/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S x:S y:S) (RULES *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ) Problem 1: Dependency Pairs Processor: -> Pairs: *#(O(x:S),y:S) -> *#(x:S,y:S) *#(O(x:S),y:S) -> O#(*(x:S,y:S)) *#(I(x:S),y:S) -> *#(x:S,y:S) *#(I(x:S),y:S) -> +#(O(*(x:S,y:S)),y:S) *#(I(x:S),y:S) -> O#(*(x:S,y:S)) +#(O(x:S),O(y:S)) -> +#(x:S,y:S) +#(O(x:S),O(y:S)) -> O#(+(x:S,y:S)) +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> O#(+(+(x:S,y:S),I(0))) -#(O(x:S),O(y:S)) -> -#(x:S,y:S) -#(O(x:S),O(y:S)) -> O#(-(x:S,y:S)) -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> O#(-(x:S,y:S)) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1: SCC Processor: -> Pairs: *#(O(x:S),y:S) -> *#(x:S,y:S) *#(O(x:S),y:S) -> O#(*(x:S,y:S)) *#(I(x:S),y:S) -> *#(x:S,y:S) *#(I(x:S),y:S) -> +#(O(*(x:S,y:S)),y:S) *#(I(x:S),y:S) -> O#(*(x:S,y:S)) +#(O(x:S),O(y:S)) -> +#(x:S,y:S) +#(O(x:S),O(y:S)) -> O#(+(x:S,y:S)) +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> O#(+(+(x:S,y:S),I(0))) -#(O(x:S),O(y:S)) -> -#(x:S,y:S) -#(O(x:S),O(y:S)) -> O#(-(x:S,y:S)) -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> O#(-(x:S,y:S)) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: -#(O(x:S),O(y:S)) -> -#(x:S,y:S) -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->->Cycle: ->->-> Pairs: +#(O(x:S),O(y:S)) -> +#(x:S,y:S) +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->->Cycle: ->->-> Pairs: *#(O(x:S),y:S) -> *#(x:S,y:S) *#(I(x:S),y:S) -> *#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 The problem is decomposed in 3 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: -#(O(x:S),O(y:S)) -> -#(x:S,y:S) -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 [O](X) = 2.X + 2 [0] = 1 [1] = 0 [I](X) = 2.X + 2 [-#](X1,X2) = X2 Problem 1.1: SCC Processor: -> Pairs: -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.1: Reduction Pair Processor: -> Pairs: -#(O(x:S),I(y:S)) -> -#(-(x:S,y:S),I(1)) -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 + X2 [O](X) = X + 2 [0] = 2 [1] = 0 [I](X) = X + 2 [-#](X1,X2) = 2.X1 + 2.X2 Problem 1.1: SCC Processor: -> Pairs: -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.1: Subterm Processor: -> Pairs: -#(O(x:S),I(y:S)) -> -#(x:S,y:S) -#(I(x:S),O(y:S)) -> -#(x:S,y:S) -#(I(x:S),I(y:S)) -> -#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Projection: pi(-#) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: +#(O(x:S),O(y:S)) -> +#(x:S,y:S) +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + 2.X2 [O](X) = X + 2 [0] = 0 [I](X) = X + 2 [+#](X1,X2) = X2 Problem 1.2: SCC Processor: -> Pairs: +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.2: Reduction Pair Processor: -> Pairs: +#(O(x:S),I(y:S)) -> +#(x:S,y:S) +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 [O](X) = X + 1 [0] = 1 [I](X) = X + 2 [+#](X1,X2) = X1 + X2 Problem 1.2: SCC Processor: -> Pairs: +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.2: Reduction Pair Processor: -> Pairs: +#(I(x:S),O(y:S)) -> +#(x:S,y:S) +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 [O](X) = X + 1 [0] = 0 [I](X) = X + 1 [+#](X1,X2) = 2.X1 + 2.X2 Problem 1.2: SCC Processor: -> Pairs: +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.2: Reduction Pair Processor: -> Pairs: +#(I(x:S),I(y:S)) -> +#(+(x:S,y:S),I(0)) +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 -> Usable rules: +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S O(0) -> 0 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X1 + X2 [O](X) = 2.X [0] = 0 [I](X) = 2.X + 2 [+#](X1,X2) = 2.X1 + 2.X2 Problem 1.2: SCC Processor: -> Pairs: +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(I(x:S),I(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 Problem 1.2: Subterm Processor: -> Pairs: +#(I(x:S),I(y:S)) -> +#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Projection: pi(+#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: *#(O(x:S),y:S) -> *#(x:S,y:S) *#(I(x:S),y:S) -> *#(x:S,y:S) -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Projection: pi(*#) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: *(O(x:S),y:S) -> O(*(x:S,y:S)) *(0,x:S) -> 0 *(I(x:S),y:S) -> +(O(*(x:S,y:S)),y:S) *(x:S,0) -> 0 +(O(x:S),O(y:S)) -> O(+(x:S,y:S)) +(O(x:S),I(y:S)) -> I(+(x:S,y:S)) +(0,x:S) -> x:S +(I(x:S),O(y:S)) -> I(+(x:S,y:S)) +(I(x:S),I(y:S)) -> O(+(+(x:S,y:S),I(0))) +(x:S,0) -> x:S -(O(x:S),O(y:S)) -> O(-(x:S,y:S)) -(O(x:S),I(y:S)) -> I(-(-(x:S,y:S),I(1))) -(0,x:S) -> 0 -(I(x:S),O(y:S)) -> I(-(x:S,y:S)) -(I(x:S),I(y:S)) -> O(-(x:S,y:S)) -(x:S,0) -> x:S O(0) -> 0 ->Strongly Connected Components: There is no strongly connected component The problem is finite.