c_fm_i )
| ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singletonD
thf(fact_763_subset__singleton__iff,axiom,
! [X6: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ X6 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( X6 = bot_bo4194595901900360558c_fm_i )
| ( X6
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% subset_singleton_iff
thf(fact_764_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_765_finite__subset__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_766_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_767_finite__subset__induct,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_768_finite__subset__induct_H,axiom,
! [F2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( ord_le5389487502678872194c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ F3 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_769_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_770_finite__subset__induct_H,axiom,
! [F2: set_Epistemic_fm_i,A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( ord_le3843937902494030498c_fm_i @ F2 @ A )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A6: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ F3 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ A6 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ A6 @ F3 ) ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_771_bot_Oextremum,axiom,
! [A2: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ bot_bo4194595901900360558c_fm_i @ A2 ) ).
% bot.extremum
thf(fact_772_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_773_bot_Oextremum__unique,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
= ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_unique
thf(fact_774_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_775_bot_Oextremum__uniqueI,axiom,
! [A2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i )
=> ( A2 = bot_bo4194595901900360558c_fm_i ) ) ).
% bot.extremum_uniqueI
thf(fact_776_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_777_insert__compr,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A3: set_Epistemic_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( X2 = A3 )
| ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_778_insert__compr,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( X2 = A3 )
| ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_779_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A3 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_780_insert__Collect,axiom,
! [A2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( insert7698009978809854162c_fm_i @ A2 @ ( collec3087743281813070829c_fm_i @ P5 ) )
= ( collec3087743281813070829c_fm_i
@ ^ [U2: set_Epistemic_fm_i] :
( ( U2 != A2 )
=> ( P5 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_781_insert__Collect,axiom,
! [A2: epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( insert7817948997695205106c_fm_i @ A2 @ ( collec4904205187116291597c_fm_i @ P5 ) )
= ( collec4904205187116291597c_fm_i
@ ^ [U2: epistemic_fm_i] :
( ( U2 != A2 )
=> ( P5 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_782_insert__Collect,axiom,
! [A2: nat,P5: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P5 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P5 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_783_empty__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_784_empty__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] : $false ) ) ).
% empty_def
thf(fact_785_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_786_mk__disjoint__insert,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ? [B7: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ A2 @ B7 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_787_mk__disjoint__insert,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ? [B7: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ A2 @ B7 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_788_singleton__iff,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_789_singleton__iff,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_790_insert__eq__iff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ B2 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= ( insert7698009978809854162c_fm_i @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ B2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ B2 @ C3 )
& ( B
= ( insert7698009978809854162c_fm_i @ A2 @ C3 ) )
& ~ ( member1104366573291651755c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_791_insert__eq__iff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i,B2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ B2 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= ( insert7817948997695205106c_fm_i @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ B2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ B2 @ C3 )
& ( B
= ( insert7817948997695205106c_fm_i @ A2 @ C3 ) )
& ~ ( member6642669606046002379c_fm_i @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_792_insert__absorb,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_793_insert__absorb,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_794_insert__ident,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ B )
=> ( ( ( insert7698009978809854162c_fm_i @ X4 @ A )
= ( insert7698009978809854162c_fm_i @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_795_insert__ident,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( ( insert7817948997695205106c_fm_i @ X4 @ A )
= ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_796_singletonD,axiom,
! [B2: set_Epistemic_fm_i,A2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ B2 @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_797_singletonD,axiom,
! [B2: epistemic_fm_i,A2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ B2 @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_798_Set_Oset__insert,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ~ ! [B7: set_se3485332733965609186c_fm_i] :
( ( A
= ( insert7698009978809854162c_fm_i @ X4 @ B7 ) )
=> ( member1104366573291651755c_fm_i @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_799_Set_Oset__insert,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ~ ! [B7: set_Epistemic_fm_i] :
( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ B7 ) )
=> ( member6642669606046002379c_fm_i @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_800_ex__in__conv,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A ) )
= ( A != bot_bo145720340923748686c_fm_i ) ) ).
% ex_in_conv
thf(fact_801_ex__in__conv,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A ) )
= ( A != bot_bo4194595901900360558c_fm_i ) ) ).
% ex_in_conv
thf(fact_802_insertI2,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,B2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ B )
=> ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ B ) ) ) ).
% insertI2
thf(fact_803_insertI2,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ B )
=> ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% insertI2
thf(fact_804_insertI1,axiom,
! [A2: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] : ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_805_insertI1,axiom,
! [A2: epistemic_fm_i,B: set_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% insertI1
thf(fact_806_equals0I,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [Y5: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ Y5 @ A )
=> ( A = bot_bo145720340923748686c_fm_i ) ) ).
% equals0I
thf(fact_807_equals0I,axiom,
! [A: set_Epistemic_fm_i] :
( ! [Y5: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ Y5 @ A )
=> ( A = bot_bo4194595901900360558c_fm_i ) ) ).
% equals0I
thf(fact_808_equals0D,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( A = bot_bo145720340923748686c_fm_i )
=> ~ ( member1104366573291651755c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_809_equals0D,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( A = bot_bo4194595901900360558c_fm_i )
=> ~ ( member6642669606046002379c_fm_i @ A2 @ A ) ) ).
% equals0D
thf(fact_810_insertE,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ ( insert7698009978809854162c_fm_i @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member1104366573291651755c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_811_insertE,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ ( insert7817948997695205106c_fm_i @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member6642669606046002379c_fm_i @ A2 @ A ) ) ) ).
% insertE
thf(fact_812_emptyE,axiom,
! [A2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ).
% emptyE
thf(fact_813_emptyE,axiom,
! [A2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ).
% emptyE
thf(fact_814_Collect__conv__if,axiom,
! [P5: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_815_Collect__conv__if,axiom,
! [P5: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if
thf(fact_816_Collect__conv__if,axiom,
! [P5: nat > $o,A2: nat] :
( ( ( P5 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A2 )
& ( P5 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_817_Collect__conv__if2,axiom,
! [P5: set_Epistemic_fm_i > $o,A2: set_Epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= bot_bo145720340923748686c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_818_Collect__conv__if2,axiom,
! [P5: epistemic_fm_i > $o,A2: epistemic_fm_i] :
( ( ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ).
% Collect_conv_if2
thf(fact_819_Collect__conv__if2,axiom,
! [P5: nat > $o,A2: nat] :
( ( ( P5 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P5 @ A2 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A2 = X2 )
& ( P5 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_820_singleton__Un__iff,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_821_Un__singleton__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
= ( ( ( A = bot_bo4194595901900360558c_fm_i )
& ( B
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
& ( B = bot_bo4194595901900360558c_fm_i ) )
| ( ( A
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
& ( B
= ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_822_insert__is__Un,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i] : ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% insert_is_Un
thf(fact_823_finite_Ocases,axiom,
! [A2: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A2 )
=> ( ( A2 != bot_bo4194595901900360558c_fm_i )
=> ~ ! [A7: set_Epistemic_fm_i] :
( ? [A6: epistemic_fm_i] :
( A2
= ( insert7817948997695205106c_fm_i @ A6 @ A7 ) )
=> ~ ( finite3304564979551393739c_fm_i @ A7 ) ) ) ) ).
% finite.cases
thf(fact_824_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A6: nat] :
( A2
= ( insert_nat @ A6 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_825_finite_Osimps,axiom,
( finite3304564979551393739c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( ( A3 = bot_bo4194595901900360558c_fm_i )
| ? [A4: set_Epistemic_fm_i,B5: epistemic_fm_i] :
( ( A3
= ( insert7817948997695205106c_fm_i @ B5 @ A4 ) )
& ( finite3304564979551393739c_fm_i @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_826_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A4: set_nat,B5: nat] :
( ( A3
= ( insert_nat @ B5 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_827_finite__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_828_finite__induct,axiom,
! [F2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_829_finite__induct,axiom,
! [F2: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_830_finite__ne__induct,axiom,
! [F2: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ F2 )
=> ( ( F2 != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i] : ( P5 @ ( insert7698009978809854162c_fm_i @ X3 @ bot_bo145720340923748686c_fm_i ) )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ( F3 != bot_bo145720340923748686c_fm_i )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_831_finite__ne__induct,axiom,
! [F2: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ F2 )
=> ( ( F2 != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i] : ( P5 @ ( insert7817948997695205106c_fm_i @ X3 @ bot_bo4194595901900360558c_fm_i ) )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ( F3 != bot_bo4194595901900360558c_fm_i )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_832_finite__ne__induct,axiom,
! [F2: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P5 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P5 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_833_infinite__finite__induct,axiom,
! [P5: set_se3485332733965609186c_fm_i > $o,A: set_se3485332733965609186c_fm_i] :
( ! [A7: set_se3485332733965609186c_fm_i] :
( ~ ( finite7933139204641697195c_fm_i @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,F3: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ F3 )
=> ( ~ ( member1104366573291651755c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_834_infinite__finite__induct,axiom,
! [P5: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ! [A7: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,F3: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ F3 )
=> ( ~ ( member6642669606046002379c_fm_i @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X3 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_835_infinite__finite__induct,axiom,
! [P5: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P5 @ A7 ) )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P5 @ F3 )
=> ( P5 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P5 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_836_set__ConsD,axiom,
! [Y2: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ Y2 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) ) )
=> ( ( Y2 = X4 )
| ( member1104366573291651755c_fm_i @ Y2 @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_837_set__ConsD,axiom,
! [Y2: epistemic_fm_i,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y2 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) ) )
=> ( ( Y2 = X4 )
| ( member6642669606046002379c_fm_i @ Y2 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_838_list_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A2: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ A2 ) )
=> ( ! [Z22: list_s8081015415394010888c_fm_i] :
( A2
!= ( cons_s4962720389763977656c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z22: list_s8081015415394010888c_fm_i] :
( ( A2
= ( cons_s4962720389763977656c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( set_se200842218512397079c_fm_i @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_839_list_Oset__cases,axiom,
! [E: epistemic_fm_i,A2: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ A2 ) )
=> ( ! [Z22: list_Epistemic_fm_i] :
( A2
!= ( cons_Epistemic_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: list_Epistemic_fm_i] :
( ( A2
= ( cons_Epistemic_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( set_Epistemic_fm_i2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_840_list_Oset__intros_I1_J,axiom,
! [X21: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i] : ( member1104366573291651755c_fm_i @ X21 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_841_list_Oset__intros_I1_J,axiom,
! [X21: epistemic_fm_i,X22: list_Epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X21 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_842_list_Oset__intros_I2_J,axiom,
! [Y2: set_Epistemic_fm_i,X22: list_s8081015415394010888c_fm_i,X21: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y2 @ ( set_se200842218512397079c_fm_i @ X22 ) )
=> ( member1104366573291651755c_fm_i @ Y2 @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_843_list_Oset__intros_I2_J,axiom,
! [Y2: epistemic_fm_i,X22: list_Epistemic_fm_i,X21: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y2 @ ( set_Epistemic_fm_i2 @ X22 ) )
=> ( member6642669606046002379c_fm_i @ Y2 @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_844_subset__insertI2,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,B2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_845_subset__insertI,axiom,
! [B: set_Epistemic_fm_i,A2: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) ).
% subset_insertI
thf(fact_846_subset__insert,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ B ) )
= ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_847_subset__insert,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ).
% subset_insert
thf(fact_848_insert__mono,axiom,
! [C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ C2 @ D )
=> ( ord_le3843937902494030498c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ C2 ) @ ( insert7817948997695205106c_fm_i @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_849_finite_OinsertI,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( insert7817948997695205106c_fm_i @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_850_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_851_infinite__imp__nonempty,axiom,
! [S: set_Epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ( S != bot_bo4194595901900360558c_fm_i ) ) ).
% infinite_imp_nonempty
thf(fact_852_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_853_finite_OemptyI,axiom,
finite3304564979551393739c_fm_i @ bot_bo4194595901900360558c_fm_i ).
% finite.emptyI
thf(fact_854_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_855_subset__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ bot_bo145720340923748686c_fm_i ) ) ).
% subset_emptyI
thf(fact_856_subset__emptyI,axiom,
! [A: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ bot_bo4194595901900360558c_fm_i ) ) ).
% subset_emptyI
thf(fact_857_Int__emptyI,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X3 @ B ) )
=> ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i ) ) ).
% Int_emptyI
thf(fact_858_Int__emptyI,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ! [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ~ ( member1104366573291651755c_fm_i @ X3 @ B ) )
=> ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i ) ) ).
% Int_emptyI
thf(fact_859_disjoint__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_860_disjoint__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_861_Int__empty__left,axiom,
! [B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ bot_bo145720340923748686c_fm_i @ B )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_left
thf(fact_862_Int__empty__right,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ bot_bo145720340923748686c_fm_i )
= bot_bo145720340923748686c_fm_i ) ).
% Int_empty_right
thf(fact_863_disjoint__iff__not__equal,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ! [Y4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_864_Un__empty__left,axiom,
! [B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ bot_bo4194595901900360558c_fm_i @ B )
= B ) ).
% Un_empty_left
thf(fact_865_Un__empty__right,axiom,
! [A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ bot_bo4194595901900360558c_fm_i )
= A ) ).
% Un_empty_right
thf(fact_866_insert__def,axiom,
( insert7698009978809854162c_fm_i
= ( ^ [A3: set_Epistemic_fm_i] :
( sup_su2582925890723967158c_fm_i
@ ( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( X2 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_867_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X2: nat] : ( X2 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_868_insert__def,axiom,
( insert7817948997695205106c_fm_i
= ( ^ [A3: epistemic_fm_i] :
( sup_su1936195050962291414c_fm_i
@ ( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] : ( X2 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_869_maximal__def,axiom,
! [A: epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima3264069618988350929c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S )
= ( ! [P4: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ).
% maximal_def
thf(fact_870_set__subset__Cons,axiom,
! [Xs: list_Epistemic_fm_i,X4: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_871_finite__has__maximal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_872_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_873_finite__has__minimal,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ? [X3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
& ! [Xa2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Xa2 @ A )
=> ( ( ord_le3843937902494030498c_fm_i @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_874_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ord_less_eq_nat @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_875_consistent__consequent,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V )
=> ( ( epistemic_AK_i @ A @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent
thf(fact_876_imply_Osimps_I2_J,axiom,
! [P2: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q )
= ( epistemic_Imp_i @ P2 @ ( epistemic_imply_i @ Ps @ Q ) ) ) ).
% imply.simps(2)
thf(fact_877_consistent__consequent_H,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ P2 @ V )
=> ( ! [G4: list_char > $o,H3: epistemic_fm_i > $o] : ( epistemic_eval_i @ G4 @ H3 @ ( epistemic_Imp_i @ P2 @ Q ) )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_consequent'
thf(fact_878_K__imply__head,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Ps: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ P2 ) ) ).
% K_imply_head
thf(fact_879_K__imply__Cons,axiom,
! [A: epistemic_fm_i > $o,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ Ps @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q ) ) ) ).
% K_imply_Cons
thf(fact_880_K__swap,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,G: list_Epistemic_fm_i,R2: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ ( cons_Epistemic_fm_i @ Q @ G ) ) @ R2 ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ ( cons_Epistemic_fm_i @ P2 @ G ) ) @ R2 ) ) ) ).
% K_swap
thf(fact_881_conjunct_Osimps_I2_J,axiom,
! [P2: epistemic_fm_i,Ps: list_Epistemic_fm_i] :
( ( stalnaker_conjunct_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) )
= ( epistemic_Con_i @ P2 @ ( stalnaker_conjunct_i @ Ps ) ) ) ).
% conjunct.simps(2)
thf(fact_882_inconsistent__subset,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [V2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ V2 ) @ V )
=> ~ ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ V2 ) @ epistemic_FF_i ) ) ) ) ) ).
% inconsistent_subset
thf(fact_883_K__ImpI,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G: list_Epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ G ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Imp_i @ P2 @ Q ) ) ) ) ).
% K_ImpI
thf(fact_884_K__mp,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i,G: list_Epistemic_fm_i] : ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ ( cons_Epistemic_fm_i @ ( epistemic_Imp_i @ P2 @ Q ) @ G ) ) @ Q ) ) ).
% K_mp
thf(fact_885_subset__code_I2_J,axiom,
! [A: set_se3485332733965609186c_fm_i,Ys: list_s8081015415394010888c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Ys ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
=> ~ ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_886_subset__code_I2_J,axiom,
! [A: set_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( coset_Epistemic_fm_i @ Ys ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
=> ~ ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_887_exists__finite__inconsistent,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,V: set_Epistemic_fm_i] :
( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ~ ! [W2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ V ) )
=> ( ~ ( member6642669606046002379c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ W2 )
=> ( ( finite3304564979551393739c_fm_i @ W2 )
=> ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ ( epistemic_Imp_i @ P2 @ epistemic_FF_i ) @ bot_bo4194595901900360558c_fm_i ) @ W2 ) ) ) ) ) ) ).
% exists_finite_inconsistent
thf(fact_888_finite__ranking__induct,axiom,
! [S: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o,F: set_Epistemic_fm_i > nat] :
( ( finite7933139204641697195c_fm_i @ S )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,S5: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ S5 )
=> ( ! [Y3: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Y3 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P5 @ S5 )
=> ( P5 @ ( insert7698009978809854162c_fm_i @ X3 @ S5 ) ) ) ) )
=> ( P5 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_889_finite__ranking__induct,axiom,
! [S: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,S5: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ S5 )
=> ( ! [Y3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Y3 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P5 @ S5 )
=> ( P5 @ ( insert7817948997695205106c_fm_i @ X3 @ S5 ) ) ) ) )
=> ( P5 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_890_finite__ranking__induct,axiom,
! [S: set_nat,P5: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [X3: nat,S5: set_nat] :
( ( finite_finite_nat @ S5 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ S5 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X3 ) ) )
=> ( ( P5 @ S5 )
=> ( P5 @ ( insert_nat @ X3 @ S5 ) ) ) ) )
=> ( P5 @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_891_extendS__def,axiom,
! [A: epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ ( episte2285483198712856234tent_i @ A ) @ S @ N @ Prev )
= Prev ) ) ) ).
% extendS_def
thf(fact_892_MCS_Omaximal__def,axiom,
! [Consistent: set_se3485332733965609186c_fm_i > $o,S: set_se3485332733965609186c_fm_i] :
( ( maxima5540402109999912942c_fm_i @ Consistent )
=> ( ( maxima7741914846562630577c_fm_i @ Consistent @ S )
= ( ! [P4: set_Epistemic_fm_i] :
( ( Consistent @ ( sup_su2582925890723967158c_fm_i @ ( insert7698009978809854162c_fm_i @ P4 @ bot_bo145720340923748686c_fm_i ) @ S ) )
=> ( member1104366573291651755c_fm_i @ P4 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_893_MCS_Omaximal__def,axiom,
! [Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima1924290493700099598c_fm_i @ Consistent )
=> ( ( maxima3264069618988350929c_fm_i @ Consistent @ S )
= ( ! [P4: epistemic_fm_i] :
( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P4 @ bot_bo4194595901900360558c_fm_i ) @ S ) )
=> ( member6642669606046002379c_fm_i @ P4 @ S ) ) ) ) ) ).
% MCS.maximal_def
thf(fact_894_bot__set__def,axiom,
( bot_bo145720340923748686c_fm_i
= ( collec3087743281813070829c_fm_i @ bot_bo6089404950617257231fm_i_o ) ) ).
% bot_set_def
thf(fact_895_bot__set__def,axiom,
( bot_bo4194595901900360558c_fm_i
= ( collec4904205187116291597c_fm_i @ bot_bo2580527446789968623fm_i_o ) ) ).
% bot_set_def
thf(fact_896_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_897_MCS__axioms,axiom,
! [A: epistemic_fm_i > $o] : ( maxima1924290493700099598c_fm_i @ ( episte2285483198712856234tent_i @ A ) ) ).
% MCS_axioms
thf(fact_898_is__singletonI_H,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X3 @ A )
=> ( ( member1104366573291651755c_fm_i @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin2358732406619996150c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_899_is__singletonI_H,axiom,
! [A: set_Epistemic_fm_i] :
( ( A != bot_bo4194595901900360558c_fm_i )
=> ( ! [X3: epistemic_fm_i,Y5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ A )
=> ( ( member6642669606046002379c_fm_i @ Y5 @ A )
=> ( X3 = Y5 ) ) )
=> ( is_sin4923117429174228502c_fm_i @ A ) ) ) ).
% is_singletonI'
thf(fact_900_arg__min__least,axiom,
! [S: set_se3485332733965609186c_fm_i,Y2: set_Epistemic_fm_i,F: set_Epistemic_fm_i > nat] :
( ( finite7933139204641697195c_fm_i @ S )
=> ( ( S != bot_bo145720340923748686c_fm_i )
=> ( ( member1104366573291651755c_fm_i @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic8104409813523524473_i_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_901_arg__min__least,axiom,
! [S: set_Epistemic_fm_i,Y2: epistemic_fm_i,F: epistemic_fm_i > nat] :
( ( finite3304564979551393739c_fm_i @ S )
=> ( ( S != bot_bo4194595901900360558c_fm_i )
=> ( ( member6642669606046002379c_fm_i @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7487310264250698137_i_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_902_arg__min__least,axiom,
! [S: set_nat,Y2: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y2 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y2 ) ) ) ) ) ).
% arg_min_least
thf(fact_903_MCS__Lim__Ord_OextendS__def,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,N: epistemic_fm_i,Prev: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) ) )
& ( ~ ( Consistent @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ N @ bot_bo4194595901900360558c_fm_i ) @ Prev ) )
=> ( ( maxima5458213620894205884c_fm_i @ Consistent @ S @ N @ Prev )
= Prev ) ) ) ) ).
% MCS_Lim_Ord.extendS_def
thf(fact_904_set__rec,axiom,
( set_Epistemic_fm_i2
= ( rec_li5834409330748533462c_fm_i @ bot_bo4194595901900360558c_fm_i
@ ^ [X2: epistemic_fm_i,Uu2: list_Epistemic_fm_i] : ( insert7817948997695205106c_fm_i @ X2 ) ) ) ).
% set_rec
thf(fact_905_Sup__fin_Oinsert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_906_Sup__fin_Oinsert,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_907_inf__Sup__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_908_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_909_inf__Sup__absorb,axiom,
! [A: set_se7339729205154126530c_fm_i,A2: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ A2 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ A2 @ ( lattic5920028816461561881c_fm_i @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_910_MCS__Lim__Ord_Oconsistent__hereditary,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i,S2: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ( Consistent @ S )
=> ( ( ord_le3843937902494030498c_fm_i @ S2 @ S )
=> ( Consistent @ S2 ) ) ) ) ).
% MCS_Lim_Ord.consistent_hereditary
thf(fact_911_Sup__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A2 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_912_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_913_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_914_Sup__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ X4 @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_915_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr1261947904930325089at_nat,Consistent: set_nat > $o,S: set_nat] :
( ( maxima4501515429357695765rd_nat @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ S )
& ( finite_finite_nat @ S3 )
& ~ ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_916_MCS__Lim__Ord_Oinconsistent__finite,axiom,
! [R2: set_Pr4658907567593863815c_fm_i,Consistent: set_Epistemic_fm_i > $o,S: set_Epistemic_fm_i] :
( ( maxima8116622286610175645c_fm_i @ R2 @ Consistent )
=> ( ~ ( Consistent @ S )
=> ? [S3: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ S3 @ S )
& ( finite3304564979551393739c_fm_i @ S3 )
& ~ ( Consistent @ S3 ) ) ) ) ).
% MCS_Lim_Ord.inconsistent_finite
thf(fact_917_Sup__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A8 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_918_Sup__fin_OboundedE,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ A8 @ X4 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_919_Sup__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A6 @ X4 ) )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_920_Sup__fin_OboundedI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ord_less_eq_nat @ A6 @ X4 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_921_Sup__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ X4 )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X2 @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_922_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ X4 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_923_Sup__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_924_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_925_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_926_Sup__fin_Osubset,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ B @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ B ) @ ( lattic3744900211830597177c_fm_i @ A ) )
= ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_927_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y5: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y5 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_928_Sup__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( sup_su1936195050962291414c_fm_i @ X3 @ Y5 ) @ ( insert7698009978809854162c_fm_i @ X3 @ ( insert7698009978809854162c_fm_i @ Y5 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_929_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_930_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_931_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_932_Sup__fin_Ounion,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ( B != bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( sup_su2582925890723967158c_fm_i @ A @ B ) )
= ( sup_su1936195050962291414c_fm_i @ ( lattic3744900211830597177c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_933_Inf__fin_Oinsert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_934_Inf__fin_Oinsert,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X4 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_935_consistent__disjuncts,axiom,
! [A: epistemic_fm_i > $o,V: set_Epistemic_fm_i,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( episte2285483198712856234tent_i @ A @ V )
=> ( ( member6642669606046002379c_fm_i @ ( epistemic_Dis_i @ P2 @ Q ) @ V )
=> ( ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ P2 @ bot_bo4194595901900360558c_fm_i ) @ V ) )
| ( episte2285483198712856234tent_i @ A @ ( sup_su1936195050962291414c_fm_i @ ( insert7817948997695205106c_fm_i @ Q @ bot_bo4194595901900360558c_fm_i ) @ V ) ) ) ) ) ).
% consistent_disjuncts
thf(fact_936_subset__singleton__iff__Uniq,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( ? [A3: set_Epistemic_fm_i] : ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A3 @ bot_bo145720340923748686c_fm_i ) ) )
= ( uniq_s5891897140743797767c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_937_subset__singleton__iff__Uniq,axiom,
! [A: set_Epistemic_fm_i] :
( ( ? [A3: epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A3 @ bot_bo4194595901900360558c_fm_i ) ) )
= ( uniq_Epistemic_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_938_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_939_sup__Inf__absorb,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_940_eval_Osimps_I3_J,axiom,
! [G2: list_char > $o,H2: epistemic_fm_i > $o,P2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_eval_i @ G2 @ H2 @ ( epistemic_Dis_i @ P2 @ Q ) )
= ( ( epistemic_eval_i @ G2 @ H2 @ P2 )
| ( epistemic_eval_i @ G2 @ H2 @ Q ) ) ) ).
% eval.simps(3)
thf(fact_941_fm_Odistinct_I19_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_942_fm_Odistinct_I3_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i] :
( epistemic_FF_i
!= ( epistemic_Dis_i @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_943_fm_Odistinct_I21_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_944_fm_Odistinct_I23_J,axiom,
! [X31: epistemic_fm_i,X32: epistemic_fm_i,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Dis_i @ X31 @ X32 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_945_Inf__fin_OcoboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,A2: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_946_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_947_Inf__fin_Oin__idem,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( inf_in3450601097109690352c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_948_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_949_Inf__fin_Oin__idem,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X4 @ A )
=> ( ( inf_in161960956874937808c_fm_i @ X4 @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_950_Inf__fin_Obounded__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) )
= ( ! [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_951_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X4 @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_952_Inf__fin_OboundedI,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [A6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ A6 ) )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_953_Inf__fin_OboundedI,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ord_less_eq_nat @ X4 @ A6 ) )
=> ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_954_Inf__fin_OboundedE,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) )
=> ! [A8: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A8 @ A )
=> ( ord_le3843937902494030498c_fm_i @ X4 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_955_Inf__fin_OboundedE,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A )
=> ( ord_less_eq_nat @ X4 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_956_K__DisE,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,G: list_Epistemic_fm_i,R2: epistemic_fm_i,Q: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ G ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ Q @ G ) @ R2 ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ ( epistemic_Dis_i @ P2 @ Q ) ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ G @ R2 ) ) ) ) ) ).
% K_DisE
thf(fact_957_K__DisL,axiom,
! [A: epistemic_fm_i > $o,P2: epistemic_fm_i,Ps: list_Epistemic_fm_i,Q: epistemic_fm_i,P7: epistemic_fm_i] :
( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P2 @ Ps ) @ Q ) )
=> ( ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ P7 @ Ps ) @ Q ) )
=> ( epistemic_AK_i @ A @ ( epistemic_imply_i @ ( cons_Epistemic_fm_i @ ( epistemic_Dis_i @ P2 @ P7 ) @ Ps ) @ Q ) ) ) ) ).
% K_DisL
thf(fact_958_Inf__fin_Osubset__imp,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ B )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( finite7933139204641697195c_fm_i @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ B ) @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_959_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_960_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_961_Inf__fin_Osubset,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( ord_le2772500130360476258c_fm_i @ B @ A )
=> ( ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ B ) @ ( lattic7860428167660714879c_fm_i @ A ) )
= ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_962_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= ( inf_in3450601097109690352c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_963_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_964_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ~ ( member1461078328125707403c_fm_i @ X4 @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X4 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( lattic7860428167660714879c_fm_i @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_965_Inf__fin_Oclosed,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ! [X3: set_Epistemic_fm_i,Y5: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ ( inf_in3450601097109690352c_fm_i @ X3 @ Y5 ) @ ( insert7698009978809854162c_fm_i @ X3 @ ( insert7698009978809854162c_fm_i @ Y5 @ bot_bo145720340923748686c_fm_i ) ) )
=> ( member1104366573291651755c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_966_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y5: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y5 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_967_Inf__fin_Oclosed,axiom,
! [A: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ! [X3: set_se3485332733965609186c_fm_i,Y5: set_se3485332733965609186c_fm_i] : ( member1461078328125707403c_fm_i @ ( inf_in161960956874937808c_fm_i @ X3 @ Y5 ) @ ( insert7111703059596746418c_fm_i @ X3 @ ( insert7111703059596746418c_fm_i @ Y5 @ bot_bo4781621276559889198c_fm_i ) ) )
=> ( member1461078328125707403c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_968_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_969_Inf__fin_Ounion,axiom,
! [A: set_se7339729205154126530c_fm_i,B: set_se7339729205154126530c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( A != bot_bo4781621276559889198c_fm_i )
=> ( ( finite9087983574947184523c_fm_i @ B )
=> ( ( B != bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( sup_su2317974602288960150c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ ( lattic7860428167660714879c_fm_i @ A ) @ ( lattic7860428167660714879c_fm_i @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_970_Inf__fin__le__Sup__fin,axiom,
! [A: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( A != bot_bo145720340923748686c_fm_i )
=> ( ord_le3843937902494030498c_fm_i @ ( lattic8495942116038605215c_fm_i @ A ) @ ( lattic3744900211830597177c_fm_i @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_971_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_972_fm_Oexhaust,axiom,
! [Y2: epistemic_fm_i] :
( ( Y2 != epistemic_FF_i )
=> ( ! [X23: list_char] :
( Y2
!= ( epistemic_Pro_i @ X23 ) )
=> ( ! [X312: epistemic_fm_i,X322: epistemic_fm_i] :
( Y2
!= ( epistemic_Dis_i @ X312 @ X322 ) )
=> ( ! [X412: epistemic_fm_i,X422: epistemic_fm_i] :
( Y2
!= ( epistemic_Con_i @ X412 @ X422 ) )
=> ( ! [X512: epistemic_fm_i,X522: epistemic_fm_i] :
( Y2
!= ( epistemic_Imp_i @ X512 @ X522 ) )
=> ~ ! [X612: i,X622: epistemic_fm_i] :
( Y2
!= ( epistemic_K_i @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_973_fm_Odistinct_I17_J,axiom,
! [X24: list_char,X61: i,X62: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_K_i @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_974_fm_Odistinct_I15_J,axiom,
! [X24: list_char,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_975_fm_Odistinct_I1_J,axiom,
! [X24: list_char] :
( epistemic_FF_i
!= ( epistemic_Pro_i @ X24 ) ) ).
% fm.distinct(1)
thf(fact_976_fm_Odistinct_I13_J,axiom,
! [X24: list_char,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_Pro_i @ X24 )
!= ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_977_eval_Osimps_I2_J,axiom,
! [G2: list_char > $o,Uw: epistemic_fm_i > $o,X4: list_char] :
( ( epistemic_eval_i @ G2 @ Uw @ ( epistemic_Pro_i @ X4 ) )
= ( G2 @ X4 ) ) ).
% eval.simps(2)
thf(fact_978_fm_Orel__induct,axiom,
! [R3: i > i > $o,X4: epistemic_fm_i,Y2: epistemic_fm_i,Q3: epistemic_fm_i > epistemic_fm_i > $o] :
( ( epistemic_rel_fm_i_i @ R3 @ X4 @ Y2 )
=> ( ( Q3 @ epistemic_FF_i @ epistemic_FF_i )
=> ( ! [A22: list_char,B22: list_char] :
( ( A22 = B22 )
=> ( Q3 @ ( epistemic_Pro_i @ A22 ) @ ( epistemic_Pro_i @ B22 ) ) )
=> ( ! [A31: epistemic_fm_i,A32: epistemic_fm_i,B31: epistemic_fm_i,B32: epistemic_fm_i] :
( ( Q3 @ A31 @ B31 )
=> ( ( Q3 @ A32 @ B32 )
=> ( Q3 @ ( epistemic_Dis_i @ A31 @ A32 ) @ ( epistemic_Dis_i @ B31 @ B32 ) ) ) )
=> ( ! [A41: epistemic_fm_i,A42: epistemic_fm_i,B41: epistemic_fm_i,B42: epistemic_fm_i] :
( ( Q3 @ A41 @ B41 )
=> ( ( Q3 @ A42 @ B42 )
=> ( Q3 @ ( epistemic_Con_i @ A41 @ A42 ) @ ( epistemic_Con_i @ B41 @ B42 ) ) ) )
=> ( ! [A51: epistemic_fm_i,A52: epistemic_fm_i,B51: epistemic_fm_i,B52: epistemic_fm_i] :
( ( Q3 @ A51 @ B51 )
=> ( ( Q3 @ A52 @ B52 )
=> ( Q3 @ ( epistemic_Imp_i @ A51 @ A52 ) @ ( epistemic_Imp_i @ B51 @ B52 ) ) ) )
=> ( ! [A61: i,A62: epistemic_fm_i,B61: i,B62: epistemic_fm_i] :
( ( R3 @ A61 @ B61 )
=> ( ( Q3 @ A62 @ B62 )
=> ( Q3 @ ( epistemic_K_i @ A61 @ A62 ) @ ( epistemic_K_i @ B61 @ B62 ) ) ) )
=> ( Q3 @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% fm.rel_induct
thf(fact_979_fm_Orel__cases,axiom,
! [R3: i > i > $o,A2: epistemic_fm_i,B2: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ A2 @ B2 )
=> ( ( ( A2 = epistemic_FF_i )
=> ( B2 != epistemic_FF_i ) )
=> ( ! [X3: list_char] :
( ( A2
= ( epistemic_Pro_i @ X3 ) )
=> ! [Y5: list_char] :
( ( B2
= ( epistemic_Pro_i @ Y5 ) )
=> ( X3 != Y5 ) ) )
=> ( ! [X12: epistemic_fm_i,X2a: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ X12 @ X2a ) )
=> ! [Y1: epistemic_fm_i,Y2a: epistemic_fm_i] :
( ( B2
= ( epistemic_Dis_i @ Y1 @ Y2a ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X12 @ Y1 )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2a @ Y2a ) ) ) )
=> ( ! [X1a: epistemic_fm_i,X2b: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ X1a @ X2b ) )
=> ! [Y1a: epistemic_fm_i,Y2b: epistemic_fm_i] :
( ( B2
= ( epistemic_Con_i @ Y1a @ Y2b ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X1a @ Y1a )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2b @ Y2b ) ) ) )
=> ( ! [X1b: epistemic_fm_i,X2c: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ X1b @ X2c ) )
=> ! [Y1b: epistemic_fm_i,Y2c: epistemic_fm_i] :
( ( B2
= ( epistemic_Imp_i @ Y1b @ Y2c ) )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X1b @ Y1b )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2c @ Y2c ) ) ) )
=> ~ ! [X1c: i,X2d: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ X1c @ X2d ) )
=> ! [Y1c: i,Y2d: epistemic_fm_i] :
( ( B2
= ( epistemic_K_i @ Y1c @ Y2d ) )
=> ( ( R3 @ X1c @ Y1c )
=> ~ ( epistemic_rel_fm_i_i @ R3 @ X2d @ Y2d ) ) ) ) ) ) ) ) ) ) ).
% fm.rel_cases
thf(fact_980_fm_Orel__inject_I4_J,axiom,
! [R3: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) )
= ( ( epistemic_rel_fm_i_i @ R3 @ X41 @ Y41 )
& ( epistemic_rel_fm_i_i @ R3 @ X42 @ Y42 ) ) ) ).
% fm.rel_inject(4)
thf(fact_981_fm_Orel__intros_I4_J,axiom,
! [R3: i > i > $o,X41: epistemic_fm_i,Y41: epistemic_fm_i,X42: epistemic_fm_i,Y42: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ X41 @ Y41 )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X42 @ Y42 )
=> ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ) ) ).
% fm.rel_intros(4)
thf(fact_982_fm_Octr__transfer_I1_J,axiom,
! [R3: i > i > $o] : ( epistemic_rel_fm_i_i @ R3 @ epistemic_FF_i @ epistemic_FF_i ) ).
% fm.ctr_transfer(1)
thf(fact_983_fm_Orel__inject_I5_J,axiom,
! [R3: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) )
= ( ( epistemic_rel_fm_i_i @ R3 @ X51 @ Y51 )
& ( epistemic_rel_fm_i_i @ R3 @ X52 @ Y52 ) ) ) ).
% fm.rel_inject(5)
thf(fact_984_fm_Orel__intros_I5_J,axiom,
! [R3: i > i > $o,X51: epistemic_fm_i,Y51: epistemic_fm_i,X52: epistemic_fm_i,Y52: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ X51 @ Y51 )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X52 @ Y52 )
=> ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ) ) ).
% fm.rel_intros(5)
thf(fact_985_fm_Orel__inject_I6_J,axiom,
! [R3: i > i > $o,X61: i,X62: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
( ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) )
= ( ( R3 @ X61 @ Y61 )
& ( epistemic_rel_fm_i_i @ R3 @ X62 @ Y62 ) ) ) ).
% fm.rel_inject(6)
thf(fact_986_fm_Orel__intros_I6_J,axiom,
! [R3: i > i > $o,X61: i,Y61: i,X62: epistemic_fm_i,Y62: epistemic_fm_i] :
( ( R3 @ X61 @ Y61 )
=> ( ( epistemic_rel_fm_i_i @ R3 @ X62 @ Y62 )
=> ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ X61 @ X62 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ) ) ).
% fm.rel_intros(6)
thf(fact_987_fm_Orel__distinct_I29_J,axiom,
! [R3: i > i > $o,X51: epistemic_fm_i,X52: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Imp_i @ X51 @ X52 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(29)
thf(fact_988_fm_Orel__distinct_I30_J,axiom,
! [R3: i > i > $o,Y61: i,Y62: epistemic_fm_i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Imp_i @ X51 @ X52 ) ) ).
% fm.rel_distinct(30)
thf(fact_989_fm_Orel__distinct_I9_J,axiom,
! [R3: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ epistemic_FF_i @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(9)
thf(fact_990_fm_Orel__distinct_I10_J,axiom,
! [R3: i > i > $o,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ Y61 @ Y62 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(10)
thf(fact_991_fm_Orel__distinct_I7_J,axiom,
! [R3: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ epistemic_FF_i @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(7)
thf(fact_992_fm_Orel__distinct_I8_J,axiom,
! [R3: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(8)
thf(fact_993_fm_Orel__distinct_I27_J,axiom,
! [R3: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y61: i,Y62: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_K_i @ Y61 @ Y62 ) ) ).
% fm.rel_distinct(27)
thf(fact_994_fm_Orel__distinct_I28_J,axiom,
! [R3: i > i > $o,Y61: i,Y62: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_K_i @ Y61 @ Y62 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(28)
thf(fact_995_fm_Orel__distinct_I25_J,axiom,
! [R3: i > i > $o,X41: epistemic_fm_i,X42: epistemic_fm_i,Y51: epistemic_fm_i,Y52: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Con_i @ X41 @ X42 ) @ ( epistemic_Imp_i @ Y51 @ Y52 ) ) ).
% fm.rel_distinct(25)
thf(fact_996_fm_Orel__distinct_I26_J,axiom,
! [R3: i > i > $o,Y51: epistemic_fm_i,Y52: epistemic_fm_i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Imp_i @ Y51 @ Y52 ) @ ( epistemic_Con_i @ X41 @ X42 ) ) ).
% fm.rel_distinct(26)
thf(fact_997_fm_Orel__distinct_I5_J,axiom,
! [R3: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ epistemic_FF_i @ ( epistemic_Con_i @ Y41 @ Y42 ) ) ).
% fm.rel_distinct(5)
thf(fact_998_fm_Orel__distinct_I6_J,axiom,
! [R3: i > i > $o,Y41: epistemic_fm_i,Y42: epistemic_fm_i] :
~ ( epistemic_rel_fm_i_i @ R3 @ ( epistemic_Con_i @ Y41 @ Y42 ) @ epistemic_FF_i ) ).
% fm.rel_distinct(6)
thf(fact_999_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1000_Sup__fin_Oinsert__remove,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1001_Sup__fin_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X4 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1002_Sup__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= X4 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic3744900211830597177c_fm_i @ A )
= ( sup_su1936195050962291414c_fm_i @ X4 @ ( lattic3744900211830597177c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1003_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1004_Inf__fin_Oinsert__remove,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X4 @ A ) )
= X4 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X4 @ A ) )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1005_Diff__iff,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( ( member1104366573291651755c_fm_i @ C @ A )
& ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1006_Diff__iff,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( ( member6642669606046002379c_fm_i @ C @ A )
& ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1007_DiffI,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ A )
=> ( ~ ( member1104366573291651755c_fm_i @ C @ B )
=> ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1008_DiffI,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ A )
=> ( ~ ( member6642669606046002379c_fm_i @ C @ B )
=> ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1009_finite__Diff2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% finite_Diff2
thf(fact_1010_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1011_finite__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1012_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1013_Diff__insert0,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ B ) )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1014_Diff__insert0,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1015_insert__Diff1,axiom,
! [X4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1016_insert__Diff1,axiom,
! [X4: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1017_Un__Diff__cancel2,axiom,
! [B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ B @ A ) @ A )
= ( sup_su1936195050962291414c_fm_i @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1018_Un__Diff__cancel,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= ( sup_su1936195050962291414c_fm_i @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1019_Diff__eq__empty__iff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ A @ B )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1020_finite__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,A2: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ B ) ) )
= ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1021_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1022_Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ A ) )
= bot_bo145720340923748686c_fm_i ) ).
% Diff_disjoint
thf(fact_1023_insert__Diff__if,axiom,
! [X4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X4 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) @ B )
= ( minus_1073301138667672009c_fm_i @ A @ B ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X4 @ B )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) @ B )
= ( insert7698009978809854162c_fm_i @ X4 @ ( minus_1073301138667672009c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1024_insert__Diff__if,axiom,
! [X4: epistemic_fm_i,B: set_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( minus_1543973939109951465c_fm_i @ A @ B ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ B )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ B )
= ( insert7817948997695205106c_fm_i @ X4 @ ( minus_1543973939109951465c_fm_i @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1025_Diff__Int__distrib2,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ C2 )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1026_Diff__Int__distrib,axiom,
! [C2: set_se3485332733965609186c_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ C2 @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ C2 @ A ) @ ( inf_in161960956874937808c_fm_i @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_1027_Diff__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= ( inf_in161960956874937808c_fm_i @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_1028_Diff__Int2,axiom,
! [A: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_1029_Int__Diff,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ C2 )
= ( inf_in161960956874937808c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_1030_set__diff__eq,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X2 @ A4 )
& ~ ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1031_set__diff__eq,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ^ [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ A4 )
& ~ ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1032_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1033_DiffD2,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( member1104366573291651755c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1034_DiffD2,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( member6642669606046002379c_fm_i @ C @ B ) ) ).
% DiffD2
thf(fact_1035_DiffD1,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ( member1104366573291651755c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1036_DiffD1,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ( member6642669606046002379c_fm_i @ C @ A ) ) ).
% DiffD1
thf(fact_1037_DiffE,axiom,
! [C: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ C @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
=> ~ ( ( member1104366573291651755c_fm_i @ C @ A )
=> ( member1104366573291651755c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1038_DiffE,axiom,
! [C: epistemic_fm_i,A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ C @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
=> ~ ( ( member6642669606046002379c_fm_i @ C @ A )
=> ( member6642669606046002379c_fm_i @ C @ B ) ) ) ).
% DiffE
thf(fact_1039_double__diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( ord_le3843937902494030498c_fm_i @ B @ C2 )
=> ( ( minus_1543973939109951465c_fm_i @ B @ ( minus_1543973939109951465c_fm_i @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1040_Diff__subset,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1041_Diff__mono,axiom,
! [A: set_Epistemic_fm_i,C2: set_Epistemic_fm_i,D: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ C2 )
=> ( ( ord_le3843937902494030498c_fm_i @ D @ B )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1042_Diff__infinite__finite,axiom,
! [T: set_Epistemic_fm_i,S: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ T )
=> ( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1043_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1044_Un__Diff,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( sup_su1936195050962291414c_fm_i @ A @ B ) @ C2 )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1045_diff__shunt__var,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( ( minus_1543973939109951465c_fm_i @ X4 @ Y2 )
= bot_bo4194595901900360558c_fm_i )
= ( ord_le3843937902494030498c_fm_i @ X4 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1046_Diff__insert__absorb,axiom,
! [X4: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( minus_1073301138667672009c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1047_Diff__insert__absorb,axiom,
! [X4: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ( minus_1543973939109951465c_fm_i @ ( insert7817948997695205106c_fm_i @ X4 @ A ) @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1048_insert__Diff,axiom,
! [A2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ A2 @ A )
=> ( ( insert7698009978809854162c_fm_i @ A2 @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ A2 @ bot_bo145720340923748686c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1049_insert__Diff,axiom,
! [A2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A2 @ A )
=> ( ( insert7817948997695205106c_fm_i @ A2 @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1050_subset__Diff__insert,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ ( insert7698009978809854162c_fm_i @ X4 @ C2 ) ) )
= ( ( ord_le5389487502678872194c_fm_i @ A @ ( minus_1073301138667672009c_fm_i @ B @ C2 ) )
& ~ ( member1104366573291651755c_fm_i @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1051_subset__Diff__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,X4: epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ ( insert7817948997695205106c_fm_i @ X4 @ C2 ) ) )
= ( ( ord_le3843937902494030498c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ C2 ) )
& ~ ( member6642669606046002379c_fm_i @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1052_Int__Diff__disjoint,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= bot_bo145720340923748686c_fm_i ) ).
% Int_Diff_disjoint
thf(fact_1053_Diff__triv,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ A @ B )
= bot_bo145720340923748686c_fm_i )
=> ( ( minus_1073301138667672009c_fm_i @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1054_Diff__partition,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ B )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ ( minus_1543973939109951465c_fm_i @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1055_Diff__subset__conv,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ C2 )
= ( ord_le3843937902494030498c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1056_Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( sup_su2582925890723967158c_fm_i @ B @ C2 ) )
= ( inf_in161960956874937808c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1057_Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( sup_su1936195050962291414c_fm_i @ B @ C2 ) )
= ( inf_in3450601097109690352c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1058_Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,C2: set_se3485332733965609186c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( inf_in161960956874937808c_fm_i @ B @ C2 ) )
= ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1059_Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,C2: set_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( inf_in3450601097109690352c_fm_i @ B @ C2 ) )
= ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1060_Int__Diff__Un,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( inf_in161960956874937808c_fm_i @ A @ B ) @ ( minus_1073301138667672009c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1061_Int__Diff__Un,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( inf_in3450601097109690352c_fm_i @ A @ B ) @ ( minus_1543973939109951465c_fm_i @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1062_Un__Diff__Int,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( sup_su2582925890723967158c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ B ) @ ( inf_in161960956874937808c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1063_Un__Diff__Int,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ B ) @ ( inf_in3450601097109690352c_fm_i @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1064_infinite__remove,axiom,
! [S: set_Epistemic_fm_i,A2: epistemic_fm_i] :
( ~ ( finite3304564979551393739c_fm_i @ S )
=> ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ S @ ( insert7817948997695205106c_fm_i @ A2 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% infinite_remove
thf(fact_1065_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1066_infinite__coinduct,axiom,
! [X6: set_Epistemic_fm_i > $o,A: set_Epistemic_fm_i] :
( ( X6 @ A )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( X6 @ A7 )
=> ? [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
& ( ( X6 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) )
| ~ ( finite3304564979551393739c_fm_i @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ~ ( finite3304564979551393739c_fm_i @ A ) ) ) ).
% infinite_coinduct
thf(fact_1067_infinite__coinduct,axiom,
! [X6: set_nat > $o,A: set_nat] :
( ( X6 @ A )
=> ( ! [A7: set_nat] :
( ( X6 @ A7 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A7 )
& ( ( X6 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1068_finite__empty__induct,axiom,
! [A: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( P5 @ A )
=> ( ! [A6: set_Epistemic_fm_i,A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( member1104366573291651755c_fm_i @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ A6 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) )
=> ( P5 @ bot_bo145720340923748686c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1069_finite__empty__induct,axiom,
! [A: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( P5 @ A )
=> ( ! [A6: epistemic_fm_i,A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( member6642669606046002379c_fm_i @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ A6 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ) )
=> ( P5 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% finite_empty_induct
thf(fact_1070_finite__empty__induct,axiom,
! [A: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P5 @ A )
=> ( ! [A6: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A6 @ A7 )
=> ( ( P5 @ A7 )
=> ( P5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P5 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1071_Diff__single__insert,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) @ B )
=> ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1072_subset__insert__iff,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i] :
( ( ord_le5389487502678872194c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ B ) )
= ( ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ord_le5389487502678872194c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) ) @ B ) )
& ( ~ ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ord_le5389487502678872194c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1073_subset__insert__iff,axiom,
! [A: set_Epistemic_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ B ) )
= ( ( ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ ( minus_1543973939109951465c_fm_i @ A @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) @ B ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ A )
=> ( ord_le3843937902494030498c_fm_i @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1074_minus__coset__filter,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( minus_1073301138667672009c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Xs ) )
= ( set_se200842218512397079c_fm_i
@ ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1075_minus__coset__filter,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( coset_Epistemic_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_1076_finite__remove__induct,axiom,
! [B: set_se3485332733965609186c_fm_i,P5: set_se3485332733965609186c_fm_i > $o] :
( ( finite7933139204641697195c_fm_i @ B )
=> ( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B )
=> ( ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X5 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1077_finite__remove__induct,axiom,
! [B: set_nat,P5: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P5 @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( P5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1078_finite__remove__induct,axiom,
! [B: set_Epistemic_fm_i,P5: set_Epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ B )
=> ( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1079_remove__induct,axiom,
! [P5: set_se3485332733965609186c_fm_i > $o,B: set_se3485332733965609186c_fm_i] :
( ( P5 @ bot_bo145720340923748686c_fm_i )
=> ( ( ~ ( finite7933139204641697195c_fm_i @ B )
=> ( P5 @ B ) )
=> ( ! [A7: set_se3485332733965609186c_fm_i] :
( ( finite7933139204641697195c_fm_i @ A7 )
=> ( ( A7 != bot_bo145720340923748686c_fm_i )
=> ( ( ord_le5389487502678872194c_fm_i @ A7 @ B )
=> ( ! [X5: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ X5 @ A7 )
=> ( P5 @ ( minus_1073301138667672009c_fm_i @ A7 @ ( insert7698009978809854162c_fm_i @ X5 @ bot_bo145720340923748686c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% remove_induct
thf(fact_1080_remove__induct,axiom,
! [P5: set_nat > $o,B: set_nat] :
( ( P5 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P5 @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( P5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% remove_induct
thf(fact_1081_remove__induct,axiom,
! [P5: set_Epistemic_fm_i > $o,B: set_Epistemic_fm_i] :
( ( P5 @ bot_bo4194595901900360558c_fm_i )
=> ( ( ~ ( finite3304564979551393739c_fm_i @ B )
=> ( P5 @ B ) )
=> ( ! [A7: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A7 )
=> ( ( A7 != bot_bo4194595901900360558c_fm_i )
=> ( ( ord_le3843937902494030498c_fm_i @ A7 @ B )
=> ( ! [X5: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X5 @ A7 )
=> ( P5 @ ( minus_1543973939109951465c_fm_i @ A7 @ ( insert7817948997695205106c_fm_i @ X5 @ bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( P5 @ A7 ) ) ) ) )
=> ( P5 @ B ) ) ) ) ).
% remove_induct
thf(fact_1082_set__minus__filter__out,axiom,
! [Xs: list_Epistemic_fm_i,Y2: epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ Y2 @ bot_bo4194595901900360558c_fm_i ) )
= ( set_Epistemic_fm_i2
@ ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] : ( X2 != Y2 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1083_Inf__fin_Oremove,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( member1104366573291651755c_fm_i @ X4 @ A )
=> ( ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= X4 ) )
& ( ( ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) )
!= bot_bo145720340923748686c_fm_i )
=> ( ( lattic8495942116038605215c_fm_i @ A )
= ( inf_in3450601097109690352c_fm_i @ X4 @ ( lattic8495942116038605215c_fm_i @ ( minus_1073301138667672009c_fm_i @ A @ ( insert7698009978809854162c_fm_i @ X4 @ bot_bo145720340923748686c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1084_Inf__fin_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X4 ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X4 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1085_Inf__fin_Oremove,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( member1461078328125707403c_fm_i @ X4 @ A )
=> ( ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) )
= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= X4 ) )
& ( ( ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) )
!= bot_bo4781621276559889198c_fm_i )
=> ( ( lattic7860428167660714879c_fm_i @ A )
= ( inf_in161960956874937808c_fm_i @ X4 @ ( lattic7860428167660714879c_fm_i @ ( minus_8476896128438495145c_fm_i @ A @ ( insert7111703059596746418c_fm_i @ X4 @ bot_bo4781621276559889198c_fm_i ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1086_set__removeAll,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X4 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ).
% set_removeAll
thf(fact_1087_removeAll__id,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remove6226062834164189966c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1088_removeAll__id,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove817376155714978094c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1089_removeAll__filter__not,axiom,
! [P5: epistemic_fm_i > $o,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( P5 @ X4 )
=> ( ( remove817376155714978094c_fm_i @ X4 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_1090_minus__set__def,axiom,
( minus_1073301138667672009c_fm_i
= ( ^ [A4: set_se3485332733965609186c_fm_i,B3: set_se3485332733965609186c_fm_i] :
( collec3087743281813070829c_fm_i
@ ( minus_5671197887077578836fm_i_o
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ A4 )
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1091_minus__set__def,axiom,
( minus_1543973939109951465c_fm_i
= ( ^ [A4: set_Epistemic_fm_i,B3: set_Epistemic_fm_i] :
( collec4904205187116291597c_fm_i
@ ( minus_6519238007984220724fm_i_o
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ A4 )
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1092_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_1093_removeAll__filter__not__eq,axiom,
( remove817376155714978094c_fm_i
= ( ^ [X2: epistemic_fm_i] :
( filter7636273843821131039c_fm_i
@ ^ [Y4: epistemic_fm_i] : ( X2 != Y4 ) ) ) ) ).
% removeAll_filter_not_eq
thf(fact_1094_List_Oset__insert,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( insert177310195918387262c_fm_i @ X4 @ Xs ) )
= ( insert7817948997695205106c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1095_member__remove,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( remove1392734035364174951c_fm_i @ Y2 @ A ) )
= ( ( member1104366573291651755c_fm_i @ X4 @ A )
& ( X4 != Y2 ) ) ) ).
% member_remove
thf(fact_1096_member__remove,axiom,
! [X4: epistemic_fm_i,Y2: epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( remove5552923401211475591c_fm_i @ Y2 @ A ) )
= ( ( member6642669606046002379c_fm_i @ X4 @ A )
& ( X4 != Y2 ) ) ) ).
% member_remove
thf(fact_1097_in__set__insert,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1098_in__set__insert,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1099_not__in__set__insert,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( insert2436375702790685214c_fm_i @ X4 @ Xs )
= ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1100_not__in__set__insert,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( insert177310195918387262c_fm_i @ X4 @ Xs )
= ( cons_Epistemic_fm_i @ X4 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1101_List_Oinsert__def,axiom,
( insert2436375702790685214c_fm_i
= ( ^ [X2: set_Epistemic_fm_i,Xs2: list_s8081015415394010888c_fm_i] : ( if_lis2088983353317256142c_fm_i @ ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs2 ) ) @ Xs2 @ ( cons_s4962720389763977656c_fm_i @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_1102_List_Oinsert__def,axiom,
( insert177310195918387262c_fm_i
= ( ^ [X2: epistemic_fm_i,Xs2: list_Epistemic_fm_i] : ( if_lis3446954105117697518c_fm_i @ ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs2 ) ) @ Xs2 @ ( cons_Epistemic_fm_i @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_1103_remove__code_I1_J,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( remove5552923401211475591c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( set_Epistemic_fm_i2 @ ( remove817376155714978094c_fm_i @ X4 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1104_is__empty__set,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( is_emp7643256682846039796c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( null_Epistemic_fm_i @ Xs ) ) ).
% is_empty_set
thf(fact_1105_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1106_filter__shuffles__disjoint2_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs3 @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_1107_set__shuffles,axiom,
! [Zs3: list_Epistemic_fm_i,Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( set_Epistemic_fm_i2 @ Zs3 )
= ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1108_partition__in__shuffles,axiom,
! [Xs: list_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( member6475149752759350353c_fm_i @ Xs
@ ( shuffl511019056595460925c_fm_i @ ( filter7636273843821131039c_fm_i @ P5 @ Xs )
@ ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] :
~ ( P5 @ X2 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_1109_filter__shuffles__disjoint1_I2_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_1110_filter__shuffles__disjoint1_I2_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs3 @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
@ Zs3 )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_1111_filter__shuffles__disjoint1_I1_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] : ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_1112_filter__shuffles__disjoint1_I1_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs3 @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] : ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Xs ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_1113_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( ( filter7636273843821131039c_fm_i
@ ^ [X2: epistemic_fm_i] :
~ ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Ys ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1114_filter__shuffles__disjoint2_I2_J,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs3 @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( ( filter3188398074982218495c_fm_i
@ ^ [X2: set_Epistemic_fm_i] :
~ ( member1104366573291651755c_fm_i @ X2 @ ( set_se200842218512397079c_fm_i @ Ys ) )
@ Zs3 )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_1115_Set__filter__fold,axiom,
! [A: set_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( filter6053540608743173075c_fm_i @ P5 @ A )
= ( finite6274917443777430184c_fm_i
@ ^ [X2: epistemic_fm_i,A9: set_Epistemic_fm_i] : ( if_set2943457721768014536c_fm_i @ ( P5 @ X2 ) @ ( insert7817948997695205106c_fm_i @ X2 @ A9 ) @ A9 )
@ bot_bo4194595901900360558c_fm_i
@ A ) ) ) ).
% Set_filter_fold
thf(fact_1116_Set__filter__fold,axiom,
! [A: set_nat,P5: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( filter_nat2 @ P5 @ A )
= ( finite5529483035118572448et_nat
@ ^ [X2: nat,A9: set_nat] : ( if_set_nat @ ( P5 @ X2 ) @ ( insert_nat @ X2 @ A9 ) @ A9 )
@ bot_bot_set_nat
@ A ) ) ) ).
% Set_filter_fold
thf(fact_1117_fm_Oset__cases,axiom,
! [E: set_Epistemic_fm_i,A2: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ A2 ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte535002356104313075c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte535002356104313075c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte64184772710425087c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte64184772710425087c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i] :
( ? [Z22: episte4276422356747265950c_fm_i] :
( A2
= ( episte5768433346120044277c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte4276422356747265950c_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte5768433346120044277c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) )
=> ( ! [Z22: episte4276422356747265950c_fm_i] :
( A2
!= ( episte3581967596712343542c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: set_Epistemic_fm_i,Z22: episte4276422356747265950c_fm_i] :
( ( A2
= ( episte3581967596712343542c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member1104366573291651755c_fm_i @ E @ ( episte6795552051664633144c_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1118_fm_Oset__cases,axiom,
! [E: epistemic_fm_i,A2: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ A2 ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte6088726798904853267c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte3685526521632971807c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i] :
( ? [Z22: episte1308613106011496894c_fm_i] :
( A2
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z1 ) ) )
=> ( ! [Z1: episte1308613106011496894c_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte260752253203357973c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) )
=> ( ! [Z22: episte1308613106011496894c_fm_i] :
( A2
!= ( episte5657488666450005526c_fm_i @ E @ Z22 ) )
=> ~ ! [Z1: epistemic_fm_i,Z22: episte1308613106011496894c_fm_i] :
( ( A2
= ( episte5657488666450005526c_fm_i @ Z1 @ Z22 ) )
=> ~ ( member6642669606046002379c_fm_i @ E @ ( episte9089240992906287960c_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1119_fm_Oset__cases,axiom,
! [E: i,A2: epistemic_fm_i] :
( ( member_i @ E @ ( epistemic_set_fm_i @ A2 ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Dis_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Con_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z1: epistemic_fm_i] :
( ? [Z22: epistemic_fm_i] :
( A2
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z1 ) ) )
=> ( ! [Z1: epistemic_fm_i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_Imp_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) )
=> ( ! [Z22: epistemic_fm_i] :
( A2
!= ( epistemic_K_i @ E @ Z22 ) )
=> ~ ! [Z1: i,Z22: epistemic_fm_i] :
( ( A2
= ( epistemic_K_i @ Z1 @ Z22 ) )
=> ~ ( member_i @ E @ ( epistemic_set_fm_i @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fm.set_cases
thf(fact_1120_fm_Osimps_I95_J,axiom,
! [X61: i,X62: epistemic_fm_i] :
( ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) )
= ( insert_i @ X61 @ ( epistemic_set_fm_i @ X62 ) ) ) ).
% fm.simps(95)
thf(fact_1121_fm_Osimps_I90_J,axiom,
( ( epistemic_set_fm_i @ epistemic_FF_i )
= bot_bot_set_i ) ).
% fm.simps(90)
thf(fact_1122_fm_Osimps_I94_J,axiom,
! [X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( epistemic_set_fm_i @ ( epistemic_Imp_i @ X51 @ X52 ) )
= ( sup_sup_set_i @ ( epistemic_set_fm_i @ X51 ) @ ( epistemic_set_fm_i @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_1123_fm_Osimps_I94_J,axiom,
! [X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X51 ) @ ( episte9089240992906287960c_fm_i @ X52 ) ) ) ).
% fm.simps(94)
thf(fact_1124_fm_Oset__intros_I8_J,axiom,
! [Yg: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i,X61: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Yg @ ( episte6795552051664633144c_fm_i @ X62 ) )
=> ( member1104366573291651755c_fm_i @ Yg @ ( episte6795552051664633144c_fm_i @ ( episte3581967596712343542c_fm_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1125_fm_Oset__intros_I8_J,axiom,
! [Yg: epistemic_fm_i,X62: episte1308613106011496894c_fm_i,X61: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ X62 ) )
=> ( member6642669606046002379c_fm_i @ Yg @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1126_fm_Oset__intros_I8_J,axiom,
! [Yg: i,X62: epistemic_fm_i,X61: i] :
( ( member_i @ Yg @ ( epistemic_set_fm_i @ X62 ) )
=> ( member_i @ Yg @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ) ).
% fm.set_intros(8)
thf(fact_1127_fm_Oset__intros_I7_J,axiom,
! [X61: set_Epistemic_fm_i,X62: episte4276422356747265950c_fm_i] : ( member1104366573291651755c_fm_i @ X61 @ ( episte6795552051664633144c_fm_i @ ( episte3581967596712343542c_fm_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1128_fm_Oset__intros_I7_J,axiom,
! [X61: epistemic_fm_i,X62: episte1308613106011496894c_fm_i] : ( member6642669606046002379c_fm_i @ X61 @ ( episte9089240992906287960c_fm_i @ ( episte5657488666450005526c_fm_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1129_fm_Oset__intros_I7_J,axiom,
! [X61: i,X62: epistemic_fm_i] : ( member_i @ X61 @ ( epistemic_set_fm_i @ ( epistemic_K_i @ X61 @ X62 ) ) ) ).
% fm.set_intros(7)
thf(fact_1130_fold__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite2237691025846545608c_fm_i @ F @ Z3 @ A )
= ( finite2237691025846545608c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1131_fold__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i] :
( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite4171219623613778664c_fm_i @ F @ Z3 @ A )
= ( finite4171219623613778664c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1132_fold__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite6274917443777430184c_fm_i @ F @ Z3 @ A )
= ( finite6274917443777430184c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1133_fold__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i] :
( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite2135258021436235976c_fm_i @ F @ Z3 @ A )
= ( finite2135258021436235976c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_closed_eq
thf(fact_1134_fm_Oset__intros_I4_J,axiom,
! [Yd: set_Epistemic_fm_i,X42: episte4276422356747265950c_fm_i,X41: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yd @ ( episte6795552051664633144c_fm_i @ X42 ) )
=> ( member1104366573291651755c_fm_i @ Yd @ ( episte6795552051664633144c_fm_i @ ( episte64184772710425087c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_1135_fm_Oset__intros_I4_J,axiom,
! [Yd: epistemic_fm_i,X42: episte1308613106011496894c_fm_i,X41: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ X42 ) )
=> ( member6642669606046002379c_fm_i @ Yd @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_1136_fm_Oset__intros_I4_J,axiom,
! [Yd: i,X42: epistemic_fm_i,X41: epistemic_fm_i] :
( ( member_i @ Yd @ ( epistemic_set_fm_i @ X42 ) )
=> ( member_i @ Yd @ ( epistemic_set_fm_i @ ( epistemic_Con_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(4)
thf(fact_1137_fm_Oset__intros_I3_J,axiom,
! [Yc: set_Epistemic_fm_i,X41: episte4276422356747265950c_fm_i,X42: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yc @ ( episte6795552051664633144c_fm_i @ X41 ) )
=> ( member1104366573291651755c_fm_i @ Yc @ ( episte6795552051664633144c_fm_i @ ( episte64184772710425087c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_1138_fm_Oset__intros_I3_J,axiom,
! [Yc: epistemic_fm_i,X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ X41 ) )
=> ( member6642669606046002379c_fm_i @ Yc @ ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_1139_fm_Oset__intros_I3_J,axiom,
! [Yc: i,X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( member_i @ Yc @ ( epistemic_set_fm_i @ X41 ) )
=> ( member_i @ Yc @ ( epistemic_set_fm_i @ ( epistemic_Con_i @ X41 @ X42 ) ) ) ) ).
% fm.set_intros(3)
thf(fact_1140_fm_Oset__intros_I6_J,axiom,
! [Yf: set_Epistemic_fm_i,X52: episte4276422356747265950c_fm_i,X51: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yf @ ( episte6795552051664633144c_fm_i @ X52 ) )
=> ( member1104366573291651755c_fm_i @ Yf @ ( episte6795552051664633144c_fm_i @ ( episte5768433346120044277c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_1141_fm_Oset__intros_I6_J,axiom,
! [Yf: epistemic_fm_i,X52: episte1308613106011496894c_fm_i,X51: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ X52 ) )
=> ( member6642669606046002379c_fm_i @ Yf @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_1142_fm_Oset__intros_I6_J,axiom,
! [Yf: i,X52: epistemic_fm_i,X51: epistemic_fm_i] :
( ( member_i @ Yf @ ( epistemic_set_fm_i @ X52 ) )
=> ( member_i @ Yf @ ( epistemic_set_fm_i @ ( epistemic_Imp_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(6)
thf(fact_1143_fm_Oset__intros_I5_J,axiom,
! [Ye: set_Epistemic_fm_i,X51: episte4276422356747265950c_fm_i,X52: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Ye @ ( episte6795552051664633144c_fm_i @ X51 ) )
=> ( member1104366573291651755c_fm_i @ Ye @ ( episte6795552051664633144c_fm_i @ ( episte5768433346120044277c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_1144_fm_Oset__intros_I5_J,axiom,
! [Ye: epistemic_fm_i,X51: episte1308613106011496894c_fm_i,X52: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ X51 ) )
=> ( member6642669606046002379c_fm_i @ Ye @ ( episte9089240992906287960c_fm_i @ ( episte260752253203357973c_fm_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_1145_fm_Oset__intros_I5_J,axiom,
! [Ye: i,X51: epistemic_fm_i,X52: epistemic_fm_i] :
( ( member_i @ Ye @ ( epistemic_set_fm_i @ X51 ) )
=> ( member_i @ Ye @ ( epistemic_set_fm_i @ ( epistemic_Imp_i @ X51 @ X52 ) ) ) ) ).
% fm.set_intros(5)
thf(fact_1146_fm_Oset__intros_I1_J,axiom,
! [Ya: set_Epistemic_fm_i,X31: episte4276422356747265950c_fm_i,X32: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Ya @ ( episte6795552051664633144c_fm_i @ X31 ) )
=> ( member1104366573291651755c_fm_i @ Ya @ ( episte6795552051664633144c_fm_i @ ( episte535002356104313075c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_1147_fm_Oset__intros_I1_J,axiom,
! [Ya: epistemic_fm_i,X31: episte1308613106011496894c_fm_i,X32: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ X31 ) )
=> ( member6642669606046002379c_fm_i @ Ya @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(1)
thf(fact_1148_fm_Oset__intros_I2_J,axiom,
! [Yb: set_Epistemic_fm_i,X32: episte4276422356747265950c_fm_i,X31: episte4276422356747265950c_fm_i] :
( ( member1104366573291651755c_fm_i @ Yb @ ( episte6795552051664633144c_fm_i @ X32 ) )
=> ( member1104366573291651755c_fm_i @ Yb @ ( episte6795552051664633144c_fm_i @ ( episte535002356104313075c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_1149_fm_Oset__intros_I2_J,axiom,
! [Yb: epistemic_fm_i,X32: episte1308613106011496894c_fm_i,X31: episte1308613106011496894c_fm_i] :
( ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ X32 ) )
=> ( member6642669606046002379c_fm_i @ Yb @ ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) ) ) ) ).
% fm.set_intros(2)
thf(fact_1150_fm_Orel__refl__strong,axiom,
! [X4: episte4276422356747265950c_fm_i,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ! [Z4: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte6795552051664633144c_fm_i @ X4 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte2362472030976073184c_fm_i @ Ra @ X4 @ X4 ) ) ).
% fm.rel_refl_strong
thf(fact_1151_fm_Orel__refl__strong,axiom,
! [X4: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ! [Z4: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte9089240992906287960c_fm_i @ X4 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X4 @ X4 ) ) ).
% fm.rel_refl_strong
thf(fact_1152_fm_Orel__mono__strong,axiom,
! [R3: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,X4: episte4276422356747265950c_fm_i,Y2: episte4276422356747265950c_fm_i,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( episte2362472030976073184c_fm_i @ R3 @ X4 @ Y2 )
=> ( ! [Z4: set_Epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte6795552051664633144c_fm_i @ X4 ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Y2 ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte2362472030976073184c_fm_i @ Ra @ X4 @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_1153_fm_Orel__mono__strong,axiom,
! [R3: set_Epistemic_fm_i > epistemic_fm_i > $o,X4: episte4276422356747265950c_fm_i,Y2: episte1308613106011496894c_fm_i,Ra: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ( episte3976198365643398144c_fm_i @ R3 @ X4 @ Y2 )
=> ( ! [Z4: set_Epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte6795552051664633144c_fm_i @ X4 ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Y2 ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte3976198365643398144c_fm_i @ Ra @ X4 @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_1154_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_i > set_Epistemic_fm_i > $o,X4: episte1308613106011496894c_fm_i,Y2: episte4276422356747265950c_fm_i,Ra: epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( episte6079896185807049664c_fm_i @ R3 @ X4 @ Y2 )
=> ( ! [Z4: epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte9089240992906287960c_fm_i @ X4 ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Y2 ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte6079896185807049664c_fm_i @ Ra @ X4 @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_1155_fm_Orel__mono__strong,axiom,
! [R3: epistemic_fm_i > epistemic_fm_i > $o,X4: episte1308613106011496894c_fm_i,Y2: episte1308613106011496894c_fm_i,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( episte3445413862081925600c_fm_i @ R3 @ X4 @ Y2 )
=> ( ! [Z4: epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte9089240992906287960c_fm_i @ X4 ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Y2 ) )
=> ( ( R3 @ Z4 @ Yb2 )
=> ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( episte3445413862081925600c_fm_i @ Ra @ X4 @ Y2 ) ) ) ).
% fm.rel_mono_strong
thf(fact_1156_fm_Orel__cong,axiom,
! [X4: episte4276422356747265950c_fm_i,Ya: episte4276422356747265950c_fm_i,Y2: episte4276422356747265950c_fm_i,Xa: episte4276422356747265950c_fm_i,R3: set_Epistemic_fm_i > set_Epistemic_fm_i > $o,Ra: set_Epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( X4 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: set_Epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte6795552051664633144c_fm_i @ Ya ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte2362472030976073184c_fm_i @ R3 @ X4 @ Y2 )
= ( episte2362472030976073184c_fm_i @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1157_fm_Orel__cong,axiom,
! [X4: episte4276422356747265950c_fm_i,Ya: episte4276422356747265950c_fm_i,Y2: episte1308613106011496894c_fm_i,Xa: episte1308613106011496894c_fm_i,R3: set_Epistemic_fm_i > epistemic_fm_i > $o,Ra: set_Epistemic_fm_i > epistemic_fm_i > $o] :
( ( X4 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: set_Epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ Z4 @ ( episte6795552051664633144c_fm_i @ Ya ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte3976198365643398144c_fm_i @ R3 @ X4 @ Y2 )
= ( episte3976198365643398144c_fm_i @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1158_fm_Orel__cong,axiom,
! [X4: episte1308613106011496894c_fm_i,Ya: episte1308613106011496894c_fm_i,Y2: episte4276422356747265950c_fm_i,Xa: episte4276422356747265950c_fm_i,R3: epistemic_fm_i > set_Epistemic_fm_i > $o,Ra: epistemic_fm_i > set_Epistemic_fm_i > $o] :
( ( X4 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: epistemic_fm_i,Yb2: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte9089240992906287960c_fm_i @ Ya ) )
=> ( ( member1104366573291651755c_fm_i @ Yb2 @ ( episte6795552051664633144c_fm_i @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte6079896185807049664c_fm_i @ R3 @ X4 @ Y2 )
= ( episte6079896185807049664c_fm_i @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1159_fm_Orel__cong,axiom,
! [X4: episte1308613106011496894c_fm_i,Ya: episte1308613106011496894c_fm_i,Y2: episte1308613106011496894c_fm_i,Xa: episte1308613106011496894c_fm_i,R3: epistemic_fm_i > epistemic_fm_i > $o,Ra: epistemic_fm_i > epistemic_fm_i > $o] :
( ( X4 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: epistemic_fm_i,Yb2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ Z4 @ ( episte9089240992906287960c_fm_i @ Ya ) )
=> ( ( member6642669606046002379c_fm_i @ Yb2 @ ( episte9089240992906287960c_fm_i @ Xa ) )
=> ( ( R3 @ Z4 @ Yb2 )
= ( Ra @ Z4 @ Yb2 ) ) ) )
=> ( ( episte3445413862081925600c_fm_i @ R3 @ X4 @ Y2 )
= ( episte3445413862081925600c_fm_i @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fm.rel_cong
thf(fact_1160_fm_Osimps_I93_J,axiom,
! [X41: epistemic_fm_i,X42: epistemic_fm_i] :
( ( epistemic_set_fm_i @ ( epistemic_Con_i @ X41 @ X42 ) )
= ( sup_sup_set_i @ ( epistemic_set_fm_i @ X41 ) @ ( epistemic_set_fm_i @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_1161_fm_Osimps_I93_J,axiom,
! [X41: episte1308613106011496894c_fm_i,X42: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte3685526521632971807c_fm_i @ X41 @ X42 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X41 ) @ ( episte9089240992906287960c_fm_i @ X42 ) ) ) ).
% fm.simps(93)
thf(fact_1162_fm_Osimps_I92_J,axiom,
! [X31: episte1308613106011496894c_fm_i,X32: episte1308613106011496894c_fm_i] :
( ( episte9089240992906287960c_fm_i @ ( episte6088726798904853267c_fm_i @ X31 @ X32 ) )
= ( sup_su1936195050962291414c_fm_i @ ( episte9089240992906287960c_fm_i @ X31 ) @ ( episte9089240992906287960c_fm_i @ X32 ) ) ) ).
% fm.simps(92)
thf(fact_1163_union__fold__insert,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= ( finite5529483035118572448et_nat @ insert_nat @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_1164_union__fold__insert,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( sup_su1936195050962291414c_fm_i @ A @ B )
= ( finite6274917443777430184c_fm_i @ insert7817948997695205106c_fm_i @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_1165_minus__fold__remove,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( ( minus_1543973939109951465c_fm_i @ B @ A )
= ( finite6274917443777430184c_fm_i @ remove5552923401211475591c_fm_i @ B @ A ) ) ) ).
% minus_fold_remove
thf(fact_1166_minus__fold__remove,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( minus_minus_set_nat @ B @ A )
= ( finite5529483035118572448et_nat @ remove_nat @ B @ A ) ) ) ).
% minus_fold_remove
thf(fact_1167_Inf__fin_Oeq__fold,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X4 @ A ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X4 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1168_Inf__fin_Oeq__fold,axiom,
! [A: set_se7339729205154126530c_fm_i,X4: set_se3485332733965609186c_fm_i] :
( ( finite9087983574947184523c_fm_i @ A )
=> ( ( lattic7860428167660714879c_fm_i @ ( insert7111703059596746418c_fm_i @ X4 @ A ) )
= ( finite436508220448094408c_fm_i @ inf_in161960956874937808c_fm_i @ X4 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1169_Sup__fin_Oeq__fold,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X4 @ A ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X4 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1170_Sup__fin_Oeq__fold,axiom,
! [A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i] :
( ( finite7933139204641697195c_fm_i @ A )
=> ( ( lattic3744900211830597177c_fm_i @ ( insert7698009978809854162c_fm_i @ X4 @ A ) )
= ( finite2237691025846545608c_fm_i @ sup_su1936195050962291414c_fm_i @ X4 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1171_fold__graph__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite5960466103032163910c_fm_i @ F @ Z3 @ A )
= ( finite5960466103032163910c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1172_fold__graph__closed__eq,axiom,
! [A: set_se3485332733965609186c_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i] :
( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite969542861950010470c_fm_i @ F @ Z3 @ A )
= ( finite969542861950010470c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1173_fold__graph__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,G2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i] :
( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite3073240682113661990c_fm_i @ F @ Z3 @ A )
= ( finite3073240682113661990c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1174_fold__graph__closed__eq,axiom,
! [A: set_Epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i] :
( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite4898207529229619782c_fm_i @ F @ Z3 @ A )
= ( finite4898207529229619782c_fm_i @ G2 @ Z3 @ A ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_1175_fold__graph__closed__lemma,axiom,
! [G2: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F: set_Epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( finite5960466103032163910c_fm_i @ G2 @ Z3 @ A @ X4 )
=> ( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite5960466103032163910c_fm_i @ F @ Z3 @ A @ X4 )
& ( member1104366573291651755c_fm_i @ X4 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1176_fold__graph__closed__lemma,axiom,
! [G2: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i,A: set_se3485332733965609186c_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i,F: set_Epistemic_fm_i > epistemic_fm_i > epistemic_fm_i] :
( ( finite969542861950010470c_fm_i @ G2 @ Z3 @ A @ X4 )
=> ( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: set_Epistemic_fm_i,B6: epistemic_fm_i] :
( ( member1104366573291651755c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite969542861950010470c_fm_i @ F @ Z3 @ A @ X4 )
& ( member6642669606046002379c_fm_i @ X4 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1177_fold__graph__closed__lemma,axiom,
! [G2: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i,Z3: set_Epistemic_fm_i,A: set_Epistemic_fm_i,X4: set_Epistemic_fm_i,B: set_se3485332733965609186c_fm_i,F: epistemic_fm_i > set_Epistemic_fm_i > set_Epistemic_fm_i] :
( ( finite3073240682113661990c_fm_i @ G2 @ Z3 @ A @ X4 )
=> ( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: set_Epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member1104366573291651755c_fm_i @ B6 @ B )
=> ( member1104366573291651755c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member1104366573291651755c_fm_i @ Z3 @ B )
=> ( ( finite3073240682113661990c_fm_i @ F @ Z3 @ A @ X4 )
& ( member1104366573291651755c_fm_i @ X4 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1178_fold__graph__closed__lemma,axiom,
! [G2: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i,Z3: epistemic_fm_i,A: set_Epistemic_fm_i,X4: epistemic_fm_i,B: set_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i > epistemic_fm_i] :
( ( finite4898207529229619782c_fm_i @ G2 @ Z3 @ A @ X4 )
=> ( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( ( F @ A6 @ B6 )
= ( G2 @ A6 @ B6 ) ) ) )
=> ( ! [A6: epistemic_fm_i,B6: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ A6 @ A )
=> ( ( member6642669606046002379c_fm_i @ B6 @ B )
=> ( member6642669606046002379c_fm_i @ ( G2 @ A6 @ B6 ) @ B ) ) )
=> ( ( member6642669606046002379c_fm_i @ Z3 @ B )
=> ( ( finite4898207529229619782c_fm_i @ F @ Z3 @ A @ X4 )
& ( member6642669606046002379c_fm_i @ X4 @ B ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_1179_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1180_finite__lists__length__le,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_s3357407047445144980c_fm_i @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1181_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ( size_size_list_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1182_finite__lists__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs2: list_Epistemic_fm_i] :
( ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs2 ) @ A )
& ( ( size_s3357407047445144980c_fm_i @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1183_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1184_length__map,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( size_s3357407047445144980c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) )
= ( size_s3357407047445144980c_fm_i @ Xs ) ) ).
% length_map
thf(fact_1185_map__eq__imp__length__eq,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,G2: epistemic_fm_i > epistemic_fm_i,Ys: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= ( map_Ep2755178516647988292c_fm_i @ G2 @ Ys ) )
=> ( ( size_s3357407047445144980c_fm_i @ Xs )
= ( size_s3357407047445144980c_fm_i @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1186_length__filter__le,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] : ( ord_less_eq_nat @ ( size_s3357407047445144980c_fm_i @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) @ ( size_s3357407047445144980c_fm_i @ Xs ) ) ).
% length_filter_le
thf(fact_1187_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1188_set__n__lists,axiom,
! [N: nat,Xs: list_Epistemic_fm_i] :
( ( set_li9010496130497932989c_fm_i @ ( n_list3185984609264368688c_fm_i @ N @ Xs ) )
= ( collec5759350117362652307c_fm_i
@ ^ [Ys2: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Ys2 )
= N )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Ys2 ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1189_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1190_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1191_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1192_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1193_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1194_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1195_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1196_Nat_Oex__has__greatest__nat,axiom,
! [P5: nat > $o,K: nat,B2: nat] :
( ( P5 @ K )
=> ( ! [Y5: nat] :
( ( P5 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X3: nat] :
( ( P5 @ X3 )
& ! [Y3: nat] :
( ( P5 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1197_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1198_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1199_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1200_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1201_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1202_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_1203_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1204_bounded__Max__nat,axiom,
! [P5: nat > $o,X4: nat,M3: nat] :
( ( P5 @ X4 )
=> ( ! [X3: nat] :
( ( P5 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P5 @ M4 )
=> ~ ! [X5: nat] :
( ( P5 @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1205_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
& ( distinct_nat @ Xs2 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1206_finite__lists__distinct__length__eq,axiom,
! [A: set_Epistemic_fm_i,N: nat] :
( ( finite3304564979551393739c_fm_i @ A )
=> ( finite3106398504192302929c_fm_i
@ ( collec5759350117362652307c_fm_i
@ ^ [Xs2: list_Epistemic_fm_i] :
( ( ( size_s3357407047445144980c_fm_i @ Xs2 )
= N )
& ( distin5379319190628067867c_fm_i @ Xs2 )
& ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ Xs2 ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_1207_distinct__filter,axiom,
! [Xs: list_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( distin5379319190628067867c_fm_i @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ).
% distinct_filter
thf(fact_1208_distinct__map__filter,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ( distin5379319190628067867c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_1209_finite__distinct__list,axiom,
! [A: set_Epistemic_fm_i] :
( ( finite3304564979551393739c_fm_i @ A )
=> ? [Xs3: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs3 )
= A )
& ( distin5379319190628067867c_fm_i @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1210_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1211_distinct_Osimps_I2_J,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( distin8272202008718213115c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) )
= ( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
& ( distin8272202008718213115c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1212_distinct_Osimps_I2_J,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
& ( distin5379319190628067867c_fm_i @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1213_union__set__fold,axiom,
! [Xs: list_Epistemic_fm_i,A: set_Epistemic_fm_i] :
( ( sup_su1936195050962291414c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ A )
= ( fold_E3768464346039846891c_fm_i @ insert7817948997695205106c_fm_i @ Xs @ A ) ) ).
% union_set_fold
thf(fact_1214_minus__set__fold,axiom,
! [A: set_Epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( minus_1543973939109951465c_fm_i @ A @ ( set_Epistemic_fm_i2 @ Xs ) )
= ( fold_E3768464346039846891c_fm_i @ remove5552923401211475591c_fm_i @ Xs @ A ) ) ).
% minus_set_fold
thf(fact_1215_distinct__disjoint__shuffles,axiom,
! [Xs: list_Epistemic_fm_i,Ys: list_Epistemic_fm_i,Zs3: list_Epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( ( distin5379319190628067867c_fm_i @ Ys )
=> ( ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( set_Epistemic_fm_i2 @ Ys ) )
= bot_bo4194595901900360558c_fm_i )
=> ( ( member6475149752759350353c_fm_i @ Zs3 @ ( shuffl511019056595460925c_fm_i @ Xs @ Ys ) )
=> ( distin5379319190628067867c_fm_i @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1216_distinct__disjoint__shuffles,axiom,
! [Xs: list_s8081015415394010888c_fm_i,Ys: list_s8081015415394010888c_fm_i,Zs3: list_s8081015415394010888c_fm_i] :
( ( distin8272202008718213115c_fm_i @ Xs )
=> ( ( distin8272202008718213115c_fm_i @ Ys )
=> ( ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Ys ) )
= bot_bo145720340923748686c_fm_i )
=> ( ( member605545779687026225c_fm_i @ Zs3 @ ( shuffl5790413406345785629c_fm_i @ Xs @ Ys ) )
=> ( distin8272202008718213115c_fm_i @ Zs3 ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_1217_inter__coset__fold,axiom,
! [A: set_se3485332733965609186c_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( inf_in161960956874937808c_fm_i @ A @ ( coset_1342939534705115381c_fm_i @ Xs ) )
= ( fold_s1063232419846502379c_fm_i @ remove1392734035364174951c_fm_i @ Xs @ A ) ) ).
% inter_coset_fold
thf(fact_1218_Inf__fin_Oset__eq__fold,axiom,
! [X4: set_se3485332733965609186c_fm_i,Xs: list_s8635415444435394792c_fm_i] :
( ( lattic7860428167660714879c_fm_i @ ( set_se5076022966804059383c_fm_i @ ( cons_s5806790718070763416c_fm_i @ X4 @ Xs ) ) )
= ( fold_s7712222339821836811c_fm_i @ inf_in161960956874937808c_fm_i @ Xs @ X4 ) ) ).
% Inf_fin.set_eq_fold
thf(fact_1219_Sup__fin_Oset__eq__fold,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( lattic3744900211830597177c_fm_i @ ( set_se200842218512397079c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) ) )
= ( fold_s1768690285866886667c_fm_i @ sup_su1936195050962291414c_fm_i @ Xs @ X4 ) ) ).
% Sup_fin.set_eq_fold
thf(fact_1220_set__remove1__eq,axiom,
! [Xs: list_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( distin5379319190628067867c_fm_i @ Xs )
=> ( ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X4 @ Xs ) )
= ( minus_1543973939109951465c_fm_i @ ( set_Epistemic_fm_i2 @ Xs ) @ ( insert7817948997695205106c_fm_i @ X4 @ bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% set_remove1_eq
thf(fact_1221_set__remdups,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( set_Epistemic_fm_i2 @ ( remdup2862625226703050045c_fm_i @ Xs ) )
= ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_remdups
thf(fact_1222_in__set__remove1,axiom,
! [A2: set_Epistemic_fm_i,B2: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( A2 != B2 )
=> ( ( member1104366573291651755c_fm_i @ A2 @ ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ B2 @ Xs ) ) )
= ( member1104366573291651755c_fm_i @ A2 @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1223_in__set__remove1,axiom,
! [A2: epistemic_fm_i,B2: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( A2 != B2 )
=> ( ( member6642669606046002379c_fm_i @ A2 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ B2 @ Xs ) ) )
= ( member6642669606046002379c_fm_i @ A2 @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1224_remove1__filter__not,axiom,
! [P5: epistemic_fm_i > $o,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( P5 @ X4 )
=> ( ( remove8001893566138066622c_fm_i @ X4 @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P5 @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_1225_set__remove1__subset,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] : ( ord_le3843937902494030498c_fm_i @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ X4 @ Xs ) ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1226_remdups_Osimps_I2_J,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ( ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remdup493799568840827677c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) )
= ( remdup493799568840827677c_fm_i @ Xs ) ) )
& ( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remdup493799568840827677c_fm_i @ ( cons_s4962720389763977656c_fm_i @ X4 @ Xs ) )
= ( cons_s4962720389763977656c_fm_i @ X4 @ ( remdup493799568840827677c_fm_i @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1227_remdups_Osimps_I2_J,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remdup2862625226703050045c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( remdup2862625226703050045c_fm_i @ Xs ) ) )
& ( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remdup2862625226703050045c_fm_i @ ( cons_Epistemic_fm_i @ X4 @ Xs ) )
= ( cons_Epistemic_fm_i @ X4 @ ( remdup2862625226703050045c_fm_i @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_1228_remove1__idem,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ( ( remove2021789187851727006c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1229_remove1__idem,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ( remove8001893566138066622c_fm_i @ X4 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1230_notin__set__remove1,axiom,
! [X4: set_Epistemic_fm_i,Xs: list_s8081015415394010888c_fm_i,Y2: set_Epistemic_fm_i] :
( ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ Xs ) )
=> ~ ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ ( remove2021789187851727006c_fm_i @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1231_notin__set__remove1,axiom,
! [X4: epistemic_fm_i,Xs: list_Epistemic_fm_i,Y2: epistemic_fm_i] :
( ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ ( remove8001893566138066622c_fm_i @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1232_remdups__map__remdups,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( remdup2862625226703050045c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ ( remdup2862625226703050045c_fm_i @ Xs ) ) )
= ( remdup2862625226703050045c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_1233_remdups__filter,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( remdup2862625226703050045c_fm_i @ ( filter7636273843821131039c_fm_i @ P5 @ Xs ) )
= ( filter7636273843821131039c_fm_i @ P5 @ ( remdup2862625226703050045c_fm_i @ Xs ) ) ) ).
% remdups_filter
thf(fact_1234_filter__remove1,axiom,
! [Q3: epistemic_fm_i > $o,X4: epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( filter7636273843821131039c_fm_i @ Q3 @ ( remove8001893566138066622c_fm_i @ X4 @ Xs ) )
= ( remove8001893566138066622c_fm_i @ X4 @ ( filter7636273843821131039c_fm_i @ Q3 @ Xs ) ) ) ).
% filter_remove1
thf(fact_1235_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X4 @ ( linord2614967742042102400et_nat @ A ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1236_distinct__concat,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin4780646717405202849c_fm_i @ Xs )
=> ( ! [Ys3: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys3 ) )
=> ( ! [Ys3: list_Epistemic_fm_i,Zs2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys3 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( member6475149752759350353c_fm_i @ Zs2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys3 ) @ ( set_Epistemic_fm_i2 @ Zs2 ) )
= bot_bo4194595901900360558c_fm_i ) ) ) )
=> ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1237_distinct__concat,axiom,
! [Xs: list_l8939085672862518926c_fm_i] :
( ( distin3563480452828447617c_fm_i @ Xs )
=> ( ! [Ys3: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys3 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( distin8272202008718213115c_fm_i @ Ys3 ) )
=> ( ! [Ys3: list_s8081015415394010888c_fm_i,Zs2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys3 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( ( member605545779687026225c_fm_i @ Zs2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Ys3 ) @ ( set_se200842218512397079c_fm_i @ Zs2 ) )
= bot_bo145720340923748686c_fm_i ) ) ) )
=> ( distin8272202008718213115c_fm_i @ ( concat3510228398265320643c_fm_i @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1238_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1239_filter__concat,axiom,
! [P2: epistemic_fm_i > $o,Xs: list_l4360594050116504750c_fm_i] :
( ( filter7636273843821131039c_fm_i @ P2 @ ( concat2780896577252151011c_fm_i @ Xs ) )
= ( concat2780896577252151011c_fm_i @ ( map_li4354675944384680516c_fm_i @ ( filter7636273843821131039c_fm_i @ P2 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_1240_map__concat,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ ( concat2780896577252151011c_fm_i @ Xs ) )
= ( concat2780896577252151011c_fm_i @ ( map_li4354675944384680516c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_1241_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1242_distinct__concat__iff,axiom,
! [Xs: list_l4360594050116504750c_fm_i] :
( ( distin5379319190628067867c_fm_i @ ( concat2780896577252151011c_fm_i @ Xs ) )
= ( ( distin4780646717405202849c_fm_i @ ( remove5890047714521709364c_fm_i @ nil_Epistemic_fm_i @ Xs ) )
& ! [Ys2: list_Epistemic_fm_i] :
( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
=> ( distin5379319190628067867c_fm_i @ Ys2 ) )
& ! [Ys2: list_Epistemic_fm_i,Zs: list_Epistemic_fm_i] :
( ( ( member6475149752759350353c_fm_i @ Ys2 @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( member6475149752759350353c_fm_i @ Zs @ ( set_li9010496130497932989c_fm_i @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in3450601097109690352c_fm_i @ ( set_Epistemic_fm_i2 @ Ys2 ) @ ( set_Epistemic_fm_i2 @ Zs ) )
= bot_bo4194595901900360558c_fm_i ) ) ) ) ).
% distinct_concat_iff
thf(fact_1243_distinct__concat__iff,axiom,
! [Xs: list_l8939085672862518926c_fm_i] :
( ( distin8272202008718213115c_fm_i @ ( concat3510228398265320643c_fm_i @ Xs ) )
= ( ( distin3563480452828447617c_fm_i @ ( remove1440612933533133076c_fm_i @ nil_se670848885773549064c_fm_i @ Xs ) )
& ! [Ys2: list_s8081015415394010888c_fm_i] :
( ( member605545779687026225c_fm_i @ Ys2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
=> ( distin8272202008718213115c_fm_i @ Ys2 ) )
& ! [Ys2: list_s8081015415394010888c_fm_i,Zs: list_s8081015415394010888c_fm_i] :
( ( ( member605545779687026225c_fm_i @ Ys2 @ ( set_li1699392436468070557c_fm_i @ Xs ) )
& ( member605545779687026225c_fm_i @ Zs @ ( set_li1699392436468070557c_fm_i @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in161960956874937808c_fm_i @ ( set_se200842218512397079c_fm_i @ Ys2 ) @ ( set_se200842218512397079c_fm_i @ Zs ) )
= bot_bo145720340923748686c_fm_i ) ) ) ) ).
% distinct_concat_iff
thf(fact_1244_map__is__Nil__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ Xs )
= nil_Epistemic_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% map_is_Nil_conv
thf(fact_1245_Nil__is__map__conv,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( nil_Epistemic_fm_i
= ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% Nil_is_map_conv
thf(fact_1246_list_Omap__disc__iff,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,A2: list_Epistemic_fm_i] :
( ( ( map_Ep2755178516647988292c_fm_i @ F @ A2 )
= nil_Epistemic_fm_i )
= ( A2 = nil_Epistemic_fm_i ) ) ).
% list.map_disc_iff
thf(fact_1247_set__empty,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( ( set_Epistemic_fm_i2 @ Xs )
= bot_bo4194595901900360558c_fm_i )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty
thf(fact_1248_set__empty2,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ Xs ) )
= ( Xs = nil_Epistemic_fm_i ) ) ).
% set_empty2
thf(fact_1249_filter__False,axiom,
! [Xs: list_Epistemic_fm_i,P5: epistemic_fm_i > $o] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P5 @ X3 ) )
=> ( ( filter7636273843821131039c_fm_i @ P5 @ Xs )
= nil_Epistemic_fm_i ) ) ).
% filter_False
thf(fact_1250_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1251_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1252_concat__map__singleton,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( concat2780896577252151011c_fm_i
@ ( map_Ep5327822066171429706c_fm_i
@ ^ [X2: epistemic_fm_i] : ( cons_Epistemic_fm_i @ ( F @ X2 ) @ nil_Epistemic_fm_i )
@ Xs ) )
= ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_1253_filter__empty__conv,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( ( filter7636273843821131039c_fm_i @ P5 @ Xs )
= nil_Epistemic_fm_i )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P5 @ X2 ) ) ) ) ).
% filter_empty_conv
thf(fact_1254_empty__filter__conv,axiom,
! [P5: epistemic_fm_i > $o,Xs: list_Epistemic_fm_i] :
( ( nil_Epistemic_fm_i
= ( filter7636273843821131039c_fm_i @ P5 @ Xs ) )
= ( ! [X2: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X2 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ~ ( P5 @ X2 ) ) ) ) ).
% empty_filter_conv
thf(fact_1255_empty__set,axiom,
( bot_bo4194595901900360558c_fm_i
= ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% empty_set
thf(fact_1256_map__rec,axiom,
( map_Ep2755178516647988292c_fm_i
= ( ^ [F4: epistemic_fm_i > epistemic_fm_i] :
( rec_li1793402101571464112c_fm_i @ nil_Epistemic_fm_i
@ ^ [X2: epistemic_fm_i,Uu2: list_Epistemic_fm_i] : ( cons_Epistemic_fm_i @ ( F4 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_1257_filter_Osimps_I1_J,axiom,
! [P5: epistemic_fm_i > $o] :
( ( filter7636273843821131039c_fm_i @ P5 @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% filter.simps(1)
thf(fact_1258_imply_Osimps_I1_J,axiom,
! [Q: epistemic_fm_i] :
( ( epistemic_imply_i @ nil_Epistemic_fm_i @ Q )
= Q ) ).
% imply.simps(1)
thf(fact_1259_list_Osimps_I8_J,axiom,
! [F: epistemic_fm_i > epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ nil_Epistemic_fm_i )
= nil_Epistemic_fm_i ) ).
% list.simps(8)
thf(fact_1260_conjunct_Osimps_I1_J,axiom,
( ( stalnaker_conjunct_i @ nil_Epistemic_fm_i )
= ( epistemic_Imp_i @ epistemic_FF_i @ epistemic_FF_i ) ) ).
% conjunct.simps(1)
thf(fact_1261_subset__code_I3_J,axiom,
~ ( ord_le3843937902494030498c_fm_i @ ( coset_Epistemic_fm_i @ nil_Epistemic_fm_i ) @ ( set_Epistemic_fm_i2 @ nil_Epistemic_fm_i ) ) ).
% subset_code(3)
thf(fact_1262_the__elem__set,axiom,
! [X4: epistemic_fm_i] :
( ( the_el2173195912186371479c_fm_i @ ( set_Epistemic_fm_i2 @ ( cons_Epistemic_fm_i @ X4 @ nil_Epistemic_fm_i ) ) )
= X4 ) ).
% the_elem_set
thf(fact_1263_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X2: nat] : X2
@ X4
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_1264_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X4 @ A )
=> ( ( linord2614967742042102400et_nat @ ( insert_nat @ X4 @ A ) )
= ( linord8961336180081300637at_nat
@ ^ [X2: nat] : X2
@ X4
@ ( linord2614967742042102400et_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_1265_insort__is__Cons,axiom,
! [Xs: list_Epistemic_fm_i,F: epistemic_fm_i > nat,A2: epistemic_fm_i] :
( ! [X3: epistemic_fm_i] :
( ( member6642669606046002379c_fm_i @ X3 @ ( set_Epistemic_fm_i2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ X3 ) ) )
=> ( ( linord2163417920900579895_i_nat @ F @ A2 @ Xs )
= ( cons_Epistemic_fm_i @ A2 @ Xs ) ) ) ).
% insort_is_Cons
thf(fact_1266_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: set_nat,X4: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= ( linord8961336180081300637at_nat
@ ^ [X2: nat] : X2
@ X4
@ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_1267_transpose__map__map,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_l4360594050116504750c_fm_i] :
( ( transp2882070894626480306c_fm_i @ ( map_li4354675944384680516c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F ) @ Xs ) )
= ( map_li4354675944384680516c_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F ) @ ( transp2882070894626480306c_fm_i @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1268_list_Omap__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( A2 != nil_Epistemic_fm_i )
=> ( ( tl_Epistemic_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ A2 ) )
= ( map_Ep2755178516647988292c_fm_i @ F @ ( tl_Epistemic_fm_i @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_1269_map__tl,axiom,
! [F: epistemic_fm_i > epistemic_fm_i,Xs: list_Epistemic_fm_i] :
( ( map_Ep2755178516647988292c_fm_i @ F @ ( tl_Epistemic_fm_i @ Xs ) )
= ( tl_Epistemic_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1270_list_Oset__sel_I2_J,axiom,
! [A2: list_s8081015415394010888c_fm_i,X4: set_Epistemic_fm_i] :
( ( A2 != nil_se670848885773549064c_fm_i )
=> ( ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ ( tl_set2575874409535201137c_fm_i @ A2 ) ) )
=> ( member1104366573291651755c_fm_i @ X4 @ ( set_se200842218512397079c_fm_i @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1271_list_Oset__sel_I2_J,axiom,
! [A2: list_Epistemic_fm_i,X4: epistemic_fm_i] :
( ( A2 != nil_Epistemic_fm_i )
=> ( ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ ( tl_Epistemic_fm_i @ A2 ) ) )
=> ( member6642669606046002379c_fm_i @ X4 @ ( set_Epistemic_fm_i2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1272_list_Oset__sel_I1_J,axiom,
! [A2: list_s8081015415394010888c_fm_i] :
( ( A2 != nil_se670848885773549064c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( hd_set3307850788051163501c_fm_i @ A2 ) @ ( set_se200842218512397079c_fm_i @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1273_list_Oset__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_i] :
( ( A2 != nil_Epistemic_fm_i )
=> ( member6642669606046002379c_fm_i @ ( hd_Epistemic_fm_i @ A2 ) @ ( set_Epistemic_fm_i2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1274_hd__in__set,axiom,
! [Xs: list_s8081015415394010888c_fm_i] :
( ( Xs != nil_se670848885773549064c_fm_i )
=> ( member1104366573291651755c_fm_i @ ( hd_set3307850788051163501c_fm_i @ Xs ) @ ( set_se200842218512397079c_fm_i @ Xs ) ) ) ).
% hd_in_set
thf(fact_1275_hd__in__set,axiom,
! [Xs: list_Epistemic_fm_i] :
( ( Xs != nil_Epistemic_fm_i )
=> ( member6642669606046002379c_fm_i @ ( hd_Epistemic_fm_i @ Xs ) @ ( set_Epistemic_fm_i2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1276_list_Omap__sel_I1_J,axiom,
! [A2: list_Epistemic_fm_i,F: epistemic_fm_i > epistemic_fm_i] :
( ( A2 != nil_Epistemic_fm_i )
=> ( ( hd_Epistemic_fm_i @ ( map_Ep2755178516647988292c_fm_i @ F @ A2 ) )
= ( F @ ( hd_Epistemic_fm_i @ A2 ) ) ) ) ).
% list.map_sel(1)
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( if_set_nat @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( if_set_nat @ $true @ X4 @ Y2 )
= X4 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: set_Epistemic_fm_i,Y2: set_Epistemic_fm_i] :
( ( if_set2943457721768014536c_fm_i @ $true @ X4 @ Y2 )
= X4 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Epistemic____Logic__Ofm_Itf__i_J_J_T,axiom,
! [X4: list_Epistemic_fm_i,Y2: list_Epistemic_fm_i] :
( ( if_lis3446954105117697518c_fm_i @ $true @ X4 @ Y2 )
= X4 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [P5: $o] :
( ( P5 = $true )
| ( P5 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X4: list_s8081015415394010888c_fm_i,Y2: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Epistemic____Logic__Ofm_Itf__i_J_J_J_T,axiom,
! [X4: list_s8081015415394010888c_fm_i,Y2: list_s8081015415394010888c_fm_i] :
( ( if_lis2088983353317256142c_fm_i @ $true @ X4 @ Y2 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------