Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_nosorts-noand_FR)

The rewrite relation of the following TRS is considered.

U11(tt,M,N)	→	U12(tt,activate(M),activate(N))	(1)
U12(tt,M,N)	→	s(plus(activate(N),activate(M)))	(2)
U21(tt,M,N)	→	U22(tt,activate(M),activate(N))	(3)
U22(tt,M,N)	→	plus(x(activate(N),activate(M)),activate(N))	(4)
plus(N,0)	→	N	(5)
plus(N,s(M))	→	U11(tt,M,N)	(6)
x(N,0)	→	0	(7)
x(N,s(M))	→	U21(tt,M,N)	(8)
activate(X)	→	X	(9)

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.

U12^#(tt,M,N)	→	activate^#(N)	(10)
U21^#(tt,M,N)	→	activate^#(M)	(11)
U22^#(tt,M,N)	→	x^#(activate(N),activate(M))	(12)
U11^#(tt,M,N)	→	activate^#(M)	(13)
x^#(N,s(M))	→	U21^#(tt,M,N)	(14)
U22^#(tt,M,N)	→	plus^#(x(activate(N),activate(M)),activate(N))	(15)
U22^#(tt,M,N)	→	activate^#(N)	(16)
U21^#(tt,M,N)	→	activate^#(N)	(17)
U11^#(tt,M,N)	→	activate^#(N)	(18)
U21^#(tt,M,N)	→	U22^#(tt,activate(M),activate(N))	(19)
U22^#(tt,M,N)	→	activate^#(N)	(16)
U12^#(tt,M,N)	→	activate^#(M)	(20)
U12^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(21)
U11^#(tt,M,N)	→	U12^#(tt,activate(M),activate(N))	(22)
U22^#(tt,M,N)	→	activate^#(M)	(23)
plus^#(N,s(M))	→	U11^#(tt,M,N)	(24)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

x^#(N,s(M))	→	U21^#(tt,M,N)	(14)
U22^#(tt,M,N)	→	x^#(activate(N),activate(M))	(12)
U21^#(tt,M,N)	→	U22^#(tt,activate(M),activate(N))	(19)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[U21(x₁, x₂, x₃)]	=	0
[U11(x₁, x₂, x₃)]	=	0
[s(x₁)]	=	x₁ + 7
[activate(x₁)]	=	x₁ + 1
[plus^#(x₁, x₂)]	=	0
[activate^#(x₁)]	=	0
[U12(x₁, x₂, x₃)]	=	0
[x(x₁, x₂)]	=	0
[U12^#(x₁, x₂, x₃)]	=	0
[0]	=	0
[x^#(x₁, x₂)]	=	x₁ + x₂ + 0
[plus(x₁, x₂)]	=	0
[U11^#(x₁, x₂, x₃)]	=	0
[U21^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 4
[U22^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 1
[tt]	=	2
[U22(x₁, x₂, x₃)]	=	0

together with the usable rule

activate(X)

→

X

(9)

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

x^#(N,s(M))	→	U21^#(tt,M,N)	(14)
U22^#(tt,M,N)	→	x^#(activate(N),activate(M))	(12)
U21^#(tt,M,N)	→	U22^#(tt,activate(M),activate(N))	(19)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

plus^#(N,s(M))	→	U11^#(tt,M,N)	(24)
U11^#(tt,M,N)	→	U12^#(tt,activate(M),activate(N))	(22)
U12^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(21)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[U21(x₁, x₂, x₃)]	=	0
[U11(x₁, x₂, x₃)]	=	0
[s(x₁)]	=	x₁ + 7
[activate(x₁)]	=	x₁ + 1
[plus^#(x₁, x₂)]	=	x₁ + x₂ + 0
[activate^#(x₁)]	=	0
[U12(x₁, x₂, x₃)]	=	0
[x(x₁, x₂)]	=	0
[U12^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[0]	=	0
[x^#(x₁, x₂)]	=	x₁ + x₂ + 0
[plus(x₁, x₂)]	=	0
[U11^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 3
[U21^#(x₁, x₂, x₃)]	=	4
[U22^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 1
[tt]	=	3
[U22(x₁, x₂, x₃)]	=	0

together with the usable rule

activate(X)

→

X

(9)

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

plus^#(N,s(M))	→	U11^#(tt,M,N)	(24)
U11^#(tt,M,N)	→	U12^#(tt,activate(M),activate(N))	(22)
U12^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(21)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.