YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(l(x0))	→	l(a(x0))
a(c(x0))	→	c(a(x0))
c(a(r(x0)))	→	r(a(x0))
l(r(a(a(x0))))	→	a(a(l(c(c(c(r(x0)))))))

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(l(x0))	→	a^#(x0)
a^#(l(x0))	→	l^#(a(x0))
a^#(c(x0))	→	a^#(x0)
a^#(c(x0))	→	c^#(a(x0))
c^#(a(r(x0)))	→	a^#(x0)
l^#(r(a(a(x0))))	→	c^#(r(x0))
l^#(r(a(a(x0))))	→	c^#(c(r(x0)))
l^#(r(a(a(x0))))	→	c^#(c(c(r(x0))))
l^#(r(a(a(x0))))	→	l^#(c(c(c(r(x0)))))
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))
l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

c^#(a(r(x0)))	→	a^#(x0)
a^#(c(x0))	→	c^#(a(x0))
a^#(c(x0))	→	a^#(x0)
a^#(l(x0))	→	l^#(a(x0))
l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(l(x0))	→	a^#(x0)
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))

1.1.1 Reduction Pair Processor with Usable Rules

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

[a^#(x₁)]	=	8 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[r(x₁)]	=	1 · x₁ + -∞
[l^#(x₁)]	=	8 · x₁ + -∞
[l(x₁)]	=	0 · x₁ + -∞
[c^#(x₁)]	=	7 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞

together with the usable rules

a(l(x0))	→	l(a(x0))
a(c(x0))	→	c(a(x0))
c(a(r(x0)))	→	r(a(x0))
l(r(a(a(x0))))	→	a(a(l(c(c(c(r(x0)))))))

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

c^#(a(r(x0)))	→	a^#(x0)
a^#(c(x0))	→	a^#(x0)
a^#(l(x0))	→	l^#(a(x0))
l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(l(x0))	→	a^#(x0)
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))

remain.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

a^#(l(x0))	→	a^#(x0)
a^#(l(x0))	→	l^#(a(x0))
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))
l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(c(x0))	→	a^#(x0)

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 + -∞
[r(x₁)]	=	0 · x₁ + -∞
[l^#(x₁)]	=	15 · x₁ + 8
[l(x₁)]	=	15 · x₁ + 8
[c(x₁)]	=	0 · x₁ + -∞

together with the usable rules

a(l(x0))	→	l(a(x0))
a(c(x0))	→	c(a(x0))
c(a(r(x0)))	→	r(a(x0))
l(r(a(a(x0))))	→	a(a(l(c(c(c(r(x0)))))))

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

a^#(l(x0))	→	l^#(a(x0))
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))
l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(c(x0))	→	a^#(x0)

remain.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

a^#(c(x0))

→

a^#(x0)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers

[a^#(x₁)]

0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[c(x₁)]

1	0	0	0
0	0	0	0
1	0	0	0
1	0	0	0

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

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

1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pairs

l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(l(x0))	→	l^#(a(x0))
l^#(r(a(a(x0))))	→	a^#(l(c(c(c(r(x0))))))

1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[a^#(x₁)]	=	5 · x₁ + -∞
[a(x₁)]	=	9 · x₁ + -∞
[r(x₁)]	=	0 · x₁ + -∞
[l^#(x₁)]	=	0 · x₁ + -∞
[l(x₁)]	=	4 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞

together with the usable rules

a(l(x0))	→	l(a(x0))
a(c(x0))	→	c(a(x0))
c(a(r(x0)))	→	r(a(x0))
l(r(a(a(x0))))	→	a(a(l(c(c(c(r(x0)))))))

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

l^#(r(a(a(x0))))	→	a^#(a(l(c(c(c(r(x0)))))))
a^#(l(x0))	→	l^#(a(x0))

remain.

1.1.1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the rationals with delta = 1/64

[a^#(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	2 · x₁ + 0
[r(x₁)]	=	0 · x₁ + 1/2
[l^#(x₁)]	=	1 · x₁ + 0
[l(x₁)]	=	2 · x₁ + 0
[c(x₁)]	=	1/2 · x₁ + 0

together with the usable rules

a(l(x0))	→	l(a(x0))
a(c(x0))	→	c(a(x0))
c(a(r(x0)))	→	r(a(x0))
l(r(a(a(x0))))	→	a(a(l(c(c(c(r(x0)))))))

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

a^#(l(x0))

→

l^#(a(x0))

remains.

1.1.1.1.1.1.2.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.