YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

g(a) f(g(a))
g(b) c
a b
f(x) h(x,x)
h(x,y) c

Proof

1 Critical Pair Closing System

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

g(b) c
f(x) h(x,x)
h(x,y) c

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c] = 0
[g(x1)] = 1 · x1 + 0
[b] = 4
[h(x1, x2)] = 2 · x1 + 2 · x2 + 0
[f(x1)] = 4 · x1 + 0
the rules
f(x) h(x,x)
h(x,y) c
remain.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c] = 0
[h(x1, x2)] = 4 · x1 + 2 · x2 + 0
[f(x1)] = 6 · x1 + 4
the rule
h(x,y) c
remains.

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c] = 0
[h(x1, x2)] = 4 · x1 + 4 · x2 + 2
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