Certification Problem

Input (TPDB SRS_Standard/Zantema_06/12)

The rewrite relation of the following TRS is considered.

a(l(x1))	→	l(a(x1))	(1)
a(c(x1))	→	c(a(x1))	(2)
c(a(r(x1)))	→	r(a(x1))	(3)
l(r(a(a(x1))))	→	a(a(l(c(c(c(r(x1)))))))	(4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(l(x1))	→	a^#(x1)	(5)
a^#(l(x1))	→	l^#(a(x1))	(6)
a^#(c(x1))	→	a^#(x1)	(7)
a^#(c(x1))	→	c^#(a(x1))	(8)
c^#(a(r(x1)))	→	a^#(x1)	(9)
l^#(r(a(a(x1))))	→	c^#(r(x1))	(10)
l^#(r(a(a(x1))))	→	c^#(c(r(x1)))	(11)
l^#(r(a(a(x1))))	→	c^#(c(c(r(x1))))	(12)
l^#(r(a(a(x1))))	→	l^#(c(c(c(r(x1)))))	(13)
l^#(r(a(a(x1))))	→	a^#(l(c(c(c(r(x1))))))	(14)
l^#(r(a(a(x1))))	→	a^#(a(l(c(c(c(r(x1)))))))	(15)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(a(r(x1)))	→	a^#(x1)	(9)
a^#(c(x1))	→	c^#(a(x1))	(8)
a^#(c(x1))	→	a^#(x1)	(7)
a^#(l(x1))	→	l^#(a(x1))	(6)
l^#(r(a(a(x1))))	→	a^#(a(l(c(c(c(r(x1)))))))	(15)
a^#(l(x1))	→	a^#(x1)	(5)
l^#(r(a(a(x1))))	→	a^#(l(c(c(c(r(x1))))))	(14)

1.1.1 Reduction Pair Processor with Usable Rules

Using the

prec(c^#)	=	stat(c^#)	=	lex
prec(l^#)	=	stat(l^#)	=	lex
prec(a^#)	=	stat(a^#)	=	lex
prec(r)	=	stat(r)	=	lex
prec(c)	=	stat(c)	=	lex
prec(a)	=	stat(a)	=	lex
prec(l)	=	stat(l)	=	lex

π(c^#)	=	1
π(l^#)	=	1
π(a^#)	=	1
π(r)	=	[1]
π(c)	=	1
π(a)	=	1
π(l)	=	1

together with the usable rules

a(l(x1))	→	l(a(x1))	(1)
a(c(x1))	→	c(a(x1))	(2)
c(a(r(x1)))	→	r(a(x1))	(3)
l(r(a(a(x1))))	→	a(a(l(c(c(c(r(x1)))))))	(4)

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

c^#(a(r(x1)))

→

a^#(x1)

(9)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

l^#(r(a(a(x1))))	→	a^#(a(l(c(c(c(r(x1)))))))	(15)
a^#(c(x1))	→	a^#(x1)	(7)
a^#(l(x1))	→	l^#(a(x1))	(6)
l^#(r(a(a(x1))))	→	a^#(l(c(c(c(r(x1))))))	(14)
a^#(l(x1))	→	a^#(x1)	(5)

1.1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(l^#)	=	{ 1, 1 }
π(a^#)	=	{ 1 }
π(r)	=	{ 1, 1 }
π(c)	=	{ 1 }
π(a)	=	{ 1 }
π(l)	=	{ 1, 1 }

to remove the pairs:

a^#(l(x1))

→

a^#(x1)

(5)

1.1.1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(l^#)	=	{ 1 }
π(a^#)	=	{ 1 }
π(r)	=	{ 1 }
π(c)	=	{ 1 }
π(a)	=	{ 1, 1 }
π(l)	=	{ 1, 1 }

to remove the pairs:

l^#(r(a(a(x1))))

→

a^#(l(c(c(c(r(x1))))))

(14)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[a^#(x₁)]	=	0 · x₁ + 0
[a(x₁)]	=	0 · x₁ + 0
[r(x₁)]	=	0 · x₁ + 0
[l^#(x₁)]	=	-∞ · x₁ + 0
[l(x₁)]	=	-∞ · x₁ + 0
[c(x₁)]	=	2 · x₁ + 2

together with the usable rules

a(l(x1))	→	l(a(x1))	(1)
a(c(x1))	→	c(a(x1))	(2)
c(a(r(x1)))	→	r(a(x1))	(3)
l(r(a(a(x1))))	→	a(a(l(c(c(c(r(x1)))))))	(4)

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

a^#(c(x1))

→

a^#(x1)

(7)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[a^#(x₁)]	=	0 · x₁ + 0
[a(x₁)]	=	2 · x₁ + 0
[r(x₁)]	=	-∞ · x₁ + 5
[l^#(x₁)]	=	-2 · x₁ + 0
[l(x₁)]	=	0 · x₁ + 0
[c(x₁)]	=	-2 · x₁ + 0

together with the usable rules

a(l(x1))	→	l(a(x1))	(1)
a(c(x1))	→	c(a(x1))	(2)
c(a(r(x1)))	→	r(a(x1))	(3)
l(r(a(a(x1))))	→	a(a(l(c(c(c(r(x1)))))))	(4)

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

l^#(r(a(a(x1))))

→

a^#(a(l(c(c(c(r(x1)))))))

(15)

could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.