WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) p(s(x)) -> x - Signature: {-/2,f/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,f,p} 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(f) = {1,2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [1] p(0) = [0] p(f) = [1] x1 + [1] x2 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: -(x,0()) = [1] x + [1] > [1] x + [0] = x Following rules are (at-least) weakly oriented: -(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = -(x,y) f(x,s(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [2] = f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) = [1] x + [1] y + [0] >= [1] x + [1] y + [2] = f(p(-(s(x),y)),p(-(y,s(x)))) p(s(x)) = [1] x + [0] >= [1] x + [0] = 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: -(s(x),s(y)) -> -(x,y) f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) p(s(x)) -> x - Weak TRS: -(x,0()) -> x - Signature: {-/2,f/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,f,p} 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(f) = {1,2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [3] p(0) = [0] p(f) = [1] x1 + [1] x2 + [3] p(p) = [1] x1 + [6] p(s) = [1] x1 + [5] Following rules are strictly oriented: -(s(x),s(y)) = [1] x + [8] > [1] x + [3] = -(x,y) p(s(x)) = [1] x + [11] > [1] x + [0] = x Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [3] >= [1] x + [0] = x f(x,s(y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [26] = f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) = [1] x + [1] y + [8] >= [1] x + [1] y + [26] = f(p(-(s(x),y)),p(-(y,s(x)))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) - Weak TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) p(s(x)) -> x - Signature: {-/2,f/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- - :: [A(0, 4) x A(0, 0)] -(0)-> A(0, 4) 0 :: [] -(0)-> A(0, 0) f :: [A(4, 4) x A(0, 4)] -(0)-> A(6, 4) p :: [A(0, 4)] -(0)-> A(4, 4) s :: [A(4, 4)] -(0)-> A(0, 4) s :: [A(8, 4)] -(4)-> A(4, 4) s :: [A(0, 0)] -(0)-> A(0, 0) Cost-free Signatures used: -------------------------- - :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) - :: [A_cf(2, 0) x A_cf(0, 0)] -(0)-> A_cf(2, 0) - :: [A_cf(4, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(0, 0) f :: [A_cf(0, 0) x A_cf(2, 0)] -(0)-> A_cf(0, 0) f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) p :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) p :: [A_cf(2, 0)] -(0)-> A_cf(2, 0) s :: [A_cf(2, 0)] -(2)-> A_cf(2, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(4, 0)] -(4)-> A_cf(4, 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: f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) 2. Weak: f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) * Step 4: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) - Weak TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) p(s(x)) -> x - Signature: {-/2,f/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,s} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- - :: [A(0, 4) x A(0, 0)] -(0)-> A(0, 4) 0 :: [] -(0)-> A(0, 0) f :: [A(0, 4) x A(1, 4)] -(8)-> A(0, 4) p :: [A(0, 4)] -(0)-> A(2, 4) s :: [A(5, 4)] -(1)-> A(1, 4) s :: [A(4, 4)] -(0)-> A(0, 4) s :: [A(0, 0)] -(0)-> A(0, 0) Cost-free Signatures used: -------------------------- - :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) - :: [A_cf(4, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) 0 :: [] -(0)-> A_cf(0, 0) f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) p :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) s :: [A_cf(4, 0)] -(4)-> A_cf(4, 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: f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) 2. Weak: * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) f(x,s(y)) -> f(p(-(x,s(y))),p(-(s(y),x))) f(s(x),y) -> f(p(-(s(x),y)),p(-(y,s(x)))) p(s(x)) -> x - Signature: {-/2,f/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,f,p} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))