/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
a -> h(b)
a -> h(c)
f(x) -> y | a -> h(y)
g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 G(x,b) -> F(c) | f(b) -> x, x -> c
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c

Conditional Termination Problem 2:
-> Pairs:
 F(x) -> A
 G(x,b) -> F(b)
-> QPairs:
 G(x,b) -> F(c) | f(b) -> x, x -> c
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(x,b) -> F(c) | f(b) -> x, x -> c
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c

Problem 1.1: 

Simplification and Narrowing on Condition Processor:
-> Pairs:
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Narrowed Pairs:
->->Original Pair:
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
->-> Narrowed pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(b), b -> x2, x2 -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(b), b -> x2, x2 -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(b), b -> x2, x2 -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(b), b -> x2, x2 -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1+x_1 >= 0) /\ (0 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[f](x_1) = 1
[b] = -1
[c] = 0
[h](x_1) = 1+x_1
[G](x_1,x_2) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
 G(x2,b) -> G(f(c),x2) | a -> h(c), c -> x2, x2 -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1+x_1 >= 0) /\ (-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[f](x_1) = 0
[b] = 0
[c] = -1
[h](x_1) = x_1
[G](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) -> c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (0 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[f](x_1) = 1+x_1
[b] = 1
[c] = 0
[h](x_1) = x_1
[G](x_1,x_2) = 2.x_1+x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(x) -> A
 G(x,b) -> F(b)
-> QPairs:
 G(x,b) -> F(c) | f(b) -> x, x -> c
 G(x,b) -> G(f(c),x) | f(b) -> x, x -> c
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x) -> y | a -> h(y)
 g(x,b) -> g(f(c),x) | f(b) -> x, x -> c
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.