YES(?,O(1)) * Step 1: TrivialSCCs WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (?,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (?,1) Signature: {(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9,12},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8},6->{3,4,5,8},7->{3,4,5,8},8->{},9->{1 ,2,9,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5,8}] + 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) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (1,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (1,1) Signature: {(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9,12},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8},6->{3,4,5,8},7->{3,4,5,8},8->{},9->{1 ,2,9,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5,8}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,12),(12,8)] * Step 3: AddSinks WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (1,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (1,1) Signature: {(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8},6->{3,4,5,8},7->{3,4,5,8},8->{},9->{1,2,9 ,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5}] + Applied Processor: AddSinks + Details: () * Step 4: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (?,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (?,1) 13. f23(A,B,C) -> exitus616(A,B,C) True (?,1) Signature: {(exitus616,3);(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9,12},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8,13},6->{3,4,5,8,13},7->{3,4,5,8,13} ,8->{},9->{1,2,9,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5,8,13},13->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,12),(12,8)] * Step 5: LooptreeTransformer WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (?,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (?,1) 13. f23(A,B,C) -> exitus616(A,B,C) True (?,1) Signature: {(exitus616,3);(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8,13},6->{3,4,5,8,13},7->{3,4,5,8,13},8->{} ,9->{1,2,9,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5,13},13->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13] | +- p:[1,9,10,2,11] c: [11] | | | `- p:[1,9,10,2] c: [10] | | | `- p:[9] c: [9] | `- p:[3,5,6,4,7] c: [7] | `- p:[3,5,6,4] c: [6] | `- p:[5] c: [5] * Step 6: SizeAbstraction WORST_CASE(?,O(1)) + Considered Problem: (Rules: 0. f0(A,B,C) -> f8(0,B,C) True (1,1) 1. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A && 999 >= D] (?,1) 2. f8(A,B,C) -> f14(A,A,C) [A >= 0 && 999 >= A] (?,1) 3. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A && 0 >= 1 + E] (?,1) 4. f23(A,B,C) -> f28(A,B,D) [A >= 0 && 999 >= A] (?,1) 5. f23(A,B,C) -> f23(1 + A,B,C) [A >= 0 && 999 >= A] (?,1) 6. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0] (?,1) 7. f28(A,B,C) -> f23(1 + A,B,C) [999 + -1*A >= 0 && A >= 0 && 998 >= D] (?,1) 8. f23(A,B,C) -> f38(A,B,C) [A >= 0 && A >= 1000] (?,1) 9. f8(A,B,C) -> f8(1 + A,A,C) [A >= 0 && 999 >= A] (?,1) 10. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0] 11. f14(A,B,C) -> f8(1 + A,B,C) [999 + -1*B >= 0 (?,1) && A + -1*B >= 0 && 1998 + -1*A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && 999 + -1*A >= 0 && A >= 0 && 998 >= D] 12. f8(A,B,C) -> f23(0,B,C) [A >= 0 && A >= 1000] (?,1) 13. f23(A,B,C) -> exitus616(A,B,C) True (?,1) Signature: {(exitus616,3);(f0,3);(f14,3);(f23,3);(f28,3);(f38,3);(f8,3)} Flow Graph: [0->{1,2,9},1->{10,11},2->{10,11},3->{6,7},4->{6,7},5->{3,4,5,8,13},6->{3,4,5,8,13},7->{3,4,5,8,13},8->{} ,9->{1,2,9,12},10->{1,2,9,12},11->{1,2,9,12},12->{3,4,5,13},13->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13] | +- p:[1,9,10,2,11] c: [11] | | | `- p:[1,9,10,2] c: [10] | | | `- p:[9] c: [9] | `- p:[3,5,6,4,7] c: [7] | `- p:[3,5,6,4] c: [6] | `- p:[5] c: [5]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 7: FlowAbstraction WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [A,B,C,0.0,0.0.0,0.0.0.0,0.1,0.1.0,0.1.0.0] f0 ~> f8 [A <= 0*K, B <= B, C <= C] f8 ~> f14 [A <= A, B <= A, C <= C] f8 ~> f14 [A <= A, B <= A, C <= C] f23 ~> f28 [A <= A, B <= B, C <= unknown] f23 ~> f28 [A <= A, B <= B, C <= unknown] f23 ~> f23 [A <= 1000*K, B <= B, C <= C] f28 ~> f23 [A <= 1000*K, B <= B, C <= C] f28 ~> f23 [A <= 1000*K, B <= B, C <= C] f23 ~> f38 [A <= A, B <= B, C <= C] f8 ~> f8 [A <= 1000*K, B <= A, C <= C] f14 ~> f8 [A <= 1000*K, B <= B, C <= C] f14 ~> f8 [A <= 1000*K, B <= B, C <= C] f8 ~> f23 [A <= 0*K, B <= B, C <= C] f23 ~> exitus616 [A <= A, B <= B, C <= C] + Loop: [0.0 <= 1000*K + A] f8 ~> f14 [A <= A, B <= A, C <= C] f8 ~> f8 [A <= 1000*K, B <= A, C <= C] f14 ~> f8 [A <= 1000*K, B <= B, C <= C] f8 ~> f14 [A <= A, B <= A, C <= C] f14 ~> f8 [A <= 1000*K, B <= B, C <= C] + Loop: [0.0.0 <= 1000*K + A] f8 ~> f14 [A <= A, B <= A, C <= C] f8 ~> f8 [A <= 1000*K, B <= A, C <= C] f14 ~> f8 [A <= 1000*K, B <= B, C <= C] f8 ~> f14 [A <= A, B <= A, C <= C] + Loop: [0.0.0.0 <= 1000*K + A] f8 ~> f8 [A <= 1000*K, B <= A, C <= C] + Loop: [0.1 <= 998002*K + 999*A] f23 ~> f28 [A <= A, B <= B, C <= unknown] f23 ~> f23 [A <= 1000*K, B <= B, C <= C] f28 ~> f23 [A <= 1000*K, B <= B, C <= C] f23 ~> f28 [A <= A, B <= B, C <= unknown] f28 ~> f23 [A <= 1000*K, B <= B, C <= C] + Loop: [0.1.0 <= 998002*K + 999*A] f23 ~> f28 [A <= A, B <= B, C <= unknown] f23 ~> f23 [A <= 1000*K, B <= B, C <= C] f28 ~> f23 [A <= 1000*K, B <= B, C <= C] f23 ~> f28 [A <= A, B <= B, C <= unknown] + Loop: [0.1.0.0 <= 1000*K + A] f23 ~> f23 [A <= 1000*K, B <= B, C <= C] + Applied Processor: FlowAbstraction + Details: () * Step 8: LareProcessor WORST_CASE(?,O(1)) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,0.0,0.0.0,0.0.0.0,0.1,0.1.0,0.1.0.0] f0 ~> f8 [K ~=> A] f8 ~> f14 [A ~=> B] f8 ~> f14 [A ~=> B] f23 ~> f28 [huge ~=> C] f23 ~> f28 [huge ~=> C] f23 ~> f23 [K ~=> A] f28 ~> f23 [K ~=> A] f28 ~> f23 [K ~=> A] f23 ~> f38 [] f8 ~> f8 [A ~=> B,K ~=> A] f14 ~> f8 [K ~=> A] f14 ~> f8 [K ~=> A] f8 ~> f23 [K ~=> A] f23 ~> exitus616 [] + Loop: [A ~+> 0.0,K ~*> 0.0] f8 ~> f14 [A ~=> B] f8 ~> f8 [A ~=> B,K ~=> A] f14 ~> f8 [K ~=> A] f8 ~> f14 [A ~=> B] f14 ~> f8 [K ~=> A] + Loop: [A ~+> 0.0.0,K ~*> 0.0.0] f8 ~> f14 [A ~=> B] f8 ~> f8 [A ~=> B,K ~=> A] f14 ~> f8 [K ~=> A] f8 ~> f14 [A ~=> B] + Loop: [A ~+> 0.0.0.0,K ~*> 0.0.0.0] f8 ~> f8 [A ~=> B,K ~=> A] + Loop: [A ~*> 0.1,K ~*> 0.1] f23 ~> f28 [huge ~=> C] f23 ~> f23 [K ~=> A] f28 ~> f23 [K ~=> A] f23 ~> f28 [huge ~=> C] f28 ~> f23 [K ~=> A] + Loop: [A ~*> 0.1.0,K ~*> 0.1.0] f23 ~> f28 [huge ~=> C] f23 ~> f23 [K ~=> A] f28 ~> f23 [K ~=> A] f23 ~> f28 [huge ~=> C] + Loop: [A ~+> 0.1.0.0,K ~*> 0.1.0.0] f23 ~> f23 [K ~=> A] + Applied Processor: LareProcessor + Details: f0 ~> f38 [K ~=> A ,K ~=> B ,huge ~=> C ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.0.0 ,K ~+> 0.1.0.0 ,K ~+> tick ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> 0.0.0.0 ,K ~*> 0.1 ,K ~*> 0.1.0 ,K ~*> 0.1.0.0 ,K ~*> tick] f0 ~> exitus616 [K ~=> A ,K ~=> B ,huge ~=> C ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.0.0 ,K ~+> 0.1.0.0 ,K ~+> tick ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> 0.0.0.0 ,K ~*> 0.1 ,K ~*> 0.1.0 ,K ~*> 0.1.0.0 ,K ~*> tick] + f8> [A ~=> B ,K ~=> A ,K ~=> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> 0.0.0 ,K ~+> 0.0.0.0 ,K ~+> tick ,A ~*> tick ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> 0.0.0.0 ,K ~*> tick] + f8> [A ~=> B ,K ~=> A ,K ~=> B ,A ~+> 0.0.0 ,A ~+> 0.0.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> 0.0.0.0 ,K ~+> tick ,A ~*> tick ,K ~*> 0.0.0 ,K ~*> 0.0.0.0 ,K ~*> tick] + f8> [A ~=> B,K ~=> A,K ~=> B,A ~+> 0.0.0.0,A ~+> tick,tick ~+> tick,K ~*> 0.0.0.0,K ~*> tick] + f23> [K ~=> A ,huge ~=> C ,A ~+> 0.1.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> 0.1.0.0 ,K ~+> tick ,A ~*> 0.1 ,A ~*> 0.1.0 ,A ~*> tick ,K ~*> 0.1 ,K ~*> 0.1.0 ,K ~*> 0.1.0.0 ,K ~*> tick] + f23> [K ~=> A ,huge ~=> C ,A ~+> 0.1.0.0 ,A ~+> tick ,tick ~+> tick ,K ~+> 0.1.0.0 ,K ~+> tick ,A ~*> 0.1.0 ,A ~*> tick ,K ~*> 0.1.0 ,K ~*> 0.1.0.0 ,K ~*> tick] + f23> [K ~=> A,A ~+> 0.1.0.0,A ~+> tick,tick ~+> tick,K ~*> 0.1.0.0,K ~*> tick] YES(?,O(1))