YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
if_quot(ffalse,x:S,y:S) -> 0
if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
le(0,y:S) -> ttrue
le(s(x:S),0) -> ffalse
le(s(x:S),s(y:S)) -> le(x:S,y:S)
minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
minus(x:S,0) -> x:S
quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> MINUS(x:S,y:S)
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(x:S,s(y:S)) -> IF_QUOT(le(s(y:S),x:S),x:S,s(y:S))
 QUOT(x:S,s(y:S)) -> LE(s(y:S),x:S)
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> MINUS(x:S,y:S)
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(x:S,s(y:S)) -> IF_QUOT(le(s(y:S),x:S),x:S,s(y:S))
 QUOT(x:S,s(y:S)) -> LE(s(y:S),x:S)
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
->->-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->->Cycle:
->->-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
->->-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->->Cycle:
->->-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 QUOT(x:S,s(y:S)) -> IF_QUOT(le(s(y:S),x:S),x:S,s(y:S))
->->-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Projection:
 pi(LE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 QUOT(x:S,s(y:S)) -> IF_QUOT(le(s(y:S),x:S),x:S,s(y:S))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Narrowed Pairs:
->->Original Pair:
 QUOT(x:S,s(y:S)) -> IF_QUOT(le(s(y:S),x:S),x:S,s(y:S))
->-> Narrowed pairs:
 QUOT(0,s(x:S)) -> IF_QUOT(ffalse,0,s(x:S))
 QUOT(s(y:S),s(x:S)) -> IF_QUOT(le(x:S,y:S),s(y:S),s(x:S))

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 QUOT(0,s(x:S)) -> IF_QUOT(ffalse,0,s(x:S))
 QUOT(s(y:S),s(x:S)) -> IF_QUOT(le(x:S,y:S),s(y:S),s(x:S))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 QUOT(s(y:S),s(x:S)) -> IF_QUOT(le(x:S,y:S),s(y:S),s(x:S))
->->-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
 QUOT(s(y:S),s(x:S)) -> IF_QUOT(le(x:S,y:S),s(y:S),s(x:S))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Narrowed Pairs:
->->Original Pair:
 IF_QUOT(ttrue,x:S,y:S) -> QUOT(minus(x:S,y:S),y:S)
->-> Narrowed pairs:
 IF_QUOT(ttrue,s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 IF_QUOT(ttrue,x:S,0) -> QUOT(x:S,0)
->->Original Pair:
 QUOT(s(y:S),s(x:S)) -> IF_QUOT(le(x:S,y:S),s(y:S),s(x:S))
->-> Narrowed pairs:
 QUOT(s(0),s(s(x:S))) -> IF_QUOT(ffalse,s(0),s(s(x:S)))
 QUOT(s(s(y:S)),s(s(x:S))) -> IF_QUOT(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 QUOT(s(y:S),s(0)) -> IF_QUOT(ttrue,s(y:S),s(0))

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_QUOT(ttrue,s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 IF_QUOT(ttrue,x:S,0) -> QUOT(x:S,0)
 QUOT(s(0),s(s(x:S))) -> IF_QUOT(ffalse,s(0),s(s(x:S)))
 QUOT(s(s(y:S)),s(s(x:S))) -> IF_QUOT(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 QUOT(s(y:S),s(0)) -> IF_QUOT(ttrue,s(y:S),s(0))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_QUOT(ttrue,s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 QUOT(s(s(y:S)),s(s(x:S))) -> IF_QUOT(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 QUOT(s(y:S),s(0)) -> IF_QUOT(ttrue,s(y:S),s(0))
->->-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 IF_QUOT(ttrue,s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 QUOT(s(s(y:S)),s(s(x:S))) -> IF_QUOT(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 QUOT(s(y:S),s(0)) -> IF_QUOT(ttrue,s(y:S),s(0))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
-> Usable rules:
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[if_quot](X1,X2,X3) = 0
[le](X1,X2) = 0
[minus](X1,X2) = 2.X1 + 1
[quot](X1,X2) = 0
[0] = 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2.X + 2
[true] = 0
[IF_QUOT](X1,X2,X3) = 2.X1 + 2.X2
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0
[QUOT](X1,X2) = 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 QUOT(s(s(y:S)),s(s(x:S))) -> IF_QUOT(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 QUOT(s(y:S),s(0)) -> IF_QUOT(ttrue,s(y:S),s(0))
-> Rules:
 if_quot(ffalse,x:S,y:S) -> 0
 if_quot(ttrue,x:S,y:S) -> s(quot(minus(x:S,y:S),y:S))
 le(0,y:S) -> ttrue
 le(s(x:S),0) -> ffalse
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(x:S,s(y:S)) -> if_quot(le(s(y:S),x:S),x:S,s(y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.