Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_nokinds_GM)

The rewrite relation of the following TRS is considered.

a__U11(tt,N)	→	mark(N)	(1)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)
a__U31(tt)	→	0	(3)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
a__and(tt,X)	→	mark(X)	(5)
a__isNat(0)	→	tt	(6)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(U31(X))	→	a__U31(mark(X))	(17)
mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
mark(isNat(X))	→	a__isNat(X)	(21)
mark(tt)	→	tt	(22)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
a__U31(X)	→	U31(X)	(28)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__x(X1,X2)	→	x(X1,X2)	(30)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__isNat(X)	→	isNat(X)	(32)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a__U21^#(tt,M,N)	→	mark^#(N)	(33)
mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(34)
mark^#(isNat(X))	→	a__isNat^#(X)	(35)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
mark^#(U41(X1,X2,X3))	→	a__U41^#(mark(X1),X2,X3)	(37)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__x^#(N,s(M))	→	a__isNat^#(M)	(41)
a__U41^#(tt,M,N)	→	mark^#(M)	(42)
a__x^#(N,0)	→	a__U31^#(a__isNat(N))	(43)
a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
a__plus^#(N,s(M))	→	a__isNat^#(M)	(45)
mark^#(U31(X))	→	a__U31^#(mark(X))	(46)
a__and^#(tt,X)	→	mark^#(X)	(47)
mark^#(s(X))	→	mark^#(X)	(48)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
a__x^#(N,0)	→	a__isNat^#(N)	(50)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
mark^#(U31(X))	→	mark^#(X)	(52)
a__isNat^#(plus(V1,V2))	→	a__isNat^#(V1)	(53)
a__isNat^#(x(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(54)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
a__plus^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(55)
a__U11^#(tt,N)	→	mark^#(N)	(56)
a__x^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(57)
mark^#(x(X1,X2))	→	mark^#(X2)	(58)
a__plus^#(N,0)	→	a__isNat^#(N)	(59)
a__isNat^#(s(V1))	→	a__isNat^#(V1)	(60)
mark^#(x(X1,X2))	→	mark^#(X1)	(61)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(62)
mark^#(plus(X1,X2))	→	mark^#(X2)	(63)
a__isNat^#(plus(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(64)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(65)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(66)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
mark^#(U11(X1,X2))	→	mark^#(X1)	(68)
a__U21^#(tt,M,N)	→	mark^#(M)	(69)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)
a__isNat^#(x(V1,V2))	→	a__isNat^#(V1)	(71)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
mark^#(and(X1,X2))	→	mark^#(X1)	(74)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(and(X1,X2))	→	mark^#(X1)	(74)
mark^#(U31(X))	→	mark^#(X)	(52)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
a__x^#(N,0)	→	a__isNat^#(N)	(50)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
mark^#(s(X))	→	mark^#(X)	(48)
a__isNat^#(x(V1,V2))	→	a__isNat^#(V1)	(71)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)
a__and^#(tt,X)	→	mark^#(X)	(47)
a__U21^#(tt,M,N)	→	mark^#(M)	(69)
mark^#(U11(X1,X2))	→	mark^#(X1)	(68)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(66)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(65)
a__plus^#(N,s(M))	→	a__isNat^#(M)	(45)
a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
mark^#(plus(X1,X2))	→	mark^#(X2)	(63)
a__isNat^#(plus(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(64)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(62)
a__U41^#(tt,M,N)	→	mark^#(M)	(42)
a__x^#(N,s(M))	→	a__isNat^#(M)	(41)
mark^#(x(X1,X2))	→	mark^#(X1)	(61)
a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__isNat^#(s(V1))	→	a__isNat^#(V1)	(60)
a__plus^#(N,0)	→	a__isNat^#(N)	(59)
mark^#(x(X1,X2))	→	mark^#(X2)	(58)
a__x^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(57)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
a__U11^#(tt,N)	→	mark^#(N)	(56)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
mark^#(U41(X1,X2,X3))	→	a__U41^#(mark(X1),X2,X3)	(37)
a__plus^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(55)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
a__isNat^#(x(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(54)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
mark^#(isNat(X))	→	a__isNat^#(X)	(35)
mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(34)
a__U21^#(tt,M,N)	→	mark^#(N)	(33)
a__isNat^#(plus(V1,V2))	→	a__isNat^#(V1)	(53)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 3, 0)
[U21(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 3, x₃ + 0, 0)
[U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	2
[and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 20591, x₂ + 5, 0)
[a__U11^#(x₁, x₂)]	=	max(x₂ + 2, 0)
[a__U31(x₁)]	=	x₁ + 10531
[a__x(x₁, x₂)]	=	max(x₁ + 20589, x₂ + 3, 0)
[a__U41(x₁, x₂, x₃)]	=	max(x₁ + 20588, x₂ + 3, x₃ + 20589, 0)
[x(x₁, x₂)]	=	max(x₁ + 20589, x₂ + 3, 0)
[a__U21^#(x₁, x₂, x₃)]	=	max(x₂ + 4, x₃ + 2, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 2, x₂ + 4, 0)
[mark^#(x₁)]	=	x₁ + 2
[0]	=	10527
[a__and^#(x₁, x₂)]	=	max(x₁ + 1, x₂ + 2, 0)
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 3, x₃ + 0, 0)
[mark(x₁)]	=	x₁ + 0
[a__U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[isNat(x₁)]	=	0
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 3, 0)
[U31(x₁)]	=	x₁ + 10531
[a__U41^#(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 5, x₃ + 20591, 0)
[tt]	=	0
[a__isNat(x₁)]	=	0
[a__and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U41(x₁, x₂, x₃)]	=	max(x₁ + 20588, x₂ + 3, x₃ + 20589, 0)

together with the usable rules

mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__U11(tt,N)	→	mark(N)	(1)
a__U31(tt)	→	0	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(isNat(X))	→	a__isNat(X)	(21)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
a__isNat(X)	→	isNat(X)	(32)
mark(U31(X))	→	a__U31(mark(X))	(17)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
mark(tt)	→	tt	(22)
a__U31(X)	→	U31(X)	(28)
a__and(tt,X)	→	mark(X)	(5)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__x(X1,X2)	→	x(X1,X2)	(30)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
a__isNat(0)	→	tt	(6)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)

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

mark^#(U31(X))	→	mark^#(X)	(52)
a__x^#(N,0)	→	a__isNat^#(N)	(50)
a__U21^#(tt,M,N)	→	mark^#(M)	(69)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(66)
a__plus^#(N,s(M))	→	a__isNat^#(M)	(45)
mark^#(plus(X1,X2))	→	mark^#(X2)	(63)
a__U41^#(tt,M,N)	→	mark^#(M)	(42)
a__x^#(N,s(M))	→	a__isNat^#(M)	(41)
mark^#(x(X1,X2))	→	mark^#(X1)	(61)
a__plus^#(N,0)	→	a__isNat^#(N)	(59)
mark^#(x(X1,X2))	→	mark^#(X2)	(58)
a__x^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(57)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
a__plus^#(N,s(M))	→	a__and^#(a__isNat(M),isNat(N))	(55)
a__U41^#(tt,M,N)	→	mark^#(N)	(38)
mark^#(U41(X1,X2,X3))	→	mark^#(X1)	(34)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U41(X1,X2,X3))	→	a__U41^#(mark(X1),X2,X3)	(37)
a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
a__isNat^#(s(V1))	→	a__isNat^#(V1)	(60)
a__U11^#(tt,N)	→	mark^#(N)	(56)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
mark^#(isNat(X))	→	a__isNat^#(X)	(35)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(65)
a__and^#(tt,X)	→	mark^#(X)	(47)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
a__isNat^#(plus(V1,V2))	→	a__isNat^#(V1)	(53)
a__isNat^#(plus(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(64)
mark^#(and(X1,X2))	→	mark^#(X1)	(74)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(62)
mark^#(U11(X1,X2))	→	mark^#(X1)	(68)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
mark^#(s(X))	→	mark^#(X)	(48)
a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
a__isNat^#(x(V1,V2))	→	a__isNat^#(V1)	(71)
a__isNat^#(x(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(54)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
a__U21^#(tt,M,N)	→	mark^#(N)	(33)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 10622, 0)
[U21(x₁, x₂, x₃)]	=	max(x₂ + 10622, x₃ + 0, 0)
[U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	x₁ + 1
[and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 3825, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 11600, x₂ + 11599, 0)
[a__U11^#(x₁, x₂)]	=	max(x₁ + 452, x₂ + 978, 0)
[a__U31(x₁)]	=	10474
[a__x(x₁, x₂)]	=	max(x₁ + 10622, x₂ + 10621, 0)
[a__U41(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 10621, x₃ + 10622, 0)
[x(x₁, x₂)]	=	max(x₁ + 10622, x₂ + 10621, 0)
[a__U21^#(x₁, x₂, x₃)]	=	max(x₃ + 978, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 978, 0)
[mark^#(x₁)]	=	x₁ + 978
[0]	=	0
[a__and^#(x₁, x₂)]	=	max(x₂ + 979, 0)
[a__U21(x₁, x₂, x₃)]	=	max(x₂ + 10622, x₃ + 0, 0)
[mark(x₁)]	=	x₁ + 0
[a__U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[isNat(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 10622, 0)
[U31(x₁)]	=	10474
[a__U41^#(x₁, x₂, x₃)]	=	max(x₂ + 11599, x₃ + 11600, 0)
[tt]	=	0
[a__isNat(x₁)]	=	x₁ + 0
[a__and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 3825, 0)
[U41(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 10621, x₃ + 10622, 0)

together with the usable rules

mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__U11(tt,N)	→	mark(N)	(1)
a__U31(tt)	→	0	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(isNat(X))	→	a__isNat(X)	(21)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
a__isNat(X)	→	isNat(X)	(32)
mark(U31(X))	→	a__U31(mark(X))	(17)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
mark(tt)	→	tt	(22)
a__U31(X)	→	U31(X)	(28)
a__and(tt,X)	→	mark(X)	(5)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__x(X1,X2)	→	x(X1,X2)	(30)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
a__isNat(0)	→	tt	(6)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)

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

mark^#(isNat(X))	→	a__isNat^#(X)	(35)
a__and^#(tt,X)	→	mark^#(X)	(47)
a__isNat^#(plus(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(64)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(62)
a__isNat^#(x(V1,V2))	→	a__isNat^#(V1)	(71)
a__isNat^#(x(V1,V2))	→	a__and^#(a__isNat(V1),isNat(V2))	(54)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

a__isNat^#(s(V1))	→	a__isNat^#(V1)	(60)
a__isNat^#(plus(V1,V2))	→	a__isNat^#(V1)	(53)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	56525
[U21(x₁, x₂, x₃)]	=	56527
[U11(x₁, x₂)]	=	56527
[s(x₁)]	=	x₁ + 2
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	x₁ + 2
[and(x₁, x₂)]	=	x₁ + 1
[a__x^#(x₁, x₂)]	=	0
[a__U11^#(x₁, x₂)]	=	2
[a__U31(x₁)]	=	x₁ + 1
[a__x(x₁, x₂)]	=	x₁ + 1
[a__U41(x₁, x₂, x₃)]	=	x₁ + x₃ + 1
[x(x₁, x₂)]	=	x₂ + 2
[a__U21^#(x₁, x₂, x₃)]	=	2
[a__plus^#(x₁, x₂)]	=	2
[mark^#(x₁)]	=	2
[0]	=	56525
[a__and^#(x₁, x₂)]	=	2
[a__U21(x₁, x₂, x₃)]	=	56526
[mark(x₁)]	=	1
[a__U11(x₁, x₂)]	=	56526
[isNat(x₁)]	=	x₁ + 24849
[plus(x₁, x₂)]	=	x₁ + 56526
[U31(x₁)]	=	2
[a__U41^#(x₁, x₂, x₃)]	=	0
[tt]	=	56523
[a__isNat(x₁)]	=	24848
[a__and(x₁, x₂)]	=	x₂ + 0
[U41(x₁, x₂, x₃)]	=	2

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

a__isNat^#(s(V1))	→	a__isNat^#(V1)	(60)
a__isNat^#(plus(V1,V2))	→	a__isNat^#(V1)	(53)

could be deleted.

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

mark^#(U41(X1,X2,X3))	→	a__U41^#(mark(X1),X2,X3)	(37)
a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
a__U11^#(tt,N)	→	mark^#(N)	(56)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(65)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
mark^#(and(X1,X2))	→	mark^#(X1)	(74)
mark^#(U11(X1,X2))	→	mark^#(X1)	(68)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
mark^#(s(X))	→	mark^#(X)	(48)
a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
a__U21^#(tt,M,N)	→	mark^#(N)	(33)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)

1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U21(x₁, x₂, x₃)]	=	max(x₂ + 0, x₃ + 0, 0)
[U11(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	1
[and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 29180, 0)
[a__U11^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U31(x₁)]	=	x₁ + 5744
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 29180, 0)
[a__U41(x₁, x₂, x₃)]	=	max(x₁ + 29178, x₂ + 29180, x₃ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 29180, 0)
[a__U21^#(x₁, x₂, x₃)]	=	max(x₂ + 0, x₃ + 0, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[mark^#(x₁)]	=	x₁ + 0
[0]	=	2
[a__and^#(x₁, x₂)]	=	max(0)
[a__U21(x₁, x₂, x₃)]	=	max(x₂ + 0, x₃ + 0, 0)
[mark(x₁)]	=	x₁ + 0
[a__U11(x₁, x₂)]	=	max(x₁ + 1, x₂ + 0, 0)
[isNat(x₁)]	=	1
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U31(x₁)]	=	x₁ + 5744
[a__U41^#(x₁, x₂, x₃)]	=	max(x₁ + 29178, x₂ + 29180, x₃ + 0, 0)
[tt]	=	1
[a__isNat(x₁)]	=	1
[a__and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U41(x₁, x₂, x₃)]	=	max(x₁ + 29178, x₂ + 29180, x₃ + 0, 0)

together with the usable rules

mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__U11(tt,N)	→	mark(N)	(1)
a__U31(tt)	→	0	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(isNat(X))	→	a__isNat(X)	(21)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
a__isNat(X)	→	isNat(X)	(32)
mark(U31(X))	→	a__U31(mark(X))	(17)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
mark(tt)	→	tt	(22)
a__U31(X)	→	U31(X)	(28)
a__and(tt,X)	→	mark(X)	(5)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__x(X1,X2)	→	x(X1,X2)	(30)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
a__isNat(0)	→	tt	(6)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)

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

mark^#(U11(X1,X2))

→

mark^#(X1)

(68)

could be deleted.

1.1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U41(X1,X2,X3))	→	a__U41^#(mark(X1),X2,X3)	(37)
a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
a__U11^#(tt,N)	→	mark^#(N)	(56)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(65)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
mark^#(and(X1,X2))	→	mark^#(X1)	(74)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
mark^#(s(X))	→	mark^#(X)	(48)
a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
a__U21^#(tt,M,N)	→	mark^#(N)	(33)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)

1.1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__U11^#)	=	2
π(mark^#)	=	1
π(a__and^#)	=	1
π(mark)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a__plus)	=	4	status(a__plus)	=	[1, 2]	list-extension(a__plus)	=	Lex
prec(U21)	=	4	status(U21)	=	[3, 2, 1]	list-extension(U21)	=	Lex
prec(U11)	=	2	status(U11)	=	[1, 2]	list-extension(U11)	=	Lex
prec(s)	=	0	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__U31^#)	=	0	status(a__U31^#)	=	[]	list-extension(a__U31^#)	=	Lex
prec(a__isNat^#)	=	0	status(a__isNat^#)	=	[]	list-extension(a__isNat^#)	=	Lex
prec(and)	=	1	status(and)	=	[2, 1]	list-extension(and)	=	Lex
prec(a__x^#)	=	5	status(a__x^#)	=	[2, 1]	list-extension(a__x^#)	=	Lex
prec(a__U31)	=	6	status(a__U31)	=	[]	list-extension(a__U31)	=	Lex
prec(a__x)	=	5	status(a__x)	=	[2, 1]	list-extension(a__x)	=	Lex
prec(a__U41)	=	5	status(a__U41)	=	[2, 3, 1]	list-extension(a__U41)	=	Lex
prec(x)	=	5	status(x)	=	[2, 1]	list-extension(x)	=	Lex
prec(a__U21^#)	=	2	status(a__U21^#)	=	[3]	list-extension(a__U21^#)	=	Lex
prec(a__plus^#)	=	2	status(a__plus^#)	=	[1]	list-extension(a__plus^#)	=	Lex
prec(0)	=	6	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__U21)	=	4	status(a__U21)	=	[3, 2, 1]	list-extension(a__U21)	=	Lex
prec(a__U11)	=	2	status(a__U11)	=	[1, 2]	list-extension(a__U11)	=	Lex
prec(isNat)	=	1	status(isNat)	=	[1]	list-extension(isNat)	=	Lex
prec(plus)	=	4	status(plus)	=	[1, 2]	list-extension(plus)	=	Lex
prec(U31)	=	6	status(U31)	=	[]	list-extension(U31)	=	Lex
prec(a__U41^#)	=	5	status(a__U41^#)	=	[2, 3, 1]	list-extension(a__U41^#)	=	Lex
prec(tt)	=	6	status(tt)	=	[]	list-extension(tt)	=	Lex
prec(a__isNat)	=	1	status(a__isNat)	=	[1]	list-extension(a__isNat)	=	Lex
prec(a__and)	=	1	status(a__and)	=	[2, 1]	list-extension(a__and)	=	Lex
prec(U41)	=	5	status(U41)	=	[2, 3, 1]	list-extension(U41)	=	Lex

and the following Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	0
[and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U31(x₁)]	=	0
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U41(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U21^#(x₁, x₂, x₃)]	=	max(x₃ + 0, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 0, 0)
[0]	=	0
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[isNat(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U31(x₁)]	=	0
[a__U41^#(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[tt]	=	0
[a__isNat(x₁)]	=	x₁ + 0
[a__and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U41(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)

together with the usable rules

mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__U11(tt,N)	→	mark(N)	(1)
a__U31(tt)	→	0	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(isNat(X))	→	a__isNat(X)	(21)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
a__isNat(X)	→	isNat(X)	(32)
mark(U31(X))	→	a__U31(mark(X))	(17)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
mark(tt)	→	tt	(22)
a__U31(X)	→	U31(X)	(28)
a__and(tt,X)	→	mark(X)	(5)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__x(X1,X2)	→	x(X1,X2)	(30)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
a__isNat(0)	→	tt	(6)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)

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

a__U41^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(40)
a__U41^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(39)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(36)
mark^#(plus(X1,X2))	→	mark^#(X1)	(49)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(51)
a__plus^#(N,0)	→	a__U11^#(a__isNat(N),N)	(67)
mark^#(and(X1,X2))	→	mark^#(X1)	(74)
mark^#(U11(X1,X2))	→	a__U11^#(mark(X1),X2)	(72)
mark^#(s(X))	→	mark^#(X)	(48)
a__x^#(N,s(M))	→	a__U41^#(a__and(a__isNat(M),isNat(N)),M,N)	(73)
a__U21^#(tt,M,N)	→	mark^#(N)	(33)

could be deleted.

1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a__plus^#(N,s(M))	→	a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)	(44)
a__U21^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(70)

1.1.1.1.1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__U11^#)	=	2
π(a__U21^#)	=	2
π(a__plus^#)	=	2
π(mark^#)	=	1
π(a__and^#)	=	1
π(mark)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a__plus)	=	4	status(a__plus)	=	[1, 2]	list-extension(a__plus)	=	Lex
prec(U21)	=	4	status(U21)	=	[3, 2, 1]	list-extension(U21)	=	Lex
prec(U11)	=	2	status(U11)	=	[1, 2]	list-extension(U11)	=	Lex
prec(s)	=	0	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__U31^#)	=	0	status(a__U31^#)	=	[]	list-extension(a__U31^#)	=	Lex
prec(a__isNat^#)	=	0	status(a__isNat^#)	=	[]	list-extension(a__isNat^#)	=	Lex
prec(and)	=	3	status(and)	=	[2, 1]	list-extension(and)	=	Lex
prec(a__x^#)	=	5	status(a__x^#)	=	[2, 1]	list-extension(a__x^#)	=	Lex
prec(a__U31)	=	7	status(a__U31)	=	[]	list-extension(a__U31)	=	Lex
prec(a__x)	=	5	status(a__x)	=	[2, 1]	list-extension(a__x)	=	Lex
prec(a__U41)	=	5	status(a__U41)	=	[2, 3, 1]	list-extension(a__U41)	=	Lex
prec(x)	=	5	status(x)	=	[2, 1]	list-extension(x)	=	Lex
prec(0)	=	7	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__U21)	=	4	status(a__U21)	=	[3, 2, 1]	list-extension(a__U21)	=	Lex
prec(a__U11)	=	2	status(a__U11)	=	[1, 2]	list-extension(a__U11)	=	Lex
prec(isNat)	=	3	status(isNat)	=	[1]	list-extension(isNat)	=	Lex
prec(plus)	=	4	status(plus)	=	[1, 2]	list-extension(plus)	=	Lex
prec(U31)	=	7	status(U31)	=	[]	list-extension(U31)	=	Lex
prec(a__U41^#)	=	5	status(a__U41^#)	=	[2, 3, 1]	list-extension(a__U41^#)	=	Lex
prec(tt)	=	7	status(tt)	=	[]	list-extension(tt)	=	Lex
prec(a__isNat)	=	3	status(a__isNat)	=	[1]	list-extension(a__isNat)	=	Lex
prec(a__and)	=	3	status(a__and)	=	[2, 1]	list-extension(a__and)	=	Lex
prec(U41)	=	5	status(U41)	=	[2, 3, 1]	list-extension(U41)	=	Lex

and the following Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__U31^#(x₁)]	=	0
[a__isNat^#(x₁)]	=	0
[and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U31(x₁)]	=	0
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U41(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[0]	=	0
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U11(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[isNat(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U31(x₁)]	=	0
[a__U41^#(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[tt]	=	0
[a__isNat(x₁)]	=	x₁ + 0
[a__and(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U41(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)

together with the usable rules

mark(U41(X1,X2,X3))	→	a__U41(mark(X1),X2,X3)	(18)
a__U41(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(15)
a__isNat(s(V1))	→	a__isNat(V1)	(8)
a__U11(tt,N)	→	mark(N)	(1)
a__U31(tt)	→	0	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(16)
mark(isNat(X))	→	a__isNat(X)	(21)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(26)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(19)
a__isNat(X)	→	isNat(X)	(32)
mark(U31(X))	→	a__U31(mark(X))	(17)
a__plus(X1,X2)	→	plus(X1,X2)	(27)
mark(tt)	→	tt	(22)
a__U31(X)	→	U31(X)	(28)
a__and(tt,X)	→	mark(X)	(5)
a__plus(N,0)	→	a__U11(a__isNat(N),N)	(10)
a__isNat(plus(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(7)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(20)
a__U11(X1,X2)	→	U11(X1,X2)	(25)
a__x(X1,X2)	→	x(X1,X2)	(30)
mark(U11(X1,X2))	→	a__U11(mark(X1),X2)	(14)
a__and(X1,X2)	→	and(X1,X2)	(31)
a__x(N,0)	→	a__U31(a__isNat(N))	(12)
mark(s(X))	→	s(mark(X))	(23)
mark(0)	→	0	(24)
a__plus(N,s(M))	→	a__U21(a__and(a__isNat(M),isNat(N)),M,N)	(11)
a__isNat(x(V1,V2))	→	a__and(a__isNat(V1),isNat(V2))	(9)
a__x(N,s(M))	→	a__U41(a__and(a__isNat(M),isNat(N)),M,N)	(13)
a__isNat(0)	→	tt	(6)
a__U41(X1,X2,X3)	→	U41(X1,X2,X3)	(29)
a__U21(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)

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

a__plus^#(N,s(M))

→

a__U21^#(a__and(a__isNat(M),isNat(N)),M,N)

(44)

could be deleted.

1.1.1.1.1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.