/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 b x y)
(RULES
div(0,s(y)) -> 0
div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
f(x,0,b) -> x
f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
minus(0,x) -> 0
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
minus(x,x) -> 0
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
 DIV(s(x),s(y)) -> MINUS(x,y)
 F(x,s(y),b) -> DIV(f(x,minus(s(y),s(0)),b),b)
 F(x,s(y),b) -> F(x,minus(s(y),s(0)),b)
 F(x,s(y),b) -> MINUS(s(y),s(0))
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0

Problem 1: 

SCC Processor:
-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
 DIV(s(x),s(y)) -> MINUS(x,y)
 F(x,s(y),b) -> DIV(f(x,minus(s(y),s(0)),b),b)
 F(x,s(y),b) -> F(x,minus(s(y),s(0)),b)
 F(x,s(y),b) -> MINUS(s(y),s(0))
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
->->-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->->Cycle:
->->-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
->->-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->->Cycle:
->->-> Pairs:
 F(x,s(y),b) -> F(x,minus(s(y),s(0)),b)
->->-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 DIV(s(x),s(y)) -> DIV(minus(x,y),s(y))
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
-> Usable rules:
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = X1
[0] = 0
[s](X) = X + 1
[DIV](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 F(x,s(y),b) -> F(x,minus(s(y),s(0)),b)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Narrowed Pairs:
->->Original Pair:
 F(x,s(y),b) -> F(x,minus(s(y),s(0)),b)
->-> Narrowed pairs:
 F(x4,s(0),x3) -> F(x4,0,x3)
 F(x4,s(x),x3) -> F(x4,minus(x,0),x3)

Problem 1.3: 

SCC Processor:
-> Pairs:
 F(x4,s(0),x3) -> F(x4,0,x3)
 F(x4,s(x),x3) -> F(x4,minus(x,0),x3)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(x4,s(x),x3) -> F(x4,minus(x,0),x3)
->->-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 F(x4,s(x),x3) -> F(x4,minus(x,0),x3)
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
-> Usable rules:
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = 2.X1 + X2 + 1
[0] = 0
[s](X) = 2.X + 2
[F](X1,X2,X3) = 2.X2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div(0,s(y)) -> 0
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 f(x,0,b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0)),b),b)
 minus(0,x) -> 0
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 minus(x,x) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.