TPTP Problem File: DAT148^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT148^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 241
% 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__241.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 316 ( 108 unt; 55 typ; 0 def)
% Number of atoms : 1044 ( 303 equ; 8 cnn)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 5131 ( 35 ~; 1 |; 33 &;4795 @)
% ( 0 <=>; 267 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 11 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 1735 (1735 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 53 usr; 3 con; 0-8 aty)
% Number of variables : 1907 ( 418 ^;1384 !; 27 ?;1907 :)
% ( 78 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:13:48.760
%------------------------------------------------------------------------------
%----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 (49)
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Coinductive__List_Oiterates,type,
coinductive_iterates:
!>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Ocase__llist,type,
coindu1381640503_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
coindu1259883913_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( 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_Opred__llist,type,
coindu543516966_llist:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
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__from__llist__setup_Opcr__stream,type,
coindu773941317stream:
!>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( coinductive_llist @ C ) > ( stream @ B ) > $o ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
coindu2010755910_llist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
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_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lifting_Orel__pred__comp,type,
rel_pred_comp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
thf(sy_c_Partial__Function_Omk__less,type,
partial_mk_less:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OBex1__rel,type,
bex1_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
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_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Stream_Ositerate,type,
siterate:
!>[A: $tType] : ( ( A > A ) > A > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocase__stream,type,
case_stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
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_Opred__stream,type,
pred_stream:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
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_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__total,type,
left_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Orev__implies,type,
rev_implies: $o > $o > $o ).
thf(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Otransfer__bforall,type,
transfer_bforall:
!>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Transfer_Otransfer__forall,type,
transfer_forall:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
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_xb,type,
xb: a > ( coinductive_llist @ b ) ).
thf(sy_v_yb,type,
yb: a > ( stream @ b ) ).
%----Relevant facts (256)
thf(fact_0_stream_ORep__inject,axiom,
! [A: $tType,X: stream @ A,Y: stream @ A] :
( ( ( coindu1724414836stream @ A @ X )
= ( coindu1724414836stream @ A @ Y ) )
= ( X = Y ) ) ).
% stream.Rep_inject
thf(fact_1_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F: B > A,G: C > B,X2: C] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_2_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
( ! [X3: A,Y2: B] :
( ( A2 @ X3 @ Y2 )
=> ( B2 @ ( F2 @ X3 ) @ ( G2 @ Y2 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 ) ) ).
% rel_funI
thf(fact_3_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G2: B > C,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sa
@ X
@ ( comp @ B @ C @ D @ G2 @ Y ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [X2: A,Y4: B] : ( Sa @ X2 @ ( G2 @ Y4 ) )
@ X
@ Y ) ) ).
% fun.rel_map(2)
thf(fact_4_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sb
@ ( comp @ A @ C @ D @ I @ X )
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y ) ) ).
% fun.rel_map(1)
thf(fact_5_o__rsp_I2_J,axiom,
! [E: $tType,F3: $tType,H: $tType,G3: $tType,R1: E > F3 > $o] :
( bNF_rel_fun @ ( G3 > H ) @ ( G3 > H ) @ ( ( E > G3 ) > E > H ) @ ( ( F3 > G3 ) > F3 > H )
@ ^ [Y3: G3 > H,Z: G3 > H] : Y3 = Z
@ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H ) @ ( F3 > H )
@ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R1
@ ^ [Y3: G3,Z: G3] : Y3 = Z )
@ ( bNF_rel_fun @ E @ F3 @ H @ H @ R1
@ ^ [Y3: H,Z: H] : Y3 = Z ) )
@ ( comp @ G3 @ H @ E )
@ ( comp @ G3 @ H @ F3 ) ) ).
% o_rsp(2)
thf(fact_6_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R3: B > D > $o,R1: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R2 @ R3 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R1 @ R3 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% o_rsp(1)
thf(fact_7_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G3 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
@ ( bNF_rel_fun @ D @ D @ A @ F3
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Rb )
@ ( bNF_rel_fun @ D @ D @ B @ G3
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F3 @ G3 @ D ) ) ).
% fun.map_transfer
thf(fact_8_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B2: A > C > $o,C2: B > D > $o,A2: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B2 @ C2 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A2 @ B2 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A2 @ C2 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% comp_transfer
thf(fact_9_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C3: B,F2: A > C] :
( ( comp @ C @ B @ A
@ ^ [X2: C] : C3
@ F2 )
= ( ^ [X2: A] : C3 ) ) ).
% K_record_comp
thf(fact_10_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T: D > A] :
( ( comp @ A @ A @ D
@ ^ [X2: A] : X2
@ T )
= T ) ).
% fun.map_ident
thf(fact_11_stream__from__llist__setup_Ocr__stream__def,axiom,
! [A: $tType] :
( ( coindu1183105481stream @ A )
= ( ^ [X2: coinductive_llist @ A,Y4: stream @ A] :
( X2
= ( coindu1724414836stream @ A @ Y4 ) ) ) ) ).
% stream_from_llist_setup.cr_stream_def
thf(fact_12_llist__of__stream__corec__stream,axiom,
! [A: $tType,B: $tType,SHD: B > A,EndORmore: B > $o,STL_more: B > ( stream @ A ),STL_end: B > B,X: B] :
( ( coindu1724414836stream @ A @ ( corec_stream @ B @ A @ SHD @ EndORmore @ STL_more @ STL_end @ X ) )
= ( coindu1259883913_llist @ B @ A
@ ^ [Uu: B] : $false
@ SHD
@ EndORmore
@ ( comp @ ( stream @ A ) @ ( coinductive_llist @ A ) @ B @ ( coindu1724414836stream @ A ) @ STL_more )
@ STL_end
@ X ) ) ).
% llist_of_stream_corec_stream
thf(fact_13_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F3 > $o,F2: A > C > D,G2: B > E > F3] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 ) @ F2 @ G2 )
= ( ! [X2: A,Y4: B] :
( ( R @ X2 @ Y4 )
=> ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T2 @ ( F2 @ X2 ) @ ( G2 @ Y4 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_14_If__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A2 @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
@ ( if @ A )
@ ( if @ B ) ) ).
% If_transfer
thf(fact_15_fun_Orel__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,D: $tType,Sa: A > C > $o,Sc: B > E > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z )
@ ( bNF_rel_fun @ D @ D @ C @ E
@ ^ [Y3: D,Z: D] : Y3 = Z ) ) ).
% fun.rel_transfer
thf(fact_16_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X: D > B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ X
@ X ) ) ).
% fun.rel_refl
thf(fact_17_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: D > A,Z: D > A] : Y3 = Z ) ) ).
% fun.rel_eq
thf(fact_18_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y5: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F2: A > C,G2: B > D] :
( ! [X3: A,Y2: B] :
( ( Y5 @ X3 @ Y2 )
=> ( X4 @ X3 @ Y2 ) )
=> ( ! [X3: C,Y2: D] :
( ( A2 @ X3 @ Y2 )
=> ( B2 @ X3 @ Y2 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% rel_fun_mono'
thf(fact_19_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X4: A > B > $o,A2: C > D > $o,F2: A > C,G2: B > D,Y5: A > B > $o,B2: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F2 @ G2 )
=> ( ! [X3: A,Y2: B] :
( ( Y5 @ X3 @ Y2 )
=> ( X4 @ X3 @ Y2 ) )
=> ( ! [X3: C,Y2: D] :
( ( A2 @ X3 @ Y2 )
=> ( B2 @ X3 @ Y2 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y5 @ B2 @ F2 @ G2 ) ) ) ) ).
% rel_fun_mono
thf(fact_20_let__rsp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
@ ^ [S2: A,F: A > C] : ( F @ S2 )
@ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% let_rsp
thf(fact_21_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
=> ( ( A2 @ X @ Y )
=> ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% rel_funD
thf(fact_22_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H2: A > C,R12: D > B,R22: A > D,F2: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H2 )
= ( comp @ D @ B @ A @ R12 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F2 @ R12 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F2 @ G2 ) @ H2 )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_23_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F2: C > B,G2: A > C,L1: D > B,L2: A > D,H2: E > A,R4: E > D] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H2 )
= R4 )
=> ( ( comp @ C @ B @ E @ F2 @ ( comp @ A @ C @ E @ G2 @ H2 ) )
= ( comp @ D @ B @ E @ L1 @ R4 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_24_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H2: A > C,R4: A > B,F2: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H2 )
= R4 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F2 @ G2 ) @ H2 )
= ( comp @ B @ D @ A @ F2 @ R4 ) ) ) ).
% rewriteR_comp_comp
thf(fact_25_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: C > B,G2: A > C,L: A > B,H2: D > A] :
( ( ( comp @ C @ B @ A @ F2 @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F2 @ ( comp @ A @ C @ D @ G2 @ H2 ) )
= ( comp @ A @ B @ D @ L @ H2 ) ) ) ).
% rewriteL_comp_comp
thf(fact_26_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F2: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F2 @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_27_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: B > A,G2: C > B,X: C,H2: D > A,K: C > D] :
( ( ( F2 @ ( G2 @ X ) )
= ( H2 @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
= ( comp @ D @ A @ C @ H2 @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_28_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= C3 )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_29_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ! [V2: A] :
( ( A3 @ ( B3 @ V2 ) )
= ( C3 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_30_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D2 ) )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_31_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: D > B,G2: C > D,H2: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F2 @ G2 ) @ H2 )
= ( comp @ D @ B @ A @ F2 @ ( comp @ C @ D @ A @ G2 @ H2 ) ) ) ).
% comp_assoc
thf(fact_32_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F: B > C,G: A > B,X2: A] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_33_Let__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ B2 )
@ ^ [S2: A,F: A > C] : ( F @ S2 )
@ ^ [S2: B,F: B > D] : ( F @ S2 ) ) ).
% Let_transfer
thf(fact_34_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( corec_stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_35_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G2: C > B,X: C,N: D > A,H2: C > D,F2: A > E] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H2 @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F2 @ M ) @ G2 @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F2 @ N ) @ H2 @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_36_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F2: B > A,G2: C > B,X: C,F4: D > A,G4: E > D,X5: E] :
( ( ( F2 @ ( G2 @ X ) )
= ( F4 @ ( G4 @ X5 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G2 @ X )
= ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% comp_cong
thf(fact_37_rel__funD2,axiom,
! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F2: A > B,G2: A > C,X: A] :
( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F2 @ G2 )
=> ( ( A2 @ X @ X )
=> ( B2 @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% rel_funD2
thf(fact_38_apply__rsp_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F2 @ G2 )
=> ( ( R1 @ X @ Y )
=> ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% apply_rsp'
thf(fact_39_rel__funE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F2: A > C,G2: B > D,X: A,Y: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F2 @ G2 )
=> ( ( A2 @ X @ Y )
=> ( B2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ).
% rel_funE
thf(fact_40_stream__from__llist__setup_Ostream_Opcr__cr__eq,axiom,
! [D: $tType] :
( ( coindu773941317stream @ D @ D
@ ^ [Y3: D,Z: D] : Y3 = Z )
= ( coindu1183105481stream @ D ) ) ).
% stream_from_llist_setup.stream.pcr_cr_eq
thf(fact_41_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_42_stream_ORep__inverse,axiom,
! [A: $tType,X: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ X ) )
= X ) ).
% stream.Rep_inverse
thf(fact_43_map__fun__parametric,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,H: $tType,G3: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G3 > $o,D3: F3 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G3 > H ) > ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G3 > H ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ E @ G3 @ F3 @ H @ C2 @ D3 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G3 ) @ ( A > F3 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G3 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H @ A2 @ D3 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G3 @ H ) ) ).
% map_fun_parametric
thf(fact_44_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F: B > C,G: D > A,H3: C > D,X2: B] : ( G @ ( H3 @ ( F @ X2 ) ) ) ) ) ).
% map_fun_apply
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G2 @ X3 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_49_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F: C > A,G: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G @ H3 ) @ F ) ) ) ).
% map_fun_def
thf(fact_50_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 ) @ ( map_fun @ C @ A @ B @ D @ H2 @ I ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) ) ) ).
% map_fun.comp
thf(fact_51_map__fun_Ocompositionality,axiom,
! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F2: E > C,G2: D > F3,H2: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F3 @ F2 @ G2 @ ( map_fun @ C @ A @ B @ D @ H2 @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F2 ) @ ( comp @ D @ F3 @ B @ G2 @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_52_fun_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true )
@ ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true ) ) ).
% fun.pred_transfer
thf(fact_53_if__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R @ ( bNF_rel_fun @ A @ A @ A @ A @ R @ R ) )
@ ( if @ A )
@ ( if @ A ) ) ) ).
% if_rsp
thf(fact_54_stream_Ocorec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > $o ) > ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( stream @ A ) ) > ( C > C ) > C > ( stream @ A ) ) @ ( ( D > ( stream @ B ) ) > ( D > D ) > D > ( stream @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( ( C > C ) > C > ( stream @ A ) ) @ ( ( D > D ) > D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( stream @ A ) ) @ ( D > ( stream @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( stream @ A ) @ ( stream @ B ) @ S @ ( stream_all2 @ A @ B @ R ) ) ) ) )
@ ( corec_stream @ C @ A )
@ ( corec_stream @ D @ B ) ) ).
% stream.corec_transfer
thf(fact_55_unfold__stream__transfer,axiom,
! [B: $tType,A: $tType] :
( bNF_rel_fun @ ( A > B ) @ ( A > B ) @ ( ( A > A ) > A > ( coinductive_llist @ B ) ) @ ( ( A > A ) > A > ( stream @ B ) )
@ ^ [Y3: A > B,Z: A > B] : Y3 = Z
@ ( bNF_rel_fun @ ( A > A ) @ ( A > A ) @ ( A > ( coinductive_llist @ B ) ) @ ( A > ( stream @ B ) )
@ ^ [Y3: A > A,Z: A > A] : Y3 = Z
@ ( bNF_rel_fun @ A @ A @ ( coinductive_llist @ B ) @ ( stream @ B )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( coindu773941317stream @ B @ B
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) )
@ ( coindu1441602521_llist @ A @ B
@ ^ [Uu: A] : $false )
@ ( coindu139217191stream @ A @ B ) ) ).
% unfold_stream_transfer
thf(fact_56_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F3: $tType,E: $tType,C2: A > B > $o,A2: C > E > $o,B2: D > F3 > $o] :
( ( bi_total @ A @ B @ C2 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F3 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F3 ) > $o )
@ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F3 > $o ) @ A2
@ ( bNF_rel_fun @ D @ F3 @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F3 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C2 @ A2 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F3 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F3 @ C2 @ B2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E @ F3 @ B ) ) ) ).
% fun_ord_parametric
thf(fact_57_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_58_cr__streamI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( coindu1183105481stream @ A @ Xs @ ( coindu2010755910_llist @ A @ Xs ) ) ) ).
% cr_streamI
thf(fact_59_stream_OAbs__inverse,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( coindu1724414836stream @ A @ ( coindu2010755910_llist @ A @ Y ) )
= Y ) ) ).
% stream.Abs_inverse
thf(fact_60_llist__of__stream__unfold__stream,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,X: B] :
( ( coindu1724414836stream @ A @ ( coindu139217191stream @ B @ A @ SHD @ STL @ X ) )
= ( coindu1441602521_llist @ B @ A
@ ^ [Uu: B] : $false
@ SHD
@ STL
@ X ) ) ).
% llist_of_stream_unfold_stream
thf(fact_61_stream_Odisc__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [Stream: stream @ A] : Stream = Stream
@ ^ [Stream: stream @ B] : Stream = Stream ) ).
% stream.disc_transfer
thf(fact_62_stream_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: stream @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( stream_all2 @ B @ B @ Ra @ X @ X ) ) ).
% stream.rel_refl
thf(fact_63_stream_Orel__eq,axiom,
! [A: $tType] :
( ( stream_all2 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: stream @ A,Z: stream @ A] : Y3 = Z ) ) ).
% stream.rel_eq
thf(fact_64_stream_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% stream.bi_total_rel
thf(fact_65_bi__total__def,axiom,
! [B: $tType,A: $tType] :
( ( bi_total @ A @ B )
= ( ^ [R5: A > B > $o] :
( ! [X2: A] :
( ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 )
@ ( R5 @ X2 ) )
& ! [Y4: B] :
? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ) ).
% bi_total_def
thf(fact_66_bi__total__eq,axiom,
! [A: $tType] :
( bi_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% bi_total_eq
thf(fact_67_Quotient3__rep__reflp,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% Quotient3_rep_reflp
thf(fact_68_Quotient3__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R4 @ R4 )
=> ( R @ ( Rep @ ( Abs @ R4 ) ) @ R4 ) ) ) ).
% Quotient3_rep_abs
thf(fact_69_Quotient3__rel__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B,B3: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
= ( A3 = B3 ) ) ) ).
% Quotient3_rel_rep
thf(fact_70_Quotient3__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R4 @ S3 )
=> ( ( Abs @ R4 )
= ( Abs @ S3 ) ) ) ) ).
% Quotient3_rel_abs
thf(fact_71_Quotient3__abs__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,A3: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( Abs @ ( Rep @ A3 ) )
= A3 ) ) ).
% Quotient3_abs_rep
thf(fact_72_rep__abs__rsp__left,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ X1 @ X22 )
=> ( R @ ( Rep @ ( Abs @ X1 ) ) @ X22 ) ) ) ).
% rep_abs_rsp_left
thf(fact_73_Quotient3__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R4 @ S3 )
=> ( R @ S3 @ S3 ) ) ) ).
% Quotient3_refl2
thf(fact_74_Quotient3__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R4 @ S3 )
=> ( R @ R4 @ R4 ) ) ) ).
% Quotient3_refl1
thf(fact_75_Quotient3__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R4: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( ( R @ R4 @ R4 )
& ( R @ S3 @ S3 )
& ( ( Abs @ R4 )
= ( Abs @ S3 ) ) )
= ( R @ R4 @ S3 ) ) ) ).
% Quotient3_rel
thf(fact_76_Quotient3__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient3 @ A @ B )
= ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A] :
( ! [A4: B] :
( ( Abs2 @ ( Rep2 @ A4 ) )
= A4 )
& ! [A4: B] : ( R5 @ ( Rep2 @ A4 ) @ ( Rep2 @ A4 ) )
& ! [R6: A,S2: A] :
( ( R5 @ R6 @ S2 )
= ( ( R5 @ R6 @ R6 )
& ( R5 @ S2 @ S2 )
& ( ( Abs2 @ R6 )
= ( Abs2 @ S2 ) ) ) ) ) ) ) ).
% Quotient3_def
thf(fact_77_rep__abs__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X22: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ X1 @ X22 )
=> ( R @ X1 @ ( Rep @ ( Abs @ X22 ) ) ) ) ) ).
% rep_abs_rsp
thf(fact_78_pred__funI,axiom,
! [B: $tType,A: $tType,A2: A > $o,B2: B > $o,F2: A > B] :
( ! [X3: A] :
( ( A2 @ X3 )
=> ( B2 @ ( F2 @ X3 ) ) )
=> ( basic_pred_fun @ A @ B @ A2 @ B2 @ F2 ) ) ).
% pred_funI
thf(fact_79_equals__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,Xa: A,Xb: A,Ya: A,Yb: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ Xa @ Xb )
=> ( ( R @ Ya @ Yb )
=> ( ( R @ Xa @ Ya )
= ( R @ Xb @ Yb ) ) ) ) ) ).
% equals_rsp
thf(fact_80_Quotient3I,axiom,
! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
( ! [A5: A] :
( ( Abs @ ( Rep @ A5 ) )
= A5 )
=> ( ! [A5: A] : ( R @ ( Rep @ A5 ) @ ( Rep @ A5 ) )
=> ( ! [R7: B,S4: B] :
( ( R @ R7 @ S4 )
= ( ( R @ R7 @ R7 )
& ( R @ S4 @ S4 )
& ( ( Abs @ R7 )
= ( Abs @ S4 ) ) ) )
=> ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% Quotient3I
thf(fact_81_cond__prs,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,A3: $o,B3: B,C3: B] :
( ( quotient3 @ A @ B @ R @ Absf @ Repf )
=> ( ( A3
=> ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
= B3 ) )
& ( ~ A3
=> ( ( Absf @ ( if @ A @ A3 @ ( Repf @ B3 ) @ ( Repf @ C3 ) ) )
= C3 ) ) ) ) ).
% cond_prs
thf(fact_82_fun_Opred__True,axiom,
! [A: $tType,D: $tType] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Uu: A] : $true )
= ( ^ [Uu: D > A] : $true ) ) ).
% fun.pred_True
thf(fact_83_fun__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
=> ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs22 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) ) ) ) ).
% fun_quotient3
thf(fact_84_apply__rspQ3_H_H,axiom,
! [C: $tType,A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,S: C > C > $o,F2: A > C,X: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F2 @ F2 )
=> ( S @ ( F2 @ ( Rep @ X ) ) @ ( F2 @ ( Rep @ X ) ) ) ) ) ).
% apply_rspQ3''
thf(fact_85_apply__rspQ3,axiom,
! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F2: A > C,G2: A > C,X: A,Y: A] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F2 @ G2 )
=> ( ( R1 @ X @ Y )
=> ( R2 @ ( F2 @ X ) @ ( G2 @ Y ) ) ) ) ) ).
% apply_rspQ3
thf(fact_86_let__prs,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
=> ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep23 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs23 @ Rep12 ) @ Abs12 )
@ ^ [S2: C,F: C > A] : ( F @ S2 ) )
= ( ^ [S2: D,F: D > B] : ( F @ S2 ) ) ) ) ) ).
% let_prs
thf(fact_87_lfinite__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lfinite @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lfinite_llist_of_stream
thf(fact_88_lambda__prs1,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
=> ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23 @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 @ F2 ) )
= F2 ) ) ) ).
% lambda_prs1
thf(fact_89_lambda__prs,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,F2: B > D] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
=> ( ( map_fun @ B @ A @ C @ D @ Rep12 @ Abs23
@ ^ [X2: A] : ( Rep23 @ ( F2 @ ( Abs12 @ X2 ) ) ) )
= F2 ) ) ) ).
% lambda_prs
thf(fact_90_stream_ORep__induct,axiom,
! [A: $tType,Y: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ! [X3: stream @ A] : ( P @ ( coindu1724414836stream @ A @ X3 ) )
=> ( P @ Y ) ) ) ).
% stream.Rep_induct
thf(fact_91_stream_ORep__cases,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ~ ! [X3: stream @ A] :
( Y
!= ( coindu1724414836stream @ A @ X3 ) ) ) ).
% stream.Rep_cases
thf(fact_92_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_93_stream_OAbs__cases,axiom,
! [A: $tType,X: stream @ A] :
~ ! [Y2: coinductive_llist @ A] :
( ( X
= ( coindu2010755910_llist @ A @ Y2 ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% stream.Abs_cases
thf(fact_94_stream_OAbs__induct,axiom,
! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
( ! [Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y2
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( P @ ( coindu2010755910_llist @ A @ Y2 ) ) )
=> ( P @ X ) ) ).
% stream.Abs_induct
thf(fact_95_stream_OAbs__inject,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) )
=> ( ( ( coindu2010755910_llist @ A @ X )
= ( coindu2010755910_llist @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% stream.Abs_inject
thf(fact_96_unfold__stream__ltl__unroll,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B3: B] :
( ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B3 ) )
= ( coindu139217191stream @ B @ A @ ( comp @ B @ A @ B @ SHD @ STL ) @ STL @ B3 ) ) ).
% unfold_stream_ltl_unroll
thf(fact_97_stream_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( stream @ A ) > ( stream @ B ) > $o ) @ ( ( stream @ C ) > ( stream @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ C ) @ ( ( stream @ B ) > $o ) @ ( ( stream @ D ) > $o ) @ ( stream_all2 @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( stream @ B ) @ ( stream @ D ) @ $o @ $o @ ( stream_all2 @ B @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( stream_all2 @ A @ B )
@ ( stream_all2 @ C @ D ) ) ).
% stream.rel_transfer
thf(fact_98_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs23: C > D,Rep23: D > C,R3: E > E > $o,Abs3: E > F3,Rep3: F3 > E] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs23 @ Rep23 )
=> ( ( quotient3 @ E @ F3 @ R3 @ Abs3 @ Rep3 )
=> ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs23 @ Rep3 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep23 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep12 @ Abs3 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F3 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_99_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F2: A > B,G2: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z2: A] :
( ( F2 @ Z2 )
= ( G2 @ Z2 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F2 @ X )
= ( comp @ A @ B @ D @ G2 @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_100_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F2: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F2 @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F2 )
@ X ) ) ).
% fun.pred_map
thf(fact_101_quot__rel__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ R
@ R ) ) ).
% quot_rel_rsp
thf(fact_102_corec__llist__never__stop,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
@ ^ [Uu: B] : $false
@ MORE
@ LTL
@ X )
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% corec_llist_never_stop
thf(fact_103_unfold__llist__ltl__unroll,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B] :
( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) )
= ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B3 ) ) ).
% unfold_llist_ltl_unroll
thf(fact_104_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( comple1396247847notone @ A @ C )
@ ( comple1396247847notone @ B @ D ) ) ) ).
% monotone_parametric
thf(fact_105_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_106_relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
( ( bi_total @ A @ B @ B2 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ( relcompp @ C @ A @ E )
@ ( relcompp @ D @ B @ F3 ) ) ) ).
% relcompp_transfer
thf(fact_107_bex1__rel__rsp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Absf: A > B,Repf: B > A] :
( ( quotient3 @ A @ B @ R @ Absf @ Repf )
=> ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bex1_rel @ A @ R )
@ ( bex1_rel @ A @ R ) ) ) ).
% bex1_rel_rsp
thf(fact_108_left__total__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( ( bi_total @ C @ D @ B2 )
=> ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( left_total @ A @ C )
@ ( left_total @ B @ D ) ) ) ) ).
% left_total_parametric
thf(fact_109_fun_Orel__compp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ( relcompp @ A @ B @ C @ R @ S ) )
= ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R )
@ ( bNF_rel_fun @ D @ D @ B @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ S ) ) ) ).
% fun.rel_compp
thf(fact_110_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G2: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ G2 )
= ( ^ [X2: A] :
( ( comp @ B @ $o @ A @ P @ G2 @ X2 )
& ( comp @ B @ $o @ A @ Q @ G2 @ X2 ) ) ) ) ).
% conj_comp_iff
thf(fact_111_stream_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( stream_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
= ( relcompp @ ( stream @ A ) @ ( stream @ B ) @ ( stream @ C ) @ ( stream_all2 @ A @ B @ R ) @ ( stream_all2 @ B @ C @ S ) ) ) ).
% stream.rel_compp
thf(fact_112_stream_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ( left_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% stream.left_total_rel
thf(fact_113_eq__comp__r,axiom,
! [A: $tType,R: A > A > $o] :
( ( relcompp @ A @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( relcompp @ A @ A @ A @ R
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
= R ) ).
% eq_comp_r
thf(fact_114_bex1__rel__aux,axiom,
! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X @ Xa2 )
= ( Y @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ X )
=> ( bex1_rel @ A @ R @ Y ) ) ) ).
% bex1_rel_aux
thf(fact_115_bex1__rel__aux2,axiom,
! [A: $tType,R: A > A > $o,X: A > $o,Y: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X @ Xa2 )
= ( Y @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ Y )
=> ( bex1_rel @ A @ R @ X ) ) ) ).
% bex1_rel_aux2
thf(fact_116_left__totalE,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ! [X7: A] :
? [X12: B] : ( R @ X7 @ X12 ) ) ).
% left_totalE
thf(fact_117_left__totalI,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ! [X3: A] :
? [X13: B] : ( R @ X3 @ X13 )
=> ( left_total @ A @ B @ R ) ) ).
% left_totalI
thf(fact_118_left__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( left_total @ A @ B @ R )
=> ( ( left_total @ B @ C @ S )
=> ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% left_total_OO
thf(fact_119_left__total__eq,axiom,
! [A: $tType] :
( left_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% left_total_eq
thf(fact_120_left__total__def,axiom,
! [B: $tType,A: $tType] :
( ( left_total @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A] :
( ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 )
@ ( R5 @ X2 ) ) ) ) ).
% left_total_def
thf(fact_121_bi__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( ( bi_total @ B @ C @ B2 )
=> ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% bi_total_OO
thf(fact_122_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( T2
= ( ^ [X2: A,Y4: B] :
( X2
= ( Rep @ Y4 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T2
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [X2: A] : X2
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_123_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: B > B > $o,Abs23: B > C,Rep23: C > B,R23: A > A > $o] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ B @ C @ R2 @ Abs23 @ Rep23 )
=> ( ! [X3: A,Y2: A] :
( ( R23 @ X3 @ Y2 )
=> ( ( R1 @ X3 @ X3 )
=> ( ( R1 @ Y2 @ Y2 )
=> ( R2 @ ( Abs12 @ X3 ) @ ( Abs12 @ Y2 ) ) ) ) )
=> ( ! [X3: B,Y2: B] :
( ( R2 @ X3 @ Y2 )
=> ( R23 @ ( Rep12 @ X3 ) @ ( Rep12 @ Y2 ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs23 @ Abs12 ) @ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_124_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,Abs23: B > C,Rep23: C > B] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ B @ C
@ ^ [Y3: B,Z: B] : Y3 = Z
@ Abs23
@ Rep23 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R1
@ ( relcompp @ A @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R1 ) )
@ ( comp @ B @ C @ A @ Abs23 @ Abs12 )
@ ( comp @ B @ A @ C @ Rep12 @ Rep23 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_125_monotone__applyI,axiom,
! [B: $tType,A: $tType,C: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,X: C] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
=> ( comple1396247847notone @ ( C > A ) @ B @ ( partial_fun_ord @ A @ A @ C @ Orda ) @ Ordb
@ ^ [F: C > A] : ( F5 @ ( F @ X ) ) ) ) ).
% monotone_applyI
thf(fact_126_monotone__fun__ord__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Orda: A > A > $o,Ordb: C > C > $o,F2: A > B > C] :
( ( comple1396247847notone @ A @ ( B > C ) @ Orda @ ( partial_fun_ord @ C @ C @ B @ Ordb ) @ F2 )
= ( ! [X2: B] :
( comple1396247847notone @ A @ C @ Orda @ Ordb
@ ^ [Y4: A] : ( F2 @ Y4 @ X2 ) ) ) ) ).
% monotone_fun_ord_apply
thf(fact_127_stream_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S: C > D > $o] : ( bNF_rel_fun @ ( A > ( stream @ A ) > C ) @ ( B > ( stream @ B ) > D ) @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( stream @ A ) > C ) @ ( ( stream @ B ) > D ) @ R @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ C @ D @ ( stream_all2 @ A @ B @ R ) @ S ) @ ( case_stream @ A @ C ) @ ( case_stream @ B @ D ) ) ).
% stream.case_transfer
thf(fact_128_monotone__if__fun,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Orda: B > B > $o,Ordb: D > D > $o,F5: ( A > B ) > C > D,G5: ( A > B ) > C > D,C3: C > $o] :
( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ F5 )
=> ( ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb ) @ G5 )
=> ( comple1396247847notone @ ( A > B ) @ ( C > D ) @ ( partial_fun_ord @ B @ B @ A @ Orda ) @ ( partial_fun_ord @ D @ D @ C @ Ordb )
@ ^ [F: A > B,N2: C] : ( if @ D @ ( C3 @ N2 ) @ ( F5 @ F @ N2 ) @ ( G5 @ F @ N2 ) ) ) ) ) ).
% monotone_if_fun
thf(fact_129_monotone__fun__apply__fun,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Ord: C > C > $o,T: A,G2: D > B] :
( comple1396247847notone @ ( A > B > C ) @ ( D > C ) @ ( partial_fun_ord @ ( B > C ) @ ( B > C ) @ A @ ( partial_fun_ord @ C @ C @ B @ Ord ) ) @ ( partial_fun_ord @ C @ C @ D @ Ord )
@ ^ [F: A > B > C,N2: D] : ( F @ T @ ( G2 @ N2 ) ) ) ).
% monotone_fun_apply_fun
thf(fact_130_stream_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F2: A > ( stream @ A ) > B,Stream2: stream @ A] :
( ( H2 @ ( case_stream @ A @ B @ F2 @ Stream2 ) )
= ( case_stream @ A @ C
@ ^ [X14: A,X23: stream @ A] : ( H2 @ ( F2 @ X14 @ X23 ) )
@ Stream2 ) ) ).
% stream.case_distrib
thf(fact_131_monotone__id_H,axiom,
! [A: $tType,Ord: A > A > $o] :
( comple1396247847notone @ A @ A @ Ord @ Ord
@ ^ [X2: A] : X2 ) ).
% monotone_id'
thf(fact_132_call__mono,axiom,
! [B: $tType,A: $tType,Ord: B > B > $o,T: A] :
( comple1396247847notone @ ( A > B ) @ B @ ( partial_fun_ord @ B @ B @ A @ Ord ) @ Ord
@ ^ [F: A > B] : ( F @ T ) ) ).
% call_mono
thf(fact_133_stream_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( stream @ A ) > $o ) @ ( ( stream @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( stream @ A ) @ ( stream @ B ) @ $o @ $o @ ( stream_all2 @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( pred_stream @ A )
@ ( pred_stream @ B ) ) ).
% stream.pred_transfer
thf(fact_134_transfer__forall__transfer_I1_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( transfer_forall @ A )
@ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(1)
thf(fact_135_Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
( ( bi_total @ A @ B @ B2 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( domainp @ C @ A )
@ ( domainp @ D @ B ) ) ) ).
% Domainp_transfer
thf(fact_136_stream_Opred__True,axiom,
! [A: $tType] :
( ( pred_stream @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: stream @ A] : $true ) ) ).
% stream.pred_True
thf(fact_137_fun_ODomainp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R: A > B > $o] :
( ( domainp @ ( C > A ) @ ( C > B )
@ ( bNF_rel_fun @ C @ C @ A @ B
@ ^ [Y3: C,Z: C] : Y3 = Z
@ R ) )
= ( basic_pred_fun @ C @ A
@ ^ [Uu: C] : $true
@ ( domainp @ A @ B @ R ) ) ) ).
% fun.Domainp_rel
thf(fact_138_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_139_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T3: A > B > $o,X2: A] :
( ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 )
@ ( T3 @ X2 ) ) ) ) ).
% Domainp_iff
thf(fact_140_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_141_pcr__Domainp__total,axiom,
! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,A2: C > A > $o,P: C > $o] :
( ( left_total @ A @ B @ B2 )
=> ( ( ( domainp @ C @ A @ A2 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
= P ) ) ) ).
% pcr_Domainp_total
thf(fact_142_stream__from__llist__setup_Ostream_Odomain__eq,axiom,
! [A: $tType] :
( ( domainp @ ( coinductive_llist @ A ) @ ( stream @ A )
@ ( coindu773941317stream @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
= ( ^ [Xs2: coinductive_llist @ A] :
~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ).
% stream_from_llist_setup.stream.domain_eq
thf(fact_143_fun__ord__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( partial_fun_ord @ A @ B @ C )
= ( ^ [Ord2: A > B > $o,F: C > A,G: C > B] :
! [X2: C] : ( Ord2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% fun_ord_def
thf(fact_144_pcr__Domainp__par__left__total,axiom,
! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P3: C > $o] :
( ( ( domainp @ A @ B @ B2 )
= P )
=> ( ( left_total @ C @ A @ A2 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P3
@ P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
= P3 ) ) ) ) ).
% pcr_Domainp_par_left_total
thf(fact_145_let__mono,axiom,
! [A: $tType,C: $tType,B: $tType,Orda: B > B > $o,Ordb: C > C > $o,B3: B > A > C,T: A] :
( ! [X3: A] :
( comple1396247847notone @ B @ C @ Orda @ Ordb
@ ^ [F: B] : ( B3 @ F @ X3 ) )
=> ( comple1396247847notone @ B @ C @ Orda @ Ordb
@ ^ [F: B] : ( B3 @ F @ T ) ) ) ).
% let_mono
thf(fact_146_if__mono,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F5: A > B,G5: A > B,C3: $o] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F5 )
=> ( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ G5 )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F: A] : ( if @ B @ C3 @ ( F5 @ F ) @ ( G5 @ F ) ) ) ) ) ).
% if_mono
thf(fact_147_transfer__forall__transfer_I4_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ rev_implies
@ ( transfer_forall @ A )
@ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(4)
thf(fact_148_transfer__forall__transfer_I5_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(5)
thf(fact_149_reflp__transfer_I1_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( reflp @ A )
@ ( reflp @ B ) ) ) ).
% reflp_transfer(1)
thf(fact_150_right__total__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( ( bi_total @ C @ D @ B2 )
=> ( bNF_rel_fun @ ( A > C > $o ) @ ( B > D > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ A2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( right_total @ A @ C )
@ ( right_total @ B @ D ) ) ) ) ).
% right_total_parametric
thf(fact_151_fun_Orel__reflp,axiom,
! [D: $tType,A: $tType,R: A > A > $o] :
( ( reflp @ A @ R )
=> ( reflp @ ( D > A )
@ ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R ) ) ) ).
% fun.rel_reflp
thf(fact_152_DEADID_Orel__reflp,axiom,
! [A: $tType] :
( reflp @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% DEADID.rel_reflp
thf(fact_153_right__total__def,axiom,
! [B: $tType,A: $tType] :
( ( right_total @ A @ B )
= ( ^ [R5: A > B > $o] :
! [Y4: B] :
? [X2: A] : ( R5 @ X2 @ Y4 ) ) ) ).
% right_total_def
thf(fact_154_rev__implies__def,axiom,
( rev_implies
= ( ^ [X2: $o,Y4: $o] :
( Y4
=> X2 ) ) ) ).
% rev_implies_def
thf(fact_155_right__total__eq,axiom,
! [A: $tType] :
( right_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% right_total_eq
thf(fact_156_right__totalI,axiom,
! [A: $tType,B: $tType,A2: B > A > $o] :
( ! [Y2: A] :
? [X7: B] : ( A2 @ X7 @ Y2 )
=> ( right_total @ B @ A @ A2 ) ) ).
% right_totalI
thf(fact_157_right__totalE,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,Y: B] :
( ( right_total @ A @ B @ A2 )
=> ~ ! [X3: A] :
~ ( A2 @ X3 @ Y ) ) ).
% right_totalE
thf(fact_158_stream__from__llist__setup_Ostream_Oright__total,axiom,
! [F3: $tType,E: $tType,T2: E > F3 > $o] :
( ( right_total @ E @ F3 @ T2 )
=> ( right_total @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) ) ) ).
% stream_from_llist_setup.stream.right_total
thf(fact_159_stream_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( right_total @ A @ B @ R )
=> ( right_total @ ( stream @ A ) @ ( stream @ B ) @ ( stream_all2 @ A @ B @ R ) ) ) ).
% stream.right_total_rel
thf(fact_160_stream_Orel__reflp,axiom,
! [A: $tType,R: A > A > $o] :
( ( reflp @ A @ R )
=> ( reflp @ ( stream @ A ) @ ( stream_all2 @ A @ A @ R ) ) ) ).
% stream.rel_reflp
thf(fact_161_right__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
( ( right_total @ A @ B @ A2 )
=> ( ( right_total @ B @ C @ B2 )
=> ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% right_total_OO
thf(fact_162_typedef__right__total,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( T2
= ( ^ [X2: A,Y4: B] :
( X2
= ( Rep @ Y4 ) ) ) )
=> ( right_total @ A @ B @ T2 ) ) ) ).
% typedef_right_total
thf(fact_163_reflp__transfer_I2_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) ) @ (=>) @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% reflp_transfer(2)
thf(fact_164_reflp__transfer_I3_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ (=>)
@ ( reflp @ A )
@ ( reflp @ B ) ) ) ).
% reflp_transfer(3)
thf(fact_165_bi__totalI,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ( ( right_total @ A @ B @ R )
=> ( bi_total @ A @ B @ R ) ) ) ).
% bi_totalI
thf(fact_166_bi__total__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( bi_total @ A @ B )
= ( ^ [A6: A > B > $o] :
( ( left_total @ A @ B @ A6 )
& ( right_total @ A @ B @ A6 ) ) ) ) ).
% bi_total_alt_def
thf(fact_167_reflp__transfer_I4_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% reflp_transfer(4)
thf(fact_168_reflp__transfer_I5_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ rev_implies
@ ( reflp @ A )
@ ( reflp @ B ) ) ) ).
% reflp_transfer(5)
thf(fact_169_transfer__forall__transfer_I3_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) ) @ (=>) @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(3)
thf(fact_170_transfer__forall__transfer_I2_J,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ (=>)
@ ( transfer_forall @ A )
@ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(2)
thf(fact_171_Domainp__forall__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( transfer_bforall @ A @ ( domainp @ A @ B @ A2 ) )
@ ( transfer_forall @ B ) ) ) ).
% Domainp_forall_transfer
thf(fact_172_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o,C2: E > F3 > $o] :
( ( right_total @ A @ B @ B2 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B2
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A2
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ^ [R5: C > A > $o,S5: A > E > $o,X2: C,Z2: E] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
& ( R5 @ X2 @ Y4 )
& ( S5 @ Y4 @ Z2 ) )
@ ( relcompp @ D @ B @ F3 ) ) ) ).
% right_total_relcompp_transfer
thf(fact_173_right__total__Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B2: A > B > $o,A2: C > D > $o] :
( ( right_total @ A @ B @ B2 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [T3: C > A > $o,X2: C] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B2 ) ) )
& ( T3 @ X2 @ Y4 ) )
@ ( domainp @ D @ B ) ) ) ).
% right_total_Domainp_transfer
thf(fact_174_bex__reg,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X7: A] :
( ( member @ A @ X7 @ R )
& ( P @ X7 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_175_right__total__Ex__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bex @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
@ ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 ) ) ) ).
% right_total_Ex_transfer
thf(fact_176_Domainp__prod__fun__eq,axiom,
! [C: $tType,B: $tType,A: $tType,T2: B > C > $o] :
( ( domainp @ ( A > B ) @ ( A > C )
@ ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y3: A,Z: A] : Y3 = Z
@ T2 ) )
= ( ^ [F: A > B] :
! [X2: A] : ( domainp @ B @ C @ T2 @ ( F @ X2 ) ) ) ) ).
% Domainp_prod_fun_eq
thf(fact_177_stream__from__llist__setup_Ostream_Odomain__par__left__total,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,P3: ( coinductive_llist @ C ) > $o] :
( ( left_total @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T2 ) )
=> ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P3
@ ^ [Xs2: coinductive_llist @ B] :
~ ( coinductive_lfinite @ B @ Xs2 ) )
=> ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
= P3 ) ) ) ).
% stream_from_llist_setup.stream.domain_par_left_total
thf(fact_178_llist_Orel__reflp,axiom,
! [A: $tType,R: A > A > $o] :
( ( reflp @ A @ R )
=> ( reflp @ ( coinductive_llist @ A ) @ ( coindu1486289336t_all2 @ A @ A @ R ) ) ) ).
% llist.rel_reflp
thf(fact_179_llist_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( right_total @ A @ B @ R )
=> ( right_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.right_total_rel
thf(fact_180_bex__reg__left,axiom,
! [A: $tType,R: set @ A,Q: A > $o,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ? [X7: A] :
( ( member @ A @ X7 @ R )
& ( Q @ X7 ) )
=> ? [X12: A] : ( P @ X12 ) ) ) ).
% bex_reg_left
thf(fact_181_rel__fun__eq__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R )
= ( ^ [F: A > B,G: A > C] :
! [X2: A] : ( R @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% rel_fun_eq_rel
thf(fact_182_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A6: A > C > $o,B4: B > D > $o,F: A > B,G: C > D] :
! [X2: A,Y4: C] :
( ( A6 @ X2 @ Y4 )
=> ( B4 @ ( F @ X2 ) @ ( G @ Y4 ) ) ) ) ) ).
% rel_fun_def
thf(fact_183_stream__from__llist__setup_Ostream_Odomain,axiom,
! [B: $tType,C: $tType,T2: C > B > $o] :
( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
= ( ^ [X2: coinductive_llist @ C] :
? [Y4: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ C @ B @ T2 @ X2 @ Y4 )
& ~ ( coinductive_lfinite @ B @ Y4 ) ) ) ) ).
% stream_from_llist_setup.stream.domain
thf(fact_184_llist__all2__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o ) @ ( coindu1486289336t_all2 @ A @ B @ R )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( coindu1486289336t_all2 @ A @ A )
@ ( coindu1486289336t_all2 @ B @ B ) ) ).
% llist_all2_transfer
thf(fact_185_llist_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( ( coinductive_llist @ B ) > $o ) @ ( ( coinductive_llist @ D ) > $o ) @ ( coindu1486289336t_all2 @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ $o @ $o @ ( coindu1486289336t_all2 @ B @ D @ Sc )
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( coindu1486289336t_all2 @ A @ B )
@ ( coindu1486289336t_all2 @ C @ D ) ) ).
% llist.rel_transfer
thf(fact_186_lfinite__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( coinductive_lfinite @ A )
@ ( coinductive_lfinite @ B ) ) ).
% lfinite_transfer
thf(fact_187_transfer__forall__def,axiom,
! [A: $tType] :
( ( transfer_forall @ A )
= ( ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 ) ) ) ).
% transfer_forall_def
thf(fact_188_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R5: A > C > $o,S5: C > B > $o,X2: A,Z2: B] :
? [Y4: C] :
( ( R5 @ X2 @ Y4 )
& ( S5 @ Y4 @ Z2 ) ) ) ) ).
% OO_def
thf(fact_189_llist_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.bi_total_rel
thf(fact_190_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_191_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coinductive_llist @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( coindu1486289336t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_192_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu1486289336t_all2 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: coinductive_llist @ A,Z: coinductive_llist @ A] : Y3 = Z ) ) ).
% llist.rel_eq
thf(fact_193_llist__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P3: A > B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X3: A,Y2: B] :
( ( P @ X3 @ Y2 )
=> ( P3 @ X3 @ Y2 ) )
=> ( coindu1486289336t_all2 @ A @ B @ P3 @ Xs @ Ys ) ) ) ).
% llist_all2_mono
thf(fact_194_llist__all2__rsp,axiom,
! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X: coinductive_llist @ A,Y: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
( ! [X3: A,Y2: B] :
( ( R @ X3 @ Y2 )
=> ! [A5: A,B5: B] :
( ( R @ A5 @ B5 )
=> ( ( S @ X3 @ A5 )
= ( T2 @ Y2 @ B5 ) ) ) )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X @ Y )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
=> ( ( coindu1486289336t_all2 @ A @ A @ S @ X @ A3 )
= ( coindu1486289336t_all2 @ B @ B @ T2 @ Y @ B3 ) ) ) ) ) ).
% llist_all2_rsp
thf(fact_195_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
@ ^ [X2: A,Y4: B] :
( ( P @ X2 @ Y4 )
& ( Q @ X2 @ Y4 ) )
@ Xs
@ Ys )
= ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
& ( coindu1486289336t_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% llist_all2_conj
thf(fact_196_llist_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ( left_total @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.left_total_rel
thf(fact_197_llist_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( coindu1486289336t_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
= ( relcompp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ C ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ B @ C @ S ) ) ) ).
% llist.rel_compp
thf(fact_198_pred__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_fun @ A @ B )
= ( ^ [A6: A > $o,B4: B > $o,F: A > B] :
! [X2: A] :
( ( A6 @ X2 )
=> ( B4 @ ( F @ X2 ) ) ) ) ) ).
% pred_fun_def
thf(fact_199_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o] :
( ( ( domainp @ A @ B @ B2 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
= ( ^ [X2: C] :
? [Y4: A] :
( ( A2 @ X2 @ Y4 )
& ( P @ Y4 ) ) ) ) ) ).
% pcr_Domainp
thf(fact_200_stream__from__llist__setup_Ostream_Orep__transfer,axiom,
! [E: $tType,F3: $tType,T2: E > F3 > $o] :
( bNF_rel_fun @ ( coinductive_llist @ E ) @ ( stream @ F3 ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ F3 ) @ ( coindu773941317stream @ E @ F3 @ T2 ) @ ( coindu1486289336t_all2 @ E @ F3 @ T2 )
@ ^ [X2: coinductive_llist @ E] : X2
@ ( coindu1724414836stream @ F3 ) ) ).
% stream_from_llist_setup.stream.rep_transfer
thf(fact_201_stream__from__llist__setup_Opcr__stream__def,axiom,
! [B: $tType,C: $tType] :
( ( coindu773941317stream @ C @ B )
= ( ^ [T3: C > B > $o] : ( relcompp @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( stream @ B ) @ ( coindu1486289336t_all2 @ C @ B @ T3 ) @ ( coindu1183105481stream @ B ) ) ) ) ).
% stream_from_llist_setup.pcr_stream_def
thf(fact_202_transfer__bforall__def,axiom,
! [A: $tType] :
( ( transfer_bforall @ A )
= ( ^ [P4: A > $o,Q2: A > $o] :
! [X2: A] :
( ( P4 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% transfer_bforall_def
thf(fact_203_unfold__llist__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > B ) > B > ( coinductive_llist @ D ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) )
@ ( coindu1441602521_llist @ A @ C )
@ ( coindu1441602521_llist @ B @ D ) ) ).
% unfold_llist_transfer
thf(fact_204_llist_Ocorec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] :
( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > B ) > ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > $o ) > ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S @ R )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > ( coinductive_llist @ B ) ) > ( D > D ) > D > ( coinductive_llist @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( ( C > C ) > C > ( coinductive_llist @ A ) ) @ ( ( D > D ) > D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( coinductive_llist @ A ) ) @ ( D > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) @ ( bNF_rel_fun @ C @ D @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ S @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ) ) )
@ ( coindu1259883913_llist @ C @ A )
@ ( coindu1259883913_llist @ D @ B ) ) ).
% llist.corec_transfer
thf(fact_205_llist__corec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > C ) > ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > D ) > ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > $o ) > ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > $o ) > ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 )
@ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( A > ( coinductive_llist @ C ) ) > ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > ( coinductive_llist @ D ) ) > ( B > B ) > B > ( coinductive_llist @ D ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( ( A > A ) > A > ( coinductive_llist @ C ) ) @ ( ( B > B ) > B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) @ ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ C ) ) @ ( B > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A2 @ ( coindu1486289336t_all2 @ C @ D @ B2 ) ) ) ) ) )
@ ( coindu1259883913_llist @ A @ C )
@ ( coindu1259883913_llist @ B @ D ) ) ).
% llist_corec_transfer
thf(fact_206_Ex__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 )
@ ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 ) ) ) ).
% Ex_transfer
thf(fact_207_All__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 )
@ ^ [P2: B > $o] :
! [X6: B] : ( P2 @ X6 ) ) ) ).
% All_transfer
thf(fact_208_bi__total__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( bi_total @ A @ B )
= ( ^ [R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 )
@ ^ [P2: B > $o] :
! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% bi_total_alt_def2
thf(fact_209_right__total__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( right_total @ A @ B )
= ( ^ [R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5 @ (=>) ) @ (=>)
@ ^ [P2: A > $o] :
! [X6: A] : ( P2 @ X6 )
@ ^ [P2: B > $o] :
! [X6: B] : ( P2 @ X6 ) ) ) ) ).
% right_total_alt_def2
thf(fact_210_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A6: set @ A,P4: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( P4 @ X2 ) ) ) ) ).
% Bex_def_raw
thf(fact_211_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R5: A > B > $o,P4: B > $o,X2: A] :
? [Y4: B] :
( ( R5 @ X2 @ Y4 )
& ( P4 @ Y4 ) ) ) ) ).
% rel_pred_comp_def
thf(fact_212_right__total__All__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( ball @ A @ ( collect @ A @ ( domainp @ A @ B @ A2 ) ) )
@ ^ [P2: B > $o] :
! [X6: B] : ( P2 @ X6 ) ) ) ).
% right_total_All_transfer
thf(fact_213_ball__reg__right,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X12: A] : ( P @ X12 )
=> ! [X7: A] :
( ( member @ A @ X7 @ R )
=> ( Q @ X7 ) ) ) ) ).
% ball_reg_right
thf(fact_214_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P4: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( P4 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_215_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A2: B > ( set @ C ),F2: C > $o,G2: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( A2 @ X2 ) )
=> ( F2 @ Y4 ) )
@ G2 )
= ( ^ [X2: A] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( comp @ B @ ( set @ C ) @ A @ A2 @ G2 @ X2 ) )
=> ( F2 @ Y4 ) ) ) ) ).
% Ball_comp_iff
thf(fact_216_ball__reg,axiom,
! [A: $tType,R: set @ A,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X7: A] :
( ( member @ A @ X7 @ R )
=> ( Q @ X7 ) ) ) ) ).
% ball_reg
thf(fact_217_iterates__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( A > A ) @ ( B > B ) @ ( A > ( coinductive_llist @ A ) ) @ ( B > ( coinductive_llist @ B ) ) @ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) @ ( bNF_rel_fun @ A @ B @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ A2 @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_iterates @ A ) @ ( coinductive_iterates @ B ) ) ).
% iterates_transfer
thf(fact_218_llist_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( coindu543516966_llist @ A )
@ ( coindu543516966_llist @ B ) ) ).
% llist.pred_transfer
thf(fact_219_lfinite__iterates,axiom,
! [A: $tType,F2: A > A,X: A] :
~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F2 @ X ) ) ).
% lfinite_iterates
thf(fact_220_llist_Opred__True,axiom,
! [A: $tType] :
( ( coindu543516966_llist @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: coinductive_llist @ A] : $true ) ) ).
% llist.pred_True
thf(fact_221_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_222_llist__all__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > $o ) @ ( ( coinductive_llist @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( coindu543516966_llist @ A )
@ ( coindu543516966_llist @ B ) ) ).
% llist_all_transfer
thf(fact_223_stream__from__llist__setup_Ostream_Odomain__par,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,DR: C > $o,P22: ( coinductive_llist @ C ) > $o] :
( ( ( domainp @ C @ B @ T2 )
= DR )
=> ( ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ C @ B @ T2 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P22
@ ^ [Xs2: coinductive_llist @ B] :
~ ( coinductive_lfinite @ B @ Xs2 ) )
=> ( ( domainp @ ( coinductive_llist @ C ) @ ( stream @ B ) @ ( coindu773941317stream @ C @ B @ T2 ) )
= ( inf_inf @ ( ( coinductive_llist @ C ) > $o ) @ ( coindu543516966_llist @ C @ DR ) @ P22 ) ) ) ) ).
% stream_from_llist_setup.stream.domain_par
thf(fact_224_case__llist__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,B2: A > B > $o,A2: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( ( C > ( coinductive_llist @ C ) > A ) > ( coinductive_llist @ C ) > A ) @ ( ( D > ( coinductive_llist @ D ) > B ) > ( coinductive_llist @ D ) > B ) @ B2 @ ( bNF_rel_fun @ ( C > ( coinductive_llist @ C ) > A ) @ ( D > ( coinductive_llist @ D ) > B ) @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ ( ( coinductive_llist @ C ) > A ) @ ( ( coinductive_llist @ D ) > B ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ C ) @ ( coinductive_llist @ D ) @ A @ B @ ( coindu1486289336t_all2 @ C @ D @ A2 ) @ B2 ) ) @ ( coindu1381640503_llist @ A @ C ) @ ( coindu1381640503_llist @ B @ D ) ) ).
% case_llist_transfer
thf(fact_225_llist__case__mono,axiom,
! [C: $tType,B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Lnil: A > B,Lcons: A > C > ( coinductive_llist @ C ) > B,X: coinductive_llist @ C] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Lnil )
=> ( ! [X3: C,Xs3: coinductive_llist @ C] :
( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F: A] : ( Lcons @ F @ X3 @ Xs3 ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F: A] : ( coindu1381640503_llist @ B @ C @ ( Lnil @ F ) @ ( Lcons @ F ) @ X ) ) ) ) ).
% llist_case_mono
thf(fact_226_llist_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( H2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( coindu1381640503_llist @ C @ A @ ( H2 @ F1 )
@ ^ [X14: A,X23: coinductive_llist @ A] : ( H2 @ ( F22 @ X14 @ X23 ) )
@ Llist ) ) ).
% llist.case_distrib
thf(fact_227_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P23: A > $o,A2: C > A > $o,P1: C > $o,P22: C > $o] :
( ( ( domainp @ A @ B @ B2 )
= P23 )
=> ( ( ( domainp @ C @ A @ A2 )
= P1 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P22
@ P23 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
= ( inf_inf @ ( C > $o ) @ P1 @ P22 ) ) ) ) ) ).
% pcr_Domainp_par
thf(fact_228_llist_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( coinductive_llist @ A ) > C ) > ( coinductive_llist @ A ) > C ) @ ( ( B > ( coinductive_llist @ B ) > D ) > ( coinductive_llist @ B ) > D ) @ S @ ( bNF_rel_fun @ ( A > ( coinductive_llist @ A ) > C ) @ ( B > ( coinductive_llist @ B ) > D ) @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > C ) @ ( ( coinductive_llist @ B ) > D ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ D @ ( coindu1486289336t_all2 @ A @ B @ R ) @ S ) ) @ ( coindu1381640503_llist @ C @ A ) @ ( coindu1381640503_llist @ D @ B ) ) ).
% llist.case_transfer
thf(fact_229_img__ord__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( partial_img_ord @ A @ C @ B )
= ( ^ [F: A > C,Ord2: C > C > B,X2: A,Y4: A] : ( Ord2 @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) ).
% img_ord_def
thf(fact_230_mk__less__def,axiom,
! [A: $tType] :
( ( partial_mk_less @ A )
= ( ^ [R5: A > A > $o,X2: A,Y4: A] :
( ( R5 @ X2 @ Y4 )
& ~ ( R5 @ Y4 @ X2 ) ) ) ) ).
% mk_less_def
thf(fact_231_type__definition_ORep,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( member @ A @ ( Rep @ X ) @ A2 ) ) ).
% type_definition.Rep
thf(fact_232_type__definition__def,axiom,
! [A: $tType,B: $tType] :
( ( type_definition @ B @ A )
= ( ^ [Rep2: B > A,Abs2: A > B,A6: set @ A] :
( ! [X2: B] : ( member @ A @ ( Rep2 @ X2 ) @ A6 )
& ! [X2: B] :
( ( Abs2 @ ( Rep2 @ X2 ) )
= X2 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ( Rep2 @ ( Abs2 @ Y4 ) )
= Y4 ) ) ) ) ) ).
% type_definition_def
thf(fact_233_type__definition_ORep__inverse,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( Abs @ ( Rep @ X ) )
= X ) ) ).
% type_definition.Rep_inverse
thf(fact_234_type__definition_OAbs__inverse,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( Rep @ ( Abs @ Y ) )
= Y ) ) ) ).
% type_definition.Abs_inverse
thf(fact_235_type__definition_ORep__inject,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B,Y: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( ( Rep @ X )
= ( Rep @ Y ) )
= ( X = Y ) ) ) ).
% type_definition.Rep_inject
thf(fact_236_type__definition_ORep__induct,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A,P: A > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ! [X3: B] : ( P @ ( Rep @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% type_definition.Rep_induct
thf(fact_237_type__definition_OAbs__inject,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( ( Abs @ X )
= ( Abs @ Y ) )
= ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
thf(fact_238_type__definition_OAbs__induct,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,P: B > $o,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( P @ ( Abs @ Y2 ) ) )
=> ( P @ X ) ) ) ).
% type_definition.Abs_induct
thf(fact_239_type__definition_ORep__cases,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,Y: A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ~ ! [X3: B] :
( Y
!= ( Rep @ X3 ) ) ) ) ).
% type_definition.Rep_cases
thf(fact_240_type__definition_OAbs__cases,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,X: B] :
( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
=> ~ ! [Y2: A] :
( ( X
= ( Abs @ Y2 ) )
=> ~ ( member @ A @ Y2 @ A2 ) ) ) ).
% type_definition.Abs_cases
thf(fact_241_type__definition_Ointro,axiom,
! [B: $tType,A: $tType,Rep: B > A,A2: set @ A,Abs: A > B] :
( ! [X3: B] : ( member @ A @ ( Rep @ X3 ) @ A2 )
=> ( ! [X3: B] :
( ( Abs @ ( Rep @ X3 ) )
= X3 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A2 )
=> ( ( Rep @ ( Abs @ Y2 ) )
= Y2 ) )
=> ( type_definition @ B @ A @ Rep @ Abs @ A2 ) ) ) ) ).
% type_definition.intro
thf(fact_242_llist__of__stream__siterates,axiom,
! [A: $tType,F2: A > A,X: A] :
( ( coindu1724414836stream @ A @ ( siterate @ A @ F2 @ X ) )
= ( coinductive_iterates @ A @ F2 @ X ) ) ).
% llist_of_stream_siterates
thf(fact_243_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: B > A,R4: C > A > $o,S3: A > D > $o,A3: C,C3: D] :
( ! [Y2: A] :
? [X7: B] :
( Y2
= ( F2 @ X7 ) )
=> ( ( relcompp @ C @ A @ D @ R4 @ S3 @ A3 @ C3 )
=> ~ ! [B5: B] :
( ( R4 @ A3 @ ( F2 @ B5 ) )
=> ~ ( S3 @ ( F2 @ B5 ) @ C3 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_244_OO__eq,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ B @ B @ R
@ ^ [Y3: B,Z: B] : Y3 = Z )
= R ) ).
% OO_eq
thf(fact_245_eq__OO,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R )
= R ) ).
% eq_OO
thf(fact_246_relcompp_OrelcompI,axiom,
! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,A3: A,B3: B,S3: B > C > $o,C3: C] :
( ( R4 @ A3 @ B3 )
=> ( ( S3 @ B3 @ C3 )
=> ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 ) ) ) ).
% relcompp.relcompI
thf(fact_247_relcompp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,X1: A,X22: C,P: A > C > $o] :
( ( relcompp @ A @ B @ C @ R4 @ S3 @ X1 @ X22 )
=> ( ! [A5: A,B5: B,C4: C] :
( ( R4 @ A5 @ B5 )
=> ( ( S3 @ B5 @ C4 )
=> ( P @ A5 @ C4 ) ) )
=> ( P @ X1 @ X22 ) ) ) ).
% relcompp.inducts
thf(fact_248_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R4: A > D > $o,S3: D > C > $o,T: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R4 @ S3 ) @ T )
= ( relcompp @ A @ D @ B @ R4 @ ( relcompp @ D @ C @ B @ S3 @ T ) ) ) ).
% relcompp_assoc
thf(fact_249_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R5: A > B > $o,S5: B > C > $o,A4: A,C5: C] :
? [B6: B] :
( ( R5 @ A4 @ B6 )
& ( S5 @ B6 @ C5 ) ) ) ) ).
% relcompp_apply
thf(fact_250_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R6: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
? [A4: A,B6: B,C5: C] :
( ( A1 = A4 )
& ( A22 = C5 )
& ( R6 @ A4 @ B6 )
& ( S2 @ B6 @ C5 ) ) ) ) ).
% relcompp.simps
thf(fact_251_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
( ( relcompp @ A @ B @ C @ R4 @ S3 @ A12 @ A23 )
=> ~ ! [B5: B] :
( ( R4 @ A12 @ B5 )
=> ~ ( S3 @ B5 @ A23 ) ) ) ).
% relcompp.cases
thf(fact_252_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R4: A > B > $o,S3: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ R4 @ S3 @ A3 @ C3 )
=> ~ ! [B5: B] :
( ( R4 @ A3 @ B5 )
=> ~ ( S3 @ B5 @ C3 ) ) ) ).
% relcomppE
thf(fact_253_Powp__def,axiom,
! [A: $tType] :
( ( powp @ A )
= ( ^ [A6: A > $o,B4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ B4 )
=> ( A6 @ X2 ) ) ) ) ).
% Powp_def
thf(fact_254_pos__fun__distr,axiom,
! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F3: $tType,R: A > E > $o,S: B > F3 > $o,R8: E > C > $o,S6: F3 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F3 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ R @ S ) @ ( bNF_rel_fun @ E @ C @ F3 @ D @ R8 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R @ R8 ) @ ( relcompp @ B @ F3 @ D @ S @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_255_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R9: A > B > $o,R4: A > B > $o,S7: B > C > $o,S3: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R9 @ R4 )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S7 @ S3 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R9 @ S7 ) @ ( relcompp @ A @ B @ C @ R4 @ S3 ) ) ) ) ).
% relcompp_mono
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( bNF_rel_fun @ a @ a @ ( coinductive_llist @ b ) @ ( stream @ b )
@ ^ [Y3: a,Z: a] : Y3 = Z
@ ^ [X2: coinductive_llist @ b,Y4: stream @ b] :
( X2
= ( coindu1724414836stream @ b @ Y4 ) )
@ xb
@ yb ) ).
thf(conj_1,conjecture,
( xb
= ( comp @ ( stream @ b ) @ ( coinductive_llist @ b ) @ a @ ( coindu1724414836stream @ b ) @ yb ) ) ).
%------------------------------------------------------------------------------