YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

f(g(x,a,b)) → x
g(f(h(c,d)),x,y) → h(k1(x),k2(y))
k1(a) → c
k2(b) → d
f(h(k1(a),k2(b))) → f(h(c,d))
f(h(c,k2(b))) → f(h(c,d))
f(h(k1(a),d)) → f(h(c,d))

Proof

1 Critical Pair Closing System

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

f(h(k1(a),k2(b))) → f(h(c,d))
f(h(c,k2(b))) → f(h(c,d))
f(h(k1(a),d)) → f(h(c,d))

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[c] = 0
[k2(x1)] = 2 · x1 + 4
[b] = 4
[f(x1)] = 1 · x1 + 1
[d] = 0
[a] = 0
[k1(x1)] = 4 · x1 + 1
[h(x1, x2)] = 1 · x1 + 1 · x2 + 7
all rules could be removed.

1.1.1 R is empty

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

Tool configuration

csi