* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {1}, uargs(g) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [10] p(0) = [0] p(1) = [2] p(f) = [8] x1 + [0] p(g) = [7] x1 + [1] x2 + [0] p(s) = [1] x1 + [3] Following rules are strictly oriented: +(x,0()) = [1] x + [10] > [1] x + [0] = x f(s(x)) = [8] x + [24] > [8] x + [3] = g(x,s(x)) g(s(x),y) = [7] x + [1] y + [21] > [7] x + [1] y + [10] = g(x,+(y,s(x))) g(s(x),y) = [7] x + [1] y + [21] > [7] x + [1] y + [13] = g(x,s(+(y,x))) Following rules are (at-least) weakly oriented: +(x,s(y)) = [1] x + [10] >= [1] x + [13] = s(+(x,y)) f(0()) = [0] >= [2] = 1() g(0(),y) = [1] y + [0] >= [1] y + [0] = y Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() g(0(),y) -> y - Weak TRS: +(x,0()) -> x f(s(x)) -> g(x,s(x)) g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {1}, uargs(g) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [0] p(0) = [0] p(1) = [0] p(f) = [1] x1 + [1] p(g) = [1] x2 + [1] p(s) = [1] x1 + [0] Following rules are strictly oriented: f(0()) = [1] > [0] = 1() g(0(),y) = [1] y + [1] > [1] y + [0] = y Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(x,s(y)) = [1] x + [0] >= [1] x + [0] = s(+(x,y)) f(s(x)) = [1] x + [1] >= [1] x + [1] = g(x,s(x)) g(s(x),y) = [1] y + [1] >= [1] y + [1] = g(x,+(y,s(x))) g(s(x),y) = [1] y + [1] >= [1] y + [1] = g(x,s(+(y,x))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) - Weak TRS: +(x,0()) -> x f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(+) = {1}, uargs(g) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = x1 + 2*x2 p(0) = 0 p(1) = 0 p(f) = 4 + x1 + 5*x1^2 p(g) = 1 + 5*x1 + 5*x1^2 + 5*x2 p(s) = 1 + x1 Following rules are strictly oriented: +(x,s(y)) = 2 + x + 2*y > 1 + x + 2*y = s(+(x,y)) Following rules are (at-least) weakly oriented: +(x,0()) = x >= x = x f(0()) = 4 >= 0 = 1() f(s(x)) = 10 + 11*x + 5*x^2 >= 6 + 10*x + 5*x^2 = g(x,s(x)) g(0(),y) = 1 + 5*y >= y = y g(s(x),y) = 11 + 15*x + 5*x^2 + 5*y >= 11 + 15*x + 5*x^2 + 5*y = g(x,+(y,s(x))) g(s(x),y) = 11 + 15*x + 5*x^2 + 5*y >= 6 + 15*x + 5*x^2 + 5*y = g(x,s(+(y,x))) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 1() f(s(x)) -> g(x,s(x)) g(0(),y) -> y g(s(x),y) -> g(x,+(y,s(x))) g(s(x),y) -> g(x,s(+(y,x))) - Signature: {+/2,f/1,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))