YES(?,POLY) * Step 1: TrivialSCCs WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] (?,1) 2. f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 3. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] 4. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] 5. f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 6. f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f19,4);(f4,4);(f7,4)} Flow Graph: [0->{1,6},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6},6->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: AddSinks WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] (?,1) 2. f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 3. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] 4. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] 5. f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 6. f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] (1,1) Signature: {(f0,4);(f19,4);(f4,4);(f7,4)} Flow Graph: [0->{1,6},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6},6->{}] + Applied Processor: AddSinks + Details: () * Step 3: LooptreeTransformer WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] (?,1) 2. f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 3. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] 4. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] 5. f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 6. f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] (?,1) 7. f4(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f19,4);(f4,4);(f7,4)} Flow Graph: [0->{1,6,7},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6,7},6->{},7->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7] | `- p:[1,5,2,3,4] c: [5] | `- p:[2,3,4] c: [4] | `- p:[2,3] c: [3] | `- p:[2] c: [2] * Step 4: SizeAbstraction WORST_CASE(?,POLY) + Considered Problem: (Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] (?,1) 2. f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 3. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] 4. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] 5. f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 6. f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] (?,1) 7. f4(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f19,4);(f4,4);(f7,4)} Flow Graph: [0->{1,6,7},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6,7},6->{},7->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7] | `- p:[1,5,2,3,4] c: [5] | `- p:[2,3,4] c: [4] | `- p:[2,3] c: [3] | `- p:[2] c: [2]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 5: FlowAbstraction WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.0.0,0.0.0.0,0.0.0.0.0] f0 ~> f4 [A <= 0*K, B <= B, C <= C, D <= D] f4 ~> f7 [A <= A, B <= B, C <= B, D <= D] f7 ~> f7 [A <= A, B <= B, C <= B, D <= 0*K] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] f7 ~> f4 [A <= C, B <= B, C <= C, D <= D] f4 ~> f19 [A <= A, B <= B, C <= C, D <= D] f4 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= A + B] f4 ~> f7 [A <= A, B <= B, C <= B, D <= D] f7 ~> f4 [A <= C, B <= B, C <= C, D <= D] f7 ~> f7 [A <= A, B <= B, C <= B, D <= 0*K] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] + Loop: [0.0.0 <= B] f7 ~> f7 [A <= A, B <= B, C <= B, D <= 0*K] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] + Loop: [0.0.0.0 <= B] f7 ~> f7 [A <= A, B <= B, C <= B, D <= 0*K] f7 ~> f7 [A <= A, B <= B, C <= C, D <= unknown] + Loop: [0.0.0.0.0 <= K + B + C] f7 ~> f7 [A <= A, B <= B, C <= B, D <= 0*K] + Applied Processor: FlowAbstraction + Details: () * Step 6: LareProcessor WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.0.0,0.0.0.0,0.0.0.0.0] f0 ~> f4 [K ~=> A] f4 ~> f7 [B ~=> C] f7 ~> f7 [B ~=> C,K ~=> D] f7 ~> f7 [huge ~=> D] f7 ~> f7 [huge ~=> D] f7 ~> f4 [C ~=> A] f4 ~> f19 [] f4 ~> exitus616 [] + Loop: [A ~+> 0.0,B ~+> 0.0] f4 ~> f7 [B ~=> C] f7 ~> f4 [C ~=> A] f7 ~> f7 [B ~=> C,K ~=> D] f7 ~> f7 [huge ~=> D] f7 ~> f7 [huge ~=> D] + Loop: [B ~=> 0.0.0] f7 ~> f7 [B ~=> C,K ~=> D] f7 ~> f7 [huge ~=> D] f7 ~> f7 [huge ~=> D] + Loop: [B ~=> 0.0.0.0] f7 ~> f7 [B ~=> C,K ~=> D] f7 ~> f7 [huge ~=> D] + Loop: [B ~+> 0.0.0.0.0,C ~+> 0.0.0.0.0,K ~+> 0.0.0.0.0] f7 ~> f7 [B ~=> C,K ~=> D] + Applied Processor: LareProcessor + Details: f0 ~> exitus616 [B ~=> A ,B ~=> C ,B ~=> 0.0.0 ,B ~=> 0.0.0.0 ,K ~=> A ,K ~=> D ,huge ~=> D ,B ~+> 0.0 ,B ~+> 0.0.0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0.0.0 ,K ~+> tick ,B ~*> 0.0.0.0.0 ,B ~*> tick ,K ~*> tick] f0 ~> f19 [B ~=> A ,B ~=> C ,B ~=> 0.0.0 ,B ~=> 0.0.0.0 ,K ~=> A ,K ~=> D ,huge ~=> D ,B ~+> 0.0 ,B ~+> 0.0.0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0.0.0 ,K ~+> tick ,B ~*> 0.0.0.0.0 ,B ~*> tick ,K ~*> tick] + f4> [B ~=> A ,B ~=> C ,B ~=> 0.0.0 ,B ~=> 0.0.0.0 ,K ~=> D ,huge ~=> D ,A ~+> 0.0 ,A ~+> tick ,B ~+> 0.0 ,B ~+> 0.0.0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0.0.0.0 ,K ~+> tick ,A ~*> tick ,B ~*> 0.0.0.0.0 ,B ~*> tick ,K ~*> tick] + f7> [B ~=> C ,B ~=> 0.0.0 ,B ~=> 0.0.0.0 ,K ~=> D ,huge ~=> D ,B ~+> 0.0.0.0.0 ,B ~+> tick ,C ~+> 0.0.0.0.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.0.0.0.0 ,K ~+> tick ,B ~*> 0.0.0.0.0 ,B ~*> tick ,C ~*> tick ,K ~*> tick] + f7> [B ~=> C ,B ~=> 0.0.0.0 ,K ~=> D ,huge ~=> D ,B ~+> 0.0.0.0.0 ,B ~+> tick ,C ~+> 0.0.0.0.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.0.0.0.0 ,K ~+> tick ,B ~*> 0.0.0.0.0 ,B ~*> tick ,C ~*> tick ,K ~*> tick] + f7> [B ~=> C ,K ~=> D ,B ~+> 0.0.0.0.0 ,B ~+> tick ,C ~+> 0.0.0.0.0 ,C ~+> tick ,tick ~+> tick ,K ~+> 0.0.0.0.0 ,K ~+> tick] YES(?,POLY)