YES(?,O(n^1)) * Step 1: ArgumentFilter WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f8(1 + A,B,0,D) [A >= 0 && B >= 1 + A] (?,1) 2. f8(A,B,C,D) -> f16(A,B,C,0) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] 3. f8(A,B,C,D) -> f16(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] 4. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] 5. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] 6. f16(A,B,C,D) -> f4(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] 7. f16(A,B,C,D) -> f4(-1 + A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] 8. f4(A,B,C,D) -> f20(A,B,C,D) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Flow Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1,8},8->{}] + Applied Processor: ArgumentFilter [3] + Details: We remove following argument positions: [3]. * Step 2: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f4(0,B,C) True (1,1) 1. f4(A,B,C) -> f8(1 + A,B,0) [A >= 0 && B >= 1 + A] (?,1) 2. f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] 3. f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] 4. f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] 5. f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] 6. f16(A,B,C) -> f4(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] 7. f16(A,B,C) -> f4(-1 + A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] 8. f4(A,B,C) -> f20(A,B,C) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Flow Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1,8},8->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(7,8)] * Step 3: FromIts WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f4(0,B,C) True (1,1) 1. f4(A,B,C) -> f8(1 + A,B,0) [A >= 0 && B >= 1 + A] (?,1) 2. f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] 3. f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] 4. f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] 5. f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] 6. f16(A,B,C) -> f4(A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] 7. f16(A,B,C) -> f4(-1 + A,B,C) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] 8. f4(A,B,C) -> f20(A,B,C) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Flow Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{}] + Applied Processor: FromIts + Details: () * Step 4: AddSinks WORST_CASE(?,O(n^1)) + Considered Problem: Rules: f0(A,B,C) -> f4(0,B,C) True f4(A,B,C) -> f8(1 + A,B,0) [A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] f16(A,B,C) -> f4(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] f16(A,B,C) -> f4(-1 + A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] f4(A,B,C) -> f20(A,B,C) [A >= 0 && A >= B] Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Rule Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{}] + Applied Processor: AddSinks + Details: () * Step 5: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: Rules: f0(A,B,C) -> f4(0,B,C) True f4(A,B,C) -> f8(1 + A,B,0) [A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] f16(A,B,C) -> f4(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] f16(A,B,C) -> f4(-1 + A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] f4(A,B,C) -> f20(A,B,C) [A >= 0 && A >= B] f20(A,B,C) -> exitus616(A,B,C) True Signature: {(exitus616,3);(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Rule Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{9}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9] | `- p:[1,6,2,4,5,3,7] c: [1,2,3,4,5,6,7] * Step 6: AbstractSize WORST_CASE(?,O(n^1)) + Considered Problem: (Rules: f0(A,B,C) -> f4(0,B,C) True f4(A,B,C) -> f8(1 + A,B,0) [A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] f8(A,B,C) -> f16(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] f8(A,B,C) -> f8(1 + A,B,1 + C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] f16(A,B,C) -> f4(A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] f16(A,B,C) -> f4(-1 + A,B,C) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] f4(A,B,C) -> f20(A,B,C) [A >= 0 && A >= B] f20(A,B,C) -> exitus616(A,B,C) True Signature: {(exitus616,3);(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Rule Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{9}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9] | `- p:[1,6,2,4,5,3,7] c: [1,2,3,4,5,6,7]) + Applied Processor: AbstractSize Minimize + Details: () * Step 7: AbstractFlow WORST_CASE(?,O(n^1)) + Considered Problem: Program: Domain: [A,B,C,0.0] f0 ~> f4 [A <= 0*K, B <= B, C <= C] f4 ~> f8 [A <= B, B <= B, C <= 0*K] f8 ~> f16 [A <= A, B <= B, C <= C] f8 ~> f16 [A <= A, B <= B, C <= C] f8 ~> f8 [A <= B, B <= B, C <= B] f8 ~> f8 [A <= B, B <= B, C <= B] f16 ~> f4 [A <= A, B <= B, C <= C] f16 ~> f4 [A <= B, B <= B, C <= C] f4 ~> f20 [A <= A, B <= B, C <= C] f20 ~> exitus616 [A <= A, B <= B, C <= C] + Loop: [0.0 <= K + A + B] f4 ~> f8 [A <= B, B <= B, C <= 0*K] f16 ~> f4 [A <= A, B <= B, C <= C] f8 ~> f16 [A <= A, B <= B, C <= C] f8 ~> f8 [A <= B, B <= B, C <= B] f8 ~> f8 [A <= B, B <= B, C <= B] f8 ~> f16 [A <= A, B <= B, C <= C] f16 ~> f4 [A <= B, B <= B, C <= C] + Applied Processor: AbstractFlow + Details: () * Step 8: Lare WORST_CASE(?,O(n^1)) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,0.0] f0 ~> f4 [K ~=> A] f4 ~> f8 [B ~=> A,K ~=> C] f8 ~> f16 [] f8 ~> f16 [] f8 ~> f8 [B ~=> A,B ~=> C] f8 ~> f8 [B ~=> A,B ~=> C] f16 ~> f4 [] f16 ~> f4 [B ~=> A] f4 ~> f20 [] f20 ~> exitus616 [] + Loop: [A ~+> 0.0,B ~+> 0.0,K ~+> 0.0] f4 ~> f8 [B ~=> A,K ~=> C] f16 ~> f4 [] f8 ~> f16 [] f8 ~> f8 [B ~=> A,B ~=> C] f8 ~> f8 [B ~=> A,B ~=> C] f8 ~> f16 [] f16 ~> f4 [B ~=> A] + Applied Processor: Lare + Details: f0 ~> exitus616 [B ~=> A ,B ~=> C ,K ~=> A ,K ~=> C ,B ~+> 0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> tick ,K ~*> 0.0 ,K ~*> tick] + f4> [B ~=> A ,B ~=> C ,K ~=> C ,A ~+> 0.0 ,A ~+> tick ,B ~+> 0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> tick] YES(?,O(n^1))