Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_nosorts_noand_GM)

The rewrite relation of the following TRS is considered.

a__U11(tt,M,N)	→	a__U12(tt,M,N)	(1)
a__U12(tt,M,N)	→	s(a__plus(mark(N),mark(M)))	(2)
a__U21(tt,M,N)	→	a__U22(tt,M,N)	(3)
a__U22(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
a__plus(N,0)	→	mark(N)	(5)
a__plus(N,s(M))	→	a__U11(tt,M,N)	(6)
a__x(N,0)	→	0	(7)
a__x(N,s(M))	→	a__U21(tt,M,N)	(8)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(9)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(10)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(11)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(12)
mark(U22(X1,X2,X3))	→	a__U22(mark(X1),X2,X3)	(13)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(14)
mark(tt)	→	tt	(15)
mark(s(X))	→	s(mark(X))	(16)
mark(0)	→	0	(17)
a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(18)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(19)
a__plus(X1,X2)	→	plus(X1,X2)	(20)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(21)
a__U22(X1,X2,X3)	→	U22(X1,X2,X3)	(22)
a__x(X1,X2)	→	x(X1,X2)	(23)

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__U12^#(tt,M,N)	→	mark^#(N)	(24)
mark^#(U22(X1,X2,X3))	→	mark^#(X1)	(25)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(26)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(27)
mark^#(plus(X1,X2))	→	mark^#(X2)	(28)
a__U22^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
a__x^#(N,s(M))	→	a__U21^#(tt,M,N)	(30)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(31)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(32)
a__U22^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(33)
mark^#(s(X))	→	mark^#(X)	(34)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(35)
mark^#(plus(X1,X2))	→	mark^#(X1)	(36)
mark^#(U22(X1,X2,X3))	→	a__U22^#(mark(X1),X2,X3)	(37)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)
a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(39)
a__U22^#(tt,M,N)	→	mark^#(M)	(40)
mark^#(U12(X1,X2,X3))	→	a__U12^#(mark(X1),X2,X3)	(41)
a__plus^#(N,0)	→	mark^#(N)	(42)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)
a__U21^#(tt,M,N)	→	a__U22^#(tt,M,N)	(43)
mark^#(x(X1,X2))	→	mark^#(X2)	(44)
a__U12^#(tt,M,N)	→	mark^#(M)	(45)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(46)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(47)
mark^#(x(X1,X2))	→	mark^#(X1)	(48)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(49)
mark^#(U12(X1,X2,X3))	→	mark^#(X1)	(50)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(U12(X1,X2,X3))	→	mark^#(X1)	(50)
a__U22^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(33)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(49)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(32)
mark^#(x(X1,X2))	→	mark^#(X1)	(48)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(31)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(47)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(46)
a__x^#(N,s(M))	→	a__U21^#(tt,M,N)	(30)
a__U12^#(tt,M,N)	→	mark^#(M)	(45)
a__U22^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
mark^#(x(X1,X2))	→	mark^#(X2)	(44)
a__U21^#(tt,M,N)	→	a__U22^#(tt,M,N)	(43)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)
mark^#(plus(X1,X2))	→	mark^#(X2)	(28)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(27)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(26)
a__plus^#(N,0)	→	mark^#(N)	(42)
mark^#(U12(X1,X2,X3))	→	a__U12^#(mark(X1),X2,X3)	(41)
mark^#(U22(X1,X2,X3))	→	mark^#(X1)	(25)
a__U22^#(tt,M,N)	→	mark^#(M)	(40)
a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(39)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)
mark^#(U22(X1,X2,X3))	→	a__U22^#(mark(X1),X2,X3)	(37)
mark^#(plus(X1,X2))	→	mark^#(X1)	(36)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(35)
a__U12^#(tt,M,N)	→	mark^#(N)	(24)
mark^#(s(X))	→	mark^#(X)	(34)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 18815, 0)
[U21(x₁, x₂, x₃)]	=	max(x₁ + 13215, x₂ + 1, x₃ + 18815, 0)
[U11(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 18815, x₃ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__x^#(x₁, x₂)]	=	max(x₁ + 75553, x₂ + 56739, 0)
[a__U22(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 1, x₃ + 18815, 0)
[a__U11^#(x₁, x₂, x₃)]	=	max(x₁ + 75552, x₂ + 75553, x₃ + 56738, 0)
[a__x(x₁, x₂)]	=	max(x₁ + 18815, x₂ + 1, 0)
[U12(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 18815, x₃ + 0, 0)
[x(x₁, x₂)]	=	max(x₁ + 18815, x₂ + 1, 0)
[a__U12^#(x₁, x₂, x₃)]	=	max(x₁ + 75551, x₂ + 75553, x₃ + 56738, 0)
[a__U21^#(x₁, x₂, x₃)]	=	max(x₂ + 56739, x₃ + 75553, 0)
[a__plus^#(x₁, x₂)]	=	max(x₁ + 56738, x₂ + 75553, 0)
[mark^#(x₁)]	=	x₁ + 56738
[0]	=	4684
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 13215, x₂ + 1, x₃ + 18815, 0)
[mark(x₁)]	=	x₁ + 0
[a__U11(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 18815, x₃ + 0, 0)
[a__U12(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 18815, x₃ + 0, 0)
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 18815, 0)
[a__U22^#(x₁, x₂, x₃)]	=	max(x₁ + 31964, x₂ + 56739, x₃ + 75553, 0)
[tt]	=	1
[U22(x₁, x₂, x₃)]	=	max(x₁ + 18814, x₂ + 1, x₃ + 18815, 0)

together with the usable rules

a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(18)
a__U22(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(tt)	→	tt	(15)
a__x(N,s(M))	→	a__U21(tt,M,N)	(8)
a__U11(tt,M,N)	→	a__U12(tt,M,N)	(1)
a__U21(tt,M,N)	→	a__U22(tt,M,N)	(3)
mark(s(X))	→	s(mark(X))	(16)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(21)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(19)
mark(0)	→	0	(17)
a__U22(X1,X2,X3)	→	U22(X1,X2,X3)	(22)
a__plus(N,0)	→	mark(N)	(5)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(10)
a__x(N,0)	→	0	(7)
a__plus(X1,X2)	→	plus(X1,X2)	(20)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(14)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(12)
a__x(X1,X2)	→	x(X1,X2)	(23)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(11)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(9)
mark(U22(X1,X2,X3))	→	a__U22(mark(X1),X2,X3)	(13)
a__plus(N,s(M))	→	a__U11(tt,M,N)	(6)
a__U12(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^#(U12(X1,X2,X3))	→	mark^#(X1)	(50)
mark^#(x(X1,X2))	→	mark^#(X1)	(48)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(47)
a__U12^#(tt,M,N)	→	mark^#(M)	(45)
mark^#(x(X1,X2))	→	mark^#(X2)	(44)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)
mark^#(plus(X1,X2))	→	mark^#(X2)	(28)
mark^#(U21(X1,X2,X3))	→	mark^#(X1)	(26)
mark^#(U22(X1,X2,X3))	→	mark^#(X1)	(25)
a__U22^#(tt,M,N)	→	mark^#(M)	(40)
a__U22^#(tt,M,N)	→	mark^#(N)	(38)

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

a__U22^#(tt,M,N)	→	a__x^#(mark(N),mark(M))	(33)
a__U22^#(tt,M,N)	→	a__plus^#(a__x(mark(N),mark(M)),mark(N))	(29)
a__x^#(N,s(M))	→	a__U21^#(tt,M,N)	(30)
a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(39)
a__U21^#(tt,M,N)	→	a__U22^#(tt,M,N)	(43)
mark^#(s(X))	→	mark^#(X)	(34)
a__plus^#(N,0)	→	mark^#(N)	(42)
mark^#(U12(X1,X2,X3))	→	a__U12^#(mark(X1),X2,X3)	(41)
mark^#(x(X1,X2))	→	a__x^#(mark(X1),mark(X2))	(32)
mark^#(U21(X1,X2,X3))	→	a__U21^#(mark(X1),X2,X3)	(31)
mark^#(plus(X1,X2))	→	mark^#(X1)	(36)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(27)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(35)
mark^#(U22(X1,X2,X3))	→	a__U22^#(mark(X1),X2,X3)	(37)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(49)
a__U12^#(tt,M,N)	→	mark^#(N)	(24)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(46)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__U11^#)	=	3
π(a__U12^#)	=	3
π(a__plus^#)	=	1
π(mark^#)	=	1
π(mark)	=	1

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

prec(a__plus)	=	3	status(a__plus)	=	[1, 2]	list-extension(a__plus)	=	Lex
prec(U21)	=	4	status(U21)	=	[3, 2, 1]	list-extension(U21)	=	Lex
prec(U11)	=	3	status(U11)	=	[3, 2, 1]	list-extension(U11)	=	Lex
prec(s)	=	0	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__x^#)	=	4	status(a__x^#)	=	[1, 2]	list-extension(a__x^#)	=	Lex
prec(a__U22)	=	4	status(a__U22)	=	[3, 2, 1]	list-extension(a__U22)	=	Lex
prec(a__x)	=	4	status(a__x)	=	[1, 2]	list-extension(a__x)	=	Lex
prec(U12)	=	3	status(U12)	=	[3, 2, 1]	list-extension(U12)	=	Lex
prec(x)	=	4	status(x)	=	[1, 2]	list-extension(x)	=	Lex
prec(a__U21^#)	=	4	status(a__U21^#)	=	[3, 2, 1]	list-extension(a__U21^#)	=	Lex
prec(0)	=	2	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__U21)	=	4	status(a__U21)	=	[3, 2, 1]	list-extension(a__U21)	=	Lex
prec(a__U11)	=	3	status(a__U11)	=	[3, 2, 1]	list-extension(a__U11)	=	Lex
prec(a__U12)	=	3	status(a__U12)	=	[3, 2, 1]	list-extension(a__U12)	=	Lex
prec(plus)	=	3	status(plus)	=	[1, 2]	list-extension(plus)	=	Lex
prec(a__U22^#)	=	4	status(a__U22^#)	=	[3, 2]	list-extension(a__U22^#)	=	Lex
prec(tt)	=	2	status(tt)	=	[]	list-extension(tt)	=	Lex
prec(U22)	=	4	status(U22)	=	[3, 2, 1]	list-extension(U22)	=	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, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U22(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U12(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(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[0]	=	0
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U11(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U12(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U22^#(x₁, x₂, x₃)]	=	max(x₂ + 0, x₃ + 0, 0)
[tt]	=	0
[U22(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)

together with the usable rules

a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(18)
a__U22(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(tt)	→	tt	(15)
a__x(N,s(M))	→	a__U21(tt,M,N)	(8)
a__U11(tt,M,N)	→	a__U12(tt,M,N)	(1)
a__U21(tt,M,N)	→	a__U22(tt,M,N)	(3)
mark(s(X))	→	s(mark(X))	(16)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(21)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(19)
mark(0)	→	0	(17)
a__U22(X1,X2,X3)	→	U22(X1,X2,X3)	(22)
a__plus(N,0)	→	mark(N)	(5)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(10)
a__x(N,0)	→	0	(7)
a__plus(X1,X2)	→	plus(X1,X2)	(20)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(14)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(12)
a__x(X1,X2)	→	x(X1,X2)	(23)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(11)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(9)
mark(U22(X1,X2,X3))	→	a__U22(mark(X1),X2,X3)	(13)
a__plus(N,s(M))	→	a__U11(tt,M,N)	(6)
a__U12(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__x^#(N,s(M))	→	a__U21^#(tt,M,N)	(30)
a__U21^#(tt,M,N)	→	a__U22^#(tt,M,N)	(43)
mark^#(s(X))	→	mark^#(X)	(34)
mark^#(U12(X1,X2,X3))	→	a__U12^#(mark(X1),X2,X3)	(41)
mark^#(plus(X1,X2))	→	mark^#(X1)	(36)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(27)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(35)
mark^#(U22(X1,X2,X3))	→	a__U22^#(mark(X1),X2,X3)	(37)

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

a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(39)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(49)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(46)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a__U11^#)	=	2
π(a__U12^#)	=	2
π(a__plus^#)	=	2
π(mark^#)	=	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)	=	5	status(U21)	=	[3, 2, 1]	list-extension(U21)	=	Lex
prec(U11)	=	4	status(U11)	=	[3, 2, 1]	list-extension(U11)	=	Lex
prec(s)	=	3	status(s)	=	[1]	list-extension(s)	=	Lex
prec(a__x^#)	=	4	status(a__x^#)	=	[1, 2]	list-extension(a__x^#)	=	Lex
prec(a__U22)	=	5	status(a__U22)	=	[3, 2, 1]	list-extension(a__U22)	=	Lex
prec(a__x)	=	5	status(a__x)	=	[1, 2]	list-extension(a__x)	=	Lex
prec(U12)	=	4	status(U12)	=	[3, 2, 1]	list-extension(U12)	=	Lex
prec(x)	=	5	status(x)	=	[1, 2]	list-extension(x)	=	Lex
prec(a__U21^#)	=	4	status(a__U21^#)	=	[3, 2, 1]	list-extension(a__U21^#)	=	Lex
prec(0)	=	1	status(0)	=	[]	list-extension(0)	=	Lex
prec(a__U21)	=	5	status(a__U21)	=	[3, 2, 1]	list-extension(a__U21)	=	Lex
prec(a__U11)	=	4	status(a__U11)	=	[3, 2, 1]	list-extension(a__U11)	=	Lex
prec(a__U12)	=	4	status(a__U12)	=	[3, 2, 1]	list-extension(a__U12)	=	Lex
prec(plus)	=	4	status(plus)	=	[1, 2]	list-extension(plus)	=	Lex
prec(a__U22^#)	=	4	status(a__U22^#)	=	[3, 2]	list-extension(a__U22^#)	=	Lex
prec(tt)	=	2	status(tt)	=	[]	list-extension(tt)	=	Lex
prec(U22)	=	5	status(U22)	=	[3, 2, 1]	list-extension(U22)	=	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, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[s(x₁)]	=	x₁ + 0
[a__x^#(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U22(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__x(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[U12(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(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[0]	=	0
[a__U21(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U11(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[a__U12(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)
[plus(x₁, x₂)]	=	max(x₁ + 0, x₂ + 0, 0)
[a__U22^#(x₁, x₂, x₃)]	=	max(x₂ + 0, x₃ + 0, 0)
[tt]	=	0
[U22(x₁, x₂, x₃)]	=	max(x₁ + 0, x₂ + 0, x₃ + 0, 0)

together with the usable rules

a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(18)
a__U22(tt,M,N)	→	a__plus(a__x(mark(N),mark(M)),mark(N))	(4)
mark(tt)	→	tt	(15)
a__x(N,s(M))	→	a__U21(tt,M,N)	(8)
a__U11(tt,M,N)	→	a__U12(tt,M,N)	(1)
a__U21(tt,M,N)	→	a__U22(tt,M,N)	(3)
mark(s(X))	→	s(mark(X))	(16)
a__U21(X1,X2,X3)	→	U21(X1,X2,X3)	(21)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(19)
mark(0)	→	0	(17)
a__U22(X1,X2,X3)	→	U22(X1,X2,X3)	(22)
a__plus(N,0)	→	mark(N)	(5)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(10)
a__x(N,0)	→	0	(7)
a__plus(X1,X2)	→	plus(X1,X2)	(20)
mark(x(X1,X2))	→	a__x(mark(X1),mark(X2))	(14)
mark(U21(X1,X2,X3))	→	a__U21(mark(X1),X2,X3)	(12)
a__x(X1,X2)	→	x(X1,X2)	(23)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(11)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(9)
mark(U22(X1,X2,X3))	→	a__U22(mark(X1),X2,X3)	(13)
a__plus(N,s(M))	→	a__U11(tt,M,N)	(6)
a__U12(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__U11^#(tt,M,N)

(49)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.