/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 x y)
(RULES
*(0,y) -> 0
*(p(x),y) -> +(*(x,y),minus(y))
*(s(x),y) -> +(*(x,y),y)
+(0,y) -> y
+(p(x),y) -> p(+(x,y))
+(s(x),y) -> s(+(x,y))
minus(0) -> 0
minus(p(x)) -> s(minus(x))
minus(s(x)) -> p(minus(x))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *#(p(x),y) -> *#(x,y)
 *#(p(x),y) -> +#(*(x,y),minus(y))
 *#(p(x),y) -> MINUS(y)
 *#(s(x),y) -> *#(x,y)
 *#(s(x),y) -> +#(*(x,y),y)
 +#(p(x),y) -> +#(x,y)
 +#(s(x),y) -> +#(x,y)
 MINUS(p(x)) -> MINUS(x)
 MINUS(s(x)) -> MINUS(x)
-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))

Problem 1: 

SCC Processor:
-> Pairs:
 *#(p(x),y) -> *#(x,y)
 *#(p(x),y) -> +#(*(x,y),minus(y))
 *#(p(x),y) -> MINUS(y)
 *#(s(x),y) -> *#(x,y)
 *#(s(x),y) -> +#(*(x,y),y)
 +#(p(x),y) -> +#(x,y)
 +#(s(x),y) -> +#(x,y)
 MINUS(p(x)) -> MINUS(x)
 MINUS(s(x)) -> MINUS(x)
-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(p(x)) -> MINUS(x)
 MINUS(s(x)) -> MINUS(x)
->->-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))
->->Cycle:
->->-> Pairs:
 +#(p(x),y) -> +#(x,y)
 +#(s(x),y) -> +#(x,y)
->->-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))
->->Cycle:
->->-> Pairs:
 *#(p(x),y) -> *#(x,y)
 *#(s(x),y) -> *#(x,y)
->->-> Rules:
 *(0,y) -> 0
 *(p(x),y) -> +(*(x,y),minus(y))
 *(s(x),y) -> +(*(x,y),y)
 +(0,y) -> y
 +(p(x),y) -> p(+(x,y))
 +(s(x),y) -> s(+(x,y))
 minus(0) -> 0
 minus(p(x)) -> s(minus(x))
 minus(s(x)) -> p(minus(x))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.