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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

R^#(c(x0))

→

R^#(a(b(x0)))

1.1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]	=	-1 · x₁ + 0
[L(x₁)]	=	-∞ · x₁ + 4
[a(x₁)]	=	2 · x₁ + 2
[R^#(x₁)]	=	0 · x₁ + 0
[c(x₁)]	=	2 · x₁ + 3

together with the usable rules

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

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

1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pairs

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

1.1.1.2 Reduction Pair Processor with Usable Rules

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

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

together with the usable rules

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

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

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

→

b^#(a(x0))

remains.

1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.