* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + 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(g) = {1,2}, uargs(h) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [0] p(c) = [0] p(d) = [0] p(e) = [0] p(f) = [0] p(g) = [1] x1 + [1] x2 + [5] p(h) = [1] x2 + [0] Following rules are strictly oriented: g(x,x) = [2] x + [5] > [1] x + [0] = h(e(),x) Following rules are (at-least) weakly oriented: f(a()) = [0] >= [0] = b() f(c()) = [0] >= [0] = d() f(g(x,y)) = [0] >= [5] = g(f(x),f(y)) f(h(x,y)) = [0] >= [5] = g(h(y,f(x)),h(x,f(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: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) - Weak TRS: g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + 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(g) = {1,2}, uargs(h) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [1] p(b) = [2] p(c) = [0] p(d) = [2] p(e) = [1] p(f) = [5] p(g) = [1] x1 + [1] x2 + [8] p(h) = [1] x2 + [0] Following rules are strictly oriented: f(a()) = [5] > [2] = b() f(c()) = [5] > [2] = d() Following rules are (at-least) weakly oriented: f(g(x,y)) = [5] >= [18] = g(f(x),f(y)) f(h(x,y)) = [5] >= [18] = g(h(y,f(x)),h(x,f(y))) g(x,x) = [2] x + [8] >= [1] x + [0] = h(e(),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: f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) - Weak TRS: f(a()) -> b() f(c()) -> d() g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + 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(g) = {1,2}, uargs(h) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 1 p(b) = 1 p(c) = 1 p(d) = 0 p(e) = 0 p(f) = 3*x1 + x1^2 p(g) = 2 + x1 + x2 p(h) = 1 + x1 + x2 Following rules are strictly oriented: f(g(x,y)) = 10 + 7*x + 2*x*y + x^2 + 7*y + y^2 > 2 + 3*x + x^2 + 3*y + y^2 = g(f(x),f(y)) Following rules are (at-least) weakly oriented: f(a()) = 4 >= 1 = b() f(c()) = 4 >= 0 = d() f(h(x,y)) = 4 + 5*x + 2*x*y + x^2 + 5*y + y^2 >= 4 + 4*x + x^2 + 4*y + y^2 = g(h(y,f(x)),h(x,f(y))) g(x,x) = 2 + 2*x >= 1 + x = h(e(),x) * Step 4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + 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(g) = {1,2}, uargs(h) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 1 p(b) = 0 p(c) = 0 p(d) = 0 p(e) = 0 p(f) = x1 + 3*x1^2 p(g) = 1 + x1 + x2 p(h) = 1 + x1 + x2 Following rules are strictly oriented: f(h(x,y)) = 4 + 7*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 > 3 + 2*x + 3*x^2 + 2*y + 3*y^2 = g(h(y,f(x)),h(x,f(y))) Following rules are (at-least) weakly oriented: f(a()) = 4 >= 0 = b() f(c()) = 0 >= 0 = d() f(g(x,y)) = 4 + 7*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 >= 1 + x + 3*x^2 + y + 3*y^2 = g(f(x),f(y)) g(x,x) = 1 + 2*x >= 1 + x = h(e(),x) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))