WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: minus(x,y){x -> s(x),y -> s(y)} = minus(s(x),s(y)) ->^+ minus(x,y) = C[minus(x,y) = minus(x,y){}] ** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s} + 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: uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {minus,plus,quot} TcT has computed the following interpretation: p(0) = [1] p(minus) = [1] x1 + [0] p(plus) = [8] x1 + [8] x2 + [1] p(quot) = [4] x1 + [4] x2 + [8] p(s) = [1] x1 + [1] Following rules are strictly oriented: minus(s(x),s(y)) = [1] x + [1] > [1] x + [0] = minus(x,y) plus(0(),y) = [8] y + [9] > [1] y + [0] = y plus(s(x),y) = [8] x + [8] y + [9] > [8] x + [8] y + [2] = s(plus(x,y)) quot(0(),s(y)) = [4] y + [16] > [1] = 0() quot(s(x),s(y)) = [4] x + [4] y + [16] > [4] x + [4] y + [13] = s(quot(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: minus(x,0()) = [1] x + [0] >= [1] x + [0] = x plus(minus(x,s(0())),minus(y,s(s(z)))) = [8] x + [8] y + [1] >= [8] x + [8] y + [1] = plus(minus(y,s(s(z))),minus(x,s(0()))) ** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(x,0()) -> x plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) - Weak TRS: minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s} + 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: uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {minus,plus,quot} TcT has computed the following interpretation: p(0) = [6] p(minus) = [1] x1 + [3] p(plus) = [2] x1 + [2] x2 + [0] p(quot) = [4] x1 + [0] p(s) = [1] x1 + [4] Following rules are strictly oriented: minus(x,0()) = [1] x + [3] > [1] x + [0] = x Following rules are (at-least) weakly oriented: minus(s(x),s(y)) = [1] x + [7] >= [1] x + [3] = minus(x,y) plus(0(),y) = [2] y + [12] >= [1] y + [0] = y plus(minus(x,s(0())),minus(y,s(s(z)))) = [2] x + [2] y + [12] >= [2] x + [2] y + [12] = plus(minus(y,s(s(z))),minus(x,s(0()))) plus(s(x),y) = [2] x + [2] y + [8] >= [2] x + [2] y + [4] = s(plus(x,y)) quot(0(),s(y)) = [24] >= [6] = 0() quot(s(x),s(y)) = [4] x + [16] >= [4] x + [16] = s(quot(minus(x,y),s(y))) ** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,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(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {minus,plus,quot} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(minus) = [1 0 0] [0 0 0] [0] [0 1 1] x1 + [1 0 0] x2 + [0] [1 1 1] [0 0 0] [0] p(plus) = [0 0 1] [1 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(quot) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [1 0 0] [0 0 0] [0] p(s) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: plus(minus(x,s(0())),minus(y,s(s(z)))) = [1 1 1] [1 1 1] [1 0 0] [3] [0 1 1] x + [0 1 1] y + [1 0 0] z + [3] [1 1 1] [1 1 1] [0 0 0] [0] > [1 1 1] [1 1 1] [0 0 0] [2] [0 1 1] x + [0 1 1] y + [1 0 0] z + [3] [1 1 1] [1 1 1] [0 0 0] [0] = plus(minus(y,s(s(z))),minus(x,s(0()))) Following rules are (at-least) weakly oriented: minus(x,0()) = [1 0 0] [0] [0 1 1] x + [0] [1 1 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x minus(s(x),s(y)) = [1 0 0] [0 0 0] [1] [0 1 1] x + [1 0 0] y + [2] [1 1 1] [0 0 0] [2] >= [1 0 0] [0 0 0] [0] [0 1 1] x + [1 0 0] y + [0] [1 1 1] [0 0 0] [0] = minus(x,y) plus(0(),y) = [1 1 0] [1] [0 1 0] y + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] y + [0] [0 0 1] [0] = y plus(s(x),y) = [0 0 1] [1 1 0] [2] [0 1 0] x + [0 1 0] y + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [1 1 0] [2] [0 1 0] x + [0 1 0] y + [0] [0 0 1] [0 0 1] [1] = s(plus(x,y)) quot(0(),s(y)) = [1 0 0] [1] [0 1 0] y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() quot(s(x),s(y)) = [1 0 0] [1 0 0] [2] [0 0 0] x + [0 1 0] y + [0] [1 0 0] [0 0 0] [1] >= [1 0 0] [1 0 0] [2] [0 0 0] x + [0 1 0] y + [0] [1 0 0] [0 0 0] [1] = s(quot(minus(x,y),s(y))) ** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0()))) plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,plus/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))