YES

Problem 1: 

(THEORY 
(AC f))
(RULES
f(g(f(h(a),a)),a) -> f(h(a),f(a,a))
f(g(h(a)),f(b,f(b,b))) -> f(g(f(h(a),a)),a)
f(h(a),a) -> f(h(a),b)
f(h(a),g(a)) -> f(g(h(a)),a)
)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.