/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
cond1(true,x,y) -> cond2(gr(y,0),x,y)
cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
gr(0,x) -> false
gr(s(x),0) -> true
gr(s(x),s(y)) -> gr(x,y)
p(0) -> 0
p(s(x)) -> x
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 COND1(true,x,y) -> COND2(gr(y,0),x,y)
 COND1(true,x,y) -> GR(y,0)
 COND2(false,x,y) -> COND1(gr(x,0),p(x),y)
 COND2(false,x,y) -> GR(x,0)
 COND2(false,x,y) -> P(x)
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
 COND2(true,x,y) -> GR(y,0)
 COND2(true,x,y) -> P(y)
 GR(s(x),s(y)) -> GR(x,y)
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x

Problem 1: 

SCC Processor:
-> Pairs:
 COND1(true,x,y) -> COND2(gr(y,0),x,y)
 COND1(true,x,y) -> GR(y,0)
 COND2(false,x,y) -> COND1(gr(x,0),p(x),y)
 COND2(false,x,y) -> GR(x,0)
 COND2(false,x,y) -> P(x)
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
 COND2(true,x,y) -> GR(y,0)
 COND2(true,x,y) -> P(y)
 GR(s(x),s(y)) -> GR(x,y)
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 GR(s(x),s(y)) -> GR(x,y)
->->-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->->Cycle:
->->-> Pairs:
 COND1(true,x,y) -> COND2(gr(y,0),x,y)
 COND2(false,x,y) -> COND1(gr(x,0),p(x),y)
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
->->-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 GR(s(x),s(y)) -> GR(x,y)
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Projection:
 pi(GR) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 COND1(true,x,y) -> COND2(gr(y,0),x,y)
 COND2(false,x,y) -> COND1(gr(x,0),p(x),y)
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
-> Usable rules:
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gr](X1,X2) = X1
[p](X) = 1/2.X + 1/2
[0] = 1
[false] = 0
[s](X) = 2.X + 2
[true] = 2
[COND1](X1,X2,X3) = 1/2.X1 + X2
[COND2](X1,X2,X3) = X2 + 1/2

Problem 1.2: 

SCC Processor:
-> Pairs:
 COND2(false,x,y) -> COND1(gr(x,0),p(x),y)
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
->->-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 COND2(true,x,y) -> COND2(gr(y,0),x,p(y))
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
-> Usable rules:
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[gr](X1,X2) = 1/2.X1
[p](X) = 1/2.X + 1/2
[0] = 1/2
[false] = 0
[s](X) = 2.X + 2
[true] = 1
[COND2](X1,X2,X3) = 2.X1 + 2.X3

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 cond1(true,x,y) -> cond2(gr(y,0),x,y)
 cond2(false,x,y) -> cond1(gr(x,0),p(x),y)
 cond2(true,x,y) -> cond2(gr(y,0),x,p(y))
 gr(0,x) -> false
 gr(s(x),0) -> true
 gr(s(x),s(y)) -> gr(x,y)
 p(0) -> 0
 p(s(x)) -> x
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.