YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.