YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

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

[c(x₁)]	=	2 · x₁ + -∞
[b(x₁)]	=	1 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[d(x₁)]	=	3 · x₁ + -∞

the rules

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

remain.

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

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

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

c^#(b(x0))	→	a^#(b(x0))
a^#(c(c(c(x0))))	→	d^#(d(x0))
d^#(x0)	→	c^#(b(x0))
d^#(x0)	→	b^#(x0)
b^#(d(x0))	→	c^#(x0)
a^#(c(c(c(x0))))	→	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)]	=	0 · x₁ + -∞
[c(x₁)]	=	4 · x₁ + -∞
[b(x₁)]	=	2 · x₁ + -∞
[d^#(x₁)]	=	2 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[c^#(x₁)]	=	0 · x₁ + -∞
[b^#(x₁)]	=	0 · x₁ + -∞
[d(x₁)]	=	6 · x₁ + -∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.