/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
plus(0,x:S) -> x:S
plus(s(x:S),y:S) -> s(plus(x:S,y:S))
plus(x:S,0) -> x:S
plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
times(x:S,0) -> 0
times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
 TIMES(x:S,s(y:S)) -> PLUS(times(x:S,y:S),x:S)
 TIMES(x:S,s(y:S)) -> TIMES(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
 TIMES(x:S,s(y:S)) -> PLUS(times(x:S,y:S),x:S)
 TIMES(x:S,s(y:S)) -> TIMES(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
->->-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->->Cycle:
->->-> Pairs:
 TIMES(x:S,s(y:S)) -> TIMES(x:S,y:S)
->->-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
->->-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(x:S,s(y:S)) -> PLUS(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->Projection:
 pi(PLUS) = 2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 TIMES(x:S,s(y:S)) -> TIMES(x:S,y:S)
-> Rules:
 plus(0,x:S) -> x:S
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
 plus(x:S,0) -> x:S
 plus(x:S,s(y:S)) -> s(plus(x:S,y:S))
 times(x:S,0) -> 0
 times(x:S,s(y:S)) -> plus(times(x:S,y:S),x:S)
->Projection:
 pi(TIMES) = 2

Problem 1.2: 

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

The problem is finite.