Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/OvConsOS_nokinds_noand_iGM)

The rewrite relation of the following TRS is considered.

There are 133 ruless (increase limit for explicit display).

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 207 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 23 components.

The 1^st component contains the pair

mark^#(cons(X1,X2))	→	active^#(cons(mark(X1),X2))	(199)
active^#(U41(tt,V2))	→	mark^#(U42(isNatIList(V2)))	(139)
mark^#(cons(X1,X2))	→	mark^#(X1)	(201)
mark^#(zeros)	→	active^#(zeros)	(198)
active^#(zeros)	→	mark^#(cons(0,zeros))	(134)
mark^#(U11(X))	→	active^#(U11(mark(X)))	(203)
active^#(U51(tt,V2))	→	mark^#(U52(isNatList(V2)))	(143)
mark^#(U11(X))	→	mark^#(X)	(205)
mark^#(U21(X))	→	active^#(U21(mark(X)))	(207)
active^#(U61(tt,V2))	→	mark^#(U62(isNatIList(V2)))	(147)
mark^#(U21(X))	→	mark^#(X)	(209)
mark^#(U31(X))	→	active^#(U31(mark(X)))	(210)
active^#(U71(tt,L,N))	→	mark^#(U72(isNat(N),L))	(151)
mark^#(U31(X))	→	mark^#(X)	(212)
mark^#(U41(X1,X2))	→	active^#(U41(mark(X1),X2))	(213)
active^#(U72(tt,L))	→	mark^#(s(length(L)))	(154)
mark^#(U41(X1,X2))	→	mark^#(X1)	(215)
mark^#(U42(X))	→	active^#(U42(mark(X)))	(216)
active^#(U91(tt,IL,M,N))	→	mark^#(U92(isNat(M),IL,M,N))	(158)
mark^#(U42(X))	→	mark^#(X)	(218)
mark^#(isNatIList(X))	→	active^#(isNatIList(X))	(219)
active^#(U92(tt,IL,M,N))	→	mark^#(U93(isNat(N),IL,M,N))	(161)
mark^#(U51(X1,X2))	→	active^#(U51(mark(X1),X2))	(220)
active^#(U93(tt,IL,M,N))	→	mark^#(cons(N,take(M,IL)))	(164)
mark^#(U51(X1,X2))	→	mark^#(X1)	(222)
mark^#(U52(X))	→	active^#(U52(mark(X)))	(223)
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	(168)
mark^#(U52(X))	→	mark^#(X)	(225)
mark^#(isNatList(X))	→	active^#(isNatList(X))	(226)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(171)
mark^#(U61(X1,X2))	→	active^#(U61(mark(X1),X2))	(227)
active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))	(174)
mark^#(U61(X1,X2))	→	mark^#(X1)	(229)
mark^#(U62(X))	→	active^#(U62(mark(X)))	(230)
active^#(isNatIList(cons(V1,V2)))	→	mark^#(U41(isNat(V1),V2))	(178)
mark^#(U62(X))	→	mark^#(X)	(232)
mark^#(U71(X1,X2,X3))	→	active^#(U71(mark(X1),X2,X3))	(233)
active^#(isNatList(cons(V1,V2)))	→	mark^#(U51(isNat(V1),V2))	(182)
mark^#(U71(X1,X2,X3))	→	mark^#(X1)	(235)
mark^#(U72(X1,X2))	→	active^#(U72(mark(X1),X2))	(236)
active^#(isNatList(take(V1,V2)))	→	mark^#(U61(isNat(V1),V2))	(185)
mark^#(U72(X1,X2))	→	mark^#(X1)	(238)
mark^#(isNat(X))	→	active^#(isNat(X))	(239)
active^#(length(cons(N,L)))	→	mark^#(U71(isNatList(L),L,N))	(189)
mark^#(s(X))	→	active^#(s(mark(X)))	(240)
active^#(take(0,IL))	→	mark^#(U81(isNatIList(IL)))	(192)
mark^#(s(X))	→	mark^#(X)	(242)
mark^#(length(X))	→	active^#(length(mark(X)))	(243)
active^#(take(s(M),cons(N,IL)))	→	mark^#(U91(isNatIList(IL),IL,M,N))	(195)
mark^#(length(X))	→	mark^#(X)	(245)
mark^#(U81(X))	→	active^#(U81(mark(X)))	(246)
mark^#(U81(X))	→	mark^#(X)	(248)
mark^#(U91(X1,X2,X3,X4))	→	active^#(U91(mark(X1),X2,X3,X4))	(250)
mark^#(U91(X1,X2,X3,X4))	→	mark^#(X1)	(252)
mark^#(U92(X1,X2,X3,X4))	→	active^#(U92(mark(X1),X2,X3,X4))	(253)
mark^#(U92(X1,X2,X3,X4))	→	mark^#(X1)	(255)
mark^#(U93(X1,X2,X3,X4))	→	active^#(U93(mark(X1),X2,X3,X4))	(256)
mark^#(U93(X1,X2,X3,X4))	→	mark^#(X1)	(258)
mark^#(take(X1,X2))	→	active^#(take(mark(X1),mark(X2)))	(259)
mark^#(take(X1,X2))	→	mark^#(X1)	(261)
mark^#(take(X1,X2))	→	mark^#(X2)	(262)

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-1 + x₁
[U11(x₁)]	=	0
[U21(x₁)]	=	-2
[U31(x₁)]	=	-2
[U41(x₁, x₂)]	=	2
[U42(x₁)]	=	-2
[U51(x₁, x₂)]	=	2
[U52(x₁)]	=	0
[U61(x₁, x₂)]	=	2
[U62(x₁)]	=	-2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[U81(x₁)]	=	-2
[U91(x₁,...,x₄)]	=	2
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2
[cons(x₁, x₂)]	=	-2
[length(x₁)]	=	2
[s(x₁)]	=	0
[take(x₁, x₂)]	=	2
[mark(x₁)]	=	-2
[active(x₁)]	=	2 · x₁
[zeros]	=	2
[0]	=	1
[tt]	=	1
[isNatIList(x₁)]	=	2
[isNatList(x₁)]	=	2
[isNat(x₁)]	=	2
[nil]	=	1
[mark^#(x₁)]	=	1

together with the usable rules

cons(X1,mark(X2))	→	cons(X1,X2)	(57)
cons(mark(X1),X2)	→	cons(X1,X2)	(56)
cons(active(X1),X2)	→	cons(X1,X2)	(58)
cons(X1,active(X2))	→	cons(X1,X2)	(59)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)
U62(active(X))	→	U62(X)	(87)
U62(mark(X))	→	U62(X)	(86)
U31(active(X))	→	U31(X)	(65)
U31(mark(X))	→	U31(X)	(64)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U92(X1,mark(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(115)
U92(mark(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(114)
U92(X1,X2,mark(X3),X4)	→	U92(X1,X2,X3,X4)	(116)
U92(X1,X2,X3,mark(X4))	→	U92(X1,X2,X3,X4)	(117)
U92(active(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(118)
U92(X1,active(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(119)
U92(X1,X2,active(X3),X4)	→	U92(X1,X2,X3,X4)	(120)
U92(X1,X2,X3,active(X4))	→	U92(X1,X2,X3,X4)	(121)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
take(X1,mark(X2))	→	take(X1,X2)	(131)
take(mark(X1),X2)	→	take(X1,X2)	(130)
take(active(X1),X2)	→	take(X1,X2)	(132)
take(X1,active(X2))	→	take(X1,X2)	(133)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U81(active(X))	→	U81(X)	(105)
U81(mark(X))	→	U81(X)	(104)
U91(X1,mark(X2),X3,X4)	→	U91(X1,X2,X3,X4)	(107)
U91(mark(X1),X2,X3,X4)	→	U91(X1,X2,X3,X4)	(106)
U91(X1,X2,mark(X3),X4)	→	U91(X1,X2,X3,X4)	(108)
U91(X1,X2,X3,mark(X4))	→	U91(X1,X2,X3,X4)	(109)
U91(active(X1),X2,X3,X4)	→	U91(X1,X2,X3,X4)	(110)
U91(X1,active(X2),X3,X4)	→	U91(X1,X2,X3,X4)	(111)
U91(X1,X2,active(X3),X4)	→	U91(X1,X2,X3,X4)	(112)
U91(X1,X2,X3,active(X4))	→	U91(X1,X2,X3,X4)	(113)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

mark^#(cons(X1,X2))	→	active^#(cons(mark(X1),X2))	(199)
mark^#(U11(X))	→	active^#(U11(mark(X)))	(203)
mark^#(U21(X))	→	active^#(U21(mark(X)))	(207)
mark^#(U31(X))	→	active^#(U31(mark(X)))	(210)
mark^#(U42(X))	→	active^#(U42(mark(X)))	(216)
mark^#(U52(X))	→	active^#(U52(mark(X)))	(223)
mark^#(U62(X))	→	active^#(U62(mark(X)))	(230)
mark^#(s(X))	→	active^#(s(mark(X)))	(240)
mark^#(U81(X))	→	active^#(U81(mark(X)))	(246)

could be deleted.

1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	1 · x₁
[U41(x₁, x₂)]	=	1 · x₁
[tt]	=	0
[mark^#(x₁)]	=	1 · x₁
[U42(x₁)]	=	1 · x₁
[isNatIList(x₁)]	=	0
[cons(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[zeros]	=	0
[0]	=	0
[U51(x₁, x₂)]	=	1 · x₁
[U52(x₁)]	=	1 · x₁
[isNatList(x₁)]	=	0
[U11(x₁)]	=	1 · x₁
[U61(x₁, x₂)]	=	1 · x₁
[U62(x₁)]	=	1 · x₁
[U21(x₁)]	=	1 · x₁
[U71(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂
[U72(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[isNat(x₁)]	=	0
[U31(x₁)]	=	1 · x₁
[mark(x₁)]	=	1 · x₁
[s(x₁)]	=	1 · x₁
[length(x₁)]	=	1 + 1 · x₁
[U91(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[U92(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[U93(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[take(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[U81(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[nil]	=	0

the pairs

mark^#(U71(X1,X2,X3))	→	mark^#(X1)	(235)
mark^#(U72(X1,X2))	→	mark^#(X1)	(238)
mark^#(length(X))	→	mark^#(X)	(245)

could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	2 · x₁
[mark^#(x₁)]	=	2 · x₁
[U42(x₁)]	=	2 · x₁
[U62(x₁)]	=	2 · x₁
[U81(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	2 + 2 · x₁ + x₃ + 2 · x₄
[isNatIList(x₁)]	=	0
[active(x₁)]	=	x₁
[mark(x₁)]	=	x₁
[U41(x₁, x₂)]	=	x₁
[U51(x₁, x₂)]	=	2 · x₁
[U61(x₁, x₂)]	=	2 · x₁
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[U92(x₁,...,x₄)]	=	2 + 2 · x₁ + 2 · x₄
[U93(x₁,...,x₄)]	=	2 + x₁ + 2 · x₄
[length(x₁)]	=	2
[take(x₁, x₂)]	=	1 + 2 · x₁ + x₂
[U11(x₁)]	=	x₁
[U31(x₁)]	=	2 · x₁
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	2 · x₁
[isNat(x₁)]	=	0
[cons(x₁, x₂)]	=	2 + 2 · x₁
[zeros]	=	2
[0]	=	0
[tt]	=	0
[s(x₁)]	=	x₁
[nil]	=	0

the pairs

mark^#(cons(X1,X2))	→	mark^#(X1)	(201)
active^#(take(0,IL))	→	mark^#(U81(isNatIList(IL)))	(192)
active^#(take(s(M),cons(N,IL)))	→	mark^#(U91(isNatIList(IL),IL,M,N))	(195)
mark^#(U91(X1,X2,X3,X4))	→	mark^#(X1)	(252)
mark^#(U92(X1,X2,X3,X4))	→	mark^#(X1)	(255)
mark^#(U93(X1,X2,X3,X4))	→	mark^#(X1)	(258)
mark^#(take(X1,X2))	→	mark^#(X1)	(261)
mark^#(take(X1,X2))	→	mark^#(X2)	(262)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U11(X))	→	mark^#(X)	(205)
mark^#(U21(X))	→	mark^#(X)	(209)
mark^#(U31(X))	→	mark^#(X)	(212)
mark^#(U41(X1,X2))	→	active^#(U41(mark(X1),X2))	(213)
active^#(U41(tt,V2))	→	mark^#(U42(isNatIList(V2)))	(139)
mark^#(U41(X1,X2))	→	mark^#(X1)	(215)
mark^#(U42(X))	→	mark^#(X)	(218)
mark^#(isNatIList(X))	→	active^#(isNatIList(X))	(219)
active^#(U51(tt,V2))	→	mark^#(U52(isNatList(V2)))	(143)
mark^#(U51(X1,X2))	→	active^#(U51(mark(X1),X2))	(220)
active^#(U61(tt,V2))	→	mark^#(U62(isNatIList(V2)))	(147)
mark^#(U51(X1,X2))	→	mark^#(X1)	(222)
mark^#(U52(X))	→	mark^#(X)	(225)
mark^#(isNatList(X))	→	active^#(isNatList(X))	(226)
active^#(U71(tt,L,N))	→	mark^#(U72(isNat(N),L))	(151)
mark^#(U61(X1,X2))	→	active^#(U61(mark(X1),X2))	(227)
active^#(U72(tt,L))	→	mark^#(s(length(L)))	(154)
mark^#(U61(X1,X2))	→	mark^#(X1)	(229)
mark^#(U62(X))	→	mark^#(X)	(232)
mark^#(U71(X1,X2,X3))	→	active^#(U71(mark(X1),X2,X3))	(233)
active^#(U91(tt,IL,M,N))	→	mark^#(U92(isNat(M),IL,M,N))	(158)
mark^#(U72(X1,X2))	→	active^#(U72(mark(X1),X2))	(236)
active^#(U92(tt,IL,M,N))	→	mark^#(U93(isNat(N),IL,M,N))	(161)
mark^#(isNat(X))	→	active^#(isNat(X))	(239)
active^#(U93(tt,IL,M,N))	→	mark^#(cons(N,take(M,IL)))	(164)
mark^#(s(X))	→	mark^#(X)	(242)
mark^#(length(X))	→	active^#(length(mark(X)))	(243)
active^#(isNat(length(V1)))	→	mark^#(U11(isNatList(V1)))	(168)
mark^#(U81(X))	→	mark^#(X)	(248)
mark^#(U91(X1,X2,X3,X4))	→	active^#(U91(mark(X1),X2,X3,X4))	(250)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(171)
mark^#(U92(X1,X2,X3,X4))	→	active^#(U92(mark(X1),X2,X3,X4))	(253)
active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))	(174)
mark^#(U93(X1,X2,X3,X4))	→	active^#(U93(mark(X1),X2,X3,X4))	(256)
active^#(isNatIList(cons(V1,V2)))	→	mark^#(U41(isNat(V1),V2))	(178)
mark^#(take(X1,X2))	→	active^#(take(mark(X1),mark(X2)))	(259)
active^#(isNatList(cons(V1,V2)))	→	mark^#(U51(isNat(V1),V2))	(182)
active^#(isNatList(take(V1,V2)))	→	mark^#(U61(isNat(V1),V2))	(185)
active^#(length(cons(N,L)))	→	mark^#(U71(isNatList(L),L,N))	(189)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	1 + 2 · x₁
[U51(x₁, x₂)]	=	1 + x₁
[U61(x₁, x₂)]	=	1 + 2 · x₁
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[U91(x₁,...,x₄)]	=	2 + x₁ + x₂ + 2 · x₃ + 2 · x₄
[U92(x₁,...,x₄)]	=	-2
[U93(x₁,...,x₄)]	=	0
[length(x₁)]	=	0
[take(x₁, x₂)]	=	0
[mark(x₁)]	=	1 + x₁
[cons(x₁, x₂)]	=	0
[active(x₁)]	=	1 + 2 · x₁
[zeros]	=	1
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	1 + 2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	1 + x₁
[U52(x₁)]	=	1 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	1 + x₁
[U62(x₁)]	=	1 + 2 · x₁
[U31(x₁)]	=	1 + x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + 2 · x₁
[U81(x₁)]	=	2 + x₁
[nil]	=	1
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U62(active(X))	→	U62(X)	(87)
U62(mark(X))	→	U62(X)	(86)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U92(X1,mark(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(115)
U92(mark(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(114)
U92(X1,X2,mark(X3),X4)	→	U92(X1,X2,X3,X4)	(116)
U92(X1,X2,X3,mark(X4))	→	U92(X1,X2,X3,X4)	(117)
U92(active(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(118)
U92(X1,active(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(119)
U92(X1,X2,active(X3),X4)	→	U92(X1,X2,X3,X4)	(120)
U92(X1,X2,X3,active(X4))	→	U92(X1,X2,X3,X4)	(121)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)
cons(X1,mark(X2))	→	cons(X1,X2)	(57)
cons(mark(X1),X2)	→	cons(X1,X2)	(56)
cons(active(X1),X2)	→	cons(X1,X2)	(58)
cons(X1,active(X2))	→	cons(X1,X2)	(59)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)
U31(active(X))	→	U31(X)	(65)
U31(mark(X))	→	U31(X)	(64)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

mark^#(U81(X))	→	mark^#(X)	(248)
mark^#(U91(X1,X2,X3,X4))	→	active^#(U91(mark(X1),X2,X3,X4))	(250)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + x₁
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	2
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2
[length(x₁)]	=	2
[take(x₁, x₂)]	=	-2
[mark(x₁)]	=	-2 + x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₂
[active(x₁)]	=	1 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	-2 + x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	-1 + x₁
[U52(x₁)]	=	-2
[isNatList(x₁)]	=	0
[U21(x₁)]	=	-1 + x₁
[U62(x₁)]	=	-2 + x₁
[U31(x₁)]	=	-2 + x₁
[isNat(x₁)]	=	2
[s(x₁)]	=	-2 + x₁
[U91(x₁,...,x₄)]	=	2 · x₁ + 2 · x₂ + 2 · x₃ + x₄
[U81(x₁)]	=	-2 + x₁
[nil]	=	0
[mark^#(x₁)]	=	-2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U92(X1,mark(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(115)
U92(mark(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(114)
U92(X1,X2,mark(X3),X4)	→	U92(X1,X2,X3,X4)	(116)
U92(X1,X2,X3,mark(X4))	→	U92(X1,X2,X3,X4)	(117)
U92(active(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(118)
U92(X1,active(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(119)
U92(X1,X2,active(X3),X4)	→	U92(X1,X2,X3,X4)	(120)
U92(X1,X2,X3,active(X4))	→	U92(X1,X2,X3,X4)	(121)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)
take(X1,mark(X2))	→	take(X1,X2)	(131)
take(mark(X1),X2)	→	take(X1,X2)	(130)
take(active(X1),X2)	→	take(X1,X2)	(132)
take(X1,active(X2))	→	take(X1,X2)	(133)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(U91(tt,IL,M,N))

→

mark^#(U92(isNat(M),IL,M,N))

(158)

could be deleted.

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + 2 · x₁
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2
[length(x₁)]	=	2
[take(x₁, x₂)]	=	0
[mark(x₁)]	=	-2
[cons(x₁, x₂)]	=	2
[active(x₁)]	=	-2
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	-2
[isNatIList(x₁)]	=	2
[U11(x₁)]	=	-2 + 2 · x₁
[U52(x₁)]	=	-2 + x₁
[isNatList(x₁)]	=	2
[U21(x₁)]	=	-2 + x₁
[U62(x₁)]	=	-2
[U31(x₁)]	=	-2 + x₁
[isNat(x₁)]	=	2
[s(x₁)]	=	0
[U91(x₁,...,x₄)]	=	-2 + x₁ + x₃
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)
take(X1,mark(X2))	→	take(X1,X2)	(131)
take(mark(X1),X2)	→	take(X1,X2)	(130)
take(active(X1),X2)	→	take(X1,X2)	(132)
take(X1,active(X2))	→	take(X1,X2)	(133)
U92(X1,mark(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(115)
U92(mark(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(114)
U92(X1,X2,mark(X3),X4)	→	U92(X1,X2,X3,X4)	(116)
U92(X1,X2,X3,mark(X4))	→	U92(X1,X2,X3,X4)	(117)
U92(active(X1),X2,X3,X4)	→	U92(X1,X2,X3,X4)	(118)
U92(X1,active(X2),X3,X4)	→	U92(X1,X2,X3,X4)	(119)
U92(X1,X2,active(X3),X4)	→	U92(X1,X2,X3,X4)	(120)
U92(X1,X2,X3,active(X4))	→	U92(X1,X2,X3,X4)	(121)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(take(X1,X2))

→

active^#(take(mark(X1),mark(X2)))

(259)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	2
[U41(x₁, x₂)]	=	2 · x₁
[U51(x₁, x₂)]	=	2 · x₁
[U61(x₁, x₂)]	=	x₁
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[U92(x₁,...,x₄)]	=	1
[U93(x₁,...,x₄)]	=	-2
[length(x₁)]	=	-2
[mark(x₁)]	=	2 · x₁
[cons(x₁, x₂)]	=	x₂
[active(x₁)]	=	2 + x₁
[zeros]	=	0
[0]	=	0
[tt]	=	1
[U42(x₁)]	=	2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	x₁
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	x₁
[U62(x₁)]	=	2 · x₁
[U31(x₁)]	=	x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	2 · x₁
[U91(x₁,...,x₄)]	=	x₃ + x₄
[take(x₁, x₂)]	=	-2
[U81(x₁)]	=	1
[nil]	=	0
[mark^#(x₁)]	=	2 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U62(active(X))	→	U62(X)	(87)
U62(mark(X))	→	U62(X)	(86)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)
take(X1,mark(X2))	→	take(X1,X2)	(131)
take(mark(X1),X2)	→	take(X1,X2)	(130)
take(active(X1),X2)	→	take(X1,X2)	(132)
take(X1,active(X2))	→	take(X1,X2)	(133)
cons(X1,mark(X2))	→	cons(X1,X2)	(57)
cons(mark(X1),X2)	→	cons(X1,X2)	(56)
cons(active(X1),X2)	→	cons(X1,X2)	(58)
cons(X1,active(X2))	→	cons(X1,X2)	(59)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)
U31(active(X))	→	U31(X)	(65)
U31(mark(X))	→	U31(X)	(64)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U92(X1,X2,X3,X4))

→

active^#(U92(mark(X1),X2,X3,X4))

(253)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + 2 · x₁
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[U93(x₁,...,x₄)]	=	2
[length(x₁)]	=	2
[mark(x₁)]	=	2
[cons(x₁, x₂)]	=	-2 + x₁
[active(x₁)]	=	-2
[zeros]	=	0
[0]	=	2
[tt]	=	2
[U42(x₁)]	=	-2 + x₁
[isNatIList(x₁)]	=	2
[U11(x₁)]	=	-2 + 2 · x₁
[U52(x₁)]	=	-2
[isNatList(x₁)]	=	2
[U21(x₁)]	=	2
[U62(x₁)]	=	-2
[U31(x₁)]	=	-2
[isNat(x₁)]	=	2
[s(x₁)]	=	2
[U91(x₁,...,x₄)]	=	-2 + 2 · x₂ + 2 · x₄
[U92(x₁,...,x₄)]	=	2 + 2 · x₁ + 2 · x₂ + x₃ + x₄
[take(x₁, x₂)]	=	-2 + x₁
[U81(x₁)]	=	2
[nil]	=	0
[mark^#(x₁)]	=	2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U93(X1,mark(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(123)
U93(mark(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(122)
U93(X1,X2,mark(X3),X4)	→	U93(X1,X2,X3,X4)	(124)
U93(X1,X2,X3,mark(X4))	→	U93(X1,X2,X3,X4)	(125)
U93(active(X1),X2,X3,X4)	→	U93(X1,X2,X3,X4)	(126)
U93(X1,active(X2),X3,X4)	→	U93(X1,X2,X3,X4)	(127)
U93(X1,X2,active(X3),X4)	→	U93(X1,X2,X3,X4)	(128)
U93(X1,X2,X3,active(X4))	→	U93(X1,X2,X3,X4)	(129)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(U92(tt,IL,M,N))

→

mark^#(U93(isNat(N),IL,M,N))

(161)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	1 + x₁
[U51(x₁, x₂)]	=	1 + x₁
[U61(x₁, x₂)]	=	1 + x₁
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[U93(x₁,...,x₄)]	=	2 + x₂ + 2 · x₃ + 2 · x₄
[length(x₁)]	=	0
[mark(x₁)]	=	2 + x₁
[cons(x₁, x₂)]	=	-2
[active(x₁)]	=	2 + 2 · x₁
[zeros]	=	2
[0]	=	2
[tt]	=	2
[U42(x₁)]	=	1 + 2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	1 + 2 · x₁
[U52(x₁)]	=	1 + x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	1 + x₁
[U62(x₁)]	=	1 + x₁
[U31(x₁)]	=	1 + x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + x₁
[U91(x₁,...,x₄)]	=	2 + 2 · x₂ + x₄
[U92(x₁,...,x₄)]	=	-2
[take(x₁, x₂)]	=	-2 + x₁
[U81(x₁)]	=	2 + 2 · x₁
[nil]	=	2
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U62(active(X))	→	U62(X)	(87)
U62(mark(X))	→	U62(X)	(86)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
cons(X1,mark(X2))	→	cons(X1,X2)	(57)
cons(mark(X1),X2)	→	cons(X1,X2)	(56)
cons(active(X1),X2)	→	cons(X1,X2)	(58)
cons(X1,active(X2))	→	cons(X1,X2)	(59)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)
U31(active(X))	→	U31(X)	(65)
U31(mark(X))	→	U31(X)	(64)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U93(X1,X2,X3,X4))

→

active^#(U93(mark(X1),X2,X3,X4))

(256)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	x₁
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	2
[U61(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[length(x₁)]	=	2
[mark(x₁)]	=	2 + 2 · x₁
[cons(x₁, x₂)]	=	-2 + x₁
[active(x₁)]	=	-2
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	2
[isNatIList(x₁)]	=	2
[U11(x₁)]	=	-2
[U52(x₁)]	=	-2
[isNatList(x₁)]	=	2
[U21(x₁)]	=	-2
[U62(x₁)]	=	-1 + x₁
[U31(x₁)]	=	2
[isNat(x₁)]	=	2
[s(x₁)]	=	-2 + x₁
[U91(x₁,...,x₄)]	=	-2 + 2 · x₄
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	-1 + 2 · x₁
[take(x₁, x₂)]	=	-2 + x₁ + x₂
[U81(x₁)]	=	1 + x₁
[nil]	=	2
[mark^#(x₁)]	=	2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
U61(X1,mark(X2))	→	U61(X1,X2)	(83)
U61(mark(X1),X2)	→	U61(X1,X2)	(82)
U61(active(X1),X2)	→	U61(X1,X2)	(84)
U61(X1,active(X2))	→	U61(X1,X2)	(85)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(U93(tt,IL,M,N))

→

mark^#(cons(N,take(M,IL)))

(164)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	x₁
[U41(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂
[U51(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[U61(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂
[U71(x₁, x₂, x₃)]	=	x₂ + 2 · x₃
[U72(x₁, x₂)]	=	x₂
[length(x₁)]	=	x₁
[mark(x₁)]	=	x₁
[cons(x₁, x₂)]	=	2 · x₁ + x₂
[active(x₁)]	=	x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	x₁
[isNatIList(x₁)]	=	1 + 2 · x₁
[U11(x₁)]	=	x₁
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	2 · x₁
[U21(x₁)]	=	x₁
[U62(x₁)]	=	x₁
[U31(x₁)]	=	x₁
[isNat(x₁)]	=	2 · x₁
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	2 + x₂ + 2 · x₃ + 2 · x₄
[U92(x₁,...,x₄)]	=	2 + x₂ + 2 · x₃ + 2 · x₄
[U93(x₁,...,x₄)]	=	2 + x₂ + 2 · x₃ + 2 · x₄
[take(x₁, x₂)]	=	2 + 2 · x₁ + x₂
[U81(x₁)]	=	2
[nil]	=	2
[mark^#(x₁)]	=	x₁

the pairs

mark^#(U41(X1,X2))	→	mark^#(X1)	(215)
active^#(U61(tt,V2))	→	mark^#(U62(isNatIList(V2)))	(147)
mark^#(U61(X1,X2))	→	mark^#(X1)	(229)
active^#(isNatIList(V))	→	mark^#(U31(isNatList(V)))	(174)
active^#(isNatList(take(V1,V2)))	→	mark^#(U61(isNat(V1),V2))	(185)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	0
[U41(x₁, x₂)]	=	0
[U51(x₁, x₂)]	=	1 + 2 · x₁
[U61(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	0
[length(x₁)]	=	0
[mark(x₁)]	=	2 · x₁
[cons(x₁, x₂)]	=	2 + x₁ + 2 · x₂
[active(x₁)]	=	2 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	1 + 2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	1 + 2 · x₁
[U52(x₁)]	=	1 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	1 + x₁
[U62(x₁)]	=	2 + x₁
[U31(x₁)]	=	2 + x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + 2 · x₁
[U91(x₁,...,x₄)]	=	-2 + x₁ + x₂ + x₄
[U92(x₁,...,x₄)]	=	2 + x₁ + 2 · x₂ + 2 · x₃
[U93(x₁,...,x₄)]	=	2 + 2 · x₂ + x₃
[take(x₁, x₂)]	=	-2 + 2 · x₁ + x₂
[U81(x₁)]	=	2 + x₁
[nil]	=	0
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

mark^#(U31(X))	→	mark^#(X)	(212)
mark^#(U61(X1,X2))	→	active^#(U61(mark(X1),X2))	(227)
mark^#(U62(X))	→	mark^#(X)	(232)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	x₁
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[U71(x₁, x₂, x₃)]	=	1 + 2 · x₂ + 2 · x₃
[U72(x₁, x₂)]	=	1 + 2 · x₂
[length(x₁)]	=	1 + 2 · x₁
[mark(x₁)]	=	x₁
[cons(x₁, x₂)]	=	2 · x₁ + x₂
[active(x₁)]	=	x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	2 + x₁
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	2 · x₁
[U21(x₁)]	=	x₁
[U61(x₁, x₂)]	=	2
[U62(x₁)]	=	0
[U31(x₁)]	=	-2
[isNat(x₁)]	=	2 · x₁
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	2 + x₂ + 2 · x₄
[U92(x₁,...,x₄)]	=	2 + x₂ + 2 · x₄
[U93(x₁,...,x₄)]	=	2 + x₂ + 2 · x₄
[take(x₁, x₂)]	=	2 + x₂
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	x₁

the pair

mark^#(U11(X))

→

mark^#(X)

(205)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	2 + 2 · x₁
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	1 + 2 · x₁
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	0
[mark(x₁)]	=	x₁
[cons(x₁, x₂)]	=	-2
[active(x₁)]	=	x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	-2
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	1
[U21(x₁)]	=	x₁
[U61(x₁, x₂)]	=	-2
[U62(x₁)]	=	-2
[U31(x₁)]	=	-2
[isNat(x₁)]	=	0
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	-2
[U92(x₁,...,x₄)]	=	0
[U93(x₁,...,x₄)]	=	-2
[take(x₁, x₂)]	=	-2
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	2 + 2 · x₁

the pair

mark^#(U51(X1,X2))

→

mark^#(X1)

(222)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + 2 · x₁
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	-2
[mark(x₁)]	=	2 · x₁
[cons(x₁, x₂)]	=	-2 + x₁
[active(x₁)]	=	2 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	1
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	2
[U21(x₁)]	=	x₁
[U61(x₁, x₂)]	=	-2 + 2 · x₁
[U62(x₁)]	=	2
[U31(x₁)]	=	-2
[isNat(x₁)]	=	2
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	2 + x₂ + x₃
[U92(x₁,...,x₄)]	=	-2 + 2 · x₁ + 2 · x₃ + x₄
[U93(x₁,...,x₄)]	=	-2 + 2 · x₂
[take(x₁, x₂)]	=	-2 + x₁
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U11(active(X))	→	U11(X)	(61)
U11(mark(X))	→	U11(X)	(60)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(isNat(length(V1)))

→

mark^#(U11(isNatList(V1)))

(168)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + x₁
[U41(x₁, x₂)]	=	0
[U51(x₁, x₂)]	=	-2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[length(x₁)]	=	-2
[mark(x₁)]	=	2 + x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[active(x₁)]	=	2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	2 + x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	-2 + x₁
[U52(x₁)]	=	2 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	2 + x₁
[U61(x₁, x₂)]	=	2 + x₁
[U62(x₁)]	=	2 + 2 · x₁
[U31(x₁)]	=	2
[isNat(x₁)]	=	2 + 2 · x₁
[s(x₁)]	=	2 + 2 · x₁
[U91(x₁,...,x₄)]	=	-2 + 2 · x₃
[U92(x₁,...,x₄)]	=	-2 + 2 · x₄
[U93(x₁,...,x₄)]	=	2
[take(x₁, x₂)]	=	-2
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	-2 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)
U21(active(X))	→	U21(X)	(63)
U21(mark(X))	→	U21(X)	(62)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(isNat(s(V1)))

→

mark^#(U21(isNat(V1)))

(171)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + 2 · x₁
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[length(x₁)]	=	2
[mark(x₁)]	=	2
[cons(x₁, x₂)]	=	-2 + 2 · x₂
[active(x₁)]	=	-2 + x₁
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	-2 + 2 · x₁
[isNatIList(x₁)]	=	2
[U11(x₁)]	=	0
[U52(x₁)]	=	-2
[isNatList(x₁)]	=	2
[U21(x₁)]	=	-2
[U61(x₁, x₂)]	=	2 + x₁ + x₂
[U62(x₁)]	=	2
[U31(x₁)]	=	1 + x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	-2
[U91(x₁,...,x₄)]	=	-2 + 2 · x₁ + 2 · x₄
[U92(x₁,...,x₄)]	=	2 + 2 · x₁ + x₄
[U93(x₁,...,x₄)]	=	-2 + x₂ + x₃ + 2 · x₄
[take(x₁, x₂)]	=	2
[U81(x₁)]	=	1
[nil]	=	0
[mark^#(x₁)]	=	2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
isNat(active(X))	→	isNat(X)	(99)
isNat(mark(X))	→	isNat(X)	(98)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(isNat(X))

→

active^#(isNat(X))

(239)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	-2
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	0
[mark(x₁)]	=	2 + 2 · x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[active(x₁)]	=	2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	1 + x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	0
[U52(x₁)]	=	1 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	2 + 2 · x₁
[U61(x₁, x₂)]	=	-2 + x₂
[U62(x₁)]	=	2 + x₁
[U31(x₁)]	=	2
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + 2 · x₁
[U91(x₁,...,x₄)]	=	2
[U92(x₁,...,x₄)]	=	-2 + x₂ + 2 · x₃
[U93(x₁,...,x₄)]	=	2
[take(x₁, x₂)]	=	-2 + x₁
[U81(x₁)]	=	2
[nil]	=	1
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U21(X))

→

mark^#(X)

(209)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	2 + x₁
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	2 · x₂
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	0
[mark(x₁)]	=	2 + 2 · x₁
[cons(x₁, x₂)]	=	1 + x₂
[active(x₁)]	=	2 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	2 + 2 · x₁
[U52(x₁)]	=	x₁
[isNatList(x₁)]	=	2 · x₁
[U21(x₁)]	=	-2
[U61(x₁, x₂)]	=	-2 + 2 · x₁
[U62(x₁)]	=	-2
[U31(x₁)]	=	-2
[isNat(x₁)]	=	0
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	-2 + 2 · x₂ + 2 · x₃ + 2 · x₄
[U92(x₁,...,x₄)]	=	-2 + 2 · x₂
[U93(x₁,...,x₄)]	=	2 + x₂
[take(x₁, x₂)]	=	-2 + 2 · x₂
[U81(x₁)]	=	0
[nil]	=	1
[mark^#(x₁)]	=	2 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(isNatList(cons(V1,V2)))

→

mark^#(U51(isNat(V1),V2))

(182)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + 2 · x₁
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[length(x₁)]	=	2
[mark(x₁)]	=	2
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[active(x₁)]	=	-2 + 2 · x₁
[zeros]	=	2
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	-2
[isNatIList(x₁)]	=	2
[U11(x₁)]	=	1 + x₁
[U52(x₁)]	=	-2 + x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	2
[U61(x₁, x₂)]	=	2 + x₂
[U62(x₁)]	=	2
[U31(x₁)]	=	2
[isNat(x₁)]	=	0
[s(x₁)]	=	2
[U91(x₁,...,x₄)]	=	-1 + x₃
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2 + 2 · x₁ + x₂ + 2 · x₃
[take(x₁, x₂)]	=	-2 + 2 · x₁
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U51(X1,mark(X2))	→	U51(X1,X2)	(75)
U51(mark(X1),X2)	→	U51(X1,X2)	(74)
U51(active(X1),X2)	→	U51(X1,X2)	(76)
U51(X1,active(X2))	→	U51(X1,X2)	(77)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(isNatList(X))

→

active^#(isNatList(X))

(226)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	-2
[U51(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	0
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	-2
[mark(x₁)]	=	2 + 2 · x₁
[cons(x₁, x₂)]	=	-2
[active(x₁)]	=	2 + x₁
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	1 + 2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	-2
[U52(x₁)]	=	1 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	2
[U61(x₁, x₂)]	=	-2 + 2 · x₂
[U62(x₁)]	=	2
[U31(x₁)]	=	-2
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + x₁
[U91(x₁,...,x₄)]	=	-2 + x₂ + x₃
[U92(x₁,...,x₄)]	=	2 + x₁ + x₂ + x₃ + x₄
[U93(x₁,...,x₄)]	=	2 + x₁ + x₃ + 2 · x₄
[take(x₁, x₂)]	=	0
[U81(x₁)]	=	-2
[nil]	=	2
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
isNatList(active(X))	→	isNatList(X)	(81)
isNatList(mark(X))	→	isNatList(X)	(80)
U52(active(X))	→	U52(X)	(79)
U52(mark(X))	→	U52(X)	(78)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U51(X1,X2))

→

active^#(U51(mark(X1),X2))

(220)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2 + x₁
[U41(x₁, x₂)]	=	-2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	2
[mark(x₁)]	=	-2 + 2 · x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₂
[active(x₁)]	=	2
[zeros]	=	2
[0]	=	0
[tt]	=	2
[U42(x₁)]	=	-2 + 2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	2 + x₁
[U51(x₁, x₂)]	=	2 + 2 · x₁
[U52(x₁)]	=	-2 + 2 · x₁
[isNatList(x₁)]	=	2 + 2 · x₁
[U21(x₁)]	=	2
[U61(x₁, x₂)]	=	-2 + 2 · x₁
[U62(x₁)]	=	-2 + 2 · x₁
[U31(x₁)]	=	-2 + x₁
[isNat(x₁)]	=	2 · x₁
[s(x₁)]	=	0
[U91(x₁,...,x₄)]	=	-1 + 2 · x₂ + x₃ + 2 · x₄
[U92(x₁,...,x₄)]	=	-2 + x₂ + 2 · x₃
[U93(x₁,...,x₄)]	=	2 + 2 · x₁ + x₃
[take(x₁, x₂)]	=	2 + 2 · x₁
[U81(x₁)]	=	x₁
[nil]	=	2
[mark^#(x₁)]	=	-2

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(U51(tt,V2))

→

mark^#(U52(isNatList(V2)))

(143)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	-2
[U71(x₁, x₂, x₃)]	=	1
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	-2
[mark(x₁)]	=	2 · x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₂
[active(x₁)]	=	x₁
[zeros]	=	2
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	1 + x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	1
[U51(x₁, x₂)]	=	-2 + x₁
[U52(x₁)]	=	2 + 2 · x₁
[isNatList(x₁)]	=	2
[U21(x₁)]	=	-2 + 2 · x₁
[U61(x₁, x₂)]	=	2 + 2 · x₁
[U62(x₁)]	=	-2 + x₁
[U31(x₁)]	=	1
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + x₁
[U91(x₁,...,x₄)]	=	2
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2 + 2 · x₃ + 2 · x₄
[take(x₁, x₂)]	=	-2 + 2 · x₁
[U81(x₁)]	=	x₁
[nil]	=	0
[mark^#(x₁)]	=	-2 + 2 · x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U52(X))

→

mark^#(X)

(225)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-1 + x₁
[U41(x₁, x₂)]	=	2 + 2 · x₂
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	-2
[mark(x₁)]	=	x₁
[cons(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂
[active(x₁)]	=	2 + x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	1 + x₁
[isNatIList(x₁)]	=	1 + 2 · x₁
[U11(x₁)]	=	-2 + x₁
[U51(x₁, x₂)]	=	-2 + x₂
[U52(x₁)]	=	-2
[isNatList(x₁)]	=	0
[U21(x₁)]	=	-2 + x₁
[U61(x₁, x₂)]	=	-1 + x₁
[U62(x₁)]	=	-2 + 2 · x₁
[U31(x₁)]	=	x₁
[isNat(x₁)]	=	2 · x₁
[s(x₁)]	=	1 + 2 · x₁
[U91(x₁,...,x₄)]	=	-1 + 2 · x₄
[U92(x₁,...,x₄)]	=	-2 + x₁ + x₂ + 2 · x₄
[U93(x₁,...,x₄)]	=	1 + 2 · x₂ + 2 · x₃
[take(x₁, x₂)]	=	0
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	-1 + x₁

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(isNatIList(cons(V1,V2)))

→

mark^#(U41(isNat(V1),V2))

(178)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-1 + x₁
[U41(x₁, x₂)]	=	2
[U71(x₁, x₂, x₃)]	=	2
[U72(x₁, x₂)]	=	2
[length(x₁)]	=	2
[mark(x₁)]	=	-2 + x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[active(x₁)]	=	-2 + 2 · x₁
[zeros]	=	2
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	0
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	-2 + x₁
[U51(x₁, x₂)]	=	-2 + x₂
[U52(x₁)]	=	-2 + 2 · x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	-2 + x₁
[U61(x₁, x₂)]	=	2 + x₁ + 2 · x₂
[U62(x₁)]	=	2
[U31(x₁)]	=	-2 + x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	-2
[U91(x₁,...,x₄)]	=	-2 + 2 · x₁ + 2 · x₂ + x₃
[U92(x₁,...,x₄)]	=	-2 + x₁ + 2 · x₄
[U93(x₁,...,x₄)]	=	-2 + x₁ + x₂
[take(x₁, x₂)]	=	-2
[U81(x₁)]	=	-2 + 2 · x₁
[nil]	=	0
[mark^#(x₁)]	=	1

together with the usable rules

U41(X1,mark(X2))	→	U41(X1,X2)	(67)
U41(mark(X1),X2)	→	U41(X1,X2)	(66)
U41(active(X1),X2)	→	U41(X1,X2)	(68)
U41(X1,active(X2))	→	U41(X1,X2)	(69)
isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(isNatIList(X))

→

active^#(isNatIList(X))

(219)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	-2
[U41(x₁, x₂)]	=	2 + x₂
[U71(x₁, x₂, x₃)]	=	-2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	0
[mark(x₁)]	=	2 + x₁
[cons(x₁, x₂)]	=	-2 + x₂
[active(x₁)]	=	x₁
[zeros]	=	1
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	2 · x₁
[isNatIList(x₁)]	=	0
[U11(x₁)]	=	2
[U51(x₁, x₂)]	=	-2 + x₁
[U52(x₁)]	=	-2 + x₁
[isNatList(x₁)]	=	0
[U21(x₁)]	=	-2 + 2 · x₁
[U61(x₁, x₂)]	=	2
[U62(x₁)]	=	2
[U31(x₁)]	=	-2 + 2 · x₁
[isNat(x₁)]	=	0
[s(x₁)]	=	x₁
[U91(x₁,...,x₄)]	=	2 + x₃ + 2 · x₄
[U92(x₁,...,x₄)]	=	2 + 2 · x₃
[U93(x₁,...,x₄)]	=	2
[take(x₁, x₂)]	=	2
[U81(x₁)]	=	-2
[nil]	=	0
[mark^#(x₁)]	=	x₁

together with the usable rules

isNatIList(active(X))	→	isNatIList(X)	(73)
isNatIList(mark(X))	→	isNatIList(X)	(72)
U42(active(X))	→	U42(X)	(71)
U42(mark(X))	→	U42(X)	(70)
U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U41(X1,X2))

→

active^#(U41(mark(X1),X2))

(213)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark^#(x₁)]	=	-2
[U42(x₁)]	=	-2 + x₁
[active^#(x₁)]	=	-2 + x₁
[U71(x₁, x₂, x₃)]	=	-2
[active(x₁)]	=	2
[U72(x₁, x₂)]	=	-2
[length(x₁)]	=	2
[mark(x₁)]	=	-2
[U62(x₁)]	=	-2
[U81(x₁)]	=	2
[U91(x₁,...,x₄)]	=	-2 + x₂
[isNatIList(x₁)]	=	0
[U21(x₁)]	=	2 + 2 · x₁
[U41(x₁, x₂)]	=	1 + 2 · x₁
[U51(x₁, x₂)]	=	2 + 2 · x₂
[U61(x₁, x₂)]	=	2 + 2 · x₂
[U92(x₁,...,x₄)]	=	-2 + 2 · x₄
[U93(x₁,...,x₄)]	=	-2 + 2 · x₂ + x₃
[isNat(x₁)]	=	0
[s(x₁)]	=	0
[cons(x₁, x₂)]	=	-1 + x₁ + 2 · x₂
[zeros]	=	0
[0]	=	0
[tt]	=	2
[U11(x₁)]	=	1
[U52(x₁)]	=	-2 + 2 · x₁
[isNatList(x₁)]	=	0
[U31(x₁)]	=	2
[take(x₁, x₂)]	=	0
[nil]	=	0

together with the usable rules

U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

active^#(U41(tt,V2))

→

mark^#(U42(isNatIList(V2)))

(139)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark^#(x₁)]	=	-2 + 2 · x₁
[U72(x₁, x₂)]	=	-2
[active^#(x₁)]	=	0
[U71(x₁, x₂, x₃)]	=	1
[active(x₁)]	=	2 · x₁
[mark(x₁)]	=	2 · x₁
[length(x₁)]	=	0
[U21(x₁)]	=	2
[U41(x₁, x₂)]	=	2
[U51(x₁, x₂)]	=	-2 + x₁ + 2 · x₂
[U61(x₁, x₂)]	=	2 + x₂
[U92(x₁,...,x₄)]	=	2
[U93(x₁,...,x₄)]	=	2 + 2 · x₂ + x₃
[isNat(x₁)]	=	0
[s(x₁)]	=	1 + x₁
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	2 + 2 · x₁
[isNatIList(x₁)]	=	1 + 2 · x₁
[U11(x₁)]	=	2
[U52(x₁)]	=	-2 + x₁
[isNatList(x₁)]	=	1 + x₁
[U62(x₁)]	=	2
[U31(x₁)]	=	2
[U91(x₁,...,x₄)]	=	-1 + x₁ + x₂ + 2 · x₄
[take(x₁, x₂)]	=	2
[U81(x₁)]	=	1
[nil]	=	0

together with the usable rules

U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
length(active(X))	→	length(X)	(103)
length(mark(X))	→	length(X)	(102)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(U42(X))

→

mark^#(X)

(218)

could be 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 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[active^#(x₁)]

· x₁

[U71(x₁, x₂, x₃)]

-∞

· x₁ +

· x₂ +

· x₃

[tt]

[mark^#(x₁)]

· x₁

[U72(x₁, x₂)]

· x₁ +

· x₂

[isNat(x₁)]

· x₁

[s(x₁)]

· x₁

[length(x₁)]

-∞

· x₁

[mark(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[isNatList(x₁)]

-∞

· x₁

[active(x₁)]

· x₁

[zeros]

[0]

[U41(x₁, x₂)]

-∞

· x₁ +

-∞

· x₂

[U42(x₁)]

-∞

· x₁

[isNatIList(x₁)]

-∞

· x₁

[U11(x₁)]

-∞

· x₁

[U51(x₁, x₂)]

-∞

· x₁ +

· x₂

[U52(x₁)]

-∞

· x₁

[U21(x₁)]

-∞

· x₁

[U61(x₁, x₂)]

· x₁ +

-∞

· x₂

[U62(x₁)]

-∞

· x₁

[U31(x₁)]

-∞

· x₁

[U91(x₁,...,x₄)]

· x₁ +

· x₂ +

· x₃ +

· x₄

[U92(x₁,...,x₄)]

-∞

· x₁ +

· x₂ +

· x₃ +

· x₄

[U93(x₁,...,x₄)]

· x₁ +

· x₂ +

· x₃ +

· x₄

[take(x₁, x₂)]

· x₁ +

· x₂

[U81(x₁)]

· x₁

[nil]

the pair

mark^#(s(X))

→

mark^#(X)

(242)

could be 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 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark^#(x₁)]	=	2 + x₁
[U72(x₁, x₂)]	=	0
[active^#(x₁)]	=	2
[U71(x₁, x₂, x₃)]	=	-2
[active(x₁)]	=	2
[mark(x₁)]	=	-2 + 2 · x₁
[length(x₁)]	=	2
[U21(x₁)]	=	2 + 2 · x₁
[U41(x₁, x₂)]	=	-2 + 2 · x₁
[U51(x₁, x₂)]	=	-2 + x₁
[U61(x₁, x₂)]	=	-2 + 2 · x₂
[U92(x₁,...,x₄)]	=	-2 + 2 · x₃ + 2 · x₄
[U93(x₁,...,x₄)]	=	-2 + x₁ + x₂
[isNat(x₁)]	=	x₁
[s(x₁)]	=	-2
[cons(x₁, x₂)]	=	-2 + 2 · x₁
[zeros]	=	0
[0]	=	0
[tt]	=	0
[U42(x₁)]	=	-2 + 2 · x₁
[isNatIList(x₁)]	=	2 · x₁
[U11(x₁)]	=	1 + 2 · x₁
[U52(x₁)]	=	2 + 2 · x₁
[isNatList(x₁)]	=	2
[U62(x₁)]	=	-2 + 2 · x₁
[U31(x₁)]	=	-2 + 2 · x₁
[U91(x₁,...,x₄)]	=	-2 + 2 · x₃ + 2 · x₄
[take(x₁, x₂)]	=	-2 + x₁
[U81(x₁)]	=	2
[nil]	=	0

together with the usable rules

U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pair

mark^#(length(X))

→

active^#(length(mark(X)))

(243)

could be 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 Reduction Pair Processor with Usable Rules

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

prec(U71)	=	3	weight(U71)	=	3
prec(U72)	=	1	weight(U72)	=	2
prec(s)	=	0	weight(s)	=	1
prec(length)	=	2	weight(length)	=	4

in combination with the following argument filter

π(active^#)	=	1
π(U71)	=	[]
π(mark^#)	=	1
π(U72)	=	[]
π(s)	=	[]
π(length)	=	[]

together with the usable rules

U72(X1,mark(X2))	→	U72(X1,X2)	(95)
U72(mark(X1),X2)	→	U72(X1,X2)	(94)
U72(active(X1),X2)	→	U72(X1,X2)	(96)
U72(X1,active(X2))	→	U72(X1,X2)	(97)
s(active(X))	→	s(X)	(101)
s(mark(X))	→	s(X)	(100)
U71(X1,mark(X2),X3)	→	U71(X1,X2,X3)	(89)
U71(mark(X1),X2,X3)	→	U71(X1,X2,X3)	(88)
U71(X1,X2,mark(X3))	→	U71(X1,X2,X3)	(90)
U71(active(X1),X2,X3)	→	U71(X1,X2,X3)	(91)
U71(X1,active(X2),X3)	→	U71(X1,X2,X3)	(92)
U71(X1,X2,active(X3))	→	U71(X1,X2,X3)	(93)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

active^#(U71(tt,L,N))	→	mark^#(U72(isNat(N),L))	(151)
active^#(U72(tt,L))	→	mark^#(s(length(L)))	(154)
active^#(length(cons(N,L)))	→	mark^#(U71(isNatList(L),L,N))	(189)

could be 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 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

cons^#(X1,mark(X2))	→	cons^#(X1,X2)	(264)
cons^#(mark(X1),X2)	→	cons^#(X1,X2)	(263)
cons^#(active(X1),X2)	→	cons^#(X1,X2)	(265)
cons^#(X1,active(X2))	→	cons^#(X1,X2)	(266)

1.1.2 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[cons^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.2.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

cons^#(X1,mark(X2))	→	cons^#(X1,X2)	(264)

1	≥	1
2	>	2
cons^#(mark(X1),X2)	→	cons^#(X1,X2)	(263)

1	>	1
2	≥	2
cons^#(active(X1),X2)	→	cons^#(X1,X2)	(265)

1	>	1
2	≥	2
cons^#(X1,active(X2))	→	cons^#(X1,X2)	(266)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 3^rd component contains the pair

U11^#(active(X)) → U11^#(X) (268)

U11^#(mark(X)) → U11^#(X) (267)

1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U11^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.3.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U11^#(active(X)) → U11^#(X) (268)

1 > 1

U11^#(mark(X)) → U11^#(X) (267)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 4^th component contains the pair

U21^#(active(X)) → U21^#(X) (270)

U21^#(mark(X)) → U21^#(X) (269)

1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U21^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.4.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U21^#(active(X)) → U21^#(X) (270)

1 > 1

U21^#(mark(X)) → U21^#(X) (269)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 5^th component contains the pair

U31^#(active(X)) → U31^#(X) (272)

U31^#(mark(X)) → U31^#(X) (271)

1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U31^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.5.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U31^#(active(X)) → U31^#(X) (272)

1 > 1

U31^#(mark(X)) → U31^#(X) (271)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

The 6^th component contains the pair

U41^#(X1,mark(X2))	→	U41^#(X1,X2)	(274)
U41^#(mark(X1),X2)	→	U41^#(X1,X2)	(273)
U41^#(active(X1),X2)	→	U41^#(X1,X2)	(275)
U41^#(X1,active(X2))	→	U41^#(X1,X2)	(276)

1.1.6 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U41^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.6.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U41^#(X1,mark(X2))	→	U41^#(X1,X2)	(274)

1	≥	1
2	>	2
U41^#(mark(X1),X2)	→	U41^#(X1,X2)	(273)

1	>	1
2	≥	2
U41^#(active(X1),X2)	→	U41^#(X1,X2)	(275)

1	>	1
2	≥	2
U41^#(X1,active(X2))	→	U41^#(X1,X2)	(276)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 7^th component contains the pair

U42^#(active(X)) → U42^#(X) (278)

U42^#(mark(X)) → U42^#(X) (277)

1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U42^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.7.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U42^#(active(X)) → U42^#(X) (278)

1 > 1

U42^#(mark(X)) → U42^#(X) (277)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 8^th component contains the pair

isNatIList^#(active(X)) → isNatIList^#(X) (280)

isNatIList^#(mark(X)) → isNatIList^#(X) (279)

1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[isNatIList^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.8.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

isNatIList^#(active(X)) → isNatIList^#(X) (280)

1 > 1

isNatIList^#(mark(X)) → isNatIList^#(X) (279)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

The 9^th component contains the pair

U51^#(X1,mark(X2))	→	U51^#(X1,X2)	(282)
U51^#(mark(X1),X2)	→	U51^#(X1,X2)	(281)
U51^#(active(X1),X2)	→	U51^#(X1,X2)	(283)
U51^#(X1,active(X2))	→	U51^#(X1,X2)	(284)

1.1.9 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U51^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.9.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U51^#(X1,mark(X2))	→	U51^#(X1,X2)	(282)

1	≥	1
2	>	2
U51^#(mark(X1),X2)	→	U51^#(X1,X2)	(281)

1	>	1
2	≥	2
U51^#(active(X1),X2)	→	U51^#(X1,X2)	(283)

1	>	1
2	≥	2
U51^#(X1,active(X2))	→	U51^#(X1,X2)	(284)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 10^th component contains the pair

U52^#(active(X)) → U52^#(X) (286)

U52^#(mark(X)) → U52^#(X) (285)

1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U52^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.10.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U52^#(active(X)) → U52^#(X) (286)

1 > 1

U52^#(mark(X)) → U52^#(X) (285)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 11^th component contains the pair

isNatList^#(active(X)) → isNatList^#(X) (288)

isNatList^#(mark(X)) → isNatList^#(X) (287)

1.1.11 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[isNatList^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.11.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

isNatList^#(active(X)) → isNatList^#(X) (288)

1 > 1

isNatList^#(mark(X)) → isNatList^#(X) (287)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

The 12^th component contains the pair

U61^#(X1,mark(X2))	→	U61^#(X1,X2)	(290)
U61^#(mark(X1),X2)	→	U61^#(X1,X2)	(289)
U61^#(active(X1),X2)	→	U61^#(X1,X2)	(291)
U61^#(X1,active(X2))	→	U61^#(X1,X2)	(292)

1.1.12 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U61^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.12.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U61^#(X1,mark(X2))	→	U61^#(X1,X2)	(290)

1	≥	1
2	>	2
U61^#(mark(X1),X2)	→	U61^#(X1,X2)	(289)

1	>	1
2	≥	2
U61^#(active(X1),X2)	→	U61^#(X1,X2)	(291)

1	>	1
2	≥	2
U61^#(X1,active(X2))	→	U61^#(X1,X2)	(292)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 13^th component contains the pair

U62^#(active(X)) → U62^#(X) (294)

U62^#(mark(X)) → U62^#(X) (293)

1.1.13 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U62^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.13.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U62^#(active(X)) → U62^#(X) (294)

1 > 1

U62^#(mark(X)) → U62^#(X) (293)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

The 14^th component contains the pair

U71^#(X1,mark(X2),X3)	→	U71^#(X1,X2,X3)	(296)
U71^#(mark(X1),X2,X3)	→	U71^#(X1,X2,X3)	(295)
U71^#(X1,X2,mark(X3))	→	U71^#(X1,X2,X3)	(297)
U71^#(active(X1),X2,X3)	→	U71^#(X1,X2,X3)	(298)
U71^#(X1,active(X2),X3)	→	U71^#(X1,X2,X3)	(299)
U71^#(X1,X2,active(X3))	→	U71^#(X1,X2,X3)	(300)

1.1.14 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U71^#(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.14.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U71^#(X1,mark(X2),X3)	→	U71^#(X1,X2,X3)	(296)

1	≥	1
2	>	2
3	≥	3
U71^#(mark(X1),X2,X3)	→	U71^#(X1,X2,X3)	(295)

1	>	1
2	≥	2
3	≥	3
U71^#(X1,X2,mark(X3))	→	U71^#(X1,X2,X3)	(297)

1	≥	1
2	≥	2
3	>	3
U71^#(active(X1),X2,X3)	→	U71^#(X1,X2,X3)	(298)

1	>	1
2	≥	2
3	≥	3
U71^#(X1,active(X2),X3)	→	U71^#(X1,X2,X3)	(299)

1	≥	1
2	>	2
3	≥	3
U71^#(X1,X2,active(X3))	→	U71^#(X1,X2,X3)	(300)

1	≥	1
2	≥	2
3	>	3

As there is no critical graph in the transitive closure, there are no infinite chains.

The 15^th component contains the pair

U72^#(X1,mark(X2))	→	U72^#(X1,X2)	(302)
U72^#(mark(X1),X2)	→	U72^#(X1,X2)	(301)
U72^#(active(X1),X2)	→	U72^#(X1,X2)	(303)
U72^#(X1,active(X2))	→	U72^#(X1,X2)	(304)

1.1.15 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U72^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.15.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U72^#(X1,mark(X2))	→	U72^#(X1,X2)	(302)

1	≥	1
2	>	2
U72^#(mark(X1),X2)	→	U72^#(X1,X2)	(301)

1	>	1
2	≥	2
U72^#(active(X1),X2)	→	U72^#(X1,X2)	(303)

1	>	1
2	≥	2
U72^#(X1,active(X2))	→	U72^#(X1,X2)	(304)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 16^th component contains the pair

isNat^#(active(X)) → isNat^#(X) (306)

isNat^#(mark(X)) → isNat^#(X) (305)

1.1.16 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[isNat^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.16.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

isNat^#(active(X)) → isNat^#(X) (306)

1 > 1

isNat^#(mark(X)) → isNat^#(X) (305)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 17^th component contains the pair

s^#(active(X)) → s^#(X) (308)

s^#(mark(X)) → s^#(X) (307)

1.1.17 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[s^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.17.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

s^#(active(X)) → s^#(X) (308)

1 > 1

s^#(mark(X)) → s^#(X) (307)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 18^th component contains the pair

length^#(active(X)) → length^#(X) (310)

length^#(mark(X)) → length^#(X) (309)

1.1.18 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[length^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.18.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

length^#(active(X)) → length^#(X) (310)

1 > 1

length^#(mark(X)) → length^#(X) (309)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 19^th component contains the pair

U81^#(active(X)) → U81^#(X) (312)

U81^#(mark(X)) → U81^#(X) (311)

1.1.19 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[active(x₁)] = 1 · x₁

[mark(x₁)] = 1 · x₁

[U81^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.19.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U81^#(active(X)) → U81^#(X) (312)

1 > 1

U81^#(mark(X)) → U81^#(X) (311)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

The 20^th component contains the pair

U91^#(X1,mark(X2),X3,X4)	→	U91^#(X1,X2,X3,X4)	(314)
U91^#(mark(X1),X2,X3,X4)	→	U91^#(X1,X2,X3,X4)	(313)
U91^#(X1,X2,mark(X3),X4)	→	U91^#(X1,X2,X3,X4)	(315)
U91^#(X1,X2,X3,mark(X4))	→	U91^#(X1,X2,X3,X4)	(316)
U91^#(active(X1),X2,X3,X4)	→	U91^#(X1,X2,X3,X4)	(317)
U91^#(X1,active(X2),X3,X4)	→	U91^#(X1,X2,X3,X4)	(318)
U91^#(X1,X2,active(X3),X4)	→	U91^#(X1,X2,X3,X4)	(319)
U91^#(X1,X2,X3,active(X4))	→	U91^#(X1,X2,X3,X4)	(320)

1.1.20 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U91^#(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.20.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U91^#(X1,mark(X2),X3,X4)	→	U91^#(X1,X2,X3,X4)	(314)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U91^#(mark(X1),X2,X3,X4)	→	U91^#(X1,X2,X3,X4)	(313)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U91^#(X1,X2,mark(X3),X4)	→	U91^#(X1,X2,X3,X4)	(315)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U91^#(X1,X2,X3,mark(X4))	→	U91^#(X1,X2,X3,X4)	(316)

1	≥	1
2	≥	2
3	≥	3
4	>	4
U91^#(active(X1),X2,X3,X4)	→	U91^#(X1,X2,X3,X4)	(317)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U91^#(X1,active(X2),X3,X4)	→	U91^#(X1,X2,X3,X4)	(318)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U91^#(X1,X2,active(X3),X4)	→	U91^#(X1,X2,X3,X4)	(319)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U91^#(X1,X2,X3,active(X4))	→	U91^#(X1,X2,X3,X4)	(320)

1	≥	1
2	≥	2
3	≥	3
4	>	4

As there is no critical graph in the transitive closure, there are no infinite chains.

The 21^th component contains the pair

U92^#(X1,mark(X2),X3,X4)	→	U92^#(X1,X2,X3,X4)	(322)
U92^#(mark(X1),X2,X3,X4)	→	U92^#(X1,X2,X3,X4)	(321)
U92^#(X1,X2,mark(X3),X4)	→	U92^#(X1,X2,X3,X4)	(323)
U92^#(X1,X2,X3,mark(X4))	→	U92^#(X1,X2,X3,X4)	(324)
U92^#(active(X1),X2,X3,X4)	→	U92^#(X1,X2,X3,X4)	(325)
U92^#(X1,active(X2),X3,X4)	→	U92^#(X1,X2,X3,X4)	(326)
U92^#(X1,X2,active(X3),X4)	→	U92^#(X1,X2,X3,X4)	(327)
U92^#(X1,X2,X3,active(X4))	→	U92^#(X1,X2,X3,X4)	(328)

1.1.21 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U92^#(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.21.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U92^#(X1,mark(X2),X3,X4)	→	U92^#(X1,X2,X3,X4)	(322)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U92^#(mark(X1),X2,X3,X4)	→	U92^#(X1,X2,X3,X4)	(321)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U92^#(X1,X2,mark(X3),X4)	→	U92^#(X1,X2,X3,X4)	(323)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U92^#(X1,X2,X3,mark(X4))	→	U92^#(X1,X2,X3,X4)	(324)

1	≥	1
2	≥	2
3	≥	3
4	>	4
U92^#(active(X1),X2,X3,X4)	→	U92^#(X1,X2,X3,X4)	(325)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U92^#(X1,active(X2),X3,X4)	→	U92^#(X1,X2,X3,X4)	(326)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U92^#(X1,X2,active(X3),X4)	→	U92^#(X1,X2,X3,X4)	(327)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U92^#(X1,X2,X3,active(X4))	→	U92^#(X1,X2,X3,X4)	(328)

1	≥	1
2	≥	2
3	≥	3
4	>	4

As there is no critical graph in the transitive closure, there are no infinite chains.

The 22^th component contains the pair

U93^#(X1,mark(X2),X3,X4)	→	U93^#(X1,X2,X3,X4)	(330)
U93^#(mark(X1),X2,X3,X4)	→	U93^#(X1,X2,X3,X4)	(329)
U93^#(X1,X2,mark(X3),X4)	→	U93^#(X1,X2,X3,X4)	(331)
U93^#(X1,X2,X3,mark(X4))	→	U93^#(X1,X2,X3,X4)	(332)
U93^#(active(X1),X2,X3,X4)	→	U93^#(X1,X2,X3,X4)	(333)
U93^#(X1,active(X2),X3,X4)	→	U93^#(X1,X2,X3,X4)	(334)
U93^#(X1,X2,active(X3),X4)	→	U93^#(X1,X2,X3,X4)	(335)
U93^#(X1,X2,X3,active(X4))	→	U93^#(X1,X2,X3,X4)	(336)

1.1.22 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[U93^#(x₁,...,x₄)]	=	1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.22.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

U93^#(X1,mark(X2),X3,X4)	→	U93^#(X1,X2,X3,X4)	(330)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U93^#(mark(X1),X2,X3,X4)	→	U93^#(X1,X2,X3,X4)	(329)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U93^#(X1,X2,mark(X3),X4)	→	U93^#(X1,X2,X3,X4)	(331)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U93^#(X1,X2,X3,mark(X4))	→	U93^#(X1,X2,X3,X4)	(332)

1	≥	1
2	≥	2
3	≥	3
4	>	4
U93^#(active(X1),X2,X3,X4)	→	U93^#(X1,X2,X3,X4)	(333)

1	>	1
2	≥	2
3	≥	3
4	≥	4
U93^#(X1,active(X2),X3,X4)	→	U93^#(X1,X2,X3,X4)	(334)

1	≥	1
2	>	2
3	≥	3
4	≥	4
U93^#(X1,X2,active(X3),X4)	→	U93^#(X1,X2,X3,X4)	(335)

1	≥	1
2	≥	2
3	>	3
4	≥	4
U93^#(X1,X2,X3,active(X4))	→	U93^#(X1,X2,X3,X4)	(336)

1	≥	1
2	≥	2
3	≥	3
4	>	4

As there is no critical graph in the transitive closure, there are no infinite chains.

The 23^th component contains the pair

take^#(X1,mark(X2))	→	take^#(X1,X2)	(338)
take^#(mark(X1),X2)	→	take^#(X1,X2)	(337)
take^#(active(X1),X2)	→	take^#(X1,X2)	(339)
take^#(X1,active(X2))	→	take^#(X1,X2)	(340)

1.1.23 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark(x₁)]	=	1 · x₁
[active(x₁)]	=	1 · x₁
[take^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.23.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

take^#(X1,mark(X2))	→	take^#(X1,X2)	(338)

1	≥	1
2	>	2
take^#(mark(X1),X2)	→	take^#(X1,X2)	(337)

1	>	1
2	≥	2
take^#(active(X1),X2)	→	take^#(X1,X2)	(339)

1	>	1
2	≥	2
take^#(X1,active(X2))	→	take^#(X1,X2)	(340)

1	≥	1
2	>	2

As there is no critical graph in the transitive closure, there are no infinite chains.

U11^#(active(X))	→	U11^#(X)	(268)
U11^#(mark(X))	→	U11^#(X)	(267)

U21^#(active(X))	→	U21^#(X)	(270)
U21^#(mark(X))	→	U21^#(X)	(269)

U31^#(active(X))	→	U31^#(X)	(272)
U31^#(mark(X))	→	U31^#(X)	(271)

U42^#(active(X))	→	U42^#(X)	(278)
U42^#(mark(X))	→	U42^#(X)	(277)

isNatIList^#(active(X))	→	isNatIList^#(X)	(280)
isNatIList^#(mark(X))	→	isNatIList^#(X)	(279)

U52^#(active(X))	→	U52^#(X)	(286)
U52^#(mark(X))	→	U52^#(X)	(285)

isNatList^#(active(X))	→	isNatList^#(X)	(288)
isNatList^#(mark(X))	→	isNatList^#(X)	(287)

U62^#(active(X))	→	U62^#(X)	(294)
U62^#(mark(X))	→	U62^#(X)	(293)

isNat^#(active(X))	→	isNat^#(X)	(306)
isNat^#(mark(X))	→	isNat^#(X)	(305)

s^#(active(X))	→	s^#(X)	(308)

1	>	1
s^#(mark(X))	→	s^#(X)	(307)

1	>	1

length^#(active(X))	→	length^#(X)	(310)
length^#(mark(X))	→	length^#(X)	(309)

U81^#(active(X))	→	U81^#(X)	(312)
U81^#(mark(X))	→	U81^#(X)	(311)

Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/OvConsOS_nokinds_noand_iGM)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1 Reduction Pair Processor

1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor

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 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

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 Dependency Graph Processor

1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.2.1 Size-Change Termination

1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.3.1 Size-Change Termination

1.1.4 Monotonic Reduction Pair Processor with Usable Rules

1.1.4.1 Size-Change Termination

1.1.5 Monotonic Reduction Pair Processor with Usable Rules

1.1.5.1 Size-Change Termination

1.1.6 Monotonic Reduction Pair Processor with Usable Rules

1.1.6.1 Size-Change Termination

1.1.7 Monotonic Reduction Pair Processor with Usable Rules

1.1.7.1 Size-Change Termination

1.1.8 Monotonic Reduction Pair Processor with Usable Rules

1.1.8.1 Size-Change Termination

1.1.9 Monotonic Reduction Pair Processor with Usable Rules

1.1.9.1 Size-Change Termination

1.1.10 Monotonic Reduction Pair Processor with Usable Rules

1.1.10.1 Size-Change Termination

1.1.11 Monotonic Reduction Pair Processor with Usable Rules

1.1.11.1 Size-Change Termination

1.1.12 Monotonic Reduction Pair Processor with Usable Rules

1.1.12.1 Size-Change Termination

1.1.13 Monotonic Reduction Pair Processor with Usable Rules

1.1.13.1 Size-Change Termination

1.1.14 Monotonic Reduction Pair Processor with Usable Rules

1.1.14.1 Size-Change Termination

1.1.15 Monotonic Reduction Pair Processor with Usable Rules

1.1.15.1 Size-Change Termination

1.1.16 Monotonic Reduction Pair Processor with Usable Rules

1.1.16.1 Size-Change Termination

1.1.17 Monotonic Reduction Pair Processor with Usable Rules

1.1.17.1 Size-Change Termination

1.1.18 Monotonic Reduction Pair Processor with Usable Rules

1.1.18.1 Size-Change Termination

1.1.19 Monotonic Reduction Pair Processor with Usable Rules

1.1.19.1 Size-Change Termination

1.1.20 Monotonic Reduction Pair Processor with Usable Rules

1.1.20.1 Size-Change Termination

1.1.21 Monotonic Reduction Pair Processor with Usable Rules

1.1.21.1 Size-Change Termination