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