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 && -1 + B >= 0] (?,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 && -1 + B >= 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] 4. l1(A,B,C,D) -> l3(A,B,C,D) [A >= 0 && 0 >= B] (?,1) 5. l3(A,B,C,D) -> l3(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] (?,1) Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,4},1->{2,3},2->{2,3},3->{1,4},4->{5},5->{5}] + 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 && -1 + B >= 0] (?,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 && -1 + B >= 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] 4. l1(A,B,C,D) -> l3(A,B,C,D) [A >= 0 && 0 >= B] (?,1) 5. l3(A,B,C,D) -> l3(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] (?,1) Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,4},1->{2},2->{2,3},3->{1,4},4->{5},5->{5}] + Applied Processor: FromIts + Details: () * Step 3: 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 && -1 + B >= 0] 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 && -1 + B >= 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] l1(A,B,C,D) -> l3(A,B,C,D) [A >= 0 && 0 >= B] l3(A,B,C,D) -> l3(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Rule Graph: [0->{1,4},1->{2},2->{2,3},3->{1,4},4->{5},5->{5}] + Applied Processor: Unfold + Details: () * Step 4: AddSinks WORST_CASE(?,POLY) + Considered Problem: Rules: l0.0(A,B,C,D) -> l1.1(0,B,C,D) True l0.0(A,B,C,D) -> l1.4(0,B,C,D) True l1.1(A,B,C,D) -> l2.2(A,B,0,0) [A >= 0 && -1 + B >= 0] 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 && -1 + B >= 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 && -1 + B >= 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) -> l1.4(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] l1.4(A,B,C,D) -> l3.5(A,B,C,D) [A >= 0 && 0 >= B] l3.5(A,B,C,D) -> l3.5(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] Signature: {(l0.0,4);(l1.1,4);(l1.4,4);(l2.2,4);(l2.3,4);(l3.5,4)} Rule Graph: [0->{2},1->{7},2->{3,4},3->{3,4},4->{5,6},5->{2},6->{7},7->{8},8->{8}] + Applied Processor: AddSinks + Details: () * Step 5: Decompose WORST_CASE(?,POLY) + Considered Problem: Rules: l0.0(A,B,C,D) -> l1.1(0,B,C,D) True l0.0(A,B,C,D) -> l1.4(0,B,C,D) True l1.1(A,B,C,D) -> l2.2(A,B,0,0) [A >= 0 && -1 + B >= 0] 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 && -1 + B >= 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 && -1 + B >= 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) -> l1.4(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] l1.4(A,B,C,D) -> l3.5(A,B,C,D) [A >= 0 && 0 >= B] l3.5(A,B,C,D) -> l3.5(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0.0,4);(l1.1,4);(l1.4,4);(l2.2,4);(l2.3,4);(l3.5,4)} Rule Graph: [0->{2},1->{7},2->{3,4},3->{3,4},4->{5,6},5->{2},6->{7},7->{8},8->{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:[2,5,4,3] c: [2,4,5] | | | `- p:[3] c: [3] | `- p:[8] c: [8] * Step 6: AbstractSize WORST_CASE(?,POLY) + Considered Problem: (Rules: l0.0(A,B,C,D) -> l1.1(0,B,C,D) True l0.0(A,B,C,D) -> l1.4(0,B,C,D) True l1.1(A,B,C,D) -> l2.2(A,B,0,0) [A >= 0 && -1 + B >= 0] 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 && -1 + B >= 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 && -1 + B >= 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) -> l1.4(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] l1.4(A,B,C,D) -> l3.5(A,B,C,D) [A >= 0 && 0 >= B] l3.5(A,B,C,D) -> l3.5(-1 + A,B,C,D) [-1*B >= 0 && A + -1*B >= 0 && A >= 0 && -1 + A >= 0] l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0.0,4);(l1.1,4);(l1.4,4);(l2.2,4);(l2.3,4);(l3.5,4)} Rule Graph: [0->{2},1->{7},2->{3,4},3->{3,4},4->{5,6},5->{2},6->{7},7->{8},8->{8,9,10}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10] | +- p:[2,5,4,3] c: [2,4,5] | | | `- p:[3] c: [3] | `- p:[8] c: [8]) + Applied Processor: AbstractSize Minimize + Details: () * Step 7: AbstractFlow WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.0.0,0.1] l0.0 ~> l1.1 [A <= 0*K, B <= B, C <= C, D <= D] l0.0 ~> l1.4 [A <= 0*K, B <= B, C <= C, D <= D] l1.1 ~> l2.2 [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] l2.3 ~> l1.1 [A <= A + D, B <= C, C <= C, D <= D] l2.3 ~> l1.4 [A <= A + D, B <= C, C <= C, D <= D] l1.4 ~> l3.5 [A <= A, B <= B, C <= C, D <= D] l3.5 ~> l3.5 [A <= A, B <= B, C <= C, D <= D] l3.5 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.5 ~> exitus616 [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] + Loop: [0.1 <= K + A] l3.5 ~> l3.5 [A <= A, B <= B, C <= C, D <= D] + Applied Processor: AbstractFlow + Details: () * Step 8: Lare WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.0.0,0.1] l0.0 ~> l1.1 [K ~=> A] l0.0 ~> l1.4 [K ~=> A] l1.1 ~> l2.2 [K ~=> C,K ~=> D] l2.2 ~> l2.2 [B ~=> C,C ~+> D,D ~+> D] l2.2 ~> l2.3 [B ~=> C,C ~+> D,D ~+> D] l2.3 ~> l1.1 [C ~=> B,A ~+> A,D ~+> A] l2.3 ~> l1.4 [C ~=> B,A ~+> A,D ~+> A] l1.4 ~> l3.5 [] l3.5 ~> l3.5 [] l3.5 ~> exitus616 [] l3.5 ~> exitus616 [] + 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] + Loop: [A ~+> 0.1,K ~+> 0.1] l3.5 ~> l3.5 [] + Applied Processor: Lare + Details: l0.0 ~> exitus616 [B ~=> C ,C ~=> B ,K ~=> A ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.1 ,B ~+> tick ,C ~+> A ,C ~+> D ,C ~+> 0.0.0 ,C ~+> 0.1 ,C ~+> tick ,D ~+> A ,D ~+> 0.1 ,D ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> D ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.1 ,K ~+> tick ,B ~*> A ,B ~*> D ,B ~*> 0.0.0 ,B ~*> 0.1 ,B ~*> tick ,C ~*> A ,C ~*> D ,C ~*> 0.0.0 ,C ~*> 0.1 ,C ~*> tick ,K ~*> A ,K ~*> D ,K ~*> 0.0.0 ,K ~*> 0.1 ,K ~*> tick] + l2.3> [B ~=> C ,C ~=> B ,A ~+> A ,B ~+> A ,B ~+> D ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> tick ,C ~+> A ,C ~+> D ,C ~+> 0.0.0 ,C ~+> tick ,D ~+> A ,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 ,C ~*> A ,C ~*> D ,C ~*> 0.0.0 ,C ~*> 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] + l3.5> [A ~+> 0.1,A ~+> tick,tick ~+> tick,K ~+> 0.1,K ~+> tick] YES(?,POLY)