/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 x y)
(RULES
+(x,0) -> x
+(x,s(y)) -> s(+(x,y))
f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> +(p(g(x)),q(g(x)))
f(s(s(x))) -> p(h(g(x)))
g(0) -> pair(s(0),s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
h(x) -> pair(+(p(x),q(x)),p(x))
p(pair(x,y)) -> x
q(pair(x,y)) -> y
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
 F(s(s(x))) -> +#(p(g(x)),q(g(x)))
 F(s(s(x))) -> G(x)
 F(s(s(x))) -> H(g(x))
 F(s(s(x))) -> P(g(x))
 F(s(s(x))) -> P(h(g(x)))
 F(s(s(x))) -> Q(g(x))
 G(s(x)) -> +#(p(g(x)),q(g(x)))
 G(s(x)) -> G(x)
 G(s(x)) -> H(g(x))
 G(s(x)) -> P(g(x))
 G(s(x)) -> Q(g(x))
 H(x) -> +#(p(x),q(x))
 H(x) -> P(x)
 H(x) -> Q(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
 F(s(s(x))) -> +#(p(g(x)),q(g(x)))
 F(s(s(x))) -> G(x)
 F(s(s(x))) -> H(g(x))
 F(s(s(x))) -> P(g(x))
 F(s(s(x))) -> P(h(g(x)))
 F(s(s(x))) -> Q(g(x))
 G(s(x)) -> +#(p(g(x)),q(g(x)))
 G(s(x)) -> G(x)
 G(s(x)) -> H(g(x))
 G(s(x)) -> P(g(x))
 G(s(x)) -> Q(g(x))
 H(x) -> +#(p(x),q(x))
 H(x) -> P(x)
 H(x) -> Q(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x,s(y)) -> +#(x,y)
->->-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->->Cycle:
->->-> Pairs:
 G(s(x)) -> G(x)
->->-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x,s(y)) -> +#(x,y)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(s(x)) -> G(x)
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->Projection:
 pi(G) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(x,0) -> x
 +(x,s(y)) -> s(+(x,y))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x))) -> +(p(g(x)),q(g(x)))
 f(s(s(x))) -> p(h(g(x)))
 g(0) -> pair(s(0),s(0))
 g(s(x)) -> h(g(x))
 g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 h(x) -> pair(+(p(x),q(x)),p(x))
 p(pair(x,y)) -> x
 q(pair(x,y)) -> y
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.