YES

Problem 1: 

(VAR x y)
(THEORY 
(AC +))
(RULES
+(g(x),g(y)) -> g(+(g(a),+(x,y)))
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(g(a),+(x,y))
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(g(a),+(x,y))
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(g(a),+(x,y))
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> FAxioms:
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
 +#(+(x2,x3),x4) -> +#(x2,+(x3,x4))
 +#(x2,x3) -> +#(x3,x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(g(a),+(x,y))
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Usable Equations:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> Usable Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[a] = 0
[g](X) = X + 1
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> FAxioms:
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
 +#(+(x2,x3),x4) -> +#(x2,+(x3,x4))
 +#(x2,x3) -> +#(x3,x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(+(g(x),g(y)),x2) -> +#(x,y)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Usable Equations:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> Usable Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[a] = 0
[g](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> FAxioms:
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
 +#(+(x2,x3),x4) -> +#(x2,+(x3,x4))
 +#(x2,x3) -> +#(x3,x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(g(a),+(x,y))
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Usable Equations:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> Usable Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[a] = 0
[g](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(x,y)
-> FAxioms:
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
 +#(+(x2,x3),x4) -> +#(x2,+(x3,x4))
 +#(x2,x3) -> +#(x3,x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
 +#(g(x),g(y)) -> +#(x,y)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Usable Equations:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> Usable Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[a] = 0
[g](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
-> FAxioms:
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
 +#(+(x2,x3),x4) -> +#(x2,+(x3,x4))
 +#(x2,x3) -> +#(x3,x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 +#(+(g(x),g(y)),x2) -> +#(g(+(g(a),+(x,y))),x2)
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Usable Equations:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> Usable Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 1
[a] = 2
[g](X) = 0
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> FAxioms:
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 Empty
-> EAxioms:
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 +(g(x),g(y)) -> g(+(g(a),+(x,y)))
-> SRules:
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.