/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
add(0,y) -> y
add(s(x),y) -> s(add(x,y))
gcd(add(x,y),y) -> gcd(x,y)
gcd(0,x) -> x
gcd(x,0) -> x
gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
gcd(y,add(x,y)) -> gcd(x,y)
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 ADD(s(x),y) -> ADD(x,y)
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)

Conditional Termination Problem 2:
-> Pairs:
 Empty
-> QPairs:
 ADD(s(x),y) -> ADD(x,y)
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 ADD(s(x),y) -> ADD(x,y)
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
->->-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->->Cycle:
->->-> Pairs:
 ADD(s(x),y) -> ADD(x,y)
->->-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)


The problem is decomposed in 2 subproblems.

Problem 1.1.1: 

Reduction Triple Processor:
-> Pairs:
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-1+x_1 >= 0) /\ (1 >= 0))
->Interpretation:
 
[delta] = 2
[add](x_1,x_2) = 2.x_1+x_2
[false] = 1
[leq](x_1,x_2) = 1
[GCD](x_1,x_2) = x_1+x_2

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
->->-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)

Problem 1.1.1: 

Reduction Triple Processor:
-> Pairs:
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1+x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[add](x_1,x_2) = x_1
[false] = 1
[leq](x_1,x_2) = 0
[GCD](x_1,x_2) = 1+x_1+x_2

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GCD(y,add(x,y)) -> GCD(x,y)
->->-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)

Problem 1.1.1: 

Conditional Subterm Processor:
-> Pairs:
 GCD(y,add(x,y)) -> GCD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Projection:
 pi(GCD) = 2

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.1.2: 

Conditional Subterm Processor:
-> Pairs:
 ADD(s(x),y) -> ADD(x,y)
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Projection:
 pi(ADD) = 1

Problem 1.1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 ADD(s(x),y) -> ADD(x,y)
 GCD(add(x,y),y) -> GCD(x,y)
 GCD(x,y) -> GCD(y,x) | leq(y,x) -> false
 GCD(y,add(x,y)) -> GCD(x,y)
-> Rules:
 add(0,y) -> y
 add(s(x),y) -> s(add(x,y))
 gcd(add(x,y),y) -> gcd(x,y)
 gcd(0,x) -> x
 gcd(x,0) -> x
 gcd(x,y) -> gcd(y,x) | leq(y,x) -> false
 gcd(y,add(x,y)) -> gcd(x,y)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.