YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
plus(0,y:S) -> y:S
plus(s(x:S),y:S) -> s(plus(x:S,y:S))
quot(0,s(y:S),s(z:S)) -> 0
quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S)
 QUOT(x:S,0,s(z:S)) -> PLUS(z:S,s(0))
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S)
 QUOT(x:S,0,s(z:S)) -> PLUS(z:S,s(0))
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
->->Cycle:
->->-> Pairs:
 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S)
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
->->-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S)
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
->Projection:
 pi(QUOT) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
->->-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 quot(0,s(y:S),s(z:S)) -> 0
 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S)
 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S)))
-> Usable rules:
 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:
 
[plus](X1,X2) = X2 + 1
[quot](X1,X2,X3) = 0
[0] = 2
[fSNonEmpty] = 0
[s](X) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 2.X2

Problem 1.2: 

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

The problem is finite.