YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S)
(RULES
a__div(0,s(Y:S)) -> 0
a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
a__div(X1:S,X2:S) -> div(X1:S,X2:S)
a__geq(0,s(Y:S)) -> ffalse
a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
a__geq(X:S,0) -> ttrue
a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
a__if(ffalse,X:S,Y:S) -> mark(Y:S)
a__if(ttrue,X:S,Y:S) -> mark(X:S)
a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
a__minus(0,Y:S) -> 0
a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
mark(0) -> 0
mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
mark(ffalse) -> ffalse
mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
mark(s(X:S)) -> s(mark(X:S))
mark(ttrue) -> ttrue
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__DIV(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
 A__DIV(s(X:S),s(Y:S)) -> A__IF(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 A__GEQ(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 A__MINUS(s(X:S),s(Y:S)) -> A__MINUS(X:S,Y:S)
 MARK(div(X1:S,X2:S)) -> A__DIV(mark(X1:S),X2:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(geq(X1:S,X2:S)) -> A__GEQ(X1:S,X2:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(minus(X1:S,X2:S)) -> A__MINUS(X1:S,X2:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue

Problem 1: 

SCC Processor:
-> Pairs:
 A__DIV(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
 A__DIV(s(X:S),s(Y:S)) -> A__IF(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 A__GEQ(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 A__MINUS(s(X:S),s(Y:S)) -> A__MINUS(X:S,Y:S)
 MARK(div(X1:S,X2:S)) -> A__DIV(mark(X1:S),X2:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(geq(X1:S,X2:S)) -> A__GEQ(X1:S,X2:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(minus(X1:S,X2:S)) -> A__MINUS(X1:S,X2:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__MINUS(s(X:S),s(Y:S)) -> A__MINUS(X:S,Y:S)
->->-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->->Cycle:
->->-> Pairs:
 A__GEQ(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
->->-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->->Cycle:
->->-> Pairs:
 A__DIV(s(X:S),s(Y:S)) -> A__IF(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> A__DIV(mark(X1:S),X2:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A__MINUS(s(X:S),s(Y:S)) -> A__MINUS(X:S,Y:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Projection:
 pi(A__MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A__GEQ(s(X:S),s(Y:S)) -> A__GEQ(X:S,Y:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Projection:
 pi(A__GEQ) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 A__DIV(s(X:S),s(Y:S)) -> A__IF(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> A__DIV(mark(X1:S),X2:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
-> Usable rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__div](X1,X2) = 2.X1 + 2.X2 + 2
[a__geq](X1,X2) = 1
[a__if](X1,X2,X3) = 2.X1 + X2 + 2.X3
[a__minus](X1,X2) = 0
[mark](X) = X
[0] = 0
[div](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[false] = 1
[geq](X1,X2) = 1
[if](X1,X2,X3) = 2.X1 + X2 + 2.X3
[minus](X1,X2) = 0
[s](X) = X + 2
[true] = 1
[A__DIV](X1,X2) = 2.X1 + 2.X2 + 2
[A__GEQ](X1,X2) = 0
[A__IF](X1,X2,X3) = X2 + 2.X3 + 1
[A__MINUS](X1,X2) = 0
[MARK](X) = X + 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> A__DIV(mark(X1:S),X2:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 A__IF(ffalse,X:S,Y:S) -> MARK(Y:S)
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
-> Usable rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__div](X1,X2) = 2.X1 + 2.X2 + 2
[a__geq](X1,X2) = 0
[a__if](X1,X2,X3) = 2.X1 + X2 + X3 + 1
[a__minus](X1,X2) = X1
[mark](X) = X
[0] = 1
[div](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[false] = 0
[geq](X1,X2) = 0
[if](X1,X2,X3) = 2.X1 + X2 + X3 + 1
[minus](X1,X2) = X1
[s](X) = X + 2
[true] = 0
[A__DIV](X1,X2) = 0
[A__GEQ](X1,X2) = 0
[A__IF](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__MINUS](X1,X2) = 0
[MARK](X) = 2.X + 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue

Problem 1.3: 

Subterm Processor:
-> Pairs:
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
 MARK(div(X1:S,X2:S)) -> MARK(X1:S)
 MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S)
 MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Projection:
 pi(A__IF) = 2
 pi(MARK) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 A__IF(ttrue,X:S,Y:S) -> MARK(X:S)
-> Rules:
 a__div(0,s(Y:S)) -> 0
 a__div(s(X:S),s(Y:S)) -> a__if(a__geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)
 a__div(X1:S,X2:S) -> div(X1:S,X2:S)
 a__geq(0,s(Y:S)) -> ffalse
 a__geq(s(X:S),s(Y:S)) -> a__geq(X:S,Y:S)
 a__geq(X:S,0) -> ttrue
 a__geq(X1:S,X2:S) -> geq(X1:S,X2:S)
 a__if(ffalse,X:S,Y:S) -> mark(Y:S)
 a__if(ttrue,X:S,Y:S) -> mark(X:S)
 a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S)
 a__minus(0,Y:S) -> 0
 a__minus(s(X:S),s(Y:S)) -> a__minus(X:S,Y:S)
 a__minus(X1:S,X2:S) -> minus(X1:S,X2:S)
 mark(0) -> 0
 mark(div(X1:S,X2:S)) -> a__div(mark(X1:S),X2:S)
 mark(ffalse) -> ffalse
 mark(geq(X1:S,X2:S)) -> a__geq(X1:S,X2:S)
 mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S)
 mark(minus(X1:S,X2:S)) -> a__minus(X1:S,X2:S)
 mark(s(X:S)) -> s(mark(X:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.