/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N V1 V2 X X1 X2 X3) (RULES a__U11(tt,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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: Dependency Pairs Processor: -> Pairs: A__U11(tt,N) -> MARK(N) A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__U11(a__isNat(N),N) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__U31(a__isNat(N)) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> A__U31(mark(X)) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> MARK(N) A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__U11(a__isNat(N),N) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__U31(a__isNat(N)) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> A__U31(mark(X)) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> MARK(N) A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__U11(a__isNat(N),N) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U11(tt,N) -> MARK(N) A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__U11(a__isNat(N),N) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 + 1 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1 + X2 + 1 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [A__U11](X1,X2) = X2 + 1 [A__U21](X1,X2,X3) = X2 + X3 + 2 [A__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 [A__AND](X1,X2) = X1.X2 + 2.X1 + X2 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = X1 + X2 [A__X](X1,X2) = 2.X1.X2 + 2.X2 [MARK](X) = X Problem 1: SCC Processor: -> Pairs: A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__U11(a__isNat(N),N) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U21(tt,M,N) -> A__PLUS(mark(N),mark(M)) A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 1 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 1 [U21](X1,X2,X3) = 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__U21](X1,X2,X3) = 2.X1.X3 + 2.X1 + X2 + X3 + 2 [A__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [A__AND](X1,X2) = 2.X1 + X2 + 2 [A__ISNAT](X) = 2 [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__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U21(tt,M,N) -> MARK(M) A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [a__U31](X) = X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [U31](X) = X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [A__U21](X1,X2,X3) = 2.X1.X2 + X2 + X3 + 2 [A__U41](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [A__AND](X1,X2) = 2.X1 + 2.X2 + 1 [A__ISNAT](X) = 1 [A__PLUS](X1,X2) = X1 + X2 + 2 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [MARK](X) = X + 1 Problem 1: SCC Processor: -> Pairs: A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U21(tt,M,N) -> MARK(N) A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = X1.X2 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 [mark](X) = X [0] = 2 [U11](X1,X2) = X1.X2 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2 + X2.X3 + 2.X1 + X2 + 2.X3 + 1 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = X1.X2 + X1 + X2 [A__U21](X1,X2,X3) = 2.X3 + 2 [A__U41](X1,X2,X3) = 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [A__AND](X1,X2) = 2.X2 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 2.X1 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X2 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__U21(a__and(a__isNat(M),isNat(N)),M,N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> A__U21(mark(X1),X2,X3) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U41(tt,M,N) -> A__PLUS(a__x(mark(N),mark(M)),mark(N)) A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1.X2 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__U41](X1,X2,X3) = X1.X2.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 1 [A__AND](X1,X2) = 2.X1.X2 + 2.X2 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 2.X2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U41(tt,M,N) -> A__X(mark(N),mark(M)) A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1 + X2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X2 + 2.X3 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2 [A__PLUS](X1,X2) = 2.X1 + X2 + 2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U41(tt,M,N) -> MARK(M) A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = X1.X2 + X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = X1.X2 + X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1.X2.X3 + X2.X3 + X1 + 2.X2 + 2.X3 + 2 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = 1 [A__PLUS](X1,X2) = 2.X1 + 2.X2 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__U41(tt,M,N) -> MARK(N) A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [mark](X) = X [0] = 1 [U11](X1,X2) = 2.X1 + X2 [U21](X1,X2,X3) = 2.X1.X2.X3 + X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X3 + 2 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = 1 [A__PLUS](X1,X2) = X1 + 2.X2 + 1 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__U41(a__and(a__isNat(M),isNat(N)),M,N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,0) -> A__ISNAT(N) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = 1 [A__PLUS](X1,X2) = X1 + X2 + 1 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 1 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 1 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__AND](X1,X2) = 2.X2 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 1 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__PLUS(N,s(M)) -> A__ISNAT(M) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + X1 + X2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [mark](X) = X [0] = 0 [U11](X1,X2) = 2.X1.X2 + X1 + X2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X1.X2 + 2.X2 + 2 [A__ISNAT](X) = 2 [A__PLUS](X1,X2) = X1 + 2.X2 + 2 [A__X](X1,X2) = 2.X1.X2 + X1 + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,0) -> A__ISNAT(N) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 1 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1.X2 + 2.X1 + X2 + 1 [U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X1.X2 + 2.X2 + 1 [A__ISNAT](X) = 1 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__AND(a__isNat(M),isNat(N)) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 1 [a__U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1.X2 + 2.X1 + X2 + 1 [U21](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X1 + 2.X2 + 2 [A__ISNAT](X) = 2 [A__X](X1,X2) = 2.X1.X2 + X1 + 2.X2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) A__X(N,s(M)) -> A__ISNAT(M) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + X1 + X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + X1 + X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = 1 [A__X](X1,X2) = 2.X1.X2 + 2.X2 + 1 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = X1.X2 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1.X2 + X1 + 2.X2 [a__isNat](X) = 0 [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) = X1.X2 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + X1.X2 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1.X2 + X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U21(X1,X2,X3)) -> MARK(X1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 [a__U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + X [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [a__isNat](X) = 0 [a__plus](X1,X2) = X1 + X2 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [mark](X) = X [0] = 0 [U11](X1,X2) = 2.X1 + X2 [U21](X1,X2,X3) = X1.X2 + 2.X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + X [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 [A__AND](X1,X2) = 2.X1 + 2.X2 + 2 [A__ISNAT](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1.X2 + 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 1 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1.X2 + 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2.X3 + 2.X1 + X2 + X3 + 1 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 1 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 1 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = 2.X1 + X2 + 2 [a__U21](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [a__U31](X) = 2.X.X + 2.X + 2 [a__U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [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) = 2.X1 + X2 + 2 [U21](X1,X2,X3) = 2.X1.X2 + X1.X3 + 2.X1 + X2 + X3 + 2 [U31](X) = 2.X.X + 2.X + 2 [U41](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2 [and](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [plus](X1,X2) = X1 + X2 [s](X) = X + 2 [tt] = 0 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X1 + 2.X2 + 2 [A__ISNAT](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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) = X2 [a__U21](X1,X2,X3) = X2 + X3 + 2 [a__U31](X) = 2 [a__U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [a__and](X1,X2) = X1 + X2 + 1 [a__isNat](X) = 2.X.X + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [0] = 2 [U11](X1,X2) = X2 [U21](X1,X2,X3) = X2 + X3 + 2 [U31](X) = 2 [U41](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2 [and](X1,X2) = X1 + X2 + 1 [isNat](X) = 2.X.X + 1 [plus](X1,X2) = X1 + X2 + 1 [s](X) = X + 1 [tt] = 2 [x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [A__AND](X1,X2) = X1 + X2 + 2 [A__ISNAT](X) = 2.X.X + 2 [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__ISNAT(plus(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__AND(a__isNat(V1),isNat(V2)) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) ->->-> Rules: a__U11(tt,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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)) ->->Cycle: ->->-> Pairs: MARK(and(X1,X2)) -> 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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)) The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__ISNAT(plus(V1,V2)) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNAT(x(V1,V2)) -> A__ISNAT(V1) -> Rules: a__U11(tt,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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(A__ISNAT) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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. Problem 1.2: Subterm Processor: -> Pairs: MARK(and(X1,X2)) -> 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,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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 1.2: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,N) -> mark(N) a__U11(X1,X2) -> U11(X1,X2) a__U21(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U31(tt) -> 0 a__U31(X) -> U31(X) a__U41(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(X) -> isNat(X) a__plus(N,0) -> a__U11(a__isNat(N),N) a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0) -> a__U31(a__isNat(N)) a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N) a__x(X1,X2) -> x(X1,X2) mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) 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.