/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
-(0,s(y:S)) -> 0
-(s(x:S),s(y:S)) -> -(x:S,y:S)
-(x:S,0) -> x:S
div(0,y:S) -> 0
div(s(x:S),s(y:S)) -> if(lt(x:S,y:S),0,s(div(-(x:S,y:S),s(y:S))))
div(x:S,0) -> 0
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
lt(0,s(y:S)) -> ttrue
lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
lt(x:S,0) -> ffalse
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 -(0,s(y:S)) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 div(0,y:S) -> 0
 div(s(x:S),s(y:S)) -> if(lt(x:S,y:S),0,s(div(-(x:S,y:S),s(y:S))))
 div(x:S,0) -> 0
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 lt(0,s(y:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse
-> 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),s(y:S)) -> -#(x:S,y:S)
 DIV(s(x:S),s(y:S)) -> -#(x:S,y:S)
 DIV(s(x:S),s(y:S)) -> DIV(-(x:S,y:S),s(y:S))
 DIV(s(x:S),s(y:S)) -> IF(lt(x:S,y:S),0,s(div(-(x:S,y:S),s(y:S))))
 DIV(s(x:S),s(y:S)) -> LT(x:S,y:S)
 LT(s(x:S),s(y:S)) -> LT(x:S,y:S)
-> Rules:
 -(0,s(y:S)) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 div(0,y:S) -> 0
 div(s(x:S),s(y:S)) -> if(lt(x:S,y:S),0,s(div(-(x:S,y:S),s(y:S))))
 div(x:S,0) -> 0
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 lt(0,s(y:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 DIV(s(x:S),s(y:S)) -> DIV(-(x:S,y:S),s(y:S))
-> Rules:
 -(0,s(y:S)) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 div(0,y:S) -> 0
 div(s(x:S),s(y:S)) -> if(lt(x:S,y:S),0,s(div(-(x:S,y:S),s(y:S))))
 div(x:S,0) -> 0
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 lt(0,s(y:S)) -> ttrue
 lt(s(x:S),s(y:S)) -> lt(x:S,y:S)
 lt(x:S,0) -> ffalse
-> Usable rules:
 -(0,s(y:S)) -> 0
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = 2.X1
[div](X1,X2) = 0
[if](X1,X2,X3) = 0
[lt](X1,X2) = 0
[0] = 2
[fSNonEmpty] = 0
[false] = 0
[s](X) = 2.X + 2
[true] = 0
[-#](X1,X2) = 0
[DIV](X1,X2) = 2.X1
[IF](X1,X2,X3) = 0
[LT](X1,X2) = 0

Problem 1.3: 

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

The problem is finite.