YES(?,O(1)) * Step 1: TrivialSCCs WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (?,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (?,1) Signature: {(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2,3,4},1->{1,2,3,4},2->{1,2,3,4},3->{},4->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (1,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (1,1) Signature: {(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2,3,4},1->{1,2,3,4},2->{1,2,3,4},3->{},4->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,3),(0,4)] * Step 3: AddSinks WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (1,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (1,1) Signature: {(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2},1->{1,2,3,4},2->{1,2,3,4},3->{},4->{}] + Applied Processor: AddSinks + Details: () * Step 4: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (?,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (?,1) 5. f8(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2,3,4,5},1->{1,2,3,4,5},2->{1,2,3,4,5},3->{},4->{},5->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,3),(0,4)] * Step 5: LooptreeTransformer WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (?,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (?,1) 5. f8(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2,5},1->{1,2,3,4,5},2->{1,2,3,4,5},3->{},4->{},5->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5] | `- p:[1,2] c: [2] | `- p:[1] c: [1] * Step 6: SizeAbstraction WORST_CASE(?,O(1)) + Considered Problem: (Rules: 0. f0(A,B,C,D) -> f8(0,B,C,D) True (1,1) 1. f8(A,B,C,D) -> f8(1 + A,B,C,D) [3 >= A] (?,1) 2. f8(A,B,C,D) -> f8(1 + A,A,1 + A,E) [3 >= A] (?,1) 3. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4 && 0 >= 1 + E] (?,1) 4. f8(A,B,C,D) -> f23(A,B,C,D) [A >= 4] (?,1) 5. f8(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f23,4);(f8,4)} Flow Graph: [0->{1,2,5},1->{1,2,3,4,5},2->{1,2,3,4,5},3->{},4->{},5->{}] ,We construct a looptree: P: [0,1,2,3,4,5] | `- p:[1,2] c: [2] | `- p:[1] c: [1]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 7: FlowAbstraction WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.0.0] f0 ~> f8 [A <= 0*K, B <= B, C <= C, D <= D] f8 ~> f8 [A <= K + A, B <= B, C <= C, D <= D] f8 ~> f8 [A <= K + A, B <= A, C <= K + A, D <= unknown] f8 ~> f23 [A <= A, B <= B, C <= C, D <= D] f8 ~> f23 [A <= A, B <= B, C <= C, D <= D] f8 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= 4*K + A] f8 ~> f8 [A <= K + A, B <= B, C <= C, D <= D] f8 ~> f8 [A <= K + A, B <= A, C <= K + A, D <= unknown] + Loop: [0.0.0 <= 4*K + A] f8 ~> f8 [A <= K + A, B <= B, C <= C, D <= D] + Applied Processor: FlowAbstraction + Details: () * Step 8: LareProcessor WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.0.0] f0 ~> f8 [K ~=> A] f8 ~> f8 [A ~+> A,K ~+> A] f8 ~> f8 [A ~=> B,huge ~=> D,A ~+> A,A ~+> C,K ~+> A,K ~+> C] f8 ~> f23 [] f8 ~> f23 [] f8 ~> exitus616 [] + Loop: [A ~+> 0.0,K ~*> 0.0] f8 ~> f8 [A ~+> A,K ~+> A] f8 ~> f8 [A ~=> B,huge ~=> D,A ~+> A,A ~+> C,K ~+> A,K ~+> C] + Loop: [A ~+> 0.0.0,K ~*> 0.0.0] f8 ~> f8 [A ~+> A,K ~+> A] + Applied Processor: LareProcessor + Details: f0 ~> exitus616 [K ~=> A ,K ~=> B ,huge ~=> D ,tick ~+> tick ,K ~+> A ,K ~+> B ,K ~+> C ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> tick ,K ~*> A ,K ~*> B ,K ~*> C ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> tick ,K ~^> A] f0 ~> f23 [K ~=> A ,K ~=> B ,huge ~=> D ,tick ~+> tick ,K ~+> A ,K ~+> B ,K ~+> C ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> tick ,K ~*> A ,K ~*> B ,K ~*> C ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> tick ,K ~^> A] + f8> [A ~=> B ,huge ~=> D ,A ~+> A ,A ~+> B ,A ~+> C ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> B ,K ~+> C ,K ~+> 0.0.0 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> C ,A ~*> 0.0.0 ,A ~*> tick ,K ~*> A ,K ~*> B ,K ~*> C ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> tick ,A ~^> A ,K ~^> A] + f8> [A ~+> A ,A ~+> 0.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> A ,A ~*> A ,K ~*> A ,K ~*> 0.0.0 ,K ~*> tick] YES(?,O(1))