YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
any(s(x:S)) -> s(s(any(x:S)))
any(x:S) -> x:S
gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
max(0,y:S) -> y:S
max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
max(x:S,0) -> x:S
min(0,y:S) -> 0
min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
min(x:S,0) -> 0
minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
minus(x:S,0) -> x:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ANY(s(x:S)) -> ANY(x:S)
 GCD(s(x:S),s(y:S)) -> GCD(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 GCD(s(x:S),s(y:S)) -> MAX(x:S,y:S)
 GCD(s(x:S),s(y:S)) -> MIN(x:S,y:S)
 GCD(s(x:S),s(y:S)) -> MINUS(max(x:S,y:S),min(x:S,y:S))
 MAX(s(x:S),s(y:S)) -> MAX(x:S,y:S)
 MIN(s(x:S),s(y:S)) -> MIN(x:S,y:S)
 MINUS(s(x:S),s(y:S)) -> ANY(y:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,any(y:S))
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 ANY(s(x:S)) -> ANY(x:S)
 GCD(s(x:S),s(y:S)) -> GCD(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 GCD(s(x:S),s(y:S)) -> MAX(x:S,y:S)
 GCD(s(x:S),s(y:S)) -> MIN(x:S,y:S)
 GCD(s(x:S),s(y:S)) -> MINUS(max(x:S,y:S),min(x:S,y:S))
 MAX(s(x:S),s(y:S)) -> MAX(x:S,y:S)
 MIN(s(x:S),s(y:S)) -> MIN(x:S,y:S)
 MINUS(s(x:S),s(y:S)) -> ANY(y:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,any(y:S))
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MIN(s(x:S),s(y:S)) -> MIN(x:S,y:S)
->->-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 MAX(s(x:S),s(y:S)) -> MAX(x:S,y:S)
->->-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 ANY(s(x:S)) -> ANY(x:S)
->->-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,any(y:S))
->->-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 GCD(s(x:S),s(y:S)) -> GCD(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
->->-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MIN(s(x:S),s(y:S)) -> MIN(x:S,y:S)
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Projection:
 pi(MIN) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 MAX(s(x:S),s(y:S)) -> MAX(x:S,y:S)
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Projection:
 pi(MAX) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 ANY(s(x:S)) -> ANY(x:S)
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Projection:
 pi(ANY) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,any(y:S))
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Projection:
 pi(MINUS) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Reduction Pair Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> GCD(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
-> Usable rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[any](X) = 2.X.X + 2.X
[max](X1,X2) = X1 + X2 + 1
[min](X1,X2) = X1
[minus](X1,X2) = X1
[0] = 0
[s](X) = 2.X + 2
[GCD](X1,X2) = 2.X1 + X2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 any(s(x:S)) -> s(s(any(x:S)))
 any(x:S) -> x:S
 gcd(s(x:S),s(y:S)) -> gcd(minus(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
 max(0,y:S) -> y:S
 max(s(x:S),s(y:S)) -> s(max(x:S,y:S))
 max(x:S,0) -> x:S
 min(0,y:S) -> 0
 min(s(x:S),s(y:S)) -> s(min(x:S,y:S))
 min(x:S,0) -> 0
 minus(s(x:S),s(y:S)) -> s(minus(x:S,any(y:S)))
 minus(x:S,0) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.