TPTP Problem File: ITP204^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP204^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_793__6341386_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : USubst/prob_793__6341386_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 438 ( 222 unt; 80 typ; 0 def)
% Number of atoms : 895 ( 538 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2963 ( 148 ~; 31 |; 71 &;2383 @)
% ( 0 <=>; 330 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 693 ( 693 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 11 con; 0-3 aty)
% Number of variables : 1209 ( 339 ^; 849 !; 21 ?;1209 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:35:38.645
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_J,type,
produc1418842292n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J,type,
produc62062515n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J,type,
produc799602994n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1739631244on_fml: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Ogame_J,type,
option_game: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Otrm_J,type,
option_trm: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Ofml_J,type,
option_fml: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Ovariable_J,type,
set_variable: $tType ).
thf(ty_n_t__Syntax__Ovariable,type,
variable: $tType ).
thf(ty_n_t__Syntax__Ogame,type,
game: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Syntax__Otrm,type,
trm: $tType ).
thf(ty_n_t__Syntax__Ofml,type,
fml: $tType ).
% Explicit typings (67)
thf(sy_c_If_001t__Option__Ooption_It__Syntax__Ofml_J,type,
if_option_fml: $o > option_fml > option_fml > option_fml ).
thf(sy_c_If_001t__Option__Ooption_It__Syntax__Otrm_J,type,
if_option_trm: $o > option_trm > option_trm > option_trm ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Syntax__Ovariable_M_Eo_J,type,
inf_inf_variable_o: ( variable > $o ) > ( variable > $o ) > variable > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Syntax__Ovariable_J,type,
inf_inf_set_variable: set_variable > set_variable > set_variable ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ofml,type,
none_fml: option_fml ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ogame,type,
none_game: option_game ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Otrm,type,
none_trm: option_trm ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ofml,type,
some_fml: fml > option_fml ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ogame,type,
some_game: game > option_game ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Otrm,type,
some_trm: trm > option_trm ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Syntax__Ofml,type,
case_option_o_fml: $o > ( fml > $o ) > option_fml > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Syntax__Ogame,type,
case_option_o_game: $o > ( game > $o ) > option_game > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Syntax__Otrm,type,
case_option_o_trm: $o > ( trm > $o ) > option_trm > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Syntax__Ofml_J_001t__Syntax__Ofml,type,
case_o209804997ml_fml: option_fml > ( fml > option_fml ) > option_fml > option_fml ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Syntax__Ofml_J_001t__Syntax__Otrm,type,
case_o325794063ml_trm: option_fml > ( trm > option_fml ) > option_trm > option_fml ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Syntax__Otrm_J_001t__Syntax__Otrm,type,
case_o487727941rm_trm: option_trm > ( trm > option_trm ) > option_trm > option_trm ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Ofml,type,
the_fml: option_fml > fml ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Ogame,type,
the_game: option_game > game ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Otrm,type,
the_trm: option_trm > trm ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Syntax__Ovariable_M_Eo_J,type,
bot_bot_variable_o: variable > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Ovariable_J,type,
bot_bot_set_variable: set_variable ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Syntax__Ovariable_M_Eo_J,type,
ord_le1407353162able_o: ( variable > $o ) > ( variable > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ovariable_J,type,
ord_le282106107riable: set_variable > set_variable > $o ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc2124676926on_fml: option_game > option_fml > produc1739631244on_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1598111045on_fml: ( ( char > option_fml ) > ( char > option_game ) > char > option_fml ) > produc799602994n_game > char > option_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc1214549391on_trm: ( ( char > option_fml ) > ( char > option_game ) > char > option_trm ) > produc799602994n_game > char > option_trm ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc1591622303on_fml: ( ( char > option_fml ) > ( char > option_game ) > option_fml ) > produc799602994n_game > option_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc2129367017on_trm: ( ( char > option_fml ) > ( char > option_game ) > option_trm ) > produc799602994n_game > option_trm ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc377919108on_fml: ( ( char > option_trm ) > produc799602994n_game > char > option_fml ) > produc62062515n_game > char > option_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc2141841102on_trm: ( ( char > option_trm ) > produc799602994n_game > char > option_trm ) > produc62062515n_game > char > option_trm ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc1903410526on_fml: ( ( char > option_trm ) > produc799602994n_game > option_fml ) > produc62062515n_game > option_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc293671592on_trm: ( ( char > option_trm ) > produc799602994n_game > option_trm ) > produc62062515n_game > option_trm ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1278328899on_fml: ( ( char > option_trm ) > produc62062515n_game > char > option_fml ) > produc1418842292n_game > char > option_fml ).
thf(sy_c_Product__Type_Ointernal__case__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc724717725on_fml: ( ( char > option_trm ) > produc62062515n_game > option_fml ) > produc1418842292n_game > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1420172399on_fml: ( ( char > option_fml ) > ( char > option_game ) > char > option_fml ) > produc799602994n_game > char > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc1036610745on_trm: ( ( char > option_fml ) > ( char > option_game ) > char > option_trm ) > produc799602994n_game > char > option_trm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc347802313on_fml: ( ( char > option_fml ) > ( char > option_game ) > option_fml ) > produc799602994n_game > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc885547027on_trm: ( ( char > option_fml ) > ( char > option_game ) > option_trm ) > produc799602994n_game > option_trm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1615133358on_fml: ( ( char > option_trm ) > produc799602994n_game > char > option_fml ) > produc62062515n_game > char > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc1231571704on_trm: ( ( char > option_trm ) > produc799602994n_game > char > option_trm ) > produc62062515n_game > char > option_trm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc154833544on_fml: ( ( char > option_trm ) > produc799602994n_game > option_fml ) > produc62062515n_game > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc692578258on_trm: ( ( char > option_trm ) > produc799602994n_game > option_trm ) > produc62062515n_game > option_trm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1250175341on_fml: ( ( char > option_trm ) > produc62062515n_game > char > option_fml ) > produc1418842292n_game > char > option_fml ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc866613687on_trm: ( ( char > option_trm ) > produc62062515n_game > char > option_trm ) > produc1418842292n_game > char > option_trm ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_001t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc1499310791on_fml: ( ( char > option_trm ) > produc62062515n_game > option_fml ) > produc1418842292n_game > option_fml ).
thf(sy_c_Set_OCollect_001t__Syntax__Ovariable,type,
collect_variable: ( variable > $o ) > set_variable ).
thf(sy_c_Set_Ois__empty_001t__Syntax__Ovariable,type,
is_empty_variable: set_variable > $o ).
thf(sy_c_Static__Semantics_OFVF,type,
static_FVF: fml > set_variable ).
thf(sy_c_Static__Semantics_OFVT,type,
static_FVT: trm > set_variable ).
thf(sy_c_Syntax_Ofml_OGeq,type,
geq: trm > trm > fml ).
thf(sy_c_Syntax_Ofml_OPred,type,
pred: char > trm > fml ).
thf(sy_c_Syntax_Otrm_OConst,type,
const: char > trm ).
thf(sy_c_Syntax_Otrm_OFunc,type,
func: char > trm > trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OGeqo,type,
uSubst152838031e_Geqo: option_trm > option_trm > option_fml ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Odotsubstt,type,
uSubst1408967625substt: trm > produc1418842292n_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousappconst,type,
uSubst1559070963pconst: produc1418842292n_game > set_variable > char > option_trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousubstappf,type,
uSubst516392804stappf: produc1418842292n_game > set_variable > fml > option_fml ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousubstappt,type,
uSubst516392818stappt: produc1418842292n_game > set_variable > trm > option_trm ).
thf(sy_c_member_001t__Syntax__Ovariable,type,
member_variable: variable > set_variable > $o ).
thf(sy_v_Ua____,type,
ua: set_variable ).
thf(sy_v_Va____,type,
va: set_variable ).
thf(sy_v__092_060sigma_062,type,
sigma: produc1418842292n_game ).
thf(sy_v__092_060theta_062____,type,
theta: trm ).
thf(sy_v_p____,type,
p: char ).
thf(sy_v_vv____,type,
vv: set_variable > set_variable > variable ).
thf(sy_v_vva____,type,
vva: set_variable > set_variable > variable ).
% Relevant facts (352)
thf(fact_0__092_060open_062_092_060forall_062x0_Ax1_O_A_I_092_060exists_062v2_O_Av2_A_092_060in_062_Ax1_A_092_060and_062_A_I_092_060exists_062v3_O_Av3_A_092_060in_062_Ax0_A_092_060and_062_Av2_A_061_Av3_J_J_A_061_A_Ivv_Ax0_Ax1_A_092_060in_062_Ax1_A_092_060and_062_A_I_092_060exists_062v3_O_Av3_A_092_060in_062_Ax0_A_092_060and_062_Avv_Ax0_Ax1_A_061_Av3_J_J_092_060close_062,axiom,
! [X0: set_variable,X1: set_variable] :
( ( ? [V2: variable] :
( ( member_variable @ V2 @ X1 )
& ? [V3: variable] :
( ( member_variable @ V3 @ X0 )
& ( V2 = V3 ) ) ) )
= ( ( member_variable @ ( vv @ X0 @ X1 ) @ X1 )
& ? [V3: variable] :
( ( member_variable @ V3 @ X0 )
& ( ( vv @ X0 @ X1 )
= V3 ) ) ) ) ).
% \<open>\<forall>x0 x1. (\<exists>v2. v2 \<in> x1 \<and> (\<exists>v3. v3 \<in> x0 \<and> v2 = v3)) = (vv x0 x1 \<in> x1 \<and> (\<exists>v3. v3 \<in> x0 \<and> vv x0 x1 = v3))\<close>
thf(fact_1_Pred_Oprems_I2_J,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= none_fml ) ).
% Pred.prems(2)
thf(fact_2__092_060open_062_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_A_092_060noteq_062_Aundeff_092_060close_062,axiom,
~ ( ( ( inf_inf_set_variable
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
@ va )
= bot_bot_set_variable )
=> ( ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
= none_fml ) ) ).
% \<open>(if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff) \<noteq> undeff\<close>
thf(fact_3__092_060open_062_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_A_092_060noteq_062_Ausubstappf_A_092_060sigma_062_AV_A_IPred_Ap_A_092_060theta_062_J_A_092_060Longrightarrow_062_ASPreds_A_092_060sigma_062_Ap_A_061_Aundeff_092_060close_062,axiom,
( ~ ( ( ( ( inf_inf_set_variable
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
@ va )
= bot_bot_set_variable )
=> ( ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
= ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) )
& ( ( ( inf_inf_set_variable
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
@ va )
!= bot_bot_set_variable )
=> ( none_fml
= ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ) )
=> ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p )
= none_fml ) ) ).
% \<open>(if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff) \<noteq> usubstappf \<sigma> V (Pred p \<theta>) \<Longrightarrow> SPreds \<sigma> p = undeff\<close>
thf(fact_4_usubstappf__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Phi: fml,V: set_variable] :
( ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
!= none_fml )
=> ( ( ( uSubst516392804stappf @ Sigma @ V @ Phi )
!= none_fml )
=> ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
= ( uSubst516392804stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_det
thf(fact_5_f8,axiom,
( ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p )
= none_fml )
=> ( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ) ).
% f8
thf(fact_6__092_060open_062vv_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060notin_062_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060or_062_Avva_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060notin_062_AV_A_092_060or_062_Avv_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060noteq_062_Avva_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_092_060close_062,axiom,
( ~ ( member_variable
@ ( vv @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
| ~ ( member_variable
@ ( vva @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
@ va )
| ( ( vv @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
!= ( vva @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) ) ) ) ).
% \<open>vv V (FVF (the (SPreds \<sigma> p))) \<notin> FVF (the (SPreds \<sigma> p)) \<or> vva V (FVF (the (SPreds \<sigma> p))) \<notin> V \<or> vv V (FVF (the (SPreds \<sigma> p))) \<noteq> vva V (FVF (the (SPreds \<sigma> p)))\<close>
thf(fact_7_fml_Oinject_I1_J,axiom,
! [X11: char,X12: trm,Y11: char,Y12: trm] :
( ( ( pred @ X11 @ X12 )
= ( pred @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fml.inject(1)
thf(fact_8_f7,axiom,
( ( uSubst516392818stappt @ sigma @ ua @ theta )
= ( uSubst516392818stappt @ sigma @ va @ theta ) ) ).
% f7
thf(fact_9_case__prod__app,axiom,
( produc1250175341on_fml
= ( ^ [F: ( char > option_trm ) > produc62062515n_game > char > option_fml,X: produc1418842292n_game,Y: char] :
( produc1499310791on_fml
@ ^ [L: char > option_trm,R: produc62062515n_game] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_10_case__prod__app,axiom,
( produc1231571704on_trm
= ( ^ [F: ( char > option_trm ) > produc799602994n_game > char > option_trm,X: produc62062515n_game,Y: char] :
( produc692578258on_trm
@ ^ [L: char > option_trm,R: produc799602994n_game] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_11_case__prod__app,axiom,
( produc1615133358on_fml
= ( ^ [F: ( char > option_trm ) > produc799602994n_game > char > option_fml,X: produc62062515n_game,Y: char] :
( produc154833544on_fml
@ ^ [L: char > option_trm,R: produc799602994n_game] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_12_case__prod__app,axiom,
( produc1036610745on_trm
= ( ^ [F: ( char > option_fml ) > ( char > option_game ) > char > option_trm,X: produc799602994n_game,Y: char] :
( produc885547027on_trm
@ ^ [L: char > option_fml,R: char > option_game] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_13_case__prod__app,axiom,
( produc1420172399on_fml
= ( ^ [F: ( char > option_fml ) > ( char > option_game ) > char > option_fml,X: produc799602994n_game,Y: char] :
( produc347802313on_fml
@ ^ [L: char > option_fml,R: char > option_game] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_14_f1,axiom,
! [V4: variable] :
( ~ ( member_variable @ V4 @ va )
| ( member_variable @ V4 @ ua ) ) ).
% f1
thf(fact_15_f10,axiom,
( ( ( ( inf_inf_set_variable
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
@ va )
!= bot_bot_set_variable )
| ! [V4: variable] :
( ~ ( member_variable @ V4
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
| ! [Va: variable] :
( ~ ( member_variable @ Va @ va )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf_set_variable
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) )
@ va )
= bot_bot_set_variable )
| ( ( member_variable
@ ( vv @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
& ( member_variable
@ ( vva @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
@ va )
& ( ( vv @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) )
= ( vva @ va
@ ( static_FVF
@ ( the_fml
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) ) ) ) ) ) ).
% f10
thf(fact_16_prod_Ocase__distrib,axiom,
! [H: option_fml > option_fml,F2: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F2 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_17_prod_Ocase__distrib,axiom,
! [H: option_fml > option_trm,F2: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F2 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_18_prod_Ocase__distrib,axiom,
! [H: option_trm > option_fml,F2: ( char > option_fml ) > ( char > option_game ) > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc885547027on_trm @ F2 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_19_prod_Ocase__distrib,axiom,
! [H: option_trm > option_trm,F2: ( char > option_fml ) > ( char > option_game ) > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc885547027on_trm @ F2 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_20_prod_Ocase__distrib,axiom,
! [H: ( char > option_trm ) > option_fml,F2: ( char > option_fml ) > ( char > option_game ) > char > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc1036610745on_trm @ F2 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_21_prod_Ocase__distrib,axiom,
! [H: ( char > option_trm ) > option_trm,F2: ( char > option_fml ) > ( char > option_game ) > char > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc1036610745on_trm @ F2 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_22_prod_Ocase__distrib,axiom,
! [H: ( char > option_fml ) > option_fml,F2: ( char > option_fml ) > ( char > option_game ) > char > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc1420172399on_fml @ F2 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_23_prod_Ocase__distrib,axiom,
! [H: ( char > option_fml ) > option_trm,F2: ( char > option_fml ) > ( char > option_game ) > char > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc1420172399on_fml @ F2 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_24_prod_Ocase__distrib,axiom,
! [H: option_fml > char > option_trm,F2: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F2 @ Prod ) )
= ( produc1036610745on_trm
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_25_prod_Ocase__distrib,axiom,
! [H: option_fml > char > option_fml,F2: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F2 @ Prod ) )
= ( produc1420172399on_fml
@ ^ [X13: char > option_fml,X2: char > option_game] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_26_usubstappf__pred2,axiom,
! [Sigma: produc1418842292n_game,P2: char,R2: fml,U: set_variable,Theta: trm] :
( ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ Sigma
@ P2 )
= ( some_fml @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVF @ R2 ) @ U )
!= bot_bot_set_variable )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= none_fml ) ) ) ).
% usubstappf_pred2
thf(fact_27_f2,axiom,
! [Z: option_fml,F3: fml > option_fml,Za: option_fml] :
( ( ( Za = none_fml )
=> ( ( case_o209804997ml_fml @ Z @ F3 @ Za )
= Z ) )
& ( ( Za != none_fml )
=> ( ( case_o209804997ml_fml @ Z @ F3 @ Za )
= ( F3 @ ( the_fml @ Za ) ) ) ) ) ).
% f2
thf(fact_28_Pred_Oprems_I1_J,axiom,
ord_le282106107riable @ va @ ua ).
% Pred.prems(1)
thf(fact_29_f9,axiom,
! [V5: set_variable,Va2: set_variable] :
( ( ( ( inf_inf_set_variable @ V5 @ Va2 )
!= bot_bot_set_variable )
| ! [V4: variable] :
( ~ ( member_variable @ V4 @ V5 )
| ! [Va: variable] :
( ~ ( member_variable @ Va @ Va2 )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf_set_variable @ V5 @ Va2 )
= bot_bot_set_variable )
| ( ( member_variable @ ( vv @ Va2 @ V5 ) @ V5 )
& ( member_variable @ ( vva @ Va2 @ V5 ) @ Va2 )
& ( ( vv @ Va2 @ V5 )
= ( vva @ Va2 @ V5 ) ) ) ) ) ).
% f9
thf(fact_30__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062vva_O_A_092_060forall_062V_AVa_O_A_IV_A_092_060inter_062_AVa_A_092_060noteq_062_A_123_125_A_092_060or_062_A_I_092_060forall_062v_O_Av_A_092_060notin_062_AV_A_092_060or_062_A_I_092_060forall_062va_O_Ava_A_092_060notin_062_AVa_A_092_060or_062_Av_A_092_060noteq_062_Ava_J_J_J_A_092_060and_062_A_IV_A_092_060inter_062_AVa_A_061_A_123_125_A_092_060or_062_Avv_AVa_AV_A_092_060in_062_AV_A_092_060and_062_Avva_AVa_AV_A_092_060in_062_AVa_A_092_060and_062_Avv_AVa_AV_A_061_Avva_AVa_AV_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Vva: set_variable > set_variable > variable] :
~ ! [V5: set_variable,Va2: set_variable] :
( ( ( ( inf_inf_set_variable @ V5 @ Va2 )
!= bot_bot_set_variable )
| ! [V4: variable] :
( ~ ( member_variable @ V4 @ V5 )
| ! [Va: variable] :
( ~ ( member_variable @ Va @ Va2 )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf_set_variable @ V5 @ Va2 )
= bot_bot_set_variable )
| ( ( member_variable @ ( vv @ Va2 @ V5 ) @ V5 )
& ( member_variable @ ( Vva @ Va2 @ V5 ) @ Va2 )
& ( ( vv @ Va2 @ V5 )
= ( Vva @ Va2 @ V5 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>vva. \<forall>V Va. (V \<inter> Va \<noteq> {} \<or> (\<forall>v. v \<notin> V \<or> (\<forall>va. va \<notin> Va \<or> v \<noteq> va))) \<and> (V \<inter> Va = {} \<or> vv Va V \<in> V \<and> vva Va V \<in> Va \<and> vv Va V = vva Va V) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_31_option_Oinject,axiom,
! [X22: fml,Y2: fml] :
( ( ( some_fml @ X22 )
= ( some_fml @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_32_option_Oinject,axiom,
! [X22: trm,Y2: trm] :
( ( ( some_trm @ X22 )
= ( some_trm @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_33_option_Oinject,axiom,
! [X22: game,Y2: game] :
( ( ( some_game @ X22 )
= ( some_game @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_34_f5,axiom,
( ( uSubst516392818stappt @ sigma @ ua @ theta )
!= none_trm ) ).
% f5
thf(fact_35_not__Some__eq,axiom,
! [X3: option_fml] :
( ( ! [Y: fml] :
( X3
!= ( some_fml @ Y ) ) )
= ( X3 = none_fml ) ) ).
% not_Some_eq
thf(fact_36_not__Some__eq,axiom,
! [X3: option_trm] :
( ( ! [Y: trm] :
( X3
!= ( some_trm @ Y ) ) )
= ( X3 = none_trm ) ) ).
% not_Some_eq
thf(fact_37_not__Some__eq,axiom,
! [X3: option_game] :
( ( ! [Y: game] :
( X3
!= ( some_game @ Y ) ) )
= ( X3 = none_game ) ) ).
% not_Some_eq
thf(fact_38_not__None__eq,axiom,
! [X3: option_fml] :
( ( X3 != none_fml )
= ( ? [Y: fml] :
( X3
= ( some_fml @ Y ) ) ) ) ).
% not_None_eq
thf(fact_39_not__None__eq,axiom,
! [X3: option_trm] :
( ( X3 != none_trm )
= ( ? [Y: trm] :
( X3
= ( some_trm @ Y ) ) ) ) ).
% not_None_eq
thf(fact_40_not__None__eq,axiom,
! [X3: option_game] :
( ( X3 != none_game )
= ( ? [Y: game] :
( X3
= ( some_game @ Y ) ) ) ) ).
% not_None_eq
thf(fact_41_option_Ocollapse,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( ( some_fml @ ( the_fml @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_42_option_Ocollapse,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( ( some_trm @ ( the_trm @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_43_option_Ocollapse,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( ( some_game @ ( the_game @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_44_option_Osimps_I5_J,axiom,
! [F1: $o,F22: game > $o,X22: game] :
( ( case_option_o_game @ F1 @ F22 @ ( some_game @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_45_option_Osimps_I5_J,axiom,
! [F1: $o,F22: trm > $o,X22: trm] :
( ( case_option_o_trm @ F1 @ F22 @ ( some_trm @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_46_option_Osimps_I5_J,axiom,
! [F1: $o,F22: fml > $o,X22: fml] :
( ( case_option_o_fml @ F1 @ F22 @ ( some_fml @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_47_option_Osimps_I5_J,axiom,
! [F1: option_fml,F22: fml > option_fml,X22: fml] :
( ( case_o209804997ml_fml @ F1 @ F22 @ ( some_fml @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_48_option_Osimps_I5_J,axiom,
! [F1: option_fml,F22: trm > option_fml,X22: trm] :
( ( case_o325794063ml_trm @ F1 @ F22 @ ( some_trm @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_49_option_Osimps_I5_J,axiom,
! [F1: option_trm,F22: trm > option_trm,X22: trm] :
( ( case_o487727941rm_trm @ F1 @ F22 @ ( some_trm @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_50_option_Osel,axiom,
! [X22: fml] :
( ( the_fml @ ( some_fml @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_51_option_Osel,axiom,
! [X22: trm] :
( ( the_trm @ ( some_trm @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_52_option_Osel,axiom,
! [X22: game] :
( ( the_game @ ( some_game @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_53_option_Osplit__sel,axiom,
! [P3: $o > $o,F1: $o,F22: game > $o,Option: option_game] :
( ( P3 @ ( case_option_o_game @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_game )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_game @ ( the_game @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_game @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_54_option_Osplit__sel,axiom,
! [P3: $o > $o,F1: $o,F22: trm > $o,Option: option_trm] :
( ( P3 @ ( case_option_o_trm @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_trm )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_55_option_Osplit__sel,axiom,
! [P3: $o > $o,F1: $o,F22: fml > $o,Option: option_fml] :
( ( P3 @ ( case_option_o_fml @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_fml )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_56_option_Osplit__sel,axiom,
! [P3: option_fml > $o,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( P3 @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_fml )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_57_option_Osplit__sel,axiom,
! [P3: option_fml > $o,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( P3 @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_trm )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_58_option_Osplit__sel,axiom,
! [P3: option_trm > $o,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( P3 @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_trm )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
=> ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_59_option_Ocase__distrib,axiom,
! [H: option_fml > option_fml,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( H @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( case_o209804997ml_fml @ ( H @ F1 )
@ ^ [X: fml] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_60_option_Ocase__distrib,axiom,
! [H: option_fml > option_fml,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( H @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( case_o325794063ml_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_61_option_Ocase__distrib,axiom,
! [H: option_fml > option_trm,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( H @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( case_o487727941rm_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_62_option_Ocase__distrib,axiom,
! [H: option_trm > option_fml,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( H @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( case_o325794063ml_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_63_option_Ocase__distrib,axiom,
! [H: option_trm > option_trm,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( H @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( case_o487727941rm_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_64_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: game > $o,Option: option_game] :
( ( H @ ( case_option_o_game @ F1 @ F22 @ Option ) )
= ( case_option_o_game @ ( H @ F1 )
@ ^ [X: game] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_65_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: trm > $o,Option: option_trm] :
( ( H @ ( case_option_o_trm @ F1 @ F22 @ Option ) )
= ( case_option_o_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_66_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: fml > $o,Option: option_fml] :
( ( H @ ( case_option_o_fml @ F1 @ F22 @ Option ) )
= ( case_option_o_fml @ ( H @ F1 )
@ ^ [X: fml] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_67_option_Ocase__distrib,axiom,
! [H: option_fml > $o,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( H @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( case_option_o_fml @ ( H @ F1 )
@ ^ [X: fml] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_68_option_Ocase__distrib,axiom,
! [H: option_fml > $o,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( H @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( case_option_o_trm @ ( H @ F1 )
@ ^ [X: trm] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_69_option_Osplit__sel__asm,axiom,
! [P3: $o > $o,F1: $o,F22: game > $o,Option: option_game] :
( ( P3 @ ( case_option_o_game @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_game )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_game @ ( the_game @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_game @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_70_option_Osplit__sel__asm,axiom,
! [P3: $o > $o,F1: $o,F22: trm > $o,Option: option_trm] :
( ( P3 @ ( case_option_o_trm @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_trm )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_71_option_Osplit__sel__asm,axiom,
! [P3: $o > $o,F1: $o,F22: fml > $o,Option: option_fml] :
( ( P3 @ ( case_option_o_fml @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_fml )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_72_option_Osplit__sel__asm,axiom,
! [P3: option_fml > $o,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( P3 @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_fml )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_73_option_Osplit__sel__asm,axiom,
! [P3: option_fml > $o,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( P3 @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_trm )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_74_option_Osplit__sel__asm,axiom,
! [P3: option_trm > $o,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( P3 @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_trm )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_75_usubstappt__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,V: set_variable] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ V @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_76_usubstappt__antimon,axiom,
! [V: set_variable,U: set_variable,Sigma: produc1418842292n_game,Theta: trm] :
( ( ord_le282106107riable @ V @ U )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_antimon
thf(fact_77_option_Oexhaust__sel,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( Option
= ( some_fml @ ( the_fml @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_78_option_Oexhaust__sel,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( Option
= ( some_trm @ ( the_trm @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_79_option_Oexhaust__sel,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( Option
= ( some_game @ ( the_game @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_80_combine__options__cases,axiom,
! [X3: option_fml,P3: option_fml > option_fml > $o,Y3: option_fml] :
( ( ( X3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: fml,B: fml] :
( ( X3
= ( some_fml @ A ) )
=> ( ( Y3
= ( some_fml @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_81_combine__options__cases,axiom,
! [X3: option_fml,P3: option_fml > option_trm > $o,Y3: option_trm] :
( ( ( X3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: fml,B: trm] :
( ( X3
= ( some_fml @ A ) )
=> ( ( Y3
= ( some_trm @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_82_combine__options__cases,axiom,
! [X3: option_fml,P3: option_fml > option_game > $o,Y3: option_game] :
( ( ( X3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: fml,B: game] :
( ( X3
= ( some_fml @ A ) )
=> ( ( Y3
= ( some_game @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_83_combine__options__cases,axiom,
! [X3: option_trm,P3: option_trm > option_fml > $o,Y3: option_fml] :
( ( ( X3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: trm,B: fml] :
( ( X3
= ( some_trm @ A ) )
=> ( ( Y3
= ( some_fml @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_84_combine__options__cases,axiom,
! [X3: option_trm,P3: option_trm > option_trm > $o,Y3: option_trm] :
( ( ( X3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: trm,B: trm] :
( ( X3
= ( some_trm @ A ) )
=> ( ( Y3
= ( some_trm @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_85_combine__options__cases,axiom,
! [X3: option_trm,P3: option_trm > option_game > $o,Y3: option_game] :
( ( ( X3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: trm,B: game] :
( ( X3
= ( some_trm @ A ) )
=> ( ( Y3
= ( some_game @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_86_combine__options__cases,axiom,
! [X3: option_game,P3: option_game > option_fml > $o,Y3: option_fml] :
( ( ( X3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_fml )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: game,B: fml] :
( ( X3
= ( some_game @ A ) )
=> ( ( Y3
= ( some_fml @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_87_combine__options__cases,axiom,
! [X3: option_game,P3: option_game > option_trm > $o,Y3: option_trm] :
( ( ( X3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_trm )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: game,B: trm] :
( ( X3
= ( some_game @ A ) )
=> ( ( Y3
= ( some_trm @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_88_combine__options__cases,axiom,
! [X3: option_game,P3: option_game > option_game > $o,Y3: option_game] :
( ( ( X3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3 = none_game )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A: game,B: game] :
( ( X3
= ( some_game @ A ) )
=> ( ( Y3
= ( some_game @ B ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_89_split__option__all,axiom,
( ( ^ [P4: option_fml > $o] :
! [X4: option_fml] : ( P4 @ X4 ) )
= ( ^ [P: option_fml > $o] :
( ( P @ none_fml )
& ! [X: fml] : ( P @ ( some_fml @ X ) ) ) ) ) ).
% split_option_all
thf(fact_90_split__option__all,axiom,
( ( ^ [P4: option_trm > $o] :
! [X4: option_trm] : ( P4 @ X4 ) )
= ( ^ [P: option_trm > $o] :
( ( P @ none_trm )
& ! [X: trm] : ( P @ ( some_trm @ X ) ) ) ) ) ).
% split_option_all
thf(fact_91_split__option__all,axiom,
( ( ^ [P4: option_game > $o] :
! [X4: option_game] : ( P4 @ X4 ) )
= ( ^ [P: option_game > $o] :
( ( P @ none_game )
& ! [X: game] : ( P @ ( some_game @ X ) ) ) ) ) ).
% split_option_all
thf(fact_92_split__option__ex,axiom,
( ( ^ [P4: option_fml > $o] :
? [X4: option_fml] : ( P4 @ X4 ) )
= ( ^ [P: option_fml > $o] :
( ( P @ none_fml )
| ? [X: fml] : ( P @ ( some_fml @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_93_split__option__ex,axiom,
( ( ^ [P4: option_trm > $o] :
? [X4: option_trm] : ( P4 @ X4 ) )
= ( ^ [P: option_trm > $o] :
( ( P @ none_trm )
| ? [X: trm] : ( P @ ( some_trm @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_94_split__option__ex,axiom,
( ( ^ [P4: option_game > $o] :
? [X4: option_game] : ( P4 @ X4 ) )
= ( ^ [P: option_game > $o] :
( ( P @ none_game )
| ? [X: game] : ( P @ ( some_game @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_95_option_Oinducts,axiom,
! [P3: option_fml > $o,Option: option_fml] :
( ( P3 @ none_fml )
=> ( ! [X5: fml] : ( P3 @ ( some_fml @ X5 ) )
=> ( P3 @ Option ) ) ) ).
% option.inducts
thf(fact_96_option_Oinducts,axiom,
! [P3: option_trm > $o,Option: option_trm] :
( ( P3 @ none_trm )
=> ( ! [X5: trm] : ( P3 @ ( some_trm @ X5 ) )
=> ( P3 @ Option ) ) ) ).
% option.inducts
thf(fact_97_option_Oinducts,axiom,
! [P3: option_game > $o,Option: option_game] :
( ( P3 @ none_game )
=> ( ! [X5: game] : ( P3 @ ( some_game @ X5 ) )
=> ( P3 @ Option ) ) ) ).
% option.inducts
thf(fact_98_option_Oexhaust,axiom,
! [Y3: option_fml] :
( ( Y3 != none_fml )
=> ~ ! [X23: fml] :
( Y3
!= ( some_fml @ X23 ) ) ) ).
% option.exhaust
thf(fact_99_option_Oexhaust,axiom,
! [Y3: option_trm] :
( ( Y3 != none_trm )
=> ~ ! [X23: trm] :
( Y3
!= ( some_trm @ X23 ) ) ) ).
% option.exhaust
thf(fact_100_option_Oexhaust,axiom,
! [Y3: option_game] :
( ( Y3 != none_game )
=> ~ ! [X23: game] :
( Y3
!= ( some_game @ X23 ) ) ) ).
% option.exhaust
thf(fact_101_option_OdiscI,axiom,
! [Option: option_fml,X22: fml] :
( ( Option
= ( some_fml @ X22 ) )
=> ( Option != none_fml ) ) ).
% option.discI
thf(fact_102_option_OdiscI,axiom,
! [Option: option_trm,X22: trm] :
( ( Option
= ( some_trm @ X22 ) )
=> ( Option != none_trm ) ) ).
% option.discI
thf(fact_103_option_OdiscI,axiom,
! [Option: option_game,X22: game] :
( ( Option
= ( some_game @ X22 ) )
=> ( Option != none_game ) ) ).
% option.discI
thf(fact_104_option_Odistinct_I1_J,axiom,
! [X22: fml] :
( none_fml
!= ( some_fml @ X22 ) ) ).
% option.distinct(1)
thf(fact_105_option_Odistinct_I1_J,axiom,
! [X22: trm] :
( none_trm
!= ( some_trm @ X22 ) ) ).
% option.distinct(1)
thf(fact_106_option_Odistinct_I1_J,axiom,
! [X22: game] :
( none_game
!= ( some_game @ X22 ) ) ).
% option.distinct(1)
thf(fact_107_option_Ocase__eq__if,axiom,
( case_option_o_game
= ( ^ [F12: $o,F23: game > $o,Option2: option_game] :
( ( ( Option2 = none_game )
=> F12 )
& ( ( Option2 != none_game )
=> ( F23 @ ( the_game @ Option2 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_108_option_Ocase__eq__if,axiom,
( case_option_o_trm
= ( ^ [F12: $o,F23: trm > $o,Option2: option_trm] :
( ( ( Option2 = none_trm )
=> F12 )
& ( ( Option2 != none_trm )
=> ( F23 @ ( the_trm @ Option2 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_109_option_Ocase__eq__if,axiom,
( case_option_o_fml
= ( ^ [F12: $o,F23: fml > $o,Option2: option_fml] :
( ( ( Option2 = none_fml )
=> F12 )
& ( ( Option2 != none_fml )
=> ( F23 @ ( the_fml @ Option2 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_110_option_Ocase__eq__if,axiom,
( case_o209804997ml_fml
= ( ^ [F12: option_fml,F23: fml > option_fml,Option2: option_fml] : ( if_option_fml @ ( Option2 = none_fml ) @ F12 @ ( F23 @ ( the_fml @ Option2 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_111_option_Ocase__eq__if,axiom,
( case_o325794063ml_trm
= ( ^ [F12: option_fml,F23: trm > option_fml,Option2: option_trm] : ( if_option_fml @ ( Option2 = none_trm ) @ F12 @ ( F23 @ ( the_trm @ Option2 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_112_option_Ocase__eq__if,axiom,
( case_o487727941rm_trm
= ( ^ [F12: option_trm,F23: trm > option_trm,Option2: option_trm] : ( if_option_trm @ ( Option2 = none_trm ) @ F12 @ ( F23 @ ( the_trm @ Option2 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_113_option_Osimps_I4_J,axiom,
! [F1: $o,F22: game > $o] :
( ( case_option_o_game @ F1 @ F22 @ none_game )
= F1 ) ).
% option.simps(4)
thf(fact_114_option_Osimps_I4_J,axiom,
! [F1: $o,F22: trm > $o] :
( ( case_option_o_trm @ F1 @ F22 @ none_trm )
= F1 ) ).
% option.simps(4)
thf(fact_115_option_Osimps_I4_J,axiom,
! [F1: $o,F22: fml > $o] :
( ( case_option_o_fml @ F1 @ F22 @ none_fml )
= F1 ) ).
% option.simps(4)
thf(fact_116_option_Osimps_I4_J,axiom,
! [F1: option_fml,F22: fml > option_fml] :
( ( case_o209804997ml_fml @ F1 @ F22 @ none_fml )
= F1 ) ).
% option.simps(4)
thf(fact_117_option_Osimps_I4_J,axiom,
! [F1: option_fml,F22: trm > option_fml] :
( ( case_o325794063ml_trm @ F1 @ F22 @ none_trm )
= F1 ) ).
% option.simps(4)
thf(fact_118_option_Osimps_I4_J,axiom,
! [F1: option_trm,F22: trm > option_trm] :
( ( case_o487727941rm_trm @ F1 @ F22 @ none_trm )
= F1 ) ).
% option.simps(4)
thf(fact_119_Existso_Oinduct,axiom,
! [P3: variable > option_fml > $o,A0: variable,A1: option_fml] :
( ! [X5: variable,Phi2: fml] : ( P3 @ X5 @ ( some_fml @ Phi2 ) )
=> ( ! [X5: variable] : ( P3 @ X5 @ none_fml )
=> ( P3 @ A0 @ A1 ) ) ) ).
% Existso.induct
thf(fact_120_undeff__equiv,axiom,
! [Phi: option_fml] :
( ( Phi != none_fml )
= ( ? [F: fml] :
( Phi
= ( some_fml @ F ) ) ) ) ).
% undeff_equiv
thf(fact_121_Testo_Oinduct,axiom,
! [P3: option_fml > $o,A0: option_fml] :
( ! [Phi2: fml] : ( P3 @ ( some_fml @ Phi2 ) )
=> ( ( P3 @ none_fml )
=> ( P3 @ A0 ) ) ) ).
% Testo.induct
thf(fact_122_Testo_Ocases,axiom,
! [X3: option_fml] :
( ! [Phi2: fml] :
( X3
!= ( some_fml @ Phi2 ) )
=> ( X3 = none_fml ) ) ).
% Testo.cases
thf(fact_123_mem__Collect__eq,axiom,
! [A2: variable,P3: variable > $o] :
( ( member_variable @ A2 @ ( collect_variable @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A3: set_variable] :
( ( collect_variable
@ ^ [X: variable] : ( member_variable @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
! [P3: variable > $o,Q: variable > $o] :
( ! [X5: variable] :
( ( P3 @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_variable @ P3 )
= ( collect_variable @ Q ) ) ) ).
% Collect_cong
thf(fact_126_Ando_Oinduct,axiom,
! [P3: option_fml > option_fml > $o,A0: option_fml,A1: option_fml] :
( ! [Phi2: fml,Psi: fml] : ( P3 @ ( some_fml @ Phi2 ) @ ( some_fml @ Psi ) )
=> ( ! [X_1: option_fml] : ( P3 @ none_fml @ X_1 )
=> ( ! [V6: fml] : ( P3 @ ( some_fml @ V6 ) @ none_fml )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Ando.induct
thf(fact_127_option_Oexpand,axiom,
! [Option: option_fml,Option3: option_fml] :
( ( ( Option = none_fml )
= ( Option3 = none_fml ) )
=> ( ( ( Option != none_fml )
=> ( ( Option3 != none_fml )
=> ( ( the_fml @ Option )
= ( the_fml @ Option3 ) ) ) )
=> ( Option = Option3 ) ) ) ).
% option.expand
thf(fact_128_option_Oexpand,axiom,
! [Option: option_trm,Option3: option_trm] :
( ( ( Option = none_trm )
= ( Option3 = none_trm ) )
=> ( ( ( Option != none_trm )
=> ( ( Option3 != none_trm )
=> ( ( the_trm @ Option )
= ( the_trm @ Option3 ) ) ) )
=> ( Option = Option3 ) ) ) ).
% option.expand
thf(fact_129_option_Oexpand,axiom,
! [Option: option_game,Option3: option_game] :
( ( ( Option = none_game )
= ( Option3 = none_game ) )
=> ( ( ( Option != none_game )
=> ( ( Option3 != none_game )
=> ( ( the_game @ Option )
= ( the_game @ Option3 ) ) ) )
=> ( Option = Option3 ) ) ) ).
% option.expand
thf(fact_130_usubstappf__pred__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,P2: char,Theta: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
!= none_fml )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ Sigma
@ P2 )
= none_fml )
| ? [R3: fml] :
( ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ Sigma
@ P2 )
= ( some_fml @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVF @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ) ).
% usubstappf_pred_conv
thf(fact_131_f6,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F )
@ none_fml )
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) )
@ ( uSubst516392818stappt @ sigma @ ua @ theta ) )
= ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ua @ theta ) ) ) )
@ ^ [F: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ua @ theta ) ) ) @ bot_bot_set_variable @ F )
@ none_fml )
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) ) ) ).
% f6
thf(fact_132_f3,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F )
@ none_fml )
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p ) )
@ ( uSubst516392818stappt @ sigma @ ua @ theta ) )
!= none_fml ) ).
% f3
thf(fact_133_usubstappf_Osimps_I1_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,P2: char,Theta: trm] :
( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= ( case_o325794063ml_trm @ none_fml
@ ^ [Sigma_theta: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ P2 @ Sigma_theta ) )
@ ^ [R: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ R ) @ U )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ Sigma_theta ) @ bot_bot_set_variable @ R )
@ none_fml )
@ ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ Sigma
@ P2 ) )
@ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappf.simps(1)
thf(fact_134_Int__subset__iff,axiom,
! [C: set_variable,A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C @ ( inf_inf_set_variable @ A3 @ B2 ) )
= ( ( ord_le282106107riable @ C @ A3 )
& ( ord_le282106107riable @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_135_inf__bot__left,axiom,
! [X3: variable > $o] :
( ( inf_inf_variable_o @ bot_bot_variable_o @ X3 )
= bot_bot_variable_o ) ).
% inf_bot_left
thf(fact_136_inf__bot__left,axiom,
! [X3: set_variable] :
( ( inf_inf_set_variable @ bot_bot_set_variable @ X3 )
= bot_bot_set_variable ) ).
% inf_bot_left
thf(fact_137_inf__bot__right,axiom,
! [X3: variable > $o] :
( ( inf_inf_variable_o @ X3 @ bot_bot_variable_o )
= bot_bot_variable_o ) ).
% inf_bot_right
thf(fact_138_inf__bot__right,axiom,
! [X3: set_variable] :
( ( inf_inf_set_variable @ X3 @ bot_bot_set_variable )
= bot_bot_set_variable ) ).
% inf_bot_right
thf(fact_139_subset__empty,axiom,
! [A3: set_variable] :
( ( ord_le282106107riable @ A3 @ bot_bot_set_variable )
= ( A3 = bot_bot_set_variable ) ) ).
% subset_empty
thf(fact_140_empty__subsetI,axiom,
! [A3: set_variable] : ( ord_le282106107riable @ bot_bot_set_variable @ A3 ) ).
% empty_subsetI
thf(fact_141_le__inf__iff,axiom,
! [X3: variable > $o,Y3: variable > $o,Z2: variable > $o] :
( ( ord_le1407353162able_o @ X3 @ ( inf_inf_variable_o @ Y3 @ Z2 ) )
= ( ( ord_le1407353162able_o @ X3 @ Y3 )
& ( ord_le1407353162able_o @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_142_le__inf__iff,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) )
= ( ( ord_le282106107riable @ X3 @ Y3 )
& ( ord_le282106107riable @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_143_inf_Obounded__iff,axiom,
! [A2: variable > $o,B3: variable > $o,C2: variable > $o] :
( ( ord_le1407353162able_o @ A2 @ ( inf_inf_variable_o @ B3 @ C2 ) )
= ( ( ord_le1407353162able_o @ A2 @ B3 )
& ( ord_le1407353162able_o @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_144_inf_Obounded__iff,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) )
= ( ( ord_le282106107riable @ A2 @ B3 )
& ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_145_f4,axiom,
! [Z: option_fml,F3: trm > option_fml,Za: option_trm] :
( ( ( Za = none_trm )
=> ( ( case_o325794063ml_trm @ Z @ F3 @ Za )
= Z ) )
& ( ( Za != none_trm )
=> ( ( case_o325794063ml_trm @ Z @ F3 @ Za )
= ( F3 @ ( the_trm @ Za ) ) ) ) ) ).
% f4
thf(fact_146_empty__Collect__eq,axiom,
! [P3: variable > $o] :
( ( bot_bot_set_variable
= ( collect_variable @ P3 ) )
= ( ! [X: variable] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_147_Collect__empty__eq,axiom,
! [P3: variable > $o] :
( ( ( collect_variable @ P3 )
= bot_bot_set_variable )
= ( ! [X: variable] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_148_all__not__in__conv,axiom,
! [A3: set_variable] :
( ( ! [X: variable] :
~ ( member_variable @ X @ A3 ) )
= ( A3 = bot_bot_set_variable ) ) ).
% all_not_in_conv
thf(fact_149_empty__iff,axiom,
! [C2: variable] :
~ ( member_variable @ C2 @ bot_bot_set_variable ) ).
% empty_iff
thf(fact_150_subset__antisym,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_151_subsetI,axiom,
! [A3: set_variable,B2: set_variable] :
( ! [X5: variable] :
( ( member_variable @ X5 @ A3 )
=> ( member_variable @ X5 @ B2 ) )
=> ( ord_le282106107riable @ A3 @ B2 ) ) ).
% subsetI
thf(fact_152_inf__right__idem,axiom,
! [X3: variable > $o,Y3: variable > $o] :
( ( inf_inf_variable_o @ ( inf_inf_variable_o @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_variable_o @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_153_inf__right__idem,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_variable @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_154_inf_Oright__idem,axiom,
! [A2: variable > $o,B3: variable > $o] :
( ( inf_inf_variable_o @ ( inf_inf_variable_o @ A2 @ B3 ) @ B3 )
= ( inf_inf_variable_o @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_155_inf_Oright__idem,axiom,
! [A2: set_variable,B3: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A2 @ B3 ) @ B3 )
= ( inf_inf_set_variable @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_156_inf__left__idem,axiom,
! [X3: variable > $o,Y3: variable > $o] :
( ( inf_inf_variable_o @ X3 @ ( inf_inf_variable_o @ X3 @ Y3 ) )
= ( inf_inf_variable_o @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_157_inf__left__idem,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ X3 @ Y3 ) )
= ( inf_inf_set_variable @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_158_inf_Oleft__idem,axiom,
! [A2: variable > $o,B3: variable > $o] :
( ( inf_inf_variable_o @ A2 @ ( inf_inf_variable_o @ A2 @ B3 ) )
= ( inf_inf_variable_o @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_159_inf_Oleft__idem,axiom,
! [A2: set_variable,B3: set_variable] :
( ( inf_inf_set_variable @ A2 @ ( inf_inf_set_variable @ A2 @ B3 ) )
= ( inf_inf_set_variable @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_160_inf__idem,axiom,
! [X3: variable > $o] :
( ( inf_inf_variable_o @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_161_inf__idem,axiom,
! [X3: set_variable] :
( ( inf_inf_set_variable @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_162_inf_Oidem,axiom,
! [A2: variable > $o] :
( ( inf_inf_variable_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_163_inf_Oidem,axiom,
! [A2: set_variable] :
( ( inf_inf_set_variable @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_164_inf__apply,axiom,
( inf_inf_variable_o
= ( ^ [F: variable > $o,G: variable > $o,X: variable] : ( inf_inf_o @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% inf_apply
thf(fact_165_Int__iff,axiom,
! [C2: variable,A3: set_variable,B2: set_variable] :
( ( member_variable @ C2 @ ( inf_inf_set_variable @ A3 @ B2 ) )
= ( ( member_variable @ C2 @ A3 )
& ( member_variable @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_166_IntI,axiom,
! [C2: variable,A3: set_variable,B2: set_variable] :
( ( member_variable @ C2 @ A3 )
=> ( ( member_variable @ C2 @ B2 )
=> ( member_variable @ C2 @ ( inf_inf_set_variable @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_167_usubstappf__pred,axiom,
! [Sigma: produc1418842292n_game,P2: char,R2: fml,U: set_variable,Theta: trm,Sigma_theta2: trm] :
( ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ Sigma
@ P2 )
= ( some_fml @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVF @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( some_trm @ Sigma_theta2 ) )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= ( uSubst516392804stappf @ ( uSubst1408967625substt @ Sigma_theta2 ) @ bot_bot_set_variable @ R2 ) ) ) ) ) ).
% usubstappf_pred
thf(fact_168_ODEo_Oinduct,axiom,
! [P3: char > option_trm > $o,A0: char,A1: option_trm] :
( ! [X5: char,Theta2: trm] : ( P3 @ X5 @ ( some_trm @ Theta2 ) )
=> ( ! [X5: char] : ( P3 @ X5 @ none_trm )
=> ( P3 @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_169_undeft__None,axiom,
none_trm = none_trm ).
% undeft_None
thf(fact_170_undeft__equiv,axiom,
! [Theta: option_trm] :
( ( Theta != none_trm )
= ( ? [T: trm] :
( Theta
= ( some_trm @ T ) ) ) ) ).
% undeft_equiv
thf(fact_171_Timeso_Oinduct,axiom,
! [P3: option_trm > option_trm > $o,A0: option_trm,A1: option_trm] :
( ! [Theta2: trm,Eta: trm] : ( P3 @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta ) )
=> ( ! [X_1: option_trm] : ( P3 @ none_trm @ X_1 )
=> ( ! [V6: trm] : ( P3 @ ( some_trm @ V6 ) @ none_trm )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_172_Assigno_Oinduct,axiom,
! [P3: variable > option_trm > $o,A0: variable,A1: option_trm] :
( ! [X5: variable,Theta2: trm] : ( P3 @ X5 @ ( some_trm @ Theta2 ) )
=> ( ! [X5: variable] : ( P3 @ X5 @ none_trm )
=> ( P3 @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_173_Differentialo_Ocases,axiom,
! [X3: option_trm] :
( ! [Theta2: trm] :
( X3
!= ( some_trm @ Theta2 ) )
=> ( X3 = none_trm ) ) ).
% Differentialo.cases
thf(fact_174_Differentialo_Oinduct,axiom,
! [P3: option_trm > $o,A0: option_trm] :
( ! [Theta2: trm] : ( P3 @ ( some_trm @ Theta2 ) )
=> ( ( P3 @ none_trm )
=> ( P3 @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_175_bot__set__def,axiom,
( bot_bot_set_variable
= ( collect_variable @ bot_bot_variable_o ) ) ).
% bot_set_def
thf(fact_176_inf__set__def,axiom,
( inf_inf_set_variable
= ( ^ [A4: set_variable,B4: set_variable] :
( collect_variable
@ ( inf_inf_variable_o
@ ^ [X: variable] : ( member_variable @ X @ A4 )
@ ^ [X: variable] : ( member_variable @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_177_less__eq__set__def,axiom,
( ord_le282106107riable
= ( ^ [A4: set_variable,B4: set_variable] :
( ord_le1407353162able_o
@ ^ [X: variable] : ( member_variable @ X @ A4 )
@ ^ [X: variable] : ( member_variable @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_178_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
= ( case_option_o_fml @ $false
@ ^ [Uu: fml] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_179_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
= ( case_option_o_trm @ $false
@ ^ [Uu: trm] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_180_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_game] :
( ( Option != none_game )
= ( case_option_o_game @ $false
@ ^ [Uu: game] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_181_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_fml] :
( ( Option = none_fml )
= ( case_option_o_fml @ $true
@ ^ [Uu: fml] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_182_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_trm] :
( ( Option = none_trm )
= ( case_option_o_trm @ $true
@ ^ [Uu: trm] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_183_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_game] :
( ( Option = none_game )
= ( case_option_o_game @ $true
@ ^ [Uu: game] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_184_Diamondo_Oinduct,axiom,
! [P3: option_game > option_fml > $o,A0: option_game,A1: option_fml] :
( ! [Alpha: game,Phi2: fml] : ( P3 @ ( some_game @ Alpha ) @ ( some_fml @ Phi2 ) )
=> ( ! [X_1: option_fml] : ( P3 @ none_game @ X_1 )
=> ( ! [V6: game] : ( P3 @ ( some_game @ V6 ) @ none_fml )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Diamondo.induct
thf(fact_185_case__optionE,axiom,
! [P3: $o,Q: fml > $o,X3: option_fml] :
( ( case_option_o_fml @ P3 @ Q @ X3 )
=> ( ( ( X3 = none_fml )
=> ~ P3 )
=> ~ ! [Y4: fml] :
( ( X3
= ( some_fml @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_186_case__optionE,axiom,
! [P3: $o,Q: trm > $o,X3: option_trm] :
( ( case_option_o_trm @ P3 @ Q @ X3 )
=> ( ( ( X3 = none_trm )
=> ~ P3 )
=> ~ ! [Y4: trm] :
( ( X3
= ( some_trm @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_187_case__optionE,axiom,
! [P3: $o,Q: game > $o,X3: option_game] :
( ( case_option_o_game @ P3 @ Q @ X3 )
=> ( ( ( X3 = none_game )
=> ~ P3 )
=> ~ ! [Y4: game] :
( ( X3
= ( some_game @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_188_ex__in__conv,axiom,
! [A3: set_variable] :
( ( ? [X: variable] : ( member_variable @ X @ A3 ) )
= ( A3 != bot_bot_set_variable ) ) ).
% ex_in_conv
thf(fact_189_equals0I,axiom,
! [A3: set_variable] :
( ! [Y4: variable] :
~ ( member_variable @ Y4 @ A3 )
=> ( A3 = bot_bot_set_variable ) ) ).
% equals0I
thf(fact_190_equals0D,axiom,
! [A3: set_variable,A2: variable] :
( ( A3 = bot_bot_set_variable )
=> ~ ( member_variable @ A2 @ A3 ) ) ).
% equals0D
thf(fact_191_emptyE,axiom,
! [A2: variable] :
~ ( member_variable @ A2 @ bot_bot_set_variable ) ).
% emptyE
thf(fact_192_Collect__mono__iff,axiom,
! [P3: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ ( collect_variable @ P3 ) @ ( collect_variable @ Q ) )
= ( ! [X: variable] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_193_set__eq__subset,axiom,
( ( ^ [Y5: set_variable,Z3: set_variable] : Y5 = Z3 )
= ( ^ [A4: set_variable,B4: set_variable] :
( ( ord_le282106107riable @ A4 @ B4 )
& ( ord_le282106107riable @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_194_subset__trans,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( ord_le282106107riable @ B2 @ C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_195_Collect__mono,axiom,
! [P3: variable > $o,Q: variable > $o] :
( ! [X5: variable] :
( ( P3 @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le282106107riable @ ( collect_variable @ P3 ) @ ( collect_variable @ Q ) ) ) ).
% Collect_mono
thf(fact_196_subset__refl,axiom,
! [A3: set_variable] : ( ord_le282106107riable @ A3 @ A3 ) ).
% subset_refl
thf(fact_197_subset__iff,axiom,
( ord_le282106107riable
= ( ^ [A4: set_variable,B4: set_variable] :
! [T: variable] :
( ( member_variable @ T @ A4 )
=> ( member_variable @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_198_equalityD2,axiom,
! [A3: set_variable,B2: set_variable] :
( ( A3 = B2 )
=> ( ord_le282106107riable @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_199_equalityD1,axiom,
! [A3: set_variable,B2: set_variable] :
( ( A3 = B2 )
=> ( ord_le282106107riable @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_200_subset__eq,axiom,
( ord_le282106107riable
= ( ^ [A4: set_variable,B4: set_variable] :
! [X: variable] :
( ( member_variable @ X @ A4 )
=> ( member_variable @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_201_equalityE,axiom,
! [A3: set_variable,B2: set_variable] :
( ( A3 = B2 )
=> ~ ( ( ord_le282106107riable @ A3 @ B2 )
=> ~ ( ord_le282106107riable @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_202_subsetD,axiom,
! [A3: set_variable,B2: set_variable,C2: variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( member_variable @ C2 @ A3 )
=> ( member_variable @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_203_in__mono,axiom,
! [A3: set_variable,B2: set_variable,X3: variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( member_variable @ X3 @ A3 )
=> ( member_variable @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_204_inf__left__commute,axiom,
! [X3: variable > $o,Y3: variable > $o,Z2: variable > $o] :
( ( inf_inf_variable_o @ X3 @ ( inf_inf_variable_o @ Y3 @ Z2 ) )
= ( inf_inf_variable_o @ Y3 @ ( inf_inf_variable_o @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_205_inf__left__commute,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) )
= ( inf_inf_set_variable @ Y3 @ ( inf_inf_set_variable @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_206_inf_Oleft__commute,axiom,
! [B3: variable > $o,A2: variable > $o,C2: variable > $o] :
( ( inf_inf_variable_o @ B3 @ ( inf_inf_variable_o @ A2 @ C2 ) )
= ( inf_inf_variable_o @ A2 @ ( inf_inf_variable_o @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_207_inf_Oleft__commute,axiom,
! [B3: set_variable,A2: set_variable,C2: set_variable] :
( ( inf_inf_set_variable @ B3 @ ( inf_inf_set_variable @ A2 @ C2 ) )
= ( inf_inf_set_variable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_208_inf__commute,axiom,
( inf_inf_variable_o
= ( ^ [X: variable > $o,Y: variable > $o] : ( inf_inf_variable_o @ Y @ X ) ) ) ).
% inf_commute
thf(fact_209_inf__commute,axiom,
( inf_inf_set_variable
= ( ^ [X: set_variable,Y: set_variable] : ( inf_inf_set_variable @ Y @ X ) ) ) ).
% inf_commute
thf(fact_210_inf_Ocommute,axiom,
( inf_inf_variable_o
= ( ^ [A5: variable > $o,B5: variable > $o] : ( inf_inf_variable_o @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_211_inf_Ocommute,axiom,
( inf_inf_set_variable
= ( ^ [A5: set_variable,B5: set_variable] : ( inf_inf_set_variable @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_212_inf__assoc,axiom,
! [X3: variable > $o,Y3: variable > $o,Z2: variable > $o] :
( ( inf_inf_variable_o @ ( inf_inf_variable_o @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_variable_o @ X3 @ ( inf_inf_variable_o @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_213_inf__assoc,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_214_inf_Oassoc,axiom,
! [A2: variable > $o,B3: variable > $o,C2: variable > $o] :
( ( inf_inf_variable_o @ ( inf_inf_variable_o @ A2 @ B3 ) @ C2 )
= ( inf_inf_variable_o @ A2 @ ( inf_inf_variable_o @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_215_inf_Oassoc,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_variable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) ) ) ).
% inf.assoc
thf(fact_216_boolean__algebra__cancel_Oinf2,axiom,
! [B2: variable > $o,K: variable > $o,B3: variable > $o,A2: variable > $o] :
( ( B2
= ( inf_inf_variable_o @ K @ B3 ) )
=> ( ( inf_inf_variable_o @ A2 @ B2 )
= ( inf_inf_variable_o @ K @ ( inf_inf_variable_o @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_217_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_variable,K: set_variable,B3: set_variable,A2: set_variable] :
( ( B2
= ( inf_inf_set_variable @ K @ B3 ) )
=> ( ( inf_inf_set_variable @ A2 @ B2 )
= ( inf_inf_set_variable @ K @ ( inf_inf_set_variable @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_218_boolean__algebra__cancel_Oinf1,axiom,
! [A3: variable > $o,K: variable > $o,A2: variable > $o,B3: variable > $o] :
( ( A3
= ( inf_inf_variable_o @ K @ A2 ) )
=> ( ( inf_inf_variable_o @ A3 @ B3 )
= ( inf_inf_variable_o @ K @ ( inf_inf_variable_o @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_219_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_variable,K: set_variable,A2: set_variable,B3: set_variable] :
( ( A3
= ( inf_inf_set_variable @ K @ A2 ) )
=> ( ( inf_inf_set_variable @ A3 @ B3 )
= ( inf_inf_set_variable @ K @ ( inf_inf_set_variable @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_220_inf__sup__aci_I1_J,axiom,
( inf_inf_set_variable
= ( ^ [X: set_variable,Y: set_variable] : ( inf_inf_set_variable @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_221_inf__sup__aci_I2_J,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_222_inf__sup__aci_I3_J,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) )
= ( inf_inf_set_variable @ Y3 @ ( inf_inf_set_variable @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_223_inf__sup__aci_I4_J,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( inf_inf_set_variable @ X3 @ ( inf_inf_set_variable @ X3 @ Y3 ) )
= ( inf_inf_set_variable @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_224_Int__left__commute,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( inf_inf_set_variable @ A3 @ ( inf_inf_set_variable @ B2 @ C ) )
= ( inf_inf_set_variable @ B2 @ ( inf_inf_set_variable @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_225_Int__left__absorb,axiom,
! [A3: set_variable,B2: set_variable] :
( ( inf_inf_set_variable @ A3 @ ( inf_inf_set_variable @ A3 @ B2 ) )
= ( inf_inf_set_variable @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_226_Int__commute,axiom,
( inf_inf_set_variable
= ( ^ [A4: set_variable,B4: set_variable] : ( inf_inf_set_variable @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_227_Int__absorb,axiom,
! [A3: set_variable] :
( ( inf_inf_set_variable @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_228_Int__assoc,axiom,
! [A3: set_variable,B2: set_variable,C: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A3 @ B2 ) @ C )
= ( inf_inf_set_variable @ A3 @ ( inf_inf_set_variable @ B2 @ C ) ) ) ).
% Int_assoc
thf(fact_229_IntD2,axiom,
! [C2: variable,A3: set_variable,B2: set_variable] :
( ( member_variable @ C2 @ ( inf_inf_set_variable @ A3 @ B2 ) )
=> ( member_variable @ C2 @ B2 ) ) ).
% IntD2
thf(fact_230_IntD1,axiom,
! [C2: variable,A3: set_variable,B2: set_variable] :
( ( member_variable @ C2 @ ( inf_inf_set_variable @ A3 @ B2 ) )
=> ( member_variable @ C2 @ A3 ) ) ).
% IntD1
thf(fact_231_IntE,axiom,
! [C2: variable,A3: set_variable,B2: set_variable] :
( ( member_variable @ C2 @ ( inf_inf_set_variable @ A3 @ B2 ) )
=> ~ ( ( member_variable @ C2 @ A3 )
=> ~ ( member_variable @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_232_empty__def,axiom,
( bot_bot_set_variable
= ( collect_variable
@ ^ [X: variable] : $false ) ) ).
% empty_def
thf(fact_233_Collect__subset,axiom,
! [A3: set_variable,P3: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ A3 )
& ( P3 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_234_Int__def,axiom,
( inf_inf_set_variable
= ( ^ [A4: set_variable,B4: set_variable] :
( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ A4 )
& ( member_variable @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_235_Int__Collect,axiom,
! [X3: variable,A3: set_variable,P3: variable > $o] :
( ( member_variable @ X3 @ ( inf_inf_set_variable @ A3 @ ( collect_variable @ P3 ) ) )
= ( ( member_variable @ X3 @ A3 )
& ( P3 @ X3 ) ) ) ).
% Int_Collect
thf(fact_236_Collect__conj__eq,axiom,
! [P3: variable > $o,Q: variable > $o] :
( ( collect_variable
@ ^ [X: variable] :
( ( P3 @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_variable @ ( collect_variable @ P3 ) @ ( collect_variable @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_237_inf_OcoboundedI2,axiom,
! [B3: set_variable,C2: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B3 @ C2 )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_238_inf_OcoboundedI1,axiom,
! [A2: set_variable,C2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A2 @ C2 )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_239_inf_Oabsorb__iff2,axiom,
( ord_le282106107riable
= ( ^ [B5: set_variable,A5: set_variable] :
( ( inf_inf_set_variable @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_240_inf_Oabsorb__iff1,axiom,
( ord_le282106107riable
= ( ^ [A5: set_variable,B5: set_variable] :
( ( inf_inf_set_variable @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_241_inf_Ocobounded2,axiom,
! [A2: set_variable,B3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_242_inf_Ocobounded1,axiom,
! [A2: set_variable,B3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_243_inf_Oorder__iff,axiom,
( ord_le282106107riable
= ( ^ [A5: set_variable,B5: set_variable] :
( A5
= ( inf_inf_set_variable @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_244_inf__greatest,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ X3 @ Y3 )
=> ( ( ord_le282106107riable @ X3 @ Z2 )
=> ( ord_le282106107riable @ X3 @ ( inf_inf_set_variable @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_245_inf_OboundedI,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( ord_le282106107riable @ A2 @ C2 )
=> ( ord_le282106107riable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_246_inf_OboundedE,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) )
=> ~ ( ( ord_le282106107riable @ A2 @ B3 )
=> ~ ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_247_inf__absorb2,axiom,
! [Y3: set_variable,X3: set_variable] :
( ( ord_le282106107riable @ Y3 @ X3 )
=> ( ( inf_inf_set_variable @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_248_inf__absorb1,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( ord_le282106107riable @ X3 @ Y3 )
=> ( ( inf_inf_set_variable @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_249_inf_Oabsorb2,axiom,
! [B3: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B3 @ A2 )
=> ( ( inf_inf_set_variable @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_250_inf_Oabsorb1,axiom,
! [A2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( inf_inf_set_variable @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_251_le__iff__inf,axiom,
( ord_le282106107riable
= ( ^ [X: set_variable,Y: set_variable] :
( ( inf_inf_set_variable @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_252_inf__unique,axiom,
! [F2: set_variable > set_variable > set_variable,X3: set_variable,Y3: set_variable] :
( ! [X5: set_variable,Y4: set_variable] : ( ord_le282106107riable @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: set_variable,Y4: set_variable] : ( ord_le282106107riable @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: set_variable,Y4: set_variable,Z4: set_variable] :
( ( ord_le282106107riable @ X5 @ Y4 )
=> ( ( ord_le282106107riable @ X5 @ Z4 )
=> ( ord_le282106107riable @ X5 @ ( F2 @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_variable @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_253_inf_OorderI,axiom,
! [A2: set_variable,B3: set_variable] :
( ( A2
= ( inf_inf_set_variable @ A2 @ B3 ) )
=> ( ord_le282106107riable @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_254_inf_OorderE,axiom,
! [A2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_variable @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_255_le__infI2,axiom,
! [B3: set_variable,X3: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B3 @ X3 )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_256_le__infI1,axiom,
! [A2: set_variable,X3: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A2 @ X3 )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_257_inf__mono,axiom,
! [A2: set_variable,C2: set_variable,B3: set_variable,D: set_variable] :
( ( ord_le282106107riable @ A2 @ C2 )
=> ( ( ord_le282106107riable @ B3 @ D )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A2 @ B3 ) @ ( inf_inf_set_variable @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_258_le__infI,axiom,
! [X3: set_variable,A2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ X3 @ A2 )
=> ( ( ord_le282106107riable @ X3 @ B3 )
=> ( ord_le282106107riable @ X3 @ ( inf_inf_set_variable @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_259_le__infE,axiom,
! [X3: set_variable,A2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ X3 @ ( inf_inf_set_variable @ A2 @ B3 ) )
=> ~ ( ( ord_le282106107riable @ X3 @ A2 )
=> ~ ( ord_le282106107riable @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_260_inf__le2,axiom,
! [X3: set_variable,Y3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_261_inf__le1,axiom,
! [X3: set_variable,Y3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_262_inf__sup__ord_I1_J,axiom,
! [X3: set_variable,Y3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_263_inf__sup__ord_I2_J,axiom,
! [X3: set_variable,Y3: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_264_disjoint__iff__not__equal,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ( inf_inf_set_variable @ A3 @ B2 )
= bot_bot_set_variable )
= ( ! [X: variable] :
( ( member_variable @ X @ A3 )
=> ! [Y: variable] :
( ( member_variable @ Y @ B2 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_265_Int__empty__right,axiom,
! [A3: set_variable] :
( ( inf_inf_set_variable @ A3 @ bot_bot_set_variable )
= bot_bot_set_variable ) ).
% Int_empty_right
thf(fact_266_Int__empty__left,axiom,
! [B2: set_variable] :
( ( inf_inf_set_variable @ bot_bot_set_variable @ B2 )
= bot_bot_set_variable ) ).
% Int_empty_left
thf(fact_267_disjoint__iff,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ( inf_inf_set_variable @ A3 @ B2 )
= bot_bot_set_variable )
= ( ! [X: variable] :
( ( member_variable @ X @ A3 )
=> ~ ( member_variable @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_268_Int__emptyI,axiom,
! [A3: set_variable,B2: set_variable] :
( ! [X5: variable] :
( ( member_variable @ X5 @ A3 )
=> ~ ( member_variable @ X5 @ B2 ) )
=> ( ( inf_inf_set_variable @ A3 @ B2 )
= bot_bot_set_variable ) ) ).
% Int_emptyI
thf(fact_269_Int__Collect__mono,axiom,
! [A3: set_variable,B2: set_variable,P3: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ! [X5: variable] :
( ( member_variable @ X5 @ A3 )
=> ( ( P3 @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A3 @ ( collect_variable @ P3 ) ) @ ( inf_inf_set_variable @ B2 @ ( collect_variable @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_270_Int__greatest,axiom,
! [C: set_variable,A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ C @ A3 )
=> ( ( ord_le282106107riable @ C @ B2 )
=> ( ord_le282106107riable @ C @ ( inf_inf_set_variable @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_271_Int__absorb2,axiom,
! [A3: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A3 @ B2 )
=> ( ( inf_inf_set_variable @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_272_Int__absorb1,axiom,
! [B2: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( inf_inf_set_variable @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_273_Int__lower2,axiom,
! [A3: set_variable,B2: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_274_Int__lower1,axiom,
! [A3: set_variable,B2: set_variable] : ( ord_le282106107riable @ ( inf_inf_set_variable @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_275_Int__mono,axiom,
! [A3: set_variable,C: set_variable,B2: set_variable,D2: set_variable] :
( ( ord_le282106107riable @ A3 @ C )
=> ( ( ord_le282106107riable @ B2 @ D2 )
=> ( ord_le282106107riable @ ( inf_inf_set_variable @ A3 @ B2 ) @ ( inf_inf_set_variable @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_276_order__refl,axiom,
! [X3: set_variable] : ( ord_le282106107riable @ X3 @ X3 ) ).
% order_refl
thf(fact_277_subset__emptyI,axiom,
! [A3: set_variable] :
( ! [X5: variable] :
~ ( member_variable @ X5 @ A3 )
=> ( ord_le282106107riable @ A3 @ bot_bot_set_variable ) ) ).
% subset_emptyI
thf(fact_278_bot_Oextremum__uniqueI,axiom,
! [A2: set_variable] :
( ( ord_le282106107riable @ A2 @ bot_bot_set_variable )
=> ( A2 = bot_bot_set_variable ) ) ).
% bot.extremum_uniqueI
thf(fact_279_bot_Oextremum__unique,axiom,
! [A2: set_variable] :
( ( ord_le282106107riable @ A2 @ bot_bot_set_variable )
= ( A2 = bot_bot_set_variable ) ) ).
% bot.extremum_unique
thf(fact_280_bot_Oextremum,axiom,
! [A2: set_variable] : ( ord_le282106107riable @ bot_bot_set_variable @ A2 ) ).
% bot.extremum
thf(fact_281_usubstappf__geqr,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= none_fml )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some_fml @ ( geq @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_282_fml_Oinject_I2_J,axiom,
! [X21: trm,X222: trm,Y21: trm,Y22: trm] :
( ( ( geq @ X21 @ X222 )
= ( geq @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% fml.inject(2)
thf(fact_283_Aterm__Some,axiom,
some_trm = some_trm ).
% Aterm_Some
thf(fact_284_Loopo_Ocases,axiom,
! [X3: option_game] :
( ! [Alpha: game] :
( X3
!= ( some_game @ Alpha ) )
=> ( X3 = none_game ) ) ).
% Loopo.cases
thf(fact_285_Loopo_Oinduct,axiom,
! [P3: option_game > $o,A0: option_game] :
( ! [Alpha: game] : ( P3 @ ( some_game @ Alpha ) )
=> ( ( P3 @ none_game )
=> ( P3 @ A0 ) ) ) ).
% Loopo.induct
thf(fact_286_undefg__equiv,axiom,
! [Alpha2: option_game] :
( ( Alpha2 != none_game )
= ( ? [G: game] :
( Alpha2
= ( some_game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_287_Composeo_Oinduct,axiom,
! [P3: option_game > option_game > $o,A0: option_game,A1: option_game] :
( ! [Alpha: game,Beta: game] : ( P3 @ ( some_game @ Alpha ) @ ( some_game @ Beta ) )
=> ( ! [Alpha: option_game] : ( P3 @ Alpha @ none_game )
=> ( ! [V6: game] : ( P3 @ none_game @ ( some_game @ V6 ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_288_fml_Odistinct_I1_J,axiom,
! [X11: char,X12: trm,X21: trm,X222: trm] :
( ( pred @ X11 @ X12 )
!= ( geq @ X21 @ X222 ) ) ).
% fml.distinct(1)
thf(fact_289_order__subst1,axiom,
! [A2: set_variable,F2: set_variable > set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_le282106107riable @ B3 @ C2 )
=> ( ! [X5: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X5 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_290_order__subst2,axiom,
! [A2: set_variable,B3: set_variable,F2: set_variable > set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( ord_le282106107riable @ ( F2 @ B3 ) @ C2 )
=> ( ! [X5: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X5 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_291_ord__eq__le__subst,axiom,
! [A2: set_variable,F2: set_variable > set_variable,B3: set_variable,C2: set_variable] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_le282106107riable @ B3 @ C2 )
=> ( ! [X5: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X5 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ A2 @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_292_ord__le__eq__subst,axiom,
! [A2: set_variable,B3: set_variable,F2: set_variable > set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X5: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X5 @ Y4 )
=> ( ord_le282106107riable @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F2 @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_293_eq__iff,axiom,
( ( ^ [Y5: set_variable,Z3: set_variable] : Y5 = Z3 )
= ( ^ [X: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ X @ Y )
& ( ord_le282106107riable @ Y @ X ) ) ) ) ).
% eq_iff
thf(fact_294_antisym,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( ord_le282106107riable @ X3 @ Y3 )
=> ( ( ord_le282106107riable @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_295_eq__refl,axiom,
! [X3: set_variable,Y3: set_variable] :
( ( X3 = Y3 )
=> ( ord_le282106107riable @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_296_order_Otrans,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( ord_le282106107riable @ B3 @ C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_297_antisym__conv,axiom,
! [Y3: set_variable,X3: set_variable] :
( ( ord_le282106107riable @ Y3 @ X3 )
=> ( ( ord_le282106107riable @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_298_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: set_variable,Z3: set_variable] : Y5 = Z3 )
= ( ^ [A5: set_variable,B5: set_variable] :
( ( ord_le282106107riable @ A5 @ B5 )
& ( ord_le282106107riable @ B5 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_299_ord__eq__le__trans,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( A2 = B3 )
=> ( ( ord_le282106107riable @ B3 @ C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_300_ord__le__eq__trans,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_le282106107riable @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_301_order__class_Oorder_Oantisym,axiom,
! [A2: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A2 @ B3 )
=> ( ( ord_le282106107riable @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_302_order__trans,axiom,
! [X3: set_variable,Y3: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ X3 @ Y3 )
=> ( ( ord_le282106107riable @ Y3 @ Z2 )
=> ( ord_le282106107riable @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_303_dual__order_Orefl,axiom,
! [A2: set_variable] : ( ord_le282106107riable @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_304_dual__order_Otrans,axiom,
! [B3: set_variable,A2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ B3 @ A2 )
=> ( ( ord_le282106107riable @ C2 @ B3 )
=> ( ord_le282106107riable @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_305_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_variable,Z3: set_variable] : Y5 = Z3 )
= ( ^ [A5: set_variable,B5: set_variable] :
( ( ord_le282106107riable @ B5 @ A5 )
& ( ord_le282106107riable @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_306_dual__order_Oantisym,axiom,
! [B3: set_variable,A2: set_variable] :
( ( ord_le282106107riable @ B3 @ A2 )
=> ( ( ord_le282106107riable @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_307_prop__restrict,axiom,
! [X3: variable,Z5: set_variable,X6: set_variable,P3: variable > $o] :
( ( member_variable @ X3 @ Z5 )
=> ( ( ord_le282106107riable @ Z5
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ X6 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X3 ) ) ) ).
% prop_restrict
thf(fact_308_Collect__restrict,axiom,
! [X6: set_variable,P3: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ X6 )
& ( P3 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_309_usubstappf__geq__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= none_fml )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta2 )
!= none_trm ) ) ) ).
% usubstappf_geq_conv
thf(fact_310_usubstappf__geq,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Eta2 )
!= none_trm )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some_fml @ ( geq @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_311_disjE__realizer2,axiom,
! [P3: $o,Q: fml > $o,X3: option_fml,R4: option_fml > $o,F2: option_fml,G2: fml > option_fml] :
( ( case_option_o_fml @ P3 @ Q @ X3 )
=> ( ( P3
=> ( R4 @ F2 ) )
=> ( ! [Q2: fml] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o209804997ml_fml @ F2 @ G2 @ X3 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_312_disjE__realizer2,axiom,
! [P3: $o,Q: trm > $o,X3: option_trm,R4: option_fml > $o,F2: option_fml,G2: trm > option_fml] :
( ( case_option_o_trm @ P3 @ Q @ X3 )
=> ( ( P3
=> ( R4 @ F2 ) )
=> ( ! [Q2: trm] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o325794063ml_trm @ F2 @ G2 @ X3 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_313_disjE__realizer2,axiom,
! [P3: $o,Q: trm > $o,X3: option_trm,R4: option_trm > $o,F2: option_trm,G2: trm > option_trm] :
( ( case_option_o_trm @ P3 @ Q @ X3 )
=> ( ( P3
=> ( R4 @ F2 ) )
=> ( ! [Q2: trm] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o487727941rm_trm @ F2 @ G2 @ X3 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_314_Geqo_Oelims,axiom,
! [X3: option_trm,Xa: option_trm,Y3: option_fml] :
( ( ( uSubst152838031e_Geqo @ X3 @ Xa )
= Y3 )
=> ( ! [Theta2: trm] :
( ( X3
= ( some_trm @ Theta2 ) )
=> ! [Eta: trm] :
( ( Xa
= ( some_trm @ Eta ) )
=> ( Y3
!= ( some_fml @ ( geq @ Theta2 @ Eta ) ) ) ) )
=> ( ( ( X3 = none_trm )
=> ( Y3 != none_fml ) )
=> ~ ( ? [V6: trm] :
( X3
= ( some_trm @ V6 ) )
=> ( ( Xa = none_trm )
=> ( Y3 != none_fml ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_315_inf__Int__eq,axiom,
! [R4: set_variable,S: set_variable] :
( ( inf_inf_variable_o
@ ^ [X: variable] : ( member_variable @ X @ R4 )
@ ^ [X: variable] : ( member_variable @ X @ S ) )
= ( ^ [X: variable] : ( member_variable @ X @ ( inf_inf_set_variable @ R4 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_316_Geqo_Osimps_I2_J,axiom,
! [Eta2: option_trm] :
( ( uSubst152838031e_Geqo @ none_trm @ Eta2 )
= none_fml ) ).
% Geqo.simps(2)
thf(fact_317_Geqo__undef,axiom,
! [Theta: option_trm,Eta2: option_trm] :
( ( ( uSubst152838031e_Geqo @ Theta @ Eta2 )
= none_fml )
= ( ( Theta = none_trm )
| ( Eta2 = none_trm ) ) ) ).
% Geqo_undef
thf(fact_318_Geqo_Osimps_I3_J,axiom,
! [V7: trm] :
( ( uSubst152838031e_Geqo @ ( some_trm @ V7 ) @ none_trm )
= none_fml ) ).
% Geqo.simps(3)
thf(fact_319_Geqo_Osimps_I1_J,axiom,
! [Theta: trm,Eta2: trm] :
( ( uSubst152838031e_Geqo @ ( some_trm @ Theta ) @ ( some_trm @ Eta2 ) )
= ( some_fml @ ( geq @ Theta @ Eta2 ) ) ) ).
% Geqo.simps(1)
thf(fact_320_usubstappf_Osimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( uSubst152838031e_Geqo @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta2 ) ) ) ).
% usubstappf.simps(2)
thf(fact_321_bot__empty__eq,axiom,
( bot_bot_variable_o
= ( ^ [X: variable] : ( member_variable @ X @ bot_bot_set_variable ) ) ) ).
% bot_empty_eq
thf(fact_322_pred__subset__eq,axiom,
! [R4: set_variable,S: set_variable] :
( ( ord_le1407353162able_o
@ ^ [X: variable] : ( member_variable @ X @ R4 )
@ ^ [X: variable] : ( member_variable @ X @ S ) )
= ( ord_le282106107riable @ R4 @ S ) ) ).
% pred_subset_eq
thf(fact_323_Collect__empty__eq__bot,axiom,
! [P3: variable > $o] :
( ( ( collect_variable @ P3 )
= bot_bot_set_variable )
= ( P3 = bot_bot_variable_o ) ) ).
% Collect_empty_eq_bot
thf(fact_324_Set_Ois__empty__def,axiom,
( is_empty_variable
= ( ^ [A4: set_variable] : A4 = bot_bot_set_variable ) ) ).
% Set.is_empty_def
thf(fact_325_internal__case__prod__def,axiom,
produc1278328899on_fml = produc1250175341on_fml ).
% internal_case_prod_def
thf(fact_326_internal__case__prod__def,axiom,
produc2141841102on_trm = produc1231571704on_trm ).
% internal_case_prod_def
thf(fact_327_internal__case__prod__def,axiom,
produc377919108on_fml = produc1615133358on_fml ).
% internal_case_prod_def
thf(fact_328_internal__case__prod__def,axiom,
produc1214549391on_trm = produc1036610745on_trm ).
% internal_case_prod_def
thf(fact_329_internal__case__prod__def,axiom,
produc1598111045on_fml = produc1420172399on_fml ).
% internal_case_prod_def
thf(fact_330_internal__case__prod__def,axiom,
produc1591622303on_fml = produc347802313on_fml ).
% internal_case_prod_def
thf(fact_331_internal__case__prod__def,axiom,
produc2129367017on_trm = produc885547027on_trm ).
% internal_case_prod_def
thf(fact_332_internal__case__prod__def,axiom,
produc1903410526on_fml = produc154833544on_fml ).
% internal_case_prod_def
thf(fact_333_internal__case__prod__def,axiom,
produc293671592on_trm = produc692578258on_trm ).
% internal_case_prod_def
thf(fact_334_internal__case__prod__def,axiom,
produc724717725on_fml = produc1499310791on_fml ).
% internal_case_prod_def
thf(fact_335_usubstappt_Osimps_I4_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F2: char,Theta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= ( case_o487727941rm_trm @ none_trm
@ ^ [Sigma_theta: trm] :
( case_o487727941rm_trm @ ( some_trm @ ( func @ F2 @ Sigma_theta ) )
@ ^ [R: trm] :
( if_option_trm
@ ( ( inf_inf_set_variable @ ( static_FVT @ R ) @ U )
= bot_bot_set_variable )
@ ( uSubst516392818stappt @ ( uSubst1408967625substt @ Sigma_theta ) @ bot_bot_set_variable @ R )
@ none_trm )
@ ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma
@ F2 ) )
@ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappt.simps(4)
thf(fact_336_trm_Oinject_I4_J,axiom,
! [X41: char,X42: trm,Y41: char,Y42: trm] :
( ( ( func @ X41 @ X42 )
= ( func @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_337_usubstappt__func2,axiom,
! [Sigma: produc1418842292n_game,F2: char,R2: trm,U: set_variable,Theta: trm] :
( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma
@ F2 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
!= bot_bot_set_variable )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= none_trm ) ) ) ).
% usubstappt_func2
thf(fact_338_usubstappt__func,axiom,
! [Sigma: produc1418842292n_game,F2: char,R2: trm,U: set_variable,Theta: trm,Sigma_theta2: trm] :
( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma
@ F2 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( some_trm @ Sigma_theta2 ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= ( uSubst516392818stappt @ ( uSubst1408967625substt @ Sigma_theta2 ) @ bot_bot_set_variable @ R2 ) ) ) ) ) ).
% usubstappt_func
thf(fact_339_usubstappt__func__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F2: char,Theta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma
@ F2 )
= none_trm )
| ? [R3: trm] :
( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma
@ F2 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ) ).
% usubstappt_func_conv
thf(fact_340_usappconst__simp,axiom,
! [Sigma: produc1418842292n_game,F2: char,R2: trm,U: set_variable] :
( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( uSubst1559070963pconst @ Sigma @ U @ F2 )
= ( some_trm @ R2 ) ) ) ) ).
% usappconst_simp
thf(fact_341_usappconst__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F2: char] :
( ( ( uSubst1559070963pconst @ Sigma @ U @ F2 )
!= none_trm )
=> ( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= none_trm )
| ? [R3: trm] :
( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ).
% usappconst_conv
thf(fact_342_usappconst__def,axiom,
( uSubst1559070963pconst
= ( ^ [Sigma2: produc1418842292n_game,U2: set_variable,F: char] :
( case_o487727941rm_trm @ ( some_trm @ ( const @ F ) )
@ ^ [R: trm] :
( if_option_trm
@ ( ( inf_inf_set_variable @ ( static_FVT @ R ) @ U2 )
= bot_bot_set_variable )
@ ( some_trm @ R )
@ none_trm )
@ ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma2
@ F ) ) ) ) ).
% usappconst_def
thf(fact_343_usubstappt__const,axiom,
! [Sigma: produc1418842292n_game,F2: char,R2: trm,U: set_variable] :
( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F2 ) )
= ( some_trm @ R2 ) ) ) ) ).
% usubstappt_const
thf(fact_344_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_345_usubstappt_Osimps_I3_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F2: char] :
( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F2 ) )
= ( uSubst1559070963pconst @ Sigma @ U @ F2 ) ) ).
% usubstappt.simps(3)
thf(fact_346_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_347_usubstappt__const__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F2: char] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F2 ) )
!= none_trm )
=> ( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= none_trm )
| ? [R3: trm] :
( ( ( produc866613687on_trm
@ ^ [F0: char > option_trm] :
( produc1231571704on_trm
@ ^ [Uu: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F0 ) )
@ Sigma
@ F2 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ).
% usubstappt_const_conv
thf(fact_348_subset__Collect__iff,axiom,
! [B2: set_variable,A3: set_variable,P3: variable > $o] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ( ord_le282106107riable @ B2
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ A3 )
& ( P3 @ X ) ) ) )
= ( ! [X: variable] :
( ( member_variable @ X @ B2 )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_349_subset__CollectI,axiom,
! [B2: set_variable,A3: set_variable,Q: variable > $o,P3: variable > $o] :
( ( ord_le282106107riable @ B2 @ A3 )
=> ( ! [X5: variable] :
( ( member_variable @ X5 @ B2 )
=> ( ( Q @ X5 )
=> ( P3 @ X5 ) ) )
=> ( ord_le282106107riable
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ A3 )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_350_conj__subset__def,axiom,
! [A3: set_variable,P3: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ A3
@ ( collect_variable
@ ^ [X: variable] :
( ( P3 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le282106107riable @ A3 @ ( collect_variable @ P3 ) )
& ( ord_le282106107riable @ A3 @ ( collect_variable @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_351_Diamondo_Ocases,axiom,
! [X3: produc1739631244on_fml] :
( ! [Alpha: game,Phi2: fml] :
( X3
!= ( produc2124676926on_fml @ ( some_game @ Alpha ) @ ( some_fml @ Phi2 ) ) )
=> ( ! [Phi2: option_fml] :
( X3
!= ( produc2124676926on_fml @ none_game @ Phi2 ) )
=> ~ ! [V6: game] :
( X3
!= ( produc2124676926on_fml @ ( some_game @ V6 ) @ none_fml ) ) ) ) ).
% Diamondo.cases
% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_It__Syntax__Ofml_J_T,axiom,
! [X3: option_fml,Y3: option_fml] :
( ( if_option_fml @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Syntax__Ofml_J_T,axiom,
! [X3: option_fml,Y3: option_fml] :
( ( if_option_fml @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [X3: option_trm,Y3: option_trm] :
( ( if_option_trm @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [X3: option_trm,Y3: option_trm] :
( ( if_option_trm @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) )
!= none_fml )
| ( ( produc1250175341on_fml
@ ^ [Uu: char > option_trm] :
( produc1615133358on_fml
@ ^ [Uv: char > option_trm] :
( produc1420172399on_fml
@ ^ [P: char > option_fml,Uw: char > option_game] : P ) )
@ sigma
@ p )
= none_fml ) ) ).
%------------------------------------------------------------------------------