TPTP Problem File: DAT149^1.p
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%------------------------------------------------------------------------------
% File : DAT149^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 295
% 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__295.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 309 ( 149 unt; 47 typ; 0 def)
% Number of atoms : 669 ( 294 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3137 ( 80 ~; 7 |; 60 &;2756 @)
% ( 0 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 516 ( 516 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 45 usr; 5 con; 0-6 aty)
% Number of variables : 1264 ( 137 ^;1052 !; 28 ?;1264 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:14:03.345
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $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 (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
coindu1486289336t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olmap,type,
coinductive_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
coindu543516966_llist:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
coindu1724414836stream:
!>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__from__llist__setup_Ocr__stream,type,
coindu1183105481stream:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
coindu2010755910_llist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Opart__equivp,type,
equiv_part_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : 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_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Ostream_Opred__stream,type,
pred_stream:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_Osmap,type,
smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
thf(sy_c_Stream_Ostream_Ostream__all2,type,
stream_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( stream @ A ) > ( stream @ B ) > $o ) ).
thf(sy_c_Stream_Ostreams,type,
streams:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( stream @ A ) ) ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > b > $o ).
thf(sy_v_xs,type,
xs: stream @ a ).
thf(sy_v_y____,type,
y: coinductive_llist @ a ).
thf(sy_v_ya____,type,
ya: coinductive_llist @ b ).
thf(sy_v_ys,type,
ys: stream @ b ).
%----Relevant facts (254)
thf(fact_0_stream_ORep__inverse,axiom,
! [A: $tType,X: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
= X ) ).
% stream.Rep_inverse
thf(fact_1_stream__of__llist__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
= Xs ) ).
% stream_of_llist_llist_of_stream
thf(fact_2_llist__of__stream__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= Xs ) ) ).
% llist_of_stream_stream_of_llist
thf(fact_3_stream_ORep,axiom,
! [A: $tType,X: stream @ A] :
( member @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A @ X )
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% stream.Rep
thf(fact_4_stream_OAbs__cases,axiom,
! [A: $tType,X: stream @ A] :
~ ! [Y: coinductive_llist @ A] :
( ( X
= ( coindu2010755910_llist @ A @ Y ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% stream.Abs_cases
thf(fact_5_stream_ORep__cases,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ~ ! [X2: stream @ A] :
( Y2
!= ( coindu1724414836stream @ A @ X2 ) ) ) ).
% stream.Rep_cases
thf(fact_6_stream_OAbs__induct,axiom,
! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( P @ ( coindu2010755910_llist @ A @ Y ) ) )
=> ( P @ X ) ) ).
% stream.Abs_induct
thf(fact_7_stream_OAbs__inject,axiom,
! [A: $tType,X: coinductive_llist @ A,Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( ( coindu2010755910_llist @ A @ X )
= ( coindu2010755910_llist @ A @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% stream.Abs_inject
thf(fact_8_stream_ORep__induct,axiom,
! [A: $tType,Y2: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ! [X2: stream @ A] : ( P @ ( coindu1724414836stream @ A @ X2 ) )
=> ( P @ Y2 ) ) ) ).
% stream.Rep_induct
thf(fact_9_stream_ORep__inject,axiom,
! [A: $tType,X: stream @ A,Y2: stream @ A] :
( ( ( coindu1724414836stream @ A @ X )
= ( coindu1724414836stream @ A @ Y2 ) )
= ( X = Y2 ) ) ).
% stream.Rep_inject
thf(fact_10_stream_OAbs__inverse,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Y2 ) )
= Y2 ) ) ).
% stream.Abs_inverse
thf(fact_11_lfinite__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lfinite_llist_of_stream
thf(fact_12_llist__all2__lfiniteD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coinductive_lfinite @ A @ Xs )
= ( coinductive_lfinite @ B @ Ys ) ) ) ).
% llist_all2_lfiniteD
thf(fact_13_cr__streamI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% cr_streamI
thf(fact_14_llist__all2__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B
@ ^ [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
& ( Q @ X3 @ Y3 ) )
@ Xs
@ Ys )
= ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
& ( coindu1486289336t_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% llist_all2_conj
thf(fact_15_stream__from__llist__setup_Ocr__stream__def,axiom,
! [A: $tType] :
( ( coindu1183105481stream @ A )
= ( ^ [X3: coinductive_llist @ A,Y3: stream @ A] :
( X3
= ( coindu1724414836stream @ A @ Y3 ) ) ) ) ).
% stream_from_llist_setup.cr_stream_def
thf(fact_16_stream_Otype__definition__axioms,axiom,
! [A: $tType] :
( type_definition @ ( stream @ A ) @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A ) @ ( coindu2010755910_llist @ A )
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% stream.type_definition_axioms
thf(fact_17_stream_Orel__eq,axiom,
! [A: $tType] :
( ( stream_all2 @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [Y4: stream @ A,Z: stream @ A] : Y4 = Z ) ) ).
% stream.rel_eq
thf(fact_18_stream_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: stream @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( stream_all2 @ B @ B @ Ra @ X @ X ) ) ).
% stream.rel_refl
thf(fact_19_llist__all2__rsp,axiom,
! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T: B > B > $o,X: coinductive_llist @ A,Y2: coinductive_llist @ B,A2: coinductive_llist @ A,B2: coinductive_llist @ B] :
( ! [X2: A,Y: B] :
( ( R @ X2 @ Y )
=> ! [A3: A,B3: B] :
( ( R @ A3 @ B3 )
=> ( ( S @ X2 @ A3 )
= ( T @ Y @ B3 ) ) ) )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X @ Y2 )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( coindu1486289336t_all2 @ A @ A @ S @ X @ A2 )
= ( coindu1486289336t_all2 @ B @ B @ T @ Y2 @ B2 ) ) ) ) ) ).
% llist_all2_rsp
thf(fact_20_llist__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P2: A > B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X2: A,Y: B] :
( ( P @ X2 @ Y )
=> ( P2 @ X2 @ Y ) )
=> ( coindu1486289336t_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% llist_all2_mono
thf(fact_21_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu1486289336t_all2 @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [Y4: coinductive_llist @ A,Z: coinductive_llist @ A] : Y4 = Z ) ) ).
% llist.rel_eq
thf(fact_22_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coinductive_llist @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( coindu1486289336t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_23_stream__from__llist__setup_OQuotient__stream,axiom,
! [A: $tType] :
( quotient @ ( coinductive_llist @ A ) @ ( stream @ A )
@ ( bNF_eq_onp @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) )
@ ( coindu2010755910_llist @ A )
@ ( coindu1724414836stream @ A )
@ ( coindu1183105481stream @ A ) ) ).
% stream_from_llist_setup.Quotient_stream
thf(fact_24_eq__onpI,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P @ X )
=> ( bNF_eq_onp @ A @ P @ X @ X ) ) ).
% eq_onpI
thf(fact_25_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: set @ B,T: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ S )
=> ( ( T
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X3: B] : ( member @ B @ X3 @ S ) )
@ Abs
@ Rep
@ T ) ) ) ).
% typedef_to_Quotient
thf(fact_26_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o,T: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( ( T
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs @ Rep @ T ) ) ) ).
% open_typedef_to_Quotient
thf(fact_27_type__definition__Quotient__not__empty,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ? [X1: A] : ( P @ X1 ) ) ).
% type_definition_Quotient_not_empty
thf(fact_28_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R2: A > $o,X3: A,Y3: A] :
( ( R2 @ X3 )
& ( X3 = Y3 ) ) ) ) ).
% eq_onp_def
thf(fact_29_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y4: A,Z: A] : Y4 = Z ) ) ).
% eq_onp_True
thf(fact_30_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R: B > B > $o,T: B > A > $o] :
( ! [A3: A] :
( ( Abs @ ( Rep @ A3 ) )
= A3 )
=> ( ! [A3: A] : ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) )
=> ( ! [R3: B,S2: B] :
( ( R @ R3 @ S2 )
= ( ( R @ R3 @ R3 )
& ( R @ S2 @ S2 )
& ( ( Abs @ R3 )
= ( Abs @ S2 ) ) ) )
=> ( ( T
= ( ^ [X3: B,Y3: A] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y3 ) ) ) )
=> ( quotient @ B @ A @ R @ Abs @ Rep @ T ) ) ) ) ) ).
% QuotientI
thf(fact_31_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: B] :
( ( Abs2 @ ( Rep2 @ A4 ) )
= A4 )
& ! [A4: B] : ( R2 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
& ! [R4: A,S3: A] :
( ( R2 @ R4 @ S3 )
= ( ( R2 @ R4 @ R4 )
& ( R2 @ S3 @ S3 )
& ( ( Abs2 @ R4 )
= ( Abs2 @ S3 ) ) ) )
& ( T2
= ( ^ [X3: A,Y3: B] :
( ( R2 @ X3 @ X3 )
& ( ( Abs2 @ X3 )
= Y3 ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_32_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( T
= ( ^ [X3: A,Y3: B] :
( ( R @ X3 @ X3 )
& ( ( Abs @ X3 )
= Y3 ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_33_llist_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( coindu1486289336t_all2 @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( coinductive_llist @ A ) @ ( coindu543516966_llist @ A @ P ) ) ) ).
% llist.rel_eq_onp
thf(fact_34_type__definition_ORep,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( member @ A @ ( Rep @ X ) @ A5 ) ) ).
% type_definition.Rep
thf(fact_35_llist_Opred__True,axiom,
! [A: $tType] :
( ( coindu543516966_llist @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: coinductive_llist @ A] : $true ) ) ).
% llist.pred_True
thf(fact_36_llist_Opred__rel,axiom,
! [A: $tType] :
( ( coindu543516966_llist @ A )
= ( ^ [P3: A > $o,X3: coinductive_llist @ A] : ( coindu1486289336t_all2 @ A @ A @ ( bNF_eq_onp @ A @ P3 ) @ X3 @ X3 ) ) ) ).
% llist.pred_rel
thf(fact_37_Quotient__rep__abs__fold__unmap,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A,Rep3: B > A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( X4
= ( Abs @ X ) )
=> ( ( R @ X @ X )
=> ( ( ( Rep @ X4 )
= ( Rep3 @ X4 ) )
=> ( R @ ( Rep3 @ X4 ) @ X ) ) ) ) ) ).
% Quotient_rep_abs_fold_unmap
thf(fact_38_Quotient__to__transfer,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,C2: A,C3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ C2 @ C2 )
=> ( ( C3
= ( Abs @ C2 ) )
=> ( T @ C2 @ C3 ) ) ) ) ).
% Quotient_to_transfer
thf(fact_39_Quotient__abs__induct,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,P: B > $o,X: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ! [Y: A] :
( ( R @ Y @ Y )
=> ( P @ ( Abs @ Y ) ) )
=> ( P @ X ) ) ) ).
% Quotient_abs_induct
thf(fact_40_Quotient__rep__reflp,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A2: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( R @ ( Rep @ A2 ) @ ( Rep @ A2 ) ) ) ).
% Quotient_rep_reflp
thf(fact_41_Quotient__rel__abs2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X: B,Y2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ ( Rep @ X ) @ Y2 )
=> ( X
= ( Abs @ Y2 ) ) ) ) ).
% Quotient_rel_abs2
thf(fact_42_Quotient__alt__def3,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T2 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T2 @ ( Rep2 @ B4 ) @ B4 )
& ! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
= ( ? [Z2: B] :
( ( T2 @ X3 @ Z2 )
& ( T2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% Quotient_alt_def3
thf(fact_43_Quotient__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T2 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T2 @ ( Rep2 @ B4 ) @ B4 )
& ! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
= ( ( T2 @ X3 @ ( Abs2 @ Y3 ) )
& ( T2 @ Y3 @ ( Abs2 @ X3 ) ) ) ) ) ) ) ).
% Quotient_alt_def2
thf(fact_44_Quotient__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R5: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R5 @ R5 )
=> ( R @ ( Rep @ ( Abs @ R5 ) ) @ R5 ) ) ) ).
% Quotient_rep_abs
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Quotient__rel__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A2: B,B2: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ ( Rep @ A2 ) @ ( Rep @ B2 ) )
= ( A2 = B2 ) ) ) ).
% Quotient_rel_rep
thf(fact_50_Quotient__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R5: A,S4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R5 @ S4 )
=> ( ( Abs @ R5 )
= ( Abs @ S4 ) ) ) ) ).
% Quotient_rel_abs
thf(fact_51_Quotient__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R2: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B4: B] :
( ( T2 @ A4 @ B4 )
=> ( ( Abs2 @ A4 )
= B4 ) )
& ! [B4: B] : ( T2 @ ( Rep2 @ B4 ) @ B4 )
& ! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
= ( ( T2 @ X3 @ ( Abs2 @ X3 ) )
& ( T2 @ Y3 @ ( Abs2 @ Y3 ) )
& ( ( Abs2 @ X3 )
= ( Abs2 @ Y3 ) ) ) ) ) ) ) ).
% Quotient_alt_def
thf(fact_52_Quotient__abs__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A2: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( Abs @ ( Rep @ A2 ) )
= A2 ) ) ).
% Quotient_abs_rep
thf(fact_53_Quotient__Rep__eq,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( X4
= ( Abs @ X ) )
=> ( ( Rep @ X4 )
= ( Rep @ X4 ) ) ) ) ).
% Quotient_Rep_eq
thf(fact_54_Quotient__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R5: A,S4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R5 @ S4 )
=> ( R @ S4 @ S4 ) ) ) ).
% Quotient_refl2
thf(fact_55_Quotient__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R5: A,S4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R5 @ S4 )
=> ( R @ R5 @ R5 ) ) ) ).
% Quotient_refl1
thf(fact_56_Quotient__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R5: A,S4: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( ( R @ R5 @ R5 )
& ( R @ S4 @ S4 )
& ( ( Abs @ R5 )
= ( Abs @ S4 ) ) )
= ( R @ R5 @ S4 ) ) ) ).
% Quotient_rel
thf(fact_57_eq__onp__same__args,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( bNF_eq_onp @ A @ P @ X @ X )
= ( P @ X ) ) ).
% eq_onp_same_args
thf(fact_58_eq__onp__to__eq,axiom,
! [A: $tType,P: A > $o,X: A,Y2: A] :
( ( bNF_eq_onp @ A @ P @ X @ Y2 )
=> ( X = Y2 ) ) ).
% eq_onp_to_eq
thf(fact_59_eq__onp__mono0,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( bNF_eq_onp @ A @ P @ X5 @ Xa )
=> ( bNF_eq_onp @ A @ Q @ X5 @ Xa ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_60_eq__onp__eqD,axiom,
! [A: $tType,P: A > $o,Q: A > A > $o,X: A] :
( ( ( bNF_eq_onp @ A @ P )
= Q )
=> ( ( P @ X )
= ( Q @ X @ X ) ) ) ).
% eq_onp_eqD
thf(fact_61_type__definition_ORep__inverse,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_62_type__definition_OAbs__inverse,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,Y2: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) ) ) ).
% type_definition.Abs_inverse
thf(fact_63_type__definition_ORep__inject,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,X: B,Y2: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( ( Rep @ X )
= ( Rep @ Y2 ) )
= ( X = Y2 ) ) ) ).
% type_definition.Rep_inject
thf(fact_64_type__definition_ORep__induct,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,Y2: A,P: A > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ! [X2: B] : ( P @ ( Rep @ X2 ) )
=> ( P @ Y2 ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_65_type__definition_OAbs__inject,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,X: A,Y2: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( ( Abs @ X )
= ( Abs @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_66_type__definition_OAbs__induct,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,P: B > $o,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( P @ ( Abs @ Y ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_67_type__definition_ORep__cases,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,Y2: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ~ ! [X2: B] :
( Y2
!= ( Rep @ X2 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_68_type__definition_OAbs__cases,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ~ ! [Y: A] :
( ( X
= ( Abs @ Y ) )
=> ~ ( member @ A @ Y @ A5 ) ) ) ).
% type_definition.Abs_cases
thf(fact_69_type__definition_Ointro,axiom,
! [B: $tType,A: $tType,Rep: B > A,A5: set @ A,Abs: A > B] :
( ! [X2: B] : ( member @ A @ ( Rep @ X2 ) @ A5 )
=> ( ! [X2: B] :
( ( Abs @ ( Rep @ X2 ) )
= X2 )
=> ( ! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) )
=> ( type_definition @ B @ A @ Rep @ Abs @ A5 ) ) ) ) ).
% type_definition.intro
thf(fact_70_type__definition__def,axiom,
! [A: $tType,B: $tType] :
( ( type_definition @ B @ A )
= ( ^ [Rep2: B > A,Abs2: A > B,A6: set @ A] :
( ! [X3: B] : ( member @ A @ ( Rep2 @ X3 ) @ A6 )
& ! [X3: B] :
( ( Abs2 @ ( Rep2 @ X3 ) )
= X3 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( Rep2 @ ( Abs2 @ Y3 ) )
= Y3 ) ) ) ) ) ).
% type_definition_def
thf(fact_71_open__typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( equiv_part_equivp @ B @ ( bNF_eq_onp @ B @ P ) ) ) ).
% open_typedef_to_part_equivp
thf(fact_72_typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: set @ B] :
( ( type_definition @ A @ B @ Rep @ Abs @ S )
=> ( equiv_part_equivp @ B
@ ( bNF_eq_onp @ B
@ ^ [X3: B] : ( member @ B @ X3 @ S ) ) ) ) ).
% typedef_to_part_equivp
thf(fact_73_eq__onp__live__step,axiom,
! [A: $tType,X: $o,Y2: $o,P: A > $o,A2: A] :
( ( X = Y2 )
=> ( ( ( bNF_eq_onp @ A @ P @ A2 @ A2 )
& X )
= ( ( P @ A2 )
& Y2 ) ) ) ).
% eq_onp_live_step
thf(fact_74_stream_Opred__rel,axiom,
! [A: $tType] :
( ( pred_stream @ A )
= ( ^ [P3: A > $o,X3: stream @ A] : ( stream_all2 @ A @ A @ ( bNF_eq_onp @ A @ P3 ) @ X3 @ X3 ) ) ) ).
% stream.pred_rel
thf(fact_75_stream_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( stream_all2 @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( stream @ A ) @ ( pred_stream @ A @ P ) ) ) ).
% stream.rel_eq_onp
thf(fact_76_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,T: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( T
= ( ^ [X3: B,Y3: A] :
( X3
= ( Rep @ Y3 ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y4: B,Z: B] : Y4 = Z
@ Abs
@ Rep
@ T ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_77_eq__onp__to__Domainp,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ( ( domainp @ A @ B @ T )
= P ) ) ).
% eq_onp_to_Domainp
thf(fact_78_type__definition__Quotient__not__empty__witness,axiom,
! [A: $tType,B: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ( P @ ( Rep @ ( undefined @ B ) ) ) ) ).
% type_definition_Quotient_not_empty_witness
thf(fact_79_stream_OQuotient,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( quotient @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ A @ R ) @ ( smap @ A @ B @ Abs ) @ ( smap @ B @ A @ Rep ) @ ( stream_all2 @ A @ B @ T ) ) ) ).
% stream.Quotient
thf(fact_80_stream_Omap__ident,axiom,
! [A: $tType,T3: stream @ A] :
( ( smap @ A @ A
@ ^ [X3: A] : X3
@ T3 )
= T3 ) ).
% stream.map_ident
thf(fact_81_stream_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: stream @ A,Ya: stream @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( pred_stream @ A
@ ^ [Z2: A] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( smap @ A @ B @ F @ X )
= ( smap @ A @ B @ G @ Ya ) ) ) ) ).
% stream.map_cong_pred
thf(fact_82_stream_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( domainp @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) )
= ( pred_stream @ A @ ( domainp @ A @ B @ R ) ) ) ).
% stream.Domainp_rel
thf(fact_83_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S4: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: B] :
( S4
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_84_stream_Opred__True,axiom,
! [A: $tType] :
( ( pred_stream @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: stream @ A] : $true ) ) ).
% stream.pred_True
thf(fact_85_llist_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( domainp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) )
= ( coindu543516966_llist @ A @ ( domainp @ A @ B @ R ) ) ) ).
% llist.Domainp_rel
thf(fact_86_Quotient__to__Domainp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( domainp @ A @ B @ T )
= ( ^ [X3: A] : ( R @ X3 @ X3 ) ) ) ) ).
% Quotient_to_Domainp
thf(fact_87_stream_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: stream @ A,Y2: stream @ B] :
( ( stream_all2 @ C @ B @ Sb @ ( smap @ A @ C @ I @ X ) @ Y2 )
= ( stream_all2 @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y2 ) ) ).
% stream.rel_map(1)
thf(fact_88_stream_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: stream @ A,G: B > C,Y2: stream @ B] :
( ( stream_all2 @ A @ C @ Sa @ X @ ( smap @ B @ C @ G @ Y2 ) )
= ( stream_all2 @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y2 ) ) ).
% stream.rel_map(2)
thf(fact_89_Quotient__part__equivp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( equiv_part_equivp @ A @ R ) ) ).
% Quotient_part_equivp
thf(fact_90_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_91_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_92_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_93_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,F2: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ^ [P4: B > $o] :
? [X6: B] : ( P4 @ X6 )
@ F2 )
= ( ? [B4: A] : ( F2 @ ( Rep @ B4 ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_94_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_95_part__equivp__def,axiom,
! [A: $tType] :
( ( equiv_part_equivp @ A )
= ( ^ [R2: A > A > $o] :
( ? [X3: A] : ( R2 @ X3 @ X3 )
& ! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
= ( ( R2 @ X3 @ X3 )
& ( R2 @ Y3 @ Y3 )
& ( ( R2 @ X3 )
= ( R2 @ Y3 ) ) ) ) ) ) ) ).
% part_equivp_def
thf(fact_96_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_97_eq__onp__top__eq__eq,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
= ( ^ [Y4: A,Z: A] : Y4 = Z ) ) ).
% eq_onp_top_eq_eq
thf(fact_98_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_99_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_100_part__equivp__transp,axiom,
! [A: $tType,R: A > A > $o,X: A,Y2: A,Z3: A] :
( ( equiv_part_equivp @ A @ R )
=> ( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) ) ) ).
% part_equivp_transp
thf(fact_101_part__equivp__symp,axiom,
! [A: $tType,R: A > A > $o,X: A,Y2: A] :
( ( equiv_part_equivp @ A @ R )
=> ( ( R @ X @ Y2 )
=> ( R @ Y2 @ X ) ) ) ).
% part_equivp_symp
thf(fact_102_Domain__eq__top,axiom,
! [A: $tType] :
( ( domainp @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( top_top @ ( A > $o ) ) ) ).
% Domain_eq_top
thf(fact_103_Domainp_Oinducts,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,X: A,P: A > $o] :
( ( domainp @ A @ B @ R5 @ X )
=> ( ! [A3: A,B3: B] :
( ( R5 @ A3 @ B3 )
=> ( P @ A3 ) )
=> ( P @ X ) ) ) ).
% Domainp.inducts
thf(fact_104_Domainp_ODomainI,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,A2: A,B2: B] :
( ( R5 @ A2 @ B2 )
=> ( domainp @ A @ B @ R5 @ A2 ) ) ).
% Domainp.DomainI
thf(fact_105_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R4: A > B > $o,A4: A] :
? [B4: A,C4: B] :
( ( A4 = B4 )
& ( R4 @ B4 @ C4 ) ) ) ) ).
% Domainp.simps
thf(fact_106_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R5 @ A2 )
=> ~ ! [B3: B] :
~ ( R5 @ A2 @ B3 ) ) ).
% Domainp.cases
thf(fact_107_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_108_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_109_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_110_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_111_DomainpE,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R5 @ A2 )
=> ~ ! [B3: B] :
~ ( R5 @ A2 @ B3 ) ) ).
% DomainpE
thf(fact_112_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T2: A > B > $o,X3: A] :
( ^ [P4: B > $o] :
? [X6: B] : ( P4 @ X6 )
@ ( T2 @ X3 ) ) ) ) ).
% Domainp_iff
thf(fact_113_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_114_smap__stream__of__llist,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,F: A > B] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( smap @ A @ B @ F @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coindu2010755910_llist @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) ) ) ) ).
% smap_stream_of_llist
thf(fact_115_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
! [A: $tType] : ( type_definition @ A @ A @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ A ) @ ( top_top @ ( set @ A ) ) ) ).
% BNF_Composition.type_definition_id_bnf_UNIV
thf(fact_116_smap__smap2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: B > A,G: C > D > B,S1: stream @ C,S22: stream @ D] :
( ( smap @ B @ A @ F @ ( smap2 @ C @ D @ B @ G @ S1 @ S22 ) )
= ( smap2 @ C @ D @ A
@ ^ [X3: C,Y3: D] : ( F @ ( G @ X3 @ Y3 ) )
@ S1
@ S22 ) ) ).
% smap_smap2
thf(fact_117_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R4: A > A > $o,F3: B > A,X3: B,Y3: B] : ( R4 @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) ).
% in_inv_imagep
thf(fact_118_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_119_llist_Omap__ident,axiom,
! [A: $tType,T3: coinductive_llist @ A] :
( ( coinductive_lmap @ A @ A
@ ^ [X3: A] : X3
@ T3 )
= T3 ) ).
% llist.map_ident
thf(fact_120_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_lmap
thf(fact_121_lmap__llist__of__stream,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: stream @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( coindu1724414836stream @ B @ Xs ) )
= ( coindu1724414836stream @ A @ ( smap @ B @ A @ F @ Xs ) ) ) ).
% lmap_llist_of_stream
thf(fact_122_BNF__Composition_Oid__bnf__def,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X3: A] : X3 ) ) ).
% BNF_Composition.id_bnf_def
thf(fact_123_id__bnf__apply,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X3: A] : X3 ) ) ).
% id_bnf_apply
thf(fact_124_ID_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( bNF_id_bnf @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ A @ ( bNF_id_bnf @ ( A > $o ) @ P ) ) ) ).
% ID.rel_eq_onp
thf(fact_125_ID_Opred__rel,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P3: A > $o,X3: A] : ( bNF_id_bnf @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P3 ) @ X3 @ X3 ) ) ) ).
% ID.pred_rel
thf(fact_126_llist_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: coinductive_llist @ A,G: B > C,Y2: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ C @ Sa @ X @ ( coinductive_lmap @ B @ C @ G @ Y2 ) )
= ( coindu1486289336t_all2 @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y2 ) ) ).
% llist.rel_map(2)
thf(fact_127_llist_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: coinductive_llist @ A,Y2: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ C @ B @ Sb @ ( coinductive_lmap @ A @ C @ I @ X ) @ Y2 )
= ( coindu1486289336t_all2 @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y2 ) ) ).
% llist.rel_map(1)
thf(fact_128_llist__all2__lmap1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,Xs: coinductive_llist @ C,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lmap @ C @ A @ F @ Xs ) @ Ys )
= ( coindu1486289336t_all2 @ C @ B
@ ^ [X3: C] : ( P @ ( F @ X3 ) )
@ Xs
@ Ys ) ) ).
% llist_all2_lmap1
thf(fact_129_llist__all2__lmap2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,Xs: coinductive_llist @ A,G: C > B,Ys: coinductive_llist @ C] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_lmap @ C @ B @ G @ Ys ) )
= ( coindu1486289336t_all2 @ A @ C
@ ^ [X3: A,Y3: C] : ( P @ X3 @ ( G @ Y3 ) )
@ Xs
@ Ys ) ) ).
% llist_all2_lmap2
thf(fact_130_lmap__eq__lmap__conv__llist__all2,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,Xs: coinductive_llist @ B,G: C > A,Ys: coinductive_llist @ C] :
( ( ( coinductive_lmap @ B @ A @ F @ Xs )
= ( coinductive_lmap @ C @ A @ G @ Ys ) )
= ( coindu1486289336t_all2 @ B @ C
@ ^ [X3: B,Y3: C] :
( ( F @ X3 )
= ( G @ Y3 ) )
@ Xs
@ Ys ) ) ).
% lmap_eq_lmap_conv_llist_all2
thf(fact_131_llist_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( coindu543516966_llist @ A
@ ^ [Z2: A] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( coinductive_lmap @ A @ B @ F @ X )
= ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong_pred
thf(fact_132_llist_OQuotient,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( quotient @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ A @ R ) @ ( coinductive_lmap @ A @ B @ Abs ) @ ( coinductive_lmap @ B @ A @ Rep ) @ ( coindu1486289336t_all2 @ A @ B @ T ) ) ) ).
% llist.Quotient
thf(fact_133_inv__imagep__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_imagep @ B @ A )
= ( ^ [R4: B > B > $o,F3: A > B,X3: A,Y3: A] : ( R4 @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) ).
% inv_imagep_def
thf(fact_134_llast__lmap,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,F: A > B] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) )
= ( F @ ( coinductive_llast @ A @ Xs ) ) ) ) ) ).
% llast_lmap
thf(fact_135_UNIV__typedef__to__equivp,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B] :
( ( type_definition @ B @ A @ Rep @ Abs @ ( top_top @ ( set @ A ) ) )
=> ( equiv_equivp @ A
@ ^ [Y4: A,Z: A] : Y4 = Z ) ) ).
% UNIV_typedef_to_equivp
thf(fact_136_stream_OAbs__image,axiom,
! [A: $tType] :
( ( image @ ( coinductive_llist @ A ) @ ( stream @ A ) @ ( coindu2010755910_llist @ A )
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
= ( top_top @ ( set @ ( stream @ A ) ) ) ) ).
% stream.Abs_image
thf(fact_137_stream_ORep__range,axiom,
! [A: $tType] :
( ( image @ ( stream @ A ) @ ( coinductive_llist @ A ) @ ( coindu1724414836stream @ A ) @ ( top_top @ ( set @ ( stream @ A ) ) ) )
= ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% stream.Rep_range
thf(fact_138_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A5: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_139_image__ident,axiom,
! [A: $tType,Y5: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : X3
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_140_llist_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: coinductive_llist @ A] :
( ( coinductive_lnull @ B @ ( coinductive_lmap @ A @ B @ F @ A2 ) )
= ( coinductive_lnull @ A @ A2 ) ) ).
% llist.map_disc_iff
thf(fact_141_ID_Orel__refl,axiom,
! [A: $tType,Ra: A > A > $o,X: A] :
( ! [X2: A] : ( Ra @ X2 @ X2 )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra @ X @ X ) ) ).
% ID.rel_refl
thf(fact_142_ID_Opred__True,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o )
@ ^ [Uu: A] : $true )
= ( ^ [Uu: A] : $true ) ) ).
% ID.pred_True
thf(fact_143_ID_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: A,Ya: A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( bNF_id_bnf @ ( A > $o )
@ ^ [Z2: A] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( bNF_id_bnf @ ( A > B ) @ F @ X )
= ( bNF_id_bnf @ ( A > B ) @ G @ Ya ) ) ) ) ).
% ID.map_cong_pred
thf(fact_144_lnull__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lnull @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lnull_llist_of_stream
thf(fact_145_identity__equivp,axiom,
! [A: $tType] :
( equiv_equivp @ A
@ ^ [Y4: A,Z: A] : Y4 = Z ) ).
% identity_equivp
thf(fact_146_equivp__transp,axiom,
! [A: $tType,R: A > A > $o,X: A,Y2: A,Z3: A] :
( ( equiv_equivp @ A @ R )
=> ( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) ) ) ).
% equivp_transp
thf(fact_147_equivp__reflp,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( equiv_equivp @ A @ R )
=> ( R @ X @ X ) ) ).
% equivp_reflp
thf(fact_148_equivp__symp,axiom,
! [A: $tType,R: A > A > $o,X: A,Y2: A] :
( ( equiv_equivp @ A @ R )
=> ( ( R @ X @ Y2 )
=> ( R @ Y2 @ X ) ) ) ).
% equivp_symp
thf(fact_149_equivp__def,axiom,
! [A: $tType] :
( ( equiv_equivp @ A )
= ( ^ [R2: A > A > $o] :
! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
= ( ( R2 @ X3 )
= ( R2 @ Y3 ) ) ) ) ) ).
% equivp_def
thf(fact_150_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_151_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_152_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A5: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A5 ) )
= ( image @ C @ A
@ ^ [X3: C] : ( F @ ( G @ X3 ) )
@ A5 ) ) ).
% image_image
thf(fact_153_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A5 ) )
=> ( P @ X2 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_154_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_155_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F @ A5 ) )
& ( P @ X5 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_156_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A5: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A5 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_157_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A5 ) ) ) ).
% imageI
thf(fact_158_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ A5 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member @ B @ X2 @ A5 ) ) ) ).
% imageE
thf(fact_159_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_160_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_161_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_162_llist__all2__lnullD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coinductive_lnull @ A @ Xs )
= ( coinductive_lnull @ B @ Ys ) ) ) ).
% llist_all2_lnullD
thf(fact_163_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_164_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_165_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X3: B] : ( F @ ( G @ X3 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_166_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X2: B] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_167_equivp__implies__part__equivp,axiom,
! [A: $tType,R: A > A > $o] :
( ( equiv_equivp @ A @ R )
=> ( equiv_part_equivp @ A @ R ) ) ).
% equivp_implies_part_equivp
thf(fact_168_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( image @ A @ B @ Abs @ A5 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_169_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A5 ) ) ).
% type_definition.Rep_range
thf(fact_170_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X3: B] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_171_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X2: A] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_172_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y2: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y2
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_173_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y2: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y2
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_174_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A5: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A5 ) )
= ( Sup @ A5 ) ) ).
% Sup.SUP_identity_eq
thf(fact_175_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A5: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X3: A] : X3
@ A5 ) )
= ( Inf @ A5 ) ) ).
% Inf.INF_identity_eq
thf(fact_176_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_177_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_178_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_179_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_180_llist__all2__lappendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Xs3: coinductive_llist @ A,Ys2: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ B @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ Ys2 ) ) )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ ( coinductive_lappend @ B @ Ys @ Ys2 ) ) ) ) ).
% llist_all2_lappendI
thf(fact_181_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Ys: coinductive_llist @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( coinductive_lappend @ B @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) @ ( coinductive_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_182_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_183_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_184_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_185_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_186_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_187_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_188_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_189_streams__UNIV,axiom,
! [A: $tType] :
( ( streams @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( stream @ A ) ) ) ) ).
% streams_UNIV
thf(fact_190_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_191_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_192_llist__all2__LCons__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coinductive_llist @ A,Y21: B,Y22: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X22 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu1486289336t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist_all2_LCons_LCons
thf(fact_193_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_194_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_195_lappend__code_I2_J,axiom,
! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_196_llast__LCons2,axiom,
! [A: $tType,X: A,Y2: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_197_llist_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: coinductive_llist @ A] :
( ( coindu543516966_llist @ A @ P @ ( coinductive_LCons @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( coindu543516966_llist @ A @ P @ Aa2 ) ) ) ).
% llist.pred_inject(2)
thf(fact_198_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y2: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) )
= ( ( X = Y2 )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_199_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_200_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_201_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_202_smap__streams,axiom,
! [A: $tType,B: $tType,S4: stream @ A,A5: set @ A,F: A > B,B5: set @ B] :
( ( member @ ( stream @ A ) @ S4 @ ( streams @ A @ A5 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ B @ ( F @ X2 ) @ B5 ) )
=> ( member @ ( stream @ B ) @ ( smap @ A @ B @ F @ S4 ) @ ( streams @ B @ B5 ) ) ) ) ).
% smap_streams
thf(fact_203_llist__all2__LCons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Y2: B,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LCons @ B @ Y2 @ Ys ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( P @ X3 @ Y2 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% llist_all2_LCons2
thf(fact_204_llist__all2__LCons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y3: B,Ys3: coinductive_llist @ B] :
( ( Ys
= ( coinductive_LCons @ B @ Y3 @ Ys3 ) )
& ( P @ X @ Y3 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% llist_all2_LCons1
thf(fact_205_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coinductive_llist @ A,Y22: coinductive_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X22 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_206_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_207_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs5: coinductive_llist @ A] :
( ! [Ys4: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys4 @ Xs5 )
=> ( P @ Ys4 ) )
=> ( P @ Xs5 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_208_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Y2: A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lmap @ B @ A @ F @ Xs )
= ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ? [X3: B,Xs4: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ B @ X3 @ Xs4 ) )
& ( Y2
= ( F @ X3 ) )
& ( Ys
= ( coinductive_lmap @ B @ A @ F @ Xs4 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_209_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lmap @ A @ B @ F @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( coinductive_LCons @ B @ ( F @ X21 ) @ ( coinductive_lmap @ A @ B @ F @ X22 ) ) ) ).
% llist.simps(13)
thf(fact_210_lmap__lstrict__prefix,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,F: A > B] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coindu1478340336prefix @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) @ ( coinductive_lmap @ A @ B @ F @ Ys ) ) ) ).
% lmap_lstrict_prefix
thf(fact_211_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_212_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_213_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs5: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) )
=> ( ( ! [Ys4: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys4 @ Xs )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_214_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( X = Y2 )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_215_llist__all2__LNil1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LNil @ A ) @ Xs )
= ( Xs
= ( coinductive_LNil @ B ) ) ) ).
% llist_all2_LNil1
thf(fact_216_llist__all2__LNil2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LNil @ B ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% llist_all2_LNil2
thf(fact_217_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_218_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_219_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_220_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_221_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_222_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y2: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_223_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_224_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_225_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_226_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs5: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
=> ~ ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ).
% lfinite.cases
thf(fact_227_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A,X3: A] :
( ( A4
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
& ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ) ).
% lfinite.simps
thf(fact_228_llist_Oexhaust,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( Y2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y2
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_229_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_230_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs5: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_231_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: coinductive_llist @ A,B2: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( ( A2
= ( coinductive_LNil @ A ) )
=> ( B2
!= ( coinductive_LNil @ B ) ) )
=> ~ ! [X1: A,X23: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y23: coinductive_llist @ B] :
( ( B2
= ( coinductive_LCons @ B @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( coindu1486289336t_all2 @ A @ B @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_232_llist__all2__cases,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( Xs
= ( coinductive_LNil @ A ) )
=> ( Ys
!= ( coinductive_LNil @ B ) ) )
=> ~ ! [X2: A,Xs6: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X2 @ Xs6 ) )
=> ! [Y: B,Ys5: coinductive_llist @ B] :
( ( Ys
= ( coinductive_LCons @ B @ Y @ Ys5 ) )
=> ( ( P @ X2 @ Y )
=> ~ ( coindu1486289336t_all2 @ A @ B @ P @ Xs6 @ Ys5 ) ) ) ) ) ) ).
% llist_all2_cases
thf(fact_233_llist__all2__LCons__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coinductive_llist @ A] :
~ ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ Y21 @ Y22 ) @ ( coinductive_LNil @ B ) ) ).
% llist_all2_LCons_LNil
thf(fact_234_llist__all2__LNil__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coinductive_llist @ B] :
~ ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) ) ).
% llist_all2_LNil_LCons
thf(fact_235_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_236_llist_Osimps_I12_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( coinductive_lmap @ A @ B @ F @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ B ) ) ).
% llist.simps(12)
thf(fact_237_LNil__eq__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( Xs
= ( coinductive_LNil @ B ) ) ) ).
% LNil_eq_lmap
thf(fact_238_lmap__eq__LNil,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( ( coinductive_lmap @ B @ A @ F @ Xs )
= ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ B ) ) ) ).
% lmap_eq_LNil
thf(fact_239_LNil__transfer,axiom,
! [A: $tType,B: $tType,P: A > B > $o] : ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ B ) ) ).
% LNil_transfer
thf(fact_240_llist__all2__LNil__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ B ) ) ).
% llist_all2_LNil_LNil
thf(fact_241_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_242_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_243_llist_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( coindu543516966_llist @ A @ P @ ( coinductive_LNil @ A ) ) ).
% llist.pred_inject(1)
thf(fact_244_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_245_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_246_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_247_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_248_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_249_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_250_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_251_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_252_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs5: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coinductive_lappend @ A @ Xs5 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_253_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Xs4 ) )
& ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
%----Type constructors (3)
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 @ ( type2 @ A8 ) )
=> ( top @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_1,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Otop_2,axiom,
top @ $o @ ( type2 @ $o ) ).
%----Conjectures (5)
thf(conj_0,hypothesis,
( xs
= ( coindu2010755910_llist @ a @ y ) ) ).
thf(conj_1,hypothesis,
( member @ ( coinductive_llist @ a ) @ y
@ ( collect @ ( coinductive_llist @ a )
@ ^ [Xs2: coinductive_llist @ a] :
~ ( coinductive_lfinite @ a @ Xs2 ) ) ) ).
thf(conj_2,hypothesis,
( ys
= ( coindu2010755910_llist @ b @ ya ) ) ).
thf(conj_3,hypothesis,
( member @ ( coinductive_llist @ b ) @ ya
@ ( collect @ ( coinductive_llist @ b )
@ ^ [Xs2: coinductive_llist @ b] :
~ ( coinductive_lfinite @ b @ Xs2 ) ) ) ).
thf(conj_4,conjecture,
( ( coindu1486289336t_all2 @ a @ b @ p @ ( coindu1724414836stream @ a @ xs ) @ ( coindu1724414836stream @ b @ ys ) )
= ( stream_all2 @ a @ b @ p @ xs @ ys ) ) ).
%------------------------------------------------------------------------------