YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

+(x,0) → x
+(x,s(y)) → s(+(x,y))
d(0) → 0
d(s(x)) → s(s(d(x)))
f(0) → 0
f(s(x)) → +(+(s(x),s(x)),s(x))
f(g(0)) → +(+(g(0),g(0)),g(0))
g(x) → s(d(x))

Proof

1 Critical Pair Closing System

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

f(s(x)) → +(+(s(x),s(x)),s(x))
g(x) → s(d(x))

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[f(x1)] = 7 · x1 + 0
[+(x1, x2)] = 2 · x1 + 1 · x2 + 0
[d(x1)] = 2 · x1 + 0
[g(x1)] = 3 · x1 + 4
[s(x1)] = 1 · x1 + 0
the rule
f(s(x)) → +(+(s(x),s(x)),s(x))
remains.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[f(x1)] = 7 · x1 + 7
[+(x1, x2)] = 1 · x1 + 2 · x2 + 5
[s(x1)] = 2 · x1 + 3
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