TPTP Problem File: ITP205^2.p
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%------------------------------------------------------------------------------
% File : ITP205^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_966__6345404_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_966__6345404_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 322 ( 119 unt; 50 typ; 0 def)
% Number of atoms : 673 ( 291 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4420 ( 83 ~; 9 |; 37 &;3977 @)
% ( 0 <=>; 314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 300 ( 300 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 4 con; 0-9 aty)
% Number of variables : 1179 ( 117 ^; 993 !; 13 ?;1179 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:34.396
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_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_Osemilattice__sup,type,
semilattice_sup:
!>[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_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
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_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_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_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_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Static__Semantics_OBVG,type,
static_BVG: game > ( set @ variable ) ).
thf(sy_c_Syntax_Ogame_OChoice,type,
choice: game > game > game ).
thf(sy_c_Syntax_Ogame_OCompose,type,
compose: game > 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_OComposeo,type,
uSubst1385204910mposeo: ( option @ game ) > ( option @ game ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OLoopo,type,
uSubst993936602_Loopo: ( option @ game ) > ( option @ game ) ).
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_Ua____,type,
ua: set @ variable ).
thf(sy_v__092_060alpha_062_H____,type,
alpha: game ).
thf(sy_v__092_060beta_062____,type,
beta: game ).
thf(sy_v__092_060sigma_062,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_062U_A_092_060union_062_ABVG_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_J_A_092_060union_062_ABVG_A_Ithe_A_Isnd_A_Iusubstappp_A_092_060sigma_062_A_Ifst_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_A_092_060beta_062_J_J_J_A_092_060subseteq_062_Afst_A_Iusubstappp_A_092_060sigma_062_A_Ifst_A_Iusubstappp_A_092_060sigma_062_AU_A_092_060alpha_062_J_J_A_092_060beta_062_J_092_060close_062,axiom,
ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ ua @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) ) ) @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
% \<open>U \<union> BVG (the (snd (usubstappp \<sigma> U \<alpha>))) \<union> BVG (the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>))) \<subseteq> fst (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>)\<close>
thf(fact_1_fact,axiom,
ord_less_eq @ ( set @ variable ) @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) ) ) ) @ ( sup_sup @ ( set @ variable ) @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) ) @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) ).
% fact
thf(fact_2_IHa,axiom,
ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ ua @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) ).
% IHa
thf(fact_3_IHb,axiom,
ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
% IHb
thf(fact_4_Compose_Oprems,axiom,
( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) )
!= ( none @ game ) ) ).
% Compose.prems
thf(fact_5_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_6_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_7_Compose_OIH_I2_J,axiom,
! [U: set @ variable] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ beta ) )
!= ( none @ game ) )
=> ( ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ U @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ beta ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ beta ) ) ) ) ).
% Compose.IH(2)
thf(fact_8_Compose_OIH_I1_J,axiom,
! [U: set @ variable] :
( ( ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ alpha ) )
!= ( none @ game ) )
=> ( ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ U @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ alpha ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ U @ alpha ) ) ) ) ).
% Compose.IH(1)
thf(fact_9_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_10_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_11_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_12_BVG__compose,axiom,
! [Alpha: game,Beta: game] : ( ord_less_eq @ ( set @ variable ) @ ( static_BVG @ ( compose @ Alpha @ Beta ) ) @ ( sup_sup @ ( set @ variable ) @ ( static_BVG @ Alpha ) @ ( static_BVG @ Beta ) ) ) ).
% BVG_compose
thf(fact_13_game_Oinject_I5_J,axiom,
! [X51: game,X52: game,Y51: game,Y52: game] :
( ( ( compose @ X51 @ X52 )
= ( compose @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% game.inject(5)
thf(fact_14_UnCI,axiom,
! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ A2 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_15_union__or,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C3 @ A2 )
| ( member @ A @ C3 @ B2 ) ) ) ).
% union_or
thf(fact_16_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_17_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_18_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_19_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_20_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_21_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_22_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_23_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_24_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_25_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_26_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_27_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_28_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_29_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_30_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_31_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_32_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_33_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_34_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_35_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_36_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_37_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_38_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C3 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_39_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_40_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_41_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_42_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C3 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_43_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,K: A,B3: A,A3: A] :
( ( B2
= ( sup_sup @ A @ K @ B3 ) )
=> ( ( sup_sup @ A @ A3 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_44_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,K: A,A3: A,B3: A] :
( ( A2
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A2 @ B3 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G2 @ X3 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F: A > B,G: A > B,X2: A] : ( sup_sup @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_50_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y3: A] : ( sup_sup @ A @ Y3 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_51_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_52_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_53_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_54_not__union__or,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( ~ ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) )
= ( ~ ( member @ A @ X @ A2 )
& ~ ( member @ A @ X @ B2 ) ) ) ).
% not_union_or
thf(fact_55_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_56_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_57_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_58_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_59_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_60_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_61_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_62_UnI2,axiom,
! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_63_UnI1,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_64_UnE,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_65_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_66_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_67_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_68_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_69_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_70_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_71_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_72_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_73_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_74_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_75_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_76_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_77_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_78_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_79_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_80_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% sup.orderI
thf(fact_81_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_82_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( sup_sup @ A @ X2 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_83_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_84_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C3: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C3 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_85_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_86_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B3: A,A3: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_87_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_88_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_89_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_90_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_91_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_92_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_93_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_94_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_95_subset__UnE,axiom,
! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ! [A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ A2 )
=> ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ B2 )
=> ( C2
!= ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_96_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_97_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_98_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_99_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_100_Un__least,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_101_Un__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_102_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_103_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_104_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y2: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y2 = Z2 )
= ( ^ [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_105_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_106_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 ) )
=> ~ ! [X3: B,Y4: A] :
~ ( P @ Y4 @ X3 ) ) ).
% exE_realizer'
thf(fact_107_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_108_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_109_BVG__choice,axiom,
! [Alpha: game,Beta: game] : ( ord_less_eq @ ( set @ variable ) @ ( static_BVG @ ( choice @ Alpha @ Beta ) ) @ ( sup_sup @ ( set @ variable ) @ ( static_BVG @ Alpha ) @ ( static_BVG @ Beta ) ) ) ).
% BVG_choice
thf(fact_110_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_111_game_Oinject_I6_J,axiom,
! [X6: game,Y6: game] :
( ( ( loop @ X6 )
= ( loop @ Y6 ) )
= ( X6 = Y6 ) ) ).
% game.inject(6)
thf(fact_112_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_113_game_Odistinct_I39_J,axiom,
! [X41: game,X42: game,X6: game] :
( ( choice @ X41 @ X42 )
!= ( loop @ X6 ) ) ).
% game.distinct(39)
thf(fact_114_game_Odistinct_I45_J,axiom,
! [X51: game,X52: game,X6: game] :
( ( compose @ X51 @ X52 )
!= ( loop @ X6 ) ) ).
% game.distinct(45)
thf(fact_115_game_Odistinct_I37_J,axiom,
! [X41: game,X42: game,X51: game,X52: game] :
( ( choice @ X41 @ X42 )
!= ( compose @ X51 @ X52 ) ) ).
% game.distinct(37)
thf(fact_116_BVG__loop,axiom,
! [Alpha: game] : ( ord_less_eq @ ( set @ variable ) @ ( static_BVG @ ( loop @ Alpha ) ) @ ( static_BVG @ Alpha ) ) ).
% BVG_loop
thf(fact_117_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_118_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_119_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_120_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_121_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F2 @ B3 ) @ C3 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_123_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B3: B,C3: B] :
( ( A3
= ( F2 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_124_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F2 @ B3 )
= C3 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_126_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_127_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_128_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_129_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_130_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_131_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_132_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_133_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_134_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_135_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_136_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_137_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_138_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_139_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: A,B7: A] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_140_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_142_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_143_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_144_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_145_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_146_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_147_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_148_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A8: A,B8: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A8 @ B8 ) )
= ( ( A3 = A8 )
& ( B3 = B8 ) ) ) ).
% old.prod.inject
thf(fact_149_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_150_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_151_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_152_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_153_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_154_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A7: A,B7: B] :
( Y
!= ( product_Pair @ A @ B @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_155_prod__induct7,axiom,
! [G3: $tType,F3: $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 @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A7: A,B7: B,C4: C,D4: D,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_156_prod__induct6,axiom,
! [F3: $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 @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A7: A,B7: B,C4: C,D4: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_157_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,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A7: A,B7: B,C4: C,D4: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_158_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A7: A,B7: B,C4: C,D4: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B7 @ ( product_Pair @ C @ D @ C4 @ D4 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_159_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A7: A,B7: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_160_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A7: A,B7: B,C4: C,D4: D,E2: E,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_161_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A7: A,B7: B,C4: C,D4: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_162_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 ) ) )] :
~ ! [A7: A,B7: B,C4: C,D4: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_163_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A7: A,B7: B,C4: C,D4: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B7 @ ( product_Pair @ C @ D @ C4 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_164_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A7: A,B7: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B7 @ C4 ) ) ) ).
% prod_cases3
thf(fact_165_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A8: A,B8: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A8 @ B8 ) )
=> ~ ( ( A3 = A8 )
=> ( B3 != B8 ) ) ) ).
% Pair_inject
thf(fact_166_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A7: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_167_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_168_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_169_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_170_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_171_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_172_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_173_fst__pair,axiom,
! [B: $tType,A: $tType,A3: A,B3: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= A3 ) ).
% fst_pair
thf(fact_174_snd__pair,axiom,
! [B: $tType,A: $tType,A3: B,B3: A] :
( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ A3 @ B3 ) )
= B3 ) ).
% snd_pair
thf(fact_175_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_176_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_177_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_178_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_179_Choiceo_Osimps_I2_J,axiom,
! [Alpha: option @ game] :
( ( uSubst1976112797hoiceo @ Alpha @ ( none @ game ) )
= ( none @ game ) ) ).
% Choiceo.simps(2)
thf(fact_180_Choiceo__undef,axiom,
! [Alpha: option @ game,Beta: option @ game] :
( ( ( uSubst1976112797hoiceo @ Alpha @ Beta )
= ( none @ game ) )
= ( ( Alpha
= ( none @ game ) )
| ( Beta
= ( none @ game ) ) ) ) ).
% Choiceo_undef
thf(fact_181_usubstappp__compose,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 @ ( compose @ Alpha @ Beta ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) @ ( uSubst1385204910mposeo @ ( product_snd @ ( 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 ) ) @ Beta ) ) ) ) ) ).
% usubstappp_compose
thf(fact_182_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_183_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_184_Composeo__undef,axiom,
! [Alpha: option @ game,Beta: option @ game] :
( ( ( uSubst1385204910mposeo @ Alpha @ Beta )
= ( none @ game ) )
= ( ( Alpha
= ( none @ game ) )
| ( Beta
= ( none @ game ) ) ) ) ).
% Composeo_undef
thf(fact_185_Composeo_Osimps_I2_J,axiom,
! [Alpha: option @ game] :
( ( uSubst1385204910mposeo @ Alpha @ ( none @ game ) )
= ( none @ game ) ) ).
% Composeo.simps(2)
thf(fact_186_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_187_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A5: B] :
( P2
= ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_188_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_189_Loopo_Osimps_I2_J,axiom,
( ( uSubst993936602_Loopo @ ( none @ game ) )
= ( none @ game ) ) ).
% Loopo.simps(2)
thf(fact_190_Loopo__undef,axiom,
! [Alpha: option @ game] :
( ( ( uSubst993936602_Loopo @ Alpha )
= ( none @ game ) )
= ( Alpha
= ( none @ game ) ) ) ).
% Loopo_undef
thf(fact_191_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_192_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P2: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B5: B] :
( P2
= ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_193_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C3 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_194_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_195_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_196_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_197_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: A > C > $o,Q: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F2 @ ( relcompp @ A @ C @ B @ P @ Q ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q ) ) ).
% csquare_fstOp_sndOp
thf(fact_198_usubstappp_Osimps_I4_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,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.simps(4)
thf(fact_199_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ P @ Q @ A3 @ C3 )
=> ( ( P @ A3 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A3 @ C3 ) )
& ( Q @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q @ A3 @ C3 ) @ C3 ) ) ) ).
% pick_middlep
thf(fact_200_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A4: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( F12 @ ( P1 @ X2 ) )
= ( F22 @ ( P22 @ X2 ) ) ) ) ) ) ).
% csquare_def
thf(fact_201_leq__OOI,axiom,
! [A: $tType,R2: A > A > $o] :
( ( R2
= ( ^ [Y2: A,Z2: A] : Y2 = Z2 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( relcompp @ A @ A @ A @ R2 @ R2 ) ) ) ).
% leq_OOI
thf(fact_202_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_203_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_204_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_205_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_206_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_207_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_208_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
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_209_usubstappp_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,Alpha: game,Beta: game] :
( ( uSubst95898988stappp @ Sigma @ U @ ( compose @ Alpha @ Beta ) )
= ( product_Pair @ ( set @ variable ) @ ( option @ game ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ Sigma @ U @ Alpha ) ) @ Beta ) ) @ ( uSubst1385204910mposeo @ ( product_snd @ ( 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 ) ) @ Beta ) ) ) ) ) ).
% usubstappp.simps(5)
thf(fact_210_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_211_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F: A > ( product_prod @ B @ C ),G: B > C > D,X2: A] : ( G @ ( product_fst @ B @ C @ ( F @ X2 ) ) @ ( product_snd @ B @ C @ ( F @ X2 ) ) ) ) ) ).
% scomp_unfold
thf(fact_212_sup__Un__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_213_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F2: A > ( product_prod @ E @ F3 ),G2: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F2 @ G2 ) @ H )
= ( product_scomp @ A @ E @ F3 @ B @ F2
@ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G2 @ X2 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_214_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_215_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F2: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F2 )
= ( F2 @ X ) ) ).
% Pair_scomp
thf(fact_216_refl__ge__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A] : ( R2 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R2 ) ) ).
% refl_ge_eq
thf(fact_217_ge__eq__refl,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R2 )
=> ( R2 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_218_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% subrelI
thf(fact_219_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_220_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_221_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R2 )
@ ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) )
= ( ^ [X2: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_222_pred__subset__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ord_less_eq @ ( set @ A ) @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_223_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_224_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_225_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S3: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_226_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X4 )
& ( P @ X2 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_227_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F2: B > A,X: B,C3: C,G2: B > C,A2: set @ B] :
( ( B3
= ( F2 @ X ) )
=> ( ( C3
= ( G2 @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_228_Loopo_Oelims,axiom,
! [X: option @ game,Y: option @ game] :
( ( ( uSubst993936602_Loopo @ X )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some @ game @ Alpha2 ) )
=> ( Y
!= ( some @ game @ ( loop @ Alpha2 ) ) ) )
=> ~ ( ( X
= ( none @ game ) )
=> ( Y
!= ( none @ game ) ) ) ) ) ).
% Loopo.elims
thf(fact_229_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_230_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y3: A] :
( X
!= ( some @ A @ Y3 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_231_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y3: A] :
( X
= ( some @ A @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_232_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_233_Composeo_Ocases,axiom,
! [X: product_prod @ ( option @ game ) @ ( option @ game )] :
( ! [Alpha2: game,Beta2: game] :
( X
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( some @ game @ Alpha2 ) @ ( some @ game @ Beta2 ) ) )
=> ( ! [Alpha2: option @ game] :
( X
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ Alpha2 @ ( none @ game ) ) )
=> ~ ! [V2: game] :
( X
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( none @ game ) @ ( some @ game @ V2 ) ) ) ) ) ).
% Composeo.cases
thf(fact_234_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_235_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X @ Y ) )
=> ( ! [A7: A,B7: B] :
( ( X
= ( some @ A @ A7 ) )
=> ( ( Y
= ( some @ B @ B7 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_236_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P5: ( option @ A ) > $o] :
! [X5: option @ A] : ( P5 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X2: A] : ( P4 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_237_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P5: ( option @ A ) > $o] :
? [X5: option @ A] : ( P5 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X2: A] : ( P4 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_238_option_Oinducts,axiom,
! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
( ( P @ ( none @ A ) )
=> ( ! [X3: A] : ( P @ ( some @ A @ X3 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_239_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_240_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_241_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_242_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_243_Composeo_Oinduct,axiom,
! [P: ( option @ game ) > ( option @ game ) > $o,A0: option @ game,A1: option @ game] :
( ! [Alpha2: game,Beta2: game] : ( P @ ( some @ game @ Alpha2 ) @ ( some @ game @ Beta2 ) )
=> ( ! [Alpha2: option @ game] : ( P @ Alpha2 @ ( none @ game ) )
=> ( ! [V2: game] : ( P @ ( none @ game ) @ ( some @ game @ V2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_244_undefg__equiv,axiom,
! [Alpha: option @ game] :
( ( Alpha
!= ( none @ game ) )
= ( ? [G: game] :
( Alpha
= ( some @ game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_245_Loopo_Oinduct,axiom,
! [P: ( option @ game ) > $o,A0: option @ game] :
( ! [Alpha2: game] : ( P @ ( some @ game @ Alpha2 ) )
=> ( ( P @ ( none @ game ) )
=> ( P @ A0 ) ) ) ).
% Loopo.induct
thf(fact_246_Loopo_Ocases,axiom,
! [X: option @ game] :
( ! [Alpha2: game] :
( X
!= ( some @ game @ Alpha2 ) )
=> ( X
= ( none @ game ) ) ) ).
% Loopo.cases
thf(fact_247_Loopo_Osimps_I1_J,axiom,
! [Alpha: game] :
( ( uSubst993936602_Loopo @ ( some @ game @ Alpha ) )
= ( some @ game @ ( loop @ Alpha ) ) ) ).
% Loopo.simps(1)
thf(fact_248_Choiceo_Osimps_I1_J,axiom,
! [Alpha: game,Beta: game] :
( ( uSubst1976112797hoiceo @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) )
= ( some @ game @ ( choice @ Alpha @ Beta ) ) ) ).
% Choiceo.simps(1)
thf(fact_249_Composeo_Osimps_I1_J,axiom,
! [Alpha: game,Beta: game] :
( ( uSubst1385204910mposeo @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) )
= ( some @ game @ ( compose @ Alpha @ Beta ) ) ) ).
% Composeo.simps(1)
thf(fact_250_Composeo_Osimps_I3_J,axiom,
! [V3: game] :
( ( uSubst1385204910mposeo @ ( none @ game ) @ ( some @ game @ V3 ) )
= ( none @ game ) ) ).
% Composeo.simps(3)
thf(fact_251_Choiceo_Osimps_I3_J,axiom,
! [V3: game] :
( ( uSubst1976112797hoiceo @ ( none @ game ) @ ( some @ game @ V3 ) )
= ( none @ game ) ) ).
% Choiceo.simps(3)
thf(fact_252_Composeo_Oelims,axiom,
! [X: option @ game,Xa: option @ game,Y: option @ game] :
( ( ( uSubst1385204910mposeo @ X @ Xa )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some @ game @ Alpha2 ) )
=> ! [Beta2: game] :
( ( Xa
= ( some @ game @ Beta2 ) )
=> ( Y
!= ( some @ game @ ( compose @ Alpha2 @ Beta2 ) ) ) ) )
=> ( ( ( Xa
= ( none @ game ) )
=> ( Y
!= ( none @ game ) ) )
=> ~ ( ( X
= ( none @ game ) )
=> ( ? [V2: game] :
( Xa
= ( some @ game @ V2 ) )
=> ( Y
!= ( none @ game ) ) ) ) ) ) ) ).
% Composeo.elims
thf(fact_253_Choiceo_Oelims,axiom,
! [X: option @ game,Xa: option @ game,Y: option @ game] :
( ( ( uSubst1976112797hoiceo @ X @ Xa )
= Y )
=> ( ! [Alpha2: game] :
( ( X
= ( some @ game @ Alpha2 ) )
=> ! [Beta2: game] :
( ( Xa
= ( some @ game @ Beta2 ) )
=> ( Y
!= ( some @ game @ ( choice @ Alpha2 @ Beta2 ) ) ) ) )
=> ( ( ( Xa
= ( none @ game ) )
=> ( Y
!= ( none @ game ) ) )
=> ~ ( ( X
= ( none @ game ) )
=> ( ? [V2: game] :
( Xa
= ( some @ game @ V2 ) )
=> ( Y
!= ( none @ game ) ) ) ) ) ) ) ).
% Choiceo.elims
thf(fact_254_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F23: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F23 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
& ~ ( P @ ( F23 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
% Type constructors (16)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( 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_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_3,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ ( set @ variable ) @ ( sup_sup @ ( set @ variable ) @ ua @ ( static_BVG @ ( the @ game @ ( product_snd @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ ( compose @ alpha @ beta ) ) ) ) ) ) @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ( product_fst @ ( set @ variable ) @ ( option @ game ) @ ( uSubst95898988stappp @ sigma @ ua @ alpha ) ) @ beta ) ) ).
%------------------------------------------------------------------------------