YES(?,POLY) * Step 1: UnsatPaths WORST_CASE(?,POLY) + Considered Problem: Rules: 0. l0(A,B,C,D) -> l1(0,B,C,D) True (1,1) 1. l1(A,B,C,D) -> l2(A,B,0,0) [A >= 0 && B >= 1] (?,1) 2. l2(A,B,C,D) -> l2(A,B,1 + C,C + D) [D >= 0 (?,1) && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && B >= 1 + C] 3. l2(A,B,C,D) -> l1(A + D,-1 + B,C,D) [D >= 0 (?,1) && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && C >= B] Signature: {(l0,4);(l1,4);(l2,4)} Flow Graph: [0->{1},1->{2,3},2->{2,3},3->{1}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,3)] * Step 2: FromIts WORST_CASE(?,POLY) + Considered Problem: Rules: 0. l0(A,B,C,D) -> l1(0,B,C,D) True (1,1) 1. l1(A,B,C,D) -> l2(A,B,0,0) [A >= 0 && B >= 1] (?,1) 2. l2(A,B,C,D) -> l2(A,B,1 + C,C + D) [D >= 0 (?,1) && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && B >= 1 + C] 3. l2(A,B,C,D) -> l1(A + D,-1 + B,C,D) [D >= 0 (?,1) && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && C >= B] Signature: {(l0,4);(l1,4);(l2,4)} Flow Graph: [0->{1},1->{2},2->{2,3},3->{1}] + Applied Processor: FromIts + Details: () * Step 3: AddSinks WORST_CASE(?,POLY) + Considered Problem: Rules: l0(A,B,C,D) -> l1(0,B,C,D) True l1(A,B,C,D) -> l2(A,B,0,0) [A >= 0 && B >= 1] l2(A,B,C,D) -> l2(A,B,1 + C,C + D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && B >= 1 + C] l2(A,B,C,D) -> l1(A + D,-1 + B,C,D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && C >= B] Signature: {(l0,4);(l1,4);(l2,4)} Rule Graph: [0->{1},1->{2},2->{2,3},3->{1}] + Applied Processor: AddSinks + Details: () * Step 4: Decompose WORST_CASE(?,POLY) + Considered Problem: Rules: l0(A,B,C,D) -> l1(0,B,C,D) True l1(A,B,C,D) -> l2(A,B,0,0) [A >= 0 && B >= 1] l2(A,B,C,D) -> l2(A,B,1 + C,C + D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && B >= 1 + C] l2(A,B,C,D) -> l1(A + D,-1 + B,C,D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && C >= B] l2(A,B,C,D) -> exitus616(A,B,C,D) True l1(A,B,C,D) -> exitus616(A,B,C,D) True l2(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0,4);(l1,4);(l2,4)} Rule Graph: [0->{1},1->{2,4},2->{2,3,6},3->{1,5}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6] | `- p:[1,3,2] c: [1,3] | `- p:[2] c: [2] * Step 5: AbstractSize WORST_CASE(?,POLY) + Considered Problem: (Rules: l0(A,B,C,D) -> l1(0,B,C,D) True l1(A,B,C,D) -> l2(A,B,0,0) [A >= 0 && B >= 1] l2(A,B,C,D) -> l2(A,B,1 + C,C + D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && B >= 1 + C] l2(A,B,C,D) -> l1(A + D,-1 + B,C,D) [D >= 0 && C + D >= 0 && -1 + B + D >= 0 && A + D >= 0 && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && A >= 0 && C >= B] l2(A,B,C,D) -> exitus616(A,B,C,D) True l1(A,B,C,D) -> exitus616(A,B,C,D) True l2(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0,4);(l1,4);(l2,4)} Rule Graph: [0->{1},1->{2,4},2->{2,3,6},3->{1,5}] ,We construct a looptree: P: [0,1,2,3,4,5,6] | `- p:[1,3,2] c: [1,3] | `- p:[2] c: [2]) + Applied Processor: AbstractSize Minimize + Details: () * Step 6: AbstractFlow WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.0.0] l0 ~> l1 [A <= 0*K, B <= B, C <= C, D <= D] l1 ~> l2 [A <= A, B <= B, C <= 0*K, D <= 0*K] l2 ~> l2 [A <= A, B <= B, C <= B, D <= C + D] l2 ~> l1 [A <= A + D, B <= C, C <= C, D <= D] l2 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l1 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l2 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= B] l1 ~> l2 [A <= A, B <= B, C <= 0*K, D <= 0*K] l2 ~> l1 [A <= A + D, B <= C, C <= C, D <= D] l2 ~> l2 [A <= A, B <= B, C <= B, D <= C + D] + Loop: [0.0.0 <= K + B + C] l2 ~> l2 [A <= A, B <= B, C <= B, D <= C + D] + Applied Processor: AbstractFlow + Details: () * Step 7: Lare WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.0.0] l0 ~> l1 [K ~=> A] l1 ~> l2 [K ~=> C,K ~=> D] l2 ~> l2 [B ~=> C,C ~+> D,D ~+> D] l2 ~> l1 [C ~=> B,A ~+> A,D ~+> A] l2 ~> exitus616 [] l1 ~> exitus616 [] l2 ~> exitus616 [] + Loop: [B ~=> 0.0] l1 ~> l2 [K ~=> C,K ~=> D] l2 ~> l1 [C ~=> B,A ~+> A,D ~+> A] l2 ~> l2 [B ~=> C,C ~+> D,D ~+> D] + Loop: [B ~+> 0.0.0,C ~+> 0.0.0,K ~+> 0.0.0] l2 ~> l2 [B ~=> C,C ~+> D,D ~+> D] + Applied Processor: Lare + Details: l0 ~> exitus616 [B ~=> C ,B ~=> 0.0 ,K ~=> A ,K ~=> B ,K ~=> C ,K ~=> D ,K ~=> 0.0 ,B ~+> A ,B ~+> D ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0.0 ,K ~+> tick ,B ~*> A ,B ~*> D ,B ~*> 0.0.0 ,B ~*> tick ,K ~*> A ,K ~*> D ,K ~*> 0.0.0 ,K ~*> tick] + l2> [B ~=> C ,B ~=> 0.0 ,K ~=> B ,K ~=> C ,K ~=> D ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0.0 ,K ~+> tick ,B ~*> A ,B ~*> D ,B ~*> 0.0.0 ,B ~*> tick ,K ~*> A ,K ~*> D ,K ~*> 0.0.0 ,K ~*> tick] l1> [B ~=> C ,B ~=> 0.0 ,K ~=> B ,K ~=> C ,K ~=> D ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0.0 ,K ~+> tick ,B ~*> A ,B ~*> D ,B ~*> 0.0.0 ,B ~*> tick ,K ~*> A ,K ~*> D ,K ~*> 0.0.0 ,K ~*> tick] + l2> [B ~=> C ,B ~+> D ,B ~+> 0.0.0 ,B ~+> tick ,C ~+> D ,C ~+> 0.0.0 ,C ~+> tick ,D ~+> D ,tick ~+> tick ,K ~+> 0.0.0 ,K ~+> tick ,B ~*> D ,C ~*> D ,K ~*> D] YES(?,POLY)