WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: {if,le,minus,p} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if) = x1 + x2 + x3 p(le) = 0 p(minus) = 9 + x1 + 8*x2 p(p) = x1 p(s) = x1 p(true) = 0 Following rules are strictly oriented: minus(x,0()) = 9 + x > x = x Following rules are (at-least) weakly oriented: if(false(),x,y) = x + y >= y = y if(true(),x,y) = x + y >= x = x le(0(),y) = 0 >= 0 = true() le(s(x),0()) = 0 >= 0 = false() le(s(x),s(y)) = 0 >= 0 = le(x,y) minus(x,s(y)) = 9 + x + 8*y >= 9 + x + 8*y = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = 0 >= 0 = 0() p(s(x)) = x >= x = x ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Weak TRS: minus(x,0()) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: {if,le,minus,p} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if) = 8*x1 + x2 + x3 p(le) = 0 p(minus) = 9 + x1 + 2*x2 p(p) = x1 p(s) = 4 + x1 p(true) = 0 Following rules are strictly oriented: p(s(x)) = 4 + x > x = x Following rules are (at-least) weakly oriented: if(false(),x,y) = x + y >= y = y if(true(),x,y) = x + y >= x = x le(0(),y) = 0 >= 0 = true() le(s(x),0()) = 0 >= 0 = false() le(s(x),s(y)) = 0 >= 0 = le(x,y) minus(x,0()) = 9 + x >= x = x minus(x,s(y)) = 17 + x + 2*y >= 17 + x + 2*y = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = 0 >= 0 = 0() ** Step 1.b:3: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() - Weak TRS: minus(x,0()) -> x p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 5, araRuleShifting = Nothing} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0, 0) 0 :: [] -(0)-> "A"(1, 0) 0 :: [] -(0)-> "A"(6, 6) 0 :: [] -(0)-> "A"(12, 6) false :: [] -(0)-> "A"(0, 0) if :: ["A"(0, 0) x "A"(6, 6) x "A"(6, 6)] -(1)-> "A"(6, 6) le :: ["A"(0, 0) x "A"(1, 0)] -(1)-> "A"(0, 0) minus :: ["A"(6, 6) x "A"(12, 6)] -(0)-> "A"(6, 6) p :: ["A"(6, 6)] -(1)-> "A"(12, 6) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(1, 0)] -(1)-> "A"(1, 0) s :: ["A"(18, 6)] -(12)-> "A"(12, 6) s :: ["A"(12, 6)] -(6)-> "A"(6, 6) true :: [] -(0)-> "A"(0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- WORST_CASE(Omega(n^1),O(n^2))