YES Problem 1: (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S X3:S) (RULES a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: SCC Processor: -> Pairs: A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U11(tt,M:S,N:S) -> A__U12(tt,M:S,N:S) A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__U11](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [A__U12](X1,X2,X3) = X1.X3 + 2.X1 + X2 + X3 + 1 [A__U21](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 1 [A__PLUS](X1,X2) = X1 + X2 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [MARK](X) = X Problem 1: SCC Processor: -> Pairs: A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U11(tt,M:S,N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [A__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__PLUS](X1,X2) = X1 + 2.X2 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M:S,N:S) -> MARK(M:S) A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 [a__U22](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 [U22](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [A__U21](X1,X2,X3) = X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2 [A__U22](X1,X2,X3) = X2.X3 + 2.X1 + 2.X2 + 2.X3 [A__PLUS](X1,X2) = X1 + 2 [A__X](X1,X2) = X1.X2 + 2.X2 + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M:S,N:S) -> MARK(N:S) A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 1 [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = X1.X2 + X1 + X3 + 2 [A__U21](X1,X2,X3) = X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2 [A__U22](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [A__PLUS](X1,X2) = X1 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X2 + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U21(tt,M:S,N:S) -> A__U22(tt,M:S,N:S) A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [a__U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 0 [A__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [A__U22](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + 2.X2 + 2 [A__PLUS](X1,X2) = X1 + 2.X2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> A__X(mark(N:S),mark(M:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) A__X(N:S,s(M:S)) -> A__U21(tt,M:S,N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> A__U21(mark(X1:S),X2:S,X3:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M:S,N:S) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 [a__U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + X1 + 2.X3 + 2 [a__plus](X1,X2) = X1 + 2.X2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [mark](X) = X [0] = 0 [U11](X1,X2,X3) = X1.X2 + X1 + X3 + 2 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 [U21](X1,X2,X3) = 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + X1 + 2.X3 + 2 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 0 [A__U21](X1,X2,X3) = 0 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + 1 [A__PLUS](X1,X2) = 2.X1 + 2 [A__X](X1,X2) = 0 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M:S,N:S) -> MARK(M:S) A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U12](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U12](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 0 [A__U21](X1,X2,X3) = 0 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [A__X](X1,X2) = 0 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1 + X2 + X3 + 2 [a__U12](X1,X2,X3) = X1 + X2 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 1 [a__plus](X1,X2) = X1 + X2 + 2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = X1 + X2 + X3 + 2 [U12](X1,X2,X3) = X1 + X2 + X3 + 2 [U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1 + 2.X3 + 1 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 0 [A__U21](X1,X2,X3) = 0 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + X1 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 2 [A__X](X1,X2) = 0 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> A__U22(mark(X1:S),X2:S,X3:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__PLUS(N:S,0) -> MARK(N:S) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) -> Usable rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1.X2 + 2.X1 + X3 + 2 [a__U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = 2.X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [fSNonEmpty] = 0 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 1 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [A__U11](X1,X2,X3) = 0 [A__U12](X1,X2,X3) = 0 [A__U21](X1,X2,X3) = 0 [A__U22](X1,X2,X3) = 0 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [A__X](X1,X2) = 0 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) Problem 1: Subterm Processor: -> Pairs: MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U21(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Projection: pi(MARK) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,M:S,N:S) -> a__U12(tt,M:S,N:S) a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) a__U12(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) a__U21(tt,M:S,N:S) -> a__U22(tt,M:S,N:S) a__U21(X1:S,X2:S,X3:S) -> U21(X1:S,X2:S,X3:S) a__U22(tt,M:S,N:S) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) a__U22(X1:S,X2:S,X3:S) -> U22(X1:S,X2:S,X3:S) a__plus(N:S,0) -> mark(N:S) a__plus(N:S,s(M:S)) -> a__U11(tt,M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) a__x(N:S,0) -> 0 a__x(N:S,s(M:S)) -> a__U21(tt,M:S,N:S) a__x(X1:S,X2:S) -> x(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) mark(U21(X1:S,X2:S,X3:S)) -> a__U21(mark(X1:S),X2:S,X3:S) mark(U22(X1:S,X2:S,X3:S)) -> a__U22(mark(X1:S),X2:S,X3:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.