YES(?,POLY) * Step 1: TrivialSCCs 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,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: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (1,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,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 3: AddSinks 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (1,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,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: AddSinks + Details: () * Step 4: 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,1) 6. l2(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) 7. l3(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,6},3->{4,5,7},4->{4,5,7},5->{3,6},6->{},7->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(3,5)] * Step 5: LooptreeTransformer 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,1) 6. l2(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) 7. l3(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,6},3->{4,7},4->{4,5,7},5->{3,6},6->{},7->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7] | +- p:[1] c: [1] | `- p:[3,5,4] c: [5] | `- p:[4] c: [4] * Step 6: SizeAbstraction 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) [B >= 1] (?,1) 2. l1(A,B,C,D) -> l2(A,B,A,D) [0 >= B] (?,1) 3. l2(A,B,C,D) -> l3(A,B,C,C) [C >= 1] (?,1) 4. l3(A,B,C,D) -> l3(A,B,C,-1 + D) [D >= 1 && C >= 1] (?,1) 5. l3(A,B,C,D) -> l2(A,B,-1 + C,D) [0 >= D && C >= 1] (?,1) 6. l2(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) 7. l3(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(l0,4);(l1,4);(l2,4);(l3,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,6},3->{4,7},4->{4,5,7},5->{3,6},6->{},7->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7] | +- p:[1] c: [1] | `- p:[3,5,4] c: [5] | `- p:[4] c: [4]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 7: FlowAbstraction WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.1,0.1.0] l0 ~> l1 [A <= 0*K, B <= B, C <= C, D <= D] l1 ~> l1 [A <= K + A, B <= B, C <= C, D <= D] l1 ~> l2 [A <= A, B <= B, C <= A, D <= D] l2 ~> l3 [A <= A, B <= B, C <= C, D <= C] l3 ~> l3 [A <= A, B <= B, C <= C, D <= D] l3 ~> l2 [A <= A, B <= B, C <= C, D <= D] l2 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] l3 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= B] l1 ~> l1 [A <= K + A, B <= B, C <= C, D <= D] + Loop: [0.1 <= C] l2 ~> l3 [A <= A, B <= B, C <= C, D <= C] l3 ~> l2 [A <= A, B <= B, C <= C, D <= D] l3 ~> l3 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.1.0 <= D] l3 ~> l3 [A <= A, B <= B, C <= C, D <= D] + Applied Processor: FlowAbstraction + Details: () * Step 8: LareProcessor WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.1,0.1.0] l0 ~> l1 [K ~=> A] l1 ~> l1 [A ~+> A,K ~+> A] l1 ~> l2 [A ~=> C] l2 ~> l3 [C ~=> D] l3 ~> l3 [] l3 ~> l2 [] l2 ~> exitus616 [] l3 ~> exitus616 [] + Loop: [B ~=> 0.0] l1 ~> l1 [A ~+> A,K ~+> A] + Loop: [C ~=> 0.1] l2 ~> l3 [C ~=> D] l3 ~> l2 [] l3 ~> l3 [] + Loop: [D ~=> 0.1.0] l3 ~> l3 [] + Applied Processor: LareProcessor + Details: l0 ~> exitus616 [B ~=> 0.0 ,K ~=> A ,K ~=> C ,K ~=> D ,K ~=> 0.1 ,K ~=> 0.1.0 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> C ,K ~+> D ,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> [B ~=> 0.0,A ~+> A,B ~+> tick,tick ~+> tick,K ~+> A,B ~*> A,K ~*> A] + l3> [C ~=> D,C ~=> 0.1,C ~=> 0.1.0,C ~+> tick,tick ~+> tick,C ~*> tick] l2> [C ~=> D,C ~=> 0.1,C ~=> 0.1.0,C ~+> tick,tick ~+> tick,C ~*> tick] + l3> [D ~=> 0.1.0,D ~+> tick,tick ~+> tick] YES(?,POLY)