Certification Problem

Input (TPDB TRS_Standard/SK90/4.03)

The rewrite relation of the following TRS is considered.

+(x,0)	→	x	(1)
+(minus(x),x)	→	0	(2)
minus(0)	→	0	(3)
minus(minus(x))	→	x	(4)
minus(+(x,y))	→	+(minus(y),minus(x))	(5)
*(x,1)	→	x	(6)
*(x,0)	→	0	(7)
*(x,+(y,z))	→	+((x,y),(x,z))	(8)
*(x,minus(y))	→	minus(*(x,y))	(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.

*^#(x,minus(y))	→	*^#(x,y)	(10)
*^#(x,+(y,z))	→	+^#((x,y),(x,z))	(11)
*^#(x,+(y,z))	→	*^#(x,y)	(12)
minus^#(+(x,y))	→	minus^#(x)	(13)
minus^#(+(x,y))	→	+^#(minus(y),minus(x))	(14)
*^#(x,minus(y))	→	minus^#(*(x,y))	(15)
*^#(x,+(y,z))	→	*^#(x,z)	(16)
minus^#(+(x,y))	→	minus^#(y)	(17)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

*^#(x,+(y,z))	→	*^#(x,z)	(16)
*^#(x,+(y,z))	→	*^#(x,y)	(12)
*^#(x,minus(y))	→	*^#(x,y)	(10)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	0
[minus(x₁)]	=	x₁ + 1
[*^#(x₁, x₂)]	=	x₂ + 0
[0]	=	0
[minus^#(x₁)]	=	0
[+(x₁, x₂)]	=	x₁ + x₂ + 1
[+^#(x₁, x₂)]	=	0
[*(x₁, x₂)]	=	0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

*^#(x,+(y,z))	→	*^#(x,z)	(16)
*^#(x,+(y,z))	→	*^#(x,y)	(12)
*^#(x,minus(y))	→	*^#(x,y)	(10)

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

minus^#(+(x,y))	→	minus^#(y)	(17)
minus^#(+(x,y))	→	minus^#(x)	(13)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	0
[minus(x₁)]	=	1
[*^#(x₁, x₂)]	=	0
[0]	=	0
[minus^#(x₁)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 1
[+^#(x₁, x₂)]	=	0
[*(x₁, x₂)]	=	0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

minus^#(+(x,y))	→	minus^#(y)	(17)
minus^#(+(x,y))	→	minus^#(x)	(13)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.