WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),C()) -> triple(a,b,s(c)) f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c) foldB(t,0()) -> t foldB(t,s(n)) -> f(foldB(t,n),B()) foldC(t,0()) -> t foldC(t,s(n)) -> f(foldC(t,n),C()) g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() - Signature: {f/2,f'/2,f''/1,foldB/2,foldC/2,g/1} / {0/0,A/0,B/0,C/0,s/1,triple/3} - Obligation: innermost runtime complexity wrt. defined symbols {f,f',f'',foldB,foldC,g} and constructors {0,A,B,C,s ,triple} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),C()) -> triple(a,b,s(c)) f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c) foldB(t,0()) -> t foldB(t,s(n)) -> f(foldB(t,n),B()) foldC(t,0()) -> t foldC(t,s(n)) -> f(foldC(t,n),C()) g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() - Signature: {f/2,f'/2,f''/1,foldB/2,foldC/2,g/1} / {0/0,A/0,B/0,C/0,s/1,triple/3} - Obligation: innermost runtime complexity wrt. defined symbols {f,f',f'',foldB,foldC,g} and constructors {0,A,B,C,s ,triple} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 2.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),C()) -> triple(a,b,s(c)) f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c) foldB(t,0()) -> t foldB(t,s(n)) -> f(foldB(t,n),B()) foldC(t,0()) -> t foldC(t,s(n)) -> f(foldC(t,n),C()) g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() - Signature: {f/2,f'/2,f''/1,foldB/2,foldC/2,g/1} / {0/0,A/0,B/0,C/0,s/1,triple/3} - Obligation: innermost runtime complexity wrt. defined symbols {f,f',f'',foldB,foldC,g} and constructors {0,A,B,C,s ,triple} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "A") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "A") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "B") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "B") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "C") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "C") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "f") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "f'") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "f''") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "foldB") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "foldC") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "g") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "main") :: ["A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "triple") :: ["A"(0, 0, 0) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),C()) -> triple(a,b,s(c)) f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c) foldB(t,0()) -> t foldB(t,s(n)) -> f(foldB(t,n),B()) foldC(t,0()) -> t foldC(t,s(n)) -> f(foldC(t,n),C()) g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() main(x1) -> g(x1) 2. Weak: ** Step 2.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),A()) -> f''(foldB(triple(s(a),0(),c),b)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),C()) -> triple(a,b,s(c)) f''(triple(a,b,c)) -> foldC(triple(a,b,0()),c) foldB(t,0()) -> t foldB(t,s(n)) -> f(foldB(t,n),B()) foldC(t,0()) -> t foldC(t,s(n)) -> f(foldC(t,n),C()) g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() - Signature: {f/2,f'/2,f''/1,foldB/2,foldC/2,g/1} / {0/0,A/0,B/0,C/0,s/1,triple/3} - Obligation: innermost runtime complexity wrt. defined symbols {f,f',f'',foldB,foldC,g} and constructors {0,A,B,C,s ,triple} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: foldB(x,y){y -> s(y)} = foldB(x,s(y)) ->^+ f(foldB(x,y),B()) = C[foldB(x,y) = foldB(x,y){}] WORST_CASE(Omega(n^1),?)