YES(?,O(1)) * Step 1: TrivialSCCs WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] Signature: {(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,4},3->{3,4},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) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (1,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (1,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] Signature: {(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,4},3->{3,4},4->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,2),(2,4)] * Step 3: AddSinks WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (1,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (1,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] Signature: {(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1},1->{1,2},2->{3},3->{3,4},4->{}] + Applied Processor: AddSinks + Details: () * Step 4: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] 5. f28(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1,2},1->{1,2},2->{3,4,5},3->{3,4,5},4->{},5->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,2),(2,4)] * Step 5: LooptreeTransformer WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] 5. f28(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1},1->{1,2},2->{3,5},3->{3,4,5},4->{},5->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5] | +- p:[1] c: [1] | `- p:[3] c: [3] * Step 6: SizeAbstraction WORST_CASE(?,O(1)) + Considered Problem: (Rules: 0. f0(A,B,C,D) -> f19(A,999,C,1) True (1,1) 1. f19(A,B,C,D) -> f19(A,-1 + B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && B >= 0] 2. f19(A,B,C,D) -> f28(A,B,999,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*B + D >= 0 && 999 + -1*B >= 0 && 0 >= 1 + B] 3. f28(A,B,C,D) -> f28(A,B,-1 + C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && C >= 0] 4. f28(A,B,C,D) -> f36(A,B,C,D) [1 + -1*D >= 0 (?,1) && 1000 + -1*C + -1*D >= 0 && -1*B + -1*D >= 0 && -1 + D >= 0 && 998 + -1*C + D >= 0 && -2 + -1*B + D >= 0 && 999 + -1*C >= 0 && 998 + -1*B + -1*C >= 0 && -1 + -1*B >= 0 && 0 >= 1 + C] 5. f28(A,B,C,D) -> exitus616(A,B,C,D) True (?,1) Signature: {(exitus616,4);(f0,4);(f19,4);(f28,4);(f36,4)} Flow Graph: [0->{1},1->{1,2},2->{3,5},3->{3,4,5},4->{},5->{}] ,We construct a looptree: P: [0,1,2,3,4,5] | +- p:[1] c: [1] | `- p:[3] c: [3]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 7: FlowAbstraction WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.1] f0 ~> f19 [A <= A, B <= 999*K, C <= C, D <= K] f19 ~> f19 [A <= A, B <= 998*K, C <= C, D <= D] f19 ~> f28 [A <= A, B <= B, C <= 999*K, D <= D] f28 ~> f28 [A <= A, B <= B, C <= 998*K, D <= D] f28 ~> f36 [A <= A, B <= B, C <= C, D <= D] f28 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= K + B] f19 ~> f19 [A <= A, B <= 998*K, C <= C, D <= D] + Loop: [0.1 <= K + C] f28 ~> f28 [A <= A, B <= B, C <= 998*K, D <= D] + Applied Processor: FlowAbstraction + Details: () * Step 8: LareProcessor WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.1] f0 ~> f19 [K ~=> B,K ~=> D] f19 ~> f19 [K ~=> B] f19 ~> f28 [K ~=> C] f28 ~> f28 [K ~=> C] f28 ~> f36 [] f28 ~> exitus616 [] + Loop: [B ~+> 0.0,K ~+> 0.0] f19 ~> f19 [K ~=> B] + Loop: [C ~+> 0.1,K ~+> 0.1] f28 ~> f28 [K ~=> C] + Applied Processor: LareProcessor + Details: f0 ~> f36 [K ~=> B ,K ~=> C ,K ~=> D ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.1 ,K ~+> tick ,K ~*> 0.0 ,K ~*> 0.1 ,K ~*> tick] f0 ~> exitus616 [K ~=> B ,K ~=> C ,K ~=> D ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.1 ,K ~+> tick ,K ~*> 0.0 ,K ~*> 0.1 ,K ~*> tick] + f19> [K ~=> B,B ~+> 0.0,B ~+> tick,tick ~+> tick,K ~+> 0.0,K ~+> tick] + f28> [K ~=> C,C ~+> 0.1,C ~+> tick,tick ~+> tick,K ~+> 0.1,K ~+> tick] YES(?,O(1))