Certification Problem

Input (TPDB TRS_Standard/SK90/4.19)

The rewrite relation of the following TRS is considered.

-(-(neg(x),neg(x)),-(neg(y),neg(y)))

→

-(-(x,y),-(x,y))

(1)

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.

-^#(-(neg(x),neg(x)),-(neg(y),neg(y)))	→	-^#(x,y)	(2)
-^#(-(neg(x),neg(x)),-(neg(y),neg(y)))	→	-^#(x,y)	(2)
-^#(-(neg(x),neg(x)),-(neg(y),neg(y)))	→	-^#(-(x,y),-(x,y))	(3)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(-(x,y),-(x,y)) (3)

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(x,y) (2)

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(x,y) (2)

1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[-(x₁, x₂)] = x₁ + x₂ + 1

[neg(x₁)] = x₁ + 36466

[-^#(x₁, x₂)] = x₁ + x₂ + 0

together with the usable rule
-(-(neg(x),neg(x)),-(neg(y),neg(y))) → -(-(x,y),-(x,y)) (1)
(w.r.t. the implicit argument filter of the reduction pair), the pairs

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(-(x,y),-(x,y)) (3)

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(x,y) (2)

-^#(-(neg(x),neg(x)),-(neg(y),neg(y))) → -^#(x,y) (2)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.