TPTP Problem File: DAT197^1.p
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%------------------------------------------------------------------------------
% File : DAT197^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 137
% 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__137.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.33 v7.3.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 353 ( 131 unt; 67 typ; 0 def)
% Number of atoms : 806 ( 309 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 4610 ( 27 ~; 6 |; 47 &;4261 @)
% ( 0 <=>; 269 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 9 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 918 ( 918 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 62 usr; 6 con; 0-8 aty)
% Number of variables : 1359 ( 122 ^;1152 !; 20 ?;1359 :)
% ( 65 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:49.835
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (59)
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__llength,type,
coindu246717305length:
!>[A: $tType] : ( nat > ( coinductive_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List_Oiterates,type,
coinductive_iterates:
!>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldrop,type,
coinductive_ldrop:
!>[A: $tType] : ( extended_enat > ( coinductive_llist @ 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_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollcp,type,
coinductive_llcp:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List_Ollength,type,
coinductive_llength:
!>[A: $tType] : ( ( coinductive_llist @ A ) > extended_enat ) ).
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_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_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
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_Olprefix,type,
coinductive_lprefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Oltake,type,
coinductive_ltake:
!>[A: $tType] : ( extended_enat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
coindu780009021istsum:
!>[A: $tType] : ( ( coinductive_llist @ A ) > 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_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_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
lMirro999291890or_aux:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > b > $o ).
thf(sy_v_acc_Ha____,type,
acc_a: coinductive_llist @ b ).
thf(sy_v_acca____,type,
acca: coinductive_llist @ a ).
thf(sy_v_xs_Ha____,type,
xs_a: coinductive_llist @ b ).
thf(sy_v_xsa____,type,
xsa: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_local_OLNil_I2_J,axiom,
coindu1486289336t_all2 @ a @ b @ p @ acca @ acc_a ).
% local.LNil(2)
thf(fact_1_local_OLNil_I3_J,axiom,
coinductive_lfinite @ a @ acca ).
% local.LNil(3)
thf(fact_2_calculation,axiom,
( ( coinductive_llength @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) )
= ( coinductive_llength @ b @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ) ) ).
% calculation
thf(fact_3_local_OLNil_I1_J,axiom,
coindu1486289336t_all2 @ a @ b @ p @ ( lMirro999291890or_aux @ a @ acca @ xsa ) @ ( lMirro999291890or_aux @ b @ acc_a @ xs_a ) ).
% local.LNil(1)
thf(fact_4_llist__all2__llengthD,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_llength @ A @ Xs )
= ( coinductive_llength @ B @ Ys ) ) ) ).
% llist_all2_llengthD
thf(fact_5_llcp__same__conv__length,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ Xs @ Xs )
= ( coinductive_llength @ A @ Xs ) ) ).
% llcp_same_conv_length
thf(fact_6_llength__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_llength @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_llength @ B @ Xs ) ) ).
% llength_lmap
thf(fact_7_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( ( coinductive_llength @ A @ Xs )
= ( coinductive_llength @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_8_llength__ltakeWhile__all,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coinductive_llength @ A @ Xs ) )
= ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs ) ) ).
% llength_ltakeWhile_all
thf(fact_9_llist__all2__llength__ltakeWhileD,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 ) ) )
=> ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ Q1 @ Xs ) )
= ( coinductive_llength @ B @ ( coindu501562517eWhile @ B @ Q2 @ Ys ) ) ) ) ) ).
% llist_all2_llength_ltakeWhileD
thf(fact_10_llength__code,axiom,
! [A: $tType] :
( ( coinductive_llength @ A )
= ( coindu246717305length @ A @ ( zero_zero @ nat ) ) ) ).
% llength_code
thf(fact_11_llength__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ extended_enat @ extended_enat @ ( coindu1486289336t_all2 @ A @ B @ A2 )
@ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = Z )
@ ( coinductive_llength @ A )
@ ( coinductive_llength @ B ) ) ).
% llength_transfer
thf(fact_12_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_13_llist_Oleq__refl,axiom,
! [A: $tType,X2: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X2 @ X2 ) ).
% llist.leq_refl
thf(fact_14_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_lmap
thf(fact_15_lfinite__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Acc ) ) ) ).
% lfinite_lmirror_aux
thf(fact_16_llcp__lprefix1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_llcp @ A @ Xs @ Ys )
= ( coinductive_llength @ A @ Xs ) ) ) ).
% llcp_lprefix1
thf(fact_17_llcp__lprefix2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( ( coinductive_llcp @ A @ Xs @ Ys )
= ( coinductive_llength @ A @ Ys ) ) ) ).
% llcp_lprefix2
thf(fact_18_llcp__commute,axiom,
! [A: $tType] :
( ( coinductive_llcp @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] : ( coinductive_llcp @ A @ Ys2 @ Xs2 ) ) ) ).
% llcp_commute
thf(fact_19_lmap__lprefix,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,F: A > B] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ B @ ( coinductive_lmap @ A @ B @ F @ Xs ) @ ( coinductive_lmap @ A @ B @ F @ Ys ) ) ) ).
% lmap_lprefix
thf(fact_20_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_21_lprefix__trans,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_22_llist__all2__rsp,axiom,
! [A: $tType,B: $tType,R: A > B > $o,S: A > A > $o,T2: B > B > $o,X2: coinductive_llist @ A,Y3: 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 )
= ( T2 @ Y @ B4 ) ) ) )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ A3 @ B3 )
=> ( ( coindu1486289336t_all2 @ A @ A @ S @ X2 @ A3 )
= ( coindu1486289336t_all2 @ B @ B @ T2 @ Y3 @ B3 ) ) ) ) ) ).
% llist_all2_rsp
thf(fact_23_llist_Oleq__trans,axiom,
! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A,Z2: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X2 @ Y3 )
=> ( ( coinductive_lprefix @ A @ Y3 @ Z2 )
=> ( coinductive_lprefix @ A @ X2 @ Z2 ) ) ) ).
% llist.leq_trans
thf(fact_24_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_25_lprefix__antisym,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_26_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_27_llist_Oleq__antisym,axiom,
! [A: $tType,X2: coinductive_llist @ A,Y3: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X2 @ Y3 )
=> ( ( coinductive_lprefix @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% llist.leq_antisym
thf(fact_28_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_29_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_30_lprefix__down__linear,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Zs )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_31_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_32_lmirror__aux__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= Xs ) ) ).
% lmirror_aux_inf
thf(fact_33_lmap__lmirror__aux,axiom,
! [A: $tType,B: $tType,F: B > A,Acc: coinductive_llist @ B,Xs: coinductive_llist @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( lMirro999291890or_aux @ B @ Acc @ Xs ) )
= ( lMirro999291890or_aux @ A @ ( coinductive_lmap @ B @ A @ F @ Acc ) @ ( coinductive_lmap @ B @ A @ F @ Xs ) ) ) ).
% lmap_lmirror_aux
thf(fact_34_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu1486289336t_all2 @ A @ A
@ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
= ( ^ [Y2: coinductive_llist @ A,Z: coinductive_llist @ A] : ( Y2 = Z ) ) ) ).
% llist.rel_eq
thf(fact_35_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_36_llist__all2__lmirror__aux,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Acc: coinductive_llist @ A,Acc2: coinductive_llist @ B,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Acc @ Acc2 )
=> ( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Xs3 )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) @ ( lMirro999291890or_aux @ B @ Acc2 @ Xs3 ) ) ) ) ).
% llist_all2_lmirror_aux
thf(fact_37_llist_Omap__transfer,axiom,
! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb: A > E > $o,Sd: B > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F2 ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) ) @ ( ( coinductive_llist @ E ) > ( coinductive_llist @ F2 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F2 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ E ) @ ( coinductive_llist @ B ) @ ( coinductive_llist @ F2 ) @ ( coindu1486289336t_all2 @ A @ E @ Rb ) @ ( coindu1486289336t_all2 @ B @ F2 @ Sd ) ) @ ( coinductive_lmap @ A @ B ) @ ( coinductive_lmap @ E @ F2 ) ) ).
% llist.map_transfer
thf(fact_38_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_39_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs2 )
=> ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs2 )
=> ( Xs2 = Ys2 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_40_llistsum__inf,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ Xs )
= ( zero_zero @ A ) ) ) ) ).
% llistsum_inf
thf(fact_41_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,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
=> ( ( A2 @ X2 @ Y3 )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_42_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,F3: A > C] : ( F3 @ S2 )
@ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% let_rsp
thf(fact_43_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,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A2 @ B2 @ F @ G )
=> ( ( A2 @ X2 @ Y3 )
=> ( B2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funE
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_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,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X2 @ Y3 )
=> ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% apply_rsp'
thf(fact_49_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_50_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X4: 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 @ X4 @ A2 @ F @ G )
=> ( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X4 @ 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_51_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: A > B > $o,X4: A > B > $o,A2: C > D > $o,B2: C > D > $o,F: A > C,G: B > D] :
( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X4 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( A2 @ X @ Y )
=> ( B2 @ X @ Y ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X4 @ A2 @ F @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B2 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_52_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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 )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
@ ( coindu1486289336t_all2 @ A @ A )
@ ( coindu1486289336t_all2 @ B @ B ) ) ).
% llist_all2_transfer
thf(fact_53_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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 )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
@ ( coindu1486289336t_all2 @ A @ B )
@ ( coindu1486289336t_all2 @ C @ D ) ) ).
% llist.rel_transfer
thf(fact_54_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_55_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 )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ ( coinductive_lfinite @ A )
@ ( coinductive_lfinite @ B ) ) ).
% lfinite_transfer
thf(fact_56_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_57_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type @ A ) )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_58_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,F3: A > C] : ( F3 @ S2 )
@ ^ [S2: B,F3: B > D] : ( F3 @ S2 ) ) ).
% Let_transfer
thf(fact_59_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,F2: $tType,A: $tType,R: A > B > $o,S: C > E > $o,T2: D > F2 > $o,F: A > C > D,G: B > E > F2] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F2 ) @ R @ ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 ) @ F @ G )
= ( ! [X3: A,Y5: B] :
( ( R @ X3 @ Y5 )
=> ( bNF_rel_fun @ C @ E @ D @ F2 @ S @ T2 @ ( F @ X3 ) @ ( G @ Y5 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_60_If__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_61_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_62_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_63_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_64_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_65_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_66_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_67_lfinite__iterates,axiom,
! [A: $tType,F: A > A,X2: A] :
~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F @ X2 ) ) ).
% lfinite_iterates
thf(fact_68_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_69_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y2: D,Z: D] : ( Y2 = Z )
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E
@ ^ [Y2: D,Z: D] : ( Y2 = Z )
@ Sc )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z: D] : ( Y2 = Z ) )
@ ( bNF_rel_fun @ D @ D @ C @ E
@ ^ [Y2: D,Z: D] : ( Y2 = Z ) ) ) ).
% fun.rel_transfer
thf(fact_70_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_71_map__fun__parametric,axiom,
! [A: $tType,B: $tType,E: $tType,F2: $tType,H: $tType,G2: $tType,D: $tType,C: $tType,A2: A > C > $o,B2: B > D > $o,C2: E > G2 > $o,D2: F2 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > F2 ) > ( B > E ) > A > F2 ) @ ( ( G2 > H ) > ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A2 @ B2 ) @ ( bNF_rel_fun @ ( E > F2 ) @ ( G2 > H ) @ ( ( B > E ) > A > F2 ) @ ( ( D > G2 ) > C > H ) @ ( bNF_rel_fun @ E @ G2 @ F2 @ H @ C2 @ D2 ) @ ( bNF_rel_fun @ ( B > E ) @ ( D > G2 ) @ ( A > F2 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E @ G2 @ B2 @ C2 ) @ ( bNF_rel_fun @ A @ C @ F2 @ H @ A2 @ D2 ) ) ) @ ( map_fun @ A @ B @ E @ F2 ) @ ( map_fun @ C @ D @ G2 @ H ) ) ).
% map_fun_parametric
thf(fact_72_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_73_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ R )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ ( coindu543516966_llist @ A )
@ ( coindu543516966_llist @ B ) ) ).
% llist.pred_transfer
thf(fact_74_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ $o @ $o @ ( coindu1486289336t_all2 @ A @ B @ A2 )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ ( coindu543516966_llist @ A )
@ ( coindu543516966_llist @ B ) ) ).
% llist_all_transfer
thf(fact_75_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 )
@ ^ [Y2: $o,Z: $o] : ( Y2 = 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_76_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_77_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_78_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,F3: C > A] : ( F3 @ S2 ) )
= ( ^ [S2: D,F3: D > B] : ( F3 @ S2 ) ) ) ) ) ).
% let_prs
thf(fact_79_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_80_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 ) )
=> ( ! [R3: B,S3: B] :
( ( R @ R3 @ S3 )
= ( ( R @ R3 @ R3 )
& ( R @ S3 @ S3 )
& ( ( Abs @ R3 )
= ( Abs @ S3 ) ) ) )
=> ( quotient3 @ B @ A @ R @ Abs @ Rep ) ) ) ) ).
% Quotient3I
thf(fact_81_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_82_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_83_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_84_Quotient3__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( ( R @ R6 @ R6 )
& ( R @ S4 @ S4 )
& ( ( Abs @ R6 )
= ( Abs @ S4 ) ) )
= ( R @ R6 @ S4 ) ) ) ).
% Quotient3_rel
thf(fact_85_Quotient3__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R6 @ S4 )
=> ( R @ R6 @ R6 ) ) ) ).
% Quotient3_refl1
thf(fact_86_Quotient3__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R6 @ S4 )
=> ( R @ S4 @ S4 ) ) ) ).
% Quotient3_refl2
thf(fact_87_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_88_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_89_Quotient3__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A,S4: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R6 @ S4 )
=> ( ( Abs @ R6 )
= ( Abs @ S4 ) ) ) ) ).
% Quotient3_rel_abs
thf(fact_90_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_91_Quotient3__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,R6: A] :
( ( quotient3 @ A @ B @ R @ Abs @ Rep )
=> ( ( R @ R6 @ R6 )
=> ( R @ ( Rep @ ( Abs @ R6 ) ) @ R6 ) ) ) ).
% Quotient3_rep_abs
thf(fact_92_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_93_lprefix__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ).
% lprefix_lfilterI
thf(fact_94_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_95_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_96_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,Y3: A] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X2 @ Y3 )
=> ( R2 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ) ).
% apply_rspQ3
thf(fact_97_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_98_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_99_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ R
@ R ) ) ).
% quot_rel_rsp
thf(fact_100_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
@ ^ [Y2: D,Z: D] : ( Y2 = Z )
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl
thf(fact_101_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y2: D,Z: D] : ( Y2 = Z )
@ ^ [Y2: A,Z: A] : ( Y2 = Z ) )
= ( ^ [Y2: D > A,Z: D > A] : ( Y2 = Z ) ) ) ).
% fun.rel_eq
thf(fact_102_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F3: B > C,G3: D > A,H2: C > D,X3: B] : ( G3 @ ( H2 @ ( F3 @ X3 ) ) ) ) ) ).
% map_fun_apply
thf(fact_103_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
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z ) )
@ ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ ( bex1_rel @ A @ R )
@ ( bex1_rel @ A @ R ) ) ) ).
% bex1_rel_rsp
thf(fact_104_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_105_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_106_ldropWhile__eq__ldrop,axiom,
! [A: $tType] :
( ( coindu218763757pWhile @ A )
= ( ^ [P3: A > $o,Xs2: coinductive_llist @ A] : ( coinductive_ldrop @ A @ ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ).
% ldropWhile_eq_ldrop
thf(fact_107_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_108_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_109_ldrop__0,axiom,
! [B: $tType,Xs: coinductive_llist @ B] :
( ( coinductive_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
= Xs ) ).
% ldrop_0
thf(fact_110_LCons__lprefix__LCons,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_111_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_112_lfinite__code_I2_J,axiom,
! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_113_lfinite__LCons,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_114_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_115_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_116_ldrop__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_ldrop @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lmap @ B @ A @ F @ ( coinductive_ldrop @ B @ N @ Xs ) ) ) ).
% ldrop_lmap
thf(fact_117_lmirror__aux__simps_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X2: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X2 ) )
= ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X2 ) ) ) ).
% lmirror_aux_simps(2)
thf(fact_118_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_119_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_120_bex1__rel__aux2,axiom,
! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X2 @ Xa2 )
= ( Y3 @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ Y3 )
=> ( bex1_rel @ A @ R @ X2 ) ) ) ).
% bex1_rel_aux2
thf(fact_121_bex1__rel__aux,axiom,
! [A: $tType,R: A > A > $o,X2: A > $o,Y3: A > $o] :
( ! [Xa2: A,Ya2: A] :
( ( R @ Xa2 @ Ya2 )
=> ( ( X2 @ Xa2 )
= ( Y3 @ Ya2 ) ) )
=> ( ( bex1_rel @ A @ R @ X2 )
=> ( bex1_rel @ A @ R @ Y3 ) ) ) ).
% bex1_rel_aux
thf(fact_122_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X2: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ X2 @ Ys ) ) ) ).
% Le_LCons
thf(fact_123_LCons__lprefix__conv,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) @ Ys )
= ( ? [Ys3: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys3 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_124_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_125_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,Ys3: coinductive_llist @ B] :
( ( Ys
= ( coinductive_LCons @ B @ Y5 @ Ys3 ) )
& ( P @ X2 @ Y5 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% llist_all2_LCons1
thf(fact_126_llist__all2__LCons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Y3: B,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( P @ X3 @ Y3 )
& ( coindu1486289336t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% llist_all2_LCons2
thf(fact_127_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_128_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_129_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Y3: A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lmap @ B @ A @ F @ Xs )
= ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ? [X3: B,Xs4: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ B @ X3 @ Xs4 ) )
& ( Y3
= ( F @ X3 ) )
& ( Ys
= ( coinductive_lmap @ B @ A @ F @ Xs4 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_130_llist__all2__ldropI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ldrop @ A @ N @ Xs ) @ ( coinductive_ldrop @ B @ N @ Ys ) ) ) ).
% llist_all2_ldropI
thf(fact_131_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_132_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_133_lfilter__LCons__seek,axiom,
! [A: $tType,P4: A > $o,X2: A,L: coinductive_llist @ A] :
( ~ ( P4 @ X2 )
=> ( ( coinductive_lfilter @ A @ P4 @ ( coinductive_LCons @ A @ X2 @ L ) )
= ( coinductive_lfilter @ A @ P4 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_134_iterates_Ocode,axiom,
! [A: $tType] :
( ( coinductive_iterates @ A )
= ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_iterates @ A @ F3 @ ( F3 @ X3 ) ) ) ) ) ).
% iterates.code
thf(fact_135_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P4: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P4 @ A3 )
=> ( ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ A3 )
= ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_136_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P4: C > $o,A3: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P4 @ A3 )
=> ( ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ A3 )
= ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_137_iterates__equality,axiom,
! [A: $tType,H3: A > ( coinductive_llist @ A ),F: A > A] :
( ! [X: A] :
( ( H3 @ X )
= ( coinductive_LCons @ A @ X @ ( coinductive_lmap @ A @ A @ F @ ( H3 @ X ) ) ) )
=> ( H3
= ( coinductive_iterates @ A @ F ) ) ) ).
% iterates_equality
thf(fact_138_iterates__lmap,axiom,
! [A: $tType] :
( ( coinductive_iterates @ A )
= ( ^ [F3: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lmap @ A @ A @ F3 @ ( coinductive_iterates @ A @ F3 @ X3 ) ) ) ) ) ).
% iterates_lmap
thf(fact_139_ldrop__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
@ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = 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_ldrop @ A )
@ ( coinductive_ldrop @ B ) ) ).
% ldrop_transfer
thf(fact_140_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_141_llistsum__LCons,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( plus_plus @ A @ X2 @ ( coindu780009021istsum @ A @ Xs ) ) ) ) ) ).
% llistsum_LCons
thf(fact_142_llist__all2__lappend1D_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
=> ( ( coinductive_lfinite @ A @ Xs )
=> ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ ( coinductive_ldrop @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ) ).
% llist_all2_lappend1D(2)
thf(fact_143_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_144_lmember__code_I2_J,axiom,
! [A: $tType,X2: A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( X2 = Y3 )
| ( coinductive_lmember @ A @ X2 @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_145_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_146_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_147_ldrop__ldrop,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
( ( coinductive_ldrop @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
= ( coinductive_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ).
% ldrop_ldrop
thf(fact_148_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_149_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A] :
( ( A3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_150_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_151_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [B3: A,A3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= A3 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_152_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_153_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_154_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_155_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.left_neutral
thf(fact_156_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X2: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X2 ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X2 @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_157_llength__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llength @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llength @ A @ Ys ) ) ) ).
% llength_lappend
thf(fact_158_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_159_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_160_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_161_lappend__ltakeWhile__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= Xs ) ).
% lappend_ltakeWhile_ldropWhile
thf(fact_162_llcp__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llcp @ A @ Ys @ Zs ) ) ) ).
% llcp_lappend_same
thf(fact_163_lprefix__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_164_lappend__lprefixE,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs2: coinductive_llist @ A] :
( Zs
!= ( coinductive_lappend @ A @ Xs @ Zs2 ) ) ) ).
% lappend_lprefixE
thf(fact_165_lprefix__lappendD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_166_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
? [Zs3: coinductive_llist @ A] :
( Ys2
= ( coinductive_lappend @ A @ Xs2 @ Zs3 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_167_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_168_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_169_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_170_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_171_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [X2: A,Y3: A] :
( ( ( plus_plus @ A @ X2 @ Y3 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_172_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_173_llistsum__lappend,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( ( coindu780009021istsum @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( coindu780009021istsum @ A @ Xs ) @ ( coindu780009021istsum @ A @ Ys ) ) ) ) ) ) ).
% llistsum_lappend
thf(fact_174_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coinductive_llist @ B,Ys: coinductive_llist @ B] :
( ( coinductive_lmap @ B @ A @ F @ ( coinductive_lappend @ B @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lmap @ B @ A @ F @ Xs ) @ ( coinductive_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_175_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_176_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_177_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_178_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_179_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_180_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_181_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% add.commute
thf(fact_182_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_183_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_184_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_185_lappend__iterates,axiom,
! [A: $tType,F: A > A,X2: A,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_iterates @ A @ F @ X2 ) @ Xs )
= ( coinductive_iterates @ A @ F @ X2 ) ) ).
% lappend_iterates
thf(fact_186_lmirror__aux__acc,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
= ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% lmirror_aux_acc
thf(fact_187_llist__all2__lappendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ B @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ Xs3 @ Ys4 ) ) )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ ( coinductive_lappend @ B @ Ys @ Ys4 ) ) ) ) ).
% llist_all2_lappendI
thf(fact_188_lappend__eq__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Us: coinductive_llist @ A,Ys: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( ( coinductive_llength @ A @ Xs )
= ( coinductive_llength @ A @ Us ) )
=> ( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_lappend @ A @ Us @ Vs ) )
= ( ( Xs = Us )
& ( ( coinductive_lfinite @ A @ Xs )
=> ( Ys = Vs ) ) ) ) ) ).
% lappend_eq_lappend_conv
thf(fact_189_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us2 @ Vs2 ) )
& ( coinductive_lfinite @ A @ Us2 )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us2 ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_190_lappend__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] : ( bNF_rel_fun @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) ) @ ( coindu1486289336t_all2 @ A @ 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_lappend @ A ) @ ( coinductive_lappend @ B ) ) ).
% lappend_transfer
thf(fact_191_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_192_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_193_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_194_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_195_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_196_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_197_ind__euclid,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% ind_euclid
thf(fact_198_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type @ A ) )
=> ! [B3: A,A3: A] :
( ( B3
= ( plus_plus @ A @ B3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_199_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% semiring_normalization_rules(6)
thf(fact_200_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% semiring_normalization_rules(5)
thf(fact_201_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_202_rec__nat__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ A2
@ ( bNF_rel_fun @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B )
@ ( bNF_rel_fun @ nat @ nat @ ( A > A ) @ ( B > B )
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ ( bNF_rel_fun @ A @ B @ A @ B @ A2 @ A2 ) )
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A2 ) )
@ ( rec_nat @ A )
@ ( rec_nat @ B ) ) ).
% rec_nat_transfer
thf(fact_203_llast__LCons2,axiom,
! [A: $tType,X2: A,Y3: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_204_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_205_case__nat__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ A2
@ ( bNF_rel_fun @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B )
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A2 )
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A2 ) )
@ ( case_nat @ A )
@ ( case_nat @ B ) ) ).
% case_nat_transfer
thf(fact_206_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_207_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_208_ltake__transfer,axiom,
! [A: $tType,B: $tType,A2: A > B > $o] :
( bNF_rel_fun @ extended_enat @ extended_enat @ ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) @ ( ( coinductive_llist @ B ) > ( coinductive_llist @ B ) )
@ ^ [Y2: extended_enat,Z: extended_enat] : ( Y2 = 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_ltake @ A )
@ ( coinductive_ltake @ B ) ) ).
% ltake_transfer
thf(fact_209_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X2: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X2 @ Xs ) )
=> ? [Us2: coinductive_llist @ A,Vs2: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us2 @ ( coinductive_LCons @ A @ X2 @ Vs2 ) ) )
& ( coinductive_lfinite @ A @ Us2 )
& ! [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Us2 ) )
=> ~ ( P @ X5 ) )
& ( P @ X2 )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs2 ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_210_ltake__is__lprefix,axiom,
! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ Xs ) ).
% ltake_is_lprefix
thf(fact_211_llist__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs: coinductive_llist @ A] :
( ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% llist_all2_same
thf(fact_212_ltake__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coinductive_llist @ B] :
( ( coinductive_ltake @ A @ N @ ( coinductive_lmap @ B @ A @ F @ Xs ) )
= ( coinductive_lmap @ B @ A @ F @ ( coinductive_ltake @ B @ N @ Xs ) ) ) ).
% ltake_lmap
thf(fact_213_ldropWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( coindu218763757pWhile @ A @ P @ Xs )
= ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_214_in__lset__ldropWhileD,axiom,
! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_215_llist_Opred__cong,axiom,
! [A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,P: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( coindu543516966_llist @ A @ P @ X2 )
= ( coindu543516966_llist @ A @ Pa @ Ya ) ) ) ) ).
% llist.pred_cong
thf(fact_216_llist_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X2: coinductive_llist @ A,Pa: A > $o] :
( ( coindu543516966_llist @ A @ P @ X2 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( coindu543516966_llist @ A @ Pa @ X2 ) ) ) ).
% llist.pred_mono_strong
thf(fact_217_lset__ltakeWhileD,axiom,
! [A: $tType,X2: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
& ( P @ X2 ) ) ) ).
% lset_ltakeWhileD
thf(fact_218_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_219_ltakeWhile__all,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_220_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_221_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lfilter_id_conv
thf(fact_222_llist_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,Xa: coinductive_llist @ A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( coinductive_lset @ A @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% llist.inj_map_strong
thf(fact_223_llist_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,F: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ G @ X2 ) ) ) ).
% llist.map_cong0
thf(fact_224_llist_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( coinductive_lmap @ A @ B @ F @ X2 )
= ( coinductive_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong
thf(fact_225_llist_Orel__refl__strong,axiom,
! [A: $tType,X2: coinductive_llist @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( coindu1486289336t_all2 @ A @ A @ Ra @ X2 @ X2 ) ) ).
% llist.rel_refl_strong
thf(fact_226_llist_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X2: coinductive_llist @ A,Y3: coinductive_llist @ B,Ra: A > B > $o] :
( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
=> ( ! [Z3: A,Yb2: B] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ X2 ) )
=> ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Y3 ) )
=> ( ( R @ Z3 @ Yb2 )
=> ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( coindu1486289336t_all2 @ A @ B @ Ra @ X2 @ Y3 ) ) ) ).
% llist.rel_mono_strong
thf(fact_227_llist_Orel__cong,axiom,
! [A: $tType,B: $tType,X2: coinductive_llist @ A,Ya: coinductive_llist @ A,Y3: coinductive_llist @ B,Xa: coinductive_llist @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa )
=> ( ! [Z3: A,Yb2: B] :
( ( member @ A @ Z3 @ ( coinductive_lset @ A @ Ya ) )
=> ( ( member @ B @ Yb2 @ ( coinductive_lset @ B @ Xa ) )
=> ( ( R @ Z3 @ Yb2 )
= ( Ra @ Z3 @ Yb2 ) ) ) )
=> ( ( coindu1486289336t_all2 @ A @ B @ R @ X2 @ Y3 )
= ( coindu1486289336t_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% llist.rel_cong
thf(fact_228_llist__all2__lsetD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Y3: B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) )
=> ? [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( P @ X @ Y3 ) ) ) ) ).
% llist_all2_lsetD2
thf(fact_229_llist__all2__lsetD1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,X2: A] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ? [X: B] :
( ( member @ B @ X @ ( coinductive_lset @ B @ Ys ) )
& ( P @ X2 @ X ) ) ) ) ).
% llist_all2_lsetD1
thf(fact_230_llist__all2__reflI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X @ X ) )
=> ( coindu1486289336t_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% llist_all2_reflI
thf(fact_231_llist__all2__ltakeI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,N: extended_enat] :
( ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ B @ N @ Ys ) ) ) ).
% llist_all2_ltakeI
thf(fact_232_lset__intros_I2_J,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,X6: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_233_lset__intros_I1_J,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] : ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_234_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X2: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_235_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_236_lset__cases,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs5: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
=> ~ ! [X7: A,Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X7 @ Xs5 ) )
=> ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% lset_cases
thf(fact_237_lset__induct,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
=> ( ! [X7: A,Xs6: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
=> ( ( X2 != X7 )
=> ( ( P @ Xs6 )
=> ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_238_lset__induct_H,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs6: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X2 @ Xs6 ) )
=> ( ! [X7: A,Xs6: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs6 ) )
=> ( ( P @ Xs6 )
=> ( P @ ( coinductive_LCons @ A @ X7 @ Xs6 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_239_in__lset__ldropD,axiom,
! [A: $tType,X2: A,N: extended_enat,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_ldrop @ A @ N @ Xs ) ) )
=> ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropD
thf(fact_240_llist_Oset__cases,axiom,
! [A: $tType,E2: A,A3: coinductive_llist @ A] :
( ( member @ A @ E2 @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z22: coinductive_llist @ A] :
( A3
!= ( coinductive_LCons @ A @ E2 @ Z22 ) )
=> ~ ! [Z1: A,Z22: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E2 @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_241_llist_Oset__induct,axiom,
! [A: $tType,X2: A,A3: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X2 @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_242_lset__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lset @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Acc ) ) ) ).
% lset_lmirror_aux
thf(fact_243_lfinite__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X3 ) )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_244_lfinite__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
| ? [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X3 ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_245_ltakeWhile__lappend1,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile_lappend1
thf(fact_246_in__lset__lappend__iff,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_247_lappend__ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ldrop @ A @ N @ Xs ) )
= Xs ) ).
% lappend_ltake_ldrop
thf(fact_248_lset__lmember,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X2 @ Xs ) ) ).
% lset_lmember
thf(fact_249_ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
( ( coinductive_ltake @ A @ N @ ( coinductive_ldrop @ A @ M @ Xs ) )
= ( coinductive_ldrop @ A @ M @ ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ) ).
% ltake_ldrop
thf(fact_250_llist__all2__lappend1D_I1_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coinductive_llist @ A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu1486289336t_all2 @ A @ B @ P @ ( coinductive_lappend @ A @ Xs @ Xs3 ) @ Ys )
=> ( coindu1486289336t_all2 @ A @ B @ P @ Xs @ ( coinductive_ltake @ B @ ( coinductive_llength @ A @ Xs ) @ Ys ) ) ) ).
% llist_all2_lappend1D(1)
thf(fact_251_split__llist__first,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
& ( coinductive_lfinite @ A @ Ys5 )
& ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys5 ) ) ) ) ).
% split_llist_first
thf(fact_252_split__llist,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys5: coinductive_llist @ A,Zs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys5 @ ( coinductive_LCons @ A @ X2 @ Zs4 ) ) )
& ( coinductive_lfinite @ A @ Ys5 ) ) ) ).
% split_llist
thf(fact_253_ltake__plus__conv__lappend,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coinductive_llist @ A] :
( ( coinductive_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs )
= ( coinductive_lappend @ A @ ( coinductive_ltake @ A @ N @ Xs ) @ ( coinductive_ltake @ A @ M @ ( coinductive_ldrop @ A @ N @ Xs ) ) ) ) ).
% ltake_plus_conv_lappend
%----Type constructors (28)
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type @ nat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_1,axiom,
canoni770627133id_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_2,axiom,
ordere779506340up_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_3,axiom,
ab_semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_4,axiom,
comm_monoid_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_5,axiom,
comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_6,axiom,
semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_7,axiom,
monoid_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_8,axiom,
zero @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_9,axiom,
ordere779506340up_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_10,axiom,
cancel1352612707id_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_11,axiom,
cancel_semigroup_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_12,axiom,
ab_semigroup_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_13,axiom,
comm_monoid_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_14,axiom,
comm_semiring_1 @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_15,axiom,
semigroup_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_16,axiom,
monoid_add @ code_natural @ ( type @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_17,axiom,
zero @ code_natural @ ( type @ code_natural ) ).
%----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,Y3: A] :
( ( if @ A @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $true @ X2 @ Y3 )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coinductive_llength @ a @ acca )
= ( coinductive_llength @ b @ acc_a ) ) ).
%------------------------------------------------------------------------------