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


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YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
*(0,y:S) -> 0
*(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
fact(0) -> s(0)
fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
p(s(x:S)) -> x:S
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(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:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
 *#(s(x:S),y:S) -> +#(*(x:S,y:S),y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 FACT(s(x:S)) -> *#(s(x:S),fact(p(s(x:S))))
 FACT(s(x:S)) -> FACT(p(s(x:S)))
 FACT(s(x:S)) -> P(s(x:S))
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
 *#(s(x:S),y:S) -> +#(*(x:S,y:S),y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 FACT(s(x:S)) -> *#(s(x:S),fact(p(s(x:S))))
 FACT(s(x:S)) -> FACT(p(s(x:S)))
 FACT(s(x:S)) -> P(s(x:S))
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->->Cycle:
->->-> Pairs:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->->Cycle:
->->-> Pairs:
 FACT(s(x:S)) -> FACT(p(s(x:S)))
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->Projection:
 pi(*#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(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:
 FACT(s(x:S)) -> FACT(p(s(x:S)))
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
-> Usable rules:
 p(s(x:S)) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = 0
[fact](X) = 0
[p](X) = 1/2.X
[0] = 0
[fSNonEmpty] = 0
[s](X) = 2.X + 1/2
[*#](X1,X2) = 0
[+#](X1,X2) = 0
[FACT](X) = 2.X
[P](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(*(x:S,y:S),y:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fact(0) -> s(0)
 fact(s(x:S)) -> *(s(x:S),fact(p(s(x:S))))
 p(s(x:S)) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.