WORST_CASE(?,O(n^1)) * Step 1: Sum WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = 4 p(U11) = 12 + x2 + x3 p(U12) = 3*x1 + x2 + x3 p(activate) = x1 p(plus) = 6 + x1 + x2 p(s) = 6 + x1 p(tt) = 4 Following rules are strictly oriented: plus(N,0()) = 10 + N > N = N Following rules are (at-least) weakly oriented: U11(tt(),M,N) = 12 + M + N >= 12 + M + N = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = 12 + M + N >= 12 + M + N = s(plus(activate(N),activate(M))) activate(X) = X >= X = X plus(N,s(M)) = 12 + M + N >= 12 + M + N = U11(tt(),M,N) * Step 3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,s(M)) -> U11(tt(),M,N) - Weak TRS: plus(N,0()) -> N - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = 1 p(U11) = 2*x1 + 8*x2 + 2*x3 p(U12) = 2 + 8*x2 + 2*x3 p(activate) = x1 p(plus) = 2*x1 + 8*x2 p(s) = 2 + x1 p(tt) = 8 Following rules are strictly oriented: U11(tt(),M,N) = 16 + 8*M + 2*N > 2 + 8*M + 2*N = U12(tt(),activate(M),activate(N)) Following rules are (at-least) weakly oriented: U12(tt(),M,N) = 2 + 8*M + 2*N >= 2 + 8*M + 2*N = s(plus(activate(N),activate(M))) activate(X) = X >= X = X plus(N,0()) = 8 + 2*N >= N = N plus(N,s(M)) = 16 + 8*M + 2*N >= 16 + 8*M + 2*N = U11(tt(),M,N) * Step 4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,s(M)) -> U11(tt(),M,N) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) plus(N,0()) -> N - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = 2 p(U11) = 2*x1 + 5*x2 + 2*x3 p(U12) = 5 + x1 + 5*x2 + 2*x3 p(activate) = x1 p(plus) = 11 + 2*x1 + 5*x2 p(s) = 1 + x1 p(tt) = 8 Following rules are strictly oriented: U12(tt(),M,N) = 13 + 5*M + 2*N > 12 + 5*M + 2*N = s(plus(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U11(tt(),M,N) = 16 + 5*M + 2*N >= 13 + 5*M + 2*N = U12(tt(),activate(M),activate(N)) activate(X) = X >= X = X plus(N,0()) = 21 + 2*N >= N = N plus(N,s(M)) = 16 + 5*M + 2*N >= 16 + 5*M + 2*N = U11(tt(),M,N) * Step 5: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X plus(N,s(M)) -> U11(tt(),M,N) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> N - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = 3 p(U11) = 3 + 8*x2 + 8*x3 p(U12) = 3 + 8*x2 + 8*x3 p(activate) = x1 p(plus) = 8*x1 + 8*x2 p(s) = 3 + x1 p(tt) = 0 Following rules are strictly oriented: plus(N,s(M)) = 24 + 8*M + 8*N > 3 + 8*M + 8*N = U11(tt(),M,N) Following rules are (at-least) weakly oriented: U11(tt(),M,N) = 3 + 8*M + 8*N >= 3 + 8*M + 8*N = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = 3 + 8*M + 8*N >= 3 + 8*M + 8*N = s(plus(activate(N),activate(M))) activate(X) = X >= X = X plus(N,0()) = 24 + 8*N >= N = N * Step 6: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = 0 p(U11) = 7 + 8*x1 + 4*x2 + x3 p(U12) = 8 + 6*x1 + 4*x2 + x3 p(activate) = 1 + x1 p(plus) = 3 + x1 + 4*x2 p(s) = 7 + x1 p(tt) = 3 Following rules are strictly oriented: activate(X) = 1 + X > X = X Following rules are (at-least) weakly oriented: U11(tt(),M,N) = 31 + 4*M + N >= 31 + 4*M + N = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = 26 + 4*M + N >= 15 + 4*M + N = s(plus(activate(N),activate(M))) plus(N,0()) = 3 + N >= N = N plus(N,s(M)) = 31 + 4*M + N >= 31 + 4*M + N = U11(tt(),M,N) * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))