YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
+(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
+(a,+(b,z:S)) -> +(b,+(a,z:S))
+(a,b) -> +(b,a)
f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
f(a,y:S) -> a
f(b,y:S) -> b
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(a,+(b,z:S)) -> +#(a,z:S)
 +#(a,+(b,z:S)) -> +#(b,+(a,z:S))
 F(+(x:S,y:S),z:S) -> +#(f(x:S,z:S),f(y:S,z:S))
 F(+(x:S,y:S),z:S) -> F(x:S,z:S)
 F(+(x:S,y:S),z:S) -> F(y:S,z:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b

Problem 1: 

SCC Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(a,+(b,z:S)) -> +#(a,z:S)
 +#(a,+(b,z:S)) -> +#(b,+(a,z:S))
 F(+(x:S,y:S),z:S) -> +#(f(x:S,z:S),f(y:S,z:S))
 F(+(x:S,y:S),z:S) -> F(x:S,z:S)
 F(+(x:S,y:S),z:S) -> F(y:S,z:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(a,+(b,z:S)) -> +#(a,z:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->->Cycle:
->->-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->->Cycle:
->->-> Pairs:
 F(+(x:S,y:S),z:S) -> F(x:S,z:S)
 F(+(x:S,y:S),z:S) -> F(y:S,z:S)
->->-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(a,+(b,z:S)) -> +#(a,z:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 +#(+(x:S,y:S),z:S) -> +#(x:S,+(y:S,z:S))
 +#(+(x:S,y:S),z:S) -> +#(y:S,z:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Projection:
 pi(+#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 F(+(x:S,y:S),z:S) -> F(x:S,z:S)
 F(+(x:S,y:S),z:S) -> F(y:S,z:S)
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Projection:
 pi(F) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(+(x:S,y:S),z:S) -> +(x:S,+(y:S,z:S))
 +(a,+(b,z:S)) -> +(b,+(a,z:S))
 +(a,b) -> +(b,a)
 f(+(x:S,y:S),z:S) -> +(f(x:S,z:S),f(y:S,z:S))
 f(a,y:S) -> a
 f(b,y:S) -> b
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.