/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 z)
(RULES
*(x,0) -> 0
*(x,s(y)) -> +(*(x,y),x)
-(s(x),s(y)) -> -(x,y)
-(x,0) -> x
f(x,0,z) -> z
f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
if(false,x,y) -> false
if(false,x,y) -> y
if(true,x,y) -> true
if(true,x,y) -> x
odd(0) -> false
odd(s(0)) -> true
odd(s(s(x))) -> odd(x)
pow(x,y) -> f(x,y,s(0))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *#(x,s(y)) -> *#(x,y)
 -#(s(x),s(y)) -> -#(x,y)
 F(x,s(y),z) -> *#(x,x)
 F(x,s(y),z) -> *#(x,z)
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
 F(x,s(y),z) -> F(x,y,*(x,z))
 F(x,s(y),z) -> HALF(s(y))
 F(x,s(y),z) -> IF(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 F(x,s(y),z) -> ODD(s(y))
 HALF(s(s(x))) -> HALF(x)
 ODD(s(s(x))) -> ODD(x)
 POW(x,y) -> F(x,y,s(0))
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))

Problem 1: 

SCC Processor:
-> Pairs:
 *#(x,s(y)) -> *#(x,y)
 -#(s(x),s(y)) -> -#(x,y)
 F(x,s(y),z) -> *#(x,x)
 F(x,s(y),z) -> *#(x,z)
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
 F(x,s(y),z) -> F(x,y,*(x,z))
 F(x,s(y),z) -> HALF(s(y))
 F(x,s(y),z) -> IF(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 F(x,s(y),z) -> ODD(s(y))
 HALF(s(s(x))) -> HALF(x)
 ODD(s(s(x))) -> ODD(x)
 POW(x,y) -> F(x,y,s(0))
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ODD(s(s(x))) -> ODD(x)
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->->Cycle:
->->-> Pairs:
 HALF(s(s(x))) -> HALF(x)
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->->Cycle:
->->-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->->Cycle:
->->-> Pairs:
 *#(x,s(y)) -> *#(x,y)
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->->Cycle:
->->-> Pairs:
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
 F(x,s(y),z) -> F(x,y,*(x,z))
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ODD(s(s(x))) -> ODD(x)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Projection:
 pi(ODD) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x))) -> HALF(x)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Projection:
 pi(HALF) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Projection:
 pi(-#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 *#(x,s(y)) -> *#(x,y)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Projection:
 pi(*#) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Reduction Pair Processor:
-> Pairs:
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
 F(x,s(y),z) -> F(x,y,*(x,z))
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
-> Usable rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = X2 + 1
[half](X) = X
[+](X1,X2) = 2.X1
[0] = 2
[s](X) = 2.X + 2
[F](X1,X2,X3) = X2 + X3

Problem 1.5: 

SCC Processor:
-> Pairs:
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
->->-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))

Problem 1.5: 

Reduction Pair Processor:
-> Pairs:
 F(x,s(y),z) -> F(*(x,x),half(s(y)),z)
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
-> Usable rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[half](X) = 1/2.X
[+](X1,X2) = 0
[0] = 0
[s](X) = 2.X + 2
[F](X1,X2,X3) = 2.X2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(x,0) -> 0
 *(x,s(y)) -> +(*(x,y),x)
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 f(x,0,z) -> z
 f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x))) -> s(half(x))
 if(false,x,y) -> false
 if(false,x,y) -> y
 if(true,x,y) -> true
 if(true,x,y) -> x
 odd(0) -> false
 odd(s(0)) -> true
 odd(s(s(x))) -> odd(x)
 pow(x,y) -> f(x,y,s(0))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.