YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
gt(s(x:S),zero) -> ttrue
gt(zero,y:S) -> ffalse
id(x:S) -> x:S
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
not(x:S) -> if(x:S,ffalse,ttrue)
plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
plus(zero,y: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:
 GT(s(x:S),s(y:S)) -> GT(x:S,y:S)
 NOT(x:S) -> IF(x:S,ffalse,ttrue)
 PLUS(s(x:S),s(y:S)) -> GT(x:S,y:S)
 PLUS(s(x:S),s(y:S)) -> ID(x:S)
 PLUS(s(x:S),s(y:S)) -> ID(y:S)
 PLUS(s(x:S),s(y:S)) -> IF(gt(x:S,y:S),x:S,y:S)
 PLUS(s(x:S),s(y:S)) -> IF(not(gt(x:S,y:S)),id(x:S),id(y:S))
 PLUS(s(x:S),s(y:S)) -> NOT(gt(x:S,y:S))
 PLUS(s(x:S),s(y:S)) -> PLUS(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))
 PLUS(s(x:S),x:S) -> GT(x:S,x:S)
 PLUS(s(x:S),x:S) -> ID(x:S)
 PLUS(s(x:S),x:S) -> IF(gt(x:S,x:S),id(x:S),id(x:S))
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 GT(s(x:S),s(y:S)) -> GT(x:S,y:S)
 NOT(x:S) -> IF(x:S,ffalse,ttrue)
 PLUS(s(x:S),s(y:S)) -> GT(x:S,y:S)
 PLUS(s(x:S),s(y:S)) -> ID(x:S)
 PLUS(s(x:S),s(y:S)) -> ID(y:S)
 PLUS(s(x:S),s(y:S)) -> IF(gt(x:S,y:S),x:S,y:S)
 PLUS(s(x:S),s(y:S)) -> IF(not(gt(x:S,y:S)),id(x:S),id(y:S))
 PLUS(s(x:S),s(y:S)) -> NOT(gt(x:S,y:S))
 PLUS(s(x:S),s(y:S)) -> PLUS(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))
 PLUS(s(x:S),x:S) -> GT(x:S,x:S)
 PLUS(s(x:S),x:S) -> ID(x:S)
 PLUS(s(x:S),x:S) -> IF(gt(x:S,x:S),id(x:S),id(x:S))
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 GT(s(x:S),s(y:S)) -> GT(x:S,y:S)
->->-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 PLUS(s(x:S),s(y:S)) -> PLUS(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
->->-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 GT(s(x:S),s(y:S)) -> GT(x:S,y:S)
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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(GT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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: 

Reduction Pairs Processor:
-> Pairs:
 PLUS(s(x:S),s(y:S)) -> PLUS(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[gt](X1,X2) = 3.X1 + 1
[id](X) = X
[if](X1,X2,X3) = X2 + X3
[not](X) = 3.X + 4
[plus](X1,X2) = 0
[quot](X1,X2,X3) = 0
[0] = 0
[fSNonEmpty] = 0
[false] = 3
[s](X) = 4.X + 3/2
[true] = 0
[zero] = 3/4
[GT](X1,X2) = 0
[ID](X) = 0
[IF](X1,X2,X3) = 0
[NOT](X) = 0
[PLUS](X1,X2) = 4.X1 + 2.X2
[QUOT](X1,X2,X3) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
->->-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 PLUS(s(x:S),x:S) -> PLUS(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gt](X1,X2) = 2.X1 + 2
[id](X) = X
[if](X1,X2,X3) = X2 + X3
[not](X) = 0
[plus](X1,X2) = 0
[quot](X1,X2,X3) = 0
[0] = 0
[fSNonEmpty] = 0
[false] = 2
[s](X) = 2.X + 1
[true] = 1
[zero] = 2
[GT](X1,X2) = 0
[ID](X) = 0
[IF](X1,X2,X3) = 0
[NOT](X) = 0
[PLUS](X1,X2) = 2.X1
[QUOT](X1,X2,X3) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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 1.3: 

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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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 1.3: 

SCC Processor:
-> Pairs:
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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 1.3: 

Reduction Pairs Processor:
-> Pairs:
 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S))
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y:S) -> y:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gt](X1,X2) = 2
[id](X) = X + 2
[if](X1,X2,X3) = 2.X2 + X3
[not](X) = 2
[plus](X1,X2) = X2
[quot](X1,X2,X3) = 0
[0] = 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 0
[true] = 2
[zero] = 2
[GT](X1,X2) = 0
[ID](X) = 0
[IF](X1,X2,X3) = 0
[NOT](X) = 0
[PLUS](X1,X2) = 0
[QUOT](X1,X2,X3) = 2.X2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 gt(s(x:S),s(y:S)) -> gt(x:S,y:S)
 gt(s(x:S),zero) -> ttrue
 gt(zero,y:S) -> ffalse
 id(x:S) -> x:S
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 not(x:S) -> if(x:S,ffalse,ttrue)
 plus(id(x:S),s(y:S)) -> s(plus(x:S,if(gt(s(y:S),y:S),y:S,s(y:S))))
 plus(s(x:S),s(y:S)) -> s(s(plus(if(gt(x:S,y:S),x:S,y:S),if(not(gt(x:S,y:S)),id(x:S),id(y:S)))))
 plus(s(x:S),x:S) -> plus(if(gt(x:S,x:S),id(x:S),id(x:S)),s(x:S))
 plus(zero,y: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.