YES

Problem 1: 

(VAR v_NonEmpty:S)
(RULES
f(a,b) -> f(a,c)
f(c,d) -> f(b,d)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(a,b) -> f(a,c)
 f(c,d) -> f(b,d)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 f(a,b) -> f(a,c)
 f(c,d) -> f(b,d)

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(a,b) -> f(a,c)
 f(c,d) -> f(b,d)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.