YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Reduction Pair Processor with Usable Rules

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

[b^#(x₁)]	=	0 · x₁ + 0
[b(x₁)]	=	0 · x₁ + 0
[a^#(x₁)]	=	0 · x₁ + 2
[a(x₁)]	=	14 · x₁ + 2
[c(x₁)]	=	14 · x₁ + 14

together with the usable rules

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

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

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

remain.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
a^#(a(x0)) → a^#(b(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

[b(x₁)] = -4 · x₁ + 0

[a^#(x₁)] = 0 · x₁ + 1

[a(x₁)] = 4 · x₁ + 2

[c(x₁)] = 4 · x₁ + 8

together with the usable rules

a(a(b(x0))) → c(b(a(x0)))

a(c(x0)) → c(a(x0))

b(c(x0)) → a(b(x0))

a(a(x0)) → a(b(a(x0)))

(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.
1.1.1.1.1.1 P is empty

There are no pairs anymore.