WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: add(0(),x) -> x add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} and constructors {0,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) 0 :: [] -(0)-> "A"(6) 0 :: [] -(0)-> "A"(12) add :: ["A"(0) x "A"(0)] -(0)-> "A"(0) p :: ["A"(6) x "A"(6) x "A"(0) x "A"(0)] -(7)-> "A"(12) q :: ["A"(6) x "A"(0)] -(8)-> "A"(12) s :: ["A"(0)] -(0)-> "A"(0) s :: ["A"(6)] -(6)-> "A"(6) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: q(s(x),y) -> p(s(x),0(),s(0()),y) 2. Weak: add(0(),x) -> x add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) * Step 2: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: add(0(),x) -> x add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) - Weak TRS: q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} and constructors {0,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0) 0 :: [] -(0)-> "A"(12) 0 :: [] -(0)-> "A"(14) add :: ["A"(0) x "A"(0)] -(8)-> "A"(0) p :: ["A"(12) x "A"(12) x "A"(0) x "A"(0)] -(6)-> "A"(12) q :: ["A"(12) x "A"(0)] -(10)-> "A"(12) s :: ["A"(0)] -(0)-> "A"(0) s :: ["A"(12)] -(12)-> "A"(12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: add(0(),x) -> x 2. Weak: add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) * Step 3: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) - Weak TRS: add(0(),x) -> x q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} and constructors {0,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(12) 0 :: [] -(0)-> "A"(0) 0 :: [] -(0)-> "A"(14) add :: ["A"(0) x "A"(0)] -(7)-> "A"(0) p :: ["A"(12) x "A"(12) x "A"(0) x "A"(0)] -(15)-> "A"(13) q :: ["A"(12) x "A"(0)] -(15)-> "A"(13) s :: ["A"(0)] -(0)-> "A"(0) s :: ["A"(12)] -(12)-> "A"(12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1) "s_A" :: ["A"(1)] -(1)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: p(0(),s(x),y,z) -> q(x,add(x,z)) 2. Weak: add(s(x),y) -> s(add(x,y)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) * Step 4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: add(s(x),y) -> s(add(x,y)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) - Weak TRS: add(0(),x) -> x p(0(),s(x),y,z) -> q(x,add(x,z)) q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} 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(q) = {2}, uargs(s) = {1} Following symbols are considered usable: {add,p,q} TcT has computed the following interpretation: p(0) = 0 p(add) = 2*x1 + x2 p(p) = 1 + 2*x1*x2 + x1^2 + x2^2 + x4 p(q) = 1 + x1^2 + x2 p(s) = 1 + x1 Following rules are strictly oriented: add(s(x),y) = 2 + 2*x + y > 1 + 2*x + y = s(add(x,y)) Following rules are (at-least) weakly oriented: add(0(),x) = x >= x = x p(0(),s(x),y,z) = 2 + 2*x + x^2 + z >= 1 + 2*x + x^2 + z = q(x,add(x,z)) p(s(x),y,z,u) = 2 + u + 2*x + 2*x*y + x^2 + 2*y + y^2 >= 2 + u + 2*x + 2*x*y + x^2 + 2*y + y^2 = p(x,s(y),s(s(z)),u) q(s(x),y) = 2 + 2*x + x^2 + y >= 2 + 2*x + x^2 + y = p(s(x),0(),s(0()),y) * Step 5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) - Weak TRS: add(0(),x) -> x add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} 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(q) = {2}, uargs(s) = {1} Following symbols are considered usable: {add,p,q} TcT has computed the following interpretation: p(0) = 0 p(add) = x1 + x2 p(p) = 5*x1 + 6*x1*x2 + 3*x1^2 + x2 + 3*x2^2 + x3 + x4 p(q) = 4 + 5*x1 + 3*x1^2 + x2 p(s) = 1 + x1 Following rules are strictly oriented: p(s(x),y,z,u) = 8 + u + 11*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 + z > 6 + u + 11*x + 6*x*y + 3*x^2 + 7*y + 3*y^2 + z = p(x,s(y),s(s(z)),u) Following rules are (at-least) weakly oriented: add(0(),x) = x >= x = x add(s(x),y) = 1 + x + y >= 1 + x + y = s(add(x,y)) p(0(),s(x),y,z) = 4 + 7*x + 3*x^2 + y + z >= 4 + 6*x + 3*x^2 + z = q(x,add(x,z)) q(s(x),y) = 12 + 11*x + 3*x^2 + y >= 9 + 11*x + 3*x^2 + y = p(s(x),0(),s(0()),y) * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),x) -> x add(s(x),y) -> s(add(x,y)) p(0(),s(x),y,z) -> q(x,add(x,z)) p(s(x),y,z,u) -> p(x,s(y),s(s(z)),u) q(s(x),y) -> p(s(x),0(),s(0()),y) - Signature: {add/2,p/4,q/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,p,q} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))