Certification Problem

Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/t012)

The rewrite relation of the following TRS is considered.

minus(minus(x))	→	x	(1)
minus(+(x,y))	→	*(minus(minus(minus(x))),minus(minus(minus(y))))	(2)
minus(*(x,y))	→	+(minus(minus(minus(x))),minus(minus(minus(y))))	(3)
f(minus(x))	→	minus(minus(minus(f(x))))	(4)

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.

minus^#(+(x,y))	→	minus^#(minus(x))	(5)
minus^#(+(x,y))	→	minus^#(y)	(6)
f^#(minus(x))	→	minus^#(minus(minus(f(x))))	(7)
minus^#(*(x,y))	→	minus^#(x)	(8)
minus^#(*(x,y))	→	minus^#(minus(y))	(9)
minus^#(*(x,y))	→	minus^#(y)	(10)
f^#(minus(x))	→	minus^#(minus(f(x)))	(11)
minus^#(*(x,y))	→	minus^#(minus(minus(x)))	(12)
minus^#(*(x,y))	→	minus^#(minus(minus(y)))	(13)
minus^#(+(x,y))	→	minus^#(minus(y))	(14)
f^#(minus(x))	→	f^#(x)	(15)
minus^#(+(x,y))	→	minus^#(x)	(16)
minus^#(+(x,y))	→	minus^#(minus(minus(x)))	(17)
minus^#(*(x,y))	→	minus^#(minus(x))	(18)
f^#(minus(x))	→	minus^#(f(x))	(19)
minus^#(+(x,y))	→	minus^#(minus(minus(y)))	(20)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair
f^#(minus(x)) → f^#(x) (15)

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

[minus(x₁)] = x₁ + 1

[f(x₁)] = 0

[f^#(x₁)] = x₁ + 0

[minus^#(x₁)] = 0

[+(x₁, x₂)] = 0

[*(x₁, x₂)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
f^#(minus(x)) → f^#(x) (15)
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^#(minus(minus(y)))	(20)
minus^#(*(x,y))	→	minus^#(y)	(10)
minus^#(*(x,y))	→	minus^#(minus(x))	(18)
minus^#(+(x,y))	→	minus^#(minus(minus(x)))	(17)
minus^#(+(x,y))	→	minus^#(x)	(16)
minus^#(*(x,y))	→	minus^#(minus(y))	(9)
minus^#(*(x,y))	→	minus^#(x)	(8)
minus^#(+(x,y))	→	minus^#(y)	(6)
minus^#(+(x,y))	→	minus^#(minus(y))	(14)
minus^#(*(x,y))	→	minus^#(minus(minus(y)))	(13)
minus^#(*(x,y))	→	minus^#(minus(minus(x)))	(12)
minus^#(+(x,y))	→	minus^#(minus(x))	(5)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[minus(x₁)]	=	x₁ + 0
[f(x₁)]	=	0
[f^#(x₁)]	=	0
[minus^#(x₁)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 21239
[*(x₁, x₂)]	=	x₁ + x₂ + 21239

together with the usable rules

minus(minus(x))	→	x	(1)
minus(*(x,y))	→	+(minus(minus(minus(x))),minus(minus(minus(y))))	(3)
minus(+(x,y))	→	*(minus(minus(minus(x))),minus(minus(minus(y))))	(2)

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

minus^#(+(x,y))	→	minus^#(minus(minus(y)))	(20)
minus^#(*(x,y))	→	minus^#(y)	(10)
minus^#(*(x,y))	→	minus^#(minus(x))	(18)
minus^#(+(x,y))	→	minus^#(minus(minus(x)))	(17)
minus^#(+(x,y))	→	minus^#(x)	(16)
minus^#(*(x,y))	→	minus^#(minus(y))	(9)
minus^#(*(x,y))	→	minus^#(x)	(8)
minus^#(+(x,y))	→	minus^#(y)	(6)
minus^#(+(x,y))	→	minus^#(minus(y))	(14)
minus^#(*(x,y))	→	minus^#(minus(minus(y)))	(13)
minus^#(*(x,y))	→	minus^#(minus(minus(x)))	(12)
minus^#(+(x,y))	→	minus^#(minus(x))	(5)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.