YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

f(a) → b
f(a) → f(c)
a → d
f(d) → b
f(c) → b
d → c

Proof

1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

d → c
f(c) → b
f(d) → b
a → d
f(a) → f(c)
f(a) → b
a → c

All redundant rules that were added or removed can be simulated in 2 steps .

1.1 Critical Pair Closing System

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

f(c) → b
d → c
f(d) → b

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[d] = 3
[f(x1)] = 1 · x1 + 0
[c] = 1
[b] = 1
the rule
f(c) → b
remains.

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[f(x1)] = 1 · x1 + 1
[c] = 4
[b] = 0
all rules could be removed.

1.1.1.1.1 R is empty

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

Tool configuration

csi