YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (?,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (?,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (?,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (?,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (?,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (?,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (?,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (?,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,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. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (1,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (?,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (1,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (?,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (?,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (?,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (?,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (1,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,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: [(7,3)] * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (1,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (?,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (1,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (?,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (?,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (?,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (?,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (1,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalEx6bb1in) = -1*x1 + x3 p(evalEx6bb2in) = -1*x1 + x3 p(evalEx6bb3in) = -1*x1 + x3 p(evalEx6bbin) = -1*x1 + x3 p(evalEx6entryin) = -1*x2 + x3 p(evalEx6returnin) = -1*x1 + x3 p(evalEx6start) = -1*x2 + x3 p(evalEx6stop) = -1*x1 + x3 Following rules are strictly oriented: [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] ==> evalEx6bb2in(A,B,C) = -1*A + C > -1 + -1*A + C = evalEx6bb3in(1 + A,B,C) Following rules are weakly oriented: True ==> evalEx6start(A,B,C) = -1*B + C >= -1*B + C = evalEx6entryin(A,B,C) True ==> evalEx6entryin(A,B,C) = -1*B + C >= -1*B + C = evalEx6bb3in(B,A,C) [C >= 1 + B] ==> evalEx6bb3in(A,B,C) = -1*A + C >= -1*A + C = evalEx6bbin(A,B,C) [B >= C] ==> evalEx6bb3in(A,B,C) = -1*A + C >= -1*A + C = evalEx6returnin(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] ==> evalEx6bbin(A,B,C) = -1*A + C >= -1*A + C = evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] ==> evalEx6bbin(A,B,C) = -1*A + C >= -1*A + C = evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] ==> evalEx6bb1in(A,B,C) = -1*A + C >= -1*A + C = evalEx6bb3in(A,1 + B,C) [B + -1*C >= 0] ==> evalEx6returnin(A,B,C) = -1*A + C >= -1*A + C = evalEx6stop(A,B,C) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (1,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (?,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (1,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (?,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (?,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (?,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (B + C,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (1,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(evalEx6bb1in) = -1*x2 + x3 p(evalEx6bb2in) = -1*x2 + x3 p(evalEx6bb3in) = -1*x2 + x3 p(evalEx6bbin) = -1*x2 + x3 p(evalEx6entryin) = -1*x1 + x3 p(evalEx6returnin) = -1*x2 + x3 p(evalEx6start) = -1*x1 + x3 p(evalEx6stop) = -1*x2 + x3 Following rules are strictly oriented: [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] ==> evalEx6bb1in(A,B,C) = -1*B + C > -1 + -1*B + C = evalEx6bb3in(A,1 + B,C) Following rules are weakly oriented: True ==> evalEx6start(A,B,C) = -1*A + C >= -1*A + C = evalEx6entryin(A,B,C) True ==> evalEx6entryin(A,B,C) = -1*A + C >= -1*A + C = evalEx6bb3in(B,A,C) [C >= 1 + B] ==> evalEx6bb3in(A,B,C) = -1*B + C >= -1*B + C = evalEx6bbin(A,B,C) [B >= C] ==> evalEx6bb3in(A,B,C) = -1*B + C >= -1*B + C = evalEx6returnin(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] ==> evalEx6bbin(A,B,C) = -1*B + C >= -1*B + C = evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] ==> evalEx6bbin(A,B,C) = -1*B + C >= -1*B + C = evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] ==> evalEx6bb2in(A,B,C) = -1*B + C >= -1*B + C = evalEx6bb3in(1 + A,B,C) [B + -1*C >= 0] ==> evalEx6returnin(A,B,C) = -1*B + C >= -1*B + C = evalEx6stop(A,B,C) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (1,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (?,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (1,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (?,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (?,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (A + C,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (B + C,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (1,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2},8->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. evalEx6start(A,B,C) -> evalEx6entryin(A,B,C) True (1,1) 1. evalEx6entryin(A,B,C) -> evalEx6bb3in(B,A,C) True (1,1) 2. evalEx6bb3in(A,B,C) -> evalEx6bbin(A,B,C) [C >= 1 + B] (1 + A + B + 2*C,1) 3. evalEx6bb3in(A,B,C) -> evalEx6returnin(A,B,C) [B >= C] (1,1) 4. evalEx6bbin(A,B,C) -> evalEx6bb1in(A,B,C) [-1 + -1*B + C >= 0 && A >= 1 + B] (1 + A + B + 2*C,1) 5. evalEx6bbin(A,B,C) -> evalEx6bb2in(A,B,C) [-1 + -1*B + C >= 0 && B >= A] (1 + A + B + 2*C,1) 6. evalEx6bb1in(A,B,C) -> evalEx6bb3in(A,1 + B,C) [-1 + -1*B + C >= 0 && -1 + A + -1*B >= 0] (A + C,1) 7. evalEx6bb2in(A,B,C) -> evalEx6bb3in(1 + A,B,C) [-1 + -1*B + C >= 0 && -1 + -1*A + C >= 0 && -1*A + B >= 0] (B + C,1) 8. evalEx6returnin(A,B,C) -> evalEx6stop(A,B,C) [B + -1*C >= 0] (1,1) Signature: {(evalEx6bb1in,3) ;(evalEx6bb2in,3) ;(evalEx6bb3in,3) ;(evalEx6bbin,3) ;(evalEx6entryin,3) ;(evalEx6returnin,3) ;(evalEx6start,3) ;(evalEx6stop,3)} Flow Graph: [0->{1},1->{2,3},2->{4,5},3->{8},4->{6},5->{7},6->{2,3},7->{2},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))