TPTP Problem File: ITP200^2.p
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%------------------------------------------------------------------------------
% File : ITP200^2 : 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 : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 352 ( 141 unt; 73 typ; 0 def)
% Number of atoms : 769 ( 414 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 4518 ( 179 ~; 9 |; 38 &;3861 @)
% ( 0 <=>; 431 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 515 ( 515 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 7 con; 0-6 aty)
% Number of variables : 1091 ( 86 ^; 957 !; 17 ?;1091 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:23:31.757
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_t_Denotational__Semantics_Ointerp,type,
denotational_interp: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Syntax_Ovariable,type,
variable: $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Syntax_Ogame,type,
game: $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType ).
thf(ty_t_Syntax_Ofml,type,
fml: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (63)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr1317331620lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr2107612801vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Denotational__Semantics_OUvariation,type,
denota1419872369iation: ( variable > real ) > ( variable > real ) > ( set @ variable ) > $o ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Othe,type,
the:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Syntax_Ofml_OGeq,type,
geq: trm > trm > fml ).
thf(sy_c_Syntax_Ogame_OAssign,type,
assign: variable > trm > game ).
thf(sy_c_Syntax_Ogame_OODE,type,
ode: char > trm > game ).
thf(sy_c_Syntax_Ogame_OTest,type,
test: fml > game ).
thf(sy_c_Syntax_Otrm_OConst,type,
const: char > trm ).
thf(sy_c_Syntax_Otrm_ODifferential,type,
differential: trm > trm ).
thf(sy_c_Syntax_Otrm_OFunc,type,
func: char > trm > trm ).
thf(sy_c_Syntax_Otrm_ONumber,type,
number: real > trm ).
thf(sy_c_Syntax_Otrm_OPlus,type,
plus: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OVar,type,
var: variable > trm ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OAssigno,type,
uSubst1326326671ssigno: variable > ( option @ trm ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OAssigno__rel,type,
uSubst1741596714no_rel: ( product_prod @ variable @ ( option @ trm ) ) > ( product_prod @ variable @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODifferentialo,type,
uSubst259074819ntialo: ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODifferentialo__rel,type,
uSubst99643830lo_rel: ( option @ trm ) > ( option @ trm ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OGeqo,type,
uSubst1556497037e_Geqo: ( option @ trm ) > ( option @ trm ) > ( option @ fml ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OGeqo__rel,type,
uSubst864323244qo_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OODEo,type,
uSubst1083227664e_ODEo: char > ( option @ trm ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OODEo__rel,type,
uSubst2007600873Eo_rel: ( product_prod @ char @ ( option @ trm ) ) > ( product_prod @ char @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OPluso,type,
uSubst1112714340_Pluso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OPluso__rel,type,
uSubst270600597so_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTesto,type,
uSubst190403692_Testo: ( option @ fml ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTesto__rel,type,
uSubst533687181to_rel: ( option @ fml ) > ( option @ fml ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTimeso,type,
uSubst277968634Timeso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTimeso__rel,type,
uSubst1377811071so_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousappconst,type,
uSubst1138577137pconst: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > char > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappf,type,
uSubst95898978stappf: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > fml > ( option @ fml ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappt,type,
uSubst95898992stappt: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > trm > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappt__rel,type,
uSubst2096773001pt_rel: ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) > ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ).
thf(sy_v__092_060theta_062____,type,
theta: trm ).
% Relevant facts (255)
thf(fact_0_vaouter,axiom,
denota1419872369iation @ nu @ omega @ u ).
% vaouter
thf(fact_1_usubstappf__geq__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta )
!= ( none @ trm ) ) ) ) ).
% usubstappf_geq_conv
thf(fact_2_Geq_Oprems_I1_J,axiom,
( ( uSubst95898978stappf @ 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: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,V: set @ variable] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ V @ Theta )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( uSubst95898992stappt @ 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: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta )
!= ( none @ trm ) ) ) ) ).
% usubstappt_plus_conv
thf(fact_7_usubstappt__times__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta )
!= ( none @ trm ) ) ) ) ).
% usubstappt_times_conv
thf(fact_8_usubstappt__antimon,axiom,
! [V: set @ variable,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Theta: trm] :
( ( ord_less_eq @ ( set @ variable ) @ V @ U )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( uSubst95898992stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_antimon
thf(fact_9_Differentialo_Osimps_I2_J,axiom,
( ( uSubst259074819ntialo @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Differentialo.simps(2)
thf(fact_10_Differentialo__undef,axiom,
! [Theta: option @ trm] :
( ( ( uSubst259074819ntialo @ Theta )
= ( none @ trm ) )
= ( Theta
= ( none @ trm ) ) ) ).
% Differentialo_undef
thf(fact_11_Pluso_Osimps_I2_J,axiom,
! [Eta: option @ trm] :
( ( uSubst1112714340_Pluso @ ( none @ trm ) @ Eta )
= ( none @ trm ) ) ).
% Pluso.simps(2)
thf(fact_12_Timeso_Osimps_I2_J,axiom,
! [Eta: option @ trm] :
( ( uSubst277968634Timeso @ ( none @ trm ) @ Eta )
= ( none @ trm ) ) ).
% Timeso.simps(2)
thf(fact_13_Pluso__undef,axiom,
! [Theta: option @ trm,Eta: option @ trm] :
( ( ( uSubst1112714340_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] :
( ( ( uSubst277968634Timeso @ Theta @ Eta )
= ( none @ trm ) )
= ( ( Theta
= ( none @ trm ) )
| ( Eta
= ( none @ trm ) ) ) ) ).
% Timeso_undef
thf(fact_15_Geqo_Osimps_I2_J,axiom,
! [Eta: option @ trm] :
( ( uSubst1556497037e_Geqo @ ( none @ trm ) @ Eta )
= ( none @ fml ) ) ).
% Geqo.simps(2)
thf(fact_16_usubstappf__antimon,axiom,
! [V: set @ variable,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Phi: fml] :
( ( ord_less_eq @ ( set @ variable ) @ V @ U )
=> ( ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
= ( uSubst95898978stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_antimon
thf(fact_17_usubstappf__det,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Phi: fml,V: set @ variable] :
( ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
!= ( none @ fml ) )
=> ( ( ( uSubst95898978stappf @ Sigma @ V @ Phi )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
= ( uSubst95898978stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_det
thf(fact_18_usubstappt_Osimps_I5_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappt.simps(5)
thf(fact_19_usubstappt_Osimps_I6_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappt.simps(6)
thf(fact_20_usubstappf_Osimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( uSubst1556497037e_Geqo @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappf.simps(2)
thf(fact_21_Geqo__undef,axiom,
! [Theta: option @ trm,Eta: option @ trm] :
( ( ( uSubst1556497037e_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_less_eq @ ( set @ variable ) @ U @ V )
=> ( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Omega @ Nu @ V ) ) ) ).
% Uvariation_mon
thf(fact_26_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_27_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_28_trm_Odistinct_I37_J,axiom,
! [X51: trm,X52: trm,X61: trm,X62: trm] :
( ( plus @ X51 @ X52 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(37)
thf(fact_29_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_30_Uvariation__univ,axiom,
! [Nu: variable > real,Nu2: variable > real] :
( denota1419872369iation @ Nu @ Nu2
@ ( collect @ variable
@ ^ [X3: variable] : $true ) ) ).
% Uvariation_univ
thf(fact_31_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_32_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A3 )
@ ^ [X3: A] : ( member @ A @ X3 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_33_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_34_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_35_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_36_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_37_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_38_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_39_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_40_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_41_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_42_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_43_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_49_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_50_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_51_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_52_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_53_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_54_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_55_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_57_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_58_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_59_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_60_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_61_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_62_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_63_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_64_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_65_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_66_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_67_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_68_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [T: A] :
( ( member @ A @ T @ A3 )
=> ( member @ A @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_69_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_70_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_71_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_72_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_73_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_74_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_75_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_76_Uvariation__refl,axiom,
! [Nu: variable > real,V: set @ variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% Uvariation_refl
thf(fact_77_Uvariation__sym,axiom,
( denota1419872369iation
= ( ^ [Omega2: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega2 ) ) ) ).
% Uvariation_sym
thf(fact_78_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_79_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_80_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_81_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_82_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_83_prop__restrict,axiom,
! [A: $tType,X2: A,Z3: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X2 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_84_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_85_Pluso_Oelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst1112714340_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_86_Timeso_Oelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst277968634Timeso @ 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_87_usubstappt_Osimps_I7_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( differential @ Theta ) )
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ Sigma
@ ( collect @ variable
@ ^ [X3: variable] : $true )
@ Theta ) ) ) ).
% usubstappt.simps(7)
thf(fact_88_option_Oinject,axiom,
! [A: $tType,X23: A,Y23: A] :
( ( ( some @ A @ X23 )
= ( some @ A @ Y23 ) )
= ( X23 = Y23 ) ) ).
% option.inject
thf(fact_89_trm_Oinject_I7_J,axiom,
! [X7: trm,Y7: trm] :
( ( ( differential @ X7 )
= ( differential @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_90_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_91_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y3: A] :
( X2
!= ( some @ A @ Y3 ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_92_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y3: A] :
( X2
= ( some @ A @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_93_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X2: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) ) ).
% predicate1D
thf(fact_94_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_95_Aterm__Some,axiom,
( ( some @ trm )
= ( some @ trm ) ) ).
% Aterm_Some
thf(fact_96_Differentialo_Osimps_I1_J,axiom,
! [Theta: trm] :
( ( uSubst259074819ntialo @ ( some @ trm @ Theta ) )
= ( some @ trm @ ( differential @ Theta ) ) ) ).
% Differentialo.simps(1)
thf(fact_97_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P @ X2 @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P @ X2 @ Y2 ) )
=> ( ! [A6: A,B6: B] :
( ( X2
= ( some @ A @ A6 ) )
=> ( ( Y2
= ( some @ B @ B6 ) )
=> ( P @ X2 @ Y2 ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_98_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_99_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X3: A] : ( P3 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_100_option_Oinducts,axiom,
! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
( ( P @ ( none @ A ) )
=> ( ! [X: A] : ( P @ ( some @ A @ X ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_101_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X24: A] :
( Y2
!= ( some @ A @ X24 ) ) ) ).
% option.exhaust
thf(fact_102_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X23: A] :
( ( Option
= ( some @ A @ X23 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_103_option_Odistinct_I1_J,axiom,
! [A: $tType,X23: A] :
( ( none @ A )
!= ( some @ A @ X23 ) ) ).
% option.distinct(1)
thf(fact_104_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_105_Differentialo_Ocases,axiom,
! [X2: option @ trm] :
( ! [Theta2: trm] :
( X2
!= ( some @ trm @ Theta2 ) )
=> ( X2
= ( none @ trm ) ) ) ).
% Differentialo.cases
thf(fact_106_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_107_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_108_undeft__equiv,axiom,
! [Theta: option @ trm] :
( ( Theta
!= ( none @ trm ) )
= ( ? [T: trm] :
( Theta
= ( some @ trm @ T ) ) ) ) ).
% undeft_equiv
thf(fact_109_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_110_Differentialo_Oelims,axiom,
! [X2: option @ trm,Y2: option @ trm] :
( ( ( uSubst259074819ntialo @ X2 )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ( Y2
!= ( some @ trm @ ( differential @ Theta2 ) ) ) )
=> ~ ( ( X2
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) ) ) ) ).
% Differentialo.elims
thf(fact_111_trm_Odistinct_I41_J,axiom,
! [X61: trm,X62: trm,X7: trm] :
( ( times @ X61 @ X62 )
!= ( differential @ X7 ) ) ).
% trm.distinct(41)
thf(fact_112_trm_Odistinct_I39_J,axiom,
! [X51: trm,X52: trm,X7: trm] :
( ( plus @ X51 @ X52 )
!= ( differential @ X7 ) ) ).
% trm.distinct(39)
thf(fact_113_Timeso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst277968634Timeso @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Timeso.simps(3)
thf(fact_114_Pluso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst1112714340_Pluso @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Pluso.simps(3)
thf(fact_115_Pluso_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst1112714340_Pluso @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) )
= ( some @ trm @ ( plus @ Theta @ Eta ) ) ) ).
% Pluso.simps(1)
thf(fact_116_Timeso_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst277968634Timeso @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) )
= ( some @ trm @ ( times @ Theta @ Eta ) ) ) ).
% Timeso.simps(1)
thf(fact_117_usubstappt__differential__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( differential @ Theta ) )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma
@ ( collect @ variable
@ ^ [X3: variable] : $true )
@ Theta )
!= ( none @ trm ) ) ) ).
% usubstappt_differential_conv
thf(fact_118_Geqo_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ fml ) ) ).
% Geqo.simps(3)
thf(fact_119_Geqo_Oelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ fml] :
( ( ( uSubst1556497037e_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_120_Differentialo_Opelims,axiom,
! [X2: option @ trm,Y2: option @ trm] :
( ( ( uSubst259074819ntialo @ X2 )
= Y2 )
=> ( ( accp @ ( option @ trm ) @ uSubst99643830lo_rel @ X2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ( ( Y2
= ( some @ trm @ ( differential @ Theta2 ) ) )
=> ~ ( accp @ ( option @ trm ) @ uSubst99643830lo_rel @ ( some @ trm @ Theta2 ) ) ) )
=> ~ ( ( X2
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( option @ trm ) @ uSubst99643830lo_rel @ ( none @ trm ) ) ) ) ) ) ) ).
% Differentialo.pelims
thf(fact_121_Powp__mono,axiom,
! [A: $tType,A2: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A2 ) @ ( powp @ A @ B2 ) ) ) ).
% Powp_mono
thf(fact_122_Geqo_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) )
= ( some @ fml @ ( geq @ Theta @ Eta ) ) ) ).
% Geqo.simps(1)
thf(fact_123_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_124_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_125_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_126_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_127_undeff__equiv,axiom,
! [Phi: option @ fml] :
( ( Phi
!= ( none @ fml ) )
= ( ? [F2: fml] :
( Phi
= ( some @ fml @ F2 ) ) ) ) ).
% undeff_equiv
thf(fact_128_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_129_Testo_Ocases,axiom,
! [X2: option @ fml] :
( ! [Phi2: fml] :
( X2
!= ( some @ fml @ Phi2 ) )
=> ( X2
= ( none @ fml ) ) ) ).
% Testo.cases
thf(fact_130_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_131_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R2: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R2 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R2 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_132_accp__subset__induct,axiom,
! [A: $tType,D: A > $o,R: A > A > $o,X2: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D @ ( accp @ A @ R ) )
=> ( ! [X: A,Z4: A] :
( ( D @ X )
=> ( ( R @ Z4 @ X )
=> ( D @ Z4 ) ) )
=> ( ( D @ X2 )
=> ( ! [X: A] :
( ( D @ X )
=> ( ! [Z5: A] :
( ( R @ Z5 @ X )
=> ( P @ Z5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_133_usubstappf__geqr,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_134_usubstappf__geq,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Eta )
!= ( none @ trm ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_135_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X: A,Y4: B] :
( ( P @ X @ Y4 )
=> ( Q @ X @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_136_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_137_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y2: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) ).
% predicate2D
thf(fact_138_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y2: B,Q: A > B > $o] :
( ( P @ X2 @ Y2 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X2 @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_139_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_140_undefg__equiv,axiom,
! [Alpha2: option @ game] :
( ( Alpha2
!= ( none @ game ) )
= ( ? [G2: game] :
( Alpha2
= ( some @ game @ G2 ) ) ) ) ).
% undefg_equiv
thf(fact_141_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_142_Loopo_Ocases,axiom,
! [X2: option @ game] :
( ! [Alpha: game] :
( X2
!= ( some @ game @ Alpha ) )
=> ( X2
= ( none @ game ) ) ) ).
% Loopo.cases
thf(fact_143_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the @ A @ Option )
= ( the @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_144_option_Osel,axiom,
! [A: $tType,X23: A] :
( ( the @ A @ ( some @ A @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_145_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_146_accp_Ocases,axiom,
! [A: $tType,R3: A > A > $o,A4: A] :
( ( accp @ A @ R3 @ A4 )
=> ! [Y5: A] :
( ( R3 @ Y5 @ A4 )
=> ( accp @ A @ R3 @ Y5 ) ) ) ).
% accp.cases
thf(fact_147_accp_Osimps,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R4: A > A > $o,A5: A] :
? [X3: A] :
( ( A5 = X3 )
& ! [Y3: A] :
( ( R4 @ Y3 @ X3 )
=> ( accp @ A @ R4 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_148_accp_Ointros,axiom,
! [A: $tType,R3: A > A > $o,X2: A] :
( ! [Y4: A] :
( ( R3 @ Y4 @ X2 )
=> ( accp @ A @ R3 @ Y4 ) )
=> ( accp @ A @ R3 @ X2 ) ) ).
% accp.intros
thf(fact_149_accp__induct,axiom,
! [A: $tType,R3: A > A > $o,A4: A,P: A > $o] :
( ( accp @ A @ R3 @ A4 )
=> ( ! [X: A] :
( ( accp @ A @ R3 @ X )
=> ( ! [Y5: A] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A4 ) ) ) ).
% accp_induct
thf(fact_150_accp_Oinducts,axiom,
! [A: $tType,R3: A > A > $o,X2: A,P: A > $o] :
( ( accp @ A @ R3 @ X2 )
=> ( ! [X: A] :
( ! [Y5: A] :
( ( R3 @ Y5 @ X )
=> ( accp @ A @ R3 @ Y5 ) )
=> ( ! [Y5: A] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ X2 ) ) ) ).
% accp.inducts
thf(fact_151_accp__downward,axiom,
! [A: $tType,R3: A > A > $o,B4: A,A4: A] :
( ( accp @ A @ R3 @ B4 )
=> ( ( R3 @ A4 @ B4 )
=> ( accp @ A @ R3 @ A4 ) ) ) ).
% accp_downward
thf(fact_152_not__accp__down,axiom,
! [A: $tType,R: A > A > $o,X2: A] :
( ~ ( accp @ A @ R @ X2 )
=> ~ ! [Z4: A] :
( ( R @ Z4 @ X2 )
=> ( accp @ A @ R @ Z4 ) ) ) ).
% not_accp_down
thf(fact_153_accp__induct__rule,axiom,
! [A: $tType,R3: A > A > $o,A4: A,P: A > $o] :
( ( accp @ A @ R3 @ A4 )
=> ( ! [X: A] :
( ( accp @ A @ R3 @ X )
=> ( ! [Y5: A] :
( ( R3 @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) ) )
=> ( P @ A4 ) ) ) ).
% accp_induct_rule
thf(fact_154_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y: A,Z: A] : Y = Z
@ ^ [A5: A,B5: A] :
( ( P @ A5 @ B5 )
| ( A5 = B5 ) ) ) ).
% eq_subset
thf(fact_155_Geqo_Opelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ fml] :
( ( ( uSubst1556497037e_Geqo @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some @ trm @ Eta2 ) )
=> ( ( Y2
= ( some @ fml @ ( geq @ Theta2 @ Eta2 ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2
= ( none @ trm ) )
=> ( ( Y2
= ( none @ fml ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some @ trm @ V2 ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ fml ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V2 ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Geqo.pelims
thf(fact_156_Diamondo_Ocases,axiom,
! [X2: product_prod @ ( option @ game ) @ ( option @ fml )] :
( ! [Alpha: game,Phi2: fml] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi2 ) ) )
=> ( ! [Phi2: option @ fml] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( none @ game ) @ Phi2 ) )
=> ~ ! [V2: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ V2 ) @ ( none @ fml ) ) ) ) ) ).
% Diamondo.cases
thf(fact_157_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_158_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_159_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_160_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_161_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X3: A] : ( H @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_162_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_163_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_164_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X23: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_165_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_166_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_167_Timeso_Ocases,axiom,
! [X2: product_prod @ ( option @ trm ) @ ( option @ trm )] :
( ! [Theta2: trm,Eta2: trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) ) )
=> ( ! [Eta2: option @ trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Eta2 ) )
=> ~ ! [V2: trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V2 ) @ ( none @ trm ) ) ) ) ) ).
% Timeso.cases
thf(fact_168_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( ( X2
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X2
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_169_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_170_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X2: A,Y2: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) ) ).
% predicate2D_conj
thf(fact_171_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_172_Timeso_Opelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst277968634Timeso @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some @ trm @ Eta2 ) )
=> ( ( Y2
= ( some @ trm @ ( times @ Theta2 @ Eta2 ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some @ trm @ V2 ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V2 ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Timeso.pelims
thf(fact_173_Pluso_Opelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst1112714340_Pluso @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some @ trm @ Eta2 ) )
=> ( ( Y2
= ( some @ trm @ ( plus @ Theta2 @ Eta2 ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) ) ) ) ) )
=> ( ( ( X2
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V2: trm] :
( ( X2
= ( some @ trm @ V2 ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V2 ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Pluso.pelims
thf(fact_174_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( P
=> ( R @ F ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R @ ( G @ Q2 ) ) )
=> ( R @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_175_ODEo_Ocases,axiom,
! [X2: product_prod @ char @ ( option @ trm )] :
( ! [X: char,Theta2: trm] :
( X2
!= ( product_Pair @ char @ ( option @ trm ) @ X @ ( some @ trm @ Theta2 ) ) )
=> ~ ! [X: char] :
( X2
!= ( product_Pair @ char @ ( option @ trm ) @ X @ ( none @ trm ) ) ) ) ).
% ODEo.cases
thf(fact_176_Assigno_Ocases,axiom,
! [X2: product_prod @ variable @ ( option @ trm )] :
( ! [X: variable,Theta2: trm] :
( X2
!= ( product_Pair @ variable @ ( option @ trm ) @ X @ ( some @ trm @ Theta2 ) ) )
=> ~ ! [X: variable] :
( X2
!= ( product_Pair @ variable @ ( option @ trm ) @ X @ ( none @ trm ) ) ) ) ).
% Assigno.cases
thf(fact_177_Composeo_Ocases,axiom,
! [X2: product_prod @ ( option @ game ) @ ( option @ game )] :
( ! [Alpha: game,Beta: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) ) )
=> ( ! [Alpha: option @ game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ Alpha @ ( none @ game ) ) )
=> ~ ! [V2: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( none @ game ) @ ( some @ game @ V2 ) ) ) ) ) ).
% Composeo.cases
thf(fact_178_Existso_Ocases,axiom,
! [X2: product_prod @ variable @ ( option @ fml )] :
( ! [X: variable,Phi2: fml] :
( X2
!= ( product_Pair @ variable @ ( option @ fml ) @ X @ ( some @ fml @ Phi2 ) ) )
=> ~ ! [X: variable] :
( X2
!= ( product_Pair @ variable @ ( option @ fml ) @ X @ ( none @ fml ) ) ) ) ).
% Existso.cases
thf(fact_179_Ando_Ocases,axiom,
! [X2: product_prod @ ( option @ fml ) @ ( option @ fml )] :
( ! [Phi2: fml,Psi: fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ Phi2 ) @ ( some @ fml @ Psi ) ) )
=> ( ! [Psi: option @ fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( none @ fml ) @ Psi ) )
=> ~ ! [V2: fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ V2 ) @ ( none @ fml ) ) ) ) ) ).
% Ando.cases
thf(fact_180_usubstappt_Opsimps_I7_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( differential @ Theta ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( differential @ Theta ) )
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ Sigma
@ ( collect @ variable
@ ^ [X3: variable] : $true )
@ Theta ) ) ) ) ).
% usubstappt.psimps(7)
thf(fact_181_usubstappt_Opsimps_I6_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( times @ Theta @ Eta ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ) ).
% usubstappt.psimps(6)
thf(fact_182_usubstappt_Opsimps_I5_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( plus @ Theta @ Eta ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( plus @ Theta @ Eta ) )
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta ) ) ) ) ).
% usubstappt.psimps(5)
thf(fact_183_usubstappt_Opsimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,X2: variable] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( var @ X2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some @ trm @ ( var @ X2 ) ) ) ) ).
% usubstappt.psimps(1)
thf(fact_184_usubstappt_Opsimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,R3: real] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( number @ R3 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( number @ R3 ) )
= ( some @ trm @ ( number @ R3 ) ) ) ) ).
% usubstappt.psimps(2)
thf(fact_185_in__lex__prod,axiom,
! [A: $tType,B: $tType,A4: A,B4: B,A7: A,B7: B,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ A7 @ B7 ) ) @ ( lex_prod @ A @ B @ R3 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A7 ) @ R3 )
| ( ( A4 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B7 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_186_trm_Oinject_I2_J,axiom,
! [X23: real,Y23: real] :
( ( ( number @ X23 )
= ( number @ Y23 ) )
= ( X23 = Y23 ) ) ).
% trm.inject(2)
thf(fact_187_trm_Oinject_I1_J,axiom,
! [X1: variable,Y1: variable] :
( ( ( var @ X1 )
= ( var @ Y1 ) )
= ( X1 = Y1 ) ) ).
% trm.inject(1)
thf(fact_188_trm_Odistinct_I17_J,axiom,
! [X23: real,X51: trm,X52: trm] :
( ( number @ X23 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_189_trm_Odistinct_I19_J,axiom,
! [X23: real,X61: trm,X62: trm] :
( ( number @ X23 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(19)
thf(fact_190_trm_Odistinct_I21_J,axiom,
! [X23: real,X7: trm] :
( ( number @ X23 )
!= ( differential @ X7 ) ) ).
% trm.distinct(21)
thf(fact_191_trm_Odistinct_I7_J,axiom,
! [X1: variable,X51: trm,X52: trm] :
( ( var @ X1 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_192_trm_Odistinct_I9_J,axiom,
! [X1: variable,X61: trm,X62: trm] :
( ( var @ X1 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(9)
thf(fact_193_trm_Odistinct_I11_J,axiom,
! [X1: variable,X7: trm] :
( ( var @ X1 )
!= ( differential @ X7 ) ) ).
% trm.distinct(11)
thf(fact_194_trm_Odistinct_I1_J,axiom,
! [X1: variable,X23: real] :
( ( var @ X1 )
!= ( number @ X23 ) ) ).
% trm.distinct(1)
thf(fact_195_usubstappt_Osimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,R3: real] :
( ( uSubst95898992stappt @ Sigma @ U @ ( number @ R3 ) )
= ( some @ trm @ ( number @ R3 ) ) ) ).
% usubstappt.simps(2)
thf(fact_196_usubstappt_Osimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,X2: variable] :
( ( uSubst95898992stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some @ trm @ ( var @ X2 ) ) ) ).
% usubstappt.simps(1)
thf(fact_197_usubstappt_Ocases,axiom,
! [X2: product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm )] :
( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,X: variable] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( var @ X ) ) ) )
=> ( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R5: real] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( number @ R5 ) ) ) )
=> ( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F3: char] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( const @ F3 ) ) ) )
=> ( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F3: char,Theta2: trm] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( func @ F3 @ Theta2 ) ) ) )
=> ( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm,Eta2: trm] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( plus @ Theta2 @ Eta2 ) ) ) )
=> ( ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm,Eta2: trm] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( times @ Theta2 @ Eta2 ) ) ) )
=> ~ ! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm] :
( X2
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.cases
thf(fact_198_Testo_Osimps_I2_J,axiom,
( ( uSubst190403692_Testo @ ( none @ fml ) )
= ( none @ game ) ) ).
% Testo.simps(2)
thf(fact_199_Testo__undef,axiom,
! [Phi: option @ fml] :
( ( ( uSubst190403692_Testo @ Phi )
= ( none @ game ) )
= ( Phi
= ( none @ fml ) ) ) ).
% Testo_undef
thf(fact_200_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_201_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_202_trm_Odistinct_I3_J,axiom,
! [X1: variable,X32: char] :
( ( var @ X1 )
!= ( const @ X32 ) ) ).
% trm.distinct(3)
thf(fact_203_trm_Odistinct_I5_J,axiom,
! [X1: variable,X41: char,X42: trm] :
( ( var @ X1 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_204_trm_Odistinct_I13_J,axiom,
! [X23: real,X32: char] :
( ( number @ X23 )
!= ( const @ X32 ) ) ).
% trm.distinct(13)
thf(fact_205_trm_Odistinct_I15_J,axiom,
! [X23: real,X41: char,X42: trm] :
( ( number @ X23 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_206_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_207_trm_Odistinct_I29_J,axiom,
! [X32: char,X7: trm] :
( ( const @ X32 )
!= ( differential @ X7 ) ) ).
% trm.distinct(29)
thf(fact_208_trm_Odistinct_I27_J,axiom,
! [X32: char,X61: trm,X62: trm] :
( ( const @ X32 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(27)
thf(fact_209_trm_Odistinct_I25_J,axiom,
! [X32: char,X51: trm,X52: trm] :
( ( const @ X32 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_210_trm_Odistinct_I35_J,axiom,
! [X41: char,X42: trm,X7: trm] :
( ( func @ X41 @ X42 )
!= ( differential @ X7 ) ) ).
% trm.distinct(35)
thf(fact_211_trm_Odistinct_I33_J,axiom,
! [X41: char,X42: trm,X61: trm,X62: trm] :
( ( func @ X41 @ X42 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(33)
thf(fact_212_trm_Odistinct_I31_J,axiom,
! [X41: char,X42: trm,X51: trm,X52: trm] :
( ( func @ X41 @ X42 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_213_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_214_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_215_term__sem_Ocases,axiom,
! [X2: product_prod @ denotational_interp @ trm] :
( ! [I2: denotational_interp,X: variable] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( var @ X ) ) )
=> ( ! [I2: denotational_interp,R5: real] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( number @ R5 ) ) )
=> ( ! [I2: denotational_interp,F3: char] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( const @ F3 ) ) )
=> ( ! [I2: denotational_interp,F3: char,Theta2: trm] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( func @ F3 @ Theta2 ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm,Eta2: trm] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( plus @ Theta2 @ Eta2 ) ) )
=> ( ! [I2: denotational_interp,Theta2: trm,Eta2: trm] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( times @ Theta2 @ Eta2 ) ) )
=> ~ ! [I2: denotational_interp,Theta2: trm] :
( X2
!= ( product_Pair @ denotational_interp @ trm @ I2 @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ).
% term_sem.cases
thf(fact_216_usubstappt_Opsimps_I3_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F: char] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( const @ F ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F ) )
= ( uSubst1138577137pconst @ Sigma @ U @ F ) ) ) ).
% usubstappt.psimps(3)
thf(fact_217_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_218_usubstappt_Osimps_I3_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F: char] :
( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F ) )
= ( uSubst1138577137pconst @ Sigma @ U @ F ) ) ).
% usubstappt.simps(3)
thf(fact_219_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A3: set @ C,F2: C > A,G2: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A5: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F2 @ A5 ) @ ( G2 @ A5 ) ) )
& ( member @ C @ A5 @ A3 ) ) ) ) ) ).
% image2_def
thf(fact_220_Testo_Oelims,axiom,
! [X2: option @ fml,Y2: option @ game] :
( ( ( uSubst190403692_Testo @ X2 )
= Y2 )
=> ( ! [Phi2: fml] :
( ( X2
= ( some @ fml @ Phi2 ) )
=> ( Y2
!= ( some @ game @ ( test @ Phi2 ) ) ) )
=> ~ ( ( X2
= ( none @ fml ) )
=> ( Y2
!= ( none @ game ) ) ) ) ) ).
% Testo.elims
thf(fact_221_game_Oinject_I3_J,axiom,
! [X32: fml,Y32: fml] :
( ( ( test @ X32 )
= ( test @ Y32 ) )
= ( X32 = Y32 ) ) ).
% game.inject(3)
thf(fact_222_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B4: A,F: B > A,X2: B,C2: C,G: B > C,A2: set @ B] :
( ( B4
= ( F @ X2 ) )
=> ( ( C2
= ( G @ X2 ) )
=> ( ( member @ B @ X2 @ A2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B4 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_223_Testo_Osimps_I1_J,axiom,
! [Phi: fml] :
( ( uSubst190403692_Testo @ ( some @ fml @ Phi ) )
= ( some @ game @ ( test @ Phi ) ) ) ).
% Testo.simps(1)
thf(fact_224_Testo_Opelims,axiom,
! [X2: option @ fml,Y2: option @ game] :
( ( ( uSubst190403692_Testo @ X2 )
= Y2 )
=> ( ( accp @ ( option @ fml ) @ uSubst533687181to_rel @ X2 )
=> ( ! [Phi2: fml] :
( ( X2
= ( some @ fml @ Phi2 ) )
=> ( ( Y2
= ( some @ game @ ( test @ Phi2 ) ) )
=> ~ ( accp @ ( option @ fml ) @ uSubst533687181to_rel @ ( some @ fml @ Phi2 ) ) ) )
=> ~ ( ( X2
= ( none @ fml ) )
=> ( ( Y2
= ( none @ game ) )
=> ~ ( accp @ ( option @ fml ) @ uSubst533687181to_rel @ ( none @ fml ) ) ) ) ) ) ) ).
% Testo.pelims
thf(fact_225_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr1317331620lImage @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ B ),F2: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: B,A22: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A22 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_226_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A ),F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr1317331620lImage @ A @ B @ R1 @ F ) @ ( bNF_Gr1317331620lImage @ A @ B @ R2 @ F ) ) ) ).
% relImage_mono
thf(fact_227_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr2107612801vImage @ A @ B )
= ( ^ [A3: set @ A,R6: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: A,A22: A] :
( ( Uu
= ( product_Pair @ A @ A @ A12 @ A22 ) )
& ( member @ A @ A12 @ A3 )
& ( member @ A @ A22 @ A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A12 ) @ ( F2 @ A22 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_228_Assigno__undef,axiom,
! [X2: variable,Theta: option @ trm] :
( ( ( uSubst1326326671ssigno @ X2 @ Theta )
= ( none @ game ) )
= ( Theta
= ( none @ trm ) ) ) ).
% Assigno_undef
thf(fact_229_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A ),A2: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr2107612801vImage @ B @ A @ A2 @ R1 @ F ) @ ( bNF_Gr2107612801vImage @ B @ A @ A2 @ R2 @ F ) ) ) ).
% relInvImage_mono
thf(fact_230_Assigno_Osimps_I2_J,axiom,
! [X2: variable] :
( ( uSubst1326326671ssigno @ X2 @ ( none @ trm ) )
= ( none @ game ) ) ).
% Assigno.simps(2)
thf(fact_231_Assigno_Oelims,axiom,
! [X2: variable,Xa: option @ trm,Y2: option @ game] :
( ( ( uSubst1326326671ssigno @ 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_232_Assigno_Opelims,axiom,
! [X2: variable,Xa: option @ trm,Y2: option @ game] :
( ( ( uSubst1326326671ssigno @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ variable @ ( option @ trm ) ) @ uSubst1741596714no_rel @ ( product_Pair @ variable @ ( option @ trm ) @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some @ trm @ Theta2 ) )
=> ( ( Y2
= ( some @ game @ ( assign @ X2 @ Theta2 ) ) )
=> ~ ( accp @ ( product_prod @ variable @ ( option @ trm ) ) @ uSubst1741596714no_rel @ ( product_Pair @ variable @ ( option @ trm ) @ X2 @ ( some @ trm @ Theta2 ) ) ) ) )
=> ~ ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ game ) )
=> ~ ( accp @ ( product_prod @ variable @ ( option @ trm ) ) @ uSubst1741596714no_rel @ ( product_Pair @ variable @ ( option @ trm ) @ X2 @ ( none @ trm ) ) ) ) ) ) ) ) ).
% Assigno.pelims
thf(fact_233_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_234_game_Odistinct_I15_J,axiom,
! [X21: variable,X22: trm,X32: fml] :
( ( assign @ X21 @ X22 )
!= ( test @ X32 ) ) ).
% game.distinct(15)
thf(fact_235_Assigno_Osimps_I1_J,axiom,
! [X2: variable,Theta: trm] :
( ( uSubst1326326671ssigno @ X2 @ ( some @ trm @ Theta ) )
= ( some @ game @ ( assign @ X2 @ Theta ) ) ) ).
% Assigno.simps(1)
thf(fact_236_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Gr2107612801vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr1317331620lImage @ A @ B @ R @ F ) @ F ) ) ).
% relInvImage_UNIV_relImage
thf(fact_237_ODEo__undef,axiom,
! [X2: char,Theta: option @ trm] :
( ( ( uSubst1083227664e_ODEo @ X2 @ Theta )
= ( none @ game ) )
= ( Theta
= ( none @ trm ) ) ) ).
% ODEo_undef
thf(fact_238_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X3: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_239_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_240_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_241_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_242_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_243_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_244_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_245_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X: A] : ( member @ A @ X @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_246_UNIV__witness,axiom,
! [A: $tType] :
? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_247_ODEo_Osimps_I2_J,axiom,
! [X2: char] :
( ( uSubst1083227664e_ODEo @ X2 @ ( none @ trm ) )
= ( none @ game ) ) ).
% ODEo.simps(2)
thf(fact_248_ODEo_Oelims,axiom,
! [X2: char,Xa: option @ trm,Y2: option @ game] :
( ( ( uSubst1083227664e_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_249_ODEo_Opelims,axiom,
! [X2: char,Xa: option @ trm,Y2: option @ game] :
( ( ( uSubst1083227664e_ODEo @ X2 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ char @ ( option @ trm ) ) @ uSubst2007600873Eo_rel @ ( product_Pair @ char @ ( option @ trm ) @ X2 @ Xa ) )
=> ( ! [Theta2: trm] :
( ( Xa
= ( some @ trm @ Theta2 ) )
=> ( ( Y2
= ( some @ game @ ( ode @ X2 @ Theta2 ) ) )
=> ~ ( accp @ ( product_prod @ char @ ( option @ trm ) ) @ uSubst2007600873Eo_rel @ ( product_Pair @ char @ ( option @ trm ) @ X2 @ ( some @ trm @ Theta2 ) ) ) ) )
=> ~ ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ game ) )
=> ~ ( accp @ ( product_prod @ char @ ( option @ trm ) ) @ uSubst2007600873Eo_rel @ ( product_Pair @ char @ ( option @ trm ) @ X2 @ ( none @ trm ) ) ) ) ) ) ) ) ).
% ODEo.pelims
thf(fact_250_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_251_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_252_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_253_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_254_game_Odistinct_I35_J,axiom,
! [X32: fml,X81: char,X82: trm] :
( ( test @ X32 )
!= ( ode @ X81 @ X82 ) ) ).
% game.distinct(35)
% Type constructors (20)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Orderings_Opreorder_11,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_12,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_13,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_14,axiom,
ord @ real ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( uSubst95898992stappt @ sigma @ ua @ theta )
!= ( none @ trm ) ) ).
%------------------------------------------------------------------------------