WORST_CASE(?,O(n^2)) * Step 1: Sum WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() p(s(x)) -> x - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() p(s(x)) -> x - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + 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(-) = {2}, uargs(if) = {2}, uargs(s) = {1} Following symbols are considered usable: {-,p} TcT has computed the following interpretation: p(-) = 2 + x1 + x2 p(0) = 15 p(greater) = 4 p(if) = x2 p(p) = x1 p(s) = x1 Following rules are strictly oriented: -(x,0()) = 17 + x > x = x -(0(),y) = 17 + y > 15 = 0() Following rules are (at-least) weakly oriented: -(x,s(y)) = 2 + x + y >= 2 + x + y = if(greater(x,s(y)),s(-(x,p(s(y)))),0()) p(0()) = 15 >= 15 = 0() p(s(x)) = x >= x = x * Step 3: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) p(0()) -> 0() p(s(x)) -> x - Weak TRS: -(x,0()) -> x -(0(),y) -> 0() - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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: uargs(-) = {2}, uargs(if) = {2}, uargs(s) = {1} Following symbols are considered usable: {-,p} TcT has computed the following interpretation: p(-) = [1 0 0] [1 0 0] [0] [1 1 0] x1 + [1 1 1] x2 + [1] [0 0 1] [0 0 0] [0] p(0) = [1] [0] [1] p(greater) = [0] [0] [0] p(if) = [1 0 0] [0] [0 0 0] x2 + [0] [0 0 0] [0] p(p) = [1 0 1] [0] [0 1 0] x1 + [1] [0 1 0] [1] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: p(0()) = [2] [1] [1] > [1] [0] [1] = 0() Following rules are (at-least) weakly oriented: -(x,0()) = [1 0 0] [1] [1 1 0] x + [3] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x -(x,s(y)) = [1 0 0] [1 0 0] [0] [1 1 0] x + [1 1 1] y + [1] [0 0 1] [0 0 0] [0] >= [1 0 0] [1 0 0] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) = [1 0 0] [1] [1 1 1] y + [2] [0 0 0] [1] >= [1] [0] [1] = 0() p(s(x)) = [1 0 0] [0] [0 1 1] x + [1] [0 1 1] [1] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x * Step 4: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) p(s(x)) -> x - Weak TRS: -(x,0()) -> x -(0(),y) -> 0() p(0()) -> 0() - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: MI {miKind = Automaton (Just 1), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 1): The following argument positions are considered usable: uargs(-) = {2}, uargs(if) = {2}, uargs(s) = {1} Following symbols are considered usable: {-,p} TcT has computed the following interpretation: p(-) = [1 0 0] [2 0 0] [0] [0 2 3] x_1 + [0 0 0] x_2 + [0] [3 0 1] [0 0 0] [0] p(0) = [0] [0] [0] p(greater) = [0 0 0] [0] [0 0 0] x_2 + [2] [0 0 1] [0] p(if) = [1 0 0] [0 0 0] [0] [0 0 0] x_2 + [0 1 1] x_3 + [0] [0 0 0] [0 0 1] [0] p(p) = [0 0 1] [0] [0 2 3] x_1 + [2] [0 2 0] [0] p(s) = [1 0 0] [2] [0 1 1] x_1 + [0] [1 0 0] [0] Following rules are strictly oriented: -(x,s(y)) = [1 0 0] [2 0 0] [4] [0 2 3] x + [0 0 0] y + [0] [3 0 1] [0 0 0] [0] > [1 0 0] [2 0 0] [2] [0 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = if(greater(x,s(y)),s(-(x,p(s(y)))),0()) Following rules are (at-least) weakly oriented: -(x,0()) = [1 0 0] [0] [0 2 3] x + [0] [3 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x -(0(),y) = [2 0 0] [0] [0 0 0] y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() p(0()) = [0] [2] [0] >= [0] [0] [0] = 0() p(s(x)) = [1 0 0] [0] [3 2 2] x + [2] [0 2 2] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x * Step 5: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: p(s(x)) -> x - Weak TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): The following argument positions are considered usable: uargs(-) = {2}, uargs(if) = {2}, uargs(s) = {1} Following symbols are considered usable: {-,p} TcT has computed the following interpretation: p(-) = [1 0 0] [2 0 0] [0] [0 2 0] x_1 + [1 0 0] x_2 + [3] [1 0 2] [0 0 0] [0] p(0) = [1] [0] [1] p(greater) = [0 0 0] [0] [0 1 0] x_1 + [2] [0 0 0] [1] p(if) = [1 0 0] [0] [0 0 0] x_2 + [0] [0 0 0] [0] p(p) = [0 0 1] [1] [3 1 0] x_1 + [0] [0 2 0] [2] p(s) = [1 0 0] [2] [1 1 1] x_1 + [1] [1 0 0] [0] Following rules are strictly oriented: p(s(x)) = [1 0 0] [1] [4 1 1] x + [7] [2 2 2] [4] > [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x Following rules are (at-least) weakly oriented: -(x,0()) = [1 0 0] [2] [0 2 0] x + [4] [1 0 2] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x -(x,s(y)) = [1 0 0] [2 0 0] [4] [0 2 0] x + [1 0 0] y + [5] [1 0 2] [0 0 0] [0] >= [1 0 0] [2 0 0] [4] [0 0 0] x + [0 0 0] y + [0] [0 0 0] [0 0 0] [0] = if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) = [2 0 0] [1] [1 0 0] y + [3] [0 0 0] [3] >= [1] [0] [1] = 0() p(0()) = [2] [3] [2] >= [1] [0] [1] = 0() * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: -(x,0()) -> x -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0()) -(0(),y) -> 0() p(0()) -> 0() p(s(x)) -> x - Signature: {-/2,p/1} / {0/0,greater/2,if/3,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))