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) -> l1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [A >= 0 && 0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [C + -1*D >= 0 (?,1) && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [C + -1*D >= 0 (?,1) && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3},3->{4,5},4->{4,5},5->{3}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(3,5)] * 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) -> l1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [A >= 0 && 0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [C + -1*D >= 0 (?,1) && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [C + -1*D >= 0 (?,1) && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3},3->{4},4->{4,5},5->{3}] + 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) -> l1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1(A,B,C,D) -> l2(A,B,A,D) [A >= 0 && 0 >= B] l2(A,B,C,D) -> l3(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] l3(A,B,C,D) -> l3(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3(A,B,C,D) -> l2(A,B,-1 + C,D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] Signature: {(l0,4);(l1,4);(l2,4);(l3,4)} Rule Graph: [0->{1,2},1->{1,2},2->{3},3->{4},4->{4,5},5->{3}] + 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) -> l1.2(0,B,C,D) True l1.1(A,B,C,D) -> l1.1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.1(A,B,C,D) -> l1.2(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.2(A,B,C,D) -> l2.3(A,B,A,D) [A >= 0 && 0 >= B] l2.3(A,B,C,D) -> l3.4(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] l3.4(A,B,C,D) -> l3.4(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.4(A,B,C,D) -> l3.5(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.5(A,B,C,D) -> l2.3(A,B,-1 + C,D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] Signature: {(l0.0,4);(l1.1,4);(l1.2,4);(l2.3,4);(l3.4,4);(l3.5,4)} Rule Graph: [0->{2,3},1->{4},2->{2,3},3->{4},4->{5},5->{6,7},6->{6,7},7->{8},8->{5}] + 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) -> l1.2(0,B,C,D) True l1.1(A,B,C,D) -> l1.1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.1(A,B,C,D) -> l1.2(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.2(A,B,C,D) -> l2.3(A,B,A,D) [A >= 0 && 0 >= B] l2.3(A,B,C,D) -> l3.4(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] l3.4(A,B,C,D) -> l3.4(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.4(A,B,C,D) -> l3.5(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.5(A,B,C,D) -> l2.3(A,B,-1 + C,D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l2.3(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l2.3(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0.0,4);(l1.1,4);(l1.2,4);(l2.3,4);(l3.4,4);(l3.5,4)} Rule Graph: [0->{2,3},1->{4},2->{2,3},3->{4},4->{5},5->{6,7,9,13},6->{6,7,12,16},7->{8,11,15},8->{5,10,14}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] | +- p:[2] c: [2] | `- p:[5,8,7,6] c: [5,7,8] | `- p:[6] c: [6] * 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) -> l1.2(0,B,C,D) True l1.1(A,B,C,D) -> l1.1(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.1(A,B,C,D) -> l1.2(1 + A,-1 + B,C,D) [A >= 0 && B >= 1] l1.2(A,B,C,D) -> l2.3(A,B,A,D) [A >= 0 && 0 >= B] l2.3(A,B,C,D) -> l3.4(A,B,C,C) [A + -1*C >= 0 && C >= 0 && -1*B + C >= 0 && A + C >= 0 && -1*B >= 0 && A + -1*B >= 0 && A >= 0 && C >= 1] l3.4(A,B,C,D) -> l3.4(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.4(A,B,C,D) -> l3.5(A,B,C,-1 + D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && D >= 1 && C >= 1] l3.5(A,B,C,D) -> l2.3(A,B,-1 + C,D) [C + -1*D >= 0 && A + -1*D >= 0 && A + -1*C >= 0 && -1 + C >= 0 && -1 + -1*B + C >= 0 && -2 + A + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && 0 >= D && C >= 1] l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l2.3(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True l2.3(A,B,C,D) -> exitus616(A,B,C,D) True l3.5(A,B,C,D) -> exitus616(A,B,C,D) True l3.4(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(l0.0,4);(l1.1,4);(l1.2,4);(l2.3,4);(l3.4,4);(l3.5,4)} Rule Graph: [0->{2,3},1->{4},2->{2,3},3->{4},4->{5},5->{6,7,9,13},6->{6,7,12,16},7->{8,11,15},8->{5,10,14}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] | +- p:[2] c: [2] | `- p:[5,8,7,6] c: [5,7,8] | `- p:[6] c: [6]) + Applied Processor: AbstractSize Minimize + Details: () * Step 7: AbstractFlow WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.1,0.1.0] l0.0 ~> l1.1 [A <= 0*K, B <= B, C <= C, D <= D] l0.0 ~> l1.2 [A <= 0*K, B <= B, C <= C, D <= D] l1.1 ~> l1.1 [A <= K + A, B <= B, C <= C, D <= D] l1.1 ~> l1.2 [A <= K + A, B <= B, C <= C, D <= D] l1.2 ~> l2.3 [A <= A, B <= B, C <= A, D <= D] l2.3 ~> l3.4 [A <= A, B <= B, C <= C, D <= C] l3.4 ~> l3.4 [A <= A, B <= B, C <= C, D <= C] l3.4 ~> l3.5 [A <= A, B <= B, C <= C, D <= C] l3.5 ~> l2.3 [A <= A, B <= B, C <= C, D <= D] l3.4 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l2.3 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.5 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.4 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.4 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l2.3 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.5 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3.4 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= K + B] l1.1 ~> l1.1 [A <= K + A, B <= B, C <= C, D <= D] + Loop: [0.1 <= K + C] l2.3 ~> l3.4 [A <= A, B <= B, C <= C, D <= C] l3.5 ~> l2.3 [A <= A, B <= B, C <= C, D <= D] l3.4 ~> l3.5 [A <= A, B <= B, C <= C, D <= C] l3.4 ~> l3.4 [A <= A, B <= B, C <= C, D <= C] + Loop: [0.1.0 <= K + D] l3.4 ~> l3.4 [A <= A, B <= B, C <= C, D <= C] + Applied Processor: AbstractFlow + Details: () * Step 8: Lare WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.1,0.1.0] l0.0 ~> l1.1 [K ~=> A] l0.0 ~> l1.2 [K ~=> A] l1.1 ~> l1.1 [A ~+> A,K ~+> A] l1.1 ~> l1.2 [A ~+> A,K ~+> A] l1.2 ~> l2.3 [A ~=> C] l2.3 ~> l3.4 [C ~=> D] l3.4 ~> l3.4 [C ~=> D] l3.4 ~> l3.5 [C ~=> D] l3.5 ~> l2.3 [] l3.4 ~> exitus616 [] l2.3 ~> exitus616 [] l3.5 ~> exitus616 [] l3.4 ~> exitus616 [] l3.4 ~> exitus616 [] l2.3 ~> exitus616 [] l3.5 ~> exitus616 [] l3.4 ~> exitus616 [] + Loop: [B ~+> 0.0,K ~+> 0.0] l1.1 ~> l1.1 [A ~+> A,K ~+> A] + Loop: [C ~+> 0.1,K ~+> 0.1] l2.3 ~> l3.4 [C ~=> D] l3.5 ~> l2.3 [] l3.4 ~> l3.5 [C ~=> D] l3.4 ~> l3.4 [C ~=> D] + Loop: [D ~+> 0.1.0,K ~+> 0.1.0] l3.4 ~> l3.4 [C ~=> D] + Applied Processor: Lare + Details: l0.0 ~> exitus616 [K ~=> A ,K ~=> C ,K ~=> D ,B ~+> 0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> C ,K ~+> D ,K ~+> 0.0 ,K ~+> 0.1 ,K ~+> 0.1.0 ,K ~+> tick ,B ~*> A ,B ~*> C ,B ~*> D ,B ~*> 0.1 ,B ~*> 0.1.0 ,B ~*> tick ,K ~*> A ,K ~*> C ,K ~*> D ,K ~*> 0.1 ,K ~*> 0.1.0 ,K ~*> tick] + l1.1> [A ~+> A ,B ~+> 0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> 0.0 ,K ~+> tick ,B ~*> A ,K ~*> A] + l3.5> [C ~=> D ,C ~+> 0.1 ,C ~+> 0.1.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.1 ,K ~+> 0.1.0 ,K ~+> tick ,C ~*> tick ,K ~*> tick] l3.4> [C ~=> D ,C ~+> 0.1 ,C ~+> 0.1.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.1 ,K ~+> 0.1.0 ,K ~+> tick ,C ~*> tick ,K ~*> tick] l2.3> [C ~=> D ,C ~+> 0.1 ,C ~+> 0.1.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.1 ,K ~+> 0.1.0 ,K ~+> tick ,C ~*> tick ,K ~*> tick] + l3.4> [C ~=> D,D ~+> 0.1.0,D ~+> tick,tick ~+> tick,K ~+> 0.1.0,K ~+> tick] YES(?,POLY)