YES Problem 1: (VAR X Y Z x y) (THEORY (AC union)) (RULES max(union(singl(s(x)),singl(s(y)))) -> s(max(union(singl(x),singl(y)))) max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(union(singl(x),singl(0))) -> x max(singl(x)) -> x union(empty,X) -> X ) Problem 1: Reduction Order Processor: -> Rules: max(union(singl(s(x)),singl(s(y)))) -> s(max(union(singl(x),singl(y)))) max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(union(singl(x),singl(0))) -> x max(singl(x)) -> x union(empty,X) -> X ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [max](X) = X [union](X1,X2) = X1 + X2 + 1 [0] = 0 [empty] = 0 [s](X) = X + 1 [singl](X) = X Problem 1: Reduction Order Processor: -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(union(singl(x),singl(0))) -> x max(singl(x)) -> x union(empty,X) -> X ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [max](X) = X [union](X1,X2) = X1 + X2 [0] = 2 [empty] = 0 [s](X) = 2.X [singl](X) = X Problem 1: Reduction Order Processor: -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x union(empty,X) -> X ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [max](X) = X [union](X1,X2) = X1 + X2 + 1 [0] = 0 [empty] = 2 [s](X) = 2.X [singl](X) = X Problem 1: Dependency Pairs Processor: -> FAxioms: UNION(union(x5,x6),x7) = UNION(x5,union(x6,x7)) UNION(x5,x6) = UNION(x6,x5) -> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) MAX(union(singl(x),union(Y,Z))) -> MAX(union(Y,Z)) MAX(union(singl(x),union(Y,Z))) -> UNION(singl(x),singl(max(union(Y,Z)))) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: UNION(union(x5,x6),x7) -> UNION(x5,x6) UNION(x5,union(x6,x7)) -> UNION(x6,x7) Problem 1: SCC Processor: -> FAxioms: UNION(union(x5,x6),x7) = UNION(x5,union(x6,x7)) UNION(x5,x6) = UNION(x6,x5) -> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) MAX(union(singl(x),union(Y,Z))) -> MAX(union(Y,Z)) MAX(union(singl(x),union(Y,Z))) -> UNION(singl(x),singl(max(union(Y,Z)))) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: UNION(union(x5,x6),x7) -> UNION(x5,x6) UNION(x5,union(x6,x7)) -> UNION(x6,x7) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) MAX(union(singl(x),union(Y,Z))) -> MAX(union(Y,Z)) -> FAxioms: union(union(x5,x6),x7) -> union(x5,union(x6,x7)) union(x5,x6) -> union(x6,x5) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) ->->-> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty Problem 1: Reduction Pairs Processor: -> FAxioms: Empty -> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) MAX(union(singl(x),union(Y,Z))) -> MAX(union(Y,Z)) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Usable Equations: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> Usable Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [max](X) = X [union](X1,X2) = X1 + X2 + 2 [0] = 0 [empty] = 0 [s](X) = 0 [singl](X) = X [MAX](X) = 2.X [UNION](X1,X2) = 0 Problem 1: SCC Processor: -> FAxioms: Empty -> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) -> FAxioms: union(union(x5,x6),x7) -> union(x5,union(x6,x7)) union(x5,x6) -> union(x6,x5) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) ->->-> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty Problem 1: Reduction Pairs Processor: -> FAxioms: Empty -> Pairs: MAX(union(singl(x),union(Y,Z))) -> MAX(union(singl(x),singl(max(union(Y,Z))))) -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Usable Equations: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> Usable Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 2 ->Bound: 1 ->Interpretation: [max](X) = [0 1;1 1].X [union](X1,X2) = [1 0;1 0].X1 + [1 0;1 0].X2 + [1;0] [0] = 0 [empty] = 0 [s](X) = 0 [singl](X) = [0 0;1 1].X [MAX](X) = [1 1;1 1].X [UNION](X1,X2) = 0 Problem 1: SCC Processor: -> FAxioms: Empty -> Pairs: Empty -> EAxioms: union(union(x5,x6),x7) = union(x5,union(x6,x7)) union(x5,x6) = union(x6,x5) -> Rules: max(union(singl(x),union(Y,Z))) -> max(union(singl(x),singl(max(union(Y,Z))))) max(singl(x)) -> x -> SRules: Empty ->Strongly Connected Components: There is no strongly connected component The problem is finite.