YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

f(a,b) → c
a → a'
b → b'
c → f(a',b)
c → f(a,b')
c → f(a,b)

Proof

1 Critical Pair Closing System

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

c → f(a',b)
c → f(a,b')
b → b'
a → a'

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a'] = 0
[b] = 2
[c] = 3
[b'] = 1
[a] = 1
[f(x1, x2)] = 1 · x1 + 1 · x2 + 1
the rules
c → f(a',b)
c → f(a,b')
remain.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a'] = 0
[b] = 3
[c] = 3
[b'] = 2
[a] = 0
[f(x1, x2)] = 1 · x1 + 1 · x2 + 0
the rule
c → f(a',b)
remains.

1.1.1.1 Rule Removal

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