Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_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__plus(N,0)	→	mark(N)	(3)
a__plus(N,s(M))	→	a__U11(tt,M,N)	(4)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(5)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(6)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
mark(tt)	→	tt	(8)
mark(s(X))	→	s(mark(X))	(9)
mark(0)	→	0	(10)
a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(11)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(12)
a__plus(X1,X2)	→	plus(X1,X2)	(13)

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)	(14)
mark^#(s(X))	→	mark^#(X)	(15)
a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(16)
mark^#(plus(X1,X2))	→	mark^#(X2)	(17)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(18)
a__plus^#(N,0)	→	mark^#(N)	(19)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(20)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(21)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(22)
mark^#(U12(X1,X2,X3))	→	mark^#(X1)	(23)
a__U12^#(tt,M,N)	→	mark^#(M)	(24)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(25)
mark^#(plus(X1,X2))	→	mark^#(X1)	(26)
mark^#(U12(X1,X2,X3))	→	a__U12^#(mark(X1),X2,X3)	(27)

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))	→	a__U12^#(mark(X1),X2,X3)	(27)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(20)
a__plus^#(N,0)	→	mark^#(N)	(19)
mark^#(plus(X1,X2))	→	mark^#(X1)	(26)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(25)
a__U12^#(tt,M,N)	→	mark^#(M)	(24)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(18)
mark^#(plus(X1,X2))	→	mark^#(X2)	(17)
a__U11^#(tt,M,N)	→	a__U12^#(tt,M,N)	(16)
mark^#(s(X))	→	mark^#(X)	(15)
mark^#(U12(X1,X2,X3))	→	mark^#(X1)	(23)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(22)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(21)
a__U12^#(tt,M,N)	→	mark^#(N)	(14)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	x₁ + x₂ + 1324
[U11(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 21658
[s(x₁)]	=	x₁ + 64698
[a__U11^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 3
[U12(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 21658
[a__U12^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 3
[a__plus^#(x₁, x₂)]	=	x₁ + x₂ + 2
[mark^#(x₁)]	=	x₁ + 3
[0]	=	2
[mark(x₁)]	=	x₁ + 0
[a__U11(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 21658
[a__U12(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 21658
[plus(x₁, x₂)]	=	x₁ + x₂ + 1324
[tt]	=	44364

together with the usable rules

a__plus(N,s(M))	→	a__U11(tt,M,N)	(4)
mark(tt)	→	tt	(8)
a__U11(tt,M,N)	→	a__U12(tt,M,N)	(1)
a__plus(N,0)	→	mark(N)	(3)
mark(U11(X1,X2,X3))	→	a__U11(mark(X1),X2,X3)	(5)
mark(0)	→	0	(10)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(7)
a__U12(X1,X2,X3)	→	U12(X1,X2,X3)	(12)
a__U11(X1,X2,X3)	→	U11(X1,X2,X3)	(11)
mark(s(X))	→	s(mark(X))	(9)
a__plus(X1,X2)	→	plus(X1,X2)	(13)
mark(U12(X1,X2,X3))	→	a__U12(mark(X1),X2,X3)	(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))	→	a__U12^#(mark(X1),X2,X3)	(27)
a__plus^#(N,s(M))	→	a__U11^#(tt,M,N)	(20)
a__plus^#(N,0)	→	mark^#(N)	(19)
mark^#(plus(X1,X2))	→	mark^#(X1)	(26)
a__U12^#(tt,M,N)	→	a__plus^#(mark(N),mark(M))	(25)
mark^#(U11(X1,X2,X3))	→	mark^#(X1)	(18)
mark^#(plus(X1,X2))	→	mark^#(X2)	(17)
mark^#(s(X))	→	mark^#(X)	(15)
mark^#(U12(X1,X2,X3))	→	mark^#(X1)	(23)
mark^#(U11(X1,X2,X3))	→	a__U11^#(mark(X1),X2,X3)	(22)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(21)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.