/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) -> +(y:S,*(x:S,y:S))
-(0,y:S) -> 0
-(s(x:S),s(y:S)) -> -(x:S,y:S)
-(x:S,0) -> x:S
exp(x:S,0) -> s(0)
exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y: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),s(y:S)) -> -#(x:S,y:S)
 EXP(x:S,s(y:S)) -> *#(x:S,exp(x:S,y:S))
 EXP(x:S,s(y:S)) -> EXP(x:S,y:S)
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 EXP(x:S,s(y:S)) -> *#(x:S,exp(x:S,y:S))
 EXP(x:S,s(y:S)) -> EXP(x:S,y:S)
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))
->->Cycle:
->->-> Pairs:
 *#(s(x:S),y:S) -> *#(x:S,y:S)
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))
->->Cycle:
->->-> Pairs:
 EXP(x:S,s(y:S)) -> EXP(x:S,y:S)
->->-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(0,y:S) -> 0
 *(s(x:S),y:S) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y: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) -> +(y:S,*(x:S,y:S))
 -(0,y:S) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 exp(x:S,0) -> s(0)
 exp(x:S,s(y:S)) -> *(x:S,exp(x:S,y:S))
->Projection:
 pi(*#) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.