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