/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
f(0) -> s(0)
f(s(x)) -> minus(s(x),g(f(x)))
g(0) -> 0
g(s(x)) -> minus(s(x),f(g(x)))
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(x)) -> F(x)
 F(s(x)) -> G(f(x))
 F(s(x)) -> MINUS(s(x),g(f(x)))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
 G(s(x)) -> MINUS(s(x),f(g(x)))
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x)) -> F(x)
 F(s(x)) -> G(f(x))
 F(s(x)) -> MINUS(s(x),g(f(x)))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
 G(s(x)) -> MINUS(s(x),f(g(x)))
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
->->-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->->Cycle:
->->-> Pairs:
 F(s(x)) -> F(x)
 F(s(x)) -> G(f(x))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
->->-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(s(x)) -> F(x)
 F(s(x)) -> G(f(x))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
-> Usable rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2.X + 2
[g](X) = 2.X + 1
[minus](X1,X2) = 2.X1 + 1
[0] = 0
[s](X) = 2.X + 2
[F](X) = X + 2
[G](X) = X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(s(x)) -> G(f(x))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x)) -> G(f(x))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
->->-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(s(x)) -> G(f(x))
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
-> Usable rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 2.X + 2
[g](X) = 2.X
[minus](X1,X2) = 2.X1
[0] = 2
[s](X) = 2.X + 2
[F](X) = X + 1
[G](X) = X

Problem 1.2: 

SCC Processor:
-> Pairs:
 G(s(x)) -> F(g(x))
 G(s(x)) -> G(x)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(s(x)) -> G(x)
->->-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(s(x)) -> G(x)
-> Rules:
 f(0) -> s(0)
 f(s(x)) -> minus(s(x),g(f(x)))
 g(0) -> 0
 g(s(x)) -> minus(s(x),f(g(x)))
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Projection:
 pi(G) = 1

Problem 1.2: 

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

The problem is finite.