/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N X X1 X2 X3) (RULES a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U11(tt,M,N) -> A__U12(tt,M,N) A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__PLUS(N,s(M)) -> A__U11(tt,M,N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> A__U11(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: SCC Processor: -> Pairs: A__U11(tt,M,N) -> A__U12(tt,M,N) A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__PLUS(N,s(M)) -> A__U11(tt,M,N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> A__U11(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U11(tt,M,N) -> A__U12(tt,M,N) A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__PLUS(N,s(M)) -> A__U11(tt,M,N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> A__U11(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U11(tt,M,N) -> A__U12(tt,M,N) A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__PLUS(N,s(M)) -> A__U11(tt,M,N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> A__U11(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->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) = X1.X2 + 2.X1 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 [a__U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [mark](X) = X [0] = 1 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 [U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [A__U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 [A__U12](X1,X2,X3) = X1.X2 + X3 + 2 [A__U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X3 [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__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__PLUS(N,s(M)) -> A__U11(tt,M,N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> A__U11(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + 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 + 2.X1 + 2.X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X3 + 2 [a__plus](X1,X2) = X1 + 2.X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + 2 [U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X3 + 2 [plus](X1,X2) = X1 + 2.X2 + 1 [s](X) = X + 2 [tt] = 1 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 + 1 [A__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [A__U22](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X2.X3 + 2.X1 + X3 [A__PLUS](X1,X2) = X1 + X2 + 1 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M,N) -> MARK(M) A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->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 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [a__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 [mark](X) = X [0] = 1 [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 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = X1.X2 + X1 + X2 [A__U12](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2 + 2.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.X2 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U12(tt,M,N) -> MARK(N) A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + 2 [a__U12](X1,X2,X3) = X1.X2 + X1.X3 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 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) = 2.X1.X2 + X1.X3 + 2.X1 + 2 [U12](X1,X2,X3) = X1.X2 + X1.X3 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X2 + 2.X3 + 2 [U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 1 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [A__U12](X1,X2,X3) = X1.X3 + 2.X1 + X2 + 1 [A__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + X2 + 2.X3 + 2 [A__U22](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [A__PLUS](X1,X2) = X1 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> A__U12(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U21(tt,M,N) -> A__U22(tt,M,N) A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = X1.X2 + X1.X3 + 2.X1 + X2 [a__U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 [a__U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 1 [a__plus](X1,X2) = X1 + 2.X2 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = X1.X2 + X1.X3 + 2.X1 + X2 [U12](X1,X2,X3) = X1.X3 + 2.X1 + 2.X2 [U21](X1,X2,X3) = 2.X1.X3 + X2.X3 + X1 + X2 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 1 [plus](X1,X2) = X1 + 2.X2 [s](X) = X + 2 [tt] = 1 [x](X1,X2) = X1.X2 + 2.X1 + X2 + 1 [A__U21](X1,X2,X3) = 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 + X1 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 1 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> A__X(mark(N),mark(M)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) A__X(N,s(M)) -> A__U21(tt,M,N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> A__X(mark(X1),mark(X2)) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->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) = 2.X1.X2 + X1.X3 + 2.X1 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X1 + X2 + 2.X3 [a__U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + X2 [mark](X) = X [0] = 2 [U11](X1,X2,X3) = 2.X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X1 + X2 + 2.X3 [U22](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + X2.X3 + 2.X1 + X2 + 2.X3 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 1 [x](X1,X2) = 2.X1.X2 + 2.X1 + X2 [A__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M,N) -> MARK(M) A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->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) = X1.X2 + X1 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [a__U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X3 + 2 [a__plus](X1,X2) = X1 + 2.X2 + 2 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [mark](X) = X [0] = 1 [U11](X1,X2,X3) = X1.X2 + 2.X1 + X3 + 2 [U12](X1,X2,X3) = X1.X2 + X1 + X3 + 2 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X3 + 2 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [A__U22](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__U22(tt,M,N) -> MARK(N) A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->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) = X1.X2 + 2.X1 + 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) = X1.X2 + 2.X1 + 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 [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__U22](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X1 + X2 + 2.X3 + 1 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> A__U22(mark(X1),X2,X3) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Reduction Pairs Processor: -> Pairs: A__PLUS(N,0) -> MARK(N) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) -> Usable rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 [a__U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [a__U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 2 [a__U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 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) = 2.X1 + 2.X2 + X3 [U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [U21](X1,X2,X3) = X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + 2.X3 + 2 [U22](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2 [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__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> A__PLUS(mark(X1),mark(X2)) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) Problem 1: Subterm Processor: -> Pairs: MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U22(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) MARK(x(X1,X2)) -> MARK(X1) MARK(x(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Projection: pi(MARK) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,M,N) -> a__U12(tt,M,N) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U21(tt,M,N) -> a__U22(tt,M,N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U22(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__plus(N,0) -> mark(N) a__plus(N,s(M)) -> a__U11(tt,M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> 0 a__x(N,s(M)) -> a__U21(tt,M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.