TPTP Problem File: DAT198^1.p
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%------------------------------------------------------------------------------
% File : DAT198^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 164
% 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 : lmirror__164.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 306 ( 108 unt; 46 typ; 0 def)
% Number of atoms : 932 ( 306 equ; 12 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 5128 ( 19 ~; 2 |; 30 &;4807 @)
% ( 0 <=>; 270 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 11 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 1562 (1562 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 44 usr; 4 con; 0-8 aty)
% Number of variables : 1739 ( 262 ^;1391 !; 18 ?;1739 :)
% ( 68 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:42:04.265
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $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 (42)
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_Coinductive__List_Oiterates,type,
coinductive_iterates:
!>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_OldropWhile,type,
coindu218763757pWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_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_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Ollist__all2,type,
coindu1486289336t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olmap,type,
coinductive_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coinductive_llist @ A ) > ( coinductive_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List_Ollist_Opred__llist,type,
coindu543516966_llist:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
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_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_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
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_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_HOL_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
lMirro427583474mirror:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
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_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_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_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[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_Oleft__unique,type,
left_unique:
!>[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_Oright__unique,type,
right_unique:
!>[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_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_v_A,type,
a2: a > b > $o ).
%----Relevant facts (256)
thf(fact_0_llist__all2__lmirror,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro427583474mirror @ A @ Xs ) @ ( lMirro427583474mirror @ B @ Ys ) )
= ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys ) ) ).
% llist_all2_lmirror
thf(fact_1_llist__all2__lmirrorD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro427583474mirror @ A @ Xs ) @ ( lMirro427583474mirror @ B @ Ys ) )
=> ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys ) ) ).
% llist_all2_lmirrorD
thf(fact_2_llist__all2__lmirrorI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro427583474mirror @ A @ Xs ) @ ( lMirro427583474mirror @ B @ Ys ) ) ) ).
% llist_all2_lmirrorI
thf(fact_3_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D] :
( ! [X: A,Y: B] :
( ( A2 @ X @ Y )
=> ( B2 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G ) ) ).
% rel_funI
thf(fact_4_llist__all2__rsp,axiom,
! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T: B > B > $o,X2: coinductive_llist @ A,Y2: coinductive_llist @ B,A3: coinductive_llist @ A,B3: coinductive_llist @ B] :
( ! [X: A,Y: B] :
( ( R @ X @ Y )
=> ! [A4: A,B4: B] :
( ( R @ A4 @ B4 )
=> ( ( S @ X @ A4 )
= ( T @ Y @ B4 ) ) ) )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y2 )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
=> ( ( coindu1486289336t_all2 @ A @ A @ S @ X2 @ A3 )
= ( coindu1486289336t_all2 @ B @ B @ T @ Y2 @ B3 ) ) ) ) ) ).
% llist_all2_rsp
thf(fact_5_llist__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,P2: A > B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X: A,Y: B] :
( ( P @ X @ Y )
=> ( P2 @ X @ Y ) )
=> ( coindu1486289336t_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% llist_all2_mono
thf(fact_6_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_7_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X2: coinductive_llist @ B] :
( ! [X: B] : ( Ra @ X @ X )
=> ( coindu1486289336t_all2 @ B @ B @ Ra @ X2 @ X2 ) ) ).
% llist.rel_refl
thf(fact_8_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y2: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
=> ( ( A2 @ X2 @ Y2 )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ) ).
% rel_funD
thf(fact_9_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,F2: A > C] : ( F2 @ S2 )
@ ^ [S2: B,F2: B > D] : ( F2 @ S2 ) ) ).
% let_rsp
thf(fact_10_rel__funE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o,F: A > C,G: B > D,X2: A,Y2: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
=> ( ( A2 @ X2 @ Y2 )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ) ).
% rel_funE
thf(fact_11_apply__rsp_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X2: A,Y2: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X2 @ Y2 )
=> ( R2 @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ) ).
% apply_rsp'
thf(fact_12_rel__funD2,axiom,
! [B: $tType,C: $tType,A: $tType,A2: A > A > $o,B2: B > C > $o,F: A > B,G: A > C,X2: A] :
( ( bNF_rel_fun @ A @ A @ B @ C @ A2 @ B2 @ F @ G )
=> ( ( A2 @ X2 @ X2 )
=> ( B2 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% rel_funD2
thf(fact_13_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X3: A > B > $o,A2: C > D > $o,F: A > C,G: B > D,Y4: A > B > $o,B2: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X3 @ A2 @ F @ G )
=> ( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X3 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( A2 @ X @ Y )
=> ( B2 @ X @ Y ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_14_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: A > B > $o,X3: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X3 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( A2 @ X @ Y )
=> ( B2 @ X @ Y ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X3 @ A2 @ F @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_15_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_16_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_17_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,F2: A > C] : ( F2 @ S2 )
@ ^ [S2: B,F2: B > D] : ( F2 @ S2 ) ) ).
% Let_transfer
thf(fact_18_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,T: D > F3 > $o,F: A > C > D,G: B > E > F3] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T ) @ F @ G )
= ( ! [X4: A,Y5: B] :
( ( R @ X4 @ Y5 )
=> ( bNF_rel_fun @ C @ E @ D @ F3 @ S @ T @ ( F @ X4 ) @ ( G @ Y5 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_19_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_20_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_21_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_22_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_23_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_24_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_25_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X2: D > B] :
( ! [X: B] : ( Ra @ X @ X )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl
thf(fact_26_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_27_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_28_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_29_map__fun__parametric,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,H: $tType,G2: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G2 > $o,D2: F3 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F3 ) > ( B > E ) > A > F3 ) @ ( ( G2 > H ) > ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F3 ) @ ( G2 > H ) @ ( ( B > E ) > A > F3 ) @ ( ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ E @ G2 @ F3 @ H @ C2 @ D2 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G2 ) @ ( A > F3 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G2 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F3 @ H @ A2 @ D2 ) ) ) @ ( map_fun @ A @ B @ E @ F3 ) @ ( map_fun @ C @ D @ G2 @ H ) ) ).
% map_fun_parametric
thf(fact_30_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_31_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_32_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_33_ldropWhile__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
@ ( coindu218763757pWhile @ A )
@ ( coindu218763757pWhile @ B ) ) ).
% ldropWhile_transfer
thf(fact_34_ltakeWhile__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
@ ( coindu501562517eWhile @ A )
@ ( coindu501562517eWhile @ B ) ) ).
% ltakeWhile_transfer
thf(fact_35_lfilter__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) )
@ ( coinductive_lfilter @ A )
@ ( coinductive_lfilter @ B ) ) ).
% lfilter_transfer
thf(fact_36_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G2: $tType,F3: $tType,Rb: A > F3 > $o,Sd: B > G2 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F3 > G2 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G2 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G2 @ Rb @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G2 )
@ ( bNF_rel_fun @ D @ D @ A @ F3
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Rb )
@ ( bNF_rel_fun @ D @ D @ B @ G2
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F3 @ G2 @ D ) ) ).
% fun.map_transfer
thf(fact_37_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_38_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_39_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,Abs2: C > D,Rep2: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs2 @ Rep2 )
=> ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs2 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep2 ) ) ) ) ).
% fun_quotient3
thf(fact_40_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_41_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H2: A > C,R12: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H2 )
= ( comp @ D @ B @ A @ R12 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R12 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H2 )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_42_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H2: E > A,R3: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H2 )
= R3 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H2 ) )
= ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_43_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H2: A > C,R3: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H2 )
= R3 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H2 )
= ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_44_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H2: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H2 ) )
= ( comp @ A @ B @ D @ L @ H2 ) ) ) ).
% rewriteL_comp_comp
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_47_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_48_Quotient3__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R3 @ R3 )
=> ( R @ ( Rep @ ( Abs @ R3 ) ) @ R3 ) ) ) ).
% Quotient3_rep_abs
thf(fact_49_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_50_Quotient3__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R3: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R3 @ S3 )
=> ( ( Abs @ R3 )
= ( Abs @ S3 ) ) ) ) ).
% Quotient3_rel_abs
thf(fact_51_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_52_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_53_Quotient3__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R3: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R3 @ S3 )
=> ( R @ S3 @ S3 ) ) ) ).
% Quotient3_refl2
thf(fact_54_Quotient3__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R3: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R3 @ S3 )
=> ( R @ R3 @ R3 ) ) ) ).
% Quotient3_refl1
thf(fact_55_Quotient3__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R3: A,S3: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( ( R @ R3 @ R3 )
& ( R @ S3 @ S3 )
& ( ( Abs @ R3 )
= ( Abs @ S3 ) ) )
= ( R @ R3 @ S3 ) ) ) ).
% Quotient3_rel
thf(fact_56_Quotient3__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient3 @ A @ B )
= ( ^ [R4: A > A > $o,Abs3: A > B,Rep3: B > A] :
( ! [A5: B] :
( ( Abs3 @ ( Rep3 @ A5 ) )
= A5 )
& ! [A5: B] : ( R4 @ ( Rep3 @ A5 ) @ ( Rep3 @ A5 ) )
& ! [R5: A,S2: A] :
( ( R4 @ R5 @ S2 )
= ( ( R4 @ R5 @ R5 )
& ( R4 @ S2 @ S2 )
& ( ( Abs3 @ R5 )
= ( Abs3 @ S2 ) ) ) ) ) ) ) ).
% Quotient3_def
thf(fact_57_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X2: C,H2: D > A,K: C > D] :
( ( ( F @ ( G @ X2 ) )
= ( H2 @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ C @ H2 @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_58_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_59_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_60_Quotient3I,axiom,
! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R: B > B > $o] :
( ! [A4: A] :
( ( Abs @ ( Rep @ A4 ) )
= A4 )
=> ( ! [A4: A] : ( R @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
=> ( ! [R6: B,S4: B] :
( ( R @ R6 @ S4 )
= ( ( R @ R6 @ R6 )
& ( R @ S4 @ S4 )
& ( ( Abs @ R6 )
= ( Abs @ S4 ) ) ) )
=> ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% Quotient3I
thf(fact_61_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_62_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,Abs22: C > D,Rep22: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
=> ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep22 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs22 @ Rep12 ) @ Abs12 )
@ ^ [S2: C,F2: C > A] : ( F2 @ S2 ) )
= ( ^ [S2: D,F2: D > B] : ( F2 @ S2 ) ) ) ) ) ).
% let_prs
thf(fact_63_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,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F3,Rep32: F3 > E] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F3 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs12 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep12 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F3 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_64_llist__all2__lfilterI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X: A,Y: B] :
( ( P @ X @ Y )
=> ( ( Q1 @ X )
= ( Q2 @ Y ) ) )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lfilter @ A @ Q1 @ Xs ) @ ( coinductive_lfilter @ B @ Q2 @ Ys ) ) ) ) ).
% llist_all2_lfilterI
thf(fact_65_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,F: A > C,X2: B] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R @ S @ F @ F )
=> ( S @ ( F @ ( Rep @ X2 ) ) @ ( F @ ( Rep @ X2 ) ) ) ) ) ).
% apply_rspQ3''
thf(fact_66_apply__rspQ3,axiom,
! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F: A > C,G: A > C,X2: A,Y2: A] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X2 @ Y2 )
=> ( R2 @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ) ) ).
% apply_rspQ3
thf(fact_67_llist__all2__ldropWhileI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Q1: A > $o,Q2: B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X: A,Y: B] :
( ( P @ X @ Y )
=> ( ( Q1 @ X )
= ( Q2 @ Y ) ) )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coindu218763757pWhile @ A @ Q1 @ Xs ) @ ( coindu218763757pWhile @ B @ Q2 @ Ys ) ) ) ) ).
% llist_all2_ldropWhileI
thf(fact_68_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_69_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R2: A > C > $o,R32: 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 @ R32 ) @ ( 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 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% o_rsp(1)
thf(fact_70_o__rsp_I2_J,axiom,
! [E: $tType,F3: $tType,H: $tType,G2: $tType,R1: E > F3 > $o] :
( bNF_rel_fun @ ( G2 > H ) @ ( G2 > H ) @ ( ( E > G2 ) > E > H ) @ ( ( F3 > G2 ) > F3 > H )
@ ^ [Y3: G2 > H,Z: G2 > H] : Y3 = Z
@ ( bNF_rel_fun @ ( E > G2 ) @ ( F3 > G2 ) @ ( E > H ) @ ( F3 > H )
@ ( bNF_rel_fun @ E @ F3 @ G2 @ G2 @ R1
@ ^ [Y3: G2,Z: G2] : Y3 = Z )
@ ( bNF_rel_fun @ E @ F3 @ H @ H @ R1
@ ^ [Y3: H,Z: H] : Y3 = Z ) )
@ ( comp @ G2 @ H @ E )
@ ( comp @ G2 @ H @ F3 ) ) ).
% o_rsp(2)
thf(fact_71_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F2: B > C,G3: D > A,H3: C > D,X4: B] : ( G3 @ ( H3 @ ( F2 @ X4 ) ) ) ) ) ).
% map_fun_apply
thf(fact_72_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G3: C > B,X4: C] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_73_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F2: C > A,G3: B > D,H3: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G3 @ H3 ) @ F2 ) ) ) ).
% map_fun_def
thf(fact_74_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F: E > C,G: D > F3,H2: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H2 @ I ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_fun.comp
thf(fact_75_map__fun_Ocompositionality,axiom,
! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F3,H2: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F3 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H2 @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H2 @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_76_llist_Omap__transfer,axiom,
! [A: $tType,B: $tType,F3: $tType,E: $tType,Rb: A > E > $o,Sd: B > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F3 ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ E ) > ( coinductive_llist @ F3 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ F3 ) @ ( coindu1486289336t_all2 @ A @ E @ Rb ) @ ( coindu1486289336t_all2 @ B @ F3 @ Sd ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ E @ F3 ) ) ).
% llist.map_transfer
thf(fact_77_lmap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ C ) > ( coinductive_llist @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ C ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ D ) @ ( coindu1486289336t_all2 @ A @ C @ A2 ) @ ( coindu1486289336t_all2 @ B @ D @ B2 ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ C @ D ) ) ).
% lmap_transfer
thf(fact_78_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_79_LCons__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ A2 @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) @ ( coindu1486289336t_all2 @ A @ B @ A2 ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% LCons_transfer
thf(fact_80_llist_Oinject,axiom,
! [A: $tType,X21: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X222 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% llist.inject
thf(fact_81_llist__all2__LCons__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: coinductive_llist @ A,Y21: B,Y22: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 ) ) ) ).
% llist_all2_LCons_LCons
thf(fact_82_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
( ( ( P @ X2 )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_83_ldropWhile__LCons,axiom,
! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
( ( ( P @ X2 )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coindu218763757pWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_84_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X2: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
= ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( ~ ( IS_LNIL @ B3 )
& ( X2
= ( LHD @ B3 ) )
& ( Xs
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_85_llist_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A,Aa2: coinductive_llist @ A] :
( ( coindu543516966_llist @ A @ P @ ( coinductive_LCons @ A @ A3 @ Aa2 ) )
= ( ( P @ A3 )
& ( coindu543516966_llist @ A @ P @ Aa2 ) ) ) ).
% llist.pred_inject(2)
thf(fact_86_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X222: coinductive_llist @ A] :
( ( coinductive_lmap @ A @ B @ F @ ( coinductive_LCons @ A @ X21 @ X222 ) )
= ( coinductive_LCons @ B @ ( F @ X21 ) @ ( coinductive_lmap @ A @ B @ F @ X222 ) ) ) ).
% llist.simps(13)
thf(fact_87_bex1__rel__aux,axiom,
! [A: $tType,R: A > A > $o,X2: A > $o,Y2: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X2 @ Xa2 )
= ( Y2 @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ X2 )
=> ( bex1_rel @ A @ R @ Y2 ) ) ) ).
% bex1_rel_aux
thf(fact_88_bex1__rel__aux2,axiom,
! [A: $tType,R: A > A > $o,X2: A > $o,Y2: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X2 @ Xa2 )
= ( Y2 @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ Y2 )
=> ( bex1_rel @ A @ R @ X2 ) ) ) ).
% bex1_rel_aux2
thf(fact_89_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Y2: A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lmap @ B @ A @ F @ Xs )
= ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ? [X4: B,Xs2: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ B @ X4 @ Xs2 ) )
& ( Y2
= ( F @ X4 ) )
& ( Ys
= ( coinductive_lmap @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_90_iterates__equality,axiom,
! [A: $tType,H2: A > ( coinductive_llist @ A ),F: A > A] :
( ! [X: A] :
( ( H2 @ X )
= ( coinductive_LCons @ A @ X @ ( coinductive_lmap @ A @ A @ F @ ( H2 @ X ) ) ) )
=> ( H2
= ( coinductive_iterates @ A @ F ) ) ) ).
% iterates_equality
thf(fact_91_iterates__lmap,axiom,
! [A: $tType] :
( ( coinductive_iterates @ A )
= ( ^ [F2: A > A,X4: A] : ( coinductive_LCons @ A @ X4 @ ( coinductive_lmap @ A @ A @ F2 @ ( coinductive_iterates @ A @ F2 @ X4 ) ) ) ) ) ).
% iterates_lmap
thf(fact_92_llist_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: coinductive_llist @ A] :
( ( coinductive_lmap @ B @ C @ G @ ( coinductive_lmap @ A @ B @ F @ V ) )
= ( coinductive_lmap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% llist.map_comp
thf(fact_93_lfilter__lmap,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lmap @ B @ A @ F @ ( coinductive_lfilter @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% lfilter_lmap
thf(fact_94_ltakeWhile__lmap,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: coinductive_llist @ B] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lmap @ B @ A @ F @ ( coindu501562517eWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% ltakeWhile_lmap
thf(fact_95_ldropWhile__lmap,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: coinductive_llist @ B] :
( ( coindu218763757pWhile @ A @ P @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lmap @ B @ A @ F @ ( coindu218763757pWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% ldropWhile_lmap
thf(fact_96_lmap__iterates,axiom,
! [A: $tType,F: A > A,X2: A] :
( ( coinductive_lmap @ A @ A @ F @ ( coinductive_iterates @ A @ F @ X2 ) )
= ( coinductive_iterates @ A @ F @ ( F @ X2 ) ) ) ).
% lmap_iterates
thf(fact_97_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: coinductive_llist @ A,Y22: coinductive_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X222 @ Y22 )
=> ( coindu1486289336t_all2 @ A @ B @ R @ ( coinductive_LCons @ A @ X21 @ X222 ) @ ( coinductive_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_98_llist__all2__LCons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
= ( ? [Y5: B,Ys2: coinductive_llist @ B] :
( ( Ys
= ( coinductive_LCons @ B @ Y5 @ Ys2 ) )
& ( P @ X2 @ Y5 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys2 ) ) ) ) ).
% llist_all2_LCons1
thf(fact_99_llist__all2__LCons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Y2: B,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LCons @ B @ Y2 @ Ys ) )
= ( ? [X4: A,Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X4 @ Xs2 ) )
& ( P @ X4 @ Y2 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs2 @ Ys ) ) ) ) ).
% llist_all2_LCons2
thf(fact_100_lmap__lmirror,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( lMirro427583474mirror @ B @ Xs ) )
= ( lMirro427583474mirror @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) ) ) ).
% lmap_lmirror
thf(fact_101_llist_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X2: coinductive_llist @ A] :
( ( coindu543516966_llist @ B @ Q @ ( coinductive_lmap @ A @ B @ F @ X2 ) )
= ( coindu543516966_llist @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X2 ) ) ).
% llist.pred_map
thf(fact_102_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coinductive_llist @ A ) > B,X21: A,X222: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ ( coinductive_LCons @ A @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 ) ) ).
% llist.simps(5)
thf(fact_103_lfilter__LCons__seek,axiom,
! [A: $tType,P3: A > $o,X2: A,L: coinductive_llist @ A] :
( ~ ( P3 @ X2 )
=> ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X2 @ L ) )
= ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_104_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X2: A,Xs: coinductive_llist @ A] :
( ( P @ X2 )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_LCons @ A @ X2 @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_105_iterates_Ocode,axiom,
! [A: $tType] :
( ( coinductive_iterates @ A )
= ( ^ [F2: A > A,X4: A] : ( coinductive_LCons @ A @ X4 @ ( coinductive_iterates @ A @ F2 @ ( F2 @ X4 ) ) ) ) ) ).
% iterates.code
thf(fact_106_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ( ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A3 )
= ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_107_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P3 @ A3 )
=> ( ( coindu1259883913_llist @ C @ A @ P3 @ G21 @ Q22 @ G221 @ G222 @ A3 )
= ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_108_lmap__unfold__llist,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,IS_LNIL: C > $o,LHD: C > B,LTL: C > C,B3: C] :
( ( coinductive_lmap @ B @ A @ F @ ( coindu1441602521_llist @ C @ B @ IS_LNIL @ LHD @ LTL @ B3 ) )
= ( coindu1441602521_llist @ C @ A @ IS_LNIL @ ( comp @ B @ A @ C @ F @ LHD ) @ LTL @ B3 ) ) ).
% lmap_unfold_llist
thf(fact_109_llist_Omap__o__corec,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > $o,Ga: C > A,Gb: C > $o,Gc: C > ( coinductive_llist @ A ),Gd: C > C] :
( ( comp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ ( coinductive_lmap @ A @ B @ F ) @ ( coindu1259883913_llist @ C @ A @ G @ Ga @ Gb @ Gc @ Gd ) )
= ( coindu1259883913_llist @ C @ B @ G @ ( comp @ A @ B @ C @ F @ Ga ) @ Gb @ ( comp @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ C @ ( coinductive_lmap @ A @ B @ F ) @ Gc ) @ Gd ) ) ).
% llist.map_o_corec
thf(fact_110_lmap__corec__llist,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,IS_LNIL: C > $o,LHD: C > B,EndORmore: C > $o,TTL_end: C > ( coinductive_llist @ B ),TTL_more: C > C,B3: C] :
( ( coinductive_lmap @ B @ A @ F @ ( coindu1259883913_llist @ C @ B @ IS_LNIL @ LHD @ EndORmore @ TTL_end @ TTL_more @ B3 ) )
= ( coindu1259883913_llist @ C @ A @ IS_LNIL @ ( comp @ B @ A @ C @ F @ LHD ) @ EndORmore @ ( comp @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ C @ ( coinductive_lmap @ B @ A @ F ) @ TTL_end ) @ TTL_more @ B3 ) ) ).
% lmap_corec_llist
thf(fact_111_lfilter__eq__LCons,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,X2: A,Xs3: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
=> ? [Xs4: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_lfilter @ A @ P @ Xs4 ) )
& ( ( coindu218763757pWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs )
= ( coinductive_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% lfilter_eq_LCons
thf(fact_112_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_113_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ! [V2: A] :
( ( A3 @ ( B3 @ V2 ) )
= ( C3 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_114_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C3: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_115_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H2: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H2 )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H2 ) ) ) ).
% comp_assoc
thf(fact_116_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G3: A > B,X4: A] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_117_llist_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ R @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) @ ( coinductive_LCons @ A ) @ ( coinductive_LCons @ B ) ) ).
% llist.ctr_transfer(2)
thf(fact_118_lmember__code_I2_J,axiom,
! [A: $tType,X2: A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
| ( coinductive_lmember @ A @ X2 @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_119_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X2: C,N: D > A,H2: C > D,F: A > E] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H2 @ X2 ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X2 )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H2 @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_120_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X2: C,F4: D > A,G4: E > D,X5: E] :
( ( ( F @ ( G @ X2 ) )
= ( F4 @ ( G4 @ X5 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ E @ F4 @ G4 @ X5 ) ) ) ).
% comp_cong
thf(fact_121_if__prs,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( map_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) @ ( id @ $o ) @ ( map_fun @ B @ A @ ( A > A ) @ ( B > B ) @ Rep @ ( map_fun @ B @ A @ A @ B @ Rep @ Abs ) ) @ ( if @ A ) )
= ( if @ B ) ) ) ).
% if_prs
thf(fact_122_o__prs_I2_J,axiom,
! [F3: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H: $tType,G2: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F3,Rep32: F3 > E] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F3 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( G2 > H ) @ ( G2 > H ) @ ( ( A > G2 ) > A > H ) @ ( ( B > G2 ) > B > H ) @ ( id @ ( G2 > H ) ) @ ( map_fun @ ( B > G2 ) @ ( A > G2 ) @ ( A > H ) @ ( B > H ) @ ( map_fun @ A @ B @ G2 @ G2 @ Abs12 @ ( id @ G2 ) ) @ ( map_fun @ B @ A @ H @ H @ Rep12 @ ( id @ H ) ) ) @ ( comp @ G2 @ H @ A ) )
= ( comp @ G2 @ H @ B ) ) ) ) ) ).
% o_prs(2)
thf(fact_123_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_124_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X4: A] : X4 ) ) ).
% id_apply
thf(fact_125_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_126_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_127_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_128_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_129_map__fun__id,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( A > B ) ) ) ).
% map_fun_id
thf(fact_130_llist_Omap__id0,axiom,
! [A: $tType] :
( ( coinductive_lmap @ A @ A @ ( id @ A ) )
= ( id @ ( coinductive_llist @ A ) ) ) ).
% llist.map_id0
thf(fact_131_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X4: A] :
( ( F @ X4 )
= X4 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_132_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X4: A] : X4 ) ) ).
% id_def
thf(fact_133_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_134_bi__total__def,axiom,
! [B: $tType,A: $tType] :
( ( bi_total @ A @ B )
= ( ^ [R4: A > B > $o] :
( ! [X4: A] :
( ^ [P4: B > $o] :
? [X6: B] : ( P4 @ X6 )
@ ( R4 @ X4 ) )
& ! [Y5: B] :
? [X4: A] : ( R4 @ X4 @ Y5 ) ) ) ) ).
% bi_total_def
thf(fact_135_bi__total__eq,axiom,
! [A: $tType] :
( bi_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% bi_total_eq
thf(fact_136_id__rsp,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ A @ B @ R @ R @ ( id @ A ) @ ( id @ B ) ) ).
% id_rsp
thf(fact_137_id__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 @ ( id @ A ) @ ( id @ B ) ) ).
% id_transfer
thf(fact_138_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_139_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B3 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C3 ) )
=> ( ( A3 @ ( B3 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_140_llist_Omap__id,axiom,
! [A: $tType,T2: coinductive_llist @ A] :
( ( coinductive_lmap @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% llist.map_id
thf(fact_141_identity__quotient3,axiom,
! [A: $tType] :
( quotient3 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( id @ A )
@ ( id @ A ) ) ).
% identity_quotient3
thf(fact_142_abs__o__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( comp @ A @ B @ B @ Abs @ Rep )
= ( id @ B ) ) ) ).
% abs_o_rep
thf(fact_143_id__prs,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( map_fun @ B @ A @ A @ B @ Rep @ Abs @ ( id @ A ) )
= ( id @ B ) ) ) ).
% id_prs
thf(fact_144_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_145_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_146_swap__comp__involutory,axiom,
! [B: $tType,A: $tType,A3: A,B3: A] :
( ( comp @ ( A > B ) @ ( A > B ) @ ( A > B ) @ ( swap @ A @ B @ A3 @ B3 ) @ ( swap @ A @ B @ A3 @ B3 ) )
= ( id @ ( A > B ) ) ) ).
% swap_comp_involutory
thf(fact_147_ex1__prs,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Absf: A > B,Repf: B > A,F: B > $o] :
( ( quotient3 @ A @ B @ R @ Absf @ Repf )
=> ( ( map_fun @ ( B > $o ) @ ( A > $o ) @ $o @ $o @ ( map_fun @ A @ B @ $o @ $o @ Absf @ ( id @ $o ) ) @ ( id @ $o ) @ ( bex1_rel @ A @ R ) @ F )
= ( ? [X4: B] :
( ( F @ X4 )
& ! [Y5: B] :
( ( F @ Y5 )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% ex1_prs
thf(fact_148_swap__apply_I3_J,axiom,
! [A: $tType,B: $tType,C3: B,A3: B,B3: B,F: B > A] :
( ( C3 != A3 )
=> ( ( C3 != B3 )
=> ( ( swap @ B @ A @ A3 @ B3 @ F @ C3 )
= ( F @ C3 ) ) ) ) ).
% swap_apply(3)
thf(fact_149_swap__apply_I2_J,axiom,
! [A: $tType,B: $tType,A3: B,B3: B,F: B > A] :
( ( swap @ B @ A @ A3 @ B3 @ F @ B3 )
= ( F @ A3 ) ) ).
% swap_apply(2)
thf(fact_150_swap__apply_I1_J,axiom,
! [A: $tType,B: $tType,A3: B,B3: B,F: B > A] :
( ( swap @ B @ A @ A3 @ B3 @ F @ A3 )
= ( F @ B3 ) ) ).
% swap_apply(1)
thf(fact_151_swap__self,axiom,
! [B: $tType,A: $tType,A3: A,F: A > B] :
( ( swap @ A @ B @ A3 @ A3 @ F )
= F ) ).
% swap_self
thf(fact_152_swap__nilpotent,axiom,
! [B: $tType,A: $tType,A3: A,B3: A,F: A > B] :
( ( swap @ A @ B @ A3 @ B3 @ ( swap @ A @ B @ A3 @ B3 @ F ) )
= F ) ).
% swap_nilpotent
thf(fact_153_swap__triple,axiom,
! [B: $tType,A: $tType,A3: A,C3: A,B3: A,F: A > B] :
( ( A3 != C3 )
=> ( ( B3 != C3 )
=> ( ( swap @ A @ B @ A3 @ B3 @ ( swap @ A @ B @ B3 @ C3 @ ( swap @ A @ B @ A3 @ B3 @ F ) ) )
= ( swap @ A @ B @ A3 @ C3 @ F ) ) ) ) ).
% swap_triple
thf(fact_154_swap__commute,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A5: A,B5: A] : ( swap @ A @ B @ B5 @ A5 ) ) ) ).
% swap_commute
thf(fact_155_comp__swap,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,A3: A,B3: A,G: A > C] :
( ( comp @ C @ B @ A @ F @ ( swap @ A @ C @ A3 @ B3 @ G ) )
= ( swap @ A @ B @ A3 @ B3 @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% comp_swap
thf(fact_156_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_157_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_158_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_159_rev__implies__def,axiom,
( rev_implies
= ( ^ [X4: $o,Y5: $o] :
( Y5
=> X4 ) ) ) ).
% rev_implies_def
thf(fact_160_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_161_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T3: A > B > $o,X4: A] :
( ^ [P4: B > $o] :
? [X6: B] : ( P4 @ X6 )
@ ( T3 @ X4 ) ) ) ) ).
% Domainp_iff
thf(fact_162_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_163_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_164_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_165_Ex1__parametric,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( 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
@ ( ex1 @ A )
@ ( ex1 @ B ) ) ) ) ).
% Ex1_parametric
thf(fact_166_bi__unique__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( ( bi_unique @ C @ D @ B2 )
=> ( bi_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% bi_unique_fun
thf(fact_167_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_168_DEADID_Orel__reflp,axiom,
! [A: $tType] :
( reflp @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% DEADID.rel_reflp
thf(fact_169_bi__uniqueDl,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,X2: A,Y2: B,Z2: A] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( A2 @ X2 @ Y2 )
=> ( ( A2 @ Z2 @ Y2 )
=> ( X2 = Z2 ) ) ) ) ).
% bi_uniqueDl
thf(fact_170_bi__uniqueDr,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,X2: A,Y2: B,Z2: B] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( A2 @ X2 @ Y2 )
=> ( ( A2 @ X2 @ Z2 )
=> ( Y2 = Z2 ) ) ) ) ).
% bi_uniqueDr
thf(fact_171_bi__unique__eq,axiom,
! [A: $tType] :
( bi_unique @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% bi_unique_eq
thf(fact_172_bi__unique__def,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R4: A > B > $o] :
( ! [X4: A,Y5: B,Z3: B] :
( ( R4 @ X4 @ Y5 )
=> ( ( R4 @ X4 @ Z3 )
=> ( Y5 = Z3 ) ) )
& ! [X4: A,Y5: A,Z3: B] :
( ( R4 @ X4 @ Z3 )
=> ( ( R4 @ Y5 @ Z3 )
=> ( X4 = Y5 ) ) ) ) ) ) ).
% bi_unique_def
thf(fact_173_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_174_llist_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_unique @ A @ B @ R )
=> ( bi_unique @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.bi_unique_rel
thf(fact_175_bi__total__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( bi_total @ C @ D @ B2 )
=> ( bi_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% bi_total_fun
thf(fact_176_eq__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( 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: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ).
% eq_transfer
thf(fact_177_bi__unique__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R4: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R4
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R4
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ) ).
% bi_unique_alt_def2
thf(fact_178_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_179_fun__upd__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ A @ B @ ( C > A > C ) @ ( D > B > D ) @ A2 @ ( bNF_rel_fun @ C @ D @ ( A > C ) @ ( B > D ) @ B2 @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) ) ).
% fun_upd_transfer
thf(fact_180_right__unique__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A2 )
=> ( ( bi_unique @ C @ D @ B2 )
=> ( ( 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_unique @ A @ C )
@ ( right_unique @ B @ D ) ) ) ) ) ).
% right_unique_parametric
thf(fact_181_left__unique__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( 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_unique @ A @ C )
@ ( left_unique @ B @ D ) ) ) ) ) ).
% left_unique_parametric
thf(fact_182_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X4: B,Y5: A,Z3: B] : ( if @ A @ ( Z3 = X4 ) @ Y5 @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_183_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X2: A] :
( ( fun_upd @ A @ B @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_184_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y2: B,Z2: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X2 @ Y2 ) @ X2 @ Z2 )
= ( fun_upd @ A @ B @ F @ X2 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_185_bi__uniqueI,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_unique @ A @ B @ R )
=> ( ( right_unique @ A @ B @ R )
=> ( bi_unique @ A @ B @ R ) ) ) ).
% bi_uniqueI
thf(fact_186_bi__unique__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [A6: A > B > $o] :
( ( left_unique @ A @ B @ A6 )
& ( right_unique @ A @ B @ A6 ) ) ) ) ).
% bi_unique_alt_def
thf(fact_187_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X2: A,Y2: C] :
( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X2 @ Y2 ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X2 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_188_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y2: B] :
( ( ( fun_upd @ A @ B @ F @ X2 @ Y2 )
= F )
= ( ( F @ X2 )
= Y2 ) ) ).
% fun_upd_idem_iff
thf(fact_189_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A3: A,C3: A,M2: A > B,B3: B,D3: B] :
( ( A3 != C3 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ A3 @ B3 ) @ C3 @ D3 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ C3 @ D3 ) @ A3 @ B3 ) ) ) ).
% fun_upd_twist
thf(fact_190_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X2: A,F: A > B,Y2: B] :
( ( Z2 != X2 )
=> ( ( fun_upd @ A @ B @ F @ X2 @ Y2 @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_191_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X2: B,Y2: A] :
( ( fun_upd @ B @ A @ F @ X2 @ Y2 @ X2 )
= Y2 ) ).
% fun_upd_same
thf(fact_192_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B,Y2: A] :
( ( ( F @ X2 )
= Y2 )
=> ( ( fun_upd @ B @ A @ F @ X2 @ Y2 )
= F ) ) ).
% fun_upd_idem
thf(fact_193_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y2: B,G: A > B,Z2: B] :
( ( ( fun_upd @ A @ B @ F @ X2 @ Y2 )
= ( fun_upd @ A @ B @ G @ X2 @ Z2 ) )
=> ( Y2 = Z2 ) ) ).
% fun_upd_eqD
thf(fact_194_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A5: A,B5: B,X4: A] : ( if @ B @ ( X4 = A5 ) @ B5 @ ( F2 @ X4 ) ) ) ) ).
% fun_upd_def
thf(fact_195_right__unique__def,axiom,
! [B: $tType,A: $tType] :
( ( right_unique @ A @ B )
= ( ^ [R4: A > B > $o] :
! [X4: A,Y5: B,Z3: B] :
( ( R4 @ X4 @ Y5 )
=> ( ( R4 @ X4 @ Z3 )
=> ( Y5 = Z3 ) ) ) ) ) ).
% right_unique_def
thf(fact_196_right__unique__eq,axiom,
! [A: $tType] :
( right_unique @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% right_unique_eq
thf(fact_197_left__unique__def,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R4: A > B > $o] :
! [X4: A,Y5: A,Z3: B] :
( ( R4 @ X4 @ Z3 )
=> ( ( R4 @ Y5 @ Z3 )
=> ( X4 = Y5 ) ) ) ) ) ).
% left_unique_def
thf(fact_198_left__unique__eq,axiom,
! [A: $tType] :
( left_unique @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% left_unique_eq
thf(fact_199_right__uniqueI,axiom,
! [B: $tType,A: $tType,A2: A > B > $o] :
( ! [X: A,Y: B,Z4: B] :
( ( A2 @ X @ Y )
=> ( ( A2 @ X @ Z4 )
=> ( Y = Z4 ) ) )
=> ( right_unique @ A @ B @ A2 ) ) ).
% right_uniqueI
thf(fact_200_right__uniqueD,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,X2: A,Y2: B,Z2: B] :
( ( right_unique @ A @ B @ A2 )
=> ( ( A2 @ X2 @ Y2 )
=> ( ( A2 @ X2 @ Z2 )
=> ( Y2 = Z2 ) ) ) ) ).
% right_uniqueD
thf(fact_201_left__uniqueI,axiom,
! [B: $tType,A: $tType,A2: A > B > $o] :
( ! [X: A,Y: A,Z4: B] :
( ( A2 @ X @ Z4 )
=> ( ( A2 @ Y @ Z4 )
=> ( X = Y ) ) )
=> ( left_unique @ A @ B @ A2 ) ) ).
% left_uniqueI
thf(fact_202_left__uniqueD,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,X2: A,Z2: B,Y2: A] :
( ( left_unique @ A @ B @ A2 )
=> ( ( A2 @ X2 @ Z2 )
=> ( ( A2 @ Y2 @ Z2 )
=> ( X2 = Y2 ) ) ) ) ).
% left_uniqueD
thf(fact_203_llist_Oright__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( right_unique @ A @ B @ R )
=> ( right_unique @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.right_unique_rel
thf(fact_204_llist_Oleft__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_unique @ A @ B @ R )
=> ( left_unique @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coindu1486289336t_all2 @ A @ B @ R ) ) ) ).
% llist.left_unique_rel
thf(fact_205_swap__def,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A5: A,B5: A,F2: A > B] : ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F2 @ A5 @ ( F2 @ B5 ) ) @ B5 @ ( F2 @ A5 ) ) ) ) ).
% swap_def
thf(fact_206_right__unique__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( right_unique @ A @ B )
= ( ^ [R4: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ R4 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R4 @ (=>) )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ) ).
% right_unique_alt_def2
thf(fact_207_eq__imp__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( right_unique @ A @ B @ A2 )
=> ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2 @ (=>) )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z ) ) ).
% eq_imp_transfer
thf(fact_208_left__unique__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( right_total @ A @ B @ A2 )
=> ( ( right_total @ C @ D @ B2 )
=> ( ( bi_unique @ A @ B @ A2 )
=> ( 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 ) )
@ (=>)
@ ( left_unique @ A @ C )
@ ( left_unique @ B @ D ) ) ) ) ) ).
% left_unique_transfer
thf(fact_209_right__unique__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( right_total @ A @ B @ A2 )
=> ( ( right_total @ C @ D @ B2 )
=> ( ( bi_unique @ 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 ) )
@ (=>)
@ ( right_unique @ A @ C )
@ ( right_unique @ B @ D ) ) ) ) ) ).
% right_unique_transfer
thf(fact_210_rtranclp__parametric,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( bi_total @ A @ B @ A2 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( A > A > $o ) @ ( B > B > $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 @ A @ B @ ( A > $o ) @ ( B > $o ) @ A2
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A2
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( transitive_rtranclp @ A )
@ ( transitive_rtranclp @ B ) ) ) ) ).
% rtranclp_parametric
thf(fact_211_right__total__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( right_unique @ A @ B @ A2 )
=> ( ( right_total @ C @ D @ B2 )
=> ( right_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% right_total_fun
thf(fact_212_functional__converse__relation,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_unique @ A @ B @ R )
=> ( ( right_total @ A @ B @ R )
=> ! [Y6: B] :
? [X: A] :
( ( R @ X @ Y6 )
& ! [Ya3: A] :
( ( R @ Ya3 @ Y6 )
=> ( Ya3 = X ) ) ) ) ) ).
% functional_converse_relation
thf(fact_213_right__totalE,axiom,
! [A: $tType,B: $tType,A2: A > B > $o,Y2: B] :
( ( right_total @ A @ B @ A2 )
=> ~ ! [X: A] :
~ ( A2 @ X @ Y2 ) ) ).
% right_totalE
thf(fact_214_right__totalI,axiom,
! [A: $tType,B: $tType,A2: B > A > $o] :
( ! [Y: A] :
? [X7: B] : ( A2 @ X7 @ Y )
=> ( right_total @ B @ A @ A2 ) ) ).
% right_totalI
thf(fact_215_right__total__eq,axiom,
! [A: $tType] :
( right_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% right_total_eq
thf(fact_216_right__total__def,axiom,
! [B: $tType,A: $tType] :
( ( right_total @ A @ B )
= ( ^ [R4: A > B > $o] :
! [Y5: B] :
? [X4: A] : ( R4 @ X4 @ Y5 ) ) ) ).
% right_total_def
thf(fact_217_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_218_right__unique__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( right_total @ A @ B @ A2 )
=> ( ( right_unique @ C @ D @ B2 )
=> ( right_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% right_unique_fun
thf(fact_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_left__total__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( left_unique @ A @ B @ A2 )
=> ( ( left_total @ C @ D @ B2 )
=> ( left_total @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% left_total_fun
thf(fact_227_functional__relation,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( ( right_unique @ A @ B @ R )
=> ( ( left_total @ A @ B @ R )
=> ! [X7: A] :
? [Xa2: B] :
( ( R @ X7 @ Xa2 )
& ! [Y6: B] :
( ( R @ X7 @ Y6 )
=> ( Y6 = Xa2 ) ) ) ) ) ).
% functional_relation
thf(fact_228_left__unique__fun,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: A > B > $o,B2: C > D > $o] :
( ( left_total @ A @ B @ A2 )
=> ( ( left_unique @ C @ D @ B2 )
=> ( left_unique @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 ) ) ) ) ).
% left_unique_fun
thf(fact_229_left__total__def,axiom,
! [B: $tType,A: $tType] :
( ( left_total @ A @ B )
= ( ^ [R4: A > B > $o] :
! [X4: A] :
( ^ [P4: B > $o] :
? [X6: B] : ( P4 @ X6 )
@ ( R4 @ X4 ) ) ) ) ).
% left_total_def
thf(fact_230_left__total__eq,axiom,
! [A: $tType] :
( left_total @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% left_total_eq
thf(fact_231_left__totalI,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ! [X: A] :
? [X12: B] : ( R @ X @ X12 )
=> ( left_total @ A @ B @ R ) ) ).
% left_totalI
thf(fact_232_left__totalE,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ! [X7: A] :
? [X13: B] : ( R @ X7 @ X13 ) ) ).
% left_totalE
thf(fact_233_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_234_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_235_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_236_lhd__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ) ).
% lhd_lfilter
thf(fact_237_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_238_pcr__Domainp__par__left__total,axiom,
! [A: $tType,B: $tType,C: $tType,B2: A > B > $o,P: A > $o,A2: C > A > $o,P2: 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
@ P2
@ P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A2 @ B2 ) )
= P2 ) ) ) ) ).
% pcr_Domainp_par_left_total
thf(fact_239_right__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
( ( right_unique @ A @ B @ A2 )
=> ( ( right_unique @ B @ C @ B2 )
=> ( right_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% right_unique_OO
thf(fact_240_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_241_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_242_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_243_bi__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A > B > $o,B2: B > C > $o] :
( ( bi_unique @ A @ B @ A2 )
=> ( ( bi_unique @ B @ C @ B2 )
=> ( bi_unique @ A @ C @ ( relcompp @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% bi_unique_OO
thf(fact_244_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_245_lhd__LCons,axiom,
! [A: $tType,X21: A,X222: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X222 ) )
= X21 ) ).
% lhd_LCons
thf(fact_246_lhd__iterates,axiom,
! [A: $tType,F: A > A,X2: A] :
( ( coinductive_lhd @ A @ ( coinductive_iterates @ A @ F @ X2 ) )
= X2 ) ).
% lhd_iterates
thf(fact_247_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A3 ) )
= ( G21 @ A3 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_248_llist_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P3 @ A3 )
=> ( ( coinductive_lhd @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G21 @ Q22 @ G221 @ G222 @ A3 ) )
= ( G21 @ A3 ) ) ) ).
% llist.corec_sel(1)
thf(fact_249_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_250_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_251_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,Abs22: B > C,Rep22: C > B] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ B @ C
@ ^ [Y3: B,Z: B] : Y3 = Z
@ Abs22
@ Rep22 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R1
@ ( relcompp @ A @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R1 ) )
@ ( comp @ B @ C @ A @ Abs22 @ Abs12 )
@ ( comp @ B @ A @ C @ Rep12 @ Rep22 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_252_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: B > B > $o,Abs22: B > C,Rep22: C > B,R23: A > A > $o] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ B @ C @ R2 @ Abs22 @ Rep22 )
=> ( ! [X: A,Y: A] :
( ( R23 @ X @ Y )
=> ( ( R1 @ X @ X )
=> ( ( R1 @ Y @ Y )
=> ( R2 @ ( Abs12 @ X ) @ ( Abs12 @ Y ) ) ) ) )
=> ( ! [X: B,Y: B] :
( ( R2 @ X @ Y )
=> ( R23 @ ( Rep12 @ X ) @ ( Rep12 @ Y ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R23 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs12 ) @ ( comp @ B @ A @ C @ Rep12 @ Rep22 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_253_left__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( left_unique @ A @ B @ R )
=> ( ( left_unique @ B @ C @ S )
=> ( left_unique @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) ) ) ) ).
% left_unique_OO
thf(fact_254_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_255_neg__fun__distr2,axiom,
! [F3: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R7: A > B > $o,S5: C > D > $o,R: E > A > $o,S: F3 > C > $o] :
( ( right_unique @ A @ B @ R7 )
=> ( ( left_total @ A @ B @ R7 )
=> ( ( left_unique @ C @ D @ S5 )
=> ( ( right_total @ C @ D @ S5 )
=> ( ord_less_eq @ ( ( E > F3 ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ E @ B @ F3 @ D @ ( relcompp @ E @ A @ B @ R @ R7 ) @ ( relcompp @ F3 @ C @ D @ S @ S5 ) ) @ ( relcompp @ ( E > F3 ) @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ E @ A @ F3 @ C @ R @ S ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R7 @ S5 ) ) ) ) ) ) ) ).
% neg_fun_distr2
%----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,X2: A,Y2: A] :
( ( if @ A @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $true @ X2 @ Y2 )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
bNF_rel_fun @ ( coinductive_llist @ a ) @ ( coinductive_llist @ b ) @ ( coinductive_llist @ a ) @ ( coinductive_llist @ b ) @ ( coindu1486289336t_all2 @ a @ b @ a2 ) @ ( coindu1486289336t_all2 @ a @ b @ a2 ) @ ( lMirro427583474mirror @ a ) @ ( lMirro427583474mirror @ b ) ).
%------------------------------------------------------------------------------