Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/OvConsOS_complete-noand_Z)

The rewrite relation of the following TRS is considered.

zeros	→	cons(0,n__zeros)	(1)
U101(tt,V1,V2)	→	U102(isNatKind(activate(V1)),activate(V1),activate(V2))	(2)
U102(tt,V1,V2)	→	U103(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(3)
U103(tt,V1,V2)	→	U104(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(4)
U104(tt,V1,V2)	→	U105(isNat(activate(V1)),activate(V2))	(5)
U105(tt,V2)	→	U106(isNatIList(activate(V2)))	(6)
U106(tt)	→	tt	(7)
U11(tt,V1)	→	U12(isNatIListKind(activate(V1)),activate(V1))	(8)
U111(tt,L,N)	→	U112(isNatIListKind(activate(L)),activate(L),activate(N))	(9)
U112(tt,L,N)	→	U113(isNat(activate(N)),activate(L),activate(N))	(10)
U113(tt,L,N)	→	U114(isNatKind(activate(N)),activate(L))	(11)
U114(tt,L)	→	s(length(activate(L)))	(12)
U12(tt,V1)	→	U13(isNatList(activate(V1)))	(13)
U121(tt,IL)	→	U122(isNatIListKind(activate(IL)))	(14)
U122(tt)	→	nil	(15)
U13(tt)	→	tt	(16)
U131(tt,IL,M,N)	→	U132(isNatIListKind(activate(IL)),activate(IL),activate(M),activate(N))	(17)
U132(tt,IL,M,N)	→	U133(isNat(activate(M)),activate(IL),activate(M),activate(N))	(18)
U133(tt,IL,M,N)	→	U134(isNatKind(activate(M)),activate(IL),activate(M),activate(N))	(19)
U134(tt,IL,M,N)	→	U135(isNat(activate(N)),activate(IL),activate(M),activate(N))	(20)
U135(tt,IL,M,N)	→	U136(isNatKind(activate(N)),activate(IL),activate(M),activate(N))	(21)
U136(tt,IL,M,N)	→	cons(activate(N),n__take(activate(M),activate(IL)))	(22)
U21(tt,V1)	→	U22(isNatKind(activate(V1)),activate(V1))	(23)
U22(tt,V1)	→	U23(isNat(activate(V1)))	(24)
U23(tt)	→	tt	(25)
U31(tt,V)	→	U32(isNatIListKind(activate(V)),activate(V))	(26)
U32(tt,V)	→	U33(isNatList(activate(V)))	(27)
U33(tt)	→	tt	(28)
U41(tt,V1,V2)	→	U42(isNatKind(activate(V1)),activate(V1),activate(V2))	(29)
U42(tt,V1,V2)	→	U43(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(30)
U43(tt,V1,V2)	→	U44(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(31)
U44(tt,V1,V2)	→	U45(isNat(activate(V1)),activate(V2))	(32)
U45(tt,V2)	→	U46(isNatIList(activate(V2)))	(33)
U46(tt)	→	tt	(34)
U51(tt,V2)	→	U52(isNatIListKind(activate(V2)))	(35)
U52(tt)	→	tt	(36)
U61(tt,V2)	→	U62(isNatIListKind(activate(V2)))	(37)
U62(tt)	→	tt	(38)
U71(tt)	→	tt	(39)
U81(tt)	→	tt	(40)
U91(tt,V1,V2)	→	U92(isNatKind(activate(V1)),activate(V1),activate(V2))	(41)
U92(tt,V1,V2)	→	U93(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(42)
U93(tt,V1,V2)	→	U94(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(43)
U94(tt,V1,V2)	→	U95(isNat(activate(V1)),activate(V2))	(44)
U95(tt,V2)	→	U96(isNatList(activate(V2)))	(45)
U96(tt)	→	tt	(46)
isNat(n__0)	→	tt	(47)
isNat(n__length(V1))	→	U11(isNatIListKind(activate(V1)),activate(V1))	(48)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(49)
isNatIList(V)	→	U31(isNatIListKind(activate(V)),activate(V))	(50)
isNatIList(n__zeros)	→	tt	(51)
isNatIList(n__cons(V1,V2))	→	U41(isNatKind(activate(V1)),activate(V1),activate(V2))	(52)
isNatIListKind(n__nil)	→	tt	(53)
isNatIListKind(n__zeros)	→	tt	(54)
isNatIListKind(n__cons(V1,V2))	→	U51(isNatKind(activate(V1)),activate(V2))	(55)
isNatIListKind(n__take(V1,V2))	→	U61(isNatKind(activate(V1)),activate(V2))	(56)
isNatKind(n__0)	→	tt	(57)
isNatKind(n__length(V1))	→	U71(isNatIListKind(activate(V1)))	(58)
isNatKind(n__s(V1))	→	U81(isNatKind(activate(V1)))	(59)
isNatList(n__nil)	→	tt	(60)
isNatList(n__cons(V1,V2))	→	U91(isNatKind(activate(V1)),activate(V1),activate(V2))	(61)
isNatList(n__take(V1,V2))	→	U101(isNatKind(activate(V1)),activate(V1),activate(V2))	(62)
length(nil)	→	0	(63)
length(cons(N,L))	→	U111(isNatList(activate(L)),activate(L),N)	(64)
take(0,IL)	→	U121(isNatIList(IL),IL)	(65)
take(s(M),cons(N,IL))	→	U131(isNatIList(activate(IL)),activate(IL),M,N)	(66)
zeros	→	n__zeros	(67)
take(X1,X2)	→	n__take(X1,X2)	(68)
0	→	n__0	(69)
length(X)	→	n__length(X)	(70)
s(X)	→	n__s(X)	(71)
cons(X1,X2)	→	n__cons(X1,X2)	(72)
nil	→	n__nil	(73)
activate(n__zeros)	→	zeros	(74)
activate(n__take(X1,X2))	→	take(X1,X2)	(75)
activate(n__0)	→	0	(76)
activate(n__length(X))	→	length(X)	(77)
activate(n__s(X))	→	s(X)	(78)
activate(n__cons(X1,X2))	→	cons(X1,X2)	(79)
activate(n__nil)	→	nil	(80)
activate(X)	→	X	(81)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 182 ruless (increase limit for explicit display).

It remains to prove infiniteness of the resulting DP problem.

1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

zeros^#	→	cons^#(0,n__zeros)	(82)
zeros^#	→	0^#	(83)
U105^#(tt,V2)	→	U106^#(isNatIList(activate(V2)))	(100)
U114^#(tt,L)	→	s^#(length(activate(L)))	(118)
U12^#(tt,V1)	→	U13^#(isNatList(activate(V1)))	(121)
U121^#(tt,IL)	→	U122^#(isNatIListKind(activate(IL)))	(124)
U122^#(tt)	→	nil^#	(127)
U136^#(tt,IL,M,N)	→	cons^#(activate(N),n__take(activate(M),activate(IL)))	(153)
U22^#(tt,V1)	→	U23^#(isNat(activate(V1)))	(160)
U32^#(tt,V)	→	U33^#(isNatList(activate(V)))	(166)
U45^#(tt,V2)	→	U46^#(isNatIList(activate(V2)))	(185)
U51^#(tt,V2)	→	U52^#(isNatIListKind(activate(V2)))	(188)
U61^#(tt,V2)	→	U62^#(isNatIListKind(activate(V2)))	(191)
U95^#(tt,V2)	→	U96^#(isNatList(activate(V2)))	(210)
isNatKind^#(n__length(V1))	→	U71^#(isNatIListKind(activate(V1)))	(234)
isNatKind^#(n__s(V1))	→	U81^#(isNatKind(activate(V1)))	(237)
length^#(nil)	→	0^#	(248)
activate^#(n__zeros)	→	zeros^#	(257)
activate^#(n__0)	→	0^#	(259)
activate^#(n__s(X))	→	s^#(X)	(261)
activate^#(n__cons(X1,X2))	→	cons^#(X1,X2)	(262)
activate^#(n__nil)	→	nil^#	(263)

and the following rules have been deleted.

1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

activate^#(n__take(X1,X2))	→	take^#(X1,X2)	(258)
isNatList^#(n__take(V1,V2))	→	U101^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(244)
isNatList^#(n__take(V1,V2))	→	isNatKind^#(activate(V1))	(245)
isNatList^#(n__take(V1,V2))	→	activate^#(V1)	(246)
isNatList^#(n__take(V1,V2))	→	activate^#(V2)	(247)
isNatIListKind^#(n__take(V1,V2))	→	U61^#(isNatKind(activate(V1)),activate(V2))	(230)
isNatIListKind^#(n__take(V1,V2))	→	isNatKind^#(activate(V1))	(231)
isNatIListKind^#(n__take(V1,V2))	→	activate^#(V1)	(232)
isNatIListKind^#(n__take(V1,V2))	→	activate^#(V2)	(233)

and the following rules have been deleted.

1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

take^#(0,IL)	→	U121^#(isNatIList(IL),IL)	(252)
U121^#(tt,IL)	→	isNatIListKind^#(activate(IL))	(125)
U101^#(tt,V1,V2)	→	U102^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(84)
U102^#(tt,V1,V2)	→	U103^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(88)
U103^#(tt,V1,V2)	→	U104^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(92)
U104^#(tt,V1,V2)	→	U105^#(isNat(activate(V1)),activate(V2))	(96)
U105^#(tt,V2)	→	isNatIList^#(activate(V2))	(101)
isNatIList^#(V)	→	U31^#(isNatIListKind(activate(V)),activate(V))	(219)
U31^#(tt,V)	→	U32^#(isNatIListKind(activate(V)),activate(V))	(163)
U32^#(tt,V)	→	isNatList^#(activate(V))	(167)
U32^#(tt,V)	→	activate^#(V)	(168)
U31^#(tt,V)	→	isNatIListKind^#(activate(V))	(164)
U61^#(tt,V2)	→	isNatIListKind^#(activate(V2))	(192)
U61^#(tt,V2)	→	activate^#(V2)	(193)
U31^#(tt,V)	→	activate^#(V)	(165)
isNatIList^#(V)	→	isNatIListKind^#(activate(V))	(220)
isNatIList^#(V)	→	activate^#(V)	(221)
isNatIList^#(n__cons(V1,V2))	→	U41^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(222)
U41^#(tt,V1,V2)	→	U42^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(169)
U42^#(tt,V1,V2)	→	U43^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(173)
U43^#(tt,V1,V2)	→	U44^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(177)
U44^#(tt,V1,V2)	→	U45^#(isNat(activate(V1)),activate(V2))	(181)
U45^#(tt,V2)	→	isNatIList^#(activate(V2))	(186)
isNatIList^#(n__cons(V1,V2))	→	isNatKind^#(activate(V1))	(223)
isNatIList^#(n__cons(V1,V2))	→	activate^#(V1)	(224)
isNatIList^#(n__cons(V1,V2))	→	activate^#(V2)	(225)
U45^#(tt,V2)	→	activate^#(V2)	(187)
U44^#(tt,V1,V2)	→	isNat^#(activate(V1))	(182)
U44^#(tt,V1,V2)	→	activate^#(V1)	(183)
U44^#(tt,V1,V2)	→	activate^#(V2)	(184)
U43^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(178)
U43^#(tt,V1,V2)	→	activate^#(V2)	(179)
U43^#(tt,V1,V2)	→	activate^#(V1)	(180)
U42^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(174)
U42^#(tt,V1,V2)	→	activate^#(V2)	(175)
U42^#(tt,V1,V2)	→	activate^#(V1)	(176)
U41^#(tt,V1,V2)	→	isNatKind^#(activate(V1))	(170)
U41^#(tt,V1,V2)	→	activate^#(V1)	(171)
U41^#(tt,V1,V2)	→	activate^#(V2)	(172)
U105^#(tt,V2)	→	activate^#(V2)	(102)
U104^#(tt,V1,V2)	→	isNat^#(activate(V1))	(97)
U104^#(tt,V1,V2)	→	activate^#(V1)	(98)
U104^#(tt,V1,V2)	→	activate^#(V2)	(99)
U103^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(93)
U103^#(tt,V1,V2)	→	activate^#(V2)	(94)
U103^#(tt,V1,V2)	→	activate^#(V1)	(95)
U102^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(89)
U102^#(tt,V1,V2)	→	activate^#(V2)	(90)
U102^#(tt,V1,V2)	→	activate^#(V1)	(91)
U101^#(tt,V1,V2)	→	isNatKind^#(activate(V1))	(85)
U101^#(tt,V1,V2)	→	activate^#(V1)	(86)
U101^#(tt,V1,V2)	→	activate^#(V2)	(87)
U121^#(tt,IL)	→	activate^#(IL)	(126)
take^#(0,IL)	→	isNatIList^#(IL)	(253)
take^#(s(M),cons(N,IL))	→	U131^#(isNatIList(activate(IL)),activate(IL),M,N)	(254)
U131^#(tt,IL,M,N)	→	U132^#(isNatIListKind(activate(IL)),activate(IL),activate(M),activate(N))	(128)
U132^#(tt,IL,M,N)	→	U133^#(isNat(activate(M)),activate(IL),activate(M),activate(N))	(133)
U133^#(tt,IL,M,N)	→	U134^#(isNatKind(activate(M)),activate(IL),activate(M),activate(N))	(138)
U134^#(tt,IL,M,N)	→	U135^#(isNat(activate(N)),activate(IL),activate(M),activate(N))	(143)
U135^#(tt,IL,M,N)	→	U136^#(isNatKind(activate(N)),activate(IL),activate(M),activate(N))	(148)
U136^#(tt,IL,M,N)	→	activate^#(N)	(154)
U136^#(tt,IL,M,N)	→	activate^#(M)	(155)
U136^#(tt,IL,M,N)	→	activate^#(IL)	(156)
U135^#(tt,IL,M,N)	→	isNatKind^#(activate(N))	(149)
U135^#(tt,IL,M,N)	→	activate^#(N)	(150)
U135^#(tt,IL,M,N)	→	activate^#(IL)	(151)
U135^#(tt,IL,M,N)	→	activate^#(M)	(152)
U134^#(tt,IL,M,N)	→	isNat^#(activate(N))	(144)
U134^#(tt,IL,M,N)	→	activate^#(N)	(145)
U134^#(tt,IL,M,N)	→	activate^#(IL)	(146)
U134^#(tt,IL,M,N)	→	activate^#(M)	(147)
U133^#(tt,IL,M,N)	→	isNatKind^#(activate(M))	(139)
U133^#(tt,IL,M,N)	→	activate^#(M)	(140)
U133^#(tt,IL,M,N)	→	activate^#(IL)	(141)
U133^#(tt,IL,M,N)	→	activate^#(N)	(142)
U132^#(tt,IL,M,N)	→	isNat^#(activate(M))	(134)
U132^#(tt,IL,M,N)	→	activate^#(M)	(135)
U132^#(tt,IL,M,N)	→	activate^#(IL)	(136)
U132^#(tt,IL,M,N)	→	activate^#(N)	(137)
U131^#(tt,IL,M,N)	→	isNatIListKind^#(activate(IL))	(129)
U131^#(tt,IL,M,N)	→	activate^#(IL)	(130)
U131^#(tt,IL,M,N)	→	activate^#(M)	(131)
U131^#(tt,IL,M,N)	→	activate^#(N)	(132)
take^#(s(M),cons(N,IL))	→	isNatIList^#(activate(IL))	(255)
take^#(s(M),cons(N,IL))	→	activate^#(IL)	(256)

and the following rules have been deleted.

1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

activate^#(n__length(X))	→	length^#(X)	(260)
isNatKind^#(n__length(V1))	→	isNatIListKind^#(activate(V1))	(235)
isNatKind^#(n__length(V1))	→	activate^#(V1)	(236)
isNat^#(n__length(V1))	→	U11^#(isNatIListKind(activate(V1)),activate(V1))	(213)
isNat^#(n__length(V1))	→	isNatIListKind^#(activate(V1))	(214)
isNat^#(n__length(V1))	→	activate^#(V1)	(215)

and the following rules have been deleted.

1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,V2)	→	activate^#(V2)	(190)
length^#(cons(N,L))	→	U111^#(isNatList(activate(L)),activate(L),N)	(249)
U111^#(tt,L,N)	→	U112^#(isNatIListKind(activate(L)),activate(L),activate(N))	(106)
U112^#(tt,L,N)	→	U113^#(isNat(activate(N)),activate(L),activate(N))	(110)
U113^#(tt,L,N)	→	U114^#(isNatKind(activate(N)),activate(L))	(114)
U114^#(tt,L)	→	length^#(activate(L))	(119)
length^#(cons(N,L))	→	isNatList^#(activate(L))	(250)
isNatList^#(n__cons(V1,V2))	→	U91^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(240)
U91^#(tt,V1,V2)	→	U92^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(194)
U92^#(tt,V1,V2)	→	U93^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(198)
U93^#(tt,V1,V2)	→	U94^#(isNatIListKind(activate(V2)),activate(V1),activate(V2))	(202)
U94^#(tt,V1,V2)	→	U95^#(isNat(activate(V1)),activate(V2))	(206)
U95^#(tt,V2)	→	isNatList^#(activate(V2))	(211)
isNatList^#(n__cons(V1,V2))	→	isNatKind^#(activate(V1))	(241)
isNatIListKind^#(n__cons(V1,V2))	→	isNatKind^#(activate(V1))	(227)
isNatKind^#(n__s(V1))	→	isNatKind^#(activate(V1))	(238)
isNatKind^#(n__s(V1))	→	activate^#(V1)	(239)
isNatIListKind^#(n__cons(V1,V2))	→	activate^#(V1)	(228)
isNatIListKind^#(n__cons(V1,V2))	→	activate^#(V2)	(229)
isNatList^#(n__cons(V1,V2))	→	activate^#(V1)	(242)
isNatList^#(n__cons(V1,V2))	→	activate^#(V2)	(243)
U95^#(tt,V2)	→	activate^#(V2)	(212)
U94^#(tt,V1,V2)	→	isNat^#(activate(V1))	(207)
U11^#(tt,V1)	→	U12^#(isNatIListKind(activate(V1)),activate(V1))	(103)
U12^#(tt,V1)	→	isNatList^#(activate(V1))	(122)
U12^#(tt,V1)	→	activate^#(V1)	(123)
U11^#(tt,V1)	→	isNatIListKind^#(activate(V1))	(104)
U11^#(tt,V1)	→	activate^#(V1)	(105)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(216)
U21^#(tt,V1)	→	U22^#(isNatKind(activate(V1)),activate(V1))	(157)
U22^#(tt,V1)	→	isNat^#(activate(V1))	(161)
isNat^#(n__s(V1))	→	isNatKind^#(activate(V1))	(217)
isNat^#(n__s(V1))	→	activate^#(V1)	(218)
U22^#(tt,V1)	→	activate^#(V1)	(162)
U21^#(tt,V1)	→	isNatKind^#(activate(V1))	(158)
U21^#(tt,V1)	→	activate^#(V1)	(159)
U94^#(tt,V1,V2)	→	activate^#(V1)	(208)
U94^#(tt,V1,V2)	→	activate^#(V2)	(209)
U93^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(203)
U93^#(tt,V1,V2)	→	activate^#(V2)	(204)
U93^#(tt,V1,V2)	→	activate^#(V1)	(205)
U92^#(tt,V1,V2)	→	isNatIListKind^#(activate(V2))	(199)
U92^#(tt,V1,V2)	→	activate^#(V2)	(200)
U92^#(tt,V1,V2)	→	activate^#(V1)	(201)
U91^#(tt,V1,V2)	→	isNatKind^#(activate(V1))	(195)
U91^#(tt,V1,V2)	→	activate^#(V1)	(196)
U91^#(tt,V1,V2)	→	activate^#(V2)	(197)
length^#(cons(N,L))	→	activate^#(L)	(251)
U114^#(tt,L)	→	activate^#(L)	(120)
U113^#(tt,L,N)	→	isNatKind^#(activate(N))	(115)
U113^#(tt,L,N)	→	activate^#(N)	(116)
U113^#(tt,L,N)	→	activate^#(L)	(117)
U112^#(tt,L,N)	→	isNat^#(activate(N))	(111)
U112^#(tt,L,N)	→	activate^#(N)	(112)
U112^#(tt,L,N)	→	activate^#(L)	(113)
U111^#(tt,L,N)	→	isNatIListKind^#(activate(L))	(107)
U111^#(tt,L,N)	→	activate^#(L)	(108)
U111^#(tt,L,N)	→	activate^#(N)	(109)

and the following rules have been deleted.

1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__zeros,y1))	→	U51^#(isNatKind(zeros),activate(y1))	(264)
isNatIListKind^#(n__cons(n__take(x0,x1),y1))	→	U51^#(isNatKind(take(x0,x1)),activate(y1))	(265)
isNatIListKind^#(n__cons(n__0,y1))	→	U51^#(isNatKind(0),activate(y1))	(266)
isNatIListKind^#(n__cons(n__length(x0),y1))	→	U51^#(isNatKind(length(x0)),activate(y1))	(267)
isNatIListKind^#(n__cons(n__s(x0),y1))	→	U51^#(isNatKind(s(x0)),activate(y1))	(268)
isNatIListKind^#(n__cons(n__cons(x0,x1),y1))	→	U51^#(isNatKind(cons(x0,x1)),activate(y1))	(269)
isNatIListKind^#(n__cons(n__nil,y1))	→	U51^#(isNatKind(nil),activate(y1))	(270)
isNatIListKind^#(n__cons(x0,y1))	→	U51^#(isNatKind(x0),activate(y1))	(271)

1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__zeros)	→	isNatIListKind^#(zeros)	(272)
U51^#(tt,n__take(x0,x1))	→	isNatIListKind^#(take(x0,x1))	(273)
U51^#(tt,n__0)	→	isNatIListKind^#(0)	(274)
U51^#(tt,n__length(x0))	→	isNatIListKind^#(length(x0))	(275)
U51^#(tt,n__s(x0))	→	isNatIListKind^#(s(x0))	(276)
U51^#(tt,n__cons(x0,x1))	→	isNatIListKind^#(cons(x0,x1))	(277)
U51^#(tt,n__nil)	→	isNatIListKind^#(nil)	(278)
U51^#(tt,x0)	→	isNatIListKind^#(x0)	(279)

1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__zeros,y0))	→	U51^#(isNatKind(cons(0,n__zeros)),activate(y0))	(280)
isNatIListKind^#(n__cons(n__zeros,y0))	→	U51^#(isNatKind(n__zeros),activate(y0))	(281)

1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__zeros,y0))

→

U51^#(isNatKind(n__zeros),activate(y0))

(281)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__zeros)	→	isNatIListKind^#(cons(0,n__zeros))	(282)
U51^#(tt,n__zeros)	→	isNatIListKind^#(n__zeros)	(283)

1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__zeros)

→

isNatIListKind^#(n__zeros)

(283)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__0)

→

isNatIListKind^#(n__0)

(284)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__0)

→

isNatIListKind^#(n__0)

(284)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__0,y0))

→

U51^#(isNatKind(n__0),activate(y0))

(285)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__s(x0))

→

isNatIListKind^#(n__s(x0))

(286)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__s(x0))

→

isNatIListKind^#(n__s(x0))

(286)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__cons(x0,x1))

→

isNatIListKind^#(n__cons(x0,x1))

(287)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__s(x0),y1))

→

U51^#(isNatKind(n__s(x0)),activate(y1))

(288)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__nil)

→

isNatIListKind^#(n__nil)

(289)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__nil)

→

isNatIListKind^#(n__nil)

(289)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__cons(x0,x1),y2))

→

U51^#(isNatKind(n__cons(x0,x1)),activate(y2))

(290)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__cons(x0,x1),y2))

→

U51^#(isNatKind(n__cons(x0,x1)),activate(y2))

(290)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__nil,y0))

→

U51^#(isNatKind(n__nil),activate(y0))

(291)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__nil,y0))

→

U51^#(isNatKind(n__nil),activate(y0))

(291)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__zeros,y0))	→	U51^#(isNatKind(n__cons(0,n__zeros)),activate(y0))	(292)
isNatIListKind^#(n__cons(n__zeros,y0))	→	U51^#(isNatKind(cons(n__0,n__zeros)),activate(y0))	(293)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__zeros,y0))

→

U51^#(isNatKind(n__cons(0,n__zeros)),activate(y0))

(292)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__zeros)	→	isNatIListKind^#(n__cons(0,n__zeros))	(294)
U51^#(tt,n__zeros)	→	isNatIListKind^#(cons(n__0,n__zeros))	(295)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

isNatIListKind^#(n__cons(n__zeros,y0))

→

U51^#(isNatKind(n__cons(n__0,n__zeros)),activate(y0))

(296)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__zeros,y0))

→

U51^#(isNatKind(n__cons(n__0,n__zeros)),activate(y0))

(296)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider narrowings of the pair below position 1 to get the following set of pairs

U51^#(tt,n__zeros)

→

isNatIListKind^#(n__cons(n__0,n__zeros))

(297)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__length(x0))

→

isNatIListKind^#(length(x0))

(275)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__take(x0,x1),y1))

→

U51^#(isNatKind(take(x0,x1)),activate(y1))

(265)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

isNatIListKind^#(n__cons(n__length(x0),y1))	→	U51^#(isNatKind(length(x0)),activate(y1))	(267)
isNatIListKind^#(n__cons(n__s(x0),y1))	→	U51^#(isNatKind(n__s(x0)),activate(y1))	(288)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted.

U51^#(tt,n__take(x0,x1))

→

isNatIListKind^#(take(x0,x1))

(273)

and the following rules have been deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Loop

The following loop proves infiniteness of the DP problem.

t₀	=	U51^#(isNatKind(n__0),activate(n__zeros))
	→_R	U51^#(isNatKind(n__0),n__zeros)
	→_R	U51^#(tt,n__zeros)
	→_P	isNatIListKind^#(n__cons(n__0,n__zeros))
	→_P	U51^#(isNatKind(n__0),activate(n__zeros))
	=	t₄

where t₄ = t₀