YES

Problem 1: 

(VAR x y)
(THEORY 
(AC f))
(RULES
f(g(f(h(x),x)),x) -> f(h(x),f(x,x))
f(g(g(x)),x) -> f(g(x),g(x))
f(g(h(x)),f(x,f(x,y))) -> f(g(f(h(x),y)),x)
f(h(x),g(x)) -> f(g(h(x)),x)
)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 F(f(x2,x3),x4) = F(x2,f(x3,x4))
 F(x2,x3) = F(x3,x2)
-> Pairs:
 F(f(h(x),g(x)),x2) -> F(f(g(h(x)),x),x2)
 F(f(h(x),g(x)),x2) -> F(g(h(x)),x)
 F(h(x),g(x)) -> F(g(h(x)),x)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)

Problem 1: 

SCC Processor:
-> FAxioms:
 F(f(x2,x3),x4) = F(x2,f(x3,x4))
 F(x2,x3) = F(x3,x2)
-> Pairs:
 F(f(h(x),g(x)),x2) -> F(f(g(h(x)),x),x2)
 F(f(h(x),g(x)),x2) -> F(g(h(x)),x)
 F(h(x),g(x)) -> F(g(h(x)),x)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(h(x),g(x)),x2) -> F(f(g(h(x)),x),x2)
 F(f(h(x),g(x)),x2) -> F(g(h(x)),x)
 F(h(x),g(x)) -> F(g(h(x)),x)
-> FAxioms:
 f(f(x2,x3),x4) -> f(x2,f(x3,x4))
 f(x2,x3) -> f(x3,x2)
 F(f(x2,x3),x4) -> F(x2,f(x3,x4))
 F(x2,x3) -> F(x3,x2)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
->->-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)

Problem 1: 

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

Problem 1: 

SCC Processor:
-> FAxioms:
 F(f(x2,x3),x4) = F(x2,f(x3,x4))
 F(x2,x3) = F(x3,x2)
-> Pairs:
 F(f(h(x),g(x)),x2) -> F(g(h(x)),x)
 F(h(x),g(x)) -> F(g(h(x)),x)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(h(x),g(x)),x2) -> F(g(h(x)),x)
 F(h(x),g(x)) -> F(g(h(x)),x)
-> FAxioms:
 f(f(x2,x3),x4) -> f(x2,f(x3,x4))
 f(x2,x3) -> f(x3,x2)
 F(f(x2,x3),x4) -> F(x2,f(x3,x4))
 F(x2,x3) -> F(x3,x2)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
->->-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)

Problem 1: 

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

Problem 1: 

SCC Processor:
-> FAxioms:
 F(f(x2,x3),x4) = F(x2,f(x3,x4))
 F(x2,x3) = F(x3,x2)
-> Pairs:
 F(h(x),g(x)) -> F(g(h(x)),x)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(h(x),g(x)) -> F(g(h(x)),x)
-> FAxioms:
 f(f(x2,x3),x4) -> f(x2,f(x3,x4))
 f(x2,x3) -> f(x3,x2)
 F(f(x2,x3),x4) -> F(x2,f(x3,x4))
 F(x2,x3) -> F(x3,x2)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
->->-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(f(x2,x3),x4) -> F(x2,x3)
 F(x2,f(x3,x4)) -> F(x3,x4)

Problem 1: 

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

Problem 1: 

SCC Processor:
-> FAxioms:
 F(f(x2,x3),x4) = F(x2,f(x3,x4))
 F(x2,x3) = F(x3,x2)
-> Pairs:
 F(h(x),g(x)) -> F(g(h(x)),x)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(x2,f(x3,x4)) -> F(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(h(x),g(x)) -> F(g(h(x)),x)
-> FAxioms:
 f(f(x2,x3),x4) -> f(x2,f(x3,x4))
 f(x2,x3) -> f(x3,x2)
 F(f(x2,x3),x4) -> F(x2,f(x3,x4))
 F(x2,x3) -> F(x3,x2)
-> EAxioms:
 f(f(x2,x3),x4) = f(x2,f(x3,x4))
 f(x2,x3) = f(x3,x2)
->->-> Rules:
 f(h(x),g(x)) -> f(g(h(x)),x)
-> SRules:
 F(x2,f(x3,x4)) -> F(x3,x4)

Problem 1: 

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

Problem 1: 

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

The problem is finite.