YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
-(s(x:S),s(y:S)) -> -(x:S,y:S)
-(x:S,0) -> x:S
gcd(0,s(x:S)) -> s(x:S)
gcd(s(x:S),0) -> s(x:S)
gcd(s(x:S),s(y:S)) -> gcd(-(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
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 gcd(0,s(x:S)) -> s(x:S)
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> gcd(-(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
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 4 subproblems.

Problem 1.1: 

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

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 gcd(0,s(x:S)) -> s(x:S)
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> gcd(-(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
->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:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 gcd(0,s(x:S)) -> s(x:S)
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> gcd(-(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
->Projection:
 pi(MAX) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 gcd(0,s(x:S)) -> s(x:S)
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> gcd(-(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
->Projection:
 pi(-#) = 1

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> GCD(-(max(x:S,y:S),min(x:S,y:S)),s(min(x:S,y:S)))
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 gcd(0,s(x:S)) -> s(x:S)
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> gcd(-(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
-> Usable rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> 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
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1
[gcd](X1,X2) = 0
[max](X1,X2) = X1 + X2
[min](X1,X2) = X1
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[-#](X1,X2) = 0
[GCD](X1,X2) = 2.X1 + X2
[MAX](X1,X2) = 0
[MIN](X1,X2) = 0

Problem 1.4: 

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

The problem is finite.