YES(?,POLY) * Step 1: UnsatRules WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f1(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] (1,1) 1. f1(A,B,C,D) -> f1(A,1 + B,C,1 + B) [A + C >= 1 + 2*B && 0 >= 2] (?,1) 2. f1(A,B,C,D) -> f1(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] (?,1) 3. f1(A,B,C,D) -> f1(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] (?,1) 4. f1(A,B,C,D) -> f1(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C && 0 >= 2] (?,1) 5. f1(A,B,C,D) -> f1(A,B,C,B) [0 >= 1 && 2*B >= A + C && 1 + A + C >= 2*B] (?,1) 6. f1(A,B,C,D) -> f1(A,B,C,B) [0 >= 1 && 2*B >= A + C && 1 + A + C >= 2*B] (?,1) Signature: {(f0,4);(f1,4)} Flow Graph: [0->{1,2,3,4,5,6},1->{1,2,3,4,5,6},2->{1,2,3,4,5,6},3->{1,2,3,4,5,6},4->{1,2,3,4,5,6},5->{1,2,3,4,5,6} ,6->{1,2,3,4,5,6}] + Applied Processor: UnsatRules + Details: Following transitions have unsatisfiable constraints and are removed: [1,4,5,6] * Step 2: UnsatPaths WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f1(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] (1,1) 2. f1(A,B,C,D) -> f1(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] (?,1) 3. f1(A,B,C,D) -> f1(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] (?,1) Signature: {(f0,4);(f1,4)} Flow Graph: [0->{2,3},2->{2,3},3->{2,3}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(2,3),(3,2)] * Step 3: FromIts WORST_CASE(?,POLY) + Considered Problem: Rules: 0. f0(A,B,C,D) -> f1(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] (1,1) 2. f1(A,B,C,D) -> f1(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] (?,1) 3. f1(A,B,C,D) -> f1(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] (?,1) Signature: {(f0,4);(f1,4)} Flow Graph: [0->{2,3},2->{2},3->{3}] + Applied Processor: FromIts + Details: () * Step 4: Unfold WORST_CASE(?,POLY) + Considered Problem: Rules: f0(A,B,C,D) -> f1(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f1(A,B,C,D) -> f1(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] f1(A,B,C,D) -> f1(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] Signature: {(f0,4);(f1,4)} Rule Graph: [0->{2,3},2->{2},3->{3}] + Applied Processor: Unfold + Details: () * Step 5: AddSinks WORST_CASE(?,POLY) + Considered Problem: Rules: f0.0(A,B,C,D) -> f1.2(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f0.0(A,B,C,D) -> f1.3(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f1.2(A,B,C,D) -> f1.2(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] f1.3(A,B,C,D) -> f1.3(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] Signature: {(f0.0,4);(f1.2,4);(f1.3,4)} Rule Graph: [0->{2},1->{3},2->{2},3->{3}] + Applied Processor: AddSinks + Details: () * Step 6: Decompose WORST_CASE(?,POLY) + Considered Problem: Rules: f0.0(A,B,C,D) -> f1.2(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f0.0(A,B,C,D) -> f1.3(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f1.2(A,B,C,D) -> f1.2(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] f1.3(A,B,C,D) -> f1.3(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] f1.3(A,B,C,D) -> exitus616(A,B,C,D) True f1.2(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(f0.0,4);(f1.2,4);(f1.3,4)} Rule Graph: [0->{2},1->{3},2->{2,5},3->{3,4}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5] | +- p:[3] c: [3] | `- p:[2] c: [2] * Step 7: AbstractSize WORST_CASE(?,POLY) + Considered Problem: (Rules: f0.0(A,B,C,D) -> f1.2(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f0.0(A,B,C,D) -> f1.3(A,B,2,D) [A >= 0 && 3 >= A && 3 >= B && B >= 0] f1.2(A,B,C,D) -> f1.2(A,1 + B,C,1 + B) [A + C >= 1 + 2*B] f1.3(A,B,C,D) -> f1.3(A,-1 + B,C,-1 + B) [2*B >= 2 + A + C] f1.3(A,B,C,D) -> exitus616(A,B,C,D) True f1.2(A,B,C,D) -> exitus616(A,B,C,D) True Signature: {(exitus616,4);(f0.0,4);(f1.2,4);(f1.3,4)} Rule Graph: [0->{2},1->{3},2->{2,5},3->{3,4}] ,We construct a looptree: P: [0,1,2,3,4,5] | +- p:[3] c: [3] | `- p:[2] c: [2]) + Applied Processor: AbstractSize Minimize + Details: () * Step 8: AbstractFlow WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [A,B,C,D,0.0,0.1] f0.0 ~> f1.2 [A <= A, B <= B, C <= 2*K, D <= D] f0.0 ~> f1.3 [A <= A, B <= B, C <= 2*K, D <= D] f1.2 ~> f1.2 [A <= A, B <= K + B, C <= C, D <= K + B] f1.3 ~> f1.3 [A <= A, B <= K + B, C <= C, D <= K + B] f1.3 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] f1.2 ~> exitus616 [A <= A, B <= B, C <= C, D <= D] + Loop: [0.0 <= 2*K + A + 2*B + C] f1.3 ~> f1.3 [A <= A, B <= K + B, C <= C, D <= K + B] + Loop: [0.1 <= K + A + 2*B + C] f1.2 ~> f1.2 [A <= A, B <= K + B, C <= C, D <= K + B] + Applied Processor: AbstractFlow + Details: () * Step 9: Lare WORST_CASE(?,POLY) + Considered Problem: Program: Domain: [tick,huge,K,A,B,C,D,0.0,0.1] f0.0 ~> f1.2 [K ~=> C] f0.0 ~> f1.3 [K ~=> C] f1.2 ~> f1.2 [B ~+> B,B ~+> D,K ~+> B,K ~+> D] f1.3 ~> f1.3 [B ~+> B,B ~+> D,K ~+> B,K ~+> D] f1.3 ~> exitus616 [] f1.2 ~> exitus616 [] + Loop: [A ~+> 0.0,C ~+> 0.0,B ~*> 0.0,K ~*> 0.0] f1.3 ~> f1.3 [B ~+> B,B ~+> D,K ~+> B,K ~+> D] + Loop: [A ~+> 0.1,C ~+> 0.1,K ~+> 0.1,B ~*> 0.1] f1.2 ~> f1.2 [B ~+> B,B ~+> D,K ~+> B,K ~+> D] + Applied Processor: Lare + Details: f0.0 ~> exitus616 [K ~=> C ,A ~+> 0.0 ,A ~+> 0.1 ,A ~+> tick ,B ~+> B ,B ~+> D ,tick ~+> tick ,K ~+> B ,K ~+> D ,K ~+> 0.0 ,K ~+> 0.1 ,K ~+> tick ,A ~*> B ,B ~*> B ,B ~*> 0.0 ,B ~*> 0.1 ,B ~*> tick ,K ~*> B ,K ~*> D ,K ~*> 0.0 ,K ~*> 0.1 ,K ~*> tick] + f1.3> [A ~+> 0.0 ,A ~+> tick ,B ~+> B ,B ~+> D ,C ~+> 0.0 ,C ~+> tick ,tick ~+> tick ,K ~+> B ,K ~+> D ,A ~*> B ,B ~*> B ,B ~*> 0.0 ,B ~*> tick ,C ~*> B ,K ~*> B ,K ~*> D ,K ~*> 0.0 ,K ~*> tick] + f1.2> [A ~+> 0.1 ,A ~+> tick ,B ~+> B ,B ~+> D ,C ~+> 0.1 ,C ~+> tick ,tick ~+> tick ,K ~+> B ,K ~+> D ,K ~+> 0.1 ,K ~+> tick ,A ~*> B ,B ~*> B ,B ~*> 0.1 ,B ~*> tick ,C ~*> B ,K ~*> B ,K ~*> D] YES(?,POLY)