Certification Problem

Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/t008)

The rewrite relation of the following TRS is considered.

half(0)	→	0	(1)
half(s(0))	→	0	(2)
half(s(s(x)))	→	s(half(x))	(3)
s(log(0))	→	s(0)	(4)
log(s(x))	→	s(log(half(s(x))))	(5)

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.

log^#(s(x))	→	log^#(half(s(x)))	(6)
s^#(log(0))	→	s^#(0)	(7)
half^#(s(s(x)))	→	half^#(x)	(8)
log^#(s(x))	→	half^#(s(x))	(9)
log^#(s(x))	→	s^#(log(half(s(x))))	(10)
half^#(s(s(x)))	→	s^#(half(x))	(11)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

log^#(s(x))

→

log^#(half(s(x)))

(6)

1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

in combination with the following Weighted Path Order with the following precedence and status

prec(s)	=	1		status(s)	=	[]		list-extension(s)	=	Lex
prec(log^#)	=	0		status(log^#)	=	[1]		list-extension(log^#)	=	Lex
prec(half^#)	=	0		status(half^#)	=	[]		list-extension(half^#)	=	Lex
prec(half)	=	0		status(half)	=	[]		list-extension(half)	=	Lex
prec(log)	=	0		status(log)	=	[]		list-extension(log)	=	Lex
prec(0)	=	0		status(0)	=	[]		list-extension(0)	=	Lex
prec(s^#)	=	0		status(s^#)	=	[]		list-extension(s^#)	=	Lex

and the following Max-polynomial interpretation

[s(x₁)]	=	x₁ + 8366
[log^#(x₁)]	=	x₁ + 1
[half^#(x₁)]	=	1
[half(x₁)]	=	x₁ + 0
[log(x₁)]	=	1
[0]	=	1
[s^#(x₁)]	=	1

together with the usable rules

s(log(0))	→	s(0)	(4)
half(0)	→	0	(1)
half(s(s(x)))	→	s(half(x))	(3)
half(s(0))	→	0	(2)

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

log^#(s(x))

→

log^#(half(s(x)))

(6)

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
half^#(s(s(x))) → half^#(x) (8)

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

[s(x₁)] = x₁ + 1

[log^#(x₁)] = 0

[half^#(x₁)] = x₁ + 0

[half(x₁)] = 11293

[log(x₁)] = x₁ + 0

[0] = 1

[s^#(x₁)] = 0

together with the usable rules

half(0) → 0 (1)

half(s(0)) → 0 (2)

(w.r.t. the implicit argument filter of the reduction pair), the pair
half^#(s(s(x))) → half^#(x) (8)
could be deleted.
1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.