/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
-(0,s(y)) -> 0
-(s(x),s(y)) -> -(x,y)
-(x,0) -> x
div(0,y) -> 0
div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
div(x,0) -> 0
if(false,x,y) -> y
if(true,x,y) -> x
lt(0,s(y)) -> true
lt(s(x),s(y)) -> lt(x,y)
lt(x,0) -> false
)

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 DIV(s(x),s(y)) -> -#(x,y)
 DIV(s(x),s(y)) -> DIV(-(x,y),s(y))
 DIV(s(x),s(y)) -> IF(lt(x,y),0,s(div(-(x,y),s(y))))
 DIV(s(x),s(y)) -> LT(x,y)
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false

Problem 1: 

SCC Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 DIV(s(x),s(y)) -> -#(x,y)
 DIV(s(x),s(y)) -> DIV(-(x,y),s(y))
 DIV(s(x),s(y)) -> IF(lt(x,y),0,s(div(-(x,y),s(y))))
 DIV(s(x),s(y)) -> LT(x,y)
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LT(s(x),s(y)) -> LT(x,y)
->->-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->->Cycle:
->->-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
->->-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->->Cycle:
->->-> Pairs:
 DIV(s(x),s(y)) -> DIV(-(x,y),s(y))
->->-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LT(s(x),s(y)) -> LT(x,y)
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Projection:
 pi(LT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Projection:
 pi(-#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(0,s(y)) -> 0
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 div(0,y) -> 0
 div(s(x),s(y)) -> if(lt(x,y),0,s(div(-(x,y),s(y))))
 div(x,0) -> 0
 if(false,x,y) -> y
 if(true,x,y) -> x
 lt(0,s(y)) -> true
 lt(s(x),s(y)) -> lt(x,y)
 lt(x,0) -> false
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.