/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR x y z) (RULES -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ) Problem 1: Dependency Pairs Processor: -> Pairs: -#(s(x),s(y)) -> -#(x,y) GCD(s(x),s(y),z) -> -#(max(x,y),min(x,y)) GCD(s(x),s(y),z) -> GCD(-(max(x,y),min(x,y)),s(min(x,y)),z) GCD(s(x),s(y),z) -> MAX(x,y) GCD(s(x),s(y),z) -> MIN(x,y) GCD(s(x),y,s(z)) -> -#(max(x,z),min(x,z)) GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(s(x),y,s(z)) -> MAX(x,z) GCD(s(x),y,s(z)) -> MIN(x,z) GCD(x,s(y),s(z)) -> -#(max(y,z),min(y,z)) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) GCD(x,s(y),s(z)) -> MAX(y,z) GCD(x,s(y),s(z)) -> MIN(y,z) MAX(s(x),s(y)) -> MAX(x,y) MIN(s(x),s(y)) -> MIN(x,y) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 Problem 1: SCC Processor: -> Pairs: -#(s(x),s(y)) -> -#(x,y) GCD(s(x),s(y),z) -> -#(max(x,y),min(x,y)) GCD(s(x),s(y),z) -> GCD(-(max(x,y),min(x,y)),s(min(x,y)),z) GCD(s(x),s(y),z) -> MAX(x,y) GCD(s(x),s(y),z) -> MIN(x,y) GCD(s(x),y,s(z)) -> -#(max(x,z),min(x,z)) GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(s(x),y,s(z)) -> MAX(x,z) GCD(s(x),y,s(z)) -> MIN(x,z) GCD(x,s(y),s(z)) -> -#(max(y,z),min(y,z)) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) GCD(x,s(y),s(z)) -> MAX(y,z) GCD(x,s(y),s(z)) -> MIN(y,z) MAX(s(x),s(y)) -> MAX(x,y) MIN(s(x),s(y)) -> MIN(x,y) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MIN(s(x),s(y)) -> MIN(x,y) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->->Cycle: ->->-> Pairs: MAX(s(x),s(y)) -> MAX(x,y) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->->Cycle: ->->-> Pairs: -#(s(x),s(y)) -> -#(x,y) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->->Cycle: ->->-> Pairs: GCD(s(x),s(y),z) -> GCD(-(max(x,y),min(x,y)),s(min(x,y)),z) GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: MIN(s(x),s(y)) -> MIN(x,y) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Projection: pi(MIN) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: MAX(s(x),s(y)) -> MAX(x,y) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Projection: pi(MAX) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: -#(s(x),s(y)) -> -#(x,y) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Projection: pi(-#) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Reduction Pair Processor: -> Pairs: GCD(s(x),s(y),z) -> GCD(-(max(x,y),min(x,y)),s(min(x,y)),z) GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 -> Usable rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 [max](X1,X2) = X1 + X2 + 1/2 [min](X1,X2) = 1/2.X1 + 1/2.X2 [0] = 0 [s](X) = 2.X + 1 [GCD](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 Problem 1.4: SCC Processor: -> Pairs: GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 Problem 1.4: Reduction Pair Processor: -> Pairs: GCD(s(x),y,s(z)) -> GCD(-(max(x,z),min(x,z)),y,s(min(x,z))) GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 -> Usable rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 [max](X1,X2) = X1 + X2 [min](X1,X2) = X1 [0] = 0 [s](X) = 2.X + 2 [GCD](X1,X2,X3) = 2.X1 + 2.X2 + X3 Problem 1.4: SCC Processor: -> Pairs: GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) ->->-> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 Problem 1.4: Reduction Pair Processor: -> Pairs: GCD(x,s(y),s(z)) -> GCD(x,-(max(y,z),min(y,z)),s(min(y,z))) -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 -> Usable rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 [max](X1,X2) = X1 + X2 + 1 [min](X1,X2) = X1 [0] = 0 [s](X) = 2.X + 2 [GCD](X1,X2,X3) = 2.X2 + X3 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: -(s(x),s(y)) -> -(x,y) -(x,0) -> x gcd(0,0,z) -> z gcd(0,y,0) -> y gcd(s(x),s(y),z) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)),z) gcd(s(x),y,s(z)) -> gcd(-(max(x,z),min(x,z)),y,s(min(x,z))) gcd(x,0,0) -> x gcd(x,s(y),s(z)) -> gcd(x,-(max(y,z),min(y,z)),s(min(y,z))) max(0,y) -> y max(s(x),s(y)) -> s(max(x,y)) max(x,0) -> x min(0,y) -> 0 min(s(x),s(y)) -> s(min(x,y)) min(x,0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite.