TPTP Problem File: ITP198^2.p
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%------------------------------------------------------------------------------
% File : ITP198^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_1023__6346468_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_1023__6346468_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 312 ( 159 unt; 52 typ; 0 def)
% Number of atoms : 534 ( 320 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 4525 ( 58 ~; 9 |; 48 &;4206 @)
% ( 0 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 547 ( 547 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 6 con; 0-7 aty)
% Number of variables : 1467 ( 191 ^;1196 !; 16 ?;1467 :)
% ( 64 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:47.305
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Denotational__Semantics_Ointerp,type,
denotational_interp: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Syntax_Ovariable,type,
variable: $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (45)
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Denotational__Semantics_OConsts,type,
denotational_Consts: denotational_interp > char > real ).
thf(sy_c_Denotational__Semantics_OFuncs,type,
denotational_Funcs: denotational_interp > char > real > real ).
thf(sy_c_Denotational__Semantics_OGames,type,
denotational_Games: denotational_interp > char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ).
thf(sy_c_Denotational__Semantics_OPreds,type,
denotational_Preds: denotational_interp > char > real > $o ).
thf(sy_c_Denotational__Semantics_Omkinterp,type,
denota1150374853interp: ( product_prod @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) ) ) > denotational_interp ).
thf(sy_c_Denotational__Semantics_Oworlds,type,
denotational_worlds: set @ ( variable > real ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OPair__Rep,type,
product_Pair_Rep:
!>[A: $tType,B: $tType] : ( A > B > A > B > $o ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
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_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
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_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_OAbs__prod,type,
product_Abs_prod:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Orepc,type,
uSubst761942615e_repc: denotational_interp > char > real > denotational_interp ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I,type,
i: denotational_interp ).
thf(sy_v_d,type,
d: real ).
thf(sy_v_f,type,
f: char ).
% Relevant facts (256)
thf(fact_0_Preds__mkinterp,axiom,
! [C2: char > real,F: char > real > real,P: char > real > $o,G: char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) )] :
( ( denotational_Preds @ ( denota1150374853interp @ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) ) @ C2 @ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ F @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ P @ G ) ) ) ) )
= P ) ).
% Preds_mkinterp
thf(fact_1_Funcs__mkinterp,axiom,
! [C2: char > real,F: char > real > real,P: char > real > $o,G: char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) )] :
( ( denotational_Funcs @ ( denota1150374853interp @ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) ) @ C2 @ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ F @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ P @ G ) ) ) ) )
= F ) ).
% Funcs_mkinterp
thf(fact_2_Consts__mkinterp,axiom,
! [C2: char > real,F: char > real > real,P: char > real > $o,G: char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) )] :
( ( denotational_Consts @ ( denota1150374853interp @ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) ) @ C2 @ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ F @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ P @ G ) ) ) ) )
= C2 ) ).
% Consts_mkinterp
thf(fact_3_mkinterp__eq,axiom,
! [I: denotational_interp,J: denotational_interp] :
( ( ( ( denotational_Consts @ I )
= ( denotational_Consts @ J ) )
& ( ( denotational_Funcs @ I )
= ( denotational_Funcs @ J ) )
& ( ( denotational_Preds @ I )
= ( denotational_Preds @ J ) )
& ( ( denotational_Games @ I )
= ( denotational_Games @ J ) ) )
= ( I = J ) ) ).
% mkinterp_eq
thf(fact_4_repc__def,axiom,
( uSubst761942615e_repc
= ( ^ [I2: denotational_interp,F2: char,D2: real] :
( denota1150374853interp
@ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) )
@ ^ [C3: char] : ( if @ real @ ( C3 = F2 ) @ D2 @ ( denotational_Consts @ I2 @ C3 ) )
@ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ ( denotational_Funcs @ I2 ) @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ ( denotational_Preds @ I2 ) @ ( denotational_Games @ I2 ) ) ) ) ) ) ) ).
% repc_def
thf(fact_5_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A2: A,B2: B,C4: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_6_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 ) ) )] :
~ ! [A2: A,B2: B,C4: C,D3: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_7_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 ) ) ) )] :
~ ! [A2: A,B2: B,C4: C,D3: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D3 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_8_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
~ ! [A2: A,B2: B,C4: C,D3: D,E2: E,F4: F3,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_9_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 ) )] :
( ! [A2: A,B2: B,C4: C,D3: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C4 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_10_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 ) ) )] :
( ! [A2: A,B2: B,C4: C,D3: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_11_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 ) ) ) )] :
( ! [A2: A,B2: B,C4: C,D3: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D3 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_12_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_13_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_repc__consts,axiom,
! [C5: char,F5: char,I: denotational_interp,D4: real] :
( ( ( C5 = F5 )
=> ( ( denotational_Consts @ ( uSubst761942615e_repc @ I @ F5 @ D4 ) @ C5 )
= D4 ) )
& ( ( C5 != F5 )
=> ( ( denotational_Consts @ ( uSubst761942615e_repc @ I @ F5 @ D4 ) @ C5 )
= ( denotational_Consts @ I @ C5 ) ) ) ) ).
% repc_consts
thf(fact_15_repc__funcs,axiom,
! [I: denotational_interp,F5: char,D4: real] :
( ( denotational_Funcs @ ( uSubst761942615e_repc @ I @ F5 @ D4 ) )
= ( denotational_Funcs @ I ) ) ).
% repc_funcs
thf(fact_16_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_17_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_18_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_19_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_20_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_21_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 )] :
( ! [A2: A,B2: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_22_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C4 ) ) ) ).
% prod_cases3
thf(fact_23_prod__induct7,axiom,
! [G2: $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 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A2: A,B2: B,C4: C,D3: D,E2: E,F4: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_24_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_25_Games__mkinterp,axiom,
! [G: char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ),C2: char > real,F: char > real > real,P: char > real > $o] :
( ! [A2: char] : ( order_mono @ ( set @ ( variable > real ) ) @ ( set @ ( variable > real ) ) @ ( G @ A2 ) )
=> ( ( denotational_Games @ ( denota1150374853interp @ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) ) @ C2 @ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ F @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ P @ G ) ) ) ) )
= G ) ) ).
% Games_mkinterp
thf(fact_26_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_27_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C5: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C5 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C5 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_28_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_29_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F2: ( product_prod @ B @ C ) > A,A5: B,B5: C] : ( F2 @ ( product_Pair @ B @ C @ A5 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_30_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C3: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_31_curryI,axiom,
! [A: $tType,B: $tType,F5: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( F5 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F5 @ A3 @ B3 ) ) ).
% curryI
thf(fact_32_worlds__def,axiom,
( denotational_worlds
= ( collect @ ( variable > real )
@ ^ [Nu: variable > real] : $true ) ) ).
% worlds_def
thf(fact_33_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_34_Pair__Rep__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair_Rep @ A @ B )
= ( ^ [A5: A,B5: B,X4: A,Y4: B] :
( ( X4 = A5 )
& ( Y4 = B5 ) ) ) ) ).
% Pair_Rep_def
thf(fact_35_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_36_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C5: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C5 )
= ( ^ [X4: A,Y4: B] : C5 ) ) ).
% curry_K
thf(fact_37_curryD,axiom,
! [A: $tType,B: $tType,F5: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F5 @ A3 @ B3 )
=> ( F5 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryD
thf(fact_38_curryE,axiom,
! [A: $tType,B: $tType,F5: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F5 @ A3 @ B3 )
=> ( F5 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% curryE
thf(fact_39_Pair__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair @ A @ B )
= ( ^ [A5: A,B5: B] : ( product_Abs_prod @ A @ B @ ( product_Pair_Rep @ A @ B @ A5 @ B5 ) ) ) ) ).
% Pair_def
thf(fact_40_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F2: B > A,X4: B,Y4: B] : ( R3 @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_41_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_42_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ B @ B ),F5: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R2 @ F5 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F5 @ X ) @ ( F5 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_43_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R3: B > B > $o,F2: A > B,X4: A,Y4: A] : ( R3 @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) ) ) ).
% inv_imagep_def
thf(fact_44_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_prod_def
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,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% 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,F5: A > B,G4: A > B] :
( ! [X3: A] :
( ( F5 @ X3 )
= ( G4 @ X3 ) )
=> ( F5 = G4 ) ) ).
% ext
thf(fact_49_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_50_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F5: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X4: C] : ( F5 @ X4 @ Y4 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F5 @ P2 ) ) ).
% case_swap
thf(fact_51_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_52_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y5: B] :
( ( X = X5 )
& ( Y = Y5 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_53_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C5: A > B > $o] :
( ! [A2: A,B2: B] :
( ( P2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( C5 @ A2 @ B2 ) )
=> ( product_case_prod @ A @ B @ $o @ C5 @ P2 ) ) ).
% case_prodI2
thf(fact_54_case__prodI,axiom,
! [A: $tType,B: $tType,F5: A > B > $o,A3: A,B3: B] :
( ( F5 @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F5 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_55_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C5: A > B > ( set @ C )] :
( ! [A2: A,B2: B] :
( ( P2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( member @ C @ Z @ ( C5 @ A2 @ B2 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C5 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_56_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C5: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z @ ( C5 @ A3 @ B3 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C5 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_57_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C5: A > B > C > $o,X: C] :
( ! [A2: A,B2: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= P2 )
=> ( C5 @ A2 @ B2 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C5 @ P2 @ X ) ) ).
% case_prodI2'
thf(fact_58_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_59_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F5: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F5 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F5 @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_60_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F5: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F5 ) )
= F5 ) ).
% curry_case_prod
thf(fact_61_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F5: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F5 ) )
= F5 ) ).
% case_prod_curry
thf(fact_62_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A6: set @ ( product_prod @ A @ B ),F5: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A6 )
=> ( member @ C @ ( F5 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F5 ) @ A6 ) ) ) ).
% pair_imageI
thf(fact_63_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F5: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F5 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X22: B] : ( H @ ( F5 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_64_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) @ R3 ) ) ) ) ) ).
% inv_image_def
thf(fact_65_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C5: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C5 @ P2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C5 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_66_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F5: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F5 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F5 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_67_case__prodE,axiom,
! [A: $tType,B: $tType,C5: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C5 @ P2 )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C5 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_68_case__prodD,axiom,
! [A: $tType,B: $tType,F5: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F5 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F5 @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_69_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C5: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C5 @ P2 @ Z )
=> ~ ! [X3: A,Y3: B] :
( ( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C5 @ X3 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_70_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C5: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C5 )
=> ( R @ A3 @ B3 @ C5 ) ) ).
% case_prodD'
thf(fact_71_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F5: A > B > C,G4: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F5 @ X3 @ Y3 )
= ( G4 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F5 )
= G4 ) ) ).
% cond_case_prod_eta
thf(fact_72_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F5: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( F5 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
= F5 ) ).
% case_prod_eta
thf(fact_73_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_74_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% old.rec_bool_def
thf(fact_75_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_76_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_77_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_78_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_79_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F5: B > A,X: B,A6: set @ B] :
( ( B3
= ( F5 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F5 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_80_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F5: A > B > C,G4: A > B > C,P2: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F5 @ X3 @ Y3 )
= ( G4 @ X3 @ Y3 ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F5 @ P2 )
= ( product_case_prod @ A @ B @ C @ G4 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_81_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_82_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] :
( P
& ( Q @ A5 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_83_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_84_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F5: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B3
= ( F5 @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F5 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_85_ball__imageD,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F5 @ A6 ) )
=> ( P @ X3 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A6 )
=> ( P @ ( F5 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_86_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F5: A > B,G4: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F5 @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image @ A @ B @ F5 @ M )
= ( image @ A @ B @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_87_bex__imageD,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F5 @ A6 ) )
& ( P @ X6 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( P @ ( F5 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_88_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F5: B > A,A6: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F5 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z
= ( F5 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_89_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F5: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F5 @ X ) @ ( image @ A @ B @ F5 @ A6 ) ) ) ).
% imageI
thf(fact_90_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F5: B > A,A6: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F5 @ A6 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F5 @ X3 ) )
=> ~ ( member @ B @ X3 @ A6 ) ) ) ).
% imageE
thf(fact_91_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F5: B > A,G4: C > B,A6: set @ C] :
( ( image @ B @ A @ F5 @ ( image @ C @ B @ G4 @ A6 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F5 @ ( G4 @ X4 ) )
@ A6 ) ) ).
% image_image
thf(fact_92_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F5 @ A6 ) )
& ( P @ X4 ) ) )
= ( image @ B @ A @ F5
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P @ ( F5 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_93_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_94_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_95_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_96_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X4: A,Y4: A] :
( the @ A
@ ^ [Z3: A] :
( ( P3
=> ( Z3 = X4 ) )
& ( ~ P3
=> ( Z3 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_97_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_98_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R4: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F2 ) @ ( inv_image @ A @ B @ S4 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_99_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R3: C] : ( F2 @ L @ R3 @ Y4 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_100_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Inf @ A6 ) ) ).
% Inf.INF_identity_eq
thf(fact_101_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Sup @ A6 ) ) ).
% Sup.SUP_identity_eq
thf(fact_102_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_103_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_104_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C2: B > A,D5: B > A,Inf: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C2 @ X3 )
= ( D5 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A6 ) )
= ( Inf @ ( image @ B @ A @ D5 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_105_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C2: B > A,D5: B > A,Sup: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ( C2 @ X3 )
= ( D5 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A6 ) )
= ( Sup @ ( image @ B @ A @ D5 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_106_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F5: B > A] :
( ( fun_reduction_pair @ A @ P )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F5 ) ) ) ).
% rp_inv_image_rp
thf(fact_107_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F5: C > A,G4: D > B,A6: set @ C,B6: set @ D] :
( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F5 @ X4 ) @ ( G4 @ Y4 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image @ C @ A @ F5 @ A6 )
@ ^ [Uu: A] : ( image @ D @ B @ G4 @ B6 ) ) ) ).
% image_paired_Times
thf(fact_108_Nitpick_OThe__psimp,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P
= ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X ) )
=> ( ( the @ A @ P )
= X ) ) ).
% Nitpick.The_psimp
thf(fact_109_theI__unique,axiom,
! [A: $tType,P: A > $o,X: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ X )
= ( X
= ( the @ A @ P ) ) ) ) ).
% theI_unique
thf(fact_110_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_111_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_112_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B3: B,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ B @ B3 @ ( B6 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_113_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( ( member @ A @ A3 @ A6 )
& ( member @ B @ B3 @ ( B6 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_114_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] :
( ( P @ A5 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_115_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_116_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ( ( member @ A @ A3 @ A6 )
=> ~ ( member @ B @ B3 @ ( B6 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_117_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ B @ B3 @ ( B6 @ A3 ) ) ) ).
% SigmaD2
thf(fact_118_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ A @ A3 @ A6 ) ) ).
% SigmaD1
thf(fact_119_SigmaE,axiom,
! [A: $tType,B: $tType,C5: product_prod @ A @ B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C5 @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B6 @ X3 ) )
=> ( C5
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_120_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C2: A > ( set @ B ),D5: A > ( set @ B )] :
( ( A6 = B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ( C2 @ X3 )
= ( D5 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A6 @ C2 )
= ( product_Sigma @ A @ B @ B6 @ D5 ) ) ) ) ).
% Sigma_cong
thf(fact_121_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C2: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X @ C2 )
=> ( ( ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C2 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C2 ) )
= ( A6 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_122_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_123_UNIV__eq__I,axiom,
! [A: $tType,A6: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A6 )
=> ( ( top_top @ ( set @ A ) )
= A6 ) ) ).
% UNIV_eq_I
thf(fact_124_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_125_rangeI,axiom,
! [A: $tType,B: $tType,F5: B > A,X: B] : ( member @ A @ ( F5 @ X ) @ ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_126_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F5: B > A,X: B] :
( ( B3
= ( F5 @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_127_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( P @ X4 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_128_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F5: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F5 @ X3 ) ) ) ).
% rangeE
thf(fact_129_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F5: C > A,G4: B > C] :
( ( image @ B @ A
@ ^ [X4: B] : ( F5 @ ( G4 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F5 @ ( image @ B @ C @ G4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_130_product__swap,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A6 ) ) ).
% product_swap
thf(fact_131_swap__product,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I3: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I3 ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A6 ) ) ).
% swap_product
thf(fact_132_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X5: A,Y5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( X5 = X4 )
& ( P3 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y5 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_133_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_134_surj__def,axiom,
! [B: $tType,A: $tType,F5: B > A] :
( ( ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F5 @ X4 ) ) ) ) ).
% surj_def
thf(fact_135_surjI,axiom,
! [B: $tType,A: $tType,G4: B > A,F5: A > B] :
( ! [X3: A] :
( ( G4 @ ( F5 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_136_surjE,axiom,
! [A: $tType,B: $tType,F5: B > A,Y: A] :
( ( ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F5 @ X3 ) ) ) ).
% surjE
thf(fact_137_surjD,axiom,
! [A: $tType,B: $tType,F5: B > A,Y: A] :
( ( ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F5 @ X3 ) ) ) ).
% surjD
thf(fact_138_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A5: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B7: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A7 ) @ Ra )
| ( ( A5 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_139_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_140_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A6 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) ) ) ).
% member_product
thf(fact_141_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F5: A > B,G4: C > D] :
( ( ( image @ A @ B @ F5 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G4 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F5 @ G4 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_142_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y4: B] : Y4 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_143_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F5: C > A,G4: D > B,A3: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F5 @ G4 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
= ( product_Pair @ A @ B @ ( F5 @ A3 ) @ ( G4 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_144_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),F5: A > C,G4: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F5 @ A3 ) @ ( G4 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F5 @ G4 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_145_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R2 )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_146_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F5: D > B,G4: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F5 @ G4 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R3: E] : ( H @ ( F5 @ L ) @ ( G4 @ R3 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_147_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C5: product_prod @ A @ B,F5: C > A,G4: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C5 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F5 @ G4 ) @ R ) )
=> ~ ! [X3: C,Y3: D] :
( ( C5
= ( product_Pair @ A @ B @ ( F5 @ X3 ) @ ( G4 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_148_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G5: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y4: B] : ( product_Pair @ C @ D @ ( F2 @ X4 ) @ ( G5 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_149_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F5: B > A,A6: set @ B,A9: set @ A,G4: D > C,B6: set @ D,B9: set @ C] :
( ( ( image @ B @ A @ F5 @ A6 )
= A9 )
=> ( ( ( image @ D @ C @ G4 @ B6 )
= B9 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F5 @ G4 )
@ ( product_Sigma @ B @ D @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ C @ A9
@ ^ [Uu: A] : B9 ) ) ) ) ).
% map_prod_surj_on
thf(fact_150_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_151_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_152_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_153_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_154_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_155_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_156_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_157_empty__iff,axiom,
! [A: $tType,C5: A] :
~ ( member @ A @ C5 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_158_image__is__empty,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B] :
( ( ( image @ B @ A @ F5 @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_159_empty__is__image,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F5 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_160_image__empty,axiom,
! [B: $tType,A: $tType,F5: B > A] :
( ( image @ B @ A @ F5 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_161_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F5: C > A,G4: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F5 @ G4 @ X ) )
= ( F5 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_162_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F5: C > B,G4: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F5 @ G4 @ X ) )
= ( G4 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_163_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S5: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_164_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_165_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_166_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_167_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_168_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_169_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_170_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A6: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= A6 ) ) ) ).
% fst_image_times
thf(fact_171_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_172_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I: set @ A,X7: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I @ X7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I )
=> ( ( X7 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_173_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A6 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_174_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_175_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_176_equals0D,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A6 ) ) ).
% equals0D
thf(fact_177_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_178_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_179_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y6: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y6 = Z2 )
= ( ^ [S5: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S5 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S5 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_180_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_181_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_182_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X22: B] : X12 ) ) ).
% fst_def
thf(fact_183_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_184_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X22: B] : X22 ) ) ).
% snd_def
thf(fact_185_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: A > ( set @ B )] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_186_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F5: A > B > C,G4: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F5 @ ( G4 @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F5 @ ( G4 @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_187_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X4: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_188_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C3: A > B > C,P4: product_prod @ A @ B] : ( C3 @ ( product_fst @ A @ B @ P4 ) @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% case_prod_unfold
thf(fact_189_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_190_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_191_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P4: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_192_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F5: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F5 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F5 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_193_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F5: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F5 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F5 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_194_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_195_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_196_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_197_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_198_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_199_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_200_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_201_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_202_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_203_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_204_times__eq__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,C2: set @ A,D5: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
= ( product_Sigma @ A @ B @ C2
@ ^ [Uu: A] : D5 ) )
= ( ( ( A6 = C2 )
& ( B6 = D5 ) )
| ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C2
= ( bot_bot @ ( set @ A ) ) )
| ( D5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_205_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A,Q: C > $o,F5: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ( ! [X3: B,Y3: A] :
( ( P @ Y3 @ X3 )
=> ( Q @ ( F5 @ X3 @ Y3 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F5 @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A6: set @ A,F5: A > B,X: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( F5 @ Y3 )
= ( F5 @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F5 @ A6 ) )
= ( F5 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_214_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
@ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
@ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% subset_fst_snd
thf(fact_215_subsetI,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% subsetI
thf(fact_216_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( A6 = B6 ) ) ) ).
% subset_antisym
thf(fact_217_subset__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_218_empty__subsetI,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% empty_subsetI
thf(fact_219_subset__UNIV,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_220_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_221_mon__mono,axiom,
! [B: $tType,A: $tType] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) )
= ( ^ [R3: ( set @ A ) > ( set @ B )] :
! [X8: set @ A,Y9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ Y9 )
=> ( ord_less_eq @ ( set @ B ) @ ( R3 @ X8 ) @ ( R3 @ Y9 ) ) ) ) ) ).
% mon_mono
thf(fact_222_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_223_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_224_Collect__subset,axiom,
! [A: $tType,A6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) )
@ A6 ) ).
% Collect_subset
thf(fact_225_in__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_226_subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C5: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C5 @ A6 )
=> ( member @ A @ C5 @ B6 ) ) ) ).
% subsetD
thf(fact_227_equalityE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% equalityE
thf(fact_228_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ A @ X4 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_229_equalityD1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% equalityD1
thf(fact_230_equalityD2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% equalityD2
thf(fact_231_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A8 )
=> ( member @ A @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_232_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_233_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_234_subset__trans,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C2 ) ) ) ).
% subset_trans
thf(fact_235_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z2: set @ A] : Y6 = Z2 )
= ( ^ [A8: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_236_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_237_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,F5: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F5 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B6
= ( image @ B @ A @ F5 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_238_image__subset__iff,axiom,
! [A: $tType,B: $tType,F5: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F5 @ A6 ) @ B6 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F5 @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_239_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set @ A,F5: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F5 @ A6 ) )
=> ~ ! [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A6 )
=> ( B6
!= ( image @ B @ A @ F5 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_240_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F5: A > B,B6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F5 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A6 ) @ B6 ) ) ).
% image_subsetI
thf(fact_241_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F5: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A6 ) @ ( image @ A @ B @ F5 @ B6 ) ) ) ).
% image_mono
thf(fact_242_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C2: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X @ C2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C2 )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C2 ) )
= ( ord_less_eq @ ( set @ B ) @ A6 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_243_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,C2: set @ A,B6: A > ( set @ B ),D5: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X3 ) @ ( D5 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) @ ( product_Sigma @ A @ B @ C2 @ D5 ) ) ) ) ).
% Sigma_mono
thf(fact_244_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: A > B,B6: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F5 @ X3 ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ ( collect @ A @ P ) ) @ B6 ) ) ).
% image_Collect_subsetI
thf(fact_245_subset__Collect__iff,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_246_subset__CollectI,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B6 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_247_subset__emptyI,axiom,
! [A: $tType,A6: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_248_less__by__empty,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( A6
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B6 ) ) ).
% less_by_empty
thf(fact_249_times__subset__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,C2: set @ B,B6: set @ A,D5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C2 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : D5 ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( C2
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ B ) @ C2 @ D5 ) ) ) ) ).
% times_subset_iff
thf(fact_250_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X7: set @ A,A6: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X7
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y7
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y7 )
=> ( ( P @ X3 @ Xa )
=> ( Q @ X3 @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_251_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,S: set @ ( product_prod @ A @ B ),X: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
@ S )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ) ).
% subset_snd_imageI
thf(fact_252_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,S: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
@ S )
=> ( ( member @ B @ Y @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ) ).
% subset_fst_imageI
thf(fact_253_range__subsetD,axiom,
! [B: $tType,A: $tType,F5: B > A,B6: set @ A,I4: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F5 @ ( top_top @ ( set @ B ) ) ) @ B6 )
=> ( member @ A @ ( F5 @ I4 ) @ B6 ) ) ).
% range_subsetD
thf(fact_254_Gr__incl,axiom,
! [A: $tType,B: $tType,A6: set @ A,F5: A > B,B6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F5 )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F5 @ A6 ) @ B6 ) ) ).
% Gr_incl
thf(fact_255_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A6: set @ A,F5: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F5 ) )
=> ( member @ A @ X @ A6 ) ) ).
% GrD1
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( denotational_Preds
@ ( denota1150374853interp
@ ( product_Pair @ ( char > real ) @ ( product_prod @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) )
@ ^ [C3: char] : ( if @ real @ ( C3 = f ) @ d @ ( denotational_Consts @ i @ C3 ) )
@ ( product_Pair @ ( char > real > real ) @ ( product_prod @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) ) @ ( denotational_Funcs @ i ) @ ( product_Pair @ ( char > real > $o ) @ ( char > ( set @ ( variable > real ) ) > ( set @ ( variable > real ) ) ) @ ( denotational_Preds @ i ) @ ( denotational_Games @ i ) ) ) ) ) )
= ( denotational_Preds @ i ) ) ).
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