WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + 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: {goal,revapp} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [0] p(Nil) = [0] p(goal) = [3] x1 + [2] x2 + [2] p(revapp) = [2] x1 + [2] x2 + [0] Following rules are strictly oriented: goal(xs,ys) = [3] xs + [2] ys + [2] > [2] xs + [2] ys + [0] = revapp(xs,ys) Following rules are (at-least) weakly oriented: revapp(Cons(x,xs),rest) = [2] rest + [2] x + [2] xs + [0] >= [2] rest + [2] x + [2] xs + [0] = revapp(xs,Cons(x,rest)) revapp(Nil(),rest) = [2] rest + [0] >= [1] rest + [0] = rest * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Weak TRS: goal(xs,ys) -> revapp(xs,ys) - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + 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: {goal,revapp} TcT has computed the following interpretation: p(Cons) = [0] p(Nil) = [0] p(goal) = [8] x1 + [3] x2 + [2] p(revapp) = [2] x2 + [2] Following rules are strictly oriented: revapp(Nil(),rest) = [2] rest + [2] > [1] rest + [0] = rest Following rules are (at-least) weakly oriented: goal(xs,ys) = [8] xs + [3] ys + [2] >= [2] ys + [2] = revapp(xs,ys) revapp(Cons(x,xs),rest) = [2] rest + [2] >= [2] = revapp(xs,Cons(x,rest)) * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + 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: {goal,revapp} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [1] p(Nil) = [1] p(goal) = [10] x1 + [6] x2 + [8] p(revapp) = [8] x1 + [4] x2 + [1] Following rules are strictly oriented: revapp(Cons(x,xs),rest) = [4] rest + [8] x + [8] xs + [9] > [4] rest + [4] x + [8] xs + [5] = revapp(xs,Cons(x,rest)) Following rules are (at-least) weakly oriented: goal(xs,ys) = [10] xs + [6] ys + [8] >= [8] xs + [4] ys + [1] = revapp(xs,ys) revapp(Nil(),rest) = [4] rest + [9] >= [1] rest + [0] = rest * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))