YES(?,PRIMREC) * Step 1: FromIts MAYBE + Considered Problem: Rules: 0. f1(A,B) -> f2(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] (?,1) 1. f2(A,B) -> f2(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] (?,1) 2. f999(A,B) -> f1(1,-1 + B) [B >= 1 && A = 0] (1,1) 3. f1(A,B) -> f1(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] (?,1) 4. f2(A,B) -> f1(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] (?,1) Signature: {(f1,2);(f2,2);(f999,2)} Flow Graph: [0->{1,4},1->{1,4},2->{0,3},3->{0,3},4->{0,3}] + Applied Processor: FromIts + Details: () * Step 2: AddSinks MAYBE + Considered Problem: Rules: f1(A,B) -> f2(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f2(A,B) -> f2(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f999(A,B) -> f1(1,-1 + B) [B >= 1 && A = 0] f1(A,B) -> f1(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f2(A,B) -> f1(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] Signature: {(f1,2);(f2,2);(f999,2)} Rule Graph: [0->{1,4},1->{1,4},2->{0,3},3->{0,3},4->{0,3}] + Applied Processor: AddSinks + Details: () * Step 3: Unfold MAYBE + Considered Problem: Rules: f1(A,B) -> f2(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f2(A,B) -> f2(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f999(A,B) -> f1(1,-1 + B) [B >= 1 && A = 0] f1(A,B) -> f1(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f2(A,B) -> f1(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2(A,B) -> exitus616(A,B) True f1(A,B) -> exitus616(A,B) True f1(A,B) -> exitus616(A,B) True f2(A,B) -> exitus616(A,B) True Signature: {(exitus616,2);(f1,2);(f2,2);(f999,2)} Rule Graph: [0->{1,4,5},1->{1,4,8},2->{0,3},3->{0,3,6},4->{0,3,7}] + Applied Processor: Unfold + Details: () * Step 4: Decompose MAYBE + Considered Problem: Rules: f1.0(A,B) -> f2.1(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f1.0(A,B) -> f2.4(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f1.0(A,B) -> f2.5(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f2.1(A,B) -> f2.1(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f2.1(A,B) -> f2.4(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f2.1(A,B) -> f2.8(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f999.2(A,B) -> f1.0(1,-1 + B) [B >= 1 && A = 0] f999.2(A,B) -> f1.3(1,-1 + B) [B >= 1 && A = 0] f1.3(A,B) -> f1.0(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f1.3(A,B) -> f1.3(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f1.3(A,B) -> f1.6(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f2.4(A,B) -> f1.0(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.4(A,B) -> f1.3(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.4(A,B) -> f1.7(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.5(A,B) -> exitus616.9(A,B) True f1.6(A,B) -> exitus616.9(A,B) True f1.7(A,B) -> exitus616.9(A,B) True f2.8(A,B) -> exitus616.9(A,B) True Signature: {(exitus616.9,2);(f1.0,2);(f1.3,2);(f1.6,2);(f1.7,2);(f2.1,2);(f2.4,2);(f2.5,2);(f2.8,2);(f999.2,2)} Rule Graph: [0->{3,4,5},1->{11,12,13},2->{14},3->{3,4,5},4->{11,12,13},5->{17},6->{0,1,2},7->{8,9,10},8->{0,1,2},9->{8 ,9,10},10->{15},11->{0,1,2},12->{8,9,10},13->{16},14->{},15->{},16->{},17->{}] + Applied Processor: Decompose Greedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] | `- p:[0,8,9,12,1,11,4,3] c: [0,1,4,8,11,12] | +- p:[9] c: [9] | `- p:[3] c: [3] * Step 5: AbstractSize MAYBE + Considered Problem: (Rules: f1.0(A,B) -> f2.1(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f1.0(A,B) -> f2.4(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f1.0(A,B) -> f2.5(-1 + A,B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] f2.1(A,B) -> f2.1(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f2.1(A,B) -> f2.4(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f2.1(A,B) -> f2.8(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1] f999.2(A,B) -> f1.0(1,-1 + B) [B >= 1 && A = 0] f999.2(A,B) -> f1.3(1,-1 + B) [B >= 1 && A = 0] f1.3(A,B) -> f1.0(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f1.3(A,B) -> f1.3(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f1.3(A,B) -> f1.6(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] f2.4(A,B) -> f1.0(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.4(A,B) -> f1.3(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.4(A,B) -> f1.7(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1] f2.5(A,B) -> exitus616.9(A,B) True f1.6(A,B) -> exitus616.9(A,B) True f1.7(A,B) -> exitus616.9(A,B) True f2.8(A,B) -> exitus616.9(A,B) True Signature: {(exitus616.9,2);(f1.0,2);(f1.3,2);(f1.6,2);(f1.7,2);(f2.1,2);(f2.4,2);(f2.5,2);(f2.8,2);(f999.2,2)} Rule Graph: [0->{3,4,5},1->{11,12,13},2->{14},3->{3,4,5},4->{11,12,13},5->{17},6->{0,1,2},7->{8,9,10},8->{0,1,2},9->{8 ,9,10},10->{15},11->{0,1,2},12->{8,9,10},13->{16},14->{},15->{},16->{},17->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] | `- p:[0,8,9,12,1,11,4,3] c: [0,1,4,8,11,12] | +- p:[9] c: [9] | `- p:[3] c: [3]) + Applied Processor: AbstractSize Minimize + Details: () * Step 6: AbstractFlow MAYBE + Considered Problem: Program: Domain: [A,B,0.0,0.0.0,0.0.1] f1.0 ~> f2.1 [A <= A, B <= B] f1.0 ~> f2.4 [A <= A, B <= B] f1.0 ~> f2.5 [A <= A, B <= B] f2.1 ~> f2.1 [A <= A, B <= A + B] f2.1 ~> f2.4 [A <= A, B <= A + B] f2.1 ~> f2.8 [A <= A, B <= A + B] f999.2 ~> f1.0 [A <= K, B <= B] f999.2 ~> f1.3 [A <= K, B <= B] f1.3 ~> f1.0 [A <= A + B, B <= B] f1.3 ~> f1.3 [A <= A + B, B <= B] f1.3 ~> f1.6 [A <= A + B, B <= B] f2.4 ~> f1.0 [A <= A + B, B <= B] f2.4 ~> f1.3 [A <= A + B, B <= B] f2.4 ~> f1.7 [A <= A + B, B <= B] f2.5 ~> exitus616.9 [A <= A, B <= B] f1.6 ~> exitus616.9 [A <= A, B <= B] f1.7 ~> exitus616.9 [A <= A, B <= B] f2.8 ~> exitus616.9 [A <= A, B <= B] + Loop: [0.0 <= K + A + B] f1.0 ~> f2.1 [A <= A, B <= B] f1.3 ~> f1.0 [A <= A + B, B <= B] f1.3 ~> f1.3 [A <= A + B, B <= B] f2.4 ~> f1.3 [A <= A + B, B <= B] f1.0 ~> f2.4 [A <= A, B <= B] f2.4 ~> f1.0 [A <= A + B, B <= B] f2.1 ~> f2.4 [A <= A, B <= A + B] f2.1 ~> f2.1 [A <= A, B <= A + B] + Loop: [0.0.0 <= K + B] f1.3 ~> f1.3 [A <= A + B, B <= B] + Loop: [0.0.1 <= K + A] f2.1 ~> f2.1 [A <= A, B <= A + B] + Applied Processor: AbstractFlow + Details: () * Step 7: Lare MAYBE + Considered Problem: Program: Domain: [tick,huge,K,A,B,0.0,0.0.0,0.0.1] f1.0 ~> f2.1 [] f1.0 ~> f2.4 [] f1.0 ~> f2.5 [] f2.1 ~> f2.1 [A ~+> B,B ~+> B] f2.1 ~> f2.4 [A ~+> B,B ~+> B] f2.1 ~> f2.8 [A ~+> B,B ~+> B] f999.2 ~> f1.0 [K ~=> A] f999.2 ~> f1.3 [K ~=> A] f1.3 ~> f1.0 [A ~+> A,B ~+> A] f1.3 ~> f1.3 [A ~+> A,B ~+> A] f1.3 ~> f1.6 [A ~+> A,B ~+> A] f2.4 ~> f1.0 [A ~+> A,B ~+> A] f2.4 ~> f1.3 [A ~+> A,B ~+> A] f2.4 ~> f1.7 [A ~+> A,B ~+> A] f2.5 ~> exitus616.9 [] f1.6 ~> exitus616.9 [] f1.7 ~> exitus616.9 [] f2.8 ~> exitus616.9 [] + Loop: [A ~+> 0.0,B ~+> 0.0,K ~+> 0.0] f1.0 ~> f2.1 [] f1.3 ~> f1.0 [A ~+> A,B ~+> A] f1.3 ~> f1.3 [A ~+> A,B ~+> A] f2.4 ~> f1.3 [A ~+> A,B ~+> A] f1.0 ~> f2.4 [] f2.4 ~> f1.0 [A ~+> A,B ~+> A] f2.1 ~> f2.4 [A ~+> B,B ~+> B] f2.1 ~> f2.1 [A ~+> B,B ~+> B] + Loop: [B ~+> 0.0.0,K ~+> 0.0.0] f1.3 ~> f1.3 [A ~+> A,B ~+> A] + Loop: [A ~+> 0.0.1,K ~+> 0.0.1] f2.1 ~> f2.1 [A ~+> B,B ~+> B] + Applied Processor: Lare + Details: f999.2 ~> exitus616.9 [K ~=> A ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> A ,K ~+> B ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0 ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0 ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0 ,B ~^> 0.0.0 ,B ~^> 0.0.1 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0 ,K ~^> 0.0.0 ,K ~^> 0.0.1 ,K ~^> tick] + f2.1> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f1.3> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f1.0> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f2.4> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f2.1> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f1.3> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f1.0> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] f2.4> [A ~+> A ,A ~+> B ,A ~+> 0.0 ,A ~+> 0.0.0 ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> A ,B ~+> B ,B ~+> 0.0 ,B ~+> 0.0.0 ,B ~+> 0.0.1 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0 ,K ~+> 0.0.0 ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> A ,A ~*> B ,A ~*> 0.0.0 ,A ~*> 0.0.1 ,A ~*> tick ,B ~*> A ,B ~*> B ,B ~*> 0.0.0 ,B ~*> 0.0.1 ,B ~*> tick ,K ~*> A ,K ~*> B ,K ~*> 0.0.0 ,K ~*> 0.0.1 ,K ~*> tick ,A ~^> A ,A ~^> B ,A ~^> 0.0.0 ,A ~^> tick ,B ~^> A ,B ~^> B ,B ~^> 0.0.0 ,B ~^> tick ,K ~^> A ,K ~^> B ,K ~^> 0.0.0 ,K ~^> tick] + f1.3> [A ~+> A ,B ~+> A ,B ~+> 0.0.0 ,B ~+> tick ,tick ~+> tick ,K ~+> 0.0.0 ,K ~+> tick ,B ~*> A ,K ~*> A] + f2.1> [A ~+> B ,A ~+> 0.0.1 ,A ~+> tick ,B ~+> B ,tick ~+> tick ,K ~+> 0.0.1 ,K ~+> tick ,A ~*> B ,K ~*> B] YES(?,PRIMREC)