YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (?,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (?,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (?,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (?,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (?,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (?,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (?,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (?,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (?,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + 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. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (?,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (?,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (?,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (?,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (?,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalSimpleSingle2bb1in) = -1*x2 + x4 p(evalSimpleSingle2bb2in) = -1*x2 + x4 p(evalSimpleSingle2bb3in) = -1*x2 + x4 p(evalSimpleSingle2bb4in) = -1*x1 + x4 p(evalSimpleSingle2bbin) = -1*x2 + x4 p(evalSimpleSingle2entryin) = x4 p(evalSimpleSingle2returnin) = -1*x1 + -1*x2 + x4 p(evalSimpleSingle2start) = x4 p(evalSimpleSingle2stop) = -1*x1 + -1*x2 + x4 Following rules are strictly oriented: [-1 + D >= 0 ==> && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb3in(A,B,C,D) = -1*B + D > -1 + -1*A + D = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) Following rules are weakly oriented: True ==> evalSimpleSingle2start(A,B,C,D) = D >= D = evalSimpleSingle2entryin(A,B,C,D) True ==> evalSimpleSingle2entryin(A,B,C,D) = D >= D = evalSimpleSingle2bb4in(0,0,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*B + D = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*B + D = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*A + -1*B + D = evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + D >= -1*B + D = evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + D >= -1*B + D = evalSimpleSingle2bb2in(A,B,C,D) [-1 + C >= 0 ==> && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb1in(A,B,C,D) = -1*B + D >= -1 + -1*A + D = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + D >= -1*B + D = evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + D >= -1*A + -1*B + D = evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2returnin(A,B,C,D) = -1*A + -1*B + D >= -1*A + -1*B + D = evalSimpleSingle2stop(A,B,C,D) * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (?,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (?,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (?,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (?,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (D,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalSimpleSingle2bb1in) = -1*x2 + x4 p(evalSimpleSingle2bb2in) = -1*x2 + x4 p(evalSimpleSingle2bb3in) = -1 + -1*x2 + x4 p(evalSimpleSingle2bb4in) = -1*x1 + x4 p(evalSimpleSingle2bbin) = -1*x2 + x4 p(evalSimpleSingle2entryin) = x4 p(evalSimpleSingle2returnin) = -1*x1 + -1*x2 + x4 p(evalSimpleSingle2start) = 1 + x4 p(evalSimpleSingle2stop) = -1*x1 + -1*x2 + x4 Following rules are strictly oriented: [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + D > -1 + -1*B + D = evalSimpleSingle2bb3in(A,B,C,D) Following rules are weakly oriented: True ==> evalSimpleSingle2start(A,B,C,D) = 1 + D >= D = evalSimpleSingle2entryin(A,B,C,D) True ==> evalSimpleSingle2entryin(A,B,C,D) = D >= D = evalSimpleSingle2bb4in(0,0,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*B + D = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*B + D = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + D >= -1*A + -1*B + D = evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + D >= -1*B + D = evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + D >= -1*B + D = evalSimpleSingle2bb2in(A,B,C,D) [-1 + C >= 0 ==> && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb1in(A,B,C,D) = -1*B + D >= -1 + -1*A + D = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + D >= -1*A + -1*B + D = evalSimpleSingle2returnin(A,B,C,D) [-1 + D >= 0 ==> && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb3in(A,B,C,D) = -1 + -1*B + D >= -1 + -1*A + D = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2returnin(A,B,C,D) = -1*A + -1*B + D >= -1*A + -1*B + D = evalSimpleSingle2stop(A,B,C,D) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (?,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (?,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (?,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (?,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (1 + D,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (D,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalSimpleSingle2bb1in) = -1*x2 + x3 p(evalSimpleSingle2bb2in) = -1*x2 + x3 p(evalSimpleSingle2bb3in) = -1*x2 + x3 p(evalSimpleSingle2bb4in) = -1*x1 + x3 p(evalSimpleSingle2bbin) = -1*x2 + x3 p(evalSimpleSingle2entryin) = x3 p(evalSimpleSingle2returnin) = -1*x1 + -1*x2 + x3 p(evalSimpleSingle2start) = x3 p(evalSimpleSingle2stop) = -1*x1 + -1*x2 + x3 Following rules are strictly oriented: [-1 + C >= 0 ==> && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb1in(A,B,C,D) = -1*B + C > -1 + -1*A + C = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) Following rules are weakly oriented: True ==> evalSimpleSingle2start(A,B,C,D) = C >= C = evalSimpleSingle2entryin(A,B,C,D) True ==> evalSimpleSingle2entryin(A,B,C,D) = C >= C = evalSimpleSingle2bb4in(0,0,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*B + C = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*B + C = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*A + -1*B + C = evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + C >= -1*B + C = evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + C >= -1*B + C = evalSimpleSingle2bb2in(A,B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + C >= -1*B + C = evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] evalSimpleSingle2bb2in(A,B,C,D) = -1*B + C >= -1*A + -1*B + C = evalSimpleSingle2returnin(A,B,C,D) [-1 + D >= 0 ==> && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb3in(A,B,C,D) = -1*B + C >= -1 + -1*A + C = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2returnin(A,B,C,D) = -1*A + -1*B + C >= -1*A + -1*B + C = evalSimpleSingle2stop(A,B,C,D) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (?,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (?,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (?,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (C,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (1 + D,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (D,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (1 + C + D,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (1 + C + D,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (?,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (2 + 2*C + 2*D,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (C,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (1 + D,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (D,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalSimpleSingle2bb1in) = -1 + -1*x2 + x3 p(evalSimpleSingle2bb2in) = -1*x1 + x3 p(evalSimpleSingle2bb3in) = -1*x1 + x3 p(evalSimpleSingle2bb4in) = -1*x1 + x3 p(evalSimpleSingle2bbin) = -1*x2 + x3 p(evalSimpleSingle2entryin) = x3 p(evalSimpleSingle2returnin) = -1*x1 + x3 p(evalSimpleSingle2start) = x3 p(evalSimpleSingle2stop) = -1*x1 + x3 Following rules are strictly oriented: [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + C > -1 + -1*B + C = evalSimpleSingle2bb1in(A,B,C,D) Following rules are weakly oriented: True ==> evalSimpleSingle2start(A,B,C,D) = C >= C = evalSimpleSingle2entryin(A,B,C,D) True ==> evalSimpleSingle2entryin(A,B,C,D) = C >= C = evalSimpleSingle2bb4in(0,0,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*B + C = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*B + C = evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2bb4in(A,B,C,D) = -1*A + C >= -1*A + C = evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] ==> evalSimpleSingle2bbin(A,B,C,D) = -1*B + C >= -1*A + C = evalSimpleSingle2bb2in(A,B,C,D) [-1 + C >= 0 ==> && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb1in(A,B,C,D) = -1 + -1*B + C >= -1 + -1*A + C = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] evalSimpleSingle2bb2in(A,B,C,D) = -1*A + C >= -1*A + C = evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 ==> && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] evalSimpleSingle2bb2in(A,B,C,D) = -1*A + C >= -1*A + C = evalSimpleSingle2returnin(A,B,C,D) [-1 + D >= 0 ==> && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] evalSimpleSingle2bb3in(A,B,C,D) = -1*A + C >= -1 + -1*A + C = evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] ==> evalSimpleSingle2returnin(A,B,C,D) = -1*A + C >= -1*A + C = evalSimpleSingle2stop(A,B,C,D) * Step 7: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalSimpleSingle2start(A,B,C,D) -> evalSimpleSingle2entryin(A,B,C,D) True (1,1) 1. evalSimpleSingle2entryin(A,B,C,D) -> evalSimpleSingle2bb4in(0,0,C,D) True (1,1) 2. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && 0 >= 1 + E] (1 + C + D,1) 3. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2bbin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && E >= 1] (1 + C + D,1) 4. evalSimpleSingle2bb4in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) 5. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb1in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && C >= 1 + B] (C,1) 6. evalSimpleSingle2bbin(A,B,C,D) -> evalSimpleSingle2bb2in(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && B >= C] (2 + 2*C + 2*D,1) 7. evalSimpleSingle2bb1in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + C >= 0 (C,1) && -1 + B + C >= 0 && -1 + -1*B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 8. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2bb3in(A,B,C,D) [B + -1*C >= 0 (1 + D,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && D >= 1 + A] 9. evalSimpleSingle2bb2in(A,B,C,D) -> evalSimpleSingle2returnin(A,B,C,D) [B + -1*C >= 0 (1,1) && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0 && A >= D] 10. evalSimpleSingle2bb3in(A,B,C,D) -> evalSimpleSingle2bb4in(1 + A,1 + B,C,D) [-1 + D >= 0 (D,1) && -1 + -1*C + D >= 0 && -1 + B + D >= 0 && -1 + -1*B + D >= 0 && -1 + A + D >= 0 && -1 + -1*A + D >= 0 && B + -1*C >= 0 && A + -1*C >= 0 && A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] 11. evalSimpleSingle2returnin(A,B,C,D) -> evalSimpleSingle2stop(A,B,C,D) [A + -1*B >= 0 && B >= 0 && A + B >= 0 && -1*A + B >= 0 && A >= 0] (1,1) Signature: {(evalSimpleSingle2bb1in,4) ;(evalSimpleSingle2bb2in,4) ;(evalSimpleSingle2bb3in,4) ;(evalSimpleSingle2bb4in,4) ;(evalSimpleSingle2bbin,4) ;(evalSimpleSingle2entryin,4) ;(evalSimpleSingle2returnin,4) ;(evalSimpleSingle2start,4) ;(evalSimpleSingle2stop,4)} Flow Graph: [0->{1},1->{2,3,4},2->{5,6},3->{5,6},4->{11},5->{7},6->{8,9},7->{2,3,4},8->{10},9->{11},10->{2,3,4} ,11->{}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))