* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),N) -> mark(N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) 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) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(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() - Signature: {a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__and,a__isNat,a__plus ,mark} and constructors {0,U11,U21,and,isNat,plus,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),N) -> mark(N) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) 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) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U21(X1,X2,X3)) -> a__U21(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() - Signature: {a__U11/2,a__U21/3,a__and/2,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U21/3,and/2,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U21,a__and,a__isNat,a__plus ,mark} and constructors {0,U11,U21,and,isNat,plus,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: a__isNat(x){x -> plus(x,y)} = a__isNat(plus(x,y)) ->^+ a__and(a__isNat(x),isNat(y)) = C[a__isNat(x) = a__isNat(x){}] WORST_CASE(Omega(n^1),?)