YES

Problem 1: 

(VAR v_NonEmpty:S X:S Y:S)
(RULES
min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
min(X:S,0) -> X:S
plus(0,Y:S) -> Y:S
plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
quot(0,s(Y:S)) -> 0
quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 MIN(min(X:S,Y:S),Z) -> MIN(X:S,plus(Y:S,Z))
 MIN(min(X:S,Y:S),Z) -> PLUS(Y:S,Z)
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 MIN(min(X:S,Y:S),Z) -> MIN(X:S,plus(Y:S,Z))
 MIN(min(X:S,Y:S),Z) -> PLUS(Y:S,Z)
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
->->-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->->Cycle:
->->-> Pairs:
 MIN(min(X:S,Y:S),Z) -> MIN(X:S,plus(Y:S,Z))
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
->->-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->->Cycle:
->->-> Pairs:
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
->->-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(X:S),Y:S) -> PLUS(X:S,Y:S)
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 MIN(min(X:S,Y:S),Z) -> MIN(X:S,plus(Y:S,Z))
 MIN(s(X:S),s(Y:S)) -> MIN(X:S,Y:S)
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Projection:
 pi(MIN) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 QUOT(s(X:S),s(Y:S)) -> QUOT(min(X:S,Y:S),s(Y:S))
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
-> Usable rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[min](X1,X2) = X1
[plus](X1,X2) = X1 + X2 + 1
[0] = 0
[Z] = 0
[s](X) = X + 1
[QUOT](X1,X2) = 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 min(min(X:S,Y:S),Z) -> min(X:S,plus(Y:S,Z))
 min(s(X:S),s(Y:S)) -> min(X:S,Y:S)
 min(X:S,0) -> X:S
 plus(0,Y:S) -> Y:S
 plus(s(X:S),Y:S) -> s(plus(X:S,Y:S))
 quot(0,s(Y:S)) -> 0
 quot(s(X:S),s(Y:S)) -> s(quot(min(X:S,Y:S),s(Y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.