/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S X:S Y:S)
(RULES
gcd(0,Y:S) -> 0
gcd(s(X:S),0) -> s(X:S)
gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
pred(s(X:S)) -> X:S
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> 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(le(Y:S,X:S),s(X:S),s(Y:S))
 GCD(s(X:S),s(Y:S)) -> LE(Y:S,X:S)
 IF(ffalse,s(X:S),s(Y:S)) -> GCD(minus(Y:S,X:S),s(X:S))
 IF(ffalse,s(X:S),s(Y:S)) -> MINUS(Y:S,X:S)
 IF(ttrue,s(X:S),s(Y:S)) -> GCD(minus(X:S,Y:S),s(Y:S))
 IF(ttrue,s(X:S),s(Y:S)) -> MINUS(X:S,Y:S)
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
 MINUS(X:S,s(Y:S)) -> MINUS(X:S,Y:S)
 MINUS(X:S,s(Y:S)) -> PRED(minus(X:S,Y:S))
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S

Problem 1: 

SCC Processor:
-> Pairs:
 GCD(s(X:S),s(Y:S)) -> IF(le(Y:S,X:S),s(X:S),s(Y:S))
 GCD(s(X:S),s(Y:S)) -> LE(Y:S,X:S)
 IF(ffalse,s(X:S),s(Y:S)) -> GCD(minus(Y:S,X:S),s(X:S))
 IF(ffalse,s(X:S),s(Y:S)) -> MINUS(Y:S,X:S)
 IF(ttrue,s(X:S),s(Y:S)) -> GCD(minus(X:S,Y:S),s(Y:S))
 IF(ttrue,s(X:S),s(Y:S)) -> MINUS(X:S,Y:S)
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
 MINUS(X:S,s(Y:S)) -> MINUS(X:S,Y:S)
 MINUS(X:S,s(Y:S)) -> PRED(minus(X:S,Y:S))
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(X:S,s(Y:S)) -> MINUS(X:S,Y:S)
->->-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->->Cycle:
->->-> Pairs:
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
->->-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->->Cycle:
->->-> Pairs:
 GCD(s(X:S),s(Y:S)) -> IF(le(Y:S,X:S),s(X:S),s(Y:S))
 IF(ffalse,s(X:S),s(Y:S)) -> GCD(minus(Y:S,X:S),s(X:S))
 IF(ttrue,s(X:S),s(Y:S)) -> GCD(minus(X:S,Y:S),s(Y:S))
->->-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(X:S,s(Y:S)) -> MINUS(X:S,Y:S)
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->Projection:
 pi(MINUS) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> 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) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->Projection:
 pi(LE) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 GCD(s(X:S),s(Y:S)) -> IF(le(Y:S,X:S),s(X:S),s(Y:S))
 IF(ffalse,s(X:S),s(Y:S)) -> GCD(minus(Y:S,X:S),s(X:S))
 IF(ttrue,s(X:S),s(Y:S)) -> GCD(minus(X:S,Y:S),s(Y:S))
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> 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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gcd](X1,X2) = 0
[if](X1,X2,X3) = 0
[le](X1,X2) = 0
[minus](X1,X2) = 2.X1 + 1
[pred](X) = X
[0] = 1
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2.X + 2
[true] = 0
[GCD](X1,X2) = X1 + X2 + 2
[IF](X1,X2,X3) = 2.X1 + X2 + X3 + 1
[LE](X1,X2) = 0
[MINUS](X1,X2) = 0
[PRED](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF(ffalse,s(X:S),s(Y:S)) -> GCD(minus(Y:S,X:S),s(X:S))
 IF(ttrue,s(X:S),s(Y:S)) -> GCD(minus(X:S,Y:S),s(Y:S))
-> Rules:
 gcd(0,Y:S) -> 0
 gcd(s(X:S),0) -> s(X:S)
 gcd(s(X:S),s(Y:S)) -> if(le(Y:S,X:S),s(X:S),s(Y:S))
 if(ffalse,s(X:S),s(Y:S)) -> gcd(minus(Y:S,X:S),s(X:S))
 if(ttrue,s(X:S),s(Y:S)) -> gcd(minus(X:S,Y:S),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(X:S,0) -> X:S
 minus(X:S,s(Y:S)) -> pred(minus(X:S,Y:S))
 pred(s(X:S)) -> X:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.