YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

a → b
a → c
a → e
b → d
c → a
d → a
d → e
g(x) → h(a)
h(x) → e

Proof

1 Critical Pair Closing System

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

c → a
a → b
b → d
d → e
a → e

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[d] = 0
[b] = 0
[e] = 0
[a] = 4
[c] = 4
the rules
c → a
b → d
d → e
remain.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[d] = 4
[b] = 4
[e] = 0
[a] = 0
[c] = 0
the rules
c → a
b → d
remain.

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[d] = 0
[b] = 1
[a] = 0
[c] = 0
the rule
c → a
remains.

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a] = 0
[c] = 1
all rules could be removed.

1.1.1.1.1.1 R is empty

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

Tool configuration

csi