WORST_CASE(?,O(n^2)) * 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: innermost 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 + [5] p(0) = [0] p(double) = [2] p(s) = [1] x1 + [8] p(sqr) = [1] x1 + [0] Following rules are strictly oriented: +(x,0()) = [1] x + [5] > [1] x + [0] = x double(0()) = [2] > [0] = 0() Following rules are (at-least) weakly oriented: +(x,s(y)) = [1] x + [1] y + [13] >= [1] x + [1] y + [13] = s(+(x,y)) double(s(x)) = [2] >= [18] = s(s(double(x))) sqr(0()) = [0] >= [0] = 0() sqr(s(x)) = [1] x + [8] >= [1] x + [15] = +(sqr(x),s(double(x))) sqr(s(x)) = [1] x + [8] >= [1] x + [15] = 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(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Weak TRS: +(x,0()) -> x double(0()) -> 0() - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost 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) = [14] p(s) = [1] x1 + [1] p(sqr) = [1] Following rules are strictly oriented: sqr(0()) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(x,s(y)) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = s(+(x,y)) double(0()) = [14] >= [0] = 0() double(s(x)) = [14] >= [16] = s(s(double(x))) sqr(s(x)) = [1] >= [16] = +(sqr(x),s(double(x))) sqr(s(x)) = [1] >= [16] = s(+(sqr(x),double(x))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: 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: innermost 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) = [12] p(s) = [1] x1 + [2] p(sqr) = [8] x1 + [13] Following rules are strictly oriented: sqr(s(x)) = [8] x + [29] > [8] x + [27] = +(sqr(x),s(double(x))) sqr(s(x)) = [8] x + [29] > [8] x + [27] = 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()) = [12] >= [0] = 0() double(s(x)) = [12] >= [16] = s(s(double(x))) sqr(0()) = [13] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: Ara 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: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- + :: [A(0, 0) x A(2, 0)] -(0)-> A(0, 0) 0 :: [] -(0)-> A(2, 0) 0 :: [] -(0)-> A(14, 0) 0 :: [] -(0)-> A(0, 15) 0 :: [] -(0)-> A(4, 7) 0 :: [] -(0)-> A(6, 7) double :: [A(14, 0)] -(9)-> A(2, 0) s :: [A(2, 0)] -(2)-> A(2, 0) s :: [A(14, 0)] -(14)-> A(14, 0) s :: [A(15, 15)] -(15)-> A(0, 15) s :: [A(0, 0)] -(0)-> A(0, 0) sqr :: [A(0, 15)] -(1)-> A(0, 0) Cost-free Signatures used: -------------------------- + :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(0, 0) double :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) sqr :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) s_A :: [A(1, 0)] -(1)-> A(1, 0) s_A :: [A(1, 1)] -(1)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: +(x,s(y)) -> s(+(x,y)) 2. Weak: double(s(x)) -> s(s(double(x))) * Step 5: Ara 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: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- + :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) 0 :: [] -(0)-> A(0, 0) 0 :: [] -(0)-> A(10, 0) 0 :: [] -(0)-> A(1, 14) 0 :: [] -(0)-> A(7, 6) double :: [A(10, 0)] -(0)-> A(0, 0) s :: [A(10, 0)] -(10)-> A(10, 0) s :: [A(0, 0)] -(0)-> A(0, 0) s :: [A(15, 14)] -(1)-> A(1, 14) sqr :: [A(1, 14)] -(8)-> A(0, 0) Cost-free Signatures used: -------------------------- + :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(4, 0) 0 :: [] -(0)-> A_cf(2, 2) double :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) double :: [A_cf(0, 0)] -(4)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(4, 0)] -(4)-> A_cf(4, 0) sqr :: [A_cf(4, 0)] -(1)-> A_cf(0, 0) sqr :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) s_A :: [A(1, 0)] -(1)-> A(1, 0) s_A :: [A(1, 1)] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: double(s(x)) -> s(s(double(x))) 2. Weak: * Step 6: 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: innermost 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))