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) -> evalfbb3in(0,0,C) True (?,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (?,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (?,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (?,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (?,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (?,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (?,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2,3},8->{}] + 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) -> evalfbb3in(0,0,C) True (1,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (?,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (1,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (?,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (?,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (?,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2,3},8->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,3),(6,3)] * 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) -> evalfbb3in(0,0,C) True (1,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (?,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (1,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (?,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (?,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (?,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2},2->{4,5},3->{8},4->{6},5->{7},6->{2},7->{2,3},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalfbb1in) = 100 + -1*x2 p(evalfbb2in) = 100 + -1*x2 p(evalfbb3in) = 100 + -1*x2 p(evalfbbin) = 100 + -1*x2 p(evalfentryin) = 100 p(evalfreturnin) = 100 + -1*x2 p(evalfstart) = 100 p(evalfstop) = 100 + -1*x2 Following rules are strictly oriented: [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] ==> evalfbb2in(A,B,C) = 100 + -1*B > 99 + -1*B = evalfbb3in(A,1 + B,C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = 100 >= 100 = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = 100 >= 100 = evalfbb3in(0,0,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] ==> evalfbb3in(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] ==> evalfbb3in(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfreturnin(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] ==> evalfbbin(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] ==> evalfbbin(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfbb2in(A,B,C) [-1 + C >= 0 ==> && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] evalfbb1in(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfbb3in(1 + A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] ==> evalfreturnin(A,B,C) = 100 + -1*B >= 100 + -1*B = evalfstop(A,B,C) * 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) -> evalfbb3in(0,0,C) True (1,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (?,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (1,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (?,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (?,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (100,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2},2->{4,5},3->{8},4->{6},5->{7},6->{2},7->{2,3},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalfbb1in) = -1*x1 + x3 p(evalfbb2in) = -1*x1 + x3 p(evalfbb3in) = -1*x1 + x3 p(evalfbbin) = -1*x1 + x3 p(evalfentryin) = x3 p(evalfreturnin) = -1*x1 + x3 p(evalfstart) = x3 p(evalfstop) = -1*x1 + x3 Following rules are strictly oriented: [-1 + C >= 0 ==> && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] evalfbb1in(A,B,C) = -1*A + C > -1 + -1*A + C = evalfbb3in(1 + A,B,C) Following rules are weakly oriented: True ==> evalfstart(A,B,C) = C >= C = evalfentryin(A,B,C) True ==> evalfentryin(A,B,C) = C >= C = evalfbb3in(0,0,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] ==> evalfbb3in(A,B,C) = -1*A + C >= -1*A + C = evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] ==> evalfbb3in(A,B,C) = -1*A + C >= -1*A + C = evalfreturnin(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] ==> evalfbbin(A,B,C) = -1*A + C >= -1*A + C = evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] ==> evalfbbin(A,B,C) = -1*A + C >= -1*A + C = evalfbb2in(A,B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] ==> evalfbb2in(A,B,C) = -1*A + C >= -1*A + C = evalfbb3in(A,1 + B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] ==> evalfreturnin(A,B,C) = -1*A + C >= -1*A + C = 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) -> evalfbb3in(0,0,C) True (1,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (?,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (1,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (?,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (?,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (C,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (100,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2},2->{4,5},3->{8},4->{6},5->{7},6->{2},7->{2,3},8->{}] + 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) -> evalfbb3in(0,0,C) True (1,1) 2. evalfbb3in(A,B,C) -> evalfbbin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && 99 >= B] (101 + C,1) 3. evalfbb3in(A,B,C) -> evalfreturnin(A,B,C) [B >= 0 && A + B >= 0 && A >= 0 && B >= 100] (1,1) 4. evalfbbin(A,B,C) -> evalfbb1in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && C >= 1 + A] (101 + C,1) 5. evalfbbin(A,B,C) -> evalfbb2in(A,B,C) [99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0 && A >= C] (101 + C,1) 6. evalfbb1in(A,B,C) -> evalfbb3in(1 + A,B,C) [-1 + C >= 0 (C,1) && -1 + B + C >= 0 && 98 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] 7. evalfbb2in(A,B,C) -> evalfbb3in(A,1 + B,C) [A + -1*C >= 0 && 99 + -1*B >= 0 && 99 + A + -1*B >= 0 && B >= 0 && A + B >= 0 && A >= 0] (100,1) 8. evalfreturnin(A,B,C) -> evalfstop(A,B,C) [-100 + B >= 0 && -100 + A + B >= 0 && A >= 0] (1,1) Signature: {(evalfbb1in,3) ;(evalfbb2in,3) ;(evalfbb3in,3) ;(evalfbbin,3) ;(evalfentryin,3) ;(evalfreturnin,3) ;(evalfstart,3) ;(evalfstop,3)} Flow Graph: [0->{1},1->{2},2->{4,5},3->{8},4->{6},5->{7},6->{2},7->{2,3},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))