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: Unfold 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: Unfold + Details: () * Step 5: Decompose WORST_CASE(?,POLY) + Considered Problem: Rules: l0.0(A,B,C,D) -> l1.1(0,B,C,D) True l1.1(A,B,C,D) -> l2.2(A,B,0,0) [A >= 0 && B >= 1] l1.1(A,B,C,D) -> l2.4(A,B,0,0) [A >= 0 && B >= 1] l2.2(A,B,C,D) -> l2.2(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] l2.2(A,B,C,D) -> l2.3(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] l2.2(A,B,C,D) -> l2.6(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] l2.3(A,B,C,D) -> l1.1(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] l2.3(A,B,C,D) -> l1.5(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] l2.4(A,B,C,D) -> exitus616.7(A,B,C,D) True l1.5(A,B,C,D) -> exitus616.7(A,B,C,D) True l2.6(A,B,C,D) -> exitus616.7(A,B,C,D) True Signature: {(exitus616.7,4);(l0.0,4);(l1.1,4);(l1.5,4);(l2.2,4);(l2.3,4);(l2.4,4);(l2.6,4)} Rule Graph: [0->{1,2},1->{3,4,5},2->{8},3->{3,4,5},4->{6,7},5->{10},6->{1,2},7->{9},8->{},9->{},10->{}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10] | `- p:[1,6,4,3] c: [1,4,6] | `- p:[3] c: [3] * Step 6: AbstractSize WORST_CASE(?,POLY) + Considered Problem: (Rules: l0.0(A,B,C,D) -> l1.1(0,B,C,D) True l1.1(A,B,C,D) -> l2.2(A,B,0,0) [A >= 0 && B >= 1] l1.1(A,B,C,D) -> l2.4(A,B,0,0) [A >= 0 && B >= 1] l2.2(A,B,C,D) -> l2.2(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] l2.2(A,B,C,D) -> l2.3(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] l2.2(A,B,C,D) -> l2.6(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] l2.3(A,B,C,D) -> l1.1(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] l2.3(A,B,C,D) -> l1.5(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] l2.4(A,B,C,D) -> exitus616.7(A,B,C,D) True l1.5(A,B,C,D) -> exitus616.7(A,B,C,D) True l2.6(A,B,C,D) -> exitus616.7(A,B,C,D) True Signature: {(exitus616.7,4);(l0.0,4);(l1.1,4);(l1.5,4);(l2.2,4);(l2.3,4);(l2.4,4);(l2.6,4)} Rule Graph: [0->{1,2},1->{3,4,5},2->{8},3->{3,4,5},4->{6,7},5->{10},6->{1,2},7->{9},8->{},9->{},10->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10] | `- p:[1,6,4,3] c: [1,4,6] | `- p:[3] c: [3]) + Applied Processor: AbstractSize Minimize + Details: () * Step 7: AbstractFlow WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.0.0] l0.0 ~> l1.1 [A <= 0*K, B <= B, C <= C, D <= D] l1.1 ~> l2.2 [A <= A, B <= B, C <= 0*K, D <= 0*K] l1.1 ~> l2.4 [A <= A, B <= B, C <= 0*K, D <= 0*K] l2.2 ~> l2.2 [A <= A, B <= B, C <= B, D <= C + D] l2.2 ~> l2.3 [A <= A, B <= B, C <= B, D <= C + D] l2.2 ~> l2.6 [A <= A, B <= B, C <= B, D <= C + D] l2.3 ~> l1.1 [A <= A + D, B <= C, C <= C, D <= D] l2.3 ~> l1.5 [A <= A + D, B <= C, C <= C, D <= D] l2.4 ~> exitus616.7 [A <= A, B <= B, C <= C, D <= D] l1.5 ~> exitus616.7 [A <= A, B <= B, C <= C, D <= D] l2.6 ~> exitus616.7 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= K + B] l1.1 ~> l2.2 [A <= A, B <= B, C <= 0*K, D <= 0*K] l2.3 ~> l1.1 [A <= A + D, B <= C, C <= C, D <= D] l2.2 ~> l2.3 [A <= A, B <= B, C <= B, D <= C + D] l2.2 ~> l2.2 [A <= A, B <= B, C <= B, D <= C + D] + Loop: [0.0.0 <= K + B + C] l2.2 ~> l2.2 [A <= A, B <= B, C <= B, D <= C + D] + Applied Processor: AbstractFlow + Details: () * Step 8: Lare WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.0.0] l0.0 ~> l1.1 [K ~=> A] l1.1 ~> l2.2 [K ~=> C,K ~=> D] l1.1 ~> l2.4 [K ~=> C,K ~=> D] l2.2 ~> l2.2 [B ~=> C,C ~+> D,D ~+> D] l2.2 ~> l2.3 [B ~=> C,C ~+> D,D ~+> D] l2.2 ~> l2.6 [B ~=> C,C ~+> D,D ~+> D] l2.3 ~> l1.1 [C ~=> B,A ~+> A,D ~+> A] l2.3 ~> l1.5 [C ~=> B,A ~+> A,D ~+> A] l2.4 ~> exitus616.7 [] l1.5 ~> exitus616.7 [] l2.6 ~> exitus616.7 [] + Loop: [B ~+> 0.0,K ~+> 0.0] l1.1 ~> l2.2 [K ~=> C,K ~=> D] l2.3 ~> l1.1 [C ~=> B,A ~+> A,D ~+> A] l2.2 ~> l2.3 [B ~=> C,C ~+> D,D ~+> D] l2.2 ~> l2.2 [B ~=> C,C ~+> D,D ~+> D] + Loop: [B ~+> 0.0.0,C ~+> 0.0.0,K ~+> 0.0.0] l2.2 ~> l2.2 [B ~=> C,C ~+> D,D ~+> D] + Applied Processor: Lare + Details: l0.0 ~> exitus616.7 [B ~=> C ,C ~=> B ,K ~=> A ,K ~=> C ,K ~=> D ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> tick ,C ~+> D ,D ~+> A ,D ~+> D ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0 ,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] + l2.3> [B ~=> C ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0 ,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] l2.2> [B ~=> C ,K ~=> C ,K ~=> D ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0 ,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] l1.1> [B ~=> C ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0 ,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] + l2.2> [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)