Certification Problem

Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/PEANO_nokinds_noand_iGM)

The rewrite relation of the following TRS is considered.

active(U11(tt,V2))	→	mark(U12(isNat(V2)))	(1)
active(U12(tt))	→	mark(tt)	(2)
active(U21(tt))	→	mark(tt)	(3)
active(U31(tt,N))	→	mark(N)	(4)
active(U41(tt,M,N))	→	mark(U42(isNat(N),M,N))	(5)
active(U42(tt,M,N))	→	mark(s(plus(N,M)))	(6)
active(isNat(0))	→	mark(tt)	(7)
active(isNat(plus(V1,V2)))	→	mark(U11(isNat(V1),V2))	(8)
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	(9)
active(plus(N,0))	→	mark(U31(isNat(N),N))	(10)
active(plus(N,s(M)))	→	mark(U41(isNat(M),M,N))	(11)
mark(U11(X1,X2))	→	active(U11(mark(X1),X2))	(12)
mark(tt)	→	active(tt)	(13)
mark(U12(X))	→	active(U12(mark(X)))	(14)
mark(isNat(X))	→	active(isNat(X))	(15)
mark(U21(X))	→	active(U21(mark(X)))	(16)
mark(U31(X1,X2))	→	active(U31(mark(X1),X2))	(17)
mark(U41(X1,X2,X3))	→	active(U41(mark(X1),X2,X3))	(18)
mark(U42(X1,X2,X3))	→	active(U42(mark(X1),X2,X3))	(19)
mark(s(X))	→	active(s(mark(X)))	(20)
mark(plus(X1,X2))	→	active(plus(mark(X1),mark(X2)))	(21)
mark(0)	→	active(0)	(22)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U12(mark(X))	→	U12(X)	(27)
U12(active(X))	→	U12(X)	(28)
isNat(mark(X))	→	isNat(X)	(29)
isNat(active(X))	→	isNat(X)	(30)
U21(mark(X))	→	U21(X)	(31)
U21(active(X))	→	U21(X)	(32)
U31(mark(X1),X2)	→	U31(X1,X2)	(33)
U31(X1,mark(X2))	→	U31(X1,X2)	(34)
U31(active(X1),X2)	→	U31(X1,X2)	(35)
U31(X1,active(X2))	→	U31(X1,X2)	(36)
U41(mark(X1),X2,X3)	→	U41(X1,X2,X3)	(37)
U41(X1,mark(X2),X3)	→	U41(X1,X2,X3)	(38)
U41(X1,X2,mark(X3))	→	U41(X1,X2,X3)	(39)
U41(active(X1),X2,X3)	→	U41(X1,X2,X3)	(40)
U41(X1,active(X2),X3)	→	U41(X1,X2,X3)	(41)
U41(X1,X2,active(X3))	→	U41(X1,X2,X3)	(42)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)
s(mark(X))	→	s(X)	(49)
s(active(X))	→	s(X)	(50)
plus(mark(X1),X2)	→	plus(X1,X2)	(51)
plus(X1,mark(X2))	→	plus(X1,X2)	(52)
plus(active(X1),X2)	→	plus(X1,X2)	(53)
plus(X1,active(X2))	→	plus(X1,X2)	(54)

The evaluation strategy is innermost.

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.

active^#(U11(tt,V2))	→	mark^#(U12(isNat(V2)))	(55)
active^#(U11(tt,V2))	→	U12^#(isNat(V2))	(56)
active^#(U11(tt,V2))	→	isNat^#(V2)	(57)
active^#(U12(tt))	→	mark^#(tt)	(58)
active^#(U21(tt))	→	mark^#(tt)	(59)
active^#(U31(tt,N))	→	mark^#(N)	(60)
active^#(U41(tt,M,N))	→	mark^#(U42(isNat(N),M,N))	(61)
active^#(U41(tt,M,N))	→	U42^#(isNat(N),M,N)	(62)
active^#(U41(tt,M,N))	→	isNat^#(N)	(63)
active^#(U42(tt,M,N))	→	mark^#(s(plus(N,M)))	(64)
active^#(U42(tt,M,N))	→	s^#(plus(N,M))	(65)
active^#(U42(tt,M,N))	→	plus^#(N,M)	(66)
active^#(isNat(0))	→	mark^#(tt)	(67)
active^#(isNat(plus(V1,V2)))	→	mark^#(U11(isNat(V1),V2))	(68)
active^#(isNat(plus(V1,V2)))	→	U11^#(isNat(V1),V2)	(69)
active^#(isNat(plus(V1,V2)))	→	isNat^#(V1)	(70)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(71)
active^#(isNat(s(V1)))	→	U21^#(isNat(V1))	(72)
active^#(isNat(s(V1)))	→	isNat^#(V1)	(73)
active^#(plus(N,0))	→	mark^#(U31(isNat(N),N))	(74)
active^#(plus(N,0))	→	U31^#(isNat(N),N)	(75)
active^#(plus(N,0))	→	isNat^#(N)	(76)
active^#(plus(N,s(M)))	→	mark^#(U41(isNat(M),M,N))	(77)
active^#(plus(N,s(M)))	→	U41^#(isNat(M),M,N)	(78)
active^#(plus(N,s(M)))	→	isNat^#(M)	(79)
mark^#(U11(X1,X2))	→	active^#(U11(mark(X1),X2))	(80)
mark^#(U11(X1,X2))	→	U11^#(mark(X1),X2)	(81)
mark^#(U11(X1,X2))	→	mark^#(X1)	(82)
mark^#(tt)	→	active^#(tt)	(83)
mark^#(U12(X))	→	active^#(U12(mark(X)))	(84)
mark^#(U12(X))	→	U12^#(mark(X))	(85)
mark^#(U12(X))	→	mark^#(X)	(86)
mark^#(isNat(X))	→	active^#(isNat(X))	(87)
mark^#(U21(X))	→	active^#(U21(mark(X)))	(88)
mark^#(U21(X))	→	U21^#(mark(X))	(89)
mark^#(U21(X))	→	mark^#(X)	(90)
mark^#(U31(X1,X2))	→	active^#(U31(mark(X1),X2))	(91)
mark^#(U31(X1,X2))	→	U31^#(mark(X1),X2)	(92)
mark^#(U31(X1,X2))	→	mark^#(X1)	(93)
mark^#(U41(X1,X2,X3))	→	active^#(U41(mark(X1),X2,X3))	(94)
mark^#(U41(X1,X2,X3))	→	U41^#(mark(X1),X2,X3)	(95)
mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(96)
mark^#(U42(X1,X2,X3))	→	active^#(U42(mark(X1),X2,X3))	(97)
mark^#(U42(X1,X2,X3))	→	U42^#(mark(X1),X2,X3)	(98)
mark^#(U42(X1,X2,X3))	→	mark^#(X1)	(99)
mark^#(s(X))	→	active^#(s(mark(X)))	(100)
mark^#(s(X))	→	s^#(mark(X))	(101)
mark^#(s(X))	→	mark^#(X)	(102)
mark^#(plus(X1,X2))	→	active^#(plus(mark(X1),mark(X2)))	(103)
mark^#(plus(X1,X2))	→	plus^#(mark(X1),mark(X2))	(104)
mark^#(plus(X1,X2))	→	mark^#(X1)	(105)
mark^#(plus(X1,X2))	→	mark^#(X2)	(106)
mark^#(0)	→	active^#(0)	(107)
U11^#(mark(X1),X2)	→	U11^#(X1,X2)	(108)
U11^#(X1,mark(X2))	→	U11^#(X1,X2)	(109)
U11^#(active(X1),X2)	→	U11^#(X1,X2)	(110)
U11^#(X1,active(X2))	→	U11^#(X1,X2)	(111)
U12^#(mark(X))	→	U12^#(X)	(112)
U12^#(active(X))	→	U12^#(X)	(113)
isNat^#(mark(X))	→	isNat^#(X)	(114)
isNat^#(active(X))	→	isNat^#(X)	(115)
U21^#(mark(X))	→	U21^#(X)	(116)
U21^#(active(X))	→	U21^#(X)	(117)
U31^#(mark(X1),X2)	→	U31^#(X1,X2)	(118)
U31^#(X1,mark(X2))	→	U31^#(X1,X2)	(119)
U31^#(active(X1),X2)	→	U31^#(X1,X2)	(120)
U31^#(X1,active(X2))	→	U31^#(X1,X2)	(121)
U41^#(mark(X1),X2,X3)	→	U41^#(X1,X2,X3)	(122)
U41^#(X1,mark(X2),X3)	→	U41^#(X1,X2,X3)	(123)
U41^#(X1,X2,mark(X3))	→	U41^#(X1,X2,X3)	(124)
U41^#(active(X1),X2,X3)	→	U41^#(X1,X2,X3)	(125)
U41^#(X1,active(X2),X3)	→	U41^#(X1,X2,X3)	(126)
U41^#(X1,X2,active(X3))	→	U41^#(X1,X2,X3)	(127)
U42^#(mark(X1),X2,X3)	→	U42^#(X1,X2,X3)	(128)
U42^#(X1,mark(X2),X3)	→	U42^#(X1,X2,X3)	(129)
U42^#(X1,X2,mark(X3))	→	U42^#(X1,X2,X3)	(130)
U42^#(active(X1),X2,X3)	→	U42^#(X1,X2,X3)	(131)
U42^#(X1,active(X2),X3)	→	U42^#(X1,X2,X3)	(132)
U42^#(X1,X2,active(X3))	→	U42^#(X1,X2,X3)	(133)
s^#(mark(X))	→	s^#(X)	(134)
s^#(active(X))	→	s^#(X)	(135)
plus^#(mark(X1),X2)	→	plus^#(X1,X2)	(136)
plus^#(X1,mark(X2))	→	plus^#(X1,X2)	(137)
plus^#(active(X1),X2)	→	plus^#(X1,X2)	(138)
plus^#(X1,active(X2))	→	plus^#(X1,X2)	(139)

1.1 Dependency Graph Processor

The dependency pairs are split into 10 components.

The 1^st component contains the pair

mark^#(U12(X))	→	active^#(U12(mark(X)))	(84)
active^#(U11(tt,V2))	→	mark^#(U12(isNat(V2)))	(55)
mark^#(U12(X))	→	mark^#(X)	(86)
mark^#(U11(X1,X2))	→	active^#(U11(mark(X1),X2))	(80)
active^#(U31(tt,N))	→	mark^#(N)	(60)
mark^#(U11(X1,X2))	→	mark^#(X1)	(82)
mark^#(isNat(X))	→	active^#(isNat(X))	(87)
active^#(isNat(plus(V1,V2)))	→	mark^#(U11(isNat(V1),V2))	(68)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(71)
mark^#(U21(X))	→	active^#(U21(mark(X)))	(88)
active^#(U41(tt,M,N))	→	mark^#(U42(isNat(N),M,N))	(61)
mark^#(U42(X1,X2,X3))	→	active^#(U42(mark(X1),X2,X3))	(97)
active^#(U42(tt,M,N))	→	mark^#(s(plus(N,M)))	(64)
mark^#(s(X))	→	active^#(s(mark(X)))	(100)
active^#(plus(N,0))	→	mark^#(U31(isNat(N),N))	(74)
mark^#(U31(X1,X2))	→	active^#(U31(mark(X1),X2))	(91)
active^#(plus(N,s(M)))	→	mark^#(U41(isNat(M),M,N))	(77)
mark^#(U41(X1,X2,X3))	→	active^#(U41(mark(X1),X2,X3))	(94)
mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(96)
mark^#(U21(X))	→	mark^#(X)	(90)
mark^#(U31(X1,X2))	→	mark^#(X1)	(93)
mark^#(U42(X1,X2,X3))	→	mark^#(X1)	(99)
mark^#(s(X))	→	mark^#(X)	(102)
mark^#(plus(X1,X2))	→	active^#(plus(mark(X1),mark(X2)))	(103)
mark^#(plus(X1,X2))	→	mark^#(X1)	(105)
mark^#(plus(X1,X2))	→	mark^#(X2)	(106)

1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

mark(U12(X))	→	active(U12(mark(X)))	(14)
active(U11(tt,V2))	→	mark(U12(isNat(V2)))	(1)
active(U31(tt,N))	→	mark(N)	(4)
mark(U11(X1,X2))	→	active(U11(mark(X1),X2))	(12)
active(U41(tt,M,N))	→	mark(U42(isNat(N),M,N))	(5)
mark(U42(X1,X2,X3))	→	active(U42(mark(X1),X2,X3))	(19)
active(U42(tt,M,N))	→	mark(s(plus(N,M)))	(6)
mark(s(X))	→	active(s(mark(X)))	(20)
active(isNat(plus(V1,V2)))	→	mark(U11(isNat(V1),V2))	(8)
active(isNat(s(V1)))	→	mark(U21(isNat(V1)))	(9)
mark(U21(X))	→	active(U21(mark(X)))	(16)
active(plus(N,0))	→	mark(U31(isNat(N),N))	(10)
mark(U31(X1,X2))	→	active(U31(mark(X1),X2))	(17)
active(plus(N,s(M)))	→	mark(U41(isNat(M),M,N))	(11)
mark(U41(X1,X2,X3))	→	active(U41(mark(X1),X2,X3))	(18)
mark(isNat(X))	→	active(isNat(X))	(15)
mark(plus(X1,X2))	→	active(plus(mark(X1),mark(X2)))	(21)
mark(tt)	→	active(tt)	(13)
mark(0)	→	active(0)	(22)
plus(X1,mark(X2))	→	plus(X1,X2)	(52)
plus(mark(X1),X2)	→	plus(X1,X2)	(51)
plus(active(X1),X2)	→	plus(X1,X2)	(53)
plus(X1,active(X2))	→	plus(X1,X2)	(54)
U12(active(X))	→	U12(X)	(28)
U12(mark(X))	→	U12(X)	(27)
active(U12(tt))	→	mark(tt)	(2)
active(U21(tt))	→	mark(tt)	(3)
active(isNat(0))	→	mark(tt)	(7)
U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)
s(active(X))	→	s(X)	(50)
s(mark(X))	→	s(X)	(49)
U21(active(X))	→	U21(X)	(32)
U21(mark(X))	→	U21(X)	(31)
U31(X1,mark(X2))	→	U31(X1,X2)	(34)
U31(mark(X1),X2)	→	U31(X1,X2)	(33)
U31(active(X1),X2)	→	U31(X1,X2)	(35)
U31(X1,active(X2))	→	U31(X1,X2)	(36)
U41(X1,mark(X2),X3)	→	U41(X1,X2,X3)	(38)
U41(mark(X1),X2,X3)	→	U41(X1,X2,X3)	(37)
U41(X1,X2,mark(X3))	→	U41(X1,X2,X3)	(39)
U41(active(X1),X2,X3)	→	U41(X1,X2,X3)	(40)
U41(X1,active(X2),X3)	→	U41(X1,X2,X3)	(41)
U41(X1,X2,active(X3))	→	U41(X1,X2,X3)	(42)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark^#(x₁)]	=	1
[U12(x₁)]	=	0
[active^#(x₁)]	=	1 · x₁
[mark(x₁)]	=	0
[U11(x₁, x₂)]	=	1
[tt]	=	0
[isNat(x₁)]	=	1
[U31(x₁, x₂)]	=	1
[plus(x₁, x₂)]	=	1
[s(x₁)]	=	0
[U21(x₁)]	=	0
[U41(x₁, x₂, x₃)]	=	1
[U42(x₁, x₂, x₃)]	=	1
[0]	=	0
[active(x₁)]	=	0

together with the usable rules

U12(active(X))	→	U12(X)	(28)
U12(mark(X))	→	U12(X)	(27)
U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U21(active(X))	→	U21(X)	(32)
U21(mark(X))	→	U21(X)	(31)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)
s(active(X))	→	s(X)	(50)
s(mark(X))	→	s(X)	(49)
U31(X1,mark(X2))	→	U31(X1,X2)	(34)
U31(mark(X1),X2)	→	U31(X1,X2)	(33)
U31(active(X1),X2)	→	U31(X1,X2)	(35)
U31(X1,active(X2))	→	U31(X1,X2)	(36)
U41(X1,mark(X2),X3)	→	U41(X1,X2,X3)	(38)
U41(mark(X1),X2,X3)	→	U41(X1,X2,X3)	(37)
U41(X1,X2,mark(X3))	→	U41(X1,X2,X3)	(39)
U41(active(X1),X2,X3)	→	U41(X1,X2,X3)	(40)
U41(X1,active(X2),X3)	→	U41(X1,X2,X3)	(41)
U41(X1,X2,active(X3))	→	U41(X1,X2,X3)	(42)
plus(X1,mark(X2))	→	plus(X1,X2)	(52)
plus(mark(X1),X2)	→	plus(X1,X2)	(51)
plus(active(X1),X2)	→	plus(X1,X2)	(53)
plus(X1,active(X2))	→	plus(X1,X2)	(54)

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

mark^#(U12(X))	→	active^#(U12(mark(X)))	(84)
mark^#(U21(X))	→	active^#(U21(mark(X)))	(88)
mark^#(s(X))	→	active^#(s(mark(X)))	(100)

could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	1 · x₁
[U11(x₁, x₂)]	=	1 · x₁
[tt]	=	0
[mark^#(x₁)]	=	1 · x₁
[U12(x₁)]	=	1 · x₁
[isNat(x₁)]	=	0
[mark(x₁)]	=	1 · x₁
[U31(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[s(x₁)]	=	1 · x₁
[U21(x₁)]	=	1 · x₁
[U41(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[U42(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[0]	=	1
[active(x₁)]	=	1 · x₁

the pair

active^#(plus(N,0))

→

mark^#(U31(isNat(N),N))

(74)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	1 · x₁
[U11(x₁, x₂)]	=	1 · x₁
[tt]	=	0
[mark^#(x₁)]	=	1 · x₁
[U12(x₁)]	=	1 · x₁
[isNat(x₁)]	=	0
[mark(x₁)]	=	1 · x₁
[U31(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[s(x₁)]	=	1 · x₁
[U21(x₁)]	=	1 · x₁
[U41(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[U42(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[active(x₁)]	=	1 · x₁
[0]	=	1

the pairs

active^#(U31(tt,N))	→	mark^#(N)	(60)
mark^#(U31(X1,X2))	→	mark^#(X1)	(93)

could be deleted.

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
[U11(x₁, x₂)]	=	1
[tt]	=	0
[mark^#(x₁)]	=	1
[U12(x₁)]	=	0
[isNat(x₁)]	=	1
[mark(x₁)]	=	0
[plus(x₁, x₂)]	=	1
[s(x₁)]	=	0
[U21(x₁)]	=	0
[U41(x₁, x₂, x₃)]	=	1
[U42(x₁, x₂, x₃)]	=	1
[U31(x₁, x₂)]	=	0
[active(x₁)]	=	0
[0]	=	0

together with the usable rules

U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)
U31(X1,mark(X2))	→	U31(X1,X2)	(34)
U31(mark(X1),X2)	→	U31(X1,X2)	(33)
U31(active(X1),X2)	→	U31(X1,X2)	(35)
U31(X1,active(X2))	→	U31(X1,X2)	(36)
U41(X1,mark(X2),X3)	→	U41(X1,X2,X3)	(38)
U41(mark(X1),X2,X3)	→	U41(X1,X2,X3)	(37)
U41(X1,X2,mark(X3))	→	U41(X1,X2,X3)	(39)
U41(active(X1),X2,X3)	→	U41(X1,X2,X3)	(40)
U41(X1,active(X2),X3)	→	U41(X1,X2,X3)	(41)
U41(X1,X2,active(X3))	→	U41(X1,X2,X3)	(42)
plus(X1,mark(X2))	→	plus(X1,X2)	(52)
plus(mark(X1),X2)	→	plus(X1,X2)	(51)
plus(active(X1),X2)	→	plus(X1,X2)	(53)
plus(X1,active(X2))	→	plus(X1,X2)	(54)

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

mark^#(U31(X1,X2))

→

active^#(U31(mark(X1),X2))

(91)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[active^#(x₁)]	=	1 · x₁
[U11(x₁, x₂)]	=	1 · x₁
[tt]	=	1
[mark^#(x₁)]	=	1 · x₁
[U12(x₁)]	=	1 · x₁
[isNat(x₁)]	=	1
[mark(x₁)]	=	1 · x₁
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[s(x₁)]	=	1 + 1 · x₁
[U21(x₁)]	=	1 · x₁
[U41(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U42(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[active(x₁)]	=	1 · x₁
[U31(x₁, x₂)]	=	1 · x₂
[0]	=	0

the pairs

mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(96)
mark^#(U42(X1,X2,X3))	→	mark^#(X1)	(99)
mark^#(s(X))	→	mark^#(X)	(102)
mark^#(plus(X1,X2))	→	mark^#(X1)	(105)
mark^#(plus(X1,X2))	→	mark^#(X2)	(106)

could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U12(X))	→	mark^#(X)	(86)
mark^#(U11(X1,X2))	→	active^#(U11(mark(X1),X2))	(80)
active^#(U11(tt,V2))	→	mark^#(U12(isNat(V2)))	(55)
active^#(U41(tt,M,N))	→	mark^#(U42(isNat(N),M,N))	(61)
mark^#(U42(X1,X2,X3))	→	active^#(U42(mark(X1),X2,X3))	(97)
active^#(plus(N,s(M)))	→	mark^#(U41(isNat(M),M,N))	(77)
mark^#(U41(X1,X2,X3))	→	active^#(U41(mark(X1),X2,X3))	(94)
mark^#(U11(X1,X2))	→	mark^#(X1)	(82)
mark^#(isNat(X))	→	active^#(isNat(X))	(87)
active^#(isNat(plus(V1,V2)))	→	mark^#(U11(isNat(V1),V2))	(68)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(71)
mark^#(U21(X))	→	mark^#(X)	(90)
mark^#(plus(X1,X2))	→	active^#(plus(mark(X1),mark(X2)))	(103)

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
[U11(x₁, x₂)]	=	1 + 2 · x₁
[U41(x₁, x₂, x₃)]	=	-2
[U42(x₁, x₂, x₃)]	=	0
[plus(x₁, x₂)]	=	2
[mark(x₁)]	=	-2
[U12(x₁)]	=	1 + 2 · x₁
[active(x₁)]	=	-2 + x₁
[tt]	=	0
[isNat(x₁)]	=	0
[U31(x₁, x₂)]	=	-2 + 2 · x₁ + 2 · x₂
[s(x₁)]	=	-2
[U21(x₁)]	=	1 + 2 · x₁
[0]	=	0
[mark^#(x₁)]	=	-2 + 2 · x₁

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

mark^#(plus(X1,X2))

→

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

(103)

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
[U11(x₁, x₂)]	=	0
[U41(x₁, x₂, x₃)]	=	-2
[U42(x₁, x₂, x₃)]	=	-2
[mark(x₁)]	=	2
[U12(x₁)]	=	-2
[active(x₁)]	=	-2 + x₁
[tt]	=	0
[isNat(x₁)]	=	0
[U31(x₁, x₂)]	=	2 + 2 · x₁
[s(x₁)]	=	-2 + x₁
[plus(x₁, x₂)]	=	2
[U21(x₁)]	=	-2
[0]	=	0
[mark^#(x₁)]	=	-2

together with the usable rules

U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)
U41(X1,mark(X2),X3)	→	U41(X1,X2,X3)	(38)
U41(mark(X1),X2,X3)	→	U41(X1,X2,X3)	(37)
U41(X1,X2,mark(X3))	→	U41(X1,X2,X3)	(39)
U41(active(X1),X2,X3)	→	U41(X1,X2,X3)	(40)
U41(X1,active(X2),X3)	→	U41(X1,X2,X3)	(41)
U41(X1,X2,active(X3))	→	U41(X1,X2,X3)	(42)

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

active^#(plus(N,s(M)))

→

mark^#(U41(isNat(M),M,N))

(77)

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
[U11(x₁, x₂)]	=	1 + 2 · x₁
[U41(x₁, x₂, x₃)]	=	2 + 2 · x₂ + 2 · x₃
[U42(x₁, x₂, x₃)]	=	-2
[mark(x₁)]	=	0
[U12(x₁)]	=	1 + x₁
[active(x₁)]	=	-2
[tt]	=	0
[isNat(x₁)]	=	0
[U31(x₁, x₂)]	=	-2 + x₁
[s(x₁)]	=	2
[plus(x₁, x₂)]	=	-2 + x₂
[U21(x₁)]	=	1 + x₁
[0]	=	0
[mark^#(x₁)]	=	-1 + x₁

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

mark^#(U41(X1,X2,X3))

→

active^#(U41(mark(X1),X2,X3))

(94)

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)]	=	-2 + 2 · x₁
[U11(x₁, x₂)]	=	2
[U42(x₁, x₂, x₃)]	=	-2
[mark(x₁)]	=	2
[U12(x₁)]	=	-2 + x₁
[active(x₁)]	=	2
[tt]	=	2
[isNat(x₁)]	=	2
[U31(x₁, x₂)]	=	-2
[U41(x₁, x₂, x₃)]	=	-1 + 2 · x₁
[s(x₁)]	=	-2
[plus(x₁, x₂)]	=	-2
[U21(x₁)]	=	2 + 2 · x₁
[0]	=	0
[mark^#(x₁)]	=	2

together with the usable rules

U11(X1,mark(X2))	→	U11(X1,X2)	(24)
U11(mark(X1),X2)	→	U11(X1,X2)	(23)
U11(active(X1),X2)	→	U11(X1,X2)	(25)
U11(X1,active(X2))	→	U11(X1,X2)	(26)
U42(X1,mark(X2),X3)	→	U42(X1,X2,X3)	(44)
U42(mark(X1),X2,X3)	→	U42(X1,X2,X3)	(43)
U42(X1,X2,mark(X3))	→	U42(X1,X2,X3)	(45)
U42(active(X1),X2,X3)	→	U42(X1,X2,X3)	(46)
U42(X1,active(X2),X3)	→	U42(X1,X2,X3)	(47)
U42(X1,X2,active(X3))	→	U42(X1,X2,X3)	(48)

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

active^#(U41(tt,M,N))	→	mark^#(U42(isNat(N),M,N))	(61)
mark^#(U42(X1,X2,X3))	→	active^#(U42(mark(X1),X2,X3))	(97)

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
[U11(x₁, x₂)]	=	x₁ + x₂
[mark(x₁)]	=	x₁
[U12(x₁)]	=	2 + x₁
[active(x₁)]	=	x₁
[tt]	=	2
[isNat(x₁)]	=	x₁
[U31(x₁, x₂)]	=	x₂
[U41(x₁, x₂, x₃)]	=	2 + x₂ + x₃
[U42(x₁, x₂, x₃)]	=	2 + x₂ + x₃
[s(x₁)]	=	2 + x₁
[plus(x₁, x₂)]	=	x₁ + x₂
[U21(x₁)]	=	x₁
[0]	=	2
[mark^#(x₁)]	=	x₁

the pairs

mark^#(U12(X))	→	mark^#(X)	(86)
active^#(isNat(s(V1)))	→	mark^#(U21(isNat(V1)))	(71)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(isNat(X))	→	active^#(isNat(X))	(87)
active^#(isNat(plus(V1,V2)))	→	mark^#(U11(isNat(V1),V2))	(68)
mark^#(U11(X1,X2))	→	mark^#(X1)	(82)
mark^#(U21(X))	→	mark^#(X)	(90)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

U11(mark(x0),x1)

U11(x0,mark(x1))

U11(active(x0),x1)

U11(x0,active(x1))

isNat(mark(x0))

isNat(active(x0))

U21(mark(x0))

U21(active(x0))

plus(mark(x0),x1)

plus(x0,mark(x1))

plus(active(x0),x1)

plus(x0,active(x1))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mark^#(x₁)]	=	2 + 2 · x₁
[isNat(x₁)]	=	1 + 1 · x₁
[active^#(x₁)]	=	2 + 2 · x₁
[plus(x₁, x₂)]	=	2 + 2 · x₁ + 1 · x₂
[U11(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[U21(x₁)]	=	1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

active^#(isNat(plus(V1,V2)))	→	mark^#(U11(isNat(V1),V2))	(68)
mark^#(U11(X1,X2))	→	mark^#(X1)	(82)
mark^#(U21(X))	→	mark^#(X)	(90)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

isNat(mark(x0))

isNat(active(x0))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

mark^#(isNat(X))	→	active^#(isNat(X))	(87)

1	≥	1

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

The 2^nd component contains the pair

U11^#(X1,mark(X2))	→	U11^#(X1,X2)	(109)
U11^#(mark(X1),X2)	→	U11^#(X1,X2)	(108)
U11^#(active(X1),X2)	→	U11^#(X1,X2)	(110)
U11^#(X1,active(X2))	→	U11^#(X1,X2)	(111)

1.1.2 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.2.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.2.1.1 Size-Change Termination

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

U11^#(X1,mark(X2))	→	U11^#(X1,X2)	(109)

1	≥	1
2	>	2
U11^#(mark(X1),X2)	→	U11^#(X1,X2)	(108)

1	>	1
2	≥	2
U11^#(active(X1),X2)	→	U11^#(X1,X2)	(110)

1	>	1
2	≥	2
U11^#(X1,active(X2))	→	U11^#(X1,X2)	(111)

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

U12^#(active(X))	→	U12^#(X)	(113)
U12^#(mark(X))	→	U12^#(X)	(112)

1.1.3 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.3.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.3.1.1 Size-Change Termination

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

U12^#(active(X))	→	U12^#(X)	(113)

1	>	1
U12^#(mark(X))	→	U12^#(X)	(112)

1	>	1

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

The 4^th component contains the pair

isNat^#(active(X))	→	isNat^#(X)	(115)
isNat^#(mark(X))	→	isNat^#(X)	(114)

1.1.4 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.4.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.4.1.1 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)	(115)

1	>	1
isNat^#(mark(X))	→	isNat^#(X)	(114)

1	>	1

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

The 5^th component contains the pair

U21^#(active(X))	→	U21^#(X)	(117)
U21^#(mark(X))	→	U21^#(X)	(116)

1.1.5 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.5.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.5.1.1 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)	(117)

1	>	1
U21^#(mark(X))	→	U21^#(X)	(116)

1	>	1

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

The 6^th component contains the pair

U31^#(X1,mark(X2))	→	U31^#(X1,X2)	(119)
U31^#(mark(X1),X2)	→	U31^#(X1,X2)	(118)
U31^#(active(X1),X2)	→	U31^#(X1,X2)	(120)
U31^#(X1,active(X2))	→	U31^#(X1,X2)	(121)

1.1.6 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.6.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.6.1.1 Size-Change Termination

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

U31^#(X1,mark(X2))	→	U31^#(X1,X2)	(119)

1	≥	1
2	>	2
U31^#(mark(X1),X2)	→	U31^#(X1,X2)	(118)

1	>	1
2	≥	2
U31^#(active(X1),X2)	→	U31^#(X1,X2)	(120)

1	>	1
2	≥	2
U31^#(X1,active(X2))	→	U31^#(X1,X2)	(121)

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

U41^#(X1,mark(X2),X3)	→	U41^#(X1,X2,X3)	(123)
U41^#(mark(X1),X2,X3)	→	U41^#(X1,X2,X3)	(122)
U41^#(X1,X2,mark(X3))	→	U41^#(X1,X2,X3)	(124)
U41^#(active(X1),X2,X3)	→	U41^#(X1,X2,X3)	(125)
U41^#(X1,active(X2),X3)	→	U41^#(X1,X2,X3)	(126)
U41^#(X1,X2,active(X3))	→	U41^#(X1,X2,X3)	(127)

1.1.7 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.7.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.7.1.1 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),X3)	→	U41^#(X1,X2,X3)	(123)

1	≥	1
2	>	2
3	≥	3
U41^#(mark(X1),X2,X3)	→	U41^#(X1,X2,X3)	(122)

1	>	1
2	≥	2
3	≥	3
U41^#(X1,X2,mark(X3))	→	U41^#(X1,X2,X3)	(124)

1	≥	1
2	≥	2
3	>	3
U41^#(active(X1),X2,X3)	→	U41^#(X1,X2,X3)	(125)

1	>	1
2	≥	2
3	≥	3
U41^#(X1,active(X2),X3)	→	U41^#(X1,X2,X3)	(126)

1	≥	1
2	>	2
3	≥	3
U41^#(X1,X2,active(X3))	→	U41^#(X1,X2,X3)	(127)

1	≥	1
2	≥	2
3	>	3

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

The 8^th component contains the pair

U42^#(X1,mark(X2),X3)	→	U42^#(X1,X2,X3)	(129)
U42^#(mark(X1),X2,X3)	→	U42^#(X1,X2,X3)	(128)
U42^#(X1,X2,mark(X3))	→	U42^#(X1,X2,X3)	(130)
U42^#(active(X1),X2,X3)	→	U42^#(X1,X2,X3)	(131)
U42^#(X1,active(X2),X3)	→	U42^#(X1,X2,X3)	(132)
U42^#(X1,X2,active(X3))	→	U42^#(X1,X2,X3)	(133)

1.1.8 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.8.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.8.1.1 Size-Change Termination

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

U42^#(X1,mark(X2),X3)	→	U42^#(X1,X2,X3)	(129)

1	≥	1
2	>	2
3	≥	3
U42^#(mark(X1),X2,X3)	→	U42^#(X1,X2,X3)	(128)

1	>	1
2	≥	2
3	≥	3
U42^#(X1,X2,mark(X3))	→	U42^#(X1,X2,X3)	(130)

1	≥	1
2	≥	2
3	>	3
U42^#(active(X1),X2,X3)	→	U42^#(X1,X2,X3)	(131)

1	>	1
2	≥	2
3	≥	3
U42^#(X1,active(X2),X3)	→	U42^#(X1,X2,X3)	(132)

1	≥	1
2	>	2
3	≥	3
U42^#(X1,X2,active(X3))	→	U42^#(X1,X2,X3)	(133)

1	≥	1
2	≥	2
3	>	3

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

The 9^th component contains the pair

s^#(active(X))	→	s^#(X)	(135)
s^#(mark(X))	→	s^#(X)	(134)

1.1.9 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.9.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.9.1.1 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)	(135)

1	>	1
s^#(mark(X))	→	s^#(X)	(134)

1	>	1

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

The 10^th component contains the pair

plus^#(X1,mark(X2))	→	plus^#(X1,X2)	(137)
plus^#(mark(X1),X2)	→	plus^#(X1,X2)	(136)
plus^#(active(X1),X2)	→	plus^#(X1,X2)	(138)
plus^#(X1,active(X2))	→	plus^#(X1,X2)	(139)

1.1.10 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.10.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

active(U11(tt,x0))

active(U12(tt))

active(U21(tt))

active(U31(tt,x0))

active(U41(tt,x0,x1))

active(U42(tt,x0,x1))

active(isNat(0))

active(isNat(plus(x0,x1)))

active(isNat(s(x0)))

active(plus(x0,0))

active(plus(x0,s(x1)))

mark(U11(x0,x1))

mark(tt)

mark(U12(x0))

mark(isNat(x0))

mark(U21(x0))

mark(U31(x0,x1))

mark(U41(x0,x1,x2))

mark(U42(x0,x1,x2))

mark(s(x0))

mark(plus(x0,x1))

mark(0)

1.1.10.1.1 Size-Change Termination

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

plus^#(X1,mark(X2))	→	plus^#(X1,X2)	(137)

1	≥	1
2	>	2
plus^#(mark(X1),X2)	→	plus^#(X1,X2)	(136)

1	>	1
2	≥	2
plus^#(active(X1),X2)	→	plus^#(X1,X2)	(138)

1	>	1
2	≥	2
plus^#(X1,active(X2))	→	plus^#(X1,X2)	(139)

1	≥	1
2	>	2

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