/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 z) (RULES +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ) Problem 1: Dependency Pairs Processor: -> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) +#(p1,+(p1,x)) -> +#(p2,x) +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1: SCC Processor: -> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) +#(p1,+(p1,x)) -> +#(p2,x) +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p1,+(p1,x)) -> +#(p2,x) +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->->Cycle: ->->-> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 The problem is decomposed in 2 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p1,+(p1,x)) -> +#(p2,x) +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 2 [p1] = 2 [p10] = 1 [p2] = 0 [p5] = 0 [+#](X1,X2) = X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p1,+(p2,+(p2,x))) -> +#(p5,x) +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = 2.X1 + X2 + 2 [p1] = 2 [p10] = 2 [p2] = 2 [p5] = 2 [+#](X1,X2) = X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p10,+(p1,x)) -> +#(p1,+(p10,x)) +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p1,+(p2,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p1,+(p5,x)) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p1,+(p5,x)) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p10,+(p1,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 1 [p1] = 0 [p10] = 1 [p2] = 1 [p5] = 2 [+#](X1,X2) = 2.X1 + 2.X2 Problem 1.1: SCC Processor: -> Pairs: +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p10,+(p2,x)) -> +#(p10,x) +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 1 [p1] = 0 [p10] = 1 [p2] = 1 [p5] = 0 [+#](X1,X2) = X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p10,+(p2,x)) -> +#(p2,+(p10,x)) +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 2 [p1] = 1 [p10] = 2 [p2] = 0 [p5] = 1 [+#](X1,X2) = 2.X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p10,+(p5,x)) -> +#(p10,x) +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 2 [p1] = 1 [p10] = 2 [p2] = 1 [p5] = 1 [+#](X1,X2) = 2.X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p10,+(p5,x)) -> +#(p5,+(p10,x)) +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 2 [p1] = 2 [p10] = 0 [p2] = 2 [p5] = 2 [+#](X1,X2) = X1 + 2.X2 Problem 1.1: SCC Processor: -> Pairs: +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) +#(p5,+(p5,x)) -> +#(p10,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p2,+(p1,x)) -> +#(p2,x) +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 + 2 [p1] = 1 [p10] = 1 [p2] = 0 [p5] = 1 [+#](X1,X2) = 2.X1 + 2.X2 Problem 1.1: SCC Processor: -> Pairs: +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Reduction Pair Processor: -> Pairs: +#(p2,+(p2,+(p2,x))) -> +#(p5,x) +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 -> Usable rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = 2.X1 + X2 + 1 [p1] = 2 [p10] = 2 [p2] = 2 [p5] = 2 [+#](X1,X2) = 2.X1 + X2 Problem 1.1: SCC Processor: -> Pairs: +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p2,+(p5,x)) +#(p5,+(p2,x)) -> +#(p5,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p5,x) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 Problem 1.1: Subterm Processor: -> Pairs: +#(p5,+(p1,x)) -> +#(p5,x) +#(p5,+(p2,x)) -> +#(p5,x) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Projection: pi(+#) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Projection: pi(+#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(p1,+(p1,x)) -> +(p2,x) +(p1,+(p2,+(p2,x))) -> +(p5,x) +(p1,+(p2,p2)) -> p5 +(p1,p1) -> p2 +(p10,+(p1,x)) -> +(p1,+(p10,x)) +(p10,+(p2,x)) -> +(p2,+(p10,x)) +(p10,+(p5,x)) -> +(p5,+(p10,x)) +(p10,p1) -> +(p1,p10) +(p10,p2) -> +(p2,p10) +(p10,p5) -> +(p5,p10) +(p2,+(p1,x)) -> +(p1,+(p2,x)) +(p2,+(p2,+(p2,x))) -> +(p1,+(p5,x)) +(p2,+(p2,p2)) -> +(p1,p5) +(p2,p1) -> +(p1,p2) +(p5,+(p1,x)) -> +(p1,+(p5,x)) +(p5,+(p2,x)) -> +(p2,+(p5,x)) +(p5,+(p5,x)) -> +(p10,x) +(p5,p1) -> +(p1,p5) +(p5,p2) -> +(p2,p5) +(p5,p5) -> p10 ->Strongly Connected Components: There is no strongly connected component The problem is finite.