* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [6] p(0) = [4] p(double) = [10] p(s) = [1] x1 + [4] p(sqr) = [5] x1 + [2] Following rules are strictly oriented: +(x,0()) = [1] x + [10] > [1] x + [0] = x double(0()) = [10] > [4] = 0() sqr(0()) = [22] > [4] = 0() Following rules are (at-least) weakly oriented: +(x,s(y)) = [1] x + [1] y + [10] >= [1] x + [1] y + [10] = s(+(x,y)) double(s(x)) = [10] >= [18] = s(s(double(x))) sqr(s(x)) = [5] x + [22] >= [5] x + [22] = +(sqr(x),s(double(x))) sqr(s(x)) = [5] x + [22] >= [5] x + [22] = s(+(sqr(x),double(x))) 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)) double(s(x)) -> s(s(double(x))) sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Weak TRS: +(x,0()) -> x double(0()) -> 0() sqr(0()) -> 0() - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [0] p(0) = [0] p(double) = [1] p(s) = [1] x1 + [2] p(sqr) = [6] x1 + [1] Following rules are strictly oriented: sqr(s(x)) = [6] x + [13] > [6] x + [4] = +(sqr(x),s(double(x))) sqr(s(x)) = [6] x + [13] > [6] x + [4] = s(+(sqr(x),double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(x,s(y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = s(+(x,y)) double(0()) = [1] >= [0] = 0() double(s(x)) = [1] >= [5] = s(s(double(x))) sqr(0()) = [1] >= [0] = 0() 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)) double(s(x)) -> s(s(double(x))) - Weak TRS: +(x,0()) -> x double(0()) -> 0() sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = x1 + 2*x2 p(0) = 0 p(double) = 1 + 2*x1 p(s) = 1 + x1 p(sqr) = 2*x1 + 2*x1^2 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 double(0()) = 1 >= 0 = 0() double(s(x)) = 3 + 2*x >= 3 + 2*x = s(s(double(x))) sqr(0()) = 0 >= 0 = 0() sqr(s(x)) = 4 + 6*x + 2*x^2 >= 4 + 6*x + 2*x^2 = +(sqr(x),s(double(x))) sqr(s(x)) = 4 + 6*x + 2*x^2 >= 3 + 6*x + 2*x^2 = s(+(sqr(x),double(x))) * Step 4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: double(s(x)) -> s(s(double(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = x1 + x2 p(0) = 0 p(double) = 3*x1 p(s) = 1 + x1 p(sqr) = 2*x1^2 Following rules are strictly oriented: double(s(x)) = 3 + 3*x > 2 + 3*x = s(s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = x >= x = x +(x,s(y)) = 1 + x + y >= 1 + x + y = s(+(x,y)) double(0()) = 0 >= 0 = 0() sqr(0()) = 0 >= 0 = 0() sqr(s(x)) = 2 + 4*x + 2*x^2 >= 1 + 3*x + 2*x^2 = +(sqr(x),s(double(x))) sqr(s(x)) = 2 + 4*x + 2*x^2 >= 1 + 3*x + 2*x^2 = s(+(sqr(x),double(x))) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))