YES

Problem 1: 

(THEORY 
(AC plus))
(RULES
f(plus(a,a)) -> plus(f(a),f(a))
plus(f(a),f(a)) -> plus(a,f(f(a)))
)

Problem 1: 

Reduction Order Processor:
-> Rules:
 f(plus(a,a)) -> plus(f(a),f(a))
 plus(f(a),f(a)) -> plus(a,f(f(a)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2.X
[plus](X1,X2) = X1 + X2 + 2
[a] = 0

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(f(a),f(a)) -> F(f(a))
 PLUS(f(a),f(a)) -> PLUS(a,f(f(a)))
 PLUS(plus(f(a),f(a)),x0) -> F(f(a))
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(f(a),f(a)) -> F(f(a))
 PLUS(f(a),f(a)) -> PLUS(a,f(f(a)))
 PLUS(plus(f(a),f(a)),x0) -> F(f(a))
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(f(a),f(a)) -> PLUS(a,f(f(a)))
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> FAxioms:
 plus(plus(x0,x1),x2) -> plus(x0,plus(x1,x2))
 plus(x0,x1) -> plus(x1,x0)
 PLUS(plus(x0,x1),x2) -> PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) -> PLUS(x1,x0)
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
->->-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(f(a),f(a)) -> PLUS(a,f(f(a)))
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Usable Equations:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> Usable Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2
[plus](X1,X2) = X1 + X2 + 1
[a] = 0
[F](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> FAxioms:
 plus(plus(x0,x1),x2) -> plus(x0,plus(x1,x2))
 plus(x0,x1) -> plus(x1,x0)
 PLUS(plus(x0,x1),x2) -> PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) -> PLUS(x1,x0)
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
->->-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(plus(a,f(f(a))),x0)
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Usable Equations:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> Usable Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2
[plus](X1,X2) = X1 + X2
[a] = 0
[F](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> FAxioms:
 plus(plus(x0,x1),x2) -> plus(x0,plus(x1,x2))
 plus(x0,x1) -> plus(x1,x0)
 PLUS(plus(x0,x1),x2) -> PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) -> PLUS(x1,x0)
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
->->-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 PLUS(plus(f(a),f(a)),x0) -> PLUS(a,f(f(a)))
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Usable Equations:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> Usable Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 1
[plus](X1,X2) = X1 + X2
[a] = 0
[F](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x0,x1),x2) = PLUS(x0,plus(x1,x2))
 PLUS(x0,x1) = PLUS(x1,x0)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x0,x1),x2) = plus(x0,plus(x1,x2))
 plus(x0,x1) = plus(x1,x0)
-> Rules:
 plus(f(a),f(a)) -> plus(a,f(f(a)))
-> SRules:
 PLUS(plus(x0,x1),x2) -> PLUS(x0,x1)
 PLUS(x0,plus(x1,x2)) -> PLUS(x1,x2)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.