TPTP Problem File: DAT145^1.p
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%------------------------------------------------------------------------------
% File : DAT145^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 82
% 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 : coinductive_stream__82.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 324 ( 200 unt; 64 typ; 0 def)
% Number of atoms : 439 ( 322 equ; 0 cnn)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 5130 ( 44 ~; 3 |; 38 &;4913 @)
% ( 0 <=>; 132 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 598 ( 598 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 59 usr; 4 con; 0-9 aty)
% Number of variables : 1692 ( 200 ^;1359 !; 24 ?;1692 :)
% ( 109 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:13:02.283
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
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_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (55)
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
thf(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr1317331620lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr2107612801vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
coindu139217191stream:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ 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_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_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_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_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Stream_Osdrop,type,
sdrop:
!>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Osmerge,type,
smerge:
!>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osnth,type,
snth:
!>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
thf(sy_c_Stream_Osproduct,type,
sproduct:
!>[A: $tType,B: $tType] : ( ( stream @ A ) > ( stream @ B ) > ( stream @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocorec__stream,type,
corec_stream:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Osmap,type,
smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
thf(sy_c_Stream_Oszip,type,
szip:
!>[A: $tType,B: $tType] : ( ( stream @ A ) > ( stream @ B ) > ( stream @ ( product_prod @ 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_g,type,
g: c > b ).
thf(sy_v_xs,type,
xs: stream @ a ).
thf(sy_v_ys,type,
ys: stream @ c ).
%----Relevant facts (256)
thf(fact_0_szip__smap1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: stream @ C,Ys: stream @ B] :
( ( szip @ A @ B @ ( smap @ C @ A @ F @ Xs ) @ Ys )
= ( smap @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ ( product_apfst @ C @ A @ B @ F ) @ ( szip @ C @ B @ Xs @ Ys ) ) ) ).
% szip_smap1
thf(fact_1_smap__eq__SCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: stream @ B,Y: A,Ys: stream @ A] :
( ( ( smap @ B @ A @ F @ Xs )
= ( sCons @ A @ Y @ Ys ) )
= ( ? [X: B,Xs2: stream @ B] :
( ( Xs
= ( sCons @ B @ X @ Xs2 ) )
& ( Y
= ( F @ X ) )
& ( Ys
= ( smap @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% smap_eq_SCons_conv
thf(fact_2_smap2__szip,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( smap2 @ B @ C @ A )
= ( ^ [F2: B > C > A,S1: stream @ B,S2: stream @ C] : ( smap @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ F2 ) @ ( szip @ B @ C @ S1 @ S2 ) ) ) ) ).
% smap2_szip
thf(fact_3_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_4_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X2: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X2 @ Y ) )
= ( product_Pair @ A @ B @ X2 @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_5_sdrop__smap,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,S: stream @ B] :
( ( sdrop @ A @ N @ ( smap @ B @ A @ F @ S ) )
= ( smap @ B @ A @ F @ ( sdrop @ B @ N @ S ) ) ) ).
% sdrop_smap
thf(fact_6_smap__szip__fst,axiom,
! [C: $tType,A: $tType,B: $tType,F: B > A,S12: stream @ B,S22: stream @ C] :
( ( smap @ ( product_prod @ B @ C ) @ A
@ ^ [X: product_prod @ B @ C] : ( F @ ( product_fst @ B @ C @ X ) )
@ ( szip @ B @ C @ S12 @ S22 ) )
= ( smap @ B @ A @ F @ S12 ) ) ).
% smap_szip_fst
thf(fact_7_smap__szip__snd,axiom,
! [B: $tType,A: $tType,C: $tType,G: C > A,S12: stream @ B,S22: stream @ C] :
( ( smap @ ( product_prod @ B @ C ) @ A
@ ^ [X: product_prod @ B @ C] : ( G @ ( product_snd @ B @ C @ X ) )
@ ( szip @ B @ C @ S12 @ S22 ) )
= ( smap @ C @ A @ G @ S22 ) ) ).
% smap_szip_snd
thf(fact_8_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X2 ) )
= ( product_fst @ A @ C @ X2 ) ) ).
% fst_apsnd
thf(fact_9_snth__smap,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream @ B,N: nat] :
( ( snth @ A @ ( smap @ B @ A @ F @ S ) @ N )
= ( F @ ( snth @ B @ S @ N ) ) ) ).
% snth_smap
thf(fact_10_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X2: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_snd @ B @ C @ X2 ) ) ) ).
% snd_apsnd
thf(fact_11_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X2 )
= ( product_apsnd @ C @ B @ A @ G @ X2 ) )
= ( ( F @ ( product_snd @ A @ C @ X2 ) )
= ( G @ ( product_snd @ A @ C @ X2 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_12_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_13_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_14_stream_Oinject,axiom,
! [A: $tType,X1: A,X22: stream @ A,Y1: A,Y2: stream @ A] :
( ( ( sCons @ A @ X1 @ X22 )
= ( sCons @ A @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% stream.inject
thf(fact_15_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( F @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_16_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C2: A > B > $o] :
( ! [A4: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P ) ) ).
% case_prodI2
thf(fact_17_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= P )
=> ( C2 @ A4 @ B4 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P @ X2 ) ) ).
% case_prodI2'
thf(fact_18_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_19_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A4: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( member @ C @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_20_stream_Omap__ident,axiom,
! [A: $tType,T2: stream @ A] :
( ( smap @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% stream.map_ident
thf(fact_21_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X2: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X2 @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X2 ) @ Y ) ) ).
% apfst_conv
thf(fact_22_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_fst @ C @ B @ X2 ) ) ) ).
% fst_apfst
thf(fact_23_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X2 )
= ( product_apfst @ C @ A @ B @ G @ X2 ) )
= ( ( F @ ( product_fst @ C @ B @ X2 ) )
= ( G @ ( product_fst @ C @ B @ X2 ) ) ) ) ).
% apfst_eq_conv
thf(fact_24_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X2: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X2 ) )
= ( product_snd @ C @ A @ X2 ) ) ).
% snd_apfst
thf(fact_25_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_26_snth__smap2,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,S12: stream @ B,S22: stream @ C,N: nat] :
( ( snth @ A @ ( smap2 @ B @ C @ A @ F @ S12 @ S22 ) @ N )
= ( F @ ( snth @ B @ S12 @ N ) @ ( snth @ C @ S22 @ N ) ) ) ).
% snth_smap2
thf(fact_27_sdrop__szip,axiom,
! [A: $tType,B: $tType,N: nat,S12: stream @ A,S22: stream @ B] :
( ( sdrop @ ( product_prod @ A @ B ) @ N @ ( szip @ A @ B @ S12 @ S22 ) )
= ( szip @ A @ B @ ( sdrop @ A @ N @ S12 ) @ ( sdrop @ B @ N @ S22 ) ) ) ).
% sdrop_szip
thf(fact_28_sdrop__smap2,axiom,
! [B: $tType,A: $tType,C: $tType,N: nat,F: B > C > A,S12: stream @ B,S22: stream @ C] :
( ( sdrop @ A @ N @ ( smap2 @ B @ C @ A @ F @ S12 @ S22 ) )
= ( smap2 @ B @ C @ A @ F @ ( sdrop @ B @ N @ S12 ) @ ( sdrop @ C @ N @ S22 ) ) ) ).
% sdrop_smap2
thf(fact_29_smap__smap2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: B > A,G: C > D > B,S12: stream @ C,S22: stream @ D] :
( ( smap @ B @ A @ F @ ( smap2 @ C @ D @ B @ G @ S12 @ S22 ) )
= ( smap2 @ C @ D @ A
@ ^ [X: C,Y3: D] : ( F @ ( G @ X @ Y3 ) )
@ S12
@ S22 ) ) ).
% smap_smap2
thf(fact_30_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_31_snth__szip,axiom,
! [A: $tType,B: $tType,S12: stream @ A,S22: stream @ B,N: nat] :
( ( snth @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S12 @ S22 ) @ N )
= ( product_Pair @ A @ B @ ( snth @ A @ S12 @ N ) @ ( snth @ B @ S22 @ N ) ) ) ).
% snth_szip
thf(fact_32_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X2: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X2 ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X2 ) ) @ ( G @ ( product_snd @ C @ D @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_33_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X2 ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X2 ) ) @ ( F @ ( product_snd @ D @ C @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_34_smap2__alt,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > C > A,S12: stream @ B,S22: stream @ C,S: stream @ A] :
( ( ( smap2 @ B @ C @ A @ F @ S12 @ S22 )
= S )
= ( ! [N2: nat] :
( ( F @ ( snth @ B @ S12 @ N2 ) @ ( snth @ C @ S22 @ N2 ) )
= ( snth @ A @ S @ N2 ) ) ) ) ).
% smap2_alt
thf(fact_35_szip__unfold,axiom,
! [A: $tType,B: $tType,A2: A,S12: stream @ A,B2: B,S22: stream @ B] :
( ( szip @ A @ B @ ( sCons @ A @ A2 @ S12 ) @ ( sCons @ B @ B2 @ S22 ) )
= ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( szip @ A @ B @ S12 @ S22 ) ) ) ).
% szip_unfold
thf(fact_36_smap2__unfold,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,S12: stream @ B,B2: C,S22: stream @ C] :
( ( smap2 @ B @ C @ A @ F @ ( sCons @ B @ A2 @ S12 ) @ ( sCons @ C @ B2 @ S22 ) )
= ( sCons @ A @ ( F @ A2 @ B2 ) @ ( smap2 @ B @ C @ A @ F @ S12 @ S22 ) ) ) ).
% smap2_unfold
thf(fact_37_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_38_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_39_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_40_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_41_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_42_stream_Oexhaust,axiom,
! [A: $tType,Y: stream @ A] :
~ ! [X12: A,X23: stream @ A] :
( Y
!= ( sCons @ A @ X12 @ X23 ) ) ).
% stream.exhaust
thf(fact_43_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y4: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_44_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P )
=> ~ ! [X3: A,Y4: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
=> ~ ( C2 @ X3 @ Y4 ) ) ) ).
% case_prodE
thf(fact_50_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_51_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_52_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_53_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 )
=> ~ ! [X3: A,Y4: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y4 ) )
=> ~ ( C2 @ X3 @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_54_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
=> ~ ! [X3: B,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X3 @ Y4 ) )
=> ~ ( Q2 @ ( P2 @ X3 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_55_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_56_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_57_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_58_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 ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_59_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 ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_60_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 ) ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: 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 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_61_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y5 = Z2 )
= ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_62_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X13: A,X24: B] : X13 ) ) ).
% fst_def
thf(fact_63_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X13: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_64_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct3
thf(fact_65_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct4
thf(fact_66_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,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct5
thf(fact_67_prod__induct6,axiom,
! [F3: $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 @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E,F4: F3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct6
thf(fact_68_prod__induct7,axiom,
! [G2: $tType,F3: $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 @ F3 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E,F4: F3,G3: G2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct7
thf(fact_69_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X: A,Y3: B] : ( F @ ( product_Pair @ A @ B @ X @ Y3 ) ) )
= F ) ).
% case_prod_eta
thf(fact_70_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_71_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X: D] : ( F @ ( G @ X ) ) ) ) ).
% split_comp_eq
thf(fact_72_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P3: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_73_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 ) )
=> ~ ! [X3: B,Y4: C] :
( ( P
= ( product_Pair @ B @ C @ X3 @ Y4 ) )
=> ~ ( member @ A @ Z @ ( C2 @ X3 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_74_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_75_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ).
% case_prod_beta'
thf(fact_76_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_77_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P3: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_78_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_79_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_80_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_81_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
@ ^ [X13: A,X24: B] : ( H @ ( F @ X13 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_82_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_83_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y4: B] :
( ( F @ X3 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_84_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_85_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_86_stream_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X1: A,X22: stream @ A] :
( ( smap @ A @ B @ F @ ( sCons @ A @ X1 @ X22 ) )
= ( sCons @ B @ ( F @ X1 ) @ ( smap @ A @ B @ F @ X22 ) ) ) ).
% stream.map
thf(fact_87_smap__alt,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream @ B,S4: stream @ A] :
( ( ( smap @ B @ A @ F @ S )
= S4 )
= ( ! [N2: nat] :
( ( F @ ( snth @ B @ S @ N2 ) )
= ( snth @ A @ S4 @ N2 ) ) ) ) ).
% smap_alt
thf(fact_88_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q2: C > $o,F: B > A > C] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ( ! [X3: B,Y4: A] :
( ( P2 @ Y4 @ X3 )
=> ( Q2 @ ( F @ X3 @ Y4 ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_89_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_90_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X2: B] :
( ( P2 @ Y @ X2 )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_91_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q2: B > $o,Q: B] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_92_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X2 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_93_smember__code,axiom,
! [A: $tType,X2: A,Y: A,S: stream @ A] :
( ( smember @ A @ X2 @ ( sCons @ A @ Y @ S ) )
= ( ( X2 != Y )
=> ( smember @ A @ X2 @ S ) ) ) ).
% smember_code
thf(fact_94_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q: product_prod @ A @ B,F: C > A,P: product_prod @ C @ B] :
( ( Q
= ( product_apfst @ C @ A @ B @ F @ P ) )
=> ~ ! [X3: C,Y4: B] :
( ( P
= ( product_Pair @ C @ B @ X3 @ Y4 ) )
=> ( Q
!= ( product_Pair @ A @ B @ ( F @ X3 ) @ Y4 ) ) ) ) ).
% apfst_convE
thf(fact_95_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ~ ! [X3: B,Y4: A] :
~ ( P2 @ Y4 @ X3 ) ) ).
% exE_realizer'
thf(fact_96_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z ) ) ).
% sndI
thf(fact_97_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A6: B] :
( P
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_98_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_99_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( P2
& ( Q2 @ A6 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_100_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_101_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B5: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_102_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R2: C] : ( F2 @ L @ R2 @ Y3 )
@ X ) ) ) ).
% case_prod_app
thf(fact_103_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P: product_prod @ A @ B] :
( ! [X3: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y4 )
= Q )
=> ( ( F @ X3 @ Y4 )
= ( G @ X3 @ Y4 ) ) )
=> ( ( P = Q )
=> ( ( product_case_prod @ A @ B @ C @ F @ P )
= ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_104_sproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( sproduct @ A @ B )
= ( ^ [S1: stream @ A,S2: stream @ B] :
( smerge @ ( product_prod @ A @ B )
@ ( smap @ A @ ( stream @ ( product_prod @ A @ B ) )
@ ^ [X: A] : ( smap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ S2 )
@ S1 ) ) ) ) ).
% sproduct_def
thf(fact_105_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G4: B > C > D,X: A] : ( G4 @ ( product_fst @ B @ C @ ( F2 @ X ) ) @ ( product_snd @ B @ C @ ( F2 @ X ) ) ) ) ) ).
% scomp_unfold
thf(fact_106_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_107_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_108_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F2: B > ( product_prod @ C @ D ),G4: C > D > A,X: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F2 @ X ) ) ) ) ).
% scomp_apply
thf(fact_109_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_110_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y3: B,X: C] : ( F @ X @ Y3 )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% case_swap
thf(fact_111_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_112_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_113_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F3 @ B @ F
@ ^ [X: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X ) @ H ) ) ) ).
% scomp_scomp
thf(fact_114_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C ) )
= X2 ) ).
% scomp_Pair
thf(fact_115_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X2 ) @ F )
= ( F @ X2 ) ) ).
% Pair_scomp
thf(fact_116_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G4: B > C > D,X: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F2 @ X ) ) ) ) ).
% scomp_def
thf(fact_117_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_118_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_119_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R ) )
= ( ^ [X: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_120_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_121_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_122_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_123_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y6: B] :
( ( X2 = X4 )
& ( Y = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% The_split_eq
thf(fact_124_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ).
% old.rec_prod_def
thf(fact_125_the__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( the @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_126_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X: A] : X = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_127_the__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the_equality
thf(fact_128_theI,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% theI
thf(fact_129_theI_H,axiom,
! [A: $tType,P2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ).
% theI'
thf(fact_130_theI2,axiom,
! [A: $tType,P2: A > $o,A2: A,Q2: A > $o] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( the @ A @ P2 ) ) ) ) ) ).
% theI2
thf(fact_131_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X: A,Y3: A] :
( the @ A
@ ^ [Z3: A] :
( ( P4
=> ( Z3 = X ) )
& ( ~ P4
=> ( Z3 = Y3 ) ) ) ) ) ) ).
% If_def
thf(fact_132_the1I2,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( the @ A @ P2 ) ) ) ) ).
% the1I2
thf(fact_133_the1__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ( P2 @ A2 )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the1_equality
thf(fact_134_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_135_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y3 ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
thf(fact_136_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_137_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_138_in__inv__image,axiom,
! [A: $tType,B: $tType,X2: A,Y: A,R3: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( inv_image @ B @ A @ R3 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_139_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_140_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_141_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_142_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_143_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X2: C,F5: D > A,G5: E > D,X6: E] :
( ( ( F @ ( G @ X2 ) )
= ( F5 @ ( G5 @ X6 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ E @ F5 @ G5 @ X6 ) ) ) ).
% comp_cong
thf(fact_144_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L2: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L2 )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L2 @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_145_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R3 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_146_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L22: A > D,H: E > A,R3: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L22 ) )
=> ( ( ( comp @ A @ D @ E @ L22 @ H )
= R3 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_147_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L2: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L2 )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L2 @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_148_stream_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: stream @ A] :
( ( smap @ B @ C @ G @ ( smap @ A @ B @ F @ V ) )
= ( smap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% stream.map_comp
thf(fact_149_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X2: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_150_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q2: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_151_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X2: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X2 ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X2 ) ) ).
% apsnd_compose
thf(fact_152_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X2: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X2 ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X2 ) ) ).
% apfst_compose
thf(fact_153_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q2: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_154_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 ),S6: 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 @ S6 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_155_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( H
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H )
= F ) ) ) ).
% convol_expand_snd'
thf(fact_156_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= F ) ) ).
% convol_expand_snd
thf(fact_157_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z: product_prod @ A @ B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ F ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z )
= ( product_snd @ A @ B @ Z ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_158_GrpI,axiom,
! [B: $tType,A: $tType,F: B > A,X2: B,Y: A,A5: set @ B] :
( ( ( F @ X2 )
= Y )
=> ( ( member @ B @ X2 @ A5 )
=> ( bNF_Grp @ B @ A @ A5 @ F @ X2 @ Y ) ) ) ).
% GrpI
thf(fact_159_GrpE,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,X2: A,Y: B] :
( ( bNF_Grp @ A @ B @ A5 @ F @ X2 @ Y )
=> ~ ( ( ( F @ X2 )
= Y )
=> ~ ( member @ A @ X2 @ A5 ) ) ) ).
% GrpE
thf(fact_160_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A7: set @ A,F2: A > B,A6: A,B5: B] :
( ( B5
= ( F2 @ A6 ) )
& ( member @ A @ A6 @ A7 ) ) ) ) ).
% Grp_def
thf(fact_161_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F2: A > B,G4: A > C,A6: A] : ( product_Pair @ B @ C @ ( F2 @ A6 ) @ ( G4 @ A6 ) ) ) ) ).
% convol_def
thf(fact_162_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% convol_o
thf(fact_163_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X2 ) )
= ( F @ X2 ) ) ).
% fst_convol'
thf(fact_164_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X2: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X2 ) )
= ( G @ X2 ) ) ).
% snd_convol'
thf(fact_165_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z: product_prod @ A @ B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ F ) ) ) )
=> ( member @ A @ ( product_fst @ A @ B @ Z ) @ A5 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_166_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X2: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X2 )
= ( F @ ( G @ ( product_fst @ B @ C @ X2 ) ) @ ( product_snd @ B @ C @ X2 ) ) ) ).
% case_prod_comp
thf(fact_167_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
= F ) ).
% fst_convol
thf(fact_168_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
= G ) ).
% snd_convol
thf(fact_169_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_170_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_171_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_172_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_173_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_174_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
( ( comp @ C @ B @ A
@ ^ [X: C] : C2
@ F )
= ( ^ [X: A] : C2 ) ) ).
% K_record_comp
thf(fact_175_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_176_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_177_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_178_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X ) ) ) ) ).
% old.rec_unit_def
thf(fact_179_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [Y3: B] : Y3 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_180_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_181_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X2: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X2 ) )
= ( F @ ( product_fst @ C @ D @ X2 ) ) ) ).
% fst_map_prod
thf(fact_182_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X2: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X2 ) )
= ( G @ ( product_snd @ C @ D @ X2 ) ) ) ).
% snd_map_prod
thf(fact_183_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_184_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F22: A > E,G1: F3 > D,G22: B > F3] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_185_map__prod_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_186_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [X: B] : X
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_187_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X2: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X2 ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L: D,R2: E] : ( H @ ( F @ L ) @ ( G @ R2 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_188_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H2: E > C,F: A > D,G: A > E] :
( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H2 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H2 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_189_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X: A,Y3: B] : ( product_Pair @ C @ D @ ( F2 @ X ) @ ( G4 @ Y3 ) ) ) ) ) ).
% map_prod_def
thf(fact_190_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: D > E > C,G1: A > D,G22: B > E] :
( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L: A,R2: B] : ( F @ ( G1 @ L ) @ ( G22 @ R2 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_191_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_192_Nitpick_OThe__psimp,axiom,
! [A: $tType,P2: A > $o,X2: A] :
( ( P2
= ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X2 ) )
=> ( ( the @ A @ P2 )
= X2 ) ) ).
% Nitpick.The_psimp
thf(fact_193_theI__unique,axiom,
! [A: $tType,P2: A > $o,X2: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ( P2 @ X2 )
= ( X2
= ( the @ A @ P2 ) ) ) ) ).
% theI_unique
thf(fact_194_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: 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 )
@ ^ [X4: A,Y6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y3: B] :
( ( X4 = X )
& ( P4 @ X )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y3 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_195_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X2: A,Y7: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X2 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R @ X2 ) )
=> ( 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 @ X2 @ Y7 ) @ ( product_Pair @ A @ B @ X2 @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_196_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 )
@ ^ [A6: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B6: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A8 ) @ Ra )
| ( ( A6 = A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_197_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: C] :
( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X2 )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_198_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_199_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_200_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_201_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_202_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R3: set @ ( product_prod @ A @ A ),S: 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 @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R3 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R3 )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_203_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [X: B] : X )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_204_stream_Omap__id0,axiom,
! [A: $tType] :
( ( smap @ A @ A @ ( id @ A ) )
= ( id @ ( stream @ A ) ) ) ).
% stream.map_id0
thf(fact_205_stream_Omap__id,axiom,
! [A: $tType,T2: stream @ A] :
( ( smap @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% stream.map_id
thf(fact_206_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_207_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_208_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_209_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,G: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_210_fst__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z )
= ( id @ A @ Z ) ) ).
% fst_diag_id
thf(fact_211_snd__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z )
= ( id @ A @ Z ) ) ).
% snd_diag_id
thf(fact_212_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q2: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_213_smap__unfold__stream,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,SHD: C > B,STL: C > C,B2: C] :
( ( smap @ B @ A @ F @ ( coindu139217191stream @ C @ B @ SHD @ STL @ B2 ) )
= ( coindu139217191stream @ C @ A @ ( comp @ B @ A @ C @ F @ SHD ) @ STL @ B2 ) ) ).
% smap_unfold_stream
thf(fact_214_unfold__stream__eq__SCons,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B2: B,X2: A,Xs: stream @ A] :
( ( ( coindu139217191stream @ B @ A @ SHD @ STL @ B2 )
= ( sCons @ A @ X2 @ Xs ) )
= ( ( X2
= ( SHD @ B2 ) )
& ( Xs
= ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B2 ) ) ) ) ) ).
% unfold_stream_eq_SCons
thf(fact_215_unfold__stream__ltl__unroll,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B2: B] :
( ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B2 ) )
= ( coindu139217191stream @ B @ A @ ( comp @ B @ A @ B @ SHD @ STL ) @ STL @ B2 ) ) ).
% unfold_stream_ltl_unroll
thf(fact_216_unfold__stream_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu139217191stream @ A @ B )
= ( ^ [G12: A > B,G23: A > A,A6: A] : ( sCons @ B @ ( G12 @ A6 ) @ ( coindu139217191stream @ A @ B @ G12 @ G23 @ ( G23 @ A6 ) ) ) ) ) ).
% unfold_stream.code
thf(fact_217_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A7: set @ A,F12: B > C,F23: D > C,P1: A > B,P22: A > D] :
! [X: A] :
( ( member @ A @ X @ A7 )
=> ( ( F12 @ ( P1 @ X ) )
= ( F23 @ ( P22 @ X ) ) ) ) ) ) ).
% csquare_def
thf(fact_218_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q2: B > C > $o,A2: A,C2: C] :
( ( relcompp @ A @ B @ C @ P2 @ Q2 @ A2 @ C2 )
=> ( ( P2 @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q2 @ A2 @ C2 ) )
& ( Q2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q2 @ A2 @ C2 ) @ C2 ) ) ) ).
% pick_middlep
thf(fact_219_relcompp_OrelcompI,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,A2: A,B2: B,S: B > C > $o,C2: C] :
( ( R3 @ A2 @ B2 )
=> ( ( S @ B2 @ C2 )
=> ( relcompp @ A @ B @ C @ R3 @ S @ A2 @ C2 ) ) ) ).
% relcompp.relcompI
thf(fact_220_relcompp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,R3: A > B > $o,S: B > C > $o,X1: A,X22: C,P2: A > C > $o] :
( ( relcompp @ A @ B @ C @ R3 @ S @ X1 @ X22 )
=> ( ! [A4: A,B4: B,C3: C] :
( ( R3 @ A4 @ B4 )
=> ( ( S @ B4 @ C3 )
=> ( P2 @ A4 @ C3 ) ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% relcompp.inducts
thf(fact_221_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R3: A > D > $o,S: D > C > $o,T2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R3 @ S ) @ T2 )
= ( relcompp @ A @ D @ B @ R3 @ ( relcompp @ D @ C @ B @ S @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_222_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R4: A > B > $o,S6: B > C > $o,A6: A,C4: C] :
? [B5: B] :
( ( R4 @ A6 @ B5 )
& ( S6 @ B5 @ C4 ) ) ) ) ).
% relcompp_apply
thf(fact_223_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R2: A > B > $o,S3: B > C > $o,A1: A,A22: C] :
? [A6: A,B5: B,C4: C] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( R2 @ A6 @ B5 )
& ( S3 @ B5 @ C4 ) ) ) ) ).
% relcompp.simps
thf(fact_224_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,S: B > C > $o,A12: A,A23: C] :
( ( relcompp @ A @ B @ C @ R3 @ S @ A12 @ A23 )
=> ~ ! [B4: B] :
( ( R3 @ A12 @ B4 )
=> ~ ( S @ B4 @ A23 ) ) ) ).
% relcompp.cases
thf(fact_225_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,S: B > C > $o,A2: A,C2: C] :
( ( relcompp @ A @ B @ C @ R3 @ S @ A2 @ C2 )
=> ~ ! [B4: B] :
( ( R3 @ A2 @ B4 )
=> ~ ( S @ B4 @ C2 ) ) ) ).
% relcomppE
thf(fact_226_eq__OO,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ R )
= R ) ).
% eq_OO
thf(fact_227_OO__eq,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ B @ B @ R
@ ^ [Y5: B,Z2: B] : Y5 = Z2 )
= R ) ).
% OO_eq
thf(fact_228_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_229_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_230_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X2: B,C2: C,G: B > C,A5: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( C2
= ( G @ X2 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A5 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_231_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X ) ) ) ) ).
% old.rec_bool_def
thf(fact_232_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_233_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_234_smap__corec__stream,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,SHD: C > B,EndORmore: C > $o,STL_end: C > ( stream @ B ),STL_more: C > C,B2: C] :
( ( smap @ B @ A @ F @ ( corec_stream @ C @ B @ SHD @ EndORmore @ STL_end @ STL_more @ B2 ) )
= ( corec_stream @ C @ A @ ( comp @ B @ A @ C @ F @ SHD ) @ EndORmore @ ( comp @ ( stream @ B ) @ ( stream @ A ) @ C @ ( smap @ B @ A @ F ) @ STL_end ) @ STL_more @ B2 ) ) ).
% smap_corec_stream
thf(fact_235_stream_Omap__o__corec,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A,Ga: C > $o,Gb: C > ( stream @ A ),Gc: C > C] :
( ( comp @ ( stream @ A ) @ ( stream @ B ) @ C @ ( smap @ A @ B @ F ) @ ( corec_stream @ C @ A @ G @ Ga @ Gb @ Gc ) )
= ( corec_stream @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ Ga @ ( comp @ ( stream @ A ) @ ( stream @ B ) @ C @ ( smap @ A @ B @ F ) @ Gb ) @ Gc ) ) ).
% stream.map_o_corec
thf(fact_236_stream_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( ^ [G12: C > A,Q22: C > $o,G21: C > ( stream @ A ),G222: C > C,A6: C] : ( sCons @ A @ ( G12 @ A6 ) @ ( if @ ( stream @ A ) @ ( Q22 @ A6 ) @ ( G21 @ A6 ) @ ( corec_stream @ C @ A @ G12 @ Q22 @ G21 @ G222 @ ( G222 @ A6 ) ) ) ) ) ) ).
% stream.corec_code
thf(fact_237_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( corec_stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_238_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R2 )
@ ^ [X: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X @ Y3 ) @ S3 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_239_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S ) @ R )
=> ( ( S4 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_240_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R3 )
@ ^ [X: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X @ Y3 ) @ S ) )
= ( ^ [X: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y3 ) @ ( relcomp @ A @ B @ C @ R3 @ S ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_241_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C2 ) @ ( relcomp @ A @ B @ C @ R3 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_242_relcomp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,X1: A,X22: C,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C ),P2: A > C > $o] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X22 ) @ ( relcomp @ A @ B @ C @ R3 @ S ) )
=> ( ! [A4: A,B4: B,C3: C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C3 ) @ S )
=> ( P2 @ A4 @ C3 ) ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% relcomp.inducts
thf(fact_243_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R3 @ S ) )
= ( ? [A6: A,B5: B,C4: C] :
( ( A12 = A6 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R3 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_244_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R3 @ S ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A23 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_245_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A2: A,C2: B,R3: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C2 ) @ ( relcomp @ A @ C @ B @ R3 @ S ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_246_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R3: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R3 @ S ) )
=> ~ ! [X3: A,Y4: C,Z4: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z4 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y4 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z4 ) @ S ) ) ) ) ).
% relcompE
thf(fact_247_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S5: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S5 ) @ T4 )
= ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S5 @ T4 ) ) ) ).
% O_assoc
thf(fact_248_relcompp__in__rel,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ C ),S5: set @ ( product_prod @ C @ B )] :
( ( relcompp @ A @ C @ B @ ( fun_in_rel @ A @ C @ R ) @ ( fun_in_rel @ C @ B @ S5 ) )
= ( fun_in_rel @ A @ B @ ( relcomp @ A @ C @ B @ R @ S5 ) ) ) ).
% relcompp_in_rel
thf(fact_249_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R2: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Z3: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y3 ) @ R2 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z3 ) @ S3 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_250_in__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_in_rel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),X: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R4 ) ) ) ).
% in_rel_def
thf(fact_251_in__rel__Collect__case__prod__eq,axiom,
! [B: $tType,A: $tType,X7: A > B > $o] :
( ( fun_in_rel @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ X7 ) ) )
= X7 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_252_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A7: set @ C,F2: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A6: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F2 @ A6 ) @ ( G4 @ A6 ) ) )
& ( member @ C @ A6 @ A7 ) ) ) ) ) ).
% image2_def
thf(fact_253_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R4: A > C > $o,S6: C > B > $o,X: A,Z3: B] :
? [Y3: C] :
( ( R4 @ X @ Y3 )
& ( S6 @ Y3 @ Z3 ) ) ) ) ).
% OO_def
thf(fact_254_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr1317331620lImage @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F2: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A1: B,A22: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A1 @ A22 ) @ R4 ) ) ) ) ) ).
% relImage_def
thf(fact_255_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr2107612801vImage @ A @ B )
= ( ^ [A7: set @ A,R4: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A1: A,A22: A] :
( ( Uu
= ( product_Pair @ A @ A @ A1 @ A22 ) )
& ( member @ A @ A1 @ A7 )
& ( member @ A @ A22 @ A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).
% relInvImage_def
%----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,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( szip @ a @ b @ xs @ ( smap @ c @ b @ g @ ys ) )
= ( smap @ ( product_prod @ a @ c ) @ ( product_prod @ a @ b ) @ ( product_apsnd @ c @ b @ a @ g ) @ ( szip @ a @ c @ xs @ ys ) ) ) ).
%------------------------------------------------------------------------------