YES

Problem 1: 

(VAR x y)
(THEORY 
(C gcd))
(RULES
gcd(0,y) -> y
gcd(s(x),0) -> s(x)
gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
if_minus(false,s(x),y) -> s(minus(x,y))
if_minus(true,s(x),y) -> 0
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 GCD(x2,x3) = GCD(x3,x2)
-> Pairs:
 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
 GCD(s(x),s(y)) -> LE(y,x)
 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
 IF_GCD(false,s(x),s(y)) -> MINUS(y,x)
 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
 IF_GCD(true,s(x),s(y)) -> MINUS(x,y)
 IF_MINUS(false,s(x),y) -> MINUS(x,y)
 LE(s(x),s(y)) -> LE(x,y)
 MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
 MINUS(s(x),y) -> LE(s(x),y)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty

Problem 1: 

SCC Processor:
-> FAxioms:
 GCD(x2,x3) = GCD(x3,x2)
-> Pairs:
 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
 GCD(s(x),s(y)) -> LE(y,x)
 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
 IF_GCD(false,s(x),s(y)) -> MINUS(y,x)
 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
 IF_GCD(true,s(x),s(y)) -> MINUS(x,y)
 IF_MINUS(false,s(x),y) -> MINUS(x,y)
 LE(s(x),s(y)) -> LE(x,y)
 MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
 MINUS(s(x),y) -> LE(s(x),y)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
-> FAxioms:
 gcd(x2,x3) -> gcd(x3,x2)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
->->-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 IF_MINUS(false,s(x),y) -> MINUS(x,y)
 MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
-> FAxioms:
 gcd(x2,x3) -> gcd(x3,x2)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
->->-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
-> FAxioms:
 gcd(x2,x3) -> gcd(x3,x2)
 GCD(x2,x3) -> GCD(x3,x2)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
->->-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Projection:
 pi(LE) = [1]

Problem 1.1: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 IF_MINUS(false,s(x),y) -> MINUS(x,y)
 MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Projection:
 pi(IF_MINUS) = [2]
 pi(MINUS) = [1]

Problem 1.2: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 MINUS(s(x),y) -> IF_MINUS(le(s(x),y),s(x),y)
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 GCD(x2,x3) = GCD(x3,x2)
-> Pairs:
 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Usable Equations:
 Empty
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> Usable Rules:
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gcd](X1,X2) = 0
[if_gcd](X1,X2,X3) = 0
[if_minus](X1,X2,X3) = 2.X2
[le](X1,X2) = 2
[minus](X1,X2) = 2.X1
[0] = 0
[false] = 2
[s](X) = 2.X + 1
[true] = 2
[GCD](X1,X2) = X1 + X2 + 2
[IF_GCD](X1,X2,X3) = X2 + X3 + 1
[IF_MINUS](X1,X2,X3) = 0
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 GCD(x2,x3) = GCD(x3,x2)
-> Pairs:
 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
-> EAxioms:
 gcd(x2,x3) = gcd(x3,x2)
-> Rules:
 gcd(0,y) -> y
 gcd(s(x),0) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_minus(false,s(x),y) -> s(minus(x,y))
 if_minus(true,s(x),y) -> 0
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(0,y) -> 0
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.