YES Problem 1: (VAR v_NonEmpty:S DDX:S DX:S DY:S X:S Y:S) (RULES din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ) Problem 1: Innermost Equivalent Processor: -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(der(X:S))) -> U41(din(der(X:S)),X:S) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U21(dout(DX:S),X:S,Y:S) -> U22(din(der(Y:S)),X:S,Y:S,DX:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> U32(din(der(Y:S)),X:S,Y:S,DX:S) U41(dout(DX:S),X:S) -> DIN(der(DX:S)) U41(dout(DX:S),X:S) -> U42(din(der(DX:S)),X:S,DX:S) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: SCC Processor: -> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(der(X:S))) -> U41(din(der(X:S)),X:S) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U21(dout(DX:S),X:S,Y:S) -> U22(din(der(Y:S)),X:S,Y:S,DX:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> U32(din(der(Y:S)),X:S,Y:S,DX:S) U41(dout(DX:S),X:S) -> DIN(der(DX:S)) U41(dout(DX:S),X:S) -> U42(din(der(DX:S)),X:S,DX:S) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(der(X:S))) -> U41(din(der(X:S)),X:S) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U41(dout(DX:S),X:S) -> DIN(der(DX:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(der(X:S))) -> U41(din(der(X:S)),X:S) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U41(dout(DX:S),X:S) -> DIN(der(DX:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = X [u21](X1,X2,X3) = 1 [u22](X1,X2,X3,X4) = X1 [u31](X1,X2,X3) = X1 [u32](X1,X2,X3,X4) = 2 [u41](X1,X2) = X1 [u42](X1,X2,X3) = 2 [der](X) = 1 [dout](X) = 2 [fSNonEmpty] = 0 [plus](X1,X2) = 0 [times](X1,X2) = 2.X1 + 2 [DIN](X) = 2.X + 2 [U21](X1,X2,X3) = 2.X1 + 2 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = 2.X1 + 2 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = X1 + 2 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U41(dout(DX:S),X:S) -> DIN(der(DX:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(der(X:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = 2.X + 2 [u21](X1,X2,X3) = 2.X1 + 2.X3 + 2 [u22](X1,X2,X3,X4) = 2.X3 + 2 [u31](X1,X2,X3) = 2.X1 + X3 + 2 [u32](X1,X2,X3,X4) = X3 + 2 [u41](X1,X2) = 2.X1 + 2 [u42](X1,X2,X3) = 1 [der](X) = 2.X + 2 [dout](X) = 0 [fSNonEmpty] = 0 [plus](X1,X2) = 2.X1 + X2 + 2 [times](X1,X2) = 2.X1 + 2.X2 + 2 [DIN](X) = X [U21](X1,X2,X3) = X2 + 2.X3 + 2 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = 0 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(plus(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = 2.X [u21](X1,X2,X3) = X2 + 2.X3 + 2 [u22](X1,X2,X3,X4) = X2 + 2.X3 + 2 [u31](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [u32](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2 [u41](X1,X2) = X2 + 2 [u42](X1,X2,X3) = 2 [der](X) = X + 2 [dout](X) = 2 [fSNonEmpty] = 0 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [times](X1,X2) = 2.X1 + 2.X2 + 2 [DIN](X) = 2.X + 2 [U21](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = 0 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(plus(X:S,Y:S))) -> U21(din(der(X:S)),X:S,Y:S) DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = X + 1 [u21](X1,X2,X3) = 2.X2 + X3 + 2 [u22](X1,X2,X3,X4) = X2 + X3 + 2 [u31](X1,X2,X3) = 2.X2 + 2 [u32](X1,X2,X3,X4) = 2 [u41](X1,X2) = X1 + 1 [u42](X1,X2,X3) = 2 [der](X) = X + 2 [dout](X) = 2 [fSNonEmpty] = 0 [plus](X1,X2) = 2.X1 + X2 + 2 [times](X1,X2) = 2.X1 + 2.X2 + 2 [DIN](X) = X + 2 [U21](X1,X2,X3) = X1 + X3 + 2 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = 2.X1 + X3 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = 0 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U21(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(times(X:S,Y:S))) -> DIN(der(X:S)) DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = 2.X + 2 [u21](X1,X2,X3) = 2.X3 + 2 [u22](X1,X2,X3,X4) = X1 [u31](X1,X2,X3) = 2.X1 + 2 [u32](X1,X2,X3,X4) = 2 [u41](X1,X2) = X1 [u42](X1,X2,X3) = 2 [der](X) = X [dout](X) = 2 [fSNonEmpty] = 0 [plus](X1,X2) = 2.X2 + 2 [times](X1,X2) = 2.X1 + 2.X2 + 2 [DIN](X) = 2.X + 2 [U21](X1,X2,X3) = 0 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = 0 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) ->->-> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) Problem 1: Reduction Pairs Processor: -> Pairs: DIN(der(times(X:S,Y:S))) -> U31(din(der(X:S)),X:S,Y:S) U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) -> Usable rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [din](X) = 0 [u21](X1,X2,X3) = 0 [u22](X1,X2,X3,X4) = 2.X1 [u31](X1,X2,X3) = 2.X1 [u32](X1,X2,X3,X4) = 2.X1 + 2 [u41](X1,X2) = 2.X1 [u42](X1,X2,X3) = 2 [der](X) = 2.X + 1 [dout](X) = 2 [fSNonEmpty] = 0 [plus](X1,X2) = 2 [times](X1,X2) = 2.X1 + 2.X2 [DIN](X) = X + 2 [U21](X1,X2,X3) = 0 [U22](X1,X2,X3,X4) = 0 [U31](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U32](X1,X2,X3,X4) = 0 [U41](X1,X2) = 0 [U42](X1,X2,X3) = 0 Problem 1: SCC Processor: -> Pairs: U31(dout(DX:S),X:S,Y:S) -> DIN(der(Y:S)) -> Rules: din(der(der(X:S))) -> u41(din(der(X:S)),X:S) din(der(plus(X:S,Y:S))) -> u21(din(der(X:S)),X:S,Y:S) din(der(times(X:S,Y:S))) -> u31(din(der(X:S)),X:S,Y:S) u21(dout(DX:S),X:S,Y:S) -> u22(din(der(Y:S)),X:S,Y:S,DX:S) u22(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(DX:S,DY:S)) u31(dout(DX:S),X:S,Y:S) -> u32(din(der(Y:S)),X:S,Y:S,DX:S) u32(dout(DY:S),X:S,Y:S,DX:S) -> dout(plus(times(X:S,DY:S),times(Y:S,DX:S))) u41(dout(DX:S),X:S) -> u42(din(der(DX:S)),X:S,DX:S) u42(dout(DDX:S),X:S,DX:S) -> dout(DDX:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.