YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (?,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (?,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (?,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (?,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{7,8},5->{7,8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12} ,11->{13},12->{7,8},13->{2,3},14->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{7,8},5->{7,8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12} ,11->{13},12->{7,8},13->{2,3},14->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(4,7),(5,7)] * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{8},5->{8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12},11->{13} ,12->{7,8},13->{2,3},14->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalfbb1in) = -1*x1 + x2 p(evalfbb2in) = -1*x1 + x2 p(evalfbb3in) = -1*x1 + x2 p(evalfbb4in) = -1*x1 + x2 p(evalfbb5in) = -1*x1 + x2 p(evalfbb6in) = -1*x1 + x2 p(evalfentryin) = x2 p(evalfreturnin) = -1*x1 + x2 p(evalfstart) = x2 p(evalfstop) = -1*x1 + x2 Following rules are strictly oriented: [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb4in(A,B,C) = -1*A + B > -1 + B + -1*C = evalfbb5in(1 + C,B,C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = B >= B = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = B >= B = evalfbb5in(0,B,C) [A >= 0 && A >= B] ==> evalfbb5in(A,B,C) = -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [A >= 0 && B >= 1 + A] ==> evalfbb5in(A,B,C) = -1*A + B >= -1*A + B = evalfbb6in(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] evalfbb2in(A,B,C) = -1*A + B >= -1*A + B = evalfbb4in(A,B,C) [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] evalfbb2in(A,B,C) = -1*A + B >= -1*A + B = evalfbb3in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] evalfbb3in(A,B,C) = -1*A + B >= -1*A + B = evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] evalfbb3in(A,B,C) = -1*A + B >= -1*A + B = evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb3in(A,B,C) = -1*A + B >= -1*A + B = evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb1in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,1 + C) [A >= 0] ==> evalfreturnin(A,B,C) = -1*A + B >= -1*A + B = evalfstop(A,B,C) * Step 4: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{8},5->{8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12},11->{13} ,12->{7,8},13->{2,3},14->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (1 + B,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (1 + B,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (1 + B,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{8},5->{8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12},11->{13} ,12->{7,8},13->{2,3},14->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalfbb1in) = x2 + -1*x3 p(evalfbb2in) = x2 + -1*x3 p(evalfbb3in) = x2 + -1*x3 p(evalfbb4in) = x2 + -1*x3 p(evalfbb5in) = -1*x1 + x2 p(evalfbb6in) = -1*x1 + x2 p(evalfentryin) = x2 p(evalfreturnin) = -1*x1 + x2 p(evalfstart) = x2 p(evalfstop) = -1*x1 + x2 Following rules are strictly oriented: [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb1in(A,B,C) = B + -1*C > -1 + B + -1*C = evalfbb2in(A,B,1 + C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = B >= B = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = B >= B = evalfbb5in(0,B,C) [A >= 0 && A >= B] ==> evalfbb5in(A,B,C) = -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [A >= 0 && B >= 1 + A] ==> evalfbb5in(A,B,C) = -1*A + B >= -1*A + B = evalfbb6in(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] evalfbb2in(A,B,C) = B + -1*C >= B + -1*C = evalfbb4in(A,B,C) [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] evalfbb2in(A,B,C) = B + -1*C >= B + -1*C = evalfbb3in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 ==> && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb4in(A,B,C) [C >= 0 ==> && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] evalfbb4in(A,B,C) = B + -1*C >= -1 + B + -1*C = evalfbb5in(1 + C,B,C) [A >= 0] ==> evalfreturnin(A,B,C) = -1*A + B >= -1*A + B = evalfstop(A,B,C) * Step 6: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (1 + B,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (1 + B,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (1 + B,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (?,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (?,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (B,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{8},5->{8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12},11->{13} ,12->{7,8},13->{2,3},14->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 7: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalfstart(A,B,C) -> evalfentryin(A,B,C) True (1,1) 1. evalfentryin(A,B,C) -> evalfbb5in(0,B,C) True (1,1) 2. evalfbb5in(A,B,C) -> evalfreturnin(A,B,C) [A >= 0 && A >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [A >= 0 && B >= 1 + A] (1 + B,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] (1 + B,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] (1 + B,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) [-1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= 0 (B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [C >= 0 (2 + 3*B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (2 + 3*B,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && 0 >= 1 + D] 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [-1 + B + -1*C >= 0 (2 + 3*B,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && D >= 1] 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) [-1 + B + -1*C >= 0 (2 + 3*B,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) [-1 + B + -1*C >= 0 (B,1) && C >= 0 && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) [C >= 0 (B,1) && -1 + B + C >= 0 && A + C >= 0 && -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0] 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbb4in,3) ;(evalfbb5in,3) ;(evalfbb6in,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{14},3->{4,5,6},4->{8},5->{8},6->{14},7->{13},8->{9,10,11},9->{12},10->{12},11->{13} ,12->{7,8},13->{2,3},14->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))