TPTP Problem File: ITP203^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP203^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_534__6335794_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_534__6335794_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 501 ( 247 unt; 143 typ; 0 def)
% Number of atoms : 842 ( 658 equ; 0 cnn)
% Maximal formula atoms : 31 ( 2 avg)
% Number of connectives : 3409 ( 189 ~; 14 |; 44 &;2816 @)
% ( 0 <=>; 346 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 5 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 993 ( 993 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 116 usr; 17 con; 0-8 aty)
% Number of variables : 1376 ( 387 ^; 971 !; 18 ?;1376 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:41.418
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Product____Type__Oprod_It__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_Mt__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
produc1699626005le_trm: $tType ).
thf(ty_n_t__Set__Oset_It__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_J,type,
set_Pr720454634n_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__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__Set__Oset_It__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,
set_Pr739694057n_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 ).
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set_Pr79606376n_game: $tType ).
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produc799602994n_game: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ogame_J_Mt__Option__Ooption_It__Syntax__Ogame_J_J,type,
produc1016592119n_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__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Otrm_J_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc1072747943on_trm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Syntax__Ofml_J_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc46605351on_fml: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_J,type,
set_Pr943832262on_trm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
set_Pr1235432374le_trm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Syntax__Ovariable_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc1024976764on_trm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Syntax__Ovariable_Mt__Option__Ooption_It__Syntax__Ofml_J_J,type,
produc1408538418on_fml: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
produc464795408on_trm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J,type,
produc1367888000le_trm: $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 ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
% Explicit typings (116)
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_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_M_Eo_J_J,type,
inf_in243546115game_o: ( ( char > option_fml ) > ( char > option_game ) > $o ) > ( ( char > option_fml ) > ( char > option_game ) > $o ) > ( char > option_fml ) > ( char > option_game ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_M_062_It__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_M_Eo_J_J,type,
inf_in1072207554game_o: ( ( char > option_trm ) > produc799602994n_game > $o ) > ( ( char > option_trm ) > produc799602994n_game > $o ) > ( char > option_trm ) > produc799602994n_game > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_M_062_It__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_M_Eo_J_J,type,
inf_in1920662529game_o: ( ( char > option_trm ) > produc62062515n_game > $o ) > ( ( char > option_trm ) > produc62062515n_game > $o ) > ( char > option_trm ) > produc62062515n_game > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Syntax__Ovariable_J_M_062_It__Syntax__Otrm_M_Eo_J_J,type,
inf_in185389155_trm_o: ( set_variable > trm > $o ) > ( set_variable > trm > $o ) > set_variable > trm > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__String__Ochar_M_062_It__Option__Ooption_It__Syntax__Otrm_J_M_Eo_J_J,type,
inf_in605351507_trm_o: ( char > option_trm > $o ) > ( char > option_trm > $o ) > char > option_trm > $o ).
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_001t__Set__Oset_It__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,
inf_in1391117306n_game: set_Pr79606376n_game > set_Pr79606376n_game > set_Pr79606376n_game ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__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,
inf_in69689467n_game: set_Pr739694057n_game > set_Pr739694057n_game > set_Pr739694057n_game ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__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_J,type,
inf_in360367996n_game: set_Pr720454634n_game > set_Pr720454634n_game > set_Pr720454634n_game ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
inf_in906025800le_trm: set_Pr1235432374le_trm > set_Pr1235432374le_trm > set_Pr1235432374le_trm ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_J,type,
inf_in1243659096on_trm: set_Pr943832262on_trm > set_Pr943832262on_trm > set_Pr943832262on_trm ).
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_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_M_Eo_J_J,type,
bot_bo1895715589game_o: ( char > option_fml ) > ( char > option_game ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_M_062_It__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_M_Eo_J_J,type,
bot_bo1417568964game_o: ( char > option_trm ) > produc799602994n_game > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_M_062_It__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_M_Eo_J_J,type,
bot_bo849148163game_o: ( char > option_trm ) > produc62062515n_game > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Syntax__Ovariable_J_M_062_It__Syntax__Otrm_M_Eo_J_J,type,
bot_bo512610661_trm_o: set_variable > trm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__String__Ochar_M_062_It__Option__Ooption_It__Syntax__Otrm_J_M_Eo_J_J,type,
bot_bo102862421_trm_o: char > option_trm > $o ).
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__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,
bot_bo456479484n_game: set_Pr79606376n_game ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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,
bot_bo1302831229n_game: set_Pr739694057n_game ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_J,type,
bot_bo1453406334n_game: set_Pr720454634n_game ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
bot_bo896421962le_trm: set_Pr1235432374le_trm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_J,type,
bot_bo18861914on_trm: set_Pr943832262on_trm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Ovariable_J,type,
bot_bot_set_variable: set_variable ).
thf(sy_c_Product__Type_OPair_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,type,
produc123610156n_game: ( char > option_fml ) > ( char > option_game ) > produc799602994n_game ).
thf(sy_c_Product__Type_OPair_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,type,
produc1048060077n_game: ( char > option_trm ) > produc799602994n_game > produc62062515n_game ).
thf(sy_c_Product__Type_OPair_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,type,
produc1618635950n_game: ( char > option_trm ) > produc62062515n_game > produc1418842292n_game ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Ofml_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc909629463on_fml: option_fml > option_fml > produc46605351on_fml ).
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_OPair_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ogame_J,type,
produc1111617711n_game: option_game > option_game > produc1016592119n_game ).
thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Syntax__Otrm_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc709594775on_trm: option_trm > option_trm > produc1072747943on_trm ).
thf(sy_c_Product__Type_OPair_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__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_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J,type,
produc2015636557le_trm: produc1418842292n_game > produc1367888000le_trm > produc1699626005le_trm ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Syntax__Otrm,type,
produc1482344242le_trm: set_variable > trm > produc1367888000le_trm ).
thf(sy_c_Product__Type_OPair_001t__String__Ochar_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc1500300866on_trm: char > option_trm > produc464795408on_trm ).
thf(sy_c_Product__Type_OPair_001t__Syntax__Ovariable_001t__Option__Ooption_It__Syntax__Ofml_J,type,
produc1205977188on_fml: variable > option_fml > produc1408538418on_fml ).
thf(sy_c_Product__Type_OPair_001t__Syntax__Ovariable_001t__Option__Ooption_It__Syntax__Otrm_J,type,
produc1743721902on_trm: variable > option_trm > produc1024976764on_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_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_001_Eo,type,
produc159680767game_o: ( ( char > option_fml ) > ( char > option_game ) > $o ) > produc799602994n_game > $o ).
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_001_Eo,type,
produc1868740670game_o: ( ( char > option_trm ) > produc799602994n_game > $o ) > produc62062515n_game > $o ).
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_001_Eo,type,
produc1707653117game_o: ( ( char > option_trm ) > produc62062515n_game > $o ) > produc1418842292n_game > $o ).
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_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Syntax__Otrm_001_Eo,type,
produc82209913_trm_o: ( set_variable > trm > $o ) > produc1367888000le_trm > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__String__Ochar_001t__Option__Ooption_It__Syntax__Otrm_J_001_Eo,type,
produc584666985_trm_o: ( char > option_trm > $o ) > produc464795408on_trm > $o ).
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_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
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_ODifferential,type,
differential: trm > trm ).
thf(sy_c_Syntax_Otrm_OFunc,type,
func: char > trm > trm ).
thf(sy_c_Syntax_Otrm_ONumber,type,
number: real > trm ).
thf(sy_c_Syntax_Otrm_OPlus,type,
plus: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OVar,type,
var: variable > trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_ODifferentialo,type,
uSubst716837505ntialo: option_trm > option_trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OGeqo,type,
uSubst152838031e_Geqo: option_trm > option_trm > option_fml ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OPluso,type,
uSubst141955042_Pluso: option_trm > option_trm > option_trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OTesto,type,
uSubst1367128042_Testo: option_fml > option_game ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OTimeso,type,
uSubst918876924Timeso: option_trm > option_trm > option_trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Odot,type,
uSubst_Mirabelle_dot: trm ).
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_USubst__Mirabelle__vidvnmlwwz_Ousubstappt__rel,type,
uSubst1917534215pt_rel: produc1699626005le_trm > produc1699626005le_trm > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__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_Mt__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
accp_P1877119646le_trm: ( produc1699626005le_trm > produc1699626005le_trm > $o ) > produc1699626005le_trm > $o ).
thf(sy_c_member_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,type,
member849814089n_game: produc799602994n_game > set_Pr79606376n_game > $o ).
thf(sy_c_member_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,type,
member774167626n_game: produc62062515n_game > set_Pr739694057n_game > $o ).
thf(sy_c_member_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__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,
member1940201675n_game: produc1418842292n_game > set_Pr720454634n_game > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J,type,
member2122665623le_trm: produc1367888000le_trm > set_Pr1235432374le_trm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J,type,
member694524327on_trm: produc464795408on_trm > set_Pr943832262on_trm > $o ).
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 ).
% Relevant facts (352)
thf(fact_0_f7,axiom,
! [Z: option_fml,F: fml > option_fml,Za: option_fml] :
( ( ( Za = none_fml )
=> ( ( case_o209804997ml_fml @ Z @ F @ Za )
= Z ) )
& ( ( Za != none_fml )
=> ( ( case_o209804997ml_fml @ Z @ F @ Za )
= ( F @ ( the_fml @ Za ) ) ) ) ) ).
% f7
thf(fact_1_Pred_Oprems_I2_J,axiom,
( ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) )
!= none_fml ) ).
% Pred.prems(2)
thf(fact_2_case__prod__app,axiom,
( produc1250175341on_fml
= ( ^ [F2: ( char > option_trm ) > produc62062515n_game > char > option_fml,X: produc1418842292n_game,Y: char] :
( produc1499310791on_fml
@ ^ [L: char > option_trm,R: produc62062515n_game] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_3_case__prod__app,axiom,
( produc1231571704on_trm
= ( ^ [F2: ( char > option_trm ) > produc799602994n_game > char > option_trm,X: produc62062515n_game,Y: char] :
( produc692578258on_trm
@ ^ [L: char > option_trm,R: produc799602994n_game] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_4_case__prod__app,axiom,
( produc1615133358on_fml
= ( ^ [F2: ( char > option_trm ) > produc799602994n_game > char > option_fml,X: produc62062515n_game,Y: char] :
( produc154833544on_fml
@ ^ [L: char > option_trm,R: produc799602994n_game] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_5_case__prod__app,axiom,
( produc1036610745on_trm
= ( ^ [F2: ( char > option_fml ) > ( char > option_game ) > char > option_trm,X: produc799602994n_game,Y: char] :
( produc885547027on_trm
@ ^ [L: char > option_fml,R: char > option_game] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_6_case__prod__app,axiom,
( produc1420172399on_fml
= ( ^ [F2: ( char > option_fml ) > ( char > option_game ) > char > option_fml,X: produc799602994n_game,Y: char] :
( produc347802313on_fml
@ ^ [L: char > option_fml,R: char > option_game] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_7_Pred_Oprems_I1_J,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= none_fml ) ).
% Pred.prems(1)
thf(fact_8__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_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_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 )
=> ( none_fml
= ( 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 ) ) ) ) ) ).
% \<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 (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p))\<close>
thf(fact_9_f9,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 ) ) ) ) ).
% f9
thf(fact_10__092_060open_062_Icase_ASPreds_A_092_060sigma_062_Ap_Aof_ANone_A_092_060Rightarrow_062_AAfml_A_IPred_Ap_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_124_ASome_Af_A_092_060Rightarrow_062_Aif_AFVF_Af_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_Af_Aelse_Aundeff_J_A_092_060noteq_062_A_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_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 )
=> ( ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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 ) )
= ( 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 ) ) ) ) )
& ( ( ( 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 )
=> ( ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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 ) ) ) ).
% \<open>(case SPreds \<sigma> p of None \<Rightarrow> Afml (Pred p (the (usubstappt \<sigma> V \<theta>))) | Some f \<Rightarrow> if FVF f \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} f else undeff) \<noteq> (if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff)\<close>
thf(fact_11_Ando_Oinduct,axiom,
! [P2: option_fml > option_fml > $o,A0: option_fml,A1: option_fml] :
( ! [Phi: fml,Psi: fml] : ( P2 @ ( some_fml @ Phi ) @ ( some_fml @ Psi ) )
=> ( ! [X_1: option_fml] : ( P2 @ none_fml @ X_1 )
=> ( ! [V: fml] : ( P2 @ ( some_fml @ V ) @ none_fml )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Ando.induct
thf(fact_12_Testo_Ocases,axiom,
! [X2: option_fml] :
( ! [Phi: fml] :
( X2
!= ( some_fml @ Phi ) )
=> ( X2 = none_fml ) ) ).
% Testo.cases
thf(fact_13_Testo_Oinduct,axiom,
! [P2: option_fml > $o,A0: option_fml] :
( ! [Phi: fml] : ( P2 @ ( some_fml @ Phi ) )
=> ( ( P2 @ none_fml )
=> ( P2 @ A0 ) ) ) ).
% Testo.induct
thf(fact_14_undeff__equiv,axiom,
! [Phi2: option_fml] :
( ( Phi2 != none_fml )
= ( ? [F2: fml] :
( Phi2
= ( some_fml @ F2 ) ) ) ) ).
% undeff_equiv
thf(fact_15_Existso_Oinduct,axiom,
! [P2: variable > option_fml > $o,A0: variable,A1: option_fml] :
( ! [X3: variable,Phi: fml] : ( P2 @ X3 @ ( some_fml @ Phi ) )
=> ( ! [X3: variable] : ( P2 @ X3 @ none_fml )
=> ( P2 @ A0 @ A1 ) ) ) ).
% Existso.induct
thf(fact_16_prod_Ocase__distrib,axiom,
! [H: option_fml > option_fml,F3: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F3 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_17_prod_Ocase__distrib,axiom,
! [H: option_fml > option_trm,F3: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F3 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_18_prod_Ocase__distrib,axiom,
! [H: option_trm > option_fml,F3: ( char > option_fml ) > ( char > option_game ) > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc885547027on_trm @ F3 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_19_prod_Ocase__distrib,axiom,
! [H: option_trm > option_trm,F3: ( char > option_fml ) > ( char > option_game ) > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc885547027on_trm @ F3 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_20_prod_Ocase__distrib,axiom,
! [H: ( char > option_trm ) > option_fml,F3: ( char > option_fml ) > ( char > option_game ) > char > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc1036610745on_trm @ F3 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_21_prod_Ocase__distrib,axiom,
! [H: ( char > option_trm ) > option_trm,F3: ( char > option_fml ) > ( char > option_game ) > char > option_trm,Prod: produc799602994n_game] :
( ( H @ ( produc1036610745on_trm @ F3 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_22_prod_Ocase__distrib,axiom,
! [H: ( char > option_fml ) > option_fml,F3: ( char > option_fml ) > ( char > option_game ) > char > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc1420172399on_fml @ F3 @ Prod ) )
= ( produc347802313on_fml
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_23_prod_Ocase__distrib,axiom,
! [H: ( char > option_fml ) > option_trm,F3: ( char > option_fml ) > ( char > option_game ) > char > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc1420172399on_fml @ F3 @ Prod ) )
= ( produc885547027on_trm
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_24_prod_Ocase__distrib,axiom,
! [H: option_fml > char > option_trm,F3: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F3 @ Prod ) )
= ( produc1036610745on_trm
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_25_prod_Ocase__distrib,axiom,
! [H: option_fml > char > option_fml,F3: ( char > option_fml ) > ( char > option_game ) > option_fml,Prod: produc799602994n_game] :
( ( H @ ( produc347802313on_fml @ F3 @ Prod ) )
= ( produc1420172399on_fml
@ ^ [X1: char > option_fml,X22: char > option_game] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_26__092_060open_062usubstappf_A_092_060sigma_062_AU_A_IPred_Ap_A_092_060theta_062_J_A_092_060noteq_062_Ausubstappf_A_092_060sigma_062_AV_A_IPred_Ap_A_092_060theta_062_J_092_060close_062,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( uSubst516392804stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ).
% \<open>usubstappf \<sigma> U (Pred p \<theta>) \<noteq> usubstappf \<sigma> V (Pred p \<theta>)\<close>
thf(fact_27_f5,axiom,
( ( uSubst516392818stappt @ sigma @ ua @ theta )
= ( uSubst516392818stappt @ sigma @ va @ theta ) ) ).
% f5
thf(fact_28_calculation_I1_J,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 @ sigma @ ua @ ( pred @ p @ theta ) )
= ( 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 ) ) ) ) )
& ( ( ( 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 @ sigma @ ua @ ( pred @ p @ theta ) )
= none_fml ) ) ) ).
% calculation(1)
thf(fact_29_calculation_I2_J,axiom,
( ( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( 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 ) ) ) )
=> ( ( 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 ) ) ).
% calculation(2)
thf(fact_30_f6,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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_31__092_060open_062usubstappf_A_092_060sigma_062_AU_A_IPred_Ap_A_092_060theta_062_J_A_092_060noteq_062_A_Icase_ASPreds_A_092_060sigma_062_Ap_Aof_ANone_A_092_060Rightarrow_062_AAfml_A_IPred_Ap_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_124_ASome_Af_A_092_060Rightarrow_062_Aif_AFVF_Af_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_Af_Aelse_Aundeff_J_092_060close_062,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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 ) ) ) ).
% \<open>usubstappf \<sigma> U (Pred p \<theta>) \<noteq> (case SPreds \<sigma> p of None \<Rightarrow> Afml (Pred p (the (usubstappt \<sigma> V \<theta>))) | Some f \<Rightarrow> if FVF f \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} f else undeff)\<close>
thf(fact_32_usubstappf__pred2,axiom,
! [Sigma: produc1418842292n_game,P3: 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
@ P3 )
= ( some_fml @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVF @ R2 ) @ U )
!= bot_bot_set_variable )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
= none_fml ) ) ) ).
% usubstappf_pred2
thf(fact_33_f3,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F2 )
@ 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 ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ua @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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 ) ) ) ).
% f3
thf(fact_34_f8,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F2 )
@ 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 @ va @ theta ) )
= ( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ ( the_trm @ ( uSubst516392818stappt @ sigma @ va @ theta ) ) ) @ bot_bot_set_variable @ F2 )
@ 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 ) ) ) ).
% f8
thf(fact_35_f1,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ ua )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F2 )
@ 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 ) ).
% f1
thf(fact_36_f4,axiom,
( ( case_o325794063ml_trm @ none_fml
@ ^ [T: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ p @ T ) )
@ ^ [F2: fml] :
( if_option_fml
@ ( ( inf_inf_set_variable @ ( static_FVF @ F2 ) @ va )
= bot_bot_set_variable )
@ ( uSubst516392804stappf @ ( uSubst1408967625substt @ T ) @ bot_bot_set_variable @ F2 )
@ 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 @ va @ theta ) )
!= none_fml ) ).
% f4
thf(fact_37_option_Ocollapse,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( ( some_fml @ ( the_fml @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_38_option_Ocollapse,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( ( some_trm @ ( the_trm @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_39_option_Ocollapse,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( ( some_game @ ( the_game @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_40_not__Some__eq,axiom,
! [X2: option_fml] :
( ( ! [Y: fml] :
( X2
!= ( some_fml @ Y ) ) )
= ( X2 = none_fml ) ) ).
% not_Some_eq
thf(fact_41_not__Some__eq,axiom,
! [X2: option_trm] :
( ( ! [Y: trm] :
( X2
!= ( some_trm @ Y ) ) )
= ( X2 = none_trm ) ) ).
% not_Some_eq
thf(fact_42_not__Some__eq,axiom,
! [X2: option_game] :
( ( ! [Y: game] :
( X2
!= ( some_game @ Y ) ) )
= ( X2 = none_game ) ) ).
% not_Some_eq
thf(fact_43_not__None__eq,axiom,
! [X2: option_fml] :
( ( X2 != none_fml )
= ( ? [Y: fml] :
( X2
= ( some_fml @ Y ) ) ) ) ).
% not_None_eq
thf(fact_44_not__None__eq,axiom,
! [X2: option_trm] :
( ( X2 != none_trm )
= ( ? [Y: trm] :
( X2
= ( some_trm @ Y ) ) ) ) ).
% not_None_eq
thf(fact_45_not__None__eq,axiom,
! [X2: option_game] :
( ( X2 != none_game )
= ( ? [Y: game] :
( X2
= ( some_game @ Y ) ) ) ) ).
% not_None_eq
thf(fact_46_usubstappf_Osimps_I1_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,P3: char,Theta: trm] :
( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
= ( case_o325794063ml_trm @ none_fml
@ ^ [Sigma_theta: trm] :
( case_o209804997ml_fml @ ( some_fml @ ( pred @ P3 @ 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
@ P3 ) )
@ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappf.simps(1)
thf(fact_47_inf__bot__left,axiom,
! [X2: set_variable] :
( ( inf_inf_set_variable @ bot_bot_set_variable @ X2 )
= bot_bot_set_variable ) ).
% inf_bot_left
thf(fact_48_inf__bot__right,axiom,
! [X2: set_variable] :
( ( inf_inf_set_variable @ X2 @ bot_bot_set_variable )
= bot_bot_set_variable ) ).
% inf_bot_right
thf(fact_49_option_Osplit__sel__asm,axiom,
! [P2: option_fml > $o,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( P2 @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_fml )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_50_option_Osplit__sel__asm,axiom,
! [P2: option_fml > $o,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( P2 @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_trm )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_51_option_Osplit__sel__asm,axiom,
! [P2: option_trm > $o,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( P2 @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option = none_trm )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_52_f2,axiom,
! [Z: option_fml,F: trm > option_fml,Za: option_trm] :
( ( ( Za = none_trm )
=> ( ( case_o325794063ml_trm @ Z @ F @ Za )
= Z ) )
& ( ( Za != none_trm )
=> ( ( case_o325794063ml_trm @ Z @ F @ Za )
= ( F @ ( the_trm @ Za ) ) ) ) ) ).
% f2
thf(fact_53_inf__right__idem,axiom,
! [X2: set_variable,Y2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_variable @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_54_inf_Oright__idem,axiom,
! [A: set_variable,B: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A @ B ) @ B )
= ( inf_inf_set_variable @ A @ B ) ) ).
% inf.right_idem
thf(fact_55_inf__left__idem,axiom,
! [X2: set_variable,Y2: set_variable] :
( ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ X2 @ Y2 ) )
= ( inf_inf_set_variable @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_56_inf_Oleft__idem,axiom,
! [A: set_variable,B: set_variable] :
( ( inf_inf_set_variable @ A @ ( inf_inf_set_variable @ A @ B ) )
= ( inf_inf_set_variable @ A @ B ) ) ).
% inf.left_idem
thf(fact_57_inf__idem,axiom,
! [X2: set_variable] :
( ( inf_inf_set_variable @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_58_inf_Oidem,axiom,
! [A: set_variable] :
( ( inf_inf_set_variable @ A @ A )
= A ) ).
% inf.idem
thf(fact_59_option_Oinject,axiom,
! [X23: fml,Y22: fml] :
( ( ( some_fml @ X23 )
= ( some_fml @ Y22 ) )
= ( X23 = Y22 ) ) ).
% option.inject
thf(fact_60_option_Oinject,axiom,
! [X23: trm,Y22: trm] :
( ( ( some_trm @ X23 )
= ( some_trm @ Y22 ) )
= ( X23 = Y22 ) ) ).
% option.inject
thf(fact_61_option_Oinject,axiom,
! [X23: game,Y22: game] :
( ( ( some_game @ X23 )
= ( some_game @ Y22 ) )
= ( X23 = Y22 ) ) ).
% option.inject
thf(fact_62_usubstappf__pred,axiom,
! [Sigma: produc1418842292n_game,P3: 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
@ P3 )
= ( 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 @ P3 @ Theta ) )
= ( uSubst516392804stappf @ ( uSubst1408967625substt @ Sigma_theta2 ) @ bot_bot_set_variable @ R2 ) ) ) ) ) ).
% usubstappf_pred
thf(fact_63_Diamondo_Oinduct,axiom,
! [P2: option_game > option_fml > $o,A0: option_game,A1: option_fml] :
( ! [Alpha: game,Phi: fml] : ( P2 @ ( some_game @ Alpha ) @ ( some_fml @ Phi ) )
=> ( ! [X_1: option_fml] : ( P2 @ none_game @ X_1 )
=> ( ! [V: game] : ( P2 @ ( some_game @ V ) @ none_fml )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Diamondo.induct
thf(fact_64_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_65_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_66_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_67_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_68_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_69_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_70_usubstappt__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,V2: set_variable] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ V2 @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V2 @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_71_mem__Collect__eq,axiom,
! [A: variable,P2: variable > $o] :
( ( member_variable @ A @ ( collect_variable @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_variable] :
( ( collect_variable
@ ^ [X: variable] : ( member_variable @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
! [P2: variable > $o,Q: variable > $o] :
( ! [X3: variable] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_variable @ P2 )
= ( collect_variable @ Q ) ) ) ).
% Collect_cong
thf(fact_74_case__optionE,axiom,
! [P2: $o,Q: fml > $o,X2: option_fml] :
( ( case_option_o_fml @ P2 @ Q @ X2 )
=> ( ( ( X2 = none_fml )
=> ~ P2 )
=> ~ ! [Y3: fml] :
( ( X2
= ( some_fml @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_75_case__optionE,axiom,
! [P2: $o,Q: trm > $o,X2: option_trm] :
( ( case_option_o_trm @ P2 @ Q @ X2 )
=> ( ( ( X2 = none_trm )
=> ~ P2 )
=> ~ ! [Y3: trm] :
( ( X2
= ( some_trm @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_76_case__optionE,axiom,
! [P2: $o,Q: game > $o,X2: option_game] :
( ( case_option_o_game @ P2 @ Q @ X2 )
=> ( ( ( X2 = none_game )
=> ~ P2 )
=> ~ ! [Y3: game] :
( ( X2
= ( some_game @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_77_inf__left__commute,axiom,
! [X2: set_variable,Y2: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ Y2 @ Z2 ) )
= ( inf_inf_set_variable @ Y2 @ ( inf_inf_set_variable @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_78_inf_Oleft__commute,axiom,
! [B: set_variable,A: set_variable,C: set_variable] :
( ( inf_inf_set_variable @ B @ ( inf_inf_set_variable @ A @ C ) )
= ( inf_inf_set_variable @ A @ ( inf_inf_set_variable @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_79_inf__commute,axiom,
( inf_inf_set_variable
= ( ^ [X: set_variable,Y: set_variable] : ( inf_inf_set_variable @ Y @ X ) ) ) ).
% inf_commute
thf(fact_80_inf_Ocommute,axiom,
( inf_inf_set_variable
= ( ^ [A3: set_variable,B2: set_variable] : ( inf_inf_set_variable @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_81_inf__assoc,axiom,
! [X2: set_variable,Y2: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X2 @ Y2 ) @ Z2 )
= ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_82_inf_Oassoc,axiom,
! [A: set_variable,B: set_variable,C: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A @ B ) @ C )
= ( inf_inf_set_variable @ A @ ( inf_inf_set_variable @ B @ C ) ) ) ).
% inf.assoc
thf(fact_83_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_variable,K: set_variable,B: set_variable,A: set_variable] :
( ( B3
= ( inf_inf_set_variable @ K @ B ) )
=> ( ( inf_inf_set_variable @ A @ B3 )
= ( inf_inf_set_variable @ K @ ( inf_inf_set_variable @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_84_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_variable,K: set_variable,A: set_variable,B: set_variable] :
( ( A2
= ( inf_inf_set_variable @ K @ A ) )
=> ( ( inf_inf_set_variable @ A2 @ B )
= ( inf_inf_set_variable @ K @ ( inf_inf_set_variable @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_85_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_86_inf__sup__aci_I2_J,axiom,
! [X2: set_variable,Y2: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X2 @ Y2 ) @ Z2 )
= ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_87_inf__sup__aci_I3_J,axiom,
! [X2: set_variable,Y2: set_variable,Z2: set_variable] :
( ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ Y2 @ Z2 ) )
= ( inf_inf_set_variable @ Y2 @ ( inf_inf_set_variable @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_88_inf__sup__aci_I4_J,axiom,
! [X2: set_variable,Y2: set_variable] :
( ( inf_inf_set_variable @ X2 @ ( inf_inf_set_variable @ X2 @ Y2 ) )
= ( inf_inf_set_variable @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_89_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_90_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_91_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_92_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_93_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_94_option_Odistinct_I1_J,axiom,
! [X23: fml] :
( none_fml
!= ( some_fml @ X23 ) ) ).
% option.distinct(1)
thf(fact_95_option_Odistinct_I1_J,axiom,
! [X23: trm] :
( none_trm
!= ( some_trm @ X23 ) ) ).
% option.distinct(1)
thf(fact_96_option_Odistinct_I1_J,axiom,
! [X23: game] :
( none_game
!= ( some_game @ X23 ) ) ).
% option.distinct(1)
thf(fact_97_option_OdiscI,axiom,
! [Option: option_fml,X23: fml] :
( ( Option
= ( some_fml @ X23 ) )
=> ( Option != none_fml ) ) ).
% option.discI
thf(fact_98_option_OdiscI,axiom,
! [Option: option_trm,X23: trm] :
( ( Option
= ( some_trm @ X23 ) )
=> ( Option != none_trm ) ) ).
% option.discI
thf(fact_99_option_OdiscI,axiom,
! [Option: option_game,X23: game] :
( ( Option
= ( some_game @ X23 ) )
=> ( Option != none_game ) ) ).
% option.discI
thf(fact_100_option_Oexhaust,axiom,
! [Y2: option_fml] :
( ( Y2 != none_fml )
=> ~ ! [X24: fml] :
( Y2
!= ( some_fml @ X24 ) ) ) ).
% option.exhaust
thf(fact_101_option_Oexhaust,axiom,
! [Y2: option_trm] :
( ( Y2 != none_trm )
=> ~ ! [X24: trm] :
( Y2
!= ( some_trm @ X24 ) ) ) ).
% option.exhaust
thf(fact_102_option_Oexhaust,axiom,
! [Y2: option_game] :
( ( Y2 != none_game )
=> ~ ! [X24: game] :
( Y2
!= ( some_game @ X24 ) ) ) ).
% option.exhaust
thf(fact_103_option_Oinducts,axiom,
! [P2: option_fml > $o,Option: option_fml] :
( ( P2 @ none_fml )
=> ( ! [X3: fml] : ( P2 @ ( some_fml @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_104_option_Oinducts,axiom,
! [P2: option_trm > $o,Option: option_trm] :
( ( P2 @ none_trm )
=> ( ! [X3: trm] : ( P2 @ ( some_trm @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_105_option_Oinducts,axiom,
! [P2: option_game > $o,Option: option_game] :
( ( P2 @ none_game )
=> ( ! [X3: game] : ( P2 @ ( some_game @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_106_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_107_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_108_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_109_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_110_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_111_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_112_combine__options__cases,axiom,
! [X2: option_fml,P2: option_fml > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: fml,B4: fml] :
( ( X2
= ( some_fml @ A4 ) )
=> ( ( Y2
= ( some_fml @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_113_combine__options__cases,axiom,
! [X2: option_fml,P2: option_fml > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: fml,B4: trm] :
( ( X2
= ( some_fml @ A4 ) )
=> ( ( Y2
= ( some_trm @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_114_combine__options__cases,axiom,
! [X2: option_fml,P2: option_fml > option_game > $o,Y2: option_game] :
( ( ( X2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: fml,B4: game] :
( ( X2
= ( some_fml @ A4 ) )
=> ( ( Y2
= ( some_game @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_115_combine__options__cases,axiom,
! [X2: option_trm,P2: option_trm > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: trm,B4: fml] :
( ( X2
= ( some_trm @ A4 ) )
=> ( ( Y2
= ( some_fml @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_116_combine__options__cases,axiom,
! [X2: option_trm,P2: option_trm > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: trm,B4: trm] :
( ( X2
= ( some_trm @ A4 ) )
=> ( ( Y2
= ( some_trm @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_117_combine__options__cases,axiom,
! [X2: option_trm,P2: option_trm > option_game > $o,Y2: option_game] :
( ( ( X2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: trm,B4: game] :
( ( X2
= ( some_trm @ A4 ) )
=> ( ( Y2
= ( some_game @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_118_combine__options__cases,axiom,
! [X2: option_game,P2: option_game > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: game,B4: fml] :
( ( X2
= ( some_game @ A4 ) )
=> ( ( Y2
= ( some_fml @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_119_combine__options__cases,axiom,
! [X2: option_game,P2: option_game > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: game,B4: trm] :
( ( X2
= ( some_game @ A4 ) )
=> ( ( Y2
= ( some_trm @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_120_combine__options__cases,axiom,
! [X2: option_game,P2: option_game > option_game > $o,Y2: option_game] :
( ( ( X2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A4: game,B4: game] :
( ( X2
= ( some_game @ A4 ) )
=> ( ( Y2
= ( some_game @ B4 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_121_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_122_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_123_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_124_option_Oexpand,axiom,
! [Option: option_fml,Option2: option_fml] :
( ( ( Option = none_fml )
= ( Option2 = none_fml ) )
=> ( ( ( Option != none_fml )
=> ( ( Option2 != none_fml )
=> ( ( the_fml @ Option )
= ( the_fml @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_125_option_Oexpand,axiom,
! [Option: option_trm,Option2: option_trm] :
( ( ( Option = none_trm )
= ( Option2 = none_trm ) )
=> ( ( ( Option != none_trm )
=> ( ( Option2 != none_trm )
=> ( ( the_trm @ Option )
= ( the_trm @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_126_option_Oexpand,axiom,
! [Option: option_game,Option2: option_game] :
( ( ( Option = none_game )
= ( Option2 = none_game ) )
=> ( ( ( Option != none_game )
=> ( ( Option2 != none_game )
=> ( ( the_game @ Option )
= ( the_game @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_127_option_Osimps_I5_J,axiom,
! [F1: option_fml,F22: fml > option_fml,X23: fml] :
( ( case_o209804997ml_fml @ F1 @ F22 @ ( some_fml @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_128_option_Osimps_I5_J,axiom,
! [F1: option_fml,F22: trm > option_fml,X23: trm] :
( ( case_o325794063ml_trm @ F1 @ F22 @ ( some_trm @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_129_option_Osimps_I5_J,axiom,
! [F1: option_trm,F22: trm > option_trm,X23: trm] :
( ( case_o487727941rm_trm @ F1 @ F22 @ ( some_trm @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_130_option_Osel,axiom,
! [X23: fml] :
( ( the_fml @ ( some_fml @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_131_option_Osel,axiom,
! [X23: trm] :
( ( the_trm @ ( some_trm @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_132_option_Osel,axiom,
! [X23: game] :
( ( the_game @ ( some_game @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_133_option_Oexhaust__sel,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( Option
= ( some_fml @ ( the_fml @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_134_option_Oexhaust__sel,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( Option
= ( some_trm @ ( the_trm @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_135_option_Oexhaust__sel,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( Option
= ( some_game @ ( the_game @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_136_option_Ocase__eq__if,axiom,
( case_o209804997ml_fml
= ( ^ [F12: option_fml,F23: fml > option_fml,Option3: option_fml] : ( if_option_fml @ ( Option3 = none_fml ) @ F12 @ ( F23 @ ( the_fml @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_137_option_Ocase__eq__if,axiom,
( case_o325794063ml_trm
= ( ^ [F12: option_fml,F23: trm > option_fml,Option3: option_trm] : ( if_option_fml @ ( Option3 = none_trm ) @ F12 @ ( F23 @ ( the_trm @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_138_option_Ocase__eq__if,axiom,
( case_o487727941rm_trm
= ( ^ [F12: option_trm,F23: trm > option_trm,Option3: option_trm] : ( if_option_trm @ ( Option3 = none_trm ) @ F12 @ ( F23 @ ( the_trm @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_139_usubstappf__pred__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,P3: char,Theta: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( pred @ P3 @ 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
@ P3 )
= 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
@ P3 )
= ( some_fml @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVF @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ) ).
% usubstappf_pred_conv
thf(fact_140_option_Osplit__sel,axiom,
! [P2: option_fml > $o,F1: option_fml,F22: fml > option_fml,Option: option_fml] :
( ( P2 @ ( case_o209804997ml_fml @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_fml )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_fml @ ( the_fml @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_fml @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_141_option_Osplit__sel,axiom,
! [P2: option_fml > $o,F1: option_fml,F22: trm > option_fml,Option: option_trm] :
( ( P2 @ ( case_o325794063ml_trm @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_trm )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_142_option_Osplit__sel,axiom,
! [P2: option_trm > $o,F1: option_trm,F22: trm > option_trm,Option: option_trm] :
( ( P2 @ ( case_o487727941rm_trm @ F1 @ F22 @ Option ) )
= ( ( ( Option = none_trm )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some_trm @ ( the_trm @ Option ) ) )
=> ( P2 @ ( F22 @ ( the_trm @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_143_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_144_Int__iff,axiom,
! [C: variable,A2: set_variable,B3: set_variable] :
( ( member_variable @ C @ ( inf_inf_set_variable @ A2 @ B3 ) )
= ( ( member_variable @ C @ A2 )
& ( member_variable @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_145_IntI,axiom,
! [C: variable,A2: set_variable,B3: set_variable] :
( ( member_variable @ C @ A2 )
=> ( ( member_variable @ C @ B3 )
=> ( member_variable @ C @ ( inf_inf_set_variable @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_146_empty__Collect__eq,axiom,
! [P2: variable > $o] :
( ( bot_bot_set_variable
= ( collect_variable @ P2 ) )
= ( ! [X: variable] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_147_Collect__empty__eq,axiom,
! [P2: variable > $o] :
( ( ( collect_variable @ P2 )
= bot_bot_set_variable )
= ( ! [X: variable] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_148_all__not__in__conv,axiom,
! [A2: set_variable] :
( ( ! [X: variable] :
~ ( member_variable @ X @ A2 ) )
= ( A2 = bot_bot_set_variable ) ) ).
% all_not_in_conv
thf(fact_149_empty__iff,axiom,
! [C: variable] :
~ ( member_variable @ C @ bot_bot_set_variable ) ).
% empty_iff
thf(fact_150_disjoint__iff__not__equal,axiom,
! [A2: set_variable,B3: set_variable] :
( ( ( inf_inf_set_variable @ A2 @ B3 )
= bot_bot_set_variable )
= ( ! [X: variable] :
( ( member_variable @ X @ A2 )
=> ! [Y: variable] :
( ( member_variable @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_151_Aterm__Some,axiom,
some_trm = some_trm ).
% Aterm_Some
thf(fact_152_Loopo_Ocases,axiom,
! [X2: option_game] :
( ! [Alpha: game] :
( X2
!= ( some_game @ Alpha ) )
=> ( X2 = none_game ) ) ).
% Loopo.cases
thf(fact_153_ODEo_Oinduct,axiom,
! [P2: char > option_trm > $o,A0: char,A1: option_trm] :
( ! [X3: char,Theta2: trm] : ( P2 @ X3 @ ( some_trm @ Theta2 ) )
=> ( ! [X3: char] : ( P2 @ X3 @ none_trm )
=> ( P2 @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_154_Loopo_Oinduct,axiom,
! [P2: option_game > $o,A0: option_game] :
( ! [Alpha: game] : ( P2 @ ( some_game @ Alpha ) )
=> ( ( P2 @ none_game )
=> ( P2 @ A0 ) ) ) ).
% Loopo.induct
thf(fact_155_undefg__equiv,axiom,
! [Alpha2: option_game] :
( ( Alpha2 != none_game )
= ( ? [G: game] :
( Alpha2
= ( some_game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_156_undeft__equiv,axiom,
! [Theta: option_trm] :
( ( Theta != none_trm )
= ( ? [T: trm] :
( Theta
= ( some_trm @ T ) ) ) ) ).
% undeft_equiv
thf(fact_157_Timeso_Oinduct,axiom,
! [P2: option_trm > option_trm > $o,A0: option_trm,A1: option_trm] :
( ! [Theta2: trm,Eta: trm] : ( P2 @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta ) )
=> ( ! [X_1: option_trm] : ( P2 @ none_trm @ X_1 )
=> ( ! [V: trm] : ( P2 @ ( some_trm @ V ) @ none_trm )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_158_Assigno_Oinduct,axiom,
! [P2: variable > option_trm > $o,A0: variable,A1: option_trm] :
( ! [X3: variable,Theta2: trm] : ( P2 @ X3 @ ( some_trm @ Theta2 ) )
=> ( ! [X3: variable] : ( P2 @ X3 @ none_trm )
=> ( P2 @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_159_Composeo_Oinduct,axiom,
! [P2: option_game > option_game > $o,A0: option_game,A1: option_game] :
( ! [Alpha: game,Beta: game] : ( P2 @ ( some_game @ Alpha ) @ ( some_game @ Beta ) )
=> ( ! [Alpha: option_game] : ( P2 @ Alpha @ none_game )
=> ( ! [V: game] : ( P2 @ none_game @ ( some_game @ V ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_160_Differentialo_Ocases,axiom,
! [X2: option_trm] :
( ! [Theta2: trm] :
( X2
!= ( some_trm @ Theta2 ) )
=> ( X2 = none_trm ) ) ).
% Differentialo.cases
thf(fact_161_Differentialo_Oinduct,axiom,
! [P2: option_trm > $o,A0: option_trm] :
( ! [Theta2: trm] : ( P2 @ ( some_trm @ Theta2 ) )
=> ( ( P2 @ none_trm )
=> ( P2 @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_162_undeft__None,axiom,
none_trm = none_trm ).
% undeft_None
thf(fact_163_emptyE,axiom,
! [A: variable] :
~ ( member_variable @ A @ bot_bot_set_variable ) ).
% emptyE
thf(fact_164_equals0D,axiom,
! [A2: set_variable,A: variable] :
( ( A2 = bot_bot_set_variable )
=> ~ ( member_variable @ A @ A2 ) ) ).
% equals0D
thf(fact_165_equals0I,axiom,
! [A2: set_variable] :
( ! [Y3: variable] :
~ ( member_variable @ Y3 @ A2 )
=> ( A2 = bot_bot_set_variable ) ) ).
% equals0I
thf(fact_166_ex__in__conv,axiom,
! [A2: set_variable] :
( ( ? [X: variable] : ( member_variable @ X @ A2 ) )
= ( A2 != bot_bot_set_variable ) ) ).
% ex_in_conv
thf(fact_167_bot__set__def,axiom,
( bot_bot_set_variable
= ( collect_variable @ bot_bot_variable_o ) ) ).
% bot_set_def
thf(fact_168_IntE,axiom,
! [C: variable,A2: set_variable,B3: set_variable] :
( ( member_variable @ C @ ( inf_inf_set_variable @ A2 @ B3 ) )
=> ~ ( ( member_variable @ C @ A2 )
=> ~ ( member_variable @ C @ B3 ) ) ) ).
% IntE
thf(fact_169_IntD1,axiom,
! [C: variable,A2: set_variable,B3: set_variable] :
( ( member_variable @ C @ ( inf_inf_set_variable @ A2 @ B3 ) )
=> ( member_variable @ C @ A2 ) ) ).
% IntD1
thf(fact_170_IntD2,axiom,
! [C: variable,A2: set_variable,B3: set_variable] :
( ( member_variable @ C @ ( inf_inf_set_variable @ A2 @ B3 ) )
=> ( member_variable @ C @ B3 ) ) ).
% IntD2
thf(fact_171_Int__assoc,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 ) ) ) ).
% Int_assoc
thf(fact_172_Int__absorb,axiom,
! [A2: set_variable] :
( ( inf_inf_set_variable @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_173_Int__commute,axiom,
( inf_inf_set_variable
= ( ^ [A5: set_variable,B5: set_variable] : ( inf_inf_set_variable @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_174_Int__left__absorb,axiom,
! [A2: set_variable,B3: set_variable] :
( ( inf_inf_set_variable @ A2 @ ( inf_inf_set_variable @ A2 @ B3 ) )
= ( inf_inf_set_variable @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_175_Int__left__commute,axiom,
! [A2: set_variable,B3: set_variable,C2: set_variable] :
( ( inf_inf_set_variable @ A2 @ ( inf_inf_set_variable @ B3 @ C2 ) )
= ( inf_inf_set_variable @ B3 @ ( inf_inf_set_variable @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_176_empty__def,axiom,
( bot_bot_set_variable
= ( collect_variable
@ ^ [X: variable] : $false ) ) ).
% empty_def
thf(fact_177_Collect__conj__eq,axiom,
! [P2: variable > $o,Q: variable > $o] :
( ( collect_variable
@ ^ [X: variable] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_variable @ ( collect_variable @ P2 ) @ ( collect_variable @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_178_Int__Collect,axiom,
! [X2: variable,A2: set_variable,P2: variable > $o] :
( ( member_variable @ X2 @ ( inf_inf_set_variable @ A2 @ ( collect_variable @ P2 ) ) )
= ( ( member_variable @ X2 @ A2 )
& ( P2 @ X2 ) ) ) ).
% Int_Collect
thf(fact_179_Int__def,axiom,
( inf_inf_set_variable
= ( ^ [A5: set_variable,B5: set_variable] :
( collect_variable
@ ^ [X: variable] :
( ( member_variable @ X @ A5 )
& ( member_variable @ X @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_180_inf__set__def,axiom,
( inf_inf_set_variable
= ( ^ [A5: set_variable,B5: set_variable] :
( collect_variable
@ ( inf_inf_variable_o
@ ^ [X: variable] : ( member_variable @ X @ A5 )
@ ^ [X: variable] : ( member_variable @ X @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_181_Int__emptyI,axiom,
! [A2: set_variable,B3: set_variable] :
( ! [X3: variable] :
( ( member_variable @ X3 @ A2 )
=> ~ ( member_variable @ X3 @ B3 ) )
=> ( ( inf_inf_set_variable @ A2 @ B3 )
= bot_bot_set_variable ) ) ).
% Int_emptyI
thf(fact_182_disjoint__iff,axiom,
! [A2: set_variable,B3: set_variable] :
( ( ( inf_inf_set_variable @ A2 @ B3 )
= bot_bot_set_variable )
= ( ! [X: variable] :
( ( member_variable @ X @ A2 )
=> ~ ( member_variable @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_183_Int__empty__left,axiom,
! [B3: set_variable] :
( ( inf_inf_set_variable @ bot_bot_set_variable @ B3 )
= bot_bot_set_variable ) ).
% Int_empty_left
thf(fact_184_Int__empty__right,axiom,
! [A2: set_variable] :
( ( inf_inf_set_variable @ A2 @ bot_bot_set_variable )
= bot_bot_set_variable ) ).
% Int_empty_right
thf(fact_185_disjE__realizer2,axiom,
! [P2: $o,Q: fml > $o,X2: option_fml,R4: option_fml > $o,F3: option_fml,G2: fml > option_fml] :
( ( case_option_o_fml @ P2 @ Q @ X2 )
=> ( ( P2
=> ( R4 @ F3 ) )
=> ( ! [Q2: fml] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o209804997ml_fml @ F3 @ G2 @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_186_disjE__realizer2,axiom,
! [P2: $o,Q: trm > $o,X2: option_trm,R4: option_fml > $o,F3: option_fml,G2: trm > option_fml] :
( ( case_option_o_trm @ P2 @ Q @ X2 )
=> ( ( P2
=> ( R4 @ F3 ) )
=> ( ! [Q2: trm] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o325794063ml_trm @ F3 @ G2 @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_187_disjE__realizer2,axiom,
! [P2: $o,Q: trm > $o,X2: option_trm,R4: option_trm > $o,F3: option_trm,G2: trm > option_trm] :
( ( case_option_o_trm @ P2 @ Q @ X2 )
=> ( ( P2
=> ( R4 @ F3 ) )
=> ( ! [Q2: trm] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_o487727941rm_trm @ F3 @ G2 @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_188_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_189_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_190_Geqo_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst152838031e_Geqo @ ( some_trm @ V3 ) @ none_trm )
= none_fml ) ).
% Geqo.simps(3)
thf(fact_191_Set_Ois__empty__def,axiom,
( is_empty_variable
= ( ^ [A5: set_variable] : A5 = bot_bot_set_variable ) ) ).
% Set.is_empty_def
thf(fact_192_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_193_fml_Oinject_I2_J,axiom,
! [X21: trm,X222: trm,Y21: trm,Y222: trm] :
( ( ( geq @ X21 @ X222 )
= ( geq @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% fml.inject(2)
thf(fact_194_fml_Odistinct_I1_J,axiom,
! [X11: char,X12: trm,X21: trm,X222: trm] :
( ( pred @ X11 @ X12 )
!= ( geq @ X21 @ X222 ) ) ).
% fml.distinct(1)
thf(fact_195_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_196_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_197_Geqo_Oelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_fml] :
( ( ( uSubst152838031e_Geqo @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta: trm] :
( ( Xa
= ( some_trm @ Eta ) )
=> ( Y2
!= ( some_fml @ ( geq @ Theta2 @ Eta ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( Y2 != none_fml ) )
=> ~ ( ? [V: trm] :
( X2
= ( some_trm @ V ) )
=> ( ( Xa = none_trm )
=> ( Y2 != none_fml ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_198_Geqo_Osimps_I2_J,axiom,
! [Eta2: option_trm] :
( ( uSubst152838031e_Geqo @ none_trm @ Eta2 )
= none_fml ) ).
% Geqo.simps(2)
thf(fact_199_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_200_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_201_Collect__empty__eq__bot,axiom,
! [P2: variable > $o] :
( ( ( collect_variable @ P2 )
= bot_bot_set_variable )
= ( P2 = bot_bot_variable_o ) ) ).
% Collect_empty_eq_bot
thf(fact_202_bot__empty__eq,axiom,
( bot_bot_variable_o
= ( ^ [X: variable] : ( member_variable @ X @ bot_bot_set_variable ) ) ) ).
% bot_empty_eq
thf(fact_203_internal__case__prod__def,axiom,
produc1278328899on_fml = produc1250175341on_fml ).
% internal_case_prod_def
thf(fact_204_internal__case__prod__def,axiom,
produc2141841102on_trm = produc1231571704on_trm ).
% internal_case_prod_def
thf(fact_205_internal__case__prod__def,axiom,
produc377919108on_fml = produc1615133358on_fml ).
% internal_case_prod_def
thf(fact_206_internal__case__prod__def,axiom,
produc1214549391on_trm = produc1036610745on_trm ).
% internal_case_prod_def
thf(fact_207_internal__case__prod__def,axiom,
produc1598111045on_fml = produc1420172399on_fml ).
% internal_case_prod_def
thf(fact_208_internal__case__prod__def,axiom,
produc1591622303on_fml = produc347802313on_fml ).
% internal_case_prod_def
thf(fact_209_internal__case__prod__def,axiom,
produc2129367017on_trm = produc885547027on_trm ).
% internal_case_prod_def
thf(fact_210_internal__case__prod__def,axiom,
produc1903410526on_fml = produc154833544on_fml ).
% internal_case_prod_def
thf(fact_211_internal__case__prod__def,axiom,
produc293671592on_trm = produc692578258on_trm ).
% internal_case_prod_def
thf(fact_212_internal__case__prod__def,axiom,
produc724717725on_fml = produc1499310791on_fml ).
% internal_case_prod_def
thf(fact_213_usubstappt_Osimps_I4_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char,Theta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( case_o487727941rm_trm @ none_trm
@ ^ [Sigma_theta: trm] :
( case_o487727941rm_trm @ ( some_trm @ ( func @ F3 @ 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
@ F3 ) )
@ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappt.simps(4)
thf(fact_214_Diamondo_Ocases,axiom,
! [X2: produc1739631244on_fml] :
( ! [Alpha: game,Phi: fml] :
( X2
!= ( produc2124676926on_fml @ ( some_game @ Alpha ) @ ( some_fml @ Phi ) ) )
=> ( ! [Phi: option_fml] :
( X2
!= ( produc2124676926on_fml @ none_game @ Phi ) )
=> ~ ! [V: game] :
( X2
!= ( produc2124676926on_fml @ ( some_game @ V ) @ none_fml ) ) ) ) ).
% Diamondo.cases
thf(fact_215_Testo__undef,axiom,
! [Phi2: option_fml] :
( ( ( uSubst1367128042_Testo @ Phi2 )
= none_game )
= ( Phi2 = none_fml ) ) ).
% Testo_undef
thf(fact_216_prod_Oinject,axiom,
! [X13: char,X23: option_trm,Y1: char,Y22: option_trm] :
( ( ( produc1500300866on_trm @ X13 @ X23 )
= ( produc1500300866on_trm @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_217_prod_Oinject,axiom,
! [X13: set_variable,X23: trm,Y1: set_variable,Y22: trm] :
( ( ( produc1482344242le_trm @ X13 @ X23 )
= ( produc1482344242le_trm @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_218_prod_Oinject,axiom,
! [X13: char > option_trm,X23: produc62062515n_game,Y1: char > option_trm,Y22: produc62062515n_game] :
( ( ( produc1618635950n_game @ X13 @ X23 )
= ( produc1618635950n_game @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_219_prod_Oinject,axiom,
! [X13: char > option_trm,X23: produc799602994n_game,Y1: char > option_trm,Y22: produc799602994n_game] :
( ( ( produc1048060077n_game @ X13 @ X23 )
= ( produc1048060077n_game @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_220_prod_Oinject,axiom,
! [X13: char > option_fml,X23: char > option_game,Y1: char > option_fml,Y22: char > option_game] :
( ( ( produc123610156n_game @ X13 @ X23 )
= ( produc123610156n_game @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_221_old_Oprod_Oinject,axiom,
! [A: char,B: option_trm,A6: char,B6: option_trm] :
( ( ( produc1500300866on_trm @ A @ B )
= ( produc1500300866on_trm @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_222_old_Oprod_Oinject,axiom,
! [A: set_variable,B: trm,A6: set_variable,B6: trm] :
( ( ( produc1482344242le_trm @ A @ B )
= ( produc1482344242le_trm @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_223_old_Oprod_Oinject,axiom,
! [A: char > option_trm,B: produc62062515n_game,A6: char > option_trm,B6: produc62062515n_game] :
( ( ( produc1618635950n_game @ A @ B )
= ( produc1618635950n_game @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_224_old_Oprod_Oinject,axiom,
! [A: char > option_trm,B: produc799602994n_game,A6: char > option_trm,B6: produc799602994n_game] :
( ( ( produc1048060077n_game @ A @ B )
= ( produc1048060077n_game @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_225_old_Oprod_Oinject,axiom,
! [A: char > option_fml,B: char > option_game,A6: char > option_fml,B6: char > option_game] :
( ( ( produc123610156n_game @ A @ B )
= ( produc123610156n_game @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_226_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_227_case__prodI,axiom,
! [F3: char > option_trm > $o,A: char,B: option_trm] :
( ( F3 @ A @ B )
=> ( produc584666985_trm_o @ F3 @ ( produc1500300866on_trm @ A @ B ) ) ) ).
% case_prodI
thf(fact_228_case__prodI,axiom,
! [F3: set_variable > trm > $o,A: set_variable,B: trm] :
( ( F3 @ A @ B )
=> ( produc82209913_trm_o @ F3 @ ( produc1482344242le_trm @ A @ B ) ) ) ).
% case_prodI
thf(fact_229_case__prodI,axiom,
! [F3: ( char > option_trm ) > produc62062515n_game > $o,A: char > option_trm,B: produc62062515n_game] :
( ( F3 @ A @ B )
=> ( produc1707653117game_o @ F3 @ ( produc1618635950n_game @ A @ B ) ) ) ).
% case_prodI
thf(fact_230_case__prodI,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > $o,A: char > option_trm,B: produc799602994n_game] :
( ( F3 @ A @ B )
=> ( produc1868740670game_o @ F3 @ ( produc1048060077n_game @ A @ B ) ) ) ).
% case_prodI
thf(fact_231_case__prodI,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > $o,A: char > option_fml,B: char > option_game] :
( ( F3 @ A @ B )
=> ( produc159680767game_o @ F3 @ ( produc123610156n_game @ A @ B ) ) ) ).
% case_prodI
thf(fact_232_case__prodI2,axiom,
! [P3: produc464795408on_trm,C: char > option_trm > $o] :
( ! [A4: char,B4: option_trm] :
( ( P3
= ( produc1500300866on_trm @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc584666985_trm_o @ C @ P3 ) ) ).
% case_prodI2
thf(fact_233_case__prodI2,axiom,
! [P3: produc1367888000le_trm,C: set_variable > trm > $o] :
( ! [A4: set_variable,B4: trm] :
( ( P3
= ( produc1482344242le_trm @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc82209913_trm_o @ C @ P3 ) ) ).
% case_prodI2
thf(fact_234_case__prodI2,axiom,
! [P3: produc1418842292n_game,C: ( char > option_trm ) > produc62062515n_game > $o] :
( ! [A4: char > option_trm,B4: produc62062515n_game] :
( ( P3
= ( produc1618635950n_game @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc1707653117game_o @ C @ P3 ) ) ).
% case_prodI2
thf(fact_235_case__prodI2,axiom,
! [P3: produc62062515n_game,C: ( char > option_trm ) > produc799602994n_game > $o] :
( ! [A4: char > option_trm,B4: produc799602994n_game] :
( ( P3
= ( produc1048060077n_game @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc1868740670game_o @ C @ P3 ) ) ).
% case_prodI2
thf(fact_236_case__prodI2,axiom,
! [P3: produc799602994n_game,C: ( char > option_fml ) > ( char > option_game ) > $o] :
( ! [A4: char > option_fml,B4: char > option_game] :
( ( P3
= ( produc123610156n_game @ A4 @ B4 ) )
=> ( C @ A4 @ B4 ) )
=> ( produc159680767game_o @ C @ P3 ) ) ).
% case_prodI2
thf(fact_237_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc62062515n_game > char > option_fml,A: char > option_trm,B: produc62062515n_game] :
( ( produc1250175341on_fml @ F3 @ ( produc1618635950n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_238_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > char > option_trm,A: char > option_trm,B: produc799602994n_game] :
( ( produc1231571704on_trm @ F3 @ ( produc1048060077n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_239_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > char > option_fml,A: char > option_trm,B: produc799602994n_game] :
( ( produc1615133358on_fml @ F3 @ ( produc1048060077n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_240_case__prod__conv,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > char > option_trm,A: char > option_fml,B: char > option_game] :
( ( produc1036610745on_trm @ F3 @ ( produc123610156n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_241_case__prod__conv,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > char > option_fml,A: char > option_fml,B: char > option_game] :
( ( produc1420172399on_fml @ F3 @ ( produc123610156n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_242_case__prod__conv,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > option_fml,A: char > option_fml,B: char > option_game] :
( ( produc347802313on_fml @ F3 @ ( produc123610156n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_243_case__prod__conv,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > option_trm,A: char > option_fml,B: char > option_game] :
( ( produc885547027on_trm @ F3 @ ( produc123610156n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_244_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > option_fml,A: char > option_trm,B: produc799602994n_game] :
( ( produc154833544on_fml @ F3 @ ( produc1048060077n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_245_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > option_trm,A: char > option_trm,B: produc799602994n_game] :
( ( produc692578258on_trm @ F3 @ ( produc1048060077n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_246_case__prod__conv,axiom,
! [F3: ( char > option_trm ) > produc62062515n_game > option_fml,A: char > option_trm,B: produc62062515n_game] :
( ( produc1499310791on_fml @ F3 @ ( produc1618635950n_game @ A @ B ) )
= ( F3 @ A @ B ) ) ).
% case_prod_conv
thf(fact_247_usubstappt__func2,axiom,
! [Sigma: produc1418842292n_game,F3: 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
@ F3 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
!= bot_bot_set_variable )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= none_trm ) ) ) ).
% usubstappt_func2
thf(fact_248_usubstappt__func,axiom,
! [Sigma: produc1418842292n_game,F3: 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
@ F3 )
= ( 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 @ F3 @ Theta ) )
= ( uSubst516392818stappt @ ( uSubst1408967625substt @ Sigma_theta2 ) @ bot_bot_set_variable @ R2 ) ) ) ) ) ).
% usubstappt_func
thf(fact_249_inf__Int__eq2,axiom,
! [R4: set_Pr943832262on_trm,S: set_Pr943832262on_trm] :
( ( inf_in605351507_trm_o
@ ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ R4 )
@ ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ S ) )
= ( ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ ( inf_in1243659096on_trm @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_250_inf__Int__eq2,axiom,
! [R4: set_Pr1235432374le_trm,S: set_Pr1235432374le_trm] :
( ( inf_in185389155_trm_o
@ ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ R4 )
@ ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ S ) )
= ( ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ ( inf_in906025800le_trm @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_251_inf__Int__eq2,axiom,
! [R4: set_Pr720454634n_game,S: set_Pr720454634n_game] :
( ( inf_in1920662529game_o
@ ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ R4 )
@ ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ S ) )
= ( ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ ( inf_in360367996n_game @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_252_inf__Int__eq2,axiom,
! [R4: set_Pr739694057n_game,S: set_Pr739694057n_game] :
( ( inf_in1072207554game_o
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ R4 )
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ S ) )
= ( ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ ( inf_in69689467n_game @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_253_inf__Int__eq2,axiom,
! [R4: set_Pr79606376n_game,S: set_Pr79606376n_game] :
( ( inf_in243546115game_o
@ ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ R4 )
@ ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ S ) )
= ( ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ ( inf_in1391117306n_game @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_254_bot__empty__eq2,axiom,
( bot_bo102862421_trm_o
= ( ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ bot_bo18861914on_trm ) ) ) ).
% bot_empty_eq2
thf(fact_255_bot__empty__eq2,axiom,
( bot_bo512610661_trm_o
= ( ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ bot_bo896421962le_trm ) ) ) ).
% bot_empty_eq2
thf(fact_256_bot__empty__eq2,axiom,
( bot_bo849148163game_o
= ( ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ bot_bo1453406334n_game ) ) ) ).
% bot_empty_eq2
thf(fact_257_bot__empty__eq2,axiom,
( bot_bo1417568964game_o
= ( ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ bot_bo1302831229n_game ) ) ) ).
% bot_empty_eq2
thf(fact_258_bot__empty__eq2,axiom,
( bot_bo1895715589game_o
= ( ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ bot_bo456479484n_game ) ) ) ).
% bot_empty_eq2
thf(fact_259_pred__equals__eq2,axiom,
! [R4: set_Pr943832262on_trm,S: set_Pr943832262on_trm] :
( ( ( ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ R4 ) )
= ( ^ [X: char,Y: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_260_pred__equals__eq2,axiom,
! [R4: set_Pr1235432374le_trm,S: set_Pr1235432374le_trm] :
( ( ( ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ R4 ) )
= ( ^ [X: set_variable,Y: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_261_pred__equals__eq2,axiom,
! [R4: set_Pr720454634n_game,S: set_Pr720454634n_game] :
( ( ( ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ R4 ) )
= ( ^ [X: char > option_trm,Y: produc62062515n_game] : ( member1940201675n_game @ ( produc1618635950n_game @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_262_pred__equals__eq2,axiom,
! [R4: set_Pr739694057n_game,S: set_Pr739694057n_game] :
( ( ( ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ R4 ) )
= ( ^ [X: char > option_trm,Y: produc799602994n_game] : ( member774167626n_game @ ( produc1048060077n_game @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_263_pred__equals__eq2,axiom,
! [R4: set_Pr79606376n_game,S: set_Pr79606376n_game] :
( ( ( ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ R4 ) )
= ( ^ [X: char > option_fml,Y: char > option_game] : ( member849814089n_game @ ( produc123610156n_game @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_264_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc62062515n_game > char > option_fml,G2: produc1418842292n_game > char > option_fml] :
( ! [X3: char > option_trm,Y3: produc62062515n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1618635950n_game @ X3 @ Y3 ) ) )
=> ( ( produc1250175341on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_265_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > char > option_trm,G2: produc62062515n_game > char > option_trm] :
( ! [X3: char > option_trm,Y3: produc799602994n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1048060077n_game @ X3 @ Y3 ) ) )
=> ( ( produc1231571704on_trm @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_266_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > char > option_fml,G2: produc62062515n_game > char > option_fml] :
( ! [X3: char > option_trm,Y3: produc799602994n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1048060077n_game @ X3 @ Y3 ) ) )
=> ( ( produc1615133358on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_267_cond__case__prod__eta,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > char > option_trm,G2: produc799602994n_game > char > option_trm] :
( ! [X3: char > option_fml,Y3: char > option_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc123610156n_game @ X3 @ Y3 ) ) )
=> ( ( produc1036610745on_trm @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_268_cond__case__prod__eta,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > char > option_fml,G2: produc799602994n_game > char > option_fml] :
( ! [X3: char > option_fml,Y3: char > option_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc123610156n_game @ X3 @ Y3 ) ) )
=> ( ( produc1420172399on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_269_cond__case__prod__eta,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > option_fml,G2: produc799602994n_game > option_fml] :
( ! [X3: char > option_fml,Y3: char > option_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc123610156n_game @ X3 @ Y3 ) ) )
=> ( ( produc347802313on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_270_cond__case__prod__eta,axiom,
! [F3: ( char > option_fml ) > ( char > option_game ) > option_trm,G2: produc799602994n_game > option_trm] :
( ! [X3: char > option_fml,Y3: char > option_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc123610156n_game @ X3 @ Y3 ) ) )
=> ( ( produc885547027on_trm @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_271_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > option_fml,G2: produc62062515n_game > option_fml] :
( ! [X3: char > option_trm,Y3: produc799602994n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1048060077n_game @ X3 @ Y3 ) ) )
=> ( ( produc154833544on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_272_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc799602994n_game > option_trm,G2: produc62062515n_game > option_trm] :
( ! [X3: char > option_trm,Y3: produc799602994n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1048060077n_game @ X3 @ Y3 ) ) )
=> ( ( produc692578258on_trm @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_273_cond__case__prod__eta,axiom,
! [F3: ( char > option_trm ) > produc62062515n_game > option_fml,G2: produc1418842292n_game > option_fml] :
( ! [X3: char > option_trm,Y3: produc62062515n_game] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( produc1618635950n_game @ X3 @ Y3 ) ) )
=> ( ( produc1499310791on_fml @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_274_case__prod__eta,axiom,
! [F3: produc1418842292n_game > char > option_fml] :
( ( produc1250175341on_fml
@ ^ [X: char > option_trm,Y: produc62062515n_game] : ( F3 @ ( produc1618635950n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_275_case__prod__eta,axiom,
! [F3: produc62062515n_game > char > option_trm] :
( ( produc1231571704on_trm
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( F3 @ ( produc1048060077n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_276_case__prod__eta,axiom,
! [F3: produc62062515n_game > char > option_fml] :
( ( produc1615133358on_fml
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( F3 @ ( produc1048060077n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_277_case__prod__eta,axiom,
! [F3: produc799602994n_game > char > option_trm] :
( ( produc1036610745on_trm
@ ^ [X: char > option_fml,Y: char > option_game] : ( F3 @ ( produc123610156n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_278_case__prod__eta,axiom,
! [F3: produc799602994n_game > char > option_fml] :
( ( produc1420172399on_fml
@ ^ [X: char > option_fml,Y: char > option_game] : ( F3 @ ( produc123610156n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_279_case__prod__eta,axiom,
! [F3: produc799602994n_game > option_fml] :
( ( produc347802313on_fml
@ ^ [X: char > option_fml,Y: char > option_game] : ( F3 @ ( produc123610156n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_280_case__prod__eta,axiom,
! [F3: produc799602994n_game > option_trm] :
( ( produc885547027on_trm
@ ^ [X: char > option_fml,Y: char > option_game] : ( F3 @ ( produc123610156n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_281_case__prod__eta,axiom,
! [F3: produc62062515n_game > option_fml] :
( ( produc154833544on_fml
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( F3 @ ( produc1048060077n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_282_case__prod__eta,axiom,
! [F3: produc62062515n_game > option_trm] :
( ( produc692578258on_trm
@ ^ [X: char > option_trm,Y: produc799602994n_game] : ( F3 @ ( produc1048060077n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_283_case__prod__eta,axiom,
! [F3: produc1418842292n_game > option_fml] :
( ( produc1499310791on_fml
@ ^ [X: char > option_trm,Y: produc62062515n_game] : ( F3 @ ( produc1618635950n_game @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_284_case__prodE2,axiom,
! [Q: option_fml > $o,P2: ( char > option_trm ) > produc62062515n_game > option_fml,Z2: produc1418842292n_game] :
( ( Q @ ( produc1499310791on_fml @ P2 @ Z2 ) )
=> ~ ! [X3: char > option_trm,Y3: produc62062515n_game] :
( ( Z2
= ( produc1618635950n_game @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_285_usubstappt__func__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char,Theta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F3 @ 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
@ F3 )
= 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
@ F3 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ) ).
% usubstappt_func_conv
thf(fact_286_Testo_Osimps_I2_J,axiom,
( ( uSubst1367128042_Testo @ none_fml )
= none_game ) ).
% Testo.simps(2)
thf(fact_287_usappconst__simp,axiom,
! [Sigma: produc1418842292n_game,F3: 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
@ F3 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( uSubst1559070963pconst @ Sigma @ U @ F3 )
= ( some_trm @ R2 ) ) ) ) ).
% usappconst_simp
thf(fact_288_usappconst__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char] :
( ( ( uSubst1559070963pconst @ Sigma @ U @ F3 )
!= 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
@ F3 )
= 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
@ F3 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ).
% usappconst_conv
thf(fact_289_usubstappt__const,axiom,
! [Sigma: produc1418842292n_game,F3: 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
@ F3 )
= ( some_trm @ R2 ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ R2 ) @ U )
= bot_bot_set_variable )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F3 ) )
= ( some_trm @ R2 ) ) ) ) ).
% usubstappt_const
thf(fact_290_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_291_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_292_usubstappt_Osimps_I3_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char] :
( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F3 ) )
= ( uSubst1559070963pconst @ Sigma @ U @ F3 ) ) ).
% usubstappt.simps(3)
thf(fact_293_ODEo_Ocases,axiom,
! [X2: produc464795408on_trm] :
( ! [X3: char,Theta2: trm] :
( X2
!= ( produc1500300866on_trm @ X3 @ ( some_trm @ Theta2 ) ) )
=> ~ ! [X3: char] :
( X2
!= ( produc1500300866on_trm @ X3 @ none_trm ) ) ) ).
% ODEo.cases
thf(fact_294_Assigno_Ocases,axiom,
! [X2: produc1024976764on_trm] :
( ! [X3: variable,Theta2: trm] :
( X2
!= ( produc1743721902on_trm @ X3 @ ( some_trm @ Theta2 ) ) )
=> ~ ! [X3: variable] :
( X2
!= ( produc1743721902on_trm @ X3 @ none_trm ) ) ) ).
% Assigno.cases
thf(fact_295_Timeso_Ocases,axiom,
! [X2: produc1072747943on_trm] :
( ! [Theta2: trm,Eta: trm] :
( X2
!= ( produc709594775on_trm @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta ) ) )
=> ( ! [Eta: option_trm] :
( X2
!= ( produc709594775on_trm @ none_trm @ Eta ) )
=> ~ ! [V: trm] :
( X2
!= ( produc709594775on_trm @ ( some_trm @ V ) @ none_trm ) ) ) ) ).
% Timeso.cases
thf(fact_296_Existso_Ocases,axiom,
! [X2: produc1408538418on_fml] :
( ! [X3: variable,Phi: fml] :
( X2
!= ( produc1205977188on_fml @ X3 @ ( some_fml @ Phi ) ) )
=> ~ ! [X3: variable] :
( X2
!= ( produc1205977188on_fml @ X3 @ none_fml ) ) ) ).
% Existso.cases
thf(fact_297_Ando_Ocases,axiom,
! [X2: produc46605351on_fml] :
( ! [Phi: fml,Psi: fml] :
( X2
!= ( produc909629463on_fml @ ( some_fml @ Phi ) @ ( some_fml @ Psi ) ) )
=> ( ! [Psi: option_fml] :
( X2
!= ( produc909629463on_fml @ none_fml @ Psi ) )
=> ~ ! [V: fml] :
( X2
!= ( produc909629463on_fml @ ( some_fml @ V ) @ none_fml ) ) ) ) ).
% Ando.cases
thf(fact_298_Composeo_Ocases,axiom,
! [X2: produc1016592119n_game] :
( ! [Alpha: game,Beta: game] :
( X2
!= ( produc1111617711n_game @ ( some_game @ Alpha ) @ ( some_game @ Beta ) ) )
=> ( ! [Alpha: option_game] :
( X2
!= ( produc1111617711n_game @ Alpha @ none_game ) )
=> ~ ! [V: game] :
( X2
!= ( produc1111617711n_game @ none_game @ ( some_game @ V ) ) ) ) ) ).
% Composeo.cases
thf(fact_299_usappconst__def,axiom,
( uSubst1559070963pconst
= ( ^ [Sigma2: produc1418842292n_game,U2: set_variable,F2: char] :
( case_o487727941rm_trm @ ( some_trm @ ( const @ F2 ) )
@ ^ [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
@ F2 ) ) ) ) ).
% usappconst_def
thf(fact_300_usubstappt__const__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F3 ) )
!= 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
@ F3 )
= 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
@ F3 )
= ( some_trm @ R3 ) )
& ( ( inf_inf_set_variable @ ( static_FVT @ R3 ) @ U )
= bot_bot_set_variable ) ) ) ) ).
% usubstappt_const_conv
thf(fact_301_usubstappt_Opsimps_I4_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char,Theta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( func @ F3 @ Theta ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( case_o487727941rm_trm @ none_trm
@ ^ [Sigma_theta: trm] :
( case_o487727941rm_trm @ ( some_trm @ ( func @ F3 @ 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
@ F3 ) )
@ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) ) ) ).
% usubstappt.psimps(4)
thf(fact_302_usubstappt_Opsimps_I3_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F3: char] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( const @ F3 ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F3 ) )
= ( uSubst1559070963pconst @ Sigma @ U @ F3 ) ) ) ).
% usubstappt.psimps(3)
thf(fact_303_usubstappt_Opsimps_I1_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,X2: variable] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( var @ X2 ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some_trm @ ( var @ X2 ) ) ) ) ).
% usubstappt.psimps(1)
thf(fact_304_usubstappt_Opsimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,R2: real] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( number @ R2 ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( number @ R2 ) )
= ( some_trm @ ( number @ R2 ) ) ) ) ).
% usubstappt.psimps(2)
thf(fact_305_trm_Oinject_I2_J,axiom,
! [X23: real,Y22: real] :
( ( ( number @ X23 )
= ( number @ Y22 ) )
= ( X23 = Y22 ) ) ).
% trm.inject(2)
thf(fact_306_trm_Oinject_I1_J,axiom,
! [X13: variable,Y1: variable] :
( ( ( var @ X13 )
= ( var @ Y1 ) )
= ( X13 = Y1 ) ) ).
% trm.inject(1)
thf(fact_307_trm_Odistinct_I15_J,axiom,
! [X23: real,X41: char,X42: trm] :
( ( number @ X23 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_308_trm_Odistinct_I5_J,axiom,
! [X13: variable,X41: char,X42: trm] :
( ( var @ X13 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_309_trm_Odistinct_I1_J,axiom,
! [X13: variable,X23: real] :
( ( var @ X13 )
!= ( number @ X23 ) ) ).
% trm.distinct(1)
thf(fact_310_trm_Odistinct_I13_J,axiom,
! [X23: real,X32: char] :
( ( number @ X23 )
!= ( const @ X32 ) ) ).
% trm.distinct(13)
thf(fact_311_trm_Odistinct_I3_J,axiom,
! [X13: variable,X32: char] :
( ( var @ X13 )
!= ( const @ X32 ) ) ).
% trm.distinct(3)
thf(fact_312_usubstappt_Osimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,R2: real] :
( ( uSubst516392818stappt @ Sigma @ U @ ( number @ R2 ) )
= ( some_trm @ ( number @ R2 ) ) ) ).
% usubstappt.simps(2)
thf(fact_313_usubstappt_Osimps_I1_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,X2: variable] :
( ( uSubst516392818stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some_trm @ ( var @ X2 ) ) ) ).
% usubstappt.simps(1)
thf(fact_314_usubstappt_Opinduct,axiom,
! [A0: produc1418842292n_game,A1: set_variable,A22: trm,P2: produc1418842292n_game > set_variable > trm > $o] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ A0 @ ( produc1482344242le_trm @ A1 @ A22 ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,X3: variable] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( var @ X3 ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( var @ X3 ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,R3: real] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( number @ R3 ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( number @ R3 ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( const @ F5 ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( const @ F5 ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char,Theta2: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( func @ F5 @ Theta2 ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ! [X25: trm] :
( ( ( uSubst516392818stappt @ Sigma3 @ U3 @ Theta2 )
= ( some_trm @ X25 ) )
=> ! [X2a: trm] :
( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma3
@ F5 )
= ( some_trm @ X2a ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ X2a ) @ U3 )
= bot_bot_set_variable )
=> ( P2 @ ( uSubst1408967625substt @ X25 ) @ bot_bot_set_variable @ X2a ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( func @ F5 @ Theta2 ) ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( plus @ Theta2 @ Eta ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( plus @ Theta2 @ Eta ) ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( times @ Theta2 @ Eta ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( times @ Theta2 @ Eta ) ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( differential @ Theta2 ) ) ) )
=> ( ( P2 @ Sigma3
@ ( collect_variable
@ ^ [X: variable] : $true )
@ Theta2 )
=> ( P2 @ Sigma3 @ U3 @ ( differential @ Theta2 ) ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).
% usubstappt.pinduct
thf(fact_315_dotsubstt__def,axiom,
( uSubst1408967625substt
= ( ^ [Theta3: trm] :
( produc1618635950n_game
@ ^ [F2: char] :
( if_option_trm
@ ( F2
= ( char2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ $false ) )
@ ( some_trm @ Theta3 )
@ none_trm )
@ ( produc1048060077n_game
@ ^ [Uu: char] : none_trm
@ ( produc123610156n_game
@ ^ [Uu: char] : none_fml
@ ^ [Uu: char] : none_game ) ) ) ) ) ).
% dotsubstt_def
thf(fact_316_trm_Oinject_I6_J,axiom,
! [X61: trm,X62: trm,Y61: trm,Y62: trm] :
( ( ( times @ X61 @ X62 )
= ( times @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% trm.inject(6)
thf(fact_317_trm_Oinject_I5_J,axiom,
! [X51: trm,X52: trm,Y51: trm,Y52: trm] :
( ( ( plus @ X51 @ X52 )
= ( plus @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_318_trm_Oinject_I7_J,axiom,
! [X7: trm,Y7: trm] :
( ( ( differential @ X7 )
= ( differential @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_319_trm_Odistinct_I7_J,axiom,
! [X13: variable,X51: trm,X52: trm] :
( ( var @ X13 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_320_trm_Odistinct_I9_J,axiom,
! [X13: variable,X61: trm,X62: trm] :
( ( var @ X13 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(9)
thf(fact_321_trm_Odistinct_I11_J,axiom,
! [X13: variable,X7: trm] :
( ( var @ X13 )
!= ( differential @ X7 ) ) ).
% trm.distinct(11)
thf(fact_322_trm_Odistinct_I17_J,axiom,
! [X23: real,X51: trm,X52: trm] :
( ( number @ X23 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_323_trm_Odistinct_I19_J,axiom,
! [X23: real,X61: trm,X62: trm] :
( ( number @ X23 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(19)
thf(fact_324_trm_Odistinct_I21_J,axiom,
! [X23: real,X7: trm] :
( ( number @ X23 )
!= ( differential @ X7 ) ) ).
% trm.distinct(21)
thf(fact_325_trm_Oexhaust,axiom,
! [Y2: trm] :
( ! [X14: variable] :
( Y2
!= ( var @ X14 ) )
=> ( ! [X24: real] :
( Y2
!= ( number @ X24 ) )
=> ( ! [X33: char] :
( Y2
!= ( const @ X33 ) )
=> ( ! [X412: char,X422: trm] :
( Y2
!= ( func @ X412 @ X422 ) )
=> ( ! [X512: trm,X522: trm] :
( Y2
!= ( plus @ X512 @ X522 ) )
=> ( ! [X612: trm,X622: trm] :
( Y2
!= ( times @ X612 @ X622 ) )
=> ~ ! [X72: trm] :
( Y2
!= ( differential @ X72 ) ) ) ) ) ) ) ) ).
% trm.exhaust
thf(fact_326_trm_Oinduct,axiom,
! [P2: trm > $o,Trm: trm] :
( ! [X3: variable] : ( P2 @ ( var @ X3 ) )
=> ( ! [X3: real] : ( P2 @ ( number @ X3 ) )
=> ( ! [X3: char] : ( P2 @ ( const @ X3 ) )
=> ( ! [X1a: char,X2a2: trm] :
( ( P2 @ X2a2 )
=> ( P2 @ ( func @ X1a @ X2a2 ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P2 @ X1a )
=> ( ( P2 @ X2a2 )
=> ( P2 @ ( plus @ X1a @ X2a2 ) ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P2 @ X1a )
=> ( ( P2 @ X2a2 )
=> ( P2 @ ( times @ X1a @ X2a2 ) ) ) )
=> ( ! [X3: trm] :
( ( P2 @ X3 )
=> ( P2 @ ( differential @ X3 ) ) )
=> ( P2 @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_327_trm_Odistinct_I25_J,axiom,
! [X32: char,X51: trm,X52: trm] :
( ( const @ X32 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_328_trm_Odistinct_I27_J,axiom,
! [X32: char,X61: trm,X62: trm] :
( ( const @ X32 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(27)
thf(fact_329_trm_Odistinct_I29_J,axiom,
! [X32: char,X7: trm] :
( ( const @ X32 )
!= ( differential @ X7 ) ) ).
% trm.distinct(29)
thf(fact_330_trm_Odistinct_I41_J,axiom,
! [X61: trm,X62: trm,X7: trm] :
( ( times @ X61 @ X62 )
!= ( differential @ X7 ) ) ).
% trm.distinct(41)
thf(fact_331_trm_Odistinct_I39_J,axiom,
! [X51: trm,X52: trm,X7: trm] :
( ( plus @ X51 @ X52 )
!= ( differential @ X7 ) ) ).
% trm.distinct(39)
thf(fact_332_trm_Odistinct_I37_J,axiom,
! [X51: trm,X52: trm,X61: trm,X62: trm] :
( ( plus @ X51 @ X52 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(37)
thf(fact_333_trm_Odistinct_I31_J,axiom,
! [X41: char,X42: trm,X51: trm,X52: trm] :
( ( func @ X41 @ X42 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_334_trm_Odistinct_I33_J,axiom,
! [X41: char,X42: trm,X61: trm,X62: trm] :
( ( func @ X41 @ X42 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(33)
thf(fact_335_trm_Odistinct_I35_J,axiom,
! [X41: char,X42: trm,X7: trm] :
( ( func @ X41 @ X42 )
!= ( differential @ X7 ) ) ).
% trm.distinct(35)
thf(fact_336_usubstappt_Ocases,axiom,
! [X2: produc1699626005le_trm] :
( ! [Sigma3: produc1418842292n_game,U3: set_variable,X3: variable] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( var @ X3 ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,R3: real] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( number @ R3 ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( const @ F5 ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char,Theta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( func @ F5 @ Theta2 ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( plus @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( times @ Theta2 @ Eta ) ) ) )
=> ~ ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma3 @ ( produc1482344242le_trm @ U3 @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.cases
thf(fact_337_usubstappt__times__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta2 ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta2 )
!= none_trm ) ) ) ).
% usubstappt_times_conv
thf(fact_338_usubstappt__plus__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( plus @ Theta @ Eta2 ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta2 )
!= none_trm ) ) ) ).
% usubstappt_plus_conv
thf(fact_339_usubstappt__differential__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( differential @ Theta ) )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma
@ ( collect_variable
@ ^ [X: variable] : $true )
@ Theta )
!= none_trm ) ) ).
% usubstappt_differential_conv
thf(fact_340_usubstappt__induct,axiom,
! [P2: produc1418842292n_game > set_variable > trm > $o,A0: produc1418842292n_game,A1: set_variable,A22: trm] :
( ! [Sigma3: produc1418842292n_game,U3: set_variable,X3: variable] : ( P2 @ Sigma3 @ U3 @ ( var @ X3 ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,R3: real] : ( P2 @ Sigma3 @ U3 @ ( number @ R3 ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char] : ( P2 @ Sigma3 @ U3 @ ( const @ F5 ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,F5: char,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ! [X25: trm] :
( ( ( uSubst516392818stappt @ Sigma3 @ U3 @ Theta2 )
= ( some_trm @ X25 ) )
=> ! [X2a: trm] :
( ( ( produc866613687on_trm
@ ^ [Uu: char > option_trm] :
( produc1231571704on_trm
@ ^ [F4: char > option_trm] :
( produc1036610745on_trm
@ ^ [Uv: char > option_fml,Uw: char > option_game] : F4 ) )
@ Sigma3
@ F5 )
= ( some_trm @ X2a ) )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ X2a ) @ U3 )
= bot_bot_set_variable )
=> ( P2 @ ( uSubst1408967625substt @ X25 ) @ bot_bot_set_variable @ X2a ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( func @ F5 @ Theta2 ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ! [Eta: trm] :
( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( plus @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ! [Eta: trm] :
( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( times @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( ( P2 @ Sigma3
@ ( collect_variable
@ ^ [X: variable] : $true )
@ Theta2 )
=> ( P2 @ Sigma3 @ U3 @ ( differential @ Theta2 ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ).
% usubstappt_induct
thf(fact_341_dot__def,axiom,
( uSubst_Mirabelle_dot
= ( const @ ( char2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ $false ) ) ) ).
% dot_def
thf(fact_342_usubstappt_Opelims,axiom,
! [X2: produc1418842292n_game,Xa: set_variable,Xb: trm,Y2: option_trm] :
( ( ( uSubst516392818stappt @ X2 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ Xb ) ) )
=> ( ! [X3: variable] :
( ( Xb
= ( var @ X3 ) )
=> ( ( Y2
= ( some_trm @ ( var @ X3 ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( var @ X3 ) ) ) ) ) )
=> ( ! [R3: real] :
( ( Xb
= ( number @ R3 ) )
=> ( ( Y2
= ( some_trm @ ( number @ R3 ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( number @ R3 ) ) ) ) ) )
=> ( ! [F5: char] :
( ( Xb
= ( const @ F5 ) )
=> ( ( Y2
= ( uSubst1559070963pconst @ X2 @ Xa @ F5 ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( const @ F5 ) ) ) ) ) )
=> ( ! [F5: char,Theta2: trm] :
( ( Xb
= ( func @ F5 @ Theta2 ) )
=> ( ( Y2
= ( case_o487727941rm_trm @ none_trm
@ ^ [Sigma_theta: trm] :
( case_o487727941rm_trm @ ( some_trm @ ( func @ F5 @ Sigma_theta ) )
@ ^ [R: trm] :
( if_option_trm
@ ( ( inf_inf_set_variable @ ( static_FVT @ R ) @ Xa )
= 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 ) )
@ X2
@ F5 ) )
@ ( uSubst516392818stappt @ X2 @ Xa @ Theta2 ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( func @ F5 @ Theta2 ) ) ) ) ) )
=> ( ! [Theta2: trm,Eta: trm] :
( ( Xb
= ( plus @ Theta2 @ Eta ) )
=> ( ( Y2
= ( uSubst141955042_Pluso @ ( uSubst516392818stappt @ X2 @ Xa @ Theta2 ) @ ( uSubst516392818stappt @ X2 @ Xa @ Eta ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( plus @ Theta2 @ Eta ) ) ) ) ) )
=> ( ! [Theta2: trm,Eta: trm] :
( ( Xb
= ( times @ Theta2 @ Eta ) )
=> ( ( Y2
= ( uSubst918876924Timeso @ ( uSubst516392818stappt @ X2 @ Xa @ Theta2 ) @ ( uSubst516392818stappt @ X2 @ Xa @ Eta ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( times @ Theta2 @ Eta ) ) ) ) ) )
=> ~ ! [Theta2: trm] :
( ( Xb
= ( differential @ Theta2 ) )
=> ( ( Y2
= ( uSubst716837505ntialo
@ ( uSubst516392818stappt @ X2
@ ( collect_variable
@ ^ [X: variable] : $true )
@ Theta2 ) ) )
=> ~ ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ X2 @ ( produc1482344242le_trm @ Xa @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.pelims
thf(fact_343_Differentialo_Osimps_I2_J,axiom,
( ( uSubst716837505ntialo @ none_trm )
= none_trm ) ).
% Differentialo.simps(2)
thf(fact_344_Differentialo__undef,axiom,
! [Theta: option_trm] :
( ( ( uSubst716837505ntialo @ Theta )
= none_trm )
= ( Theta = none_trm ) ) ).
% Differentialo_undef
thf(fact_345_Pluso_Osimps_I2_J,axiom,
! [Eta2: option_trm] :
( ( uSubst141955042_Pluso @ none_trm @ Eta2 )
= none_trm ) ).
% Pluso.simps(2)
thf(fact_346_Timeso_Osimps_I2_J,axiom,
! [Eta2: option_trm] :
( ( uSubst918876924Timeso @ none_trm @ Eta2 )
= none_trm ) ).
% Timeso.simps(2)
thf(fact_347_Pluso__undef,axiom,
! [Theta: option_trm,Eta2: option_trm] :
( ( ( uSubst141955042_Pluso @ Theta @ Eta2 )
= none_trm )
= ( ( Theta = none_trm )
| ( Eta2 = none_trm ) ) ) ).
% Pluso_undef
thf(fact_348_Timeso__undef,axiom,
! [Theta: option_trm,Eta2: option_trm] :
( ( ( uSubst918876924Timeso @ Theta @ Eta2 )
= none_trm )
= ( ( Theta = none_trm )
| ( Eta2 = none_trm ) ) ) ).
% Timeso_undef
thf(fact_349_Pluso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst141955042_Pluso @ ( some_trm @ V3 ) @ none_trm )
= none_trm ) ).
% Pluso.simps(3)
thf(fact_350_Timeso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst918876924Timeso @ ( some_trm @ V3 ) @ none_trm )
= none_trm ) ).
% Timeso.simps(3)
thf(fact_351_usubstappt_Osimps_I5_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta2: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( plus @ Theta @ Eta2 ) )
= ( uSubst141955042_Pluso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta2 ) ) ) ).
% usubstappt.simps(5)
% Helper facts (5)
thf(help_If_2_1_If_001t__Option__Ooption_It__Syntax__Ofml_J_T,axiom,
! [X2: option_fml,Y2: option_fml] :
( ( if_option_fml @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Syntax__Ofml_J_T,axiom,
! [X2: option_fml,Y2: option_fml] :
( ( if_option_fml @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [X2: option_trm,Y2: option_trm] :
( ( if_option_trm @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Syntax__Otrm_J_T,axiom,
! [X2: option_trm,Y2: option_trm] :
( ( if_option_trm @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( 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 ) ).
%------------------------------------------------------------------------------