YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval_ndecr_start(v_0,v_i_0,v_n) -> eval_ndecr_bb0_in(v_0,v_i_0,v_n) True (1,1) 1. eval_ndecr_bb0_in(v_0,v_i_0,v_n) -> eval_ndecr_0(v_0,v_i_0,v_n) True (?,1) 2. eval_ndecr_0(v_0,v_i_0,v_n) -> eval_ndecr_1(v_0,v_i_0,v_n) True (?,1) 3. eval_ndecr_1(v_0,v_i_0,v_n) -> eval_ndecr_2(-1 + v_n,v_i_0,v_n) True (?,1) 4. eval_ndecr_2(v_0,v_i_0,v_n) -> eval_ndecr_3(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 5. eval_ndecr_3(v_0,v_i_0,v_n) -> eval_ndecr_4(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 6. eval_ndecr_4(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,v_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 7. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb2_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -1 + v_i_0 >= 1] 8. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb3_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && 1 >= v_i_0] 9. eval_ndecr_bb2_in(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,-1 + v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -3 + v_n >= 0 && -5 + v_i_0 + v_n >= 0 && -1 + -1*v_i_0 + v_n >= 0 && -5 + v_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -2 + v_i_0 >= 0 && -4 + v_0 + v_i_0 >= 0 && -2 + v_0 >= 0] 10. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> eval_ndecr_stop(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && 1 + -1*v_i_0 >= 0 && v_0 + -1*v_i_0 >= 0] Signature: {(eval_ndecr_0,3) ;(eval_ndecr_1,3) ;(eval_ndecr_2,3) ;(eval_ndecr_3,3) ;(eval_ndecr_4,3) ;(eval_ndecr_bb0_in,3) ;(eval_ndecr_bb1_in,3) ;(eval_ndecr_bb2_in,3) ;(eval_ndecr_bb3_in,3) ;(eval_ndecr_start,3) ;(eval_ndecr_stop,3)} Flow Graph: [0->{1},1->{2},2->{3},3->{4},4->{5},5->{6},6->{7,8},7->{9},8->{10},9->{7,8},10->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: AddSinks WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval_ndecr_start(v_0,v_i_0,v_n) -> eval_ndecr_bb0_in(v_0,v_i_0,v_n) True (1,1) 1. eval_ndecr_bb0_in(v_0,v_i_0,v_n) -> eval_ndecr_0(v_0,v_i_0,v_n) True (1,1) 2. eval_ndecr_0(v_0,v_i_0,v_n) -> eval_ndecr_1(v_0,v_i_0,v_n) True (1,1) 3. eval_ndecr_1(v_0,v_i_0,v_n) -> eval_ndecr_2(-1 + v_n,v_i_0,v_n) True (1,1) 4. eval_ndecr_2(v_0,v_i_0,v_n) -> eval_ndecr_3(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (1,1) 5. eval_ndecr_3(v_0,v_i_0,v_n) -> eval_ndecr_4(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (1,1) 6. eval_ndecr_4(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,v_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (1,1) 7. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb2_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -1 + v_i_0 >= 1] 8. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb3_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (1,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && 1 >= v_i_0] 9. eval_ndecr_bb2_in(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,-1 + v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -3 + v_n >= 0 && -5 + v_i_0 + v_n >= 0 && -1 + -1*v_i_0 + v_n >= 0 && -5 + v_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -2 + v_i_0 >= 0 && -4 + v_0 + v_i_0 >= 0 && -2 + v_0 >= 0] 10. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> eval_ndecr_stop(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (1,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && 1 + -1*v_i_0 >= 0 && v_0 + -1*v_i_0 >= 0] Signature: {(eval_ndecr_0,3) ;(eval_ndecr_1,3) ;(eval_ndecr_2,3) ;(eval_ndecr_3,3) ;(eval_ndecr_4,3) ;(eval_ndecr_bb0_in,3) ;(eval_ndecr_bb1_in,3) ;(eval_ndecr_bb2_in,3) ;(eval_ndecr_bb3_in,3) ;(eval_ndecr_start,3) ;(eval_ndecr_stop,3)} Flow Graph: [0->{1},1->{2},2->{3},3->{4},4->{5},5->{6},6->{7,8},7->{9},8->{10},9->{7,8},10->{}] + Applied Processor: AddSinks + Details: () * Step 3: LooptreeTransformer WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval_ndecr_start(v_0,v_i_0,v_n) -> eval_ndecr_bb0_in(v_0,v_i_0,v_n) True (1,1) 1. eval_ndecr_bb0_in(v_0,v_i_0,v_n) -> eval_ndecr_0(v_0,v_i_0,v_n) True (?,1) 2. eval_ndecr_0(v_0,v_i_0,v_n) -> eval_ndecr_1(v_0,v_i_0,v_n) True (?,1) 3. eval_ndecr_1(v_0,v_i_0,v_n) -> eval_ndecr_2(-1 + v_n,v_i_0,v_n) True (?,1) 4. eval_ndecr_2(v_0,v_i_0,v_n) -> eval_ndecr_3(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 5. eval_ndecr_3(v_0,v_i_0,v_n) -> eval_ndecr_4(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 6. eval_ndecr_4(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,v_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 7. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb2_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -1 + v_i_0 >= 1] 8. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb3_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && 1 >= v_i_0] 9. eval_ndecr_bb2_in(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,-1 + v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -3 + v_n >= 0 && -5 + v_i_0 + v_n >= 0 && -1 + -1*v_i_0 + v_n >= 0 && -5 + v_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -2 + v_i_0 >= 0 && -4 + v_0 + v_i_0 >= 0 && -2 + v_0 >= 0] 10. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> eval_ndecr_stop(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && 1 + -1*v_i_0 >= 0 && v_0 + -1*v_i_0 >= 0] 11. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> exitus616(v_0,v_i_0,v_n) True (?,1) Signature: {(eval_ndecr_0,3) ;(eval_ndecr_1,3) ;(eval_ndecr_2,3) ;(eval_ndecr_3,3) ;(eval_ndecr_4,3) ;(eval_ndecr_bb0_in,3) ;(eval_ndecr_bb1_in,3) ;(eval_ndecr_bb2_in,3) ;(eval_ndecr_bb3_in,3) ;(eval_ndecr_start,3) ;(eval_ndecr_stop,3) ;(exitus616,3)} Flow Graph: [0->{1},1->{2},2->{3},3->{4},4->{5},5->{6},6->{7,8},7->{9},8->{10,11},9->{7,8},10->{},11->{}] + Applied Processor: LooptreeTransformer + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11] | `- p:[7,9] c: [9] * Step 4: SizeAbstraction WORST_CASE(?,O(n^1)) + Considered Problem: (Rules: 0. eval_ndecr_start(v_0,v_i_0,v_n) -> eval_ndecr_bb0_in(v_0,v_i_0,v_n) True (1,1) 1. eval_ndecr_bb0_in(v_0,v_i_0,v_n) -> eval_ndecr_0(v_0,v_i_0,v_n) True (?,1) 2. eval_ndecr_0(v_0,v_i_0,v_n) -> eval_ndecr_1(v_0,v_i_0,v_n) True (?,1) 3. eval_ndecr_1(v_0,v_i_0,v_n) -> eval_ndecr_2(-1 + v_n,v_i_0,v_n) True (?,1) 4. eval_ndecr_2(v_0,v_i_0,v_n) -> eval_ndecr_3(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 5. eval_ndecr_3(v_0,v_i_0,v_n) -> eval_ndecr_4(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 6. eval_ndecr_4(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,v_0,v_n) [1 + v_0 + -1*v_n >= 0 && -1 + -1*v_0 + v_n >= 0] (?,1) 7. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb2_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -1 + v_i_0 >= 1] 8. eval_ndecr_bb1_in(v_0,v_i_0,v_n) -> eval_ndecr_bb3_in(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && 1 >= v_i_0] 9. eval_ndecr_bb2_in(v_0,v_i_0,v_n) -> eval_ndecr_bb1_in(v_0,-1 + v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -3 + v_n >= 0 && -5 + v_i_0 + v_n >= 0 && -1 + -1*v_i_0 + v_n >= 0 && -5 + v_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && v_0 + -1*v_i_0 >= 0 && -2 + v_i_0 >= 0 && -4 + v_0 + v_i_0 >= 0 && -2 + v_0 >= 0] 10. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> eval_ndecr_stop(v_0,v_i_0,v_n) [1 + v_0 + -1*v_n >= 0 (?,1) && -1 + -1*v_i_0 + v_n >= 0 && -1 + -1*v_0 + v_n >= 0 && 1 + -1*v_i_0 >= 0 && v_0 + -1*v_i_0 >= 0] 11. eval_ndecr_bb3_in(v_0,v_i_0,v_n) -> exitus616(v_0,v_i_0,v_n) True (?,1) Signature: {(eval_ndecr_0,3) ;(eval_ndecr_1,3) ;(eval_ndecr_2,3) ;(eval_ndecr_3,3) ;(eval_ndecr_4,3) ;(eval_ndecr_bb0_in,3) ;(eval_ndecr_bb1_in,3) ;(eval_ndecr_bb2_in,3) ;(eval_ndecr_bb3_in,3) ;(eval_ndecr_start,3) ;(eval_ndecr_stop,3) ;(exitus616,3)} Flow Graph: [0->{1},1->{2},2->{3},3->{4},4->{5},5->{6},6->{7,8},7->{9},8->{10,11},9->{7,8},10->{},11->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11] | `- p:[7,9] c: [9]) + Applied Processor: SizeAbstraction UseCFG Minimize + Details: () * Step 5: FlowAbstraction WORST_CASE(?,O(n^1)) + Considered Problem: Program: Domain: [v_0,v_i_0,v_n,0.0] eval_ndecr_start ~> eval_ndecr_bb0_in [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_bb0_in ~> eval_ndecr_0 [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_0 ~> eval_ndecr_1 [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_1 ~> eval_ndecr_2 [v_0 <= K + v_n, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_2 ~> eval_ndecr_3 [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_3 ~> eval_ndecr_4 [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_4 ~> eval_ndecr_bb1_in [v_0 <= v_0, v_i_0 <= v_0, v_n <= v_n] eval_ndecr_bb1_in ~> eval_ndecr_bb2_in [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_bb1_in ~> eval_ndecr_bb3_in [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_bb2_in ~> eval_ndecr_bb1_in [v_0 <= v_0, v_i_0 <= v_0, v_n <= v_n] eval_ndecr_bb3_in ~> eval_ndecr_stop [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_bb3_in ~> exitus616 [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] + Loop: [0.0 <= v_i_0] eval_ndecr_bb1_in ~> eval_ndecr_bb2_in [v_0 <= v_0, v_i_0 <= v_i_0, v_n <= v_n] eval_ndecr_bb2_in ~> eval_ndecr_bb1_in [v_0 <= v_0, v_i_0 <= v_0, v_n <= v_n] + Applied Processor: FlowAbstraction + Details: () * Step 6: LareProcessor WORST_CASE(?,O(n^1)) + Considered Problem: Program: Domain: [tick,huge,K,v_0,v_i_0,v_n,0.0] eval_ndecr_start ~> eval_ndecr_bb0_in [] eval_ndecr_bb0_in ~> eval_ndecr_0 [] eval_ndecr_0 ~> eval_ndecr_1 [] eval_ndecr_1 ~> eval_ndecr_2 [v_n ~+> v_0,K ~+> v_0] eval_ndecr_2 ~> eval_ndecr_3 [] eval_ndecr_3 ~> eval_ndecr_4 [] eval_ndecr_4 ~> eval_ndecr_bb1_in [v_0 ~=> v_i_0] eval_ndecr_bb1_in ~> eval_ndecr_bb2_in [] eval_ndecr_bb1_in ~> eval_ndecr_bb3_in [] eval_ndecr_bb2_in ~> eval_ndecr_bb1_in [v_0 ~=> v_i_0] eval_ndecr_bb3_in ~> eval_ndecr_stop [] eval_ndecr_bb3_in ~> exitus616 [] + Loop: [v_i_0 ~=> 0.0] eval_ndecr_bb1_in ~> eval_ndecr_bb2_in [] eval_ndecr_bb2_in ~> eval_ndecr_bb1_in [v_0 ~=> v_i_0] + Applied Processor: LareProcessor + Details: eval_ndecr_start ~> eval_ndecr_stop [v_n ~+> v_0 ,v_n ~+> v_i_0 ,v_n ~+> 0.0 ,v_n ~+> tick ,tick ~+> tick ,K ~+> v_0 ,K ~+> v_i_0 ,K ~+> 0.0 ,K ~+> tick] eval_ndecr_start ~> exitus616 [v_n ~+> v_0 ,v_n ~+> v_i_0 ,v_n ~+> 0.0 ,v_n ~+> tick ,tick ~+> tick ,K ~+> v_0 ,K ~+> v_i_0 ,K ~+> 0.0 ,K ~+> tick] + eval_ndecr_bb1_in> [v_0 ~=> v_i_0,v_i_0 ~=> 0.0,v_i_0 ~+> tick,tick ~+> tick] YES(?,O(n^1))