Certification Problem

Input (COPS 307)

The rewrite relation of the following conditional TRS is considered.

plus(0,X) → X
plus(s(X),Y) → plus(X,s(Y))
fib(0) → pair(s(0),0)
fib(s(X)) → pair(W,Y) | fib(X) ≈ pair(Y,Z), plus(Y,Z) ≈ W

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by ConCon @ CoCo 2020)

1 Inlining of Conditions

Inlining of conditions results in the following transformed CTRS having the same multistep rewrite relation.
plus(0,X) → X
plus(s(X),Y) → plus(X,s(Y))
fib(0) → pair(s(0),0)
fib(s(X)) → pair(plus(Y,Z),Y) | fib(X) ≈ pair(Y,Z)

1.1 Almost-orthogonal

The given (extended) properly oriented, right-stable, oriented 3-CTRS is almost-orthogonal, since there are no conditional critical pairs.