TPTP Problem File: DAT222^1.p
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%------------------------------------------------------------------------------
% File : DAT222^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Tllist 370
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : tllist__370.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 321 ( 156 unt; 58 typ; 0 def)
% Number of atoms : 565 ( 315 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4819 ( 42 ~; 3 |; 68 &;4466 @)
% ( 0 <=>; 240 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 562 ( 562 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 5 con; 0-7 aty)
% Number of variables : 1657 ( 255 ^;1304 !; 17 ?;1657 :)
% ( 81 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:51:42.139
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_TLList__Mirabelle__qhjoikztpd_Otllist,type,
tLList446370796tllist: $tType > $tType > $tType ).
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $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_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (50)
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
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_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $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_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( 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_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_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_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_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
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_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $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_TLList__Mirabelle__qhjoikztpd_Ocr__tllist,type,
tLList47617868tllist:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > ( tLList446370796tllist @ A @ B ) > $o ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Ollist__of__tllist,type,
tLList798109904tllist:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Oterminal0,type,
tLList1825092077minal0:
!>[A: $tType] : A ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Ocase__tllist,type,
tLList200813139tllist:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > ( tLList446370796tllist @ A @ B ) > C ) > ( tLList446370796tllist @ A @ B ) > C ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist_Oterminal,type,
tLList2110128105rminal:
!>[A: $tType,B: $tType] : ( ( tLList446370796tllist @ A @ B ) > B ) ).
thf(sy_c_TLList__Mirabelle__qhjoikztpd_Otllist__of__llist,type,
tLList1672613558_llist:
!>[B: $tType,A: $tType] : ( B > ( coinductive_llist @ A ) > ( tLList446370796tllist @ A @ B ) ) ).
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_b,type,
b2: b ).
thf(sy_v_fun,type,
fun: a > $o ).
thf(sy_v_prod1,type,
prod1: product_prod @ ( coinductive_llist @ a ) @ b ).
thf(sy_v_prod2,type,
prod2: product_prod @ ( coinductive_llist @ a ) @ b ).
%----Relevant facts (256)
thf(fact_0_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A2: A,B2: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= P )
=> ( C2 @ A2 @ B2 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X ) ) ).
% case_prodI2'
thf(fact_1_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( F @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_2_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
( ! [A2: A,B2: B] :
( ( P
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( C2 @ A2 @ B2 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_3_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_4_lfilter__K__True,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% lfilter_K_True
thf(fact_5_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( P2
& ( Q @ A4 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_6_lfilter__idem,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ P2 @ Xs ) )
= ( coinductive_lfilter @ A @ P2 @ Xs ) ) ).
% lfilter_idem
thf(fact_7_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
=> ( R @ A3 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_8_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ Z )
=> ~ ! [X2: A,Y: B] :
( ( P
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ~ ( C2 @ X2 @ Y @ Z ) ) ) ).
% case_prodE'
thf(fact_9_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_10_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A3 = A5 )
& ( B3 = B5 ) ) ) ).
% old.prod.inject
thf(fact_12_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A2: A,B2: B] :
( ( P
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( member @ C @ Z @ ( C2 @ A2 @ B2 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_13_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z @ ( C2 @ A3 @ B3 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_14_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P ) )
=> ~ ! [X2: B,Y: C] :
( ( P
= ( product_Pair @ B @ C @ X2 @ Y ) )
=> ~ ( member @ A @ Z @ ( C2 @ X2 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_15_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_16_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y3
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_17_prod__induct7,axiom,
! [G: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_18_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_19_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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,C3: C,D2: D,E2: E] : ( P2 @ ( 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 ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_20_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_21_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A2: A,B2: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_22_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2,G2: G] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F3 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_23_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E,F3: F2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_24_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D,E2: E] :
( Y3
!= ( 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 ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_25_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A2: A,B2: B,C3: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_26_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C3: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_27_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A3 = A5 )
=> ( B3 != B5 ) ) ) ).
% Pair_inject
thf(fact_28_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_29_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X2: A,Y: B] :
( P
= ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% surj_pair
thf(fact_30_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_31_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_32_lfilter__lfilter,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P2 @ ( coinductive_lfilter @ A @ Q @ Xs ) )
= ( coinductive_lfilter @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ( Q @ X3 ) )
@ Xs ) ) ).
% lfilter_lfilter
thf(fact_33_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_34_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P2 @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_35_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X2: A,Y: B] :
( ( F @ X2 @ Y )
= ( G3 @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_36_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_37_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
=> ~ ! [X2: B,Y: C] :
( ( Z
= ( product_Pair @ B @ C @ X2 @ Y ) )
=> ~ ( Q @ ( P2 @ X2 @ Y ) ) ) ) ).
% case_prodE2
thf(fact_38_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X3: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R2: C] : ( F4 @ L @ R2 @ Y4 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_39_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P )
=> ~ ! [X2: A,Y: B] :
( ( P
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ~ ( C2 @ X2 @ Y ) ) ) ).
% case_prodE
thf(fact_40_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_41_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_42_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
( ! [X2: A,Y: B] :
( ( ( product_Pair @ A @ B @ X2 @ Y )
= Q2 )
=> ( ( F @ X2 @ Y )
= ( G3 @ X2 @ Y ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_43_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R3: 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 )
@ ^ [X4: A,Y5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y4: B] :
( ( X4 = X3 )
& ( P3 @ X3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y5 @ Y4 ) @ ( R3 @ X3 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_44_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 )
@ ^ [A4: A,B4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B6: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A6 ) @ Ra )
| ( ( A4 = A6 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A7: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G3: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G3 @ X2 ) )
=> ( F = G3 ) ) ).
% 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_tllist_Oabs__eq__iff,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( coinductive_llist @ A ) @ B,Y3: product_prod @ ( coinductive_llist @ A ) @ B] :
( ( ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
@ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
@ X )
= ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
@ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
@ Y3 ) )
= ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
@ ^ [Xs2: coinductive_llist @ A,A4: B] :
( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
@ ^ [Ys: coinductive_llist @ A,B4: B] :
( ( Xs2 = Ys )
& ( ( coinductive_lfinite @ A @ Ys )
=> ( A4 = B4 ) ) ) )
@ X
@ Y3 ) ) ).
% tllist.abs_eq_iff
thf(fact_51_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R ) )
= ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_52_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C2 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_53_reflp__tllist,axiom,
! [B: $tType,A: $tType] :
( reflp @ ( product_prod @ ( coinductive_llist @ A ) @ B )
@ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
@ ^ [Xs2: coinductive_llist @ A,A4: B] :
( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
@ ^ [Ys: coinductive_llist @ A,B4: B] :
( ( Xs2 = Ys )
& ( ( coinductive_lfinite @ A @ Ys )
=> ( A4 = B4 ) ) ) ) ) ) ).
% reflp_tllist
thf(fact_54_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R4: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R4 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_55_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( X = X4 )
& ( Y3 = Y5 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% The_split_eq
thf(fact_56_tllist__of__llist__inject,axiom,
! [A: $tType,B: $tType,B3: B,Xs: coinductive_llist @ A,C2: B,Ys2: coinductive_llist @ A] :
( ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
= ( tLList1672613558_llist @ B @ A @ C2 @ Ys2 ) )
= ( ( Xs = Ys2 )
& ( ( coinductive_lfinite @ A @ Ys2 )
=> ( B3 = C2 ) ) ) ) ).
% tllist_of_llist_inject
thf(fact_57_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B5: B,R4: 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 @ A5 @ B5 ) ) @ ( lex_prod @ A @ B @ R4 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A5 ) @ R4 )
| ( ( A3 = A5 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B5 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_58_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y6: B,Y3: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y3 ) @ ( 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 @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_59_reflp__mono,axiom,
! [A: $tType,R: A > A > $o,Q: A > A > $o] :
( ( reflp @ A @ R )
=> ( ! [X2: A,Y: A] :
( ( R @ X2 @ Y )
=> ( Q @ X2 @ Y ) )
=> ( reflp @ A @ Q ) ) ) ).
% reflp_mono
thf(fact_60_reflp__def,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ^ [R2: A > A > $o] :
! [X3: A] : ( R2 @ X3 @ X3 ) ) ) ).
% reflp_def
thf(fact_61_theI__unique,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( ( P2 @ X )
= ( X
= ( the @ A @ P2 ) ) ) ) ).
% theI_unique
thf(fact_62_reflpI,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [X2: A] : ( R4 @ X2 @ X2 )
=> ( reflp @ A @ R4 ) ) ).
% reflpI
thf(fact_63_reflpE,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( reflp @ A @ R4 )
=> ( R4 @ X @ X ) ) ).
% reflpE
thf(fact_64_reflpD,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( reflp @ A @ R4 )
=> ( R4 @ X @ X ) ) ).
% reflpD
thf(fact_65_tllist_Oabs__induct,axiom,
! [B: $tType,A: $tType,P2: ( tLList446370796tllist @ A @ B ) > $o,X: tLList446370796tllist @ A @ B] :
( ! [Y: product_prod @ ( coinductive_llist @ A ) @ B] :
( P2
@ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
@ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 )
@ Y ) )
=> ( P2 @ X ) ) ).
% tllist.abs_induct
thf(fact_66_tllist__of__llist__cong,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,B3: B,B5: B] :
( ( Xs = Xs3 )
=> ( ( ( coinductive_lfinite @ A @ Xs3 )
=> ( B3 = B5 ) )
=> ( ( tLList1672613558_llist @ B @ A @ B3 @ Xs )
= ( tLList1672613558_llist @ B @ A @ B5 @ Xs3 ) ) ) ) ).
% tllist_of_llist_cong
thf(fact_67_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_prod_def
thf(fact_68_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y7: A,Z2: A] : Y7 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_69_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X3: A] : X3 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_70_the__equality,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ( P2 @ A3 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( X2 = A3 ) )
=> ( ( the @ A @ P2 )
= A3 ) ) ) ).
% the_equality
thf(fact_71_the1__equality,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( ( P2 @ A3 )
=> ( ( the @ A @ P2 )
= A3 ) ) ) ).
% the1_equality
thf(fact_72_the1I2,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ).
% the1I2
thf(fact_73_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X3: A,Y4: A] :
( the @ A
@ ^ [Z3: A] :
( ( P3
=> ( Z3 = X3 ) )
& ( ~ P3
=> ( Z3 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_74_theI2,axiom,
! [A: $tType,P2: A > $o,A3: A,Q: A > $o] :
( ( P2 @ A3 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( X2 = A3 ) )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% theI2
thf(fact_75_theI,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ( P2 @ A3 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( X2 = A3 ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% theI
thf(fact_76_theI_H,axiom,
! [A: $tType,P2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ).
% theI'
thf(fact_77_cr__tllist__def,axiom,
! [B: $tType,A: $tType] :
( ( tLList47617868tllist @ A @ B )
= ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( tLList446370796tllist @ A @ B ) > $o )
@ ^ [Xs2: coinductive_llist @ A,B4: B] :
( ^ [Y7: tLList446370796tllist @ A @ B,Z2: tLList446370796tllist @ A @ B] : Y7 = Z2
@ ( tLList1672613558_llist @ B @ A @ B4 @ Xs2 ) ) ) ) ).
% cr_tllist_def
thf(fact_78_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
thf(fact_79_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_unit_def
thf(fact_80_DEADID_Orel__reflp,axiom,
! [A: $tType] :
( reflp @ A
@ ^ [Y7: A,Z2: A] : Y7 = Z2 ) ).
% DEADID.rel_reflp
thf(fact_81_Nitpick_OThe__psimp,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ( P2
= ( ^ [Y7: A,Z2: A] : Y7 = Z2
@ X ) )
=> ( ( the @ A @ P2 )
= X ) ) ).
% Nitpick.The_psimp
thf(fact_82_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F22: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F22 @ X3 ) ) ) ) ).
% old.rec_bool_def
thf(fact_83_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_84_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $false )
= F23 ) ).
% old.bool.simps(6)
thf(fact_85_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y3: A,R4: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R4 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) @ R4 ) ) ).
% in_inv_image
thf(fact_86_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 ) ) ) )
@ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R3 @ F4 ) @ ( inv_image @ A @ B @ S4 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_87_Quotient__tllist,axiom,
! [B: $tType,A: $tType] :
( quotient @ ( product_prod @ ( coinductive_llist @ A ) @ B ) @ ( tLList446370796tllist @ A @ B )
@ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( ( product_prod @ ( coinductive_llist @ A ) @ B ) > $o )
@ ^ [Xs2: coinductive_llist @ A,A4: B] :
( product_case_prod @ ( coinductive_llist @ A ) @ B @ $o
@ ^ [Ys: coinductive_llist @ A,B4: B] :
( ( Xs2 = Ys )
& ( ( coinductive_lfinite @ A @ Ys )
=> ( A4 = B4 ) ) ) ) )
@ ( product_case_prod @ ( coinductive_llist @ A ) @ B @ ( tLList446370796tllist @ A @ B )
@ ^ [Xs2: coinductive_llist @ A,A4: B] : ( tLList1672613558_llist @ B @ A @ A4 @ Xs2 ) )
@ ^ [Ys: tLList446370796tllist @ A @ B] : ( product_Pair @ ( coinductive_llist @ A ) @ B @ ( tLList798109904tllist @ A @ B @ Ys ) @ ( tLList2110128105rminal @ A @ B @ Ys ) )
@ ( tLList47617868tllist @ A @ B ) ) ).
% Quotient_tllist
thf(fact_88_terminal__tllist__of__llist__lfinite,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,B3: B] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( tLList2110128105rminal @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
= B3 ) ) ).
% terminal_tllist_of_llist_lfinite
thf(fact_89_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R2: A > A > $o,F4: B > A,X3: B,Y4: B] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_90_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X3: C] : ( F @ X3 @ Y4 )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% case_swap
thf(fact_91_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A7: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A7 )
=> ( member @ C @ ( F @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A7 ) ) ) ).
% pair_imageI
thf(fact_92_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_93_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y3: A,X: B,A7: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A7 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A7 ) ) ).
% pair_in_swap_image
thf(fact_94_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y3: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% swap_simp
thf(fact_95_tllist__of__llist__inverse,axiom,
! [B: $tType,A: $tType,B3: B,Xs: coinductive_llist @ A] :
( ( tLList798109904tllist @ A @ B @ ( tLList1672613558_llist @ B @ A @ B3 @ Xs ) )
= Xs ) ).
% tllist_of_llist_inverse
thf(fact_96_llist__of__tllist__inverse,axiom,
! [B: $tType,A: $tType,B3: tLList446370796tllist @ A @ B] :
( ( tLList1672613558_llist @ B @ A @ ( tLList2110128105rminal @ A @ B @ B3 ) @ ( tLList798109904tllist @ A @ B @ B3 ) )
= B3 ) ).
% llist_of_tllist_inverse
thf(fact_97_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R2: B > B > $o,F4: A > B,X3: A,Y4: A] : ( R2 @ ( F4 @ X3 ) @ ( F4 @ Y4 ) ) ) ) ).
% inv_imagep_def
thf(fact_98_image__ident,axiom,
! [A: $tType,Y8: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : X3
@ Y8 )
= Y8 ) ).
% image_ident
thf(fact_99_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A7: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( member @ B @ X @ A7 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) ) ) ) ).
% image_eqI
thf(fact_100_Quotient__total__abs__eq__iff,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: A,Y3: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( reflp @ A @ R )
=> ( ( ( Abs @ X )
= ( Abs @ Y3 ) )
= ( R @ X @ Y3 ) ) ) ) ).
% Quotient_total_abs_eq_iff
thf(fact_101_Quotient__total__abs__induct,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( reflp @ A @ R )
=> ( ! [Y: A] : ( P2 @ ( Abs @ Y ) )
=> ( P2 @ X ) ) ) ) ).
% Quotient_total_abs_induct
thf(fact_102_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P2 )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_103_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A7: set @ A,B3: B,F: A > B] :
( ( member @ A @ X @ A7 )
=> ( ( B3
= ( F @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A7 ) ) ) ) ).
% rev_image_eqI
thf(fact_104_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A7 ) )
=> ( P2 @ X2 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A7 )
=> ( P2 @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_105_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_106_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A7 ) )
& ( P2 @ X5 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A7 )
& ( P2 @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_107_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A7: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A7 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A7 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_108_imageI,axiom,
! [B: $tType,A: $tType,X: A,A7: set @ A,F: A > B] :
( ( member @ A @ X @ A7 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A7 ) ) ) ).
% imageI
thf(fact_109_Quotient__rep__abs__fold__unmap,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A,Rep2: B > A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( X6
= ( Abs @ X ) )
=> ( ( R @ X @ X )
=> ( ( ( Rep @ X6 )
= ( Rep2 @ X6 ) )
=> ( R @ ( Rep2 @ X6 ) @ X ) ) ) ) ) ).
% Quotient_rep_abs_fold_unmap
thf(fact_110_Quotient__to__transfer,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,C2: A,C4: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ C2 @ C2 )
=> ( ( C4
= ( Abs @ C2 ) )
=> ( T2 @ C2 @ C4 ) ) ) ) ).
% Quotient_to_transfer
thf(fact_111_Quotient__abs__induct,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,P2: B > $o,X: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ! [Y: A] :
( ( R @ Y @ Y )
=> ( P2 @ ( Abs @ Y ) ) )
=> ( P2 @ X ) ) ) ).
% Quotient_abs_induct
thf(fact_112_Quotient__rep__reflp,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% Quotient_rep_reflp
thf(fact_113_Quotient__rel__abs2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X: B,Y3: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ ( Rep @ X ) @ Y3 )
=> ( X
= ( Abs @ Y3 ) ) ) ) ).
% Quotient_rel_abs2
thf(fact_114_Quotient__alt__def3,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T3 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
& ! [X3: A,Y4: A] :
( ( R3 @ X3 @ Y4 )
= ( ? [Z3: B] :
( ( T3 @ X3 @ Z3 )
& ( T3 @ Y4 @ Z3 ) ) ) ) ) ) ) ).
% Quotient_alt_def3
thf(fact_115_Quotient__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T3 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
& ! [X3: A,Y4: A] :
( ( R3 @ X3 @ Y4 )
= ( ( T3 @ X3 @ ( Abs2 @ Y4 ) )
& ( T3 @ Y4 @ ( Abs2 @ X3 ) ) ) ) ) ) ) ).
% Quotient_alt_def2
thf(fact_116_Quotient__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ R4 @ R4 )
=> ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% Quotient_rep_abs
thf(fact_117_Quotient__rel__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B,B3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
= ( A3 = B3 ) ) ) ).
% Quotient_rel_rep
thf(fact_118_Quotient__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ R4 @ S2 )
=> ( ( Abs @ R4 )
= ( Abs @ S2 ) ) ) ) ).
% Quotient_rel_abs
thf(fact_119_Quotient__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T3 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T3 @ ( Rep3 @ B4 ) @ B4 )
& ! [X3: A,Y4: A] :
( ( R3 @ X3 @ Y4 )
= ( ( T3 @ X3 @ ( Abs2 @ X3 ) )
& ( T3 @ Y4 @ ( Abs2 @ Y4 ) )
& ( ( Abs2 @ X3 )
= ( Abs2 @ Y4 ) ) ) ) ) ) ) ).
% Quotient_alt_def
thf(fact_120_Quotient__abs__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,A3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( Abs @ ( Rep @ A3 ) )
= A3 ) ) ).
% Quotient_abs_rep
thf(fact_121_Quotient__Rep__eq,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,X6: B,X: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( X6
= ( Abs @ X ) )
=> ( ( Rep @ X6 )
= ( Rep @ X6 ) ) ) ) ).
% Quotient_Rep_eq
thf(fact_122_Quotient__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ R4 @ S2 )
=> ( R @ S2 @ S2 ) ) ) ).
% Quotient_refl2
thf(fact_123_Quotient__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ R4 @ S2 )
=> ( R @ R4 @ R4 ) ) ) ).
% Quotient_refl1
thf(fact_124_Quotient__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,R4: A,S2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( ( R @ R4 @ R4 )
& ( R @ S2 @ S2 )
& ( ( Abs @ R4 )
= ( Abs @ S2 ) ) )
= ( R @ R4 @ S2 ) ) ) ).
% Quotient_rel
thf(fact_125_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A7: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A7 ) )
& ( P2 @ X3 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A7 )
& ( P2 @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_126_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G3: C > B,A7: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G3 @ A7 ) )
= ( image @ C @ A
@ ^ [X3: C] : ( F @ ( G3 @ X3 ) )
@ A7 ) ) ).
% image_image
thf(fact_127_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A7: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ A7 ) )
=> ~ ! [X2: B] :
( ( B3
= ( F @ X2 ) )
=> ~ ( member @ B @ X2 @ A7 ) ) ) ).
% imageE
thf(fact_128_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( T2
= ( ^ [X3: A,Y4: B] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y4 ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_129_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep3: B > A,T3: A > B > $o] :
( ! [A4: B] :
( ( Abs2 @ ( Rep3 @ A4 ) )
= A4 )
& ! [A4: B] : ( R3 @ ( Rep3 @ A4 ) @ ( Rep3 @ A4 ) )
& ! [R2: A,S5: A] :
( ( R3 @ R2 @ S5 )
= ( ( R3 @ R2 @ R2 )
& ( R3 @ S5 @ S5 )
& ( ( Abs2 @ R2 )
= ( Abs2 @ S5 ) ) ) )
& ( T3
= ( ^ [X3: A,Y4: B] :
( ( R3 @ X3 @ X3 )
& ( ( Abs2 @ X3 )
= Y4 ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_130_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R: B > B > $o,T2: B > A > $o] :
( ! [A2: A] :
( ( Abs @ ( Rep @ A2 ) )
= A2 )
=> ( ! [A2: A] : ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) )
=> ( ! [R5: B,S6: B] :
( ( R @ R5 @ S6 )
= ( ( R @ R5 @ R5 )
& ( R @ S6 @ S6 )
& ( ( Abs @ R5 )
= ( Abs @ S6 ) ) ) )
=> ( ( T2
= ( ^ [X3: B,Y4: A] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y4 ) ) ) )
=> ( quotient @ B @ A @ R @ Abs @ Rep @ T2 ) ) ) ) ) ).
% QuotientI
thf(fact_131_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A7: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A7 ) )
= ( Sup @ A7 ) ) ).
% Sup.SUP_identity_eq
thf(fact_132_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A7: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A7 ) )
= ( Inf @ A7 ) ) ).
% Inf.INF_identity_eq
thf(fact_133_terminal0__terminal,axiom,
! [B: $tType,A: $tType] :
( ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) )
= ( tLList2110128105rminal @ A @ B ) ) ).
% terminal0_terminal
thf(fact_134_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_135_swap__product,axiom,
! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
@ ( product_Sigma @ B @ A @ A7
@ ^ [Uu: B] : B7 ) )
= ( product_Sigma @ A @ B @ B7
@ ^ [Uu: A] : A7 ) ) ).
% swap_product
thf(fact_136_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
= ( ( member @ A @ A3 @ A7 )
& ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_137_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A7: set @ A,B3: B,B7: A > ( set @ B )] :
( ( member @ A @ A3 @ A7 )
=> ( ( member @ B @ B3 @ ( B7 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) ) ) ) ).
% SigmaI
thf(fact_138_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( ( P2 @ A4 )
& ( Q @ B4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_139_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_140_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_141_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A7: set @ A,B7: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ! [Y: B] :
( ( member @ B @ Y @ ( B7 @ X2 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X2 @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_142_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
=> ( member @ A @ A3 @ A7 ) ) ).
% SigmaD1
thf(fact_143_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
=> ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ).
% SigmaD2
thf(fact_144_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,A7: set @ A,B7: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) )
=> ~ ( ( member @ A @ A3 @ A7 )
=> ~ ( member @ B @ B3 @ ( B7 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_145_Sigma__cong,axiom,
! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A7 = B7 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( C5 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( product_Sigma @ A @ B @ A7 @ C5 )
= ( product_Sigma @ A @ B @ B7 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_146_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A7
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B7
@ ^ [Uu: B] : C5 ) )
= ( A7 = B7 ) ) ) ).
% Times_eq_cancel2
thf(fact_147_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
( ( B3
= ( F @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_148_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_149_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y4: B] :
( ( P2 @ X3 )
& ( Q @ X3 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_150_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G3: B > C] :
( ( image @ B @ A
@ ^ [X3: B] : ( F @ ( G3 @ X3 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_151_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X2: B] :
( B3
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_152_product__swap,axiom,
! [B: $tType,A: $tType,A7: set @ B,B7: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A7
@ ^ [Uu: B] : B7 ) )
= ( product_Sigma @ A @ B @ B7
@ ^ [Uu: A] : A7 ) ) ).
% product_swap
thf(fact_153_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A7 = B7 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B7 )
=> ( ( C5 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C5 @ A7 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_154_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A7: set @ B,B7: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A7 = B7 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ B7 )
=> ( ( C5 @ X2 )
= ( D3 @ X2 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C5 @ A7 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B7 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_155_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y3
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_156_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y3
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_157_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F: A > B] :
( ! [X2: A] :
( ( G3 @ ( F @ X2 ) )
= X2 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_158_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X3: B] :
( Y4
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_159_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A7 @ B7 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 ) ) ) ).
% member_product
thf(fact_160_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_161_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G3: C > D] :
( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_162_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [Y4: B] : Y4 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_163_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A3: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
= ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G3 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_164_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R: set @ ( product_prod @ A @ B ),F: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_165_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G3: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G3 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G3 @ R2 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_166_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 ) @ R ) )
=> ~ ! [X2: C,Y: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G3 @ Y ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_167_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X3: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X3 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_168_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A7: set @ B,A9: set @ A,G3: D > C,B7: set @ D,B9: set @ C] :
( ( ( image @ B @ A @ F @ A7 )
= A9 )
=> ( ( ( image @ D @ C @ G3 @ B7 )
= B9 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G3 )
@ ( product_Sigma @ B @ D @ A7
@ ^ [Uu: B] : B7 ) )
= ( product_Sigma @ A @ C @ A9
@ ^ [Uu: A] : B9 ) ) ) ) ).
% map_prod_surj_on
thf(fact_169_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3
@ T4 )
= T4 ) ).
% prod.map_ident
thf(fact_170_terminal__def,axiom,
! [B: $tType,A: $tType] :
( ( tLList2110128105rminal @ A @ B )
= ( tLList200813139tllist @ B @ B @ A
@ ^ [X12: B] : X12
@ ^ [X_dummy: A] : ( tLList1825092077minal0 @ ( ( tLList446370796tllist @ A @ B ) > B ) ) ) ) ).
% terminal_def
thf(fact_171_reflp__refl__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X3: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R4 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R4 ) ) ).
% reflp_refl_eq
thf(fact_172_refl__onD2,axiom,
! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( refl_on @ A @ A7 @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
=> ( member @ A @ Y3 @ A7 ) ) ) ).
% refl_onD2
thf(fact_173_refl__onD1,axiom,
! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( refl_on @ A @ A7 @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R4 )
=> ( member @ A @ X @ A7 ) ) ) ).
% refl_onD1
thf(fact_174_refl__onD,axiom,
! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A7 @ R4 )
=> ( ( member @ A @ A3 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R4 ) ) ) ).
% refl_onD
thf(fact_175_tllist_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F1: B > C,F23: A > ( tLList446370796tllist @ A @ B ) > C,Tllist: tLList446370796tllist @ A @ B] :
( ( H @ ( tLList200813139tllist @ B @ C @ A @ F1 @ F23 @ Tllist ) )
= ( tLList200813139tllist @ B @ D @ A
@ ^ [X3: B] : ( H @ ( F1 @ X3 ) )
@ ^ [X12: A,X23: tLList446370796tllist @ A @ B] : ( H @ ( F23 @ X12 @ X23 ) )
@ Tllist ) ) ).
% tllist.case_distrib
thf(fact_176_refl__on__domain,axiom,
! [A: $tType,A7: set @ A,R4: set @ ( product_prod @ A @ A ),A3: A,B3: A] :
( ( refl_on @ A @ A7 @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R4 )
=> ( ( member @ A @ A3 @ A7 )
& ( member @ A @ B3 @ A7 ) ) ) ) ).
% refl_on_domain
thf(fact_177_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_178_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_179_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_180_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_181_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_182_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_183_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_184_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y7: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y7 = Z2 )
= ( ^ [S5: product_prod @ A @ B,T5: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S5 )
= ( product_fst @ A @ B @ T5 ) )
& ( ( product_snd @ A @ B @ S5 )
= ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_185_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_186_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_187_split__fst,axiom,
! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
( ( R @ ( product_fst @ A @ B @ P ) )
= ( ! [X3: A,Y4: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
=> ( R @ X3 ) ) ) ) ).
% split_fst
thf(fact_188_split__fst__asm,axiom,
! [B: $tType,A: $tType,R: A > $o,P: product_prod @ A @ B] :
( ( R @ ( product_fst @ A @ B @ P ) )
= ( ~ ? [X3: A,Y4: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
& ~ ( R @ X3 ) ) ) ) ).
% split_fst_asm
thf(fact_189_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y3: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_190_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_191_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y3: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= A3 )
=> ( Y3 = A3 ) ) ).
% snd_eqD
thf(fact_192_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_193_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_194_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_195_surjective__pairing,axiom,
! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
( T4
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% surjective_pairing
thf(fact_196_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P4: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_197_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_198_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A7: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) )
=> ( A7 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_199_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G3: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G3 @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X3: D] : ( F @ ( G3 @ X3 ) ) ) ) ).
% split_comp_eq
thf(fact_200_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% case_prod_beta'
thf(fact_201_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C6: A > B > C,P4: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P4 ) @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% case_prod_unfold
thf(fact_202_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_203_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_204_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X23: B] : X12 ) ) ).
% fst_def
thf(fact_205_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_206_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_207_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A7: set @ A,B7: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A7 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B7 ) ) ) ).
% mem_Times_iff
thf(fact_208_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ( ! [X2: B,Y: A] :
( ( P2 @ Y @ X2 )
=> ( Q @ ( F @ X2 @ Y ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_209_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X: B] :
( ( P2 @ Y3 @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_210_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_211_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y3: B,A3: product_prod @ A @ B] :
( ( P2 @ X @ Y3 )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y3 ) )
=> ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_212_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ).
% above_def
thf(fact_213_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),A4: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) ) ) ) ).
% aboveS_def
thf(fact_214_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_215_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y3 ) ) ).
% fstI
thf(fact_216_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P ) )
= ( ? [A4: B] :
( P
= ( product_Pair @ B @ A @ A4 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_217_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P ) )
= ( ? [B4: B] :
( P
= ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_218_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A7: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7
@ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 )
@ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) ) ) ).
% subset_fst_snd
thf(fact_219_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X3 ) ) @ ( product_snd @ B @ C @ ( F4 @ X3 ) ) ) ) ) ).
% scomp_unfold
thf(fact_220_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X3 ) ) ) ) ).
% scomp_apply
thf(fact_221_subset__Collect__iff,axiom,
! [A: $tType,B7: set @ A,A7: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A7 )
& ( P2 @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B7 )
=> ( P2 @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_222_subset__CollectI,axiom,
! [A: $tType,B7: set @ A,A7: set @ A,Q: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B7 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A7 )
& ( P2 @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_223_Sigma__mono,axiom,
! [B: $tType,A: $tType,A7: set @ A,C5: set @ A,B7: A > ( set @ B ),D3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ C5 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ord_less_eq @ ( set @ B ) @ ( B7 @ X2 ) @ ( D3 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A7 @ B7 ) @ ( product_Sigma @ A @ B @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_224_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A7: set @ B,B7: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A7
@ ^ [Uu: B] : C5 )
@ ( product_Sigma @ B @ A @ B7
@ ^ [Uu: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A7 @ B7 ) ) ) ).
% Times_subset_cancel2
thf(fact_225_subrelI,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S2 ) ) ).
% subrelI
thf(fact_226_subset__image__iff,axiom,
! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A7 )
& ( B7
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_227_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A7: set @ B,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A7 ) @ B7 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A7 )
=> ( member @ A @ ( F @ X3 ) @ B7 ) ) ) ) ).
% image_subset_iff
thf(fact_228_subset__imageE,axiom,
! [A: $tType,B: $tType,B7: set @ A,F: B > A,A7: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A7 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A7 )
=> ( B7
!= ( image @ B @ A @ F @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_229_image__subsetI,axiom,
! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( member @ B @ ( F @ X2 ) @ B7 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% image_subsetI
thf(fact_230_image__mono,axiom,
! [B: $tType,A: $tType,A7: set @ A,B7: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ ( image @ A @ B @ F @ B7 ) ) ) ).
% image_mono
thf(fact_231_conj__subset__def,axiom,
! [A: $tType,A7: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A7
@ ( collect @ A
@ ^ [X3: A] :
( ( P2 @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_232_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( P2 @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_233_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X7 )
& ( P2 @ X3 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_234_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G3: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G3 ) @ H )
= ( product_scomp @ A @ E @ F2 @ B @ F
@ ^ [X3: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G3 @ X3 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_235_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type @ A ) )
=> ! [P2: A > A > $o,B3: A,A3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P2 @ A2 @ B2 ) )
=> ( ( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_236_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_237_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: A > B,B7: set @ B] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( member @ B @ ( F @ X2 ) @ B7 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B7 ) ) ).
% image_Collect_subsetI
thf(fact_238_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X3 ) ) ) ) ).
% scomp_def
thf(fact_239_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
= ( F @ X ) ) ).
% Pair_scomp
thf(fact_240_refl__onI,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu: A] : A7 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R4 ) )
=> ( refl_on @ A @ A7 @ R4 ) ) ) ).
% refl_onI
thf(fact_241_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R2 ) ) ) ) ) ).
% refl_on_def
thf(fact_242_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X7: set @ A,A7: set @ ( product_prod @ A @ B ),Y8: set @ B,P2: A > B > $o,Q: A > B > $o] :
( ( X7
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A7 ) )
=> ( ( Y8
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A7 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X7 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y8 )
=> ( ( P2 @ X2 @ Xa )
=> ( Q @ X2 @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_243_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),X: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 )
@ S )
=> ( ( member @ A @ X @ A7 )
=> ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ) ).
% subset_snd_imageI
thf(fact_244_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A7: set @ A,B7: set @ B,S: set @ ( product_prod @ A @ B ),Y3: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 )
@ S )
=> ( ( member @ B @ Y3 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A7 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ) ).
% subset_fst_imageI
thf(fact_245_Quotient__rep__abs__eq,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T2: A > B > $o,T4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T2 )
=> ( ( R @ T4 @ T4 )
=> ( ( ord_less_eq @ ( A > A > $o ) @ R
@ ^ [Y7: A,Z2: A] : Y7 = Z2 )
=> ( ( Rep @ ( Abs @ T4 ) )
= T4 ) ) ) ) ).
% Quotient_rep_abs_eq
thf(fact_246_reflp__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ( reflp @ A @ R )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y7: A,Z2: A] : Y7 = Z2
@ R ) ) ).
% reflp_ge_eq
thf(fact_247_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A7: A > B > $o,B7: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A7 @ B7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A7 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B7 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_248_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_249_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R )
@ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_250_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A8 )
@ ^ [X3: A] : ( member @ A @ X3 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_251_reflp__eq,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ord_less_eq @ ( A > A > $o )
@ ^ [Y7: A,Z2: A] : Y7 = Z2 ) ) ).
% reflp_eq
thf(fact_252_Gr__incl,axiom,
! [A: $tType,B: $tType,A7: set @ A,F: A > B,B7: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A7 @ F )
@ ( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B7 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A7 ) @ B7 ) ) ).
% Gr_incl
thf(fact_253_flip__pred,axiom,
! [A: $tType,B: $tType,A7: set @ ( product_prod @ A @ B ),R: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X3: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X3 ) )
@ A7 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% flip_pred
thf(fact_254_conversep__eq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [Y7: A,Z2: A] : Y7 = Z2 )
= ( ^ [Y7: A,Z2: A] : Y7 = Z2 ) ) ).
% conversep_eq
thf(fact_255_conversep__iff,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R2: A > B > $o,A4: B,B4: A] : ( R2 @ B4 @ A4 ) ) ) ).
% conversep_iff
%----Type constructors (2)
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type @ $o ) ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_1,axiom,
linorder @ product_unit @ ( type @ product_unit ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $true @ X @ Y3 )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
@ ^ [Xs2: coinductive_llist @ a,A4: b] :
( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
@ ^ [Ys: coinductive_llist @ a,B4: b] :
( ( Xs2 = Ys )
& ( ( coinductive_lfinite @ a @ Ys )
=> ( A4 = B4 ) ) ) )
@ prod1
@ prod2 ) ).
thf(conj_1,conjecture,
( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( ( product_prod @ ( coinductive_llist @ a ) @ b ) > $o )
@ ^ [Xs2: coinductive_llist @ a,A4: b] :
( product_case_prod @ ( coinductive_llist @ a ) @ b @ $o
@ ^ [Ys: coinductive_llist @ a,B4: b] :
( ( Xs2 = Ys )
& ( ( coinductive_lfinite @ a @ Ys )
=> ( A4 = B4 ) ) ) )
@ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
@ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
@ prod1 )
@ ( product_case_prod @ ( coinductive_llist @ a ) @ b @ ( product_prod @ ( coinductive_llist @ a ) @ b )
@ ^ [Xs2: coinductive_llist @ a,B6: b] : ( product_Pair @ ( coinductive_llist @ a ) @ b @ ( coinductive_lfilter @ a @ fun @ Xs2 ) @ ( if @ b @ ( coinductive_lfinite @ a @ Xs2 ) @ B6 @ b2 ) )
@ prod2 ) ) ).
%------------------------------------------------------------------------------