Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_nosorts_GM)

The rewrite relation of the following TRS is considered.

a__and(tt,X)	→	mark(X)	(1)
a__plus(N,0)	→	mark(N)	(2)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
mark(and(X1,X2))	→	a__and(mark(X1),X2)	(4)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(5)
mark(tt)	→	tt	(6)
mark(0)	→	0	(7)
mark(s(X))	→	s(mark(X))	(8)
a__and(X1,X2)	→	and(X1,X2)	(9)
a__plus(X1,X2)	→	plus(X1,X2)	(10)

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.

mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(11)
a__plus^#(N,s(M))	→	mark^#(N)	(12)
mark^#(s(X))	→	mark^#(X)	(13)
mark^#(and(X1,X2))	→	mark^#(X1)	(14)
a__plus^#(N,s(M))	→	mark^#(M)	(15)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(16)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
mark^#(plus(X1,X2))	→	mark^#(X1)	(18)
a__plus^#(N,0)	→	mark^#(N)	(19)
a__and^#(tt,X)	→	mark^#(X)	(20)
mark^#(plus(X1,X2))	→	mark^#(X2)	(21)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(plus(X1,X2))	→	mark^#(X2)	(21)
a__and^#(tt,X)	→	mark^#(X)	(20)
a__plus^#(N,0)	→	mark^#(N)	(19)
mark^#(plus(X1,X2))	→	mark^#(X1)	(18)
mark^#(and(X1,X2))	→	mark^#(X1)	(14)
mark^#(s(X))	→	mark^#(X)	(13)
a__plus^#(N,s(M))	→	mark^#(N)	(12)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(16)
a__plus^#(N,s(M))	→	mark^#(M)	(15)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(11)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__plus(x₁, x₂)]	=	x₁ + x₂ + 1
[s(x₁)]	=	x₁ + 1
[and(x₁, x₂)]	=	x₁ + x₂ + 1
[a__plus^#(x₁, x₂)]	=	x₁ + x₂ + 0
[mark^#(x₁)]	=	x₁ + 0
[0]	=	8366
[a__and^#(x₁, x₂)]	=	x₁ + x₂ + 0
[mark(x₁)]	=	x₁ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[tt]	=	1
[a__and(x₁, x₂)]	=	x₁ + x₂ + 1

together with the usable rules

mark(and(X1,X2))	→	a__and(mark(X1),X2)	(4)
mark(s(X))	→	s(mark(X))	(8)
a__and(tt,X)	→	mark(X)	(1)
a__plus(N,s(M))	→	s(a__plus(mark(N),mark(M)))	(3)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(5)
a__plus(X1,X2)	→	plus(X1,X2)	(10)
mark(0)	→	0	(7)
a__and(X1,X2)	→	and(X1,X2)	(9)
mark(tt)	→	tt	(6)
a__plus(N,0)	→	mark(N)	(2)

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

mark^#(plus(X1,X2))	→	mark^#(X2)	(21)
a__and^#(tt,X)	→	mark^#(X)	(20)
a__plus^#(N,0)	→	mark^#(N)	(19)
mark^#(plus(X1,X2))	→	mark^#(X1)	(18)
mark^#(and(X1,X2))	→	mark^#(X1)	(14)
mark^#(s(X))	→	mark^#(X)	(13)
a__plus^#(N,s(M))	→	mark^#(N)	(12)
a__plus^#(N,s(M))	→	a__plus^#(mark(N),mark(M))	(17)
mark^#(and(X1,X2))	→	a__and^#(mark(X1),X2)	(16)
a__plus^#(N,s(M))	→	mark^#(M)	(15)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(11)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.