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

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

L^#(a(a(x0)))

→

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

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
[d(x₁)]	=	1 · x₁ + 2
[R(x₁)]	=	11 · x₁ + 0
[L^#(x₁)]	=	0 · x₁ + 3
[a(x₁)]	=	2 · x₁ + 2
[c(x₁)]	=	2 · x₁ + 1

together with the usable rules

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

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

1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pairs

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

1.1.2 Reduction Pair Processor with Usable Rules

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

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

together with the usable rules

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

remain.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

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)))
b^#(c(a(x0)))	→	a^#(b(c(x0)))

1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

[a^#(x₁)]	=	0 · x₁ + 2
[b(x₁)]	=	-∞ · x₁ + 0
[d(x₁)]	=	-∞ · x₁ + 0
[R(x₁)]	=	0 · x₁ + 2
[a(x₁)]	=	-∞ · x₁ + 0
[d^#(x₁)]	=	1 · x₁ + 2
[b^#(x₁)]	=	2 · x₁ + 0
[c(x₁)]	=	5 · x₁ + 3

together with the usable rules

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

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

remain.

1.1.2.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[a^#(x₁)]

-∞	1
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

-∞	-∞
-∞	-∞

· x₁ +

0	-∞
0	-∞

[d(x₁)]

0	0
0	1

· x₁ +

0	-∞
1	-∞

[R(x₁)]

0	0
0	-∞

· x₁ +

-∞	-∞
2	-∞

[a(x₁)]

-∞	0
-∞	0

· x₁ +

0	-∞
-∞	-∞

[d^#(x₁)]

2	2
-∞	-∞

· x₁ +

1	-∞
-∞	-∞

[b^#(x₁)]

1	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[c(x₁)]

0	0
0	0

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

1.1.2.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a^#(x₁)]	=	0 · x₁ + 0
[b(x₁)]	=	-∞ · x₁ + 0
[d(x₁)]	=	9 · x₁ + 4
[R(x₁)]	=	11 · x₁ + 11
[a(x₁)]	=	0 · x₁ + -∞
[d^#(x₁)]	=	4 · x₁ + 3
[b^#(x₁)]	=	0 · x₁ + 0
[c(x₁)]	=	15 · x₁ + 15

together with the usable rules

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

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

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

→

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

remains.

1.1.2.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.