YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(x0))	→	b(c(c(c(x0))))
b(c(x0))	→	d(d(d(d(x0))))
a(x0)	→	d(c(d(x0)))
b(b(x0))	→	c(c(c(x0)))
c(c(x0))	→	d(d(d(x0)))
c(d(d(x0)))	→	a(x0)

Proof

1 Rule Removal

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

[c(x₁)]	=	3 · x₁ + -∞
[d(x₁)]	=	2 · x₁ + -∞
[a(x₁)]	=	7 · x₁ + -∞
[b(x₁)]	=	5 · x₁ + -∞

the rules

a(a(x0))	→	b(c(c(c(x0))))
b(c(x0))	→	d(d(d(d(x0))))
a(x0)	→	d(c(d(x0)))
c(c(x0))	→	d(d(d(x0)))
c(d(d(x0)))	→	a(x0)

remain.

1.1 Rule Removal

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

[c(x₁)]	=	4 · x₁ + -∞
[d(x₁)]	=	2 · x₁ + -∞
[a(x₁)]	=	8 · x₁ + -∞
[b(x₁)]	=	4 · x₁ + -∞

the rules

a(a(x0))	→	b(c(c(c(x0))))
b(c(x0))	→	d(d(d(d(x0))))
a(x0)	→	d(c(d(x0)))
c(d(d(x0)))	→	a(x0)

remain.

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(x0))	→	c^#(x0)
a^#(a(x0))	→	c^#(c(x0))
a^#(a(x0))	→	c^#(c(c(x0)))
a^#(a(x0))	→	b^#(c(c(c(x0))))
a^#(x0)	→	c^#(d(x0))
c^#(d(d(x0)))	→	a^#(x0)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

c^#(d(d(x0)))	→	a^#(x0)
a^#(a(x0))	→	c^#(x0)
a^#(a(x0))	→	c^#(c(x0))
a^#(a(x0))	→	c^#(c(c(x0)))
a^#(x0)	→	c^#(d(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)]	=	4 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[d(x₁)]	=	2 · x₁ + -∞
[c^#(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	4 · x₁ + -∞
[b(x₁)]	=	8 · x₁ + -∞

together with the usable rules

a(a(x0))	→	b(c(c(c(x0))))
b(c(x0))	→	d(d(d(d(x0))))
a(x0)	→	d(c(d(x0)))
c(d(d(x0)))	→	a(x0)

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

c^#(d(d(x0)))

→

a^#(x0)

remains.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.