YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
minus(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(minus(x:S,y:S),s(y:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 MINUS(minus(x:S,y:S),z:S) -> MINUS(x:S,plus(y:S,z:S))
 MINUS(minus(x:S,y:S),z:S) -> PLUS(y:S,z:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 MINUS(minus(x:S,y:S),z:S) -> MINUS(x:S,plus(y:S,z:S))
 MINUS(minus(x:S,y:S),z:S) -> PLUS(y:S,z:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
->->Cycle:
->->-> Pairs:
 MINUS(minus(x:S,y:S),z:S) -> MINUS(x:S,plus(y:S,z:S))
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
->->-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
->->Cycle:
->->-> Pairs:
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
->->-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(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:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(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:
 MINUS(minus(x:S,y:S),z:S) -> MINUS(x:S,plus(y:S,z:S))
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
->Projection:
 pi(MINUS) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(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(minus(x:S,y:S),s(y:S))
-> Rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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(minus(x:S,y:S),s(y:S)))
-> Usable rules:
 minus(minus(x:S,y:S),z:S) -> minus(x:S,plus(y:S,z:S))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(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:
 
[minus](X1,X2) = X1
[plus](X1,X2) = 2.X1 + X2 + 2
[0] = 0
[s](X) = X + 2
[QUOT](X1,X2) = 2.X1

Problem 1.3: 

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

The problem is finite.