YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Redundant Rules Transformation

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

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

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:

b → d
c → d
f(g(c,c)) → f(g(d,d))
f(g(b,b)) → f(g(d,d))

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c] = 4
[f(x1)] = 1 · x1 + 0
[g(x1, x2)] = 1 · x1 + 1 · x2 + 1
[d] = 0
[b] = 1
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