TPTP Problem File: ITP201^2.p
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%------------------------------------------------------------------------------
% File : ITP201^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_1580__6355392_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_1580__6355392_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 346 ( 119 unt; 57 typ; 0 def)
% Number of atoms : 714 ( 253 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4059 ( 60 ~; 5 |; 37 &;3645 @)
% ( 0 <=>; 312 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 305 ( 305 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 50 usr; 8 con; 0-7 aty)
% Number of variables : 1135 ( 102 ^; 986 !; 9 ?;1135 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:22.378
%------------------------------------------------------------------------------
% 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 (47)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[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_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Denotational__Semantics_OUvariation,type,
denota1419872369iation: ( variable > real ) > ( variable > real ) > ( set @ variable ) > $o ).
thf(sy_c_Denotational__Semantics_Ogame__sem,type,
denota1245701238me_sem: denotational_interp > game > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Syntax_OSkip,type,
skip: game ).
thf(sy_c_Syntax_Ogame_OChoice,type,
choice: game > game > game ).
thf(sy_c_Syntax_Ogame_ODual,type,
dual: game > game ).
thf(sy_c_Syntax_Ogame_OLoop,type,
loop: game > game ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OChoiceo,type,
uSubst1976112797hoiceo: ( option @ game ) > ( option @ game ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODualo,type,
uSubst739989314_Dualo: ( option @ game ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OLoopo,type,
uSubst993936602_Loopo: ( option @ game ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Oadjoint,type,
uSubst2091380086djoint: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > denotational_interp > ( variable > real ) > denotational_interp ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappp,type,
uSubst95898988stappp: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > game > ( product_prod @ ( set @ variable ) @ ( option @ game ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I,type,
i: denotational_interp ).
thf(sy_v_U,type,
u: set @ variable ).
thf(sy_v_Ua____,type,
ua: set @ variable ).
thf(sy_v_Xa____,type,
xa: set @ ( variable > real ) ).
thf(sy_v__092_060alpha_062_H_H____,type,
alpha: game ).
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 ) ) ) ) ).
% Relevant facts (255)
thf(fact_0__092_060open_062_092_060And_062X_O_A_I_092_060nu_062_A_092_060in_062_Agame__sem_AI_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_AX_J_A_061_A_I_092_060nu_062_A_092_060in_062_Agame__sem_A_IUSubst__Mirabelle__nnnzepxswx_Oadjoint_A_092_060sigma_062_AI_A_092_060omega_062_J_A_092_060alpha_062_AX_J_092_060close_062,axiom,
! [X: set @ ( variable > real )] :
( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ).
% \<open>\<And>X. (\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> U \<alpha>))) X) = (\<nu> \<in> game_sem (USubst_Mirabelle_nnnzepxswx.adjoint \<sigma> I \<omega>) \<alpha> X)\<close>
thf(fact_1_IH_092_060alpha_062,axiom,
! [X: set @ ( variable > real )] :
( ( denota1419872369iation @ nu2 @ omega2 @ ua )
=> ( ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member @ ( variable > real ) @ nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ X ) ) ) ) ).
% IH\<alpha>
thf(fact_2_uv,axiom,
denota1419872369iation @ nu2 @ omega2 @ ua ).
% uv
thf(fact_3_snd__uminus,axiom,
! [A: $tType,B: $tType] :
( ( ( uminus @ B )
& ( uminus @ A ) )
=> ! [X2: product_prod @ B @ A] :
( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X2 ) )
= ( uminus_uminus @ A @ ( product_snd @ B @ A @ X2 ) ) ) ) ).
% snd_uminus
thf(fact_4_def,axiom,
( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( dual @ alpha ) ) )
!= ( none @ game ) ) ).
% def
thf(fact_5_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_6_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_7_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A2 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_8_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B3 ) )
= B3 ) ) ).
% verit_minus_simplify(4)
thf(fact_9_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_10_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_11_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B3 ) )
= ( A4 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_12_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X2 ) )
= X2 ) ) ).
% double_compl
thf(fact_13_vaouter,axiom,
denota1419872369iation @ nu @ omega @ u ).
% vaouter
thf(fact_14_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ( uminus_uminus @ A @ X2 )
= ( uminus_uminus @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_15_usubstappp__det,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,V: set @ variable] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
!= ( none @ game ) )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) )
!= ( none @ game ) )
=> ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
= ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_det
thf(fact_16_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A4 ) ) ) ).
% minus_equation_iff
thf(fact_17_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% equation_minus_iff
thf(fact_18_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X3: A] : ( uminus_uminus @ B @ ( A3 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_19_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
=> ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B3 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_20_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_21_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_22_game__sem_Osimps_I7_J,axiom,
! [I: denotational_interp,Alpha: game] :
( ( denota1245701238me_sem @ I @ ( dual @ Alpha ) )
= ( ^ [X4: set @ ( variable > real )] : ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ X4 ) ) ) ) ) ).
% game_sem.simps(7)
thf(fact_23_Dual_OIH,axiom,
! [Nu: variable > real,Omega: variable > real,X: set @ ( variable > real )] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) )
!= ( none @ game ) )
=> ( ( denota1419872369iation @ Nu @ Omega @ ua )
=> ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ X ) )
= ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ Omega ) @ alpha @ X ) ) ) ) ) ).
% Dual.IH
thf(fact_24_game_Oinject_I7_J,axiom,
! [X7: game,Y7: game] :
( ( ( dual @ X7 )
= ( dual @ Y7 ) )
= ( X7 = Y7 ) ) ).
% game.inject(7)
thf(fact_25_Dual_Oprems_I2_J,axiom,
denota1419872369iation @ nu2 @ omega2 @ ua ).
% Dual.prems(2)
thf(fact_26_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_27_usubst__game__loop,axiom,
! [Nu: variable > real,Omega: variable > real,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game,I: denotational_interp,X: set @ ( variable > real )] :
( ( denota1419872369iation @ Nu @ Omega @ U )
=> ( ! [Nu2: variable > real,Omega2: variable > real,X5: set @ ( variable > real )] :
( ( denota1419872369iation @ Nu2 @ Omega2 @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
!= ( none @ game ) )
=> ( ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) @ X5 ) )
= ( member @ ( variable > real ) @ Nu2 @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega2 ) @ Alpha @ X5 ) ) ) ) )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
!= ( none @ game ) )
=> ( ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ I @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) ) ) @ X ) )
= ( member @ ( variable > real ) @ Nu @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ Sigma @ I @ Omega ) @ ( loop @ Alpha ) @ X ) ) ) ) ) ) ).
% usubst_game_loop
thf(fact_28_Uvariation__def,axiom,
( denota1419872369iation
= ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set @ variable] :
! [I2: variable] :
( ~ ( member @ variable @ I2 @ U2 )
=> ( ( Nu3 @ I2 )
= ( Nu4 @ I2 ) ) ) ) ) ).
% Uvariation_def
thf(fact_29_Uvariation__sym,axiom,
( denota1419872369iation
= ( ^ [Omega3: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega3 ) ) ) ).
% Uvariation_sym
thf(fact_30_Uvariation__refl,axiom,
! [Nu: variable > real,V: set @ variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% Uvariation_refl
thf(fact_31_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_32_usubstappp__choice__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) ) )
!= ( none @ game ) )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
!= ( none @ game ) )
& ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) )
!= ( none @ game ) ) ) ) ).
% usubstappp_choice_conv
thf(fact_33_game_Oinject_I6_J,axiom,
! [X6: game,Y6: game] :
( ( ( loop @ X6 )
= ( loop @ Y6 ) )
= ( X6 = Y6 ) ) ).
% game.inject(6)
thf(fact_34_game_Oinject_I4_J,axiom,
! [X41: game,X42: game,Y41: game,Y42: game] :
( ( ( choice @ X41 @ X42 )
= ( choice @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% game.inject(4)
thf(fact_35_fst__uminus,axiom,
! [B: $tType,A: $tType] :
( ( ( uminus @ A )
& ( uminus @ B ) )
=> ! [X2: product_prod @ A @ B] :
( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X2 ) )
= ( uminus_uminus @ A @ ( product_fst @ A @ B @ X2 ) ) ) ) ).
% fst_uminus
thf(fact_36_Uvariation__univ,axiom,
! [Nu: variable > real,Nu5: variable > real] :
( denota1419872369iation @ Nu @ Nu5
@ ( collect @ variable
@ ^ [X3: variable] : $true ) ) ).
% Uvariation_univ
thf(fact_37_game_Odistinct_I39_J,axiom,
! [X41: game,X42: game,X6: game] :
( ( choice @ X41 @ X42 )
!= ( loop @ X6 ) ) ).
% game.distinct(39)
thf(fact_38_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A3: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_39_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_40_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A3: set @ A] :
( collect @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_41_game_Odistinct_I51_J,axiom,
! [X6: game,X7: game] :
( ( loop @ X6 )
!= ( dual @ X7 ) ) ).
% game.distinct(51)
thf(fact_42_game_Odistinct_I41_J,axiom,
! [X41: game,X42: game,X7: game] :
( ( choice @ X41 @ X42 )
!= ( dual @ X7 ) ) ).
% game.distinct(41)
thf(fact_43_usubstappp__loop__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) ) )
!= ( none @ game ) )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
!= ( none @ game ) )
& ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) )
!= ( none @ game ) ) ) ) ).
% usubstappp_loop_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] :
( ! [X8: A] :
( ( P @ X8 )
= ( Q @ X8 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X8: A] :
( ( F @ X8 )
= ( G @ X8 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_skip__id,axiom,
! [I: denotational_interp,X: set @ ( variable > real )] :
( ( denota1245701238me_sem @ I @ skip @ X )
= X ) ).
% skip_id
thf(fact_49_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_50_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X8: B,Y2: A] :
~ ( P @ Y2 @ X8 ) ) ).
% exE_realizer'
thf(fact_51_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_52_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : Y3 = Z )
= ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_53_uminus__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( uminus @ A )
& ( uminus @ B ) )
=> ( ( uminus_uminus @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% uminus_prod_def
thf(fact_54_usubstappp__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 ) ) ) ),Alpha: game] :
( ( ord_less_eq @ ( set @ variable ) @ V @ U )
=> ( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
!= ( none @ game ) )
=> ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) )
= ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ) ).
% usubstappp_antimon
thf(fact_55_exists__diff,axiom,
! [A: $tType,P: ( set @ A ) > $o] :
( ( ? [S2: set @ A] : ( P @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) )
= ( ? [X9: set @ A] : ( P @ X9 ) ) ) ).
% exists_diff
thf(fact_56_uminus__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( uminus @ B )
& ( uminus @ A ) )
=> ! [A4: A,B3: B] :
( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ B @ B3 ) ) ) ) ).
% uminus_Pair
thf(fact_57_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
= ( ( A4 = A5 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_59_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_60_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X8: A] :
( ( member @ A @ X8 @ A2 )
=> ( member @ A @ X8 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_61_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_62_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_63_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_64_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_65_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_66_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X8: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X8 @ Y2 ) ) ).
% surj_pair
thf(fact_67_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_68_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( ( A4 = A5 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_69_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_70_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_71_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_72_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_73_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_74_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_75_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_76_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_77_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_78_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_79_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_80_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_81_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
| ~ ( ord_less_eq @ A @ A4 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% verit_la_disequality
thf(fact_82_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_83_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_84_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_85_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X8: A] :
( ( P @ X8 )
=> ( Q @ X8 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_86_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_87_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A3 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_88_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_89_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_90_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_91_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_92_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_93_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_94_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_95_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A4: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A4 )
=> ( X2 = A4 ) ) ).
% fst_eqD
thf(fact_96_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_97_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_98_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A4: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_99_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_imp_neg_le
thf(fact_100_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% minus_le_iff
thf(fact_101_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_minus_iff
thf(fact_102_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_103_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_104_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_105_fst__pair,axiom,
! [B: $tType,A: $tType,A4: A,B3: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= A4 ) ).
% fst_pair
thf(fact_106_snd__pair,axiom,
! [B: $tType,A: $tType,A4: B,B3: A] :
( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ A4 @ B3 ) )
= B3 ) ).
% snd_pair
thf(fact_107_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_108_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_109_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_110_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
( ( P @ Y @ X2 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_111_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_112_usubstappp__fst__mon,axiom,
! [U: set @ variable,V: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] :
( ( ord_less_eq @ ( set @ variable ) @ U @ V )
=> ( ord_less_eq @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ V @ Alpha ) ) ) ) ).
% usubstappp_fst_mon
thf(fact_113_usubst__taboos__mon,axiom,
! [U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Alpha: game] : ( ord_less_eq @ ( set @ variable ) @ U @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ).
% usubst_taboos_mon
thf(fact_114_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( F1 @ A4 @ B3 ) ) ).
% old.prod.rec
thf(fact_115_usubstappp__dual,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% usubstappp_dual
thf(fact_116_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A4: product_prod @ A @ B] :
( ( P @ X2 @ Y )
=> ( ( A4
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_snd @ A @ B @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_117_usubstappp_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,Alpha: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( dual @ Alpha ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst739989314_Dualo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) ) ) ) ).
% usubstappp.simps(7)
thf(fact_118_usubstappp_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,Alpha: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp.simps(6)
thf(fact_119_usubstappp__loop,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( loop @ Alpha ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( uSubst993936602_Loopo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Alpha ) ) ) ) ) ).
% usubstappp_loop
thf(fact_120_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_121_monotone,axiom,
! [X: set @ ( variable > real ),Y4: set @ ( variable > real ),I: denotational_interp,Alpha: game] :
( ( ord_less_eq @ ( set @ ( variable > real ) ) @ X @ Y4 )
=> ( ord_less_eq @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ I @ Alpha @ X ) @ ( denota1245701238me_sem @ I @ Alpha @ Y4 ) ) ) ).
% monotone
thf(fact_122_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A3 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_123_Loopo__undef,axiom,
! [Alpha: option @ game] :
( ( ( uSubst993936602_Loopo @ Alpha )
= ( none @ game ) )
= ( Alpha
= ( none @ game ) ) ) ).
% Loopo_undef
thf(fact_124_Dualo__undef,axiom,
! [Alpha: option @ game] :
( ( ( uSubst739989314_Dualo @ Alpha )
= ( none @ game ) )
= ( Alpha
= ( none @ game ) ) ) ).
% Dualo_undef
thf(fact_125_Loopo_Osimps_I2_J,axiom,
( ( uSubst993936602_Loopo @ ( none @ game ) )
= ( none @ game ) ) ).
% Loopo.simps(2)
thf(fact_126_Dualo_Osimps_I2_J,axiom,
( ( uSubst739989314_Dualo @ ( none @ game ) )
= ( none @ game ) ) ).
% Dualo.simps(2)
thf(fact_127_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_128_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ B7 @ A7 )
& ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_129_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_130_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_131_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_132_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_133_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_134_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_135_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_136_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
& ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_137_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_138_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_139_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_140_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_141_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_142_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_143_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_144_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_145_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X8: A,Y2: A] :
( ( ord_less_eq @ A @ X8 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X8: B,Y2: B] :
( ( ord_less_eq @ B @ X8 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X8: A,Y2: A] :
( ( ord_less_eq @ A @ X8 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_148_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X8: B,Y2: B] :
( ( ord_less_eq @ B @ X8 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X8 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_149_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_150_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X8: A] : ( ord_less_eq @ B @ ( F @ X8 ) @ ( G @ X8 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_151_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_152_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_153_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ B2 )
=> ( ( Q @ X8 )
=> ( P @ X8 ) ) )
=> ( 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_154_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_155_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_156_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A7: B] :
( P2
= ( product_Pair @ B @ A @ A7 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_157_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A4: A,P2: product_prod @ A @ B] :
( ( A4
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B7: B] :
( P2
= ( product_Pair @ A @ B @ A4 @ B7 ) ) ) ) ).
% eq_fst_iff
thf(fact_158_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z2: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_159_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X8: A] :
( ( P @ X8 )
=> ( Q @ X8 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_160_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_161_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_162_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ).
% pred_subset_eq
thf(fact_163_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% scomp_unfold
thf(fact_164_Collect__restrict,axiom,
! [A: $tType,X: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X )
& ( P @ X3 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_165_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C ) )
= X2 ) ).
% scomp_Pair
thf(fact_166_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F2 @ B @ F
@ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G @ X3 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_167_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X2 ) @ F )
= ( F @ X2 ) ) ).
% Pair_scomp
thf(fact_168_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
( ! [X8: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X8 @ Y2 ) @ S4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S4 ) ) ).
% subrelI
thf(fact_169_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S4: B,R: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R )
=> ( ( S5 = S4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_170_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R )
@ ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% pred_subset_eq2
thf(fact_171_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R ) )
= ( ^ [X3: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) ) )
= ( R = S3 ) ) ).
% pred_equals_eq2
thf(fact_172_prop__restrict,axiom,
! [A: $tType,X2: A,Z3: set @ A,X: set @ A,P: A > $o] :
( ( member @ A @ X2 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_173_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_174_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( C2 @ A4 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_175_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_176_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X8: A,Y2: B] :
( ( P @ X8 @ Y2 )
=> ( Q @ X8 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_177_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_178_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_179_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_180_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_181_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X2: A,Y: B,Q: A > B > $o] :
( ( P @ X2 @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X2 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_182_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X2: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) ).
% predicate2D
thf(fact_183_refl__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X8: A] : ( R @ X8 @ X8 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R ) ) ).
% refl_ge_eq
thf(fact_184_ge__eq__refl,axiom,
! [A: $tType,R: A > A > $o,X2: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R )
=> ( R @ X2 @ X2 ) ) ).
% ge_eq_refl
thf(fact_185_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [A7: A,B7: A] :
( ( P @ A7 @ B7 )
| ( A7 = B7 ) ) ) ).
% eq_subset
thf(fact_186_usubstappp__choice,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Alpha: game,Beta: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( choice @ Alpha @ Beta ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( sup_sup @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) @ ( uSubst1976112797hoiceo @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Beta ) ) ) ) ) ).
% usubstappp_choice
thf(fact_187_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_188_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_189_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_190_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ X2 )
= X2 ) ) ).
% sup_idem
thf(fact_191_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_192_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_left_idem
thf(fact_193_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_194_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_195_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_196_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X2 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_197_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_198_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_199_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B2 )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_200_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_201_Un__absorb1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_202_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_203_Un__upper1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_204_Un__least,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 ) ) ) ).
% Un_least
thf(fact_205_Un__mono,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_206_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_207_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_208_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B7: A] :
( ( sup_sup @ A @ A7 @ B7 )
= B7 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_209_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A7: A] :
( ( sup_sup @ A @ A7 @ B7 )
= A7 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_210_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_211_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_212_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B7: A,A7: A] :
( A7
= ( sup_sup @ A @ A7 @ B7 ) ) ) ) ) ).
% sup.order_iff
thf(fact_213_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% sup.boundedI
thf(fact_214_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A4 )
=> ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.boundedE
thf(fact_215_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( sup_sup @ A @ X2 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_216_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( sup_sup @ A @ X2 @ Y )
= X2 ) ) ) ).
% sup_absorb1
thf(fact_217_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_218_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb1
thf(fact_219_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X2: A,Y: A] :
( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ X8 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X8 @ Y2 ) )
=> ( ! [X8: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ Y2 @ X8 )
=> ( ( ord_less_eq @ A @ Z4 @ X8 )
=> ( ord_less_eq @ A @ ( F @ Y2 @ Z4 ) @ X8 ) ) )
=> ( ( sup_sup @ A @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_220_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% sup.orderI
thf(fact_221_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4
= ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_222_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y5: A] :
( ( sup_sup @ A @ X3 @ Y5 )
= Y5 ) ) ) ) ).
% le_iff_sup
thf(fact_223_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X2 ) ) ) ) ).
% sup_least
thf(fact_224_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,C2: A,B3: A,D4: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% sup_mono
thf(fact_225_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,D4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ A @ D4 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_226_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ X2 @ B3 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_227_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X2 @ A4 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_228_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge2
thf(fact_229_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge1
thf(fact_230_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,X2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ X2 )
=> ( ( ord_less_eq @ A @ B3 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 ) ) ) ) ).
% le_supI
thf(fact_231_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,X2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X2 )
=> ~ ( ( ord_less_eq @ A @ A4 @ X2 )
=> ~ ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ).
% le_supE
thf(fact_232_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_233_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_234_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
= ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_235_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_236_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_237_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_238_UnE,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_239_UnI1,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_240_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_241_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B2 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_242_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_243_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_assoc
thf(fact_244_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_245_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_246_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_247_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_248_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,K: A,A4: A,B3: A] :
( ( A2
= ( sup_sup @ A @ K @ A4 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_249_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A4: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A4 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_250_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C2 )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_251_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_252_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A7: A,B7: A] : ( sup_sup @ A @ B7 @ A7 ) ) ) ) ).
% sup.commute
thf(fact_253_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y5: A] : ( sup_sup @ A @ Y5 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_254_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C2 ) )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
% Type constructors (33)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A10: $tType] :
( ( boolean_algebra @ A10 )
=> ( boolean_algebra @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A10: $tType] :
( ( lattice @ A10 )
=> ( lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A10: $tType] :
( ( uminus @ A10 )
=> ( uminus @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_7,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_8,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_11,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_15,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_16,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_17,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_18,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_19,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_20,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_21,axiom,
uminus @ real ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_22,axiom,
! [A9: $tType,A10: $tType] :
( ( ( group_add @ A9 )
& ( group_add @ A10 ) )
=> ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ouminus_23,axiom,
! [A9: $tType,A10: $tType] :
( ( ( uminus @ A9 )
& ( uminus @ A10 ) )
=> ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ i @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) )
= ( member @ ( variable > real ) @ nu2 @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ ( denota1245701238me_sem @ ( uSubst2091380086djoint @ sigma @ i @ omega2 ) @ alpha @ ( uminus_uminus @ ( set @ ( variable > real ) ) @ xa ) ) ) ) ) ).
%------------------------------------------------------------------------------