/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR x y)
(RULES
f(0) -> true
f(1) -> false
f(s(x)) -> f(x)
g(s(x),s(y)) -> if(f(x),s(x),s(y))
g(x,c(y)) -> g(x,g(s(c(y)),y))
if(false,x,y) -> y
if(true,x,y) -> x
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(x)) -> F(x)
 G(s(x),s(y)) -> F(x)
 G(s(x),s(y)) -> IF(f(x),s(x),s(y))
 G(x,c(y)) -> G(s(c(y)),y)
 G(x,c(y)) -> G(x,g(s(c(y)),y))
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x)) -> F(x)
 G(s(x),s(y)) -> F(x)
 G(s(x),s(y)) -> IF(f(x),s(x),s(y))
 G(x,c(y)) -> G(s(c(y)),y)
 G(x,c(y)) -> G(x,g(s(c(y)),y))
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x)) -> F(x)
->->-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->->Cycle:
->->-> Pairs:
 G(x,c(y)) -> G(s(c(y)),y)
 G(x,c(y)) -> G(x,g(s(c(y)),y))
->->-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(s(x)) -> F(x)
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Projection:
 pi(F) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(x,c(y)) -> G(s(c(y)),y)
 G(x,c(y)) -> G(x,g(s(c(y)),y))
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
-> Usable rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 0
[g](X1,X2) = 2.X2 + 2
[if](X1,X2,X3) = 2.X1 + X2 + X3 + 2
[0] = 2
[1] = 0
[c](X) = 2.X + 2
[false] = 0
[s](X) = 2
[true] = 0
[G](X1,X2) = X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 G(x,c(y)) -> G(x,g(s(c(y)),y))
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(x,c(y)) -> G(x,g(s(c(y)),y))
->->-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 G(x,c(y)) -> G(x,g(s(c(y)),y))
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
-> Usable rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 0
[g](X1,X2) = X1 + 2.X2
[if](X1,X2,X3) = 2.X1 + 2.X2 + X3
[0] = 2
[1] = 2
[c](X) = 2.X + 2
[false] = 0
[s](X) = 1
[true] = 0
[G](X1,X2) = 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> true
 f(1) -> false
 f(s(x)) -> f(x)
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))
 if(false,x,y) -> y
 if(true,x,y) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.