YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
gcd(0,y:S) -> y:S
gcd(s(x:S),0) -> s(x:S)
gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
minus(x:S,0) -> x:S
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 GCD(s(x:S),s(y:S)) -> LE(y:S,x:S)
 IF_GCD(ffalse,x:S,y:S) -> GCD(minus(y:S,x:S),x:S)
 IF_GCD(ffalse,x:S,y:S) -> MINUS(y:S,x:S)
 IF_GCD(ttrue,x:S,y:S) -> GCD(minus(x:S,y:S),y:S)
 IF_GCD(ttrue,x:S,y:S) -> MINUS(x: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)
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 GCD(s(x:S),s(y:S)) -> LE(y:S,x:S)
 IF_GCD(ffalse,x:S,y:S) -> GCD(minus(y:S,x:S),x:S)
 IF_GCD(ffalse,x:S,y:S) -> MINUS(y:S,x:S)
 IF_GCD(ttrue,x:S,y:S) -> GCD(minus(x:S,y:S),y:S)
 IF_GCD(ttrue,x:S,y:S) -> MINUS(x: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)
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->->Cycle:
->->-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 IF_GCD(ffalse,x:S,y:S) -> GCD(minus(y:S,x:S),x:S)
 IF_GCD(ttrue,x:S,y:S) -> GCD(minus(x:S,y:S),y:S)
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x: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:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x: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:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Projection:
 pi(LE) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Instantiation Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 IF_GCD(ffalse,x:S,y:S) -> GCD(minus(y:S,x:S),x:S)
 IF_GCD(ttrue,x:S,y:S) -> GCD(minus(x:S,y:S),y:S)
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Instantiated Pairs:
->->Original Pair:
 IF_GCD(ffalse,x:S,y:S) -> GCD(minus(y:S,x:S),x:S)
->-> Instantiated pairs:
 IF_GCD(ffalse,s(x4:S),s(x5:S)) -> GCD(minus(s(x5:S),s(x4:S)),s(x4:S))
->->Original Pair:
 IF_GCD(ttrue,x:S,y:S) -> GCD(minus(x:S,y:S),y:S)
->-> Instantiated pairs:
 IF_GCD(ttrue,s(x4:S),s(x5:S)) -> GCD(minus(s(x4:S),s(x5:S)),s(x5:S))

Problem 1.3: 

SCC Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 IF_GCD(ffalse,s(x4:S),s(x5:S)) -> GCD(minus(s(x5:S),s(x4:S)),s(x4:S))
 IF_GCD(ttrue,s(x4:S),s(x5:S)) -> GCD(minus(s(x4:S),s(x5:S)),s(x5:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 IF_GCD(ffalse,s(x4:S),s(x5:S)) -> GCD(minus(s(x5:S),s(x4:S)),s(x4:S))
 IF_GCD(ttrue,s(x4:S),s(x5:S)) -> GCD(minus(s(x4:S),s(x5:S)),s(x5:S))
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
 IF_GCD(ffalse,s(x4:S),s(x5:S)) -> GCD(minus(s(x5:S),s(x4:S)),s(x4:S))
 IF_GCD(ttrue,s(x4:S),s(x5:S)) -> GCD(minus(s(x4:S),s(x5:S)),s(x5:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Narrowed Pairs:
->->Original Pair:
 GCD(s(x:S),s(y:S)) -> IF_GCD(le(y:S,x:S),s(x:S),s(y:S))
->-> Narrowed pairs:
 GCD(s(0),s(s(x:S))) -> IF_GCD(ffalse,s(0),s(s(x:S)))
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
->->Original Pair:
 IF_GCD(ffalse,s(x4:S),s(x5:S)) -> GCD(minus(s(x5:S),s(x4:S)),s(x4:S))
->-> Narrowed pairs:
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
->->Original Pair:
 IF_GCD(ttrue,s(x4:S),s(x5:S)) -> GCD(minus(s(x4:S),s(x5:S)),s(x5:S))
->-> Narrowed pairs:
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))

Problem 1.3: 

SCC Processor:
-> Pairs:
 GCD(s(0),s(s(x:S))) -> IF_GCD(ffalse,s(0),s(s(x:S)))
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(s(0),s(s(x:S))) -> IF_GCD(ffalse,s(0),s(s(x:S)))
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 GCD(s(0),s(s(x:S))) -> IF_GCD(ffalse,s(0),s(s(x:S)))
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x: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(0,x:S) -> 0
 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:
 
[gcd](X1,X2) = 0
[if_gcd](X1,X2,X3) = 0
[le](X1,X2) = 2.X1 + 2.X2
[minus](X1,X2) = X1
[0] = 2
[fSNonEmpty] = 0
[false] = 2
[s](X) = 2.X + 2
[true] = 2
[GCD](X1,X2) = 2.X1 + X2 + 1
[IF_GCD](X1,X2,X3) = X2 + X3 + 2
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 GCD(s(s(y:S)),s(s(x:S))) -> IF_GCD(le(x:S,y:S),s(s(y:S)),s(s(x:S)))
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x: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(0,x:S) -> 0
 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:
 
[gcd](X1,X2) = 0
[if_gcd](X1,X2,X3) = 0
[le](X1,X2) = 0
[minus](X1,X2) = X1 + 1
[0] = 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2.X + 2
[true] = 0
[GCD](X1,X2) = 2.X1 + 2.X2 + 1
[IF_GCD](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ffalse,s(y:S),s(x:S)) -> GCD(minus(x:S,y:S),s(y:S))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
->->-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 GCD(s(y:S),s(0)) -> IF_GCD(ttrue,s(y:S),s(0))
 IF_GCD(ttrue,s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S))
-> Rules:
 gcd(0,y:S) -> y:S
 gcd(s(x:S),0) -> s(x:S)
 gcd(s(x:S),s(y:S)) -> if_gcd(le(y:S,x:S),s(x:S),s(y:S))
 if_gcd(ffalse,x:S,y:S) -> gcd(minus(y:S,x:S),x:S)
 if_gcd(ttrue,x:S,y:S) -> gcd(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(0,x:S) -> 0
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
-> Usable rules:
 minus(0,x:S) -> 0
 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:
 
[gcd](X1,X2) = 0
[if_gcd](X1,X2,X3) = 0
[le](X1,X2) = 0
[minus](X1,X2) = X1
[0] = 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2.X + 1
[true] = 2
[GCD](X1,X2) = 2.X1 + 2.X2 + 2
[IF_GCD](X1,X2,X3) = X2 + 2.X3 + 2
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0

Problem 1.3: 

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

The problem is finite.