YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

p(x) → q(x)
p(x) → r(x)
q(x) → s(p(x))
r(x) → s(p(x))
s(x) → f(p(x))

Proof

1 Critical Pair Closing System

Confluence is proven using the following terminating critical-pair-closing-system R:

q(x) → s(p(x))
r(x) → s(p(x))

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[q(x1)] = 1 · x1 + 0
[s(x1)] = 1 · x1 + 0
[p(x1)] = 1 · x1 + 0
[r(x1)] = 1 · x1 + 1
the rule
q(x) → s(p(x))
remains.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[q(x1)] = 2 · x1 + 1
[s(x1)] = 1 · x1 + 0
[p(x1)] = 2 · x1 + 0
all rules could be removed.

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

Tool configuration

csi