/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 u:S x:S y:S z:S) (RULES -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ) Problem 1: Innermost Equivalent Processor: -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) -> 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) <=#(s(x:S),s(y:S)) -> <=#(x:S,y:S) F(s(x:S),0,z:S,u:S) -> -#(z:S,s(x:S)) F(s(x:S),0,z:S,u:S) -> F(x:S,u:S,-(z:S,s(x:S)),u:S) F(s(x:S),s(y:S),z:S,u:S) -> -#(y:S,x:S) F(s(x:S),s(y:S),z:S,u:S) -> <=#(x:S,y:S) F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(x:S,u:S,z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> IF(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) PERFECTP(s(x:S)) -> F(x:S,s(0),s(x:S),s(x:S)) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) Problem 1: SCC Processor: -> Pairs: -#(s(x:S),s(y:S)) -> -#(x:S,y:S) <=#(s(x:S),s(y:S)) -> <=#(x:S,y:S) F(s(x:S),0,z:S,u:S) -> -#(z:S,s(x:S)) F(s(x:S),0,z:S,u:S) -> F(x:S,u:S,-(z:S,s(x:S)),u:S) F(s(x:S),s(y:S),z:S,u:S) -> -#(y:S,x:S) F(s(x:S),s(y:S),z:S,u:S) -> <=#(x:S,y:S) F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(x:S,u:S,z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> IF(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) PERFECTP(s(x:S)) -> F(x:S,s(0),s(x:S),s(x:S)) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: <=#(s(x:S),s(y:S)) -> <=#(x:S,y:S) ->->-> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->->Cycle: ->->-> Pairs: -#(s(x:S),s(y:S)) -> -#(x:S,y:S) ->->-> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->->Cycle: ->->-> Pairs: F(s(x:S),0,z:S,u:S) -> F(x:S,u:S,-(z:S,s(x:S)),u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(x:S,u:S,z:S,u:S) ->->-> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: <=#(s(x:S),s(y:S)) -> <=#(x:S,y:S) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Projection: pi(<=#) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->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: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Projection: pi(-#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: F(s(x:S),0,z:S,u:S) -> F(x:S,u:S,-(z:S,s(x:S)),u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) F(s(x:S),s(y:S),z:S,u:S) -> F(x:S,u:S,z:S,u:S) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Projection: pi(F) = 1 Problem 1.3: SCC Processor: -> Pairs: F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) ->->-> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) Problem 1.3: Reduction Pairs Processor: -> Pairs: F(s(x:S),s(y:S),z:S,u:S) -> F(s(x:S),-(y:S,x:S),z:S,u:S) -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) -> Usable rules: -(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) = X1 [<=](X1,X2) = 0 [f](X1,X2,X3,X4) = 0 [if](X1,X2,X3) = 0 [perfectp](X) = 0 [0] = 0 [fSNonEmpty] = 0 [false] = 0 [s](X) = X + 2 [true] = 0 [-#](X1,X2) = 0 [<=#](X1,X2) = 0 [F](X1,X2,X3,X4) = 2.X2 [IF](X1,X2,X3) = 0 [PERFECTP](X) = 0 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S <=(0,y:S) -> ttrue <=(s(x:S),0) -> ffalse <=(s(x:S),s(y:S)) -> <=(x:S,y:S) f(0,y:S,0,u:S) -> ttrue f(0,y:S,s(z:S),u:S) -> ffalse f(s(x:S),0,z:S,u:S) -> f(x:S,u:S,-(z:S,s(x:S)),u:S) f(s(x:S),s(y:S),z:S,u:S) -> if(<=(x:S,y:S),f(s(x:S),-(y:S,x:S),z:S,u:S),f(x:S,u:S,z:S,u:S)) if(ffalse,x:S,y:S) -> y:S if(ttrue,x:S,y:S) -> x:S perfectp(0) -> ffalse perfectp(s(x:S)) -> f(x:S,s(0),s(x:S),s(x:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.