WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(a,cons(x,k)) -> f(cons(x,a),k) f(a,empty()) -> g(a,empty()) g(cons(x,k),d) -> g(k,cons(x,d)) g(empty(),d) -> d - Signature: {f/2,g/2} / {cons/2,empty/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: none Following symbols are considered usable: {f,g} TcT has computed the following interpretation: p(cons) = [1] x2 + [4] p(empty) = [1] p(f) = [1] x1 + [4] x2 + [0] p(g) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: f(a,cons(x,k)) = [1] a + [4] k + [16] > [1] a + [4] k + [4] = f(cons(x,a),k) f(a,empty()) = [1] a + [4] > [1] a + [2] = g(a,empty()) g(empty(),d) = [1] d + [2] > [1] d + [0] = d Following rules are (at-least) weakly oriented: g(cons(x,k),d) = [1] d + [1] k + [5] >= [1] d + [1] k + [5] = g(k,cons(x,d)) * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: g(cons(x,k),d) -> g(k,cons(x,d)) - Weak TRS: f(a,cons(x,k)) -> f(cons(x,a),k) f(a,empty()) -> g(a,empty()) g(empty(),d) -> d - Signature: {f/2,g/2} / {cons/2,empty/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: none Following symbols are considered usable: {f,g} TcT has computed the following interpretation: p(cons) = [1] x2 + [1] p(empty) = [4] p(f) = [3] x1 + [4] x2 + [13] p(g) = [2] x1 + [1] x2 + [0] Following rules are strictly oriented: g(cons(x,k),d) = [1] d + [2] k + [2] > [1] d + [2] k + [1] = g(k,cons(x,d)) Following rules are (at-least) weakly oriented: f(a,cons(x,k)) = [3] a + [4] k + [17] >= [3] a + [4] k + [16] = f(cons(x,a),k) f(a,empty()) = [3] a + [29] >= [2] a + [4] = g(a,empty()) g(empty(),d) = [1] d + [8] >= [1] d + [0] = d * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(a,cons(x,k)) -> f(cons(x,a),k) f(a,empty()) -> g(a,empty()) g(cons(x,k),d) -> g(k,cons(x,d)) g(empty(),d) -> d - Signature: {f/2,g/2} / {cons/2,empty/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))