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 >= B] (?,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (?,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (?,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (?,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (?,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (?,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (?,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (?,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (?,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (?,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: 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 >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (?,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (?,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (?,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (?,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (?,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (?,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (?,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalfbb1in) = 1 + x2 + -1*x3 p(evalfbb2in) = 2 + x2 + -1*x3 p(evalfbb3in) = 1 + x2 + -1*x3 p(evalfbb4in) = 1 + x2 + -1*x3 p(evalfbb5in) = 2 + -1*x1 + x2 p(evalfbb6in) = 2 + -1*x1 + x2 p(evalfentryin) = 2 + x2 p(evalfreturnin) = -1*x1 + x2 p(evalfstart) = 2 + x2 p(evalfstop) = -1*x1 + x2 Following rules are strictly oriented: [B >= 1 + C] ==> evalfbb2in(A,B,C) = 2 + B + -1*C > 1 + B + -1*C = evalfbb3in(A,B,C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = 2 + B >= 2 + B = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = 2 + B >= 2 + B = evalfbb5in(0,B,C) [A >= B] ==> evalfbb5in(A,B,C) = 2 + -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [B >= 1 + A] ==> evalfbb5in(A,B,C) = 2 + -1*A + B >= 2 + -1*A + B = evalfbb6in(A,B,C) [0 >= 1 + D] ==> evalfbb6in(A,B,C) = 2 + -1*A + B >= 2 + -1*A + B = evalfbb2in(A,B,A) [D >= 1] ==> evalfbb6in(A,B,C) = 2 + -1*A + B >= 2 + -1*A + B = evalfbb2in(A,B,A) True ==> evalfbb6in(A,B,C) = 2 + -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [C >= B] ==> evalfbb2in(A,B,C) = 2 + B + -1*C >= 1 + B + -1*C = evalfbb4in(A,B,C) [0 >= 1 + D] ==> evalfbb3in(A,B,C) = 1 + B + -1*C >= 1 + B + -1*C = evalfbb1in(A,B,C) [D >= 1] ==> evalfbb3in(A,B,C) = 1 + B + -1*C >= 1 + B + -1*C = evalfbb1in(A,B,C) True ==> evalfbb3in(A,B,C) = 1 + B + -1*C >= 1 + B + -1*C = evalfbb4in(A,B,C) True ==> evalfbb1in(A,B,C) = 1 + B + -1*C >= 1 + B + -1*C = evalfbb2in(A,B,1 + C) True ==> evalfbb4in(A,B,C) = 1 + B + -1*C >= 1 + B + -1*C = evalfbb5in(1 + C,B,C) True ==> evalfreturnin(A,B,C) = -1*A + B >= -1*A + B = evalfstop(A,B,C) * Step 3: 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 >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (?,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (2 + B,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (?,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (?,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (?,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (?,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (?,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (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: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 4: 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 >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (?,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (?,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (2 + B,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (2 + B,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (2 + B,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (2 + B,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (4 + 2*B,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (?,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (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: 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 + -1*x1 + x2 p(evalfbb6in) = -1*x1 + x2 p(evalfentryin) = 1 + x2 p(evalfreturnin) = -1*x1 + x2 p(evalfstart) = 1 + x2 p(evalfstop) = -1*x1 + x2 Following rules are strictly oriented: [B >= 1 + A] ==> evalfbb5in(A,B,C) = 1 + -1*A + B > -1*A + B = evalfbb6in(A,B,C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = 1 + B >= 1 + B = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = 1 + B >= 1 + B = evalfbb5in(0,B,C) [A >= B] ==> evalfbb5in(A,B,C) = 1 + -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [0 >= 1 + D] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) [D >= 1] ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfbb2in(A,B,A) True ==> evalfbb6in(A,B,C) = -1*A + B >= -1*A + B = evalfreturnin(A,B,C) [C >= B] ==> evalfbb2in(A,B,C) = B + -1*C >= B + -1*C = evalfbb4in(A,B,C) [B >= 1 + C] ==> evalfbb2in(A,B,C) = B + -1*C >= B + -1*C = evalfbb3in(A,B,C) [0 >= 1 + D] ==> evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb1in(A,B,C) [D >= 1] ==> evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb1in(A,B,C) True ==> evalfbb3in(A,B,C) = B + -1*C >= B + -1*C = evalfbb4in(A,B,C) True ==> evalfbb1in(A,B,C) = B + -1*C >= -1 + B + -1*C = evalfbb2in(A,B,1 + C) True ==> evalfbb4in(A,B,C) = B + -1*C >= B + -1*C = evalfbb5in(1 + C,B,C) True ==> evalfreturnin(A,B,C) = -1*A + B >= -1*A + B = evalfstop(A,B,C) * Step 5: 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 >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (1 + B,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (?,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (?,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (?,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (2 + B,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (2 + B,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (2 + B,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (2 + B,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (4 + 2*B,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (?,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (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: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: 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 >= B] (1,1) 3. evalfbb5in(A,B,C) -> evalfbb6in(A,B,C) [B >= 1 + A] (1 + B,1) 4. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [0 >= 1 + D] (1 + B,1) 5. evalfbb6in(A,B,C) -> evalfbb2in(A,B,A) [D >= 1] (1 + B,1) 6. evalfbb6in(A,B,C) -> evalfreturnin(A,B,C) True (1,1) 7. evalfbb2in(A,B,C) -> evalfbb4in(A,B,C) [C >= B] (6 + 4*B,1) 8. evalfbb2in(A,B,C) -> evalfbb3in(A,B,C) [B >= 1 + C] (2 + B,1) 9. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [0 >= 1 + D] (2 + B,1) 10. evalfbb3in(A,B,C) -> evalfbb1in(A,B,C) [D >= 1] (2 + B,1) 11. evalfbb3in(A,B,C) -> evalfbb4in(A,B,C) True (2 + B,1) 12. evalfbb1in(A,B,C) -> evalfbb2in(A,B,1 + C) True (4 + 2*B,1) 13. evalfbb4in(A,B,C) -> evalfbb5in(1 + C,B,C) True (8 + 5*B,1) 14. evalfreturnin(A,B,C) -> evalfstop(A,B,C) True (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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))