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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Reduction Pair Processor with Usable Rules

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

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

together with the usable rules

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

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

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

remain.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pairs

d^#(x0)	→	a^#(x0)
a^#(d(x0))	→	d^#(a(x0))
d^#(x0)	→	b^#(a(x0))
b^#(a(a(x0)))	→	a^#(b(c(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)]	=	0 · x₁ + 0
[b(x₁)]	=	-∞ · x₁ + 0
[d(x₁)]	=	2 · x₁ + 2
[a(x₁)]	=	0 · x₁ + 0
[d^#(x₁)]	=	2 · x₁ + 0
[a^#(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	1 · x₁ + -∞

together with the usable rules

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

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

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

remain.

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)]	=	6 · x₁ + 0
[b(x₁)]	=	-∞ · x₁ + 8
[d(x₁)]	=	7 · x₁ + 15
[a(x₁)]	=	0 · x₁ + 8
[d^#(x₁)]	=	11 · x₁ + 14
[a^#(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + -∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

a^#(a(x0))

→

a^#(b(a(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)]	=	-4 · x₁ + 0
[d(x₁)]	=	8 · x₁ + 4
[a(x₁)]	=	12 · x₁ + 8
[a^#(x₁)]	=	5 · x₁ + 12
[c(x₁)]	=	12 · x₁ + 8

together with the usable rules

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

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

1.1.1.2.1 P is empty

There are no pairs anymore.