TPTP Problem File: ITP200^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP200^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_1407__6352118_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_1407__6352118_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 549 ( 161 unt; 196 typ; 0 def)
% Number of atoms : 1044 ( 570 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 3358 ( 206 ~; 13 |; 55 &;2450 @)
% ( 0 <=>; 634 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 52 ( 51 usr)
% Number of type conns : 638 ( 638 >; 0 *; 0 +; 0 <<)
% Number of symbols : 148 ( 145 usr; 10 con; 0-3 aty)
% Number of variables : 1316 ( 173 ^;1110 !; 33 ?;1316 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:23.912
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
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produc1367888000le_trm: $tType ).
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thf(ty_n_t__Option__Ooption_It__Syntax__Ogame_J,type,
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thf(ty_n_t__Option__Ooption_It__Syntax__Otrm_J,type,
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thf(ty_n_t__Option__Ooption_It__Syntax__Ofml_J,type,
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thf(ty_n_t__Denotational____Semantics__Ointerp,type,
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thf(ty_n_t__Set__Oset_It__Syntax__Ovariable_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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thf(ty_n_t__Syntax__Ovariable,type,
variable: $tType ).
thf(ty_n_t__Syntax__Ogame,type,
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thf(ty_n_t__String__Ochar,type,
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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,
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% Explicit typings (145)
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bNF_Gr852774302le_trm: set_char > ( char > set_variable ) > ( char > trm ) > set_Pr1235432374le_trm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__String__Ochar_001t__String__Ochar_001t__Option__Ooption_It__Syntax__Otrm_J,type,
bNF_Gr951178926on_trm: set_char > ( char > char ) > ( char > option_trm ) > set_Pr943832262on_trm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ovariable_001t__Option__Ooption_It__Syntax__Ofml_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
bNF_Gr600956991on_fml: set_variable > ( variable > option_fml ) > ( variable > option_fml ) > set_Pr927595527on_fml ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ovariable_001t__Option__Ooption_It__Syntax__Ogame_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
bNF_Gr2142503190on_fml: set_variable > ( variable > option_game ) > ( variable > option_fml ) > set_Pr622314946on_fml ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ovariable_001t__Option__Ooption_It__Syntax__Otrm_J_001t__Option__Ooption_It__Syntax__Otrm_J,type,
bNF_Gr400922303on_trm: set_variable > ( variable > option_trm ) > ( variable > option_trm ) > set_Pr1471344007on_trm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ovariable_001t__Set__Oset_It__Syntax__Ovariable_J_001t__Syntax__Otrm,type,
bNF_Gr1330108682le_trm: set_variable > ( variable > set_variable ) > ( variable > trm ) > set_Pr1235432374le_trm ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Syntax__Ovariable_001t__String__Ochar_001t__Option__Ooption_It__Syntax__Otrm_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Option__Ooption_It__Syntax__Ofml_J_001t__Option__Ooption_It__Syntax__Ofml_J,type,
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thf(sy_c_Denotational__Semantics_OUvariation,type,
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thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ofml,type,
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thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ogame,type,
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thf(sy_c_Option_Ooption_ONone_001t__Syntax__Otrm,type,
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thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ofml,type,
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thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ogame,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Syntax__Ofml,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Syntax__Otrm_J_J,type,
member1150310282le_trm: produc1279577057le_trm > set_Pr1969262145le_trm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_J,type,
member1950256170on_trm: produc2018724225on_trm > set_Pr1481565921on_trm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Syntax__Ovariable_J_Mt__Set__Oset_It__Syntax__Ovariable_J_J,type,
member1978718144riable: produc432717079riable > set_Pr181266295riable > $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__Product____Type__Oprod_It__String__Ochar_Mt__String__Ochar_J,type,
member1866856748r_char: produc1593312387r_char > set_Pr604539619r_char > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Syntax__Otrm_Mt__Syntax__Otrm_J,type,
member1735386832rm_trm: product_prod_trm_trm > set_Pr491793543rm_trm > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_c_member_001t__Syntax__Ovariable,type,
member_variable: variable > set_variable > $o ).
thf(sy_v_U,type,
u: set_variable ).
thf(sy_v_Ua____,type,
ua: set_variable ).
thf(sy_v__092_060eta_062____,type,
eta: trm ).
thf(sy_v__092_060nu_062,type,
nu: variable > real ).
thf(sy_v__092_060nu_062_H____,type,
nu2: variable > real ).
thf(sy_v__092_060omega_062,type,
omega: variable > real ).
thf(sy_v__092_060omega_062_H____,type,
omega2: variable > real ).
thf(sy_v__092_060sigma_062_H_H____,type,
sigma: produc1418842292n_game ).
thf(sy_v__092_060theta_062____,type,
theta: trm ).
% Relevant facts (352)
thf(fact_0_vaouter,axiom,
denota1419872369iation @ nu @ omega @ u ).
% vaouter
thf(fact_1_usubstappf__geq__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
!= none_fml )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta )
!= none_trm ) ) ) ).
% usubstappf_geq_conv
thf(fact_2_Geq_Oprems_I1_J,axiom,
( ( uSubst516392804stappf @ sigma @ ua @ ( geq @ theta @ eta ) )
!= none_fml ) ).
% Geq.prems(1)
thf(fact_3_undeft__None,axiom,
none_trm = none_trm ).
% undeft_None
thf(fact_4_usubstappt__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,V: set_variable] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ V @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_5_Geq_Oprems_I2_J,axiom,
denota1419872369iation @ nu2 @ omega2 @ ua ).
% Geq.prems(2)
thf(fact_6_usubstappt__plus__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta )
!= none_trm ) ) ) ).
% usubstappt_plus_conv
thf(fact_7_usubstappt__times__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta )
!= none_trm ) ) ) ).
% usubstappt_times_conv
thf(fact_8_usubstappt__antimon,axiom,
! [V: set_variable,U: set_variable,Sigma: produc1418842292n_game,Theta: trm] :
( ( ord_le282106107riable @ V @ U )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_antimon
thf(fact_9_Differentialo_Osimps_I2_J,axiom,
( ( uSubst716837505ntialo @ none_trm )
= none_trm ) ).
% Differentialo.simps(2)
thf(fact_10_Differentialo__undef,axiom,
! [Theta: option_trm] :
( ( ( uSubst716837505ntialo @ Theta )
= none_trm )
= ( Theta = none_trm ) ) ).
% Differentialo_undef
thf(fact_11_Pluso_Osimps_I2_J,axiom,
! [Eta: option_trm] :
( ( uSubst141955042_Pluso @ none_trm @ Eta )
= none_trm ) ).
% Pluso.simps(2)
thf(fact_12_Timeso_Osimps_I2_J,axiom,
! [Eta: option_trm] :
( ( uSubst918876924Timeso @ none_trm @ Eta )
= none_trm ) ).
% Timeso.simps(2)
thf(fact_13_Pluso__undef,axiom,
! [Theta: option_trm,Eta: option_trm] :
( ( ( uSubst141955042_Pluso @ Theta @ Eta )
= none_trm )
= ( ( Theta = none_trm )
| ( Eta = none_trm ) ) ) ).
% Pluso_undef
thf(fact_14_Timeso__undef,axiom,
! [Theta: option_trm,Eta: option_trm] :
( ( ( uSubst918876924Timeso @ Theta @ Eta )
= none_trm )
= ( ( Theta = none_trm )
| ( Eta = none_trm ) ) ) ).
% Timeso_undef
thf(fact_15_Geqo_Osimps_I2_J,axiom,
! [Eta: option_trm] :
( ( uSubst152838031e_Geqo @ none_trm @ Eta )
= none_fml ) ).
% Geqo.simps(2)
thf(fact_16_usubstappf__antimon,axiom,
! [V: set_variable,U: set_variable,Sigma: produc1418842292n_game,Phi: fml] :
( ( ord_le282106107riable @ V @ U )
=> ( ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
!= none_fml )
=> ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
= ( uSubst516392804stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_antimon
thf(fact_17_usubstappf__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Phi: fml,V: set_variable] :
( ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
!= none_fml )
=> ( ( ( uSubst516392804stappf @ Sigma @ V @ Phi )
!= none_fml )
=> ( ( uSubst516392804stappf @ Sigma @ U @ Phi )
= ( uSubst516392804stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_det
thf(fact_18_usubstappt_Osimps_I5_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
= ( uSubst141955042_Pluso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappt.simps(5)
thf(fact_19_usubstappt_Osimps_I6_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
= ( uSubst918876924Timeso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappt.simps(6)
thf(fact_20_usubstappf_Osimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( uSubst152838031e_Geqo @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappf.simps(2)
thf(fact_21_Geqo__undef,axiom,
! [Theta: option_trm,Eta: option_trm] :
( ( ( uSubst152838031e_Geqo @ Theta @ Eta )
= none_fml )
= ( ( Theta = none_trm )
| ( Eta = none_trm ) ) ) ).
% Geqo_undef
thf(fact_22_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_23_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_24_fml_Oinject_I2_J,axiom,
! [X21: trm,X22: trm,Y21: trm,Y22: trm] :
( ( ( geq @ X21 @ X22 )
= ( geq @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fml.inject(2)
thf(fact_25_Uvariation__mon,axiom,
! [U: set_variable,V: set_variable,Omega: variable > real,Nu: variable > real] :
( ( ord_le282106107riable @ U @ V )
=> ( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Omega @ Nu @ V ) ) ) ).
% Uvariation_mon
thf(fact_26_subsetI,axiom,
! [A: set_char,B: set_char] :
( ! [X: char] :
( ( member_char @ X @ A )
=> ( member_char @ X @ B ) )
=> ( ord_less_eq_set_char @ A @ B ) ) ).
% subsetI
thf(fact_27_subsetI,axiom,
! [A: set_variable,B: set_variable] :
( ! [X: variable] :
( ( member_variable @ X @ A )
=> ( member_variable @ X @ B ) )
=> ( ord_le282106107riable @ A @ B ) ) ).
% subsetI
thf(fact_28_subset__antisym,axiom,
! [A: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_29_trm_Odistinct_I37_J,axiom,
! [X51: trm,X52: trm,X61: trm,X62: trm] :
( ( plus @ X51 @ X52 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(37)
thf(fact_30_order__refl,axiom,
! [X2: set_variable] : ( ord_le282106107riable @ X2 @ X2 ) ).
% order_refl
thf(fact_31_Uvariation__univ,axiom,
! [Nu: variable > real,Nu2: variable > real] :
( denota1419872369iation @ Nu @ Nu2
@ ( collect_variable
@ ^ [X3: variable] : $true ) ) ).
% Uvariation_univ
thf(fact_32_Collect__subset,axiom,
! [A: set_char,P: char > $o] :
( ord_less_eq_set_char
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_33_Collect__subset,axiom,
! [A: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_34_less__eq__set__def,axiom,
( ord_less_eq_set_char
= ( ^ [A2: set_char,B2: set_char] :
( ord_less_eq_char_o
@ ^ [X3: char] : ( member_char @ X3 @ A2 )
@ ^ [X3: char] : ( member_char @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_35_less__eq__set__def,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B2: set_variable] :
( ord_le1407353162able_o
@ ^ [X3: variable] : ( member_variable @ X3 @ A2 )
@ ^ [X3: variable] : ( member_variable @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_36_dual__order_Oantisym,axiom,
! [B3: set_variable,A3: set_variable] :
( ( ord_le282106107riable @ B3 @ A3 )
=> ( ( ord_le282106107riable @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_37_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_variable,Z: set_variable] : Y = Z )
= ( ^ [A4: set_variable,B4: set_variable] :
( ( ord_le282106107riable @ B4 @ A4 )
& ( ord_le282106107riable @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_38_dual__order_Otrans,axiom,
! [B3: set_variable,A3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ B3 @ A3 )
=> ( ( ord_le282106107riable @ C @ B3 )
=> ( ord_le282106107riable @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_39_dual__order_Orefl,axiom,
! [A3: set_variable] : ( ord_le282106107riable @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_40_order__trans,axiom,
! [X2: set_variable,Y2: set_variable,Z2: set_variable] :
( ( ord_le282106107riable @ X2 @ Y2 )
=> ( ( ord_le282106107riable @ Y2 @ Z2 )
=> ( ord_le282106107riable @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_41_order__class_Oorder_Oantisym,axiom,
! [A3: set_variable,B3: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
=> ( ( ord_le282106107riable @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% order_class.order.antisym
thf(fact_42_ord__le__eq__trans,axiom,
! [A3: set_variable,B3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_43_ord__eq__le__trans,axiom,
! [A3: set_variable,B3: set_variable,C: set_variable] :
( ( A3 = B3 )
=> ( ( ord_le282106107riable @ B3 @ C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_44_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_variable,Z: set_variable] : Y = Z )
= ( ^ [A4: set_variable,B4: set_variable] :
( ( ord_le282106107riable @ A4 @ B4 )
& ( ord_le282106107riable @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_45_antisym__conv,axiom,
! [Y2: set_variable,X2: set_variable] :
( ( ord_le282106107riable @ Y2 @ X2 )
=> ( ( ord_le282106107riable @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_46_mem__Collect__eq,axiom,
! [A3: variable,P: variable > $o] :
( ( member_variable @ A3 @ ( collect_variable @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A3: char,P: char > $o] :
( ( member_char @ A3 @ ( collect_char @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_variable] :
( ( collect_variable
@ ^ [X3: variable] : ( member_variable @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_char] :
( ( collect_char
@ ^ [X3: char] : ( member_char @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
! [P: variable > $o,Q: variable > $o] :
( ! [X: variable] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_variable @ P )
= ( collect_variable @ Q ) ) ) ).
% Collect_cong
thf(fact_51_Collect__cong,axiom,
! [P: char > $o,Q: char > $o] :
( ! [X: char] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_char @ P )
= ( collect_char @ Q ) ) ) ).
% Collect_cong
thf(fact_52_order_Otrans,axiom,
! [A3: set_variable,B3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
=> ( ( ord_le282106107riable @ B3 @ C )
=> ( ord_le282106107riable @ A3 @ C ) ) ) ).
% order.trans
thf(fact_53_eq__refl,axiom,
! [X2: set_variable,Y2: set_variable] :
( ( X2 = Y2 )
=> ( ord_le282106107riable @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_54_antisym,axiom,
! [X2: set_variable,Y2: set_variable] :
( ( ord_le282106107riable @ X2 @ Y2 )
=> ( ( ord_le282106107riable @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_55_eq__iff,axiom,
( ( ^ [Y: set_variable,Z: set_variable] : Y = Z )
= ( ^ [X3: set_variable,Y3: set_variable] :
( ( ord_le282106107riable @ X3 @ Y3 )
& ( ord_le282106107riable @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_56_ord__le__eq__subst,axiom,
! [A3: set_variable,B3: set_variable,F: set_variable > set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X @ Y4 )
=> ( ord_le282106107riable @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__eq__le__subst,axiom,
! [A3: set_variable,F: set_variable > set_variable,B3: set_variable,C: set_variable] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le282106107riable @ B3 @ C )
=> ( ! [X: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X @ Y4 )
=> ( ord_le282106107riable @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_58_order__subst2,axiom,
! [A3: set_variable,B3: set_variable,F: set_variable > set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ B3 )
=> ( ( ord_le282106107riable @ ( F @ B3 ) @ C )
=> ( ! [X: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X @ Y4 )
=> ( ord_le282106107riable @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le282106107riable @ ( F @ A3 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_59_order__subst1,axiom,
! [A3: set_variable,F: set_variable > set_variable,B3: set_variable,C: set_variable] :
( ( ord_le282106107riable @ A3 @ ( F @ B3 ) )
=> ( ( ord_le282106107riable @ B3 @ C )
=> ( ! [X: set_variable,Y4: set_variable] :
( ( ord_le282106107riable @ X @ Y4 )
=> ( ord_le282106107riable @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le282106107riable @ A3 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_60_Collect__mono__iff,axiom,
! [P: char > $o,Q: char > $o] :
( ( ord_less_eq_set_char @ ( collect_char @ P ) @ ( collect_char @ Q ) )
= ( ! [X3: char] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_61_Collect__mono__iff,axiom,
! [P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) )
= ( ! [X3: variable] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_62_set__eq__subset,axiom,
( ( ^ [Y: set_variable,Z: set_variable] : Y = Z )
= ( ^ [A2: set_variable,B2: set_variable] :
( ( ord_le282106107riable @ A2 @ B2 )
& ( ord_le282106107riable @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_63_subset__trans,axiom,
! [A: set_variable,B: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( ord_le282106107riable @ B @ C2 )
=> ( ord_le282106107riable @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_64_Collect__mono,axiom,
! [P: char > $o,Q: char > $o] :
( ! [X: char] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_char @ ( collect_char @ P ) @ ( collect_char @ Q ) ) ) ).
% Collect_mono
thf(fact_65_Collect__mono,axiom,
! [P: variable > $o,Q: variable > $o] :
( ! [X: variable] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le282106107riable @ ( collect_variable @ P ) @ ( collect_variable @ Q ) ) ) ).
% Collect_mono
thf(fact_66_subset__refl,axiom,
! [A: set_variable] : ( ord_le282106107riable @ A @ A ) ).
% subset_refl
thf(fact_67_subset__iff,axiom,
( ord_less_eq_set_char
= ( ^ [A2: set_char,B2: set_char] :
! [T: char] :
( ( member_char @ T @ A2 )
=> ( member_char @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_68_subset__iff,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B2: set_variable] :
! [T: variable] :
( ( member_variable @ T @ A2 )
=> ( member_variable @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_69_equalityD2,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ B @ A ) ) ).
% equalityD2
thf(fact_70_equalityD1,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ( ord_le282106107riable @ A @ B ) ) ).
% equalityD1
thf(fact_71_subset__eq,axiom,
( ord_less_eq_set_char
= ( ^ [A2: set_char,B2: set_char] :
! [X3: char] :
( ( member_char @ X3 @ A2 )
=> ( member_char @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_72_subset__eq,axiom,
( ord_le282106107riable
= ( ^ [A2: set_variable,B2: set_variable] :
! [X3: variable] :
( ( member_variable @ X3 @ A2 )
=> ( member_variable @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_73_equalityE,axiom,
! [A: set_variable,B: set_variable] :
( ( A = B )
=> ~ ( ( ord_le282106107riable @ A @ B )
=> ~ ( ord_le282106107riable @ B @ A ) ) ) ).
% equalityE
thf(fact_74_subsetD,axiom,
! [A: set_char,B: set_char,C: char] :
( ( ord_less_eq_set_char @ A @ B )
=> ( ( member_char @ C @ A )
=> ( member_char @ C @ B ) ) ) ).
% subsetD
thf(fact_75_subsetD,axiom,
! [A: set_variable,B: set_variable,C: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ C @ A )
=> ( member_variable @ C @ B ) ) ) ).
% subsetD
thf(fact_76_in__mono,axiom,
! [A: set_char,B: set_char,X2: char] :
( ( ord_less_eq_set_char @ A @ B )
=> ( ( member_char @ X2 @ A )
=> ( member_char @ X2 @ B ) ) ) ).
% in_mono
thf(fact_77_in__mono,axiom,
! [A: set_variable,B: set_variable,X2: variable] :
( ( ord_le282106107riable @ A @ B )
=> ( ( member_variable @ X2 @ A )
=> ( member_variable @ X2 @ B ) ) ) ).
% in_mono
thf(fact_78_Uvariation__sym__rel,axiom,
! [Omega: variable > real,Nu: variable > real,U: set_variable] :
( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Nu @ Omega @ U ) ) ).
% Uvariation_sym_rel
thf(fact_79_Uvariation__refl,axiom,
! [Nu: variable > real,V: set_variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% Uvariation_refl
thf(fact_80_Uvariation__sym,axiom,
( denota1419872369iation
= ( ^ [Omega2: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega2 ) ) ) ).
% Uvariation_sym
thf(fact_81_Uvariation__def,axiom,
( denota1419872369iation
= ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set_variable] :
! [I: variable] :
( ~ ( member_variable @ I @ U2 )
=> ( ( Nu3 @ I )
= ( Nu4 @ I ) ) ) ) ) ).
% Uvariation_def
thf(fact_82_pred__subset__eq,axiom,
! [R: set_char,S: set_char] :
( ( ord_less_eq_char_o
@ ^ [X3: char] : ( member_char @ X3 @ R )
@ ^ [X3: char] : ( member_char @ X3 @ S ) )
= ( ord_less_eq_set_char @ R @ S ) ) ).
% pred_subset_eq
thf(fact_83_pred__subset__eq,axiom,
! [R: set_variable,S: set_variable] :
( ( ord_le1407353162able_o
@ ^ [X3: variable] : ( member_variable @ X3 @ R )
@ ^ [X3: variable] : ( member_variable @ X3 @ S ) )
= ( ord_le282106107riable @ R @ S ) ) ).
% pred_subset_eq
thf(fact_84_subset__Collect__iff,axiom,
! [B: set_char,A: set_char,P: char > $o] :
( ( ord_less_eq_set_char @ B @ A )
=> ( ( ord_less_eq_set_char @ B
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: char] :
( ( member_char @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_85_subset__Collect__iff,axiom,
! [B: set_variable,A: set_variable,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ( ord_le282106107riable @ B
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: variable] :
( ( member_variable @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_86_subset__CollectI,axiom,
! [B: set_char,A: set_char,Q: char > $o,P: char > $o] :
( ( ord_less_eq_set_char @ B @ A )
=> ( ! [X: char] :
( ( member_char @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_char
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_87_subset__CollectI,axiom,
! [B: set_variable,A: set_variable,Q: variable > $o,P: variable > $o] :
( ( ord_le282106107riable @ B @ A )
=> ( ! [X: variable] :
( ( member_variable @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_88_Collect__restrict,axiom,
! [X4: set_char,P: char > $o] :
( ord_less_eq_set_char
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_89_Collect__restrict,axiom,
! [X4: set_variable,P: variable > $o] :
( ord_le282106107riable
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_90_prop__restrict,axiom,
! [X2: char,Z3: set_char,X4: set_char,P: char > $o] :
( ( member_char @ X2 @ Z3 )
=> ( ( ord_less_eq_set_char @ Z3
@ ( collect_char
@ ^ [X3: char] :
( ( member_char @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_91_prop__restrict,axiom,
! [X2: variable,Z3: set_variable,X4: set_variable,P: variable > $o] :
( ( member_variable @ X2 @ Z3 )
=> ( ( ord_le282106107riable @ Z3
@ ( collect_variable
@ ^ [X3: variable] :
( ( member_variable @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_92_conj__subset__def,axiom,
! [A: set_char,P: char > $o,Q: char > $o] :
( ( ord_less_eq_set_char @ A
@ ( collect_char
@ ^ [X3: char] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_char @ A @ ( collect_char @ P ) )
& ( ord_less_eq_set_char @ A @ ( collect_char @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_93_conj__subset__def,axiom,
! [A: set_variable,P: variable > $o,Q: variable > $o] :
( ( ord_le282106107riable @ A
@ ( collect_variable
@ ^ [X3: variable] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le282106107riable @ A @ ( collect_variable @ P ) )
& ( ord_le282106107riable @ A @ ( collect_variable @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_94_Pluso_Oelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_trm] :
( ( ( uSubst141955042_Pluso @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( Y2
!= ( some_trm @ ( plus @ Theta2 @ Eta2 ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( Y2 != none_trm ) )
=> ~ ( ? [V2: trm] :
( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( Y2 != none_trm ) ) ) ) ) ) ).
% Pluso.elims
thf(fact_95_Timeso_Oelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_trm] :
( ( ( uSubst918876924Timeso @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( Y2
!= ( some_trm @ ( times @ Theta2 @ Eta2 ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( Y2 != none_trm ) )
=> ~ ( ? [V2: trm] :
( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( Y2 != none_trm ) ) ) ) ) ) ).
% Timeso.elims
thf(fact_96_usubstappt_Osimps_I7_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( differential @ Theta ) )
= ( uSubst716837505ntialo
@ ( uSubst516392818stappt @ Sigma
@ ( collect_variable
@ ^ [X3: variable] : $true )
@ Theta ) ) ) ).
% usubstappt.simps(7)
thf(fact_97_option_Oinject,axiom,
! [X23: trm,Y23: trm] :
( ( ( some_trm @ X23 )
= ( some_trm @ Y23 ) )
= ( X23 = Y23 ) ) ).
% option.inject
thf(fact_98_option_Oinject,axiom,
! [X23: fml,Y23: fml] :
( ( ( some_fml @ X23 )
= ( some_fml @ Y23 ) )
= ( X23 = Y23 ) ) ).
% option.inject
thf(fact_99_option_Oinject,axiom,
! [X23: game,Y23: game] :
( ( ( some_game @ X23 )
= ( some_game @ Y23 ) )
= ( X23 = Y23 ) ) ).
% option.inject
thf(fact_100_trm_Oinject_I7_J,axiom,
! [X7: trm,Y7: trm] :
( ( ( differential @ X7 )
= ( differential @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_101_not__Some__eq,axiom,
! [X2: option_trm] :
( ( ! [Y3: trm] :
( X2
!= ( some_trm @ Y3 ) ) )
= ( X2 = none_trm ) ) ).
% not_Some_eq
thf(fact_102_not__Some__eq,axiom,
! [X2: option_fml] :
( ( ! [Y3: fml] :
( X2
!= ( some_fml @ Y3 ) ) )
= ( X2 = none_fml ) ) ).
% not_Some_eq
thf(fact_103_not__Some__eq,axiom,
! [X2: option_game] :
( ( ! [Y3: game] :
( X2
!= ( some_game @ Y3 ) ) )
= ( X2 = none_game ) ) ).
% not_Some_eq
thf(fact_104_not__None__eq,axiom,
! [X2: option_trm] :
( ( X2 != none_trm )
= ( ? [Y3: trm] :
( X2
= ( some_trm @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_105_not__None__eq,axiom,
! [X2: option_fml] :
( ( X2 != none_fml )
= ( ? [Y3: fml] :
( X2
= ( some_fml @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_106_not__None__eq,axiom,
! [X2: option_game] :
( ( X2 != none_game )
= ( ? [Y3: game] :
( X2
= ( some_game @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_107_Aterm__Some,axiom,
some_trm = some_trm ).
% Aterm_Some
thf(fact_108_Differentialo_Osimps_I1_J,axiom,
! [Theta: trm] :
( ( uSubst716837505ntialo @ ( some_trm @ Theta ) )
= ( some_trm @ ( differential @ Theta ) ) ) ).
% Differentialo.simps(1)
thf(fact_109_combine__options__cases,axiom,
! [X2: option_trm,P: option_trm > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: trm,B5: trm] :
( ( X2
= ( some_trm @ A5 ) )
=> ( ( Y2
= ( some_trm @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_110_combine__options__cases,axiom,
! [X2: option_trm,P: option_trm > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: trm,B5: fml] :
( ( X2
= ( some_trm @ A5 ) )
=> ( ( Y2
= ( some_fml @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_111_combine__options__cases,axiom,
! [X2: option_trm,P: option_trm > option_game > $o,Y2: option_game] :
( ( ( X2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: trm,B5: game] :
( ( X2
= ( some_trm @ A5 ) )
=> ( ( Y2
= ( some_game @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_112_combine__options__cases,axiom,
! [X2: option_fml,P: option_fml > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: fml,B5: trm] :
( ( X2
= ( some_fml @ A5 ) )
=> ( ( Y2
= ( some_trm @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_113_combine__options__cases,axiom,
! [X2: option_fml,P: option_fml > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: fml,B5: fml] :
( ( X2
= ( some_fml @ A5 ) )
=> ( ( Y2
= ( some_fml @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_114_combine__options__cases,axiom,
! [X2: option_fml,P: option_fml > option_game > $o,Y2: option_game] :
( ( ( X2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: fml,B5: game] :
( ( X2
= ( some_fml @ A5 ) )
=> ( ( Y2
= ( some_game @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_115_combine__options__cases,axiom,
! [X2: option_game,P: option_game > option_trm > $o,Y2: option_trm] :
( ( ( X2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_trm )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: game,B5: trm] :
( ( X2
= ( some_game @ A5 ) )
=> ( ( Y2
= ( some_trm @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_116_combine__options__cases,axiom,
! [X2: option_game,P: option_game > option_fml > $o,Y2: option_fml] :
( ( ( X2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_fml )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: game,B5: fml] :
( ( X2
= ( some_game @ A5 ) )
=> ( ( Y2
= ( some_fml @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_117_combine__options__cases,axiom,
! [X2: option_game,P: option_game > option_game > $o,Y2: option_game] :
( ( ( X2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2 = none_game )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A5: game,B5: game] :
( ( X2
= ( some_game @ A5 ) )
=> ( ( Y2
= ( some_game @ B5 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_118_split__option__all,axiom,
( ( ^ [P2: option_trm > $o] :
! [X5: option_trm] : ( P2 @ X5 ) )
= ( ^ [P3: option_trm > $o] :
( ( P3 @ none_trm )
& ! [X3: trm] : ( P3 @ ( some_trm @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_119_split__option__all,axiom,
( ( ^ [P2: option_fml > $o] :
! [X5: option_fml] : ( P2 @ X5 ) )
= ( ^ [P3: option_fml > $o] :
( ( P3 @ none_fml )
& ! [X3: fml] : ( P3 @ ( some_fml @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_120_split__option__all,axiom,
( ( ^ [P2: option_game > $o] :
! [X5: option_game] : ( P2 @ X5 ) )
= ( ^ [P3: option_game > $o] :
( ( P3 @ none_game )
& ! [X3: game] : ( P3 @ ( some_game @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_121_split__option__ex,axiom,
( ( ^ [P2: option_trm > $o] :
? [X5: option_trm] : ( P2 @ X5 ) )
= ( ^ [P3: option_trm > $o] :
( ( P3 @ none_trm )
| ? [X3: trm] : ( P3 @ ( some_trm @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_122_split__option__ex,axiom,
( ( ^ [P2: option_fml > $o] :
? [X5: option_fml] : ( P2 @ X5 ) )
= ( ^ [P3: option_fml > $o] :
( ( P3 @ none_fml )
| ? [X3: fml] : ( P3 @ ( some_fml @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_123_split__option__ex,axiom,
( ( ^ [P2: option_game > $o] :
? [X5: option_game] : ( P2 @ X5 ) )
= ( ^ [P3: option_game > $o] :
( ( P3 @ none_game )
| ? [X3: game] : ( P3 @ ( some_game @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_124_option_Oinducts,axiom,
! [P: option_trm > $o,Option: option_trm] :
( ( P @ none_trm )
=> ( ! [X: trm] : ( P @ ( some_trm @ X ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_125_option_Oinducts,axiom,
! [P: option_fml > $o,Option: option_fml] :
( ( P @ none_fml )
=> ( ! [X: fml] : ( P @ ( some_fml @ X ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_126_option_Oinducts,axiom,
! [P: option_game > $o,Option: option_game] :
( ( P @ none_game )
=> ( ! [X: game] : ( P @ ( some_game @ X ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_127_option_Oexhaust,axiom,
! [Y2: option_trm] :
( ( Y2 != none_trm )
=> ~ ! [X24: trm] :
( Y2
!= ( some_trm @ X24 ) ) ) ).
% option.exhaust
thf(fact_128_option_Oexhaust,axiom,
! [Y2: option_fml] :
( ( Y2 != none_fml )
=> ~ ! [X24: fml] :
( Y2
!= ( some_fml @ X24 ) ) ) ).
% option.exhaust
thf(fact_129_option_Oexhaust,axiom,
! [Y2: option_game] :
( ( Y2 != none_game )
=> ~ ! [X24: game] :
( Y2
!= ( some_game @ X24 ) ) ) ).
% option.exhaust
thf(fact_130_option_OdiscI,axiom,
! [Option: option_trm,X23: trm] :
( ( Option
= ( some_trm @ X23 ) )
=> ( Option != none_trm ) ) ).
% option.discI
thf(fact_131_option_OdiscI,axiom,
! [Option: option_fml,X23: fml] :
( ( Option
= ( some_fml @ X23 ) )
=> ( Option != none_fml ) ) ).
% option.discI
thf(fact_132_option_OdiscI,axiom,
! [Option: option_game,X23: game] :
( ( Option
= ( some_game @ X23 ) )
=> ( Option != none_game ) ) ).
% option.discI
thf(fact_133_option_Odistinct_I1_J,axiom,
! [X23: trm] :
( none_trm
!= ( some_trm @ X23 ) ) ).
% option.distinct(1)
thf(fact_134_option_Odistinct_I1_J,axiom,
! [X23: fml] :
( none_fml
!= ( some_fml @ X23 ) ) ).
% option.distinct(1)
thf(fact_135_option_Odistinct_I1_J,axiom,
! [X23: game] :
( none_game
!= ( some_game @ X23 ) ) ).
% option.distinct(1)
thf(fact_136_Differentialo_Oinduct,axiom,
! [P: option_trm > $o,A0: option_trm] :
( ! [Theta2: trm] : ( P @ ( some_trm @ Theta2 ) )
=> ( ( P @ none_trm )
=> ( P @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_137_Differentialo_Ocases,axiom,
! [X2: option_trm] :
( ! [Theta2: trm] :
( X2
!= ( some_trm @ Theta2 ) )
=> ( X2 = none_trm ) ) ).
% Differentialo.cases
thf(fact_138_Assigno_Oinduct,axiom,
! [P: variable > option_trm > $o,A0: variable,A1: option_trm] :
( ! [X: variable,Theta2: trm] : ( P @ X @ ( some_trm @ Theta2 ) )
=> ( ! [X: variable] : ( P @ X @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_139_Timeso_Oinduct,axiom,
! [P: option_trm > option_trm > $o,A0: option_trm,A1: option_trm] :
( ! [Theta2: trm,Eta2: trm] : ( P @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) )
=> ( ! [X_1: option_trm] : ( P @ none_trm @ X_1 )
=> ( ! [V2: trm] : ( P @ ( some_trm @ V2 ) @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_140_undeft__equiv,axiom,
! [Theta: option_trm] :
( ( Theta != none_trm )
= ( ? [T: trm] :
( Theta
= ( some_trm @ T ) ) ) ) ).
% undeft_equiv
thf(fact_141_ODEo_Oinduct,axiom,
! [P: char > option_trm > $o,A0: char,A1: option_trm] :
( ! [X: char,Theta2: trm] : ( P @ X @ ( some_trm @ Theta2 ) )
=> ( ! [X: char] : ( P @ X @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_142_Differentialo_Oelims,axiom,
! [X2: option_trm,Y2: option_trm] :
( ( ( uSubst716837505ntialo @ X2 )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ( Y2
!= ( some_trm @ ( differential @ Theta2 ) ) ) )
=> ~ ( ( X2 = none_trm )
=> ( Y2 != none_trm ) ) ) ) ).
% Differentialo.elims
thf(fact_143_trm_Odistinct_I41_J,axiom,
! [X61: trm,X62: trm,X7: trm] :
( ( times @ X61 @ X62 )
!= ( differential @ X7 ) ) ).
% trm.distinct(41)
thf(fact_144_trm_Odistinct_I39_J,axiom,
! [X51: trm,X52: trm,X7: trm] :
( ( plus @ X51 @ X52 )
!= ( differential @ X7 ) ) ).
% trm.distinct(39)
thf(fact_145_Timeso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst918876924Timeso @ ( some_trm @ V3 ) @ none_trm )
= none_trm ) ).
% Timeso.simps(3)
thf(fact_146_Pluso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst141955042_Pluso @ ( some_trm @ V3 ) @ none_trm )
= none_trm ) ).
% Pluso.simps(3)
thf(fact_147_Pluso_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst141955042_Pluso @ ( some_trm @ Theta ) @ ( some_trm @ Eta ) )
= ( some_trm @ ( plus @ Theta @ Eta ) ) ) ).
% Pluso.simps(1)
thf(fact_148_Timeso_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst918876924Timeso @ ( some_trm @ Theta ) @ ( some_trm @ Eta ) )
= ( some_trm @ ( times @ Theta @ Eta ) ) ) ).
% Timeso.simps(1)
thf(fact_149_usubstappt__differential__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( differential @ Theta ) )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma
@ ( collect_variable
@ ^ [X3: variable] : $true )
@ Theta )
!= none_trm ) ) ).
% usubstappt_differential_conv
thf(fact_150_Geqo_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst152838031e_Geqo @ ( some_trm @ V3 ) @ none_trm )
= none_fml ) ).
% Geqo.simps(3)
thf(fact_151_Geqo_Oelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_fml] :
( ( ( uSubst152838031e_Geqo @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( Y2
!= ( some_fml @ ( geq @ Theta2 @ Eta2 ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( Y2 != none_fml ) )
=> ~ ( ? [V2: trm] :
( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( Y2 != none_fml ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_152_Differentialo_Opelims,axiom,
! [X2: option_trm,Y2: option_trm] :
( ( ( uSubst716837505ntialo @ X2 )
= Y2 )
=> ( ( accp_option_trm @ uSubst2051024824lo_rel @ X2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ( ( Y2
= ( some_trm @ ( differential @ Theta2 ) ) )
=> ~ ( accp_option_trm @ uSubst2051024824lo_rel @ ( some_trm @ Theta2 ) ) ) )
=> ~ ( ( X2 = none_trm )
=> ( ( Y2 = none_trm )
=> ~ ( accp_option_trm @ uSubst2051024824lo_rel @ none_trm ) ) ) ) ) ) ).
% Differentialo.pelims
thf(fact_153_Geqo_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst152838031e_Geqo @ ( some_trm @ Theta ) @ ( some_trm @ Eta ) )
= ( some_fml @ ( geq @ Theta @ Eta ) ) ) ).
% Geqo.simps(1)
thf(fact_154_GreatestI2__order,axiom,
! [P: set_variable > $o,X2: set_variable,Q: set_variable > $o] :
( ( P @ X2 )
=> ( ! [Y4: set_variable] :
( ( P @ Y4 )
=> ( ord_le282106107riable @ Y4 @ X2 ) )
=> ( ! [X: set_variable] :
( ( P @ X )
=> ( ! [Y5: set_variable] :
( ( P @ Y5 )
=> ( ord_le282106107riable @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_272405634riable @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_155_Greatest__equality,axiom,
! [P: set_variable > $o,X2: set_variable] :
( ( P @ X2 )
=> ( ! [Y4: set_variable] :
( ( P @ Y4 )
=> ( ord_le282106107riable @ Y4 @ X2 ) )
=> ( ( order_272405634riable @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_156_Diamondo_Oinduct,axiom,
! [P: option_game > option_fml > $o,A0: option_game,A1: option_fml] :
( ! [Alpha: game,Phi2: fml] : ( P @ ( some_game @ Alpha ) @ ( some_fml @ Phi2 ) )
=> ( ! [X_1: option_fml] : ( P @ none_game @ X_1 )
=> ( ! [V2: game] : ( P @ ( some_game @ V2 ) @ none_fml )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Diamondo.induct
thf(fact_157_Existso_Oinduct,axiom,
! [P: variable > option_fml > $o,A0: variable,A1: option_fml] :
( ! [X: variable,Phi2: fml] : ( P @ X @ ( some_fml @ Phi2 ) )
=> ( ! [X: variable] : ( P @ X @ none_fml )
=> ( P @ A0 @ A1 ) ) ) ).
% Existso.induct
thf(fact_158_undeff__equiv,axiom,
! [Phi: option_fml] :
( ( Phi != none_fml )
= ( ? [F2: fml] :
( Phi
= ( some_fml @ F2 ) ) ) ) ).
% undeff_equiv
thf(fact_159_Testo_Oinduct,axiom,
! [P: option_fml > $o,A0: option_fml] :
( ! [Phi2: fml] : ( P @ ( some_fml @ Phi2 ) )
=> ( ( P @ none_fml )
=> ( P @ A0 ) ) ) ).
% Testo.induct
thf(fact_160_Testo_Ocases,axiom,
! [X2: option_fml] :
( ! [Phi2: fml] :
( X2
!= ( some_fml @ Phi2 ) )
=> ( X2 = none_fml ) ) ).
% Testo.cases
thf(fact_161_Ando_Oinduct,axiom,
! [P: option_fml > option_fml > $o,A0: option_fml,A1: option_fml] :
( ! [Phi2: fml,Psi: fml] : ( P @ ( some_fml @ Phi2 ) @ ( some_fml @ Psi ) )
=> ( ! [X_1: option_fml] : ( P @ none_fml @ X_1 )
=> ( ! [V2: fml] : ( P @ ( some_fml @ V2 ) @ none_fml )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Ando.induct
thf(fact_162_accp__subset,axiom,
! [R1: produc1024976764on_trm > produc1024976764on_trm > $o,R2: produc1024976764on_trm > produc1024976764on_trm > $o] :
( ( ord_le370603054_trm_o @ R1 @ R2 )
=> ( ord_le373274227_trm_o @ ( accp_P394694195on_trm @ R2 ) @ ( accp_P394694195on_trm @ R1 ) ) ) ).
% accp_subset
thf(fact_163_accp__subset,axiom,
! [R1: produc1072747943on_trm > produc1072747943on_trm > $o,R2: produc1072747943on_trm > produc1072747943on_trm > $o] :
( ( ord_le90123726_trm_o @ R1 @ R2 )
=> ( ord_le1721128566_trm_o @ ( accp_P758790320on_trm @ R2 ) @ ( accp_P758790320on_trm @ R1 ) ) ) ).
% accp_subset
thf(fact_164_accp__subset,axiom,
! [R1: produc464795408on_trm > produc464795408on_trm > $o,R2: produc464795408on_trm > produc464795408on_trm > $o] :
( ( ord_le157637894_trm_o @ R1 @ R2 )
=> ( ord_le1222882015_trm_o @ ( accp_P419951815on_trm @ R2 ) @ ( accp_P419951815on_trm @ R1 ) ) ) ).
% accp_subset
thf(fact_165_accp__subset,axiom,
! [R1: option_trm > option_trm > $o,R2: option_trm > option_trm > $o] :
( ( ord_le847028878_trm_o @ R1 @ R2 )
=> ( ord_le1957133745_trm_o @ ( accp_option_trm @ R2 ) @ ( accp_option_trm @ R1 ) ) ) ).
% accp_subset
thf(fact_166_accp__subset,axiom,
! [R1: option_fml > option_fml > $o,R2: option_fml > option_fml > $o] :
( ( ord_le802833166_fml_o @ R1 @ R2 )
=> ( ord_le181580155_fml_o @ ( accp_option_fml @ R2 ) @ ( accp_option_fml @ R1 ) ) ) ).
% accp_subset
thf(fact_167_accp__subset__induct,axiom,
! [D: produc1024976764on_trm > $o,R: produc1024976764on_trm > produc1024976764on_trm > $o,X2: produc1024976764on_trm,P: produc1024976764on_trm > $o] :
( ( ord_le373274227_trm_o @ D @ ( accp_P394694195on_trm @ R ) )
=> ( ! [X: produc1024976764on_trm,Z4: produc1024976764on_trm] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: produc1024976764on_trm] :
( ( D @ X )
=> ( ! [Z5: produc1024976764on_trm] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_168_accp__subset__induct,axiom,
! [D: produc1072747943on_trm > $o,R: produc1072747943on_trm > produc1072747943on_trm > $o,X2: produc1072747943on_trm,P: produc1072747943on_trm > $o] :
( ( ord_le1721128566_trm_o @ D @ ( accp_P758790320on_trm @ R ) )
=> ( ! [X: produc1072747943on_trm,Z4: produc1072747943on_trm] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: produc1072747943on_trm] :
( ( D @ X )
=> ( ! [Z5: produc1072747943on_trm] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_169_accp__subset__induct,axiom,
! [D: produc464795408on_trm > $o,R: produc464795408on_trm > produc464795408on_trm > $o,X2: produc464795408on_trm,P: produc464795408on_trm > $o] :
( ( ord_le1222882015_trm_o @ D @ ( accp_P419951815on_trm @ R ) )
=> ( ! [X: produc464795408on_trm,Z4: produc464795408on_trm] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: produc464795408on_trm] :
( ( D @ X )
=> ( ! [Z5: produc464795408on_trm] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_170_accp__subset__induct,axiom,
! [D: option_trm > $o,R: option_trm > option_trm > $o,X2: option_trm,P: option_trm > $o] :
( ( ord_le1957133745_trm_o @ D @ ( accp_option_trm @ R ) )
=> ( ! [X: option_trm,Z4: option_trm] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: option_trm] :
( ( D @ X )
=> ( ! [Z5: option_trm] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_171_accp__subset__induct,axiom,
! [D: option_fml > $o,R: option_fml > option_fml > $o,X2: option_fml,P: option_fml > $o] :
( ( ord_le181580155_fml_o @ D @ ( accp_option_fml @ R ) )
=> ( ! [X: option_fml,Z4: option_fml] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: option_fml] :
( ( D @ X )
=> ( ! [Z5: option_fml] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_172_usubstappf__geqr,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
!= none_fml )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( some_fml @ ( geq @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_173_usubstappf__geq,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Eta )
!= none_trm )
=> ( ( uSubst516392804stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( some_fml @ ( geq @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_174_option_Ocollapse,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( ( some_trm @ ( the_trm @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_175_option_Ocollapse,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( ( some_fml @ ( the_fml @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_176_option_Ocollapse,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( ( some_game @ ( the_game @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_177_Composeo_Oinduct,axiom,
! [P: option_game > option_game > $o,A0: option_game,A1: option_game] :
( ! [Alpha: game,Beta: game] : ( P @ ( some_game @ Alpha ) @ ( some_game @ Beta ) )
=> ( ! [Alpha: option_game] : ( P @ Alpha @ none_game )
=> ( ! [V2: game] : ( P @ none_game @ ( some_game @ V2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_178_undefg__equiv,axiom,
! [Alpha2: option_game] :
( ( Alpha2 != none_game )
= ( ? [G: game] :
( Alpha2
= ( some_game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_179_Loopo_Oinduct,axiom,
! [P: option_game > $o,A0: option_game] :
( ! [Alpha: game] : ( P @ ( some_game @ Alpha ) )
=> ( ( P @ none_game )
=> ( P @ A0 ) ) ) ).
% Loopo.induct
thf(fact_180_Loopo_Ocases,axiom,
! [X2: option_game] :
( ! [Alpha: game] :
( X2
!= ( some_game @ Alpha ) )
=> ( X2 = none_game ) ) ).
% Loopo.cases
thf(fact_181_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_182_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_183_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_184_option_Osel,axiom,
! [X23: trm] :
( ( the_trm @ ( some_trm @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_185_option_Osel,axiom,
! [X23: fml] :
( ( the_fml @ ( some_fml @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_186_option_Osel,axiom,
! [X23: game] :
( ( the_game @ ( some_game @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_187_option_Oexhaust__sel,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( Option
= ( some_trm @ ( the_trm @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_188_option_Oexhaust__sel,axiom,
! [Option: option_fml] :
( ( Option != none_fml )
=> ( Option
= ( some_fml @ ( the_fml @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_189_option_Oexhaust__sel,axiom,
! [Option: option_game] :
( ( Option != none_game )
=> ( Option
= ( some_game @ ( the_game @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_190_accp_Ocases,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,A3: produc1024976764on_trm] :
( ( accp_P394694195on_trm @ R3 @ A3 )
=> ! [Y5: produc1024976764on_trm] :
( ( R3 @ Y5 @ A3 )
=> ( accp_P394694195on_trm @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_191_accp_Ocases,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,A3: produc1072747943on_trm] :
( ( accp_P758790320on_trm @ R3 @ A3 )
=> ! [Y5: produc1072747943on_trm] :
( ( R3 @ Y5 @ A3 )
=> ( accp_P758790320on_trm @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_192_accp_Ocases,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,A3: produc464795408on_trm] :
( ( accp_P419951815on_trm @ R3 @ A3 )
=> ! [Y5: produc464795408on_trm] :
( ( R3 @ Y5 @ A3 )
=> ( accp_P419951815on_trm @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_193_accp_Ocases,axiom,
! [R3: option_trm > option_trm > $o,A3: option_trm] :
( ( accp_option_trm @ R3 @ A3 )
=> ! [Y5: option_trm] :
( ( R3 @ Y5 @ A3 )
=> ( accp_option_trm @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_194_accp_Ocases,axiom,
! [R3: option_fml > option_fml > $o,A3: option_fml] :
( ( accp_option_fml @ R3 @ A3 )
=> ! [Y5: option_fml] :
( ( R3 @ Y5 @ A3 )
=> ( accp_option_fml @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_195_accp_Osimps,axiom,
( accp_P394694195on_trm
= ( ^ [R4: produc1024976764on_trm > produc1024976764on_trm > $o,A4: produc1024976764on_trm] :
? [X3: produc1024976764on_trm] :
( ( A4 = X3 )
& ! [Y3: produc1024976764on_trm] :
( ( R4 @ Y3 @ X3 )
=> ( accp_P394694195on_trm @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_196_accp_Osimps,axiom,
( accp_P758790320on_trm
= ( ^ [R4: produc1072747943on_trm > produc1072747943on_trm > $o,A4: produc1072747943on_trm] :
? [X3: produc1072747943on_trm] :
( ( A4 = X3 )
& ! [Y3: produc1072747943on_trm] :
( ( R4 @ Y3 @ X3 )
=> ( accp_P758790320on_trm @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_197_accp_Osimps,axiom,
( accp_P419951815on_trm
= ( ^ [R4: produc464795408on_trm > produc464795408on_trm > $o,A4: produc464795408on_trm] :
? [X3: produc464795408on_trm] :
( ( A4 = X3 )
& ! [Y3: produc464795408on_trm] :
( ( R4 @ Y3 @ X3 )
=> ( accp_P419951815on_trm @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_198_accp_Osimps,axiom,
( accp_option_trm
= ( ^ [R4: option_trm > option_trm > $o,A4: option_trm] :
? [X3: option_trm] :
( ( A4 = X3 )
& ! [Y3: option_trm] :
( ( R4 @ Y3 @ X3 )
=> ( accp_option_trm @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_199_accp_Osimps,axiom,
( accp_option_fml
= ( ^ [R4: option_fml > option_fml > $o,A4: option_fml] :
? [X3: option_fml] :
( ( A4 = X3 )
& ! [Y3: option_fml] :
( ( R4 @ Y3 @ X3 )
=> ( accp_option_fml @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_200_accp_Ointros,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,X2: produc1024976764on_trm] :
( ! [Y4: produc1024976764on_trm] :
( ( R3 @ Y4 @ X2 )
=> ( accp_P394694195on_trm @ R3 @ Y4 ) )
=> ( accp_P394694195on_trm @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_201_accp_Ointros,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,X2: produc1072747943on_trm] :
( ! [Y4: produc1072747943on_trm] :
( ( R3 @ Y4 @ X2 )
=> ( accp_P758790320on_trm @ R3 @ Y4 ) )
=> ( accp_P758790320on_trm @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_202_accp_Ointros,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,X2: produc464795408on_trm] :
( ! [Y4: produc464795408on_trm] :
( ( R3 @ Y4 @ X2 )
=> ( accp_P419951815on_trm @ R3 @ Y4 ) )
=> ( accp_P419951815on_trm @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_203_accp_Ointros,axiom,
! [R3: option_trm > option_trm > $o,X2: option_trm] :
( ! [Y4: option_trm] :
( ( R3 @ Y4 @ X2 )
=> ( accp_option_trm @ R3 @ Y4 ) )
=> ( accp_option_trm @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_204_accp_Ointros,axiom,
! [R3: option_fml > option_fml > $o,X2: option_fml] :
( ! [Y4: option_fml] :
( ( R3 @ Y4 @ X2 )
=> ( accp_option_fml @ R3 @ Y4 ) )
=> ( accp_option_fml @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_205_accp__induct,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,A3: produc1024976764on_trm,P: produc1024976764on_trm > $o] :
( ( accp_P394694195on_trm @ R3 @ A3 )
=> ( ! [X: produc1024976764on_trm] :
( ( accp_P394694195on_trm @ R3 @ X )
=> ( ! [Y5: produc1024976764on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_206_accp__induct,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,A3: produc1072747943on_trm,P: produc1072747943on_trm > $o] :
( ( accp_P758790320on_trm @ R3 @ A3 )
=> ( ! [X: produc1072747943on_trm] :
( ( accp_P758790320on_trm @ R3 @ X )
=> ( ! [Y5: produc1072747943on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_207_accp__induct,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,A3: produc464795408on_trm,P: produc464795408on_trm > $o] :
( ( accp_P419951815on_trm @ R3 @ A3 )
=> ( ! [X: produc464795408on_trm] :
( ( accp_P419951815on_trm @ R3 @ X )
=> ( ! [Y5: produc464795408on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_208_accp__induct,axiom,
! [R3: option_trm > option_trm > $o,A3: option_trm,P: option_trm > $o] :
( ( accp_option_trm @ R3 @ A3 )
=> ( ! [X: option_trm] :
( ( accp_option_trm @ R3 @ X )
=> ( ! [Y5: option_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_209_accp__induct,axiom,
! [R3: option_fml > option_fml > $o,A3: option_fml,P: option_fml > $o] :
( ( accp_option_fml @ R3 @ A3 )
=> ( ! [X: option_fml] :
( ( accp_option_fml @ R3 @ X )
=> ( ! [Y5: option_fml] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_210_accp_Oinducts,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,X2: produc1024976764on_trm,P: produc1024976764on_trm > $o] :
( ( accp_P394694195on_trm @ R3 @ X2 )
=> ( ! [X: produc1024976764on_trm] :
( ! [Y5: produc1024976764on_trm] :
( ( R3 @ Y5 @ X )
=> ( accp_P394694195on_trm @ R3 @ Y5 ) )
=> ( ! [Y5: produc1024976764on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_211_accp_Oinducts,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,X2: produc1072747943on_trm,P: produc1072747943on_trm > $o] :
( ( accp_P758790320on_trm @ R3 @ X2 )
=> ( ! [X: produc1072747943on_trm] :
( ! [Y5: produc1072747943on_trm] :
( ( R3 @ Y5 @ X )
=> ( accp_P758790320on_trm @ R3 @ Y5 ) )
=> ( ! [Y5: produc1072747943on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_212_accp_Oinducts,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,X2: produc464795408on_trm,P: produc464795408on_trm > $o] :
( ( accp_P419951815on_trm @ R3 @ X2 )
=> ( ! [X: produc464795408on_trm] :
( ! [Y5: produc464795408on_trm] :
( ( R3 @ Y5 @ X )
=> ( accp_P419951815on_trm @ R3 @ Y5 ) )
=> ( ! [Y5: produc464795408on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_213_accp_Oinducts,axiom,
! [R3: option_trm > option_trm > $o,X2: option_trm,P: option_trm > $o] :
( ( accp_option_trm @ R3 @ X2 )
=> ( ! [X: option_trm] :
( ! [Y5: option_trm] :
( ( R3 @ Y5 @ X )
=> ( accp_option_trm @ R3 @ Y5 ) )
=> ( ! [Y5: option_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_214_accp_Oinducts,axiom,
! [R3: option_fml > option_fml > $o,X2: option_fml,P: option_fml > $o] :
( ( accp_option_fml @ R3 @ X2 )
=> ( ! [X: option_fml] :
( ! [Y5: option_fml] :
( ( R3 @ Y5 @ X )
=> ( accp_option_fml @ R3 @ Y5 ) )
=> ( ! [Y5: option_fml] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_215_accp__downward,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,B3: produc1024976764on_trm,A3: produc1024976764on_trm] :
( ( accp_P394694195on_trm @ R3 @ B3 )
=> ( ( R3 @ A3 @ B3 )
=> ( accp_P394694195on_trm @ R3 @ A3 ) ) ) ).
% accp_downward
thf(fact_216_accp__downward,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,B3: produc1072747943on_trm,A3: produc1072747943on_trm] :
( ( accp_P758790320on_trm @ R3 @ B3 )
=> ( ( R3 @ A3 @ B3 )
=> ( accp_P758790320on_trm @ R3 @ A3 ) ) ) ).
% accp_downward
thf(fact_217_accp__downward,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,B3: produc464795408on_trm,A3: produc464795408on_trm] :
( ( accp_P419951815on_trm @ R3 @ B3 )
=> ( ( R3 @ A3 @ B3 )
=> ( accp_P419951815on_trm @ R3 @ A3 ) ) ) ).
% accp_downward
thf(fact_218_accp__downward,axiom,
! [R3: option_trm > option_trm > $o,B3: option_trm,A3: option_trm] :
( ( accp_option_trm @ R3 @ B3 )
=> ( ( R3 @ A3 @ B3 )
=> ( accp_option_trm @ R3 @ A3 ) ) ) ).
% accp_downward
thf(fact_219_accp__downward,axiom,
! [R3: option_fml > option_fml > $o,B3: option_fml,A3: option_fml] :
( ( accp_option_fml @ R3 @ B3 )
=> ( ( R3 @ A3 @ B3 )
=> ( accp_option_fml @ R3 @ A3 ) ) ) ).
% accp_downward
thf(fact_220_not__accp__down,axiom,
! [R: produc1024976764on_trm > produc1024976764on_trm > $o,X2: produc1024976764on_trm] :
( ~ ( accp_P394694195on_trm @ R @ X2 )
=> ~ ! [Z4: produc1024976764on_trm] :
( ( R @ Z4 @ X2 )
=> ( accp_P394694195on_trm @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_221_not__accp__down,axiom,
! [R: produc1072747943on_trm > produc1072747943on_trm > $o,X2: produc1072747943on_trm] :
( ~ ( accp_P758790320on_trm @ R @ X2 )
=> ~ ! [Z4: produc1072747943on_trm] :
( ( R @ Z4 @ X2 )
=> ( accp_P758790320on_trm @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_222_not__accp__down,axiom,
! [R: produc464795408on_trm > produc464795408on_trm > $o,X2: produc464795408on_trm] :
( ~ ( accp_P419951815on_trm @ R @ X2 )
=> ~ ! [Z4: produc464795408on_trm] :
( ( R @ Z4 @ X2 )
=> ( accp_P419951815on_trm @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_223_not__accp__down,axiom,
! [R: option_trm > option_trm > $o,X2: option_trm] :
( ~ ( accp_option_trm @ R @ X2 )
=> ~ ! [Z4: option_trm] :
( ( R @ Z4 @ X2 )
=> ( accp_option_trm @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_224_not__accp__down,axiom,
! [R: option_fml > option_fml > $o,X2: option_fml] :
( ~ ( accp_option_fml @ R @ X2 )
=> ~ ! [Z4: option_fml] :
( ( R @ Z4 @ X2 )
=> ( accp_option_fml @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_225_accp__induct__rule,axiom,
! [R3: produc1024976764on_trm > produc1024976764on_trm > $o,A3: produc1024976764on_trm,P: produc1024976764on_trm > $o] :
( ( accp_P394694195on_trm @ R3 @ A3 )
=> ( ! [X: produc1024976764on_trm] :
( ( accp_P394694195on_trm @ R3 @ X )
=> ( ! [Y5: produc1024976764on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_226_accp__induct__rule,axiom,
! [R3: produc1072747943on_trm > produc1072747943on_trm > $o,A3: produc1072747943on_trm,P: produc1072747943on_trm > $o] :
( ( accp_P758790320on_trm @ R3 @ A3 )
=> ( ! [X: produc1072747943on_trm] :
( ( accp_P758790320on_trm @ R3 @ X )
=> ( ! [Y5: produc1072747943on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_227_accp__induct__rule,axiom,
! [R3: produc464795408on_trm > produc464795408on_trm > $o,A3: produc464795408on_trm,P: produc464795408on_trm > $o] :
( ( accp_P419951815on_trm @ R3 @ A3 )
=> ( ! [X: produc464795408on_trm] :
( ( accp_P419951815on_trm @ R3 @ X )
=> ( ! [Y5: produc464795408on_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_228_accp__induct__rule,axiom,
! [R3: option_trm > option_trm > $o,A3: option_trm,P: option_trm > $o] :
( ( accp_option_trm @ R3 @ A3 )
=> ( ! [X: option_trm] :
( ( accp_option_trm @ R3 @ X )
=> ( ! [Y5: option_trm] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_229_accp__induct__rule,axiom,
! [R3: option_fml > option_fml > $o,A3: option_fml,P: option_fml > $o] :
( ( accp_option_fml @ R3 @ A3 )
=> ( ! [X: option_fml] :
( ( accp_option_fml @ R3 @ X )
=> ( ! [Y5: option_fml] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_230_Geqo_Opelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_fml] :
( ( ( uSubst152838031e_Geqo @ X2 @ Xa )
= Y2 )
=> ( ( accp_P758790320on_trm @ uSubst1304144938qo_rel @ ( produc709594775on_trm @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( ( Y2
= ( some_fml @ ( geq @ Theta2 @ Eta2 ) ) )
=> ~ ( accp_P758790320on_trm @ uSubst1304144938qo_rel @ ( produc709594775on_trm @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( ( Y2 = none_fml )
=> ~ ( accp_P758790320on_trm @ uSubst1304144938qo_rel @ ( produc709594775on_trm @ none_trm @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( ( Y2 = none_fml )
=> ~ ( accp_P758790320on_trm @ uSubst1304144938qo_rel @ ( produc709594775on_trm @ ( some_trm @ V2 ) @ none_trm ) ) ) ) ) ) ) ) ) ).
% Geqo.pelims
thf(fact_231_Diamondo_Ocases,axiom,
! [X2: produc1739631244on_fml] :
( ! [Alpha: game,Phi2: fml] :
( X2
!= ( produc2124676926on_fml @ ( some_game @ Alpha ) @ ( some_fml @ Phi2 ) ) )
=> ( ! [Phi2: option_fml] :
( X2
!= ( produc2124676926on_fml @ none_game @ Phi2 ) )
=> ~ ! [V2: game] :
( X2
!= ( produc2124676926on_fml @ ( some_game @ V2 ) @ none_fml ) ) ) ) ).
% Diamondo.cases
thf(fact_232_subrelI,axiom,
! [R3: set_Pr622314946on_fml,S2: set_Pr622314946on_fml] :
( ! [X: option_game,Y4: option_fml] :
( ( member1883102883on_fml @ ( produc2124676926on_fml @ X @ Y4 ) @ R3 )
=> ( member1883102883on_fml @ ( produc2124676926on_fml @ X @ Y4 ) @ S2 ) )
=> ( ord_le782333474on_fml @ R3 @ S2 ) ) ).
% subrelI
thf(fact_233_subrelI,axiom,
! [R3: set_Pr1471344007on_trm,S2: set_Pr1471344007on_trm] :
( ! [X: option_trm,Y4: option_trm] :
( ( member165817296on_trm @ ( produc709594775on_trm @ X @ Y4 ) @ R3 )
=> ( member165817296on_trm @ ( produc709594775on_trm @ X @ Y4 ) @ S2 ) )
=> ( ord_le1322722599on_trm @ R3 @ S2 ) ) ).
% subrelI
thf(fact_234_subrelI,axiom,
! [R3: set_Pr927595527on_fml,S2: set_Pr927595527on_fml] :
( ! [X: option_fml,Y4: option_fml] :
( ( member1287158352on_fml @ ( produc909629463on_fml @ X @ Y4 ) @ R3 )
=> ( member1287158352on_fml @ ( produc909629463on_fml @ X @ Y4 ) @ S2 ) )
=> ( ord_le778974119on_fml @ R3 @ S2 ) ) ).
% subrelI
thf(fact_235_subrelI,axiom,
! [R3: set_Pr943832262on_trm,S2: set_Pr943832262on_trm] :
( ! [X: char,Y4: option_trm] :
( ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y4 ) @ R3 )
=> ( member694524327on_trm @ ( produc1500300866on_trm @ X @ Y4 ) @ S2 ) )
=> ( ord_le541293862on_trm @ R3 @ S2 ) ) ).
% subrelI
thf(fact_236_subrelI,axiom,
! [R3: set_Pr1235432374le_trm,S2: set_Pr1235432374le_trm] :
( ! [X: set_variable,Y4: trm] :
( ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y4 ) @ R3 )
=> ( member2122665623le_trm @ ( produc1482344242le_trm @ X @ Y4 ) @ S2 ) )
=> ( ord_le2102862870le_trm @ R3 @ S2 ) ) ).
% subrelI
thf(fact_237_ssubst__Pair__rhs,axiom,
! [R3: option_game,S2: option_fml,R: set_Pr622314946on_fml,S3: option_fml] :
( ( member1883102883on_fml @ ( produc2124676926on_fml @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member1883102883on_fml @ ( produc2124676926on_fml @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_238_ssubst__Pair__rhs,axiom,
! [R3: option_trm,S2: option_trm,R: set_Pr1471344007on_trm,S3: option_trm] :
( ( member165817296on_trm @ ( produc709594775on_trm @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member165817296on_trm @ ( produc709594775on_trm @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_239_ssubst__Pair__rhs,axiom,
! [R3: option_fml,S2: option_fml,R: set_Pr927595527on_fml,S3: option_fml] :
( ( member1287158352on_fml @ ( produc909629463on_fml @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member1287158352on_fml @ ( produc909629463on_fml @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_240_ssubst__Pair__rhs,axiom,
! [R3: char,S2: option_trm,R: set_Pr943832262on_trm,S3: option_trm] :
( ( member694524327on_trm @ ( produc1500300866on_trm @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member694524327on_trm @ ( produc1500300866on_trm @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_241_ssubst__Pair__rhs,axiom,
! [R3: set_variable,S2: trm,R: set_Pr1235432374le_trm,S3: trm] :
( ( member2122665623le_trm @ ( produc1482344242le_trm @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member2122665623le_trm @ ( produc1482344242le_trm @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_242_pred__equals__eq2,axiom,
! [R: set_Pr622314946on_fml,S: set_Pr622314946on_fml] :
( ( ( ^ [X3: option_game,Y3: option_fml] : ( member1883102883on_fml @ ( produc2124676926on_fml @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: option_game,Y3: option_fml] : ( member1883102883on_fml @ ( produc2124676926on_fml @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_243_pred__equals__eq2,axiom,
! [R: set_Pr1471344007on_trm,S: set_Pr1471344007on_trm] :
( ( ( ^ [X3: option_trm,Y3: option_trm] : ( member165817296on_trm @ ( produc709594775on_trm @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: option_trm,Y3: option_trm] : ( member165817296on_trm @ ( produc709594775on_trm @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_244_pred__equals__eq2,axiom,
! [R: set_Pr927595527on_fml,S: set_Pr927595527on_fml] :
( ( ( ^ [X3: option_fml,Y3: option_fml] : ( member1287158352on_fml @ ( produc909629463on_fml @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: option_fml,Y3: option_fml] : ( member1287158352on_fml @ ( produc909629463on_fml @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_245_pred__equals__eq2,axiom,
! [R: set_Pr943832262on_trm,S: set_Pr943832262on_trm] :
( ( ( ^ [X3: char,Y3: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: char,Y3: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_246_pred__equals__eq2,axiom,
! [R: set_Pr1235432374le_trm,S: set_Pr1235432374le_trm] :
( ( ( ^ [X3: set_variable,Y3: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: set_variable,Y3: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_247_pred__subset__eq2,axiom,
! [R: set_Pr622314946on_fml,S: set_Pr622314946on_fml] :
( ( ord_le1588489637_fml_o
@ ^ [X3: option_game,Y3: option_fml] : ( member1883102883on_fml @ ( produc2124676926on_fml @ X3 @ Y3 ) @ R )
@ ^ [X3: option_game,Y3: option_fml] : ( member1883102883on_fml @ ( produc2124676926on_fml @ X3 @ Y3 ) @ S ) )
= ( ord_le782333474on_fml @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_248_pred__subset__eq2,axiom,
! [R: set_Pr1471344007on_trm,S: set_Pr1471344007on_trm] :
( ( ord_le847028878_trm_o
@ ^ [X3: option_trm,Y3: option_trm] : ( member165817296on_trm @ ( produc709594775on_trm @ X3 @ Y3 ) @ R )
@ ^ [X3: option_trm,Y3: option_trm] : ( member165817296on_trm @ ( produc709594775on_trm @ X3 @ Y3 ) @ S ) )
= ( ord_le1322722599on_trm @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_249_pred__subset__eq2,axiom,
! [R: set_Pr927595527on_fml,S: set_Pr927595527on_fml] :
( ( ord_le802833166_fml_o
@ ^ [X3: option_fml,Y3: option_fml] : ( member1287158352on_fml @ ( produc909629463on_fml @ X3 @ Y3 ) @ R )
@ ^ [X3: option_fml,Y3: option_fml] : ( member1287158352on_fml @ ( produc909629463on_fml @ X3 @ Y3 ) @ S ) )
= ( ord_le778974119on_fml @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_250_pred__subset__eq2,axiom,
! [R: set_Pr943832262on_trm,S: set_Pr943832262on_trm] :
( ( ord_le1890124833_trm_o
@ ^ [X3: char,Y3: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X3 @ Y3 ) @ R )
@ ^ [X3: char,Y3: option_trm] : ( member694524327on_trm @ ( produc1500300866on_trm @ X3 @ Y3 ) @ S ) )
= ( ord_le541293862on_trm @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_251_pred__subset__eq2,axiom,
! [R: set_Pr1235432374le_trm,S: set_Pr1235432374le_trm] :
( ( ord_le691957041_trm_o
@ ^ [X3: set_variable,Y3: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X3 @ Y3 ) @ R )
@ ^ [X3: set_variable,Y3: trm] : ( member2122665623le_trm @ ( produc1482344242le_trm @ X3 @ Y3 ) @ S ) )
= ( ord_le2102862870le_trm @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_252_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_253_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_254_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_255_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_256_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_257_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_258_Timeso_Ocases,axiom,
! [X2: produc1072747943on_trm] :
( ! [Theta2: trm,Eta2: trm] :
( X2
!= ( produc709594775on_trm @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) ) )
=> ( ! [Eta2: option_trm] :
( X2
!= ( produc709594775on_trm @ none_trm @ Eta2 ) )
=> ~ ! [V2: trm] :
( X2
!= ( produc709594775on_trm @ ( some_trm @ V2 ) @ none_trm ) ) ) ) ).
% Timeso.cases
thf(fact_259_case__optionE,axiom,
! [P: $o,Q: trm > $o,X2: option_trm] :
( ( case_option_o_trm @ P @ Q @ X2 )
=> ( ( ( X2 = none_trm )
=> ~ P )
=> ~ ! [Y4: trm] :
( ( X2
= ( some_trm @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_260_case__optionE,axiom,
! [P: $o,Q: fml > $o,X2: option_fml] :
( ( case_option_o_fml @ P @ Q @ X2 )
=> ( ( ( X2 = none_fml )
=> ~ P )
=> ~ ! [Y4: fml] :
( ( X2
= ( some_fml @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_261_case__optionE,axiom,
! [P: $o,Q: game > $o,X2: option_game] :
( ( case_option_o_game @ P @ Q @ X2 )
=> ( ( ( X2 = none_game )
=> ~ P )
=> ~ ! [Y4: game] :
( ( X2
= ( some_game @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_262_Timeso_Opelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_trm] :
( ( ( uSubst918876924Timeso @ X2 @ Xa )
= Y2 )
=> ( ( accp_P758790320on_trm @ uSubst924844285so_rel @ ( produc709594775on_trm @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( ( Y2
= ( some_trm @ ( times @ Theta2 @ Eta2 ) ) )
=> ~ ( accp_P758790320on_trm @ uSubst924844285so_rel @ ( produc709594775on_trm @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( ( Y2 = none_trm )
=> ~ ( accp_P758790320on_trm @ uSubst924844285so_rel @ ( produc709594775on_trm @ none_trm @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( ( Y2 = none_trm )
=> ~ ( accp_P758790320on_trm @ uSubst924844285so_rel @ ( produc709594775on_trm @ ( some_trm @ V2 ) @ none_trm ) ) ) ) ) ) ) ) ) ).
% Timeso.pelims
thf(fact_263_Pluso_Opelims,axiom,
! [X2: option_trm,Xa: option_trm,Y2: option_trm] :
( ( ( uSubst141955042_Pluso @ X2 @ Xa )
= Y2 )
=> ( ( accp_P758790320on_trm @ uSubst691094423so_rel @ ( produc709594775on_trm @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( ( Y2
= ( some_trm @ ( plus @ Theta2 @ Eta2 ) ) )
=> ~ ( accp_P758790320on_trm @ uSubst691094423so_rel @ ( produc709594775on_trm @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2 = none_trm )
=> ( ( Y2 = none_trm )
=> ~ ( accp_P758790320on_trm @ uSubst691094423so_rel @ ( produc709594775on_trm @ none_trm @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( ( Y2 = none_trm )
=> ~ ( accp_P758790320on_trm @ uSubst691094423so_rel @ ( produc709594775on_trm @ ( some_trm @ V2 ) @ none_trm ) ) ) ) ) ) ) ) ) ).
% Pluso.pelims
thf(fact_264_ODEo_Ocases,axiom,
! [X2: produc464795408on_trm] :
( ! [X: char,Theta2: trm] :
( X2
!= ( produc1500300866on_trm @ X @ ( some_trm @ Theta2 ) ) )
=> ~ ! [X: char] :
( X2
!= ( produc1500300866on_trm @ X @ none_trm ) ) ) ).
% ODEo.cases
thf(fact_265_Assigno_Ocases,axiom,
! [X2: produc1024976764on_trm] :
( ! [X: variable,Theta2: trm] :
( X2
!= ( produc1743721902on_trm @ X @ ( some_trm @ Theta2 ) ) )
=> ~ ! [X: variable] :
( X2
!= ( produc1743721902on_trm @ X @ none_trm ) ) ) ).
% Assigno.cases
thf(fact_266_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 ) )
=> ~ ! [V2: game] :
( X2
!= ( produc1111617711n_game @ none_game @ ( some_game @ V2 ) ) ) ) ) ).
% Composeo.cases
thf(fact_267_Existso_Ocases,axiom,
! [X2: produc1408538418on_fml] :
( ! [X: variable,Phi2: fml] :
( X2
!= ( produc1205977188on_fml @ X @ ( some_fml @ Phi2 ) ) )
=> ~ ! [X: variable] :
( X2
!= ( produc1205977188on_fml @ X @ none_fml ) ) ) ).
% Existso.cases
thf(fact_268_Ando_Ocases,axiom,
! [X2: produc46605351on_fml] :
( ! [Phi2: fml,Psi: fml] :
( X2
!= ( produc909629463on_fml @ ( some_fml @ Phi2 ) @ ( some_fml @ Psi ) ) )
=> ( ! [Psi: option_fml] :
( X2
!= ( produc909629463on_fml @ none_fml @ Psi ) )
=> ~ ! [V2: fml] :
( X2
!= ( produc909629463on_fml @ ( some_fml @ V2 ) @ none_fml ) ) ) ) ).
% Ando.cases
thf(fact_269_usubstappt_Opsimps_I7_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( differential @ Theta ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( differential @ Theta ) )
= ( uSubst716837505ntialo
@ ( uSubst516392818stappt @ Sigma
@ ( collect_variable
@ ^ [X3: variable] : $true )
@ Theta ) ) ) ) ).
% usubstappt.psimps(7)
thf(fact_270_usubstappt_Opsimps_I6_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( times @ Theta @ Eta ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
= ( uSubst918876924Timeso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ) ).
% usubstappt.psimps(6)
thf(fact_271_usubstappt_Opsimps_I5_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( plus @ Theta @ Eta ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
= ( uSubst141955042_Pluso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ) ).
% usubstappt.psimps(5)
thf(fact_272_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_273_usubstappt_Opsimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,R3: real] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( number @ R3 ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( number @ R3 ) )
= ( some_trm @ ( number @ R3 ) ) ) ) ).
% usubstappt.psimps(2)
thf(fact_274_in__lex__prod,axiom,
! [A3: option_trm,B3: option_fml,A6: option_trm,B6: option_fml,R3: set_Pr1471344007on_trm,S2: set_Pr927595527on_fml] :
( ( member1316917136on_fml @ ( produc1235825367on_fml @ ( produc171850061on_fml @ A3 @ B3 ) @ ( produc171850061on_fml @ A6 @ B6 ) ) @ ( lex_pr1724583269on_fml @ R3 @ S2 ) )
= ( ( member165817296on_trm @ ( produc709594775on_trm @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member1287158352on_fml @ ( produc909629463on_fml @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_275_in__lex__prod,axiom,
! [A3: option_fml,B3: option_trm,A6: option_fml,B6: option_trm,R3: set_Pr927595527on_fml,S2: set_Pr1471344007on_trm] :
( ( member1883874960on_trm @ ( produc2132972503on_trm @ ( produc1447374177on_trm @ A3 @ B3 ) @ ( produc1447374177on_trm @ A6 @ B6 ) ) @ ( lex_pr852623737on_trm @ R3 @ S2 ) )
= ( ( member1287158352on_fml @ ( produc909629463on_fml @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member165817296on_trm @ ( produc709594775on_trm @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_276_in__lex__prod,axiom,
! [A3: option_game,B3: option_fml,A6: option_game,B6: option_fml,R3: set_Pr1625152599n_game,S2: set_Pr927595527on_fml] :
( ( member113767842on_fml @ ( produc556056113on_fml @ ( produc2124676926on_fml @ A3 @ B3 ) @ ( produc2124676926on_fml @ A6 @ B6 ) ) @ ( lex_pr425996582on_fml @ R3 @ S2 ) )
= ( ( member191809696n_game @ ( produc1111617711n_game @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member1287158352on_fml @ ( produc909629463on_fml @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_277_in__lex__prod,axiom,
! [A3: option_trm,B3: option_trm,A6: option_trm,B6: option_trm,R3: set_Pr1471344007on_trm,S2: set_Pr1471344007on_trm] :
( ( member602811152on_trm @ ( produc2001752919on_trm @ ( produc709594775on_trm @ A3 @ B3 ) @ ( produc709594775on_trm @ A6 @ B6 ) ) @ ( lex_pr114844335on_trm @ R3 @ S2 ) )
= ( ( member165817296on_trm @ ( produc709594775on_trm @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member165817296on_trm @ ( produc709594775on_trm @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_278_in__lex__prod,axiom,
! [A3: option_fml,B3: option_fml,A6: option_fml,B6: option_fml,R3: set_Pr927595527on_fml,S2: set_Pr927595527on_fml] :
( ( member450497296on_fml @ ( produc1367044951on_fml @ ( produc909629463on_fml @ A3 @ B3 ) @ ( produc909629463on_fml @ A6 @ B6 ) ) @ ( lex_pr314879023on_fml @ R3 @ S2 ) )
= ( ( member1287158352on_fml @ ( produc909629463on_fml @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member1287158352on_fml @ ( produc909629463on_fml @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_279_in__lex__prod,axiom,
! [A3: char,B3: option_trm,A6: char,B6: option_trm,R3: set_Pr604539619r_char,S2: set_Pr1471344007on_trm] :
( ( member1950256170on_trm @ ( produc444919481on_trm @ ( produc1500300866on_trm @ A3 @ B3 ) @ ( produc1500300866on_trm @ A6 @ B6 ) ) @ ( lex_pr1901327402on_trm @ R3 @ S2 ) )
= ( ( member1866856748r_char @ ( produc19545147r_char @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member165817296on_trm @ ( produc709594775on_trm @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_280_in__lex__prod,axiom,
! [A3: set_variable,B3: trm,A6: set_variable,B6: trm,R3: set_Pr181266295riable,S2: set_Pr491793543rm_trm] :
( ( member1150310282le_trm @ ( produc1056455193le_trm @ ( produc1482344242le_trm @ A3 @ B3 ) @ ( produc1482344242le_trm @ A6 @ B6 ) ) @ ( lex_pr1522821402le_trm @ R3 @ S2 ) )
= ( ( member1978718144riable @ ( produc1766592463riable @ A3 @ A6 ) @ R3 )
| ( ( A3 = A6 )
& ( member1735386832rm_trm @ ( product_Pair_trm_trm @ B3 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_281_trm_Oinject_I2_J,axiom,
! [X23: real,Y23: real] :
( ( ( number @ X23 )
= ( number @ Y23 ) )
= ( X23 = Y23 ) ) ).
% trm.inject(2)
thf(fact_282_trm_Oinject_I1_J,axiom,
! [X1: variable,Y1: variable] :
( ( ( var @ X1 )
= ( var @ Y1 ) )
= ( X1 = Y1 ) ) ).
% trm.inject(1)
thf(fact_283_trm_Odistinct_I17_J,axiom,
! [X23: real,X51: trm,X52: trm] :
( ( number @ X23 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_284_trm_Odistinct_I19_J,axiom,
! [X23: real,X61: trm,X62: trm] :
( ( number @ X23 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(19)
thf(fact_285_trm_Odistinct_I21_J,axiom,
! [X23: real,X7: trm] :
( ( number @ X23 )
!= ( differential @ X7 ) ) ).
% trm.distinct(21)
thf(fact_286_trm_Odistinct_I7_J,axiom,
! [X1: variable,X51: trm,X52: trm] :
( ( var @ X1 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_287_trm_Odistinct_I9_J,axiom,
! [X1: variable,X61: trm,X62: trm] :
( ( var @ X1 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(9)
thf(fact_288_trm_Odistinct_I11_J,axiom,
! [X1: variable,X7: trm] :
( ( var @ X1 )
!= ( differential @ X7 ) ) ).
% trm.distinct(11)
thf(fact_289_trm_Odistinct_I1_J,axiom,
! [X1: variable,X23: real] :
( ( var @ X1 )
!= ( number @ X23 ) ) ).
% trm.distinct(1)
thf(fact_290_usubstappt_Osimps_I2_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,R3: real] :
( ( uSubst516392818stappt @ Sigma @ U @ ( number @ R3 ) )
= ( some_trm @ ( number @ R3 ) ) ) ).
% usubstappt.simps(2)
thf(fact_291_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_292_usubstappt_Ocases,axiom,
! [X2: produc1699626005le_trm] :
( ! [Sigma2: produc1418842292n_game,U3: set_variable,X: variable] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( var @ X ) ) ) )
=> ( ! [Sigma2: produc1418842292n_game,U3: set_variable,R5: real] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( number @ R5 ) ) ) )
=> ( ! [Sigma2: produc1418842292n_game,U3: set_variable,F3: char] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( const @ F3 ) ) ) )
=> ( ! [Sigma2: produc1418842292n_game,U3: set_variable,F3: char,Theta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( func @ F3 @ Theta2 ) ) ) )
=> ( ! [Sigma2: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( plus @ Theta2 @ Eta2 ) ) ) )
=> ( ! [Sigma2: produc1418842292n_game,U3: set_variable,Theta2: trm,Eta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( times @ Theta2 @ Eta2 ) ) ) )
=> ~ ! [Sigma2: produc1418842292n_game,U3: set_variable,Theta2: trm] :
( X2
!= ( produc2015636557le_trm @ Sigma2 @ ( produc1482344242le_trm @ U3 @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.cases
thf(fact_293_Testo_Osimps_I2_J,axiom,
( ( uSubst1367128042_Testo @ none_fml )
= none_game ) ).
% Testo.simps(2)
thf(fact_294_Testo__undef,axiom,
! [Phi: option_fml] :
( ( ( uSubst1367128042_Testo @ Phi )
= none_game )
= ( Phi = none_fml ) ) ).
% Testo_undef
thf(fact_295_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_296_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_297_trm_Odistinct_I3_J,axiom,
! [X1: variable,X32: char] :
( ( var @ X1 )
!= ( const @ X32 ) ) ).
% trm.distinct(3)
thf(fact_298_trm_Odistinct_I5_J,axiom,
! [X1: variable,X41: char,X42: trm] :
( ( var @ X1 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_299_trm_Odistinct_I13_J,axiom,
! [X23: real,X32: char] :
( ( number @ X23 )
!= ( const @ X32 ) ) ).
% trm.distinct(13)
thf(fact_300_trm_Odistinct_I15_J,axiom,
! [X23: real,X41: char,X42: trm] :
( ( number @ X23 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_301_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_302_trm_Odistinct_I29_J,axiom,
! [X32: char,X7: trm] :
( ( const @ X32 )
!= ( differential @ X7 ) ) ).
% trm.distinct(29)
thf(fact_303_trm_Odistinct_I27_J,axiom,
! [X32: char,X61: trm,X62: trm] :
( ( const @ X32 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(27)
thf(fact_304_trm_Odistinct_I25_J,axiom,
! [X32: char,X51: trm,X52: trm] :
( ( const @ X32 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_305_trm_Odistinct_I35_J,axiom,
! [X41: char,X42: trm,X7: trm] :
( ( func @ X41 @ X42 )
!= ( differential @ X7 ) ) ).
% trm.distinct(35)
thf(fact_306_trm_Odistinct_I33_J,axiom,
! [X41: char,X42: trm,X61: trm,X62: trm] :
( ( func @ X41 @ X42 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(33)
thf(fact_307_trm_Odistinct_I31_J,axiom,
! [X41: char,X42: trm,X51: trm,X52: trm] :
( ( func @ X41 @ X42 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_308_trm_Oinduct,axiom,
! [P: trm > $o,Trm: trm] :
( ! [X: variable] : ( P @ ( var @ X ) )
=> ( ! [X: real] : ( P @ ( number @ X ) )
=> ( ! [X: char] : ( P @ ( const @ X ) )
=> ( ! [X1a: char,X2a: trm] :
( ( P @ X2a )
=> ( P @ ( func @ X1a @ X2a ) ) )
=> ( ! [X1a: trm,X2a: trm] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( plus @ X1a @ X2a ) ) ) )
=> ( ! [X1a: trm,X2a: trm] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( times @ X1a @ X2a ) ) ) )
=> ( ! [X: trm] :
( ( P @ X )
=> ( P @ ( differential @ X ) ) )
=> ( P @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_309_trm_Oexhaust,axiom,
! [Y2: trm] :
( ! [X12: variable] :
( Y2
!= ( var @ X12 ) )
=> ( ! [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_310_term__sem_Ocases,axiom,
! [X2: produc833882726rp_trm] :
( ! [I2: denotational_interp,X: variable] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( var @ X ) ) )
=> ( ! [I2: denotational_interp,R5: real] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( number @ R5 ) ) )
=> ( ! [I2: denotational_interp,F3: char] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( const @ F3 ) ) )
=> ( ! [I2: denotational_interp,F3: char,Theta2: trm] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( func @ F3 @ Theta2 ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm,Eta2: trm] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( plus @ Theta2 @ Eta2 ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm,Eta2: trm] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( times @ Theta2 @ Eta2 ) ) )
=> ~ ! [I2: denotational_interp,Theta2: trm] :
( X2
!= ( produc1132922070rp_trm @ I2 @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ).
% term_sem.cases
thf(fact_311_usubstappt_Opsimps_I3_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F: char] :
( ( accp_P1877119646le_trm @ uSubst1917534215pt_rel @ ( produc2015636557le_trm @ Sigma @ ( produc1482344242le_trm @ U @ ( const @ F ) ) ) )
=> ( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F ) )
= ( uSubst1559070963pconst @ Sigma @ U @ F ) ) ) ).
% usubstappt.psimps(3)
thf(fact_312_term__sem_Oinduct,axiom,
! [P: denotational_interp > trm > $o,A0: denotational_interp,A1: trm] :
( ! [I2: denotational_interp,X: variable] : ( P @ I2 @ ( var @ X ) )
=> ( ! [I2: denotational_interp,R5: real] : ( P @ I2 @ ( number @ R5 ) )
=> ( ! [I2: denotational_interp,F3: char] : ( P @ I2 @ ( const @ F3 ) )
=> ( ! [I2: denotational_interp,F3: char,Theta2: trm] :
( ( P @ I2 @ Theta2 )
=> ( P @ I2 @ ( func @ F3 @ Theta2 ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm] :
( ( P @ I2 @ Theta2 )
=> ! [Eta2: trm] :
( ( P @ I2 @ Eta2 )
=> ( P @ I2 @ ( plus @ Theta2 @ Eta2 ) ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm] :
( ( P @ I2 @ Theta2 )
=> ! [Eta2: trm] :
( ( P @ I2 @ Eta2 )
=> ( P @ I2 @ ( times @ Theta2 @ Eta2 ) ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm] :
( ( ? [Xa2: char] :
( member_char @ Xa2
@ ( collect_char
@ ^ [Uu: char] : $true ) )
=> ( P @ I2 @ Theta2 ) )
=> ( P @ I2 @ ( differential @ Theta2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% term_sem.induct
thf(fact_313_usubstappt_Osimps_I3_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,F: char] :
( ( uSubst516392818stappt @ Sigma @ U @ ( const @ F ) )
= ( uSubst1559070963pconst @ Sigma @ U @ F ) ) ).
% usubstappt.simps(3)
thf(fact_314_image2__def,axiom,
( bNF_Gr2142503190on_fml
= ( ^ [A2: set_variable,F2: variable > option_game,G: variable > option_fml] :
( collec749447393on_fml
@ ^ [Uu: produc1739631244on_fml] :
? [A4: variable] :
( ( Uu
= ( produc2124676926on_fml @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_variable @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_315_image2__def,axiom,
( bNF_Gr756940202on_fml
= ( ^ [A2: set_char,F2: char > option_game,G: char > option_fml] :
( collec749447393on_fml
@ ^ [Uu: produc1739631244on_fml] :
? [A4: char] :
( ( Uu
= ( produc2124676926on_fml @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_char @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_316_image2__def,axiom,
( bNF_Gr400922303on_trm
= ( ^ [A2: set_variable,F2: variable > option_trm,G: variable > option_trm] :
( collec1697116690on_trm
@ ^ [Uu: produc1072747943on_trm] :
? [A4: variable] :
( ( Uu
= ( produc709594775on_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_variable @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_317_image2__def,axiom,
( bNF_Gr1666135339on_trm
= ( ^ [A2: set_char,F2: char > option_trm,G: char > option_trm] :
( collec1697116690on_trm
@ ^ [Uu: produc1072747943on_trm] :
? [A4: char] :
( ( Uu
= ( produc709594775on_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_char @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_318_image2__def,axiom,
( bNF_Gr600956991on_fml
= ( ^ [A2: set_variable,F2: variable > option_fml,G: variable > option_fml] :
( collec670974098on_fml
@ ^ [Uu: produc46605351on_fml] :
? [A4: variable] :
( ( Uu
= ( produc909629463on_fml @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_variable @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_319_image2__def,axiom,
( bNF_Gr1866170027on_fml
= ( ^ [A2: set_char,F2: char > option_fml,G: char > option_fml] :
( collec670974098on_fml
@ ^ [Uu: produc46605351on_fml] :
? [A4: char] :
( ( Uu
= ( produc909629463on_fml @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_char @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_320_image2__def,axiom,
( bNF_Gr1933470746on_trm
= ( ^ [A2: set_variable,F2: variable > char,G: variable > option_trm] :
( collec1756633317on_trm
@ ^ [Uu: produc464795408on_trm] :
? [A4: variable] :
( ( Uu
= ( produc1500300866on_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_variable @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_321_image2__def,axiom,
( bNF_Gr951178926on_trm
= ( ^ [A2: set_char,F2: char > char,G: char > option_trm] :
( collec1756633317on_trm
@ ^ [Uu: produc464795408on_trm] :
? [A4: char] :
( ( Uu
= ( produc1500300866on_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_char @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_322_image2__def,axiom,
( bNF_Gr1330108682le_trm
= ( ^ [A2: set_variable,F2: variable > set_variable,G: variable > trm] :
( collec575343829le_trm
@ ^ [Uu: produc1367888000le_trm] :
? [A4: variable] :
( ( Uu
= ( produc1482344242le_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_variable @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_323_image2__def,axiom,
( bNF_Gr852774302le_trm
= ( ^ [A2: set_char,F2: char > set_variable,G: char > trm] :
( collec575343829le_trm
@ ^ [Uu: produc1367888000le_trm] :
? [A4: char] :
( ( Uu
= ( produc1482344242le_trm @ ( F2 @ A4 ) @ ( G @ A4 ) ) )
& ( member_char @ A4 @ A2 ) ) ) ) ) ).
% image2_def
thf(fact_324_Testo_Oelims,axiom,
! [X2: option_fml,Y2: option_game] :
( ( ( uSubst1367128042_Testo @ X2 )
= Y2 )
=> ( ! [Phi2: fml] :
( ( X2
= ( some_fml @ Phi2 ) )
=> ( Y2
!= ( some_game @ ( test @ Phi2 ) ) ) )
=> ~ ( ( X2 = none_fml )
=> ( Y2 != none_game ) ) ) ) ).
% Testo.elims
thf(fact_325_game_Oinject_I3_J,axiom,
! [X32: fml,Y32: fml] :
( ( ( test @ X32 )
= ( test @ Y32 ) )
= ( X32 = Y32 ) ) ).
% game.inject(3)
thf(fact_326_image2__eqI,axiom,
! [B3: option_game,F: variable > option_game,X2: variable,C: option_fml,G2: variable > option_fml,A: set_variable] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_variable @ X2 @ A )
=> ( member1883102883on_fml @ ( produc2124676926on_fml @ B3 @ C ) @ ( bNF_Gr2142503190on_fml @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_327_image2__eqI,axiom,
! [B3: option_game,F: char > option_game,X2: char,C: option_fml,G2: char > option_fml,A: set_char] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_char @ X2 @ A )
=> ( member1883102883on_fml @ ( produc2124676926on_fml @ B3 @ C ) @ ( bNF_Gr756940202on_fml @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_328_image2__eqI,axiom,
! [B3: option_trm,F: variable > option_trm,X2: variable,C: option_trm,G2: variable > option_trm,A: set_variable] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_variable @ X2 @ A )
=> ( member165817296on_trm @ ( produc709594775on_trm @ B3 @ C ) @ ( bNF_Gr400922303on_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_329_image2__eqI,axiom,
! [B3: option_trm,F: char > option_trm,X2: char,C: option_trm,G2: char > option_trm,A: set_char] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_char @ X2 @ A )
=> ( member165817296on_trm @ ( produc709594775on_trm @ B3 @ C ) @ ( bNF_Gr1666135339on_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_330_image2__eqI,axiom,
! [B3: option_fml,F: variable > option_fml,X2: variable,C: option_fml,G2: variable > option_fml,A: set_variable] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_variable @ X2 @ A )
=> ( member1287158352on_fml @ ( produc909629463on_fml @ B3 @ C ) @ ( bNF_Gr600956991on_fml @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_331_image2__eqI,axiom,
! [B3: option_fml,F: char > option_fml,X2: char,C: option_fml,G2: char > option_fml,A: set_char] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_char @ X2 @ A )
=> ( member1287158352on_fml @ ( produc909629463on_fml @ B3 @ C ) @ ( bNF_Gr1866170027on_fml @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_332_image2__eqI,axiom,
! [B3: char,F: variable > char,X2: variable,C: option_trm,G2: variable > option_trm,A: set_variable] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_variable @ X2 @ A )
=> ( member694524327on_trm @ ( produc1500300866on_trm @ B3 @ C ) @ ( bNF_Gr1933470746on_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_333_image2__eqI,axiom,
! [B3: char,F: char > char,X2: char,C: option_trm,G2: char > option_trm,A: set_char] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_char @ X2 @ A )
=> ( member694524327on_trm @ ( produc1500300866on_trm @ B3 @ C ) @ ( bNF_Gr951178926on_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_334_image2__eqI,axiom,
! [B3: set_variable,F: variable > set_variable,X2: variable,C: trm,G2: variable > trm,A: set_variable] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_variable @ X2 @ A )
=> ( member2122665623le_trm @ ( produc1482344242le_trm @ B3 @ C ) @ ( bNF_Gr1330108682le_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_335_image2__eqI,axiom,
! [B3: set_variable,F: char > set_variable,X2: char,C: trm,G2: char > trm,A: set_char] :
( ( B3
= ( F @ X2 ) )
=> ( ( C
= ( G2 @ X2 ) )
=> ( ( member_char @ X2 @ A )
=> ( member2122665623le_trm @ ( produc1482344242le_trm @ B3 @ C ) @ ( bNF_Gr852774302le_trm @ A @ F @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_336_Testo_Osimps_I1_J,axiom,
! [Phi: fml] :
( ( uSubst1367128042_Testo @ ( some_fml @ Phi ) )
= ( some_game @ ( test @ Phi ) ) ) ).
% Testo.simps(1)
thf(fact_337_Testo_Opelims,axiom,
! [X2: option_fml,Y2: option_game] :
( ( ( uSubst1367128042_Testo @ X2 )
= Y2 )
=> ( ( accp_option_fml @ uSubst954181007to_rel @ X2 )
=> ( ! [Phi2: fml] :
( ( X2
= ( some_fml @ Phi2 ) )
=> ( ( Y2
= ( some_game @ ( test @ Phi2 ) ) )
=> ~ ( accp_option_fml @ uSubst954181007to_rel @ ( some_fml @ Phi2 ) ) ) )
=> ~ ( ( X2 = none_fml )
=> ( ( Y2 = none_game )
=> ~ ( accp_option_fml @ uSubst954181007to_rel @ none_fml ) ) ) ) ) ) ).
% Testo.pelims
thf(fact_338_relImage__def,axiom,
( bNF_Gr806120164on_fml
= ( ^ [R6: set_Pr927595527on_fml,F2: option_fml > option_fml] :
( collec670974098on_fml
@ ^ [Uu: produc46605351on_fml] :
? [A12: option_fml,A22: option_fml] :
( ( Uu
= ( produc909629463on_fml @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) )
& ( member1287158352on_fml @ ( produc909629463on_fml @ A12 @ A22 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_339_Assigno__undef,axiom,
! [X2: variable,Theta: option_trm] :
( ( ( uSubst834381837ssigno @ X2 @ Theta )
= none_game )
= ( Theta = none_trm ) ) ).
% Assigno_undef
thf(fact_340_Assigno_Osimps_I2_J,axiom,
! [X2: variable] :
( ( uSubst834381837ssigno @ X2 @ none_trm )
= none_game ) ).
% Assigno.simps(2)
thf(fact_341_Assigno_Oelims,axiom,
! [X2: variable,Xa: option_trm,Y2: option_game] :
( ( ( uSubst834381837ssigno @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some_trm @ Theta2 ) )
=> ( Y2
!= ( some_game @ ( assign @ X2 @ Theta2 ) ) ) )
=> ~ ( ( Xa = none_trm )
=> ( Y2 != none_game ) ) ) ) ).
% Assigno.elims
thf(fact_342_Assigno_Opelims,axiom,
! [X2: variable,Xa: option_trm,Y2: option_game] :
( ( ( uSubst834381837ssigno @ X2 @ Xa )
= Y2 )
=> ( ( accp_P394694195on_trm @ uSubst157155628no_rel @ ( produc1743721902on_trm @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some_trm @ Theta2 ) )
=> ( ( Y2
= ( some_game @ ( assign @ X2 @ Theta2 ) ) )
=> ~ ( accp_P394694195on_trm @ uSubst157155628no_rel @ ( produc1743721902on_trm @ X2 @ ( some_trm @ Theta2 ) ) ) ) )
=> ~ ( ( Xa = none_trm )
=> ( ( Y2 = none_game )
=> ~ ( accp_P394694195on_trm @ uSubst157155628no_rel @ ( produc1743721902on_trm @ X2 @ none_trm ) ) ) ) ) ) ) ).
% Assigno.pelims
thf(fact_343_game_Oinject_I2_J,axiom,
! [X21: variable,X22: trm,Y21: variable,Y22: trm] :
( ( ( assign @ X21 @ X22 )
= ( assign @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% game.inject(2)
thf(fact_344_game_Odistinct_I15_J,axiom,
! [X21: variable,X22: trm,X32: fml] :
( ( assign @ X21 @ X22 )
!= ( test @ X32 ) ) ).
% game.distinct(15)
thf(fact_345_Assigno_Osimps_I1_J,axiom,
! [X2: variable,Theta: trm] :
( ( uSubst834381837ssigno @ X2 @ ( some_trm @ Theta ) )
= ( some_game @ ( assign @ X2 @ Theta ) ) ) ).
% Assigno.simps(1)
thf(fact_346_ODEo__undef,axiom,
! [X2: char,Theta: option_trm] :
( ( ( uSubst1827052306e_ODEo @ X2 @ Theta )
= none_game )
= ( Theta = none_trm ) ) ).
% ODEo_undef
thf(fact_347_ODEo_Osimps_I2_J,axiom,
! [X2: char] :
( ( uSubst1827052306e_ODEo @ X2 @ none_trm )
= none_game ) ).
% ODEo.simps(2)
thf(fact_348_ODEo_Oelims,axiom,
! [X2: char,Xa: option_trm,Y2: option_game] :
( ( ( uSubst1827052306e_ODEo @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some_trm @ Theta2 ) )
=> ( Y2
!= ( some_game @ ( ode @ X2 @ Theta2 ) ) ) )
=> ~ ( ( Xa = none_trm )
=> ( Y2 != none_game ) ) ) ) ).
% ODEo.elims
thf(fact_349_ODEo_Opelims,axiom,
! [X2: char,Xa: option_trm,Y2: option_game] :
( ( ( uSubst1827052306e_ODEo @ X2 @ Xa )
= Y2 )
=> ( ( accp_P419951815on_trm @ uSubst299938919Eo_rel @ ( produc1500300866on_trm @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some_trm @ Theta2 ) )
=> ( ( Y2
= ( some_game @ ( ode @ X2 @ Theta2 ) ) )
=> ~ ( accp_P419951815on_trm @ uSubst299938919Eo_rel @ ( produc1500300866on_trm @ X2 @ ( some_trm @ Theta2 ) ) ) ) )
=> ~ ( ( Xa = none_trm )
=> ( ( Y2 = none_game )
=> ~ ( accp_P419951815on_trm @ uSubst299938919Eo_rel @ ( produc1500300866on_trm @ X2 @ none_trm ) ) ) ) ) ) ) ).
% ODEo.pelims
thf(fact_350_game_Oinject_I8_J,axiom,
! [X81: char,X82: trm,Y81: char,Y82: trm] :
( ( ( ode @ X81 @ X82 )
= ( ode @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 ) ) ) ).
% game.inject(8)
thf(fact_351_game_Odistinct_I35_J,axiom,
! [X32: fml,X81: char,X82: trm] :
( ( test @ X32 )
!= ( ode @ X81 @ X82 ) ) ).
% game.distinct(35)
% Conjectures (1)
thf(conj_0,conjecture,
( ( uSubst516392818stappt @ sigma @ ua @ theta )
!= none_trm ) ).
%------------------------------------------------------------------------------