WORST_CASE(?,O(n^1)) * Step 1: LocalSizeboundsProc WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 5. m1(A,B) -> m6(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 8. m3(A,B) -> m7(A,B) True (?,1) 9. n00(A,B) -> m5(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (?,1) 11. n02(A,B) -> m2(B,A) True (?,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (?,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (?,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4,5},2->{7,8},3->{1},4->{3},5->{},6->{2},7->{6},8->{},9->{},10->{1},11->{2},12->{9,10,11} ,13->{12}] + Applied Processor: LocalSizeboundsProc + Details: LocalSizebounds generated; rvgraph (< 0,0,A>, A, .= 0) (< 0,0,B>, B, .= 0) (< 1,0,A>, A, .= 0) (< 1,0,B>, B, .= 0) (< 2,0,A>, A, .= 0) (< 2,0,B>, B, .= 0) (< 3,0,A>, 1 + A + B, .* 1) (< 3,0,B>, B, .= 0) (< 4,0,A>, A, .= 0) (< 4,0,B>, B, .= 0) (< 5,0,A>, A, .= 0) (< 5,0,B>, B, .= 0) (< 6,0,A>, 1 + A + B, .* 1) (< 6,0,B>, B, .= 0) (< 7,0,A>, A, .= 0) (< 7,0,B>, B, .= 0) (< 8,0,A>, A, .= 0) (< 8,0,B>, B, .= 0) (< 9,0,A>, A, .= 0) (< 9,0,B>, B, .= 0) (<10,0,A>, 0, .= 0) (<10,0,B>, B, .= 0) (<11,0,A>, B, .= 0) (<11,0,B>, A, .= 0) (<12,0,A>, A + B, .* 0) (<12,0,B>, B, .= 0) (<12,1,A>, A + B, .* 0) (<12,1,B>, B, .= 0) (<12,2,A>, A + B, .* 0) (<12,2,B>, B, .= 0) (<13,0,A>, A, .= 0) (<13,0,B>, B, .= 0) * Step 2: SizeboundsProc WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 5. m1(A,B) -> m6(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 8. m3(A,B) -> m7(A,B) True (?,1) 9. n00(A,B) -> m5(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (?,1) 11. n02(A,B) -> m2(B,A) True (?,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (?,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (?,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4,5},2->{7,8},3->{1},4->{3},5->{},6->{2},7->{6},8->{},9->{},10->{1},11->{2},12->{9,10,11} ,13->{12}] Sizebounds: (< 0,0,A>, ?) (< 0,0,B>, ?) (< 1,0,A>, ?) (< 1,0,B>, ?) (< 2,0,A>, ?) (< 2,0,B>, ?) (< 3,0,A>, ?) (< 3,0,B>, ?) (< 4,0,A>, ?) (< 4,0,B>, ?) (< 5,0,A>, ?) (< 5,0,B>, ?) (< 6,0,A>, ?) (< 6,0,B>, ?) (< 7,0,A>, ?) (< 7,0,B>, ?) (< 8,0,A>, ?) (< 8,0,B>, ?) (< 9,0,A>, ?) (< 9,0,B>, ?) (<10,0,A>, ?) (<10,0,B>, ?) (<11,0,A>, ?) (<11,0,B>, ?) (<12,0,A>, ?) (<12,0,B>, ?) (<12,1,A>, ?) (<12,1,B>, ?) (<12,2,A>, ?) (<12,2,B>, ?) (<13,0,A>, ?) (<13,0,B>, ?) + Applied Processor: SizeboundsProc + Details: Sizebounds computed: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 5,0,A>, B) (< 5,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (< 8,0,A>, A + B) (< 8,0,B>, A + B) (< 9,0,A>, A + B) (< 9,0,B>, B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) * Step 3: LeafRules WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 5. m1(A,B) -> m6(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 8. m3(A,B) -> m7(A,B) True (?,1) 9. n00(A,B) -> m5(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (?,1) 11. n02(A,B) -> m2(B,A) True (?,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (?,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (?,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4,5},2->{7,8},3->{1},4->{3},5->{},6->{2},7->{6},8->{},9->{},10->{1},11->{2},12->{9,10,11} ,13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 5,0,A>, B) (< 5,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (< 8,0,A>, A + B) (< 8,0,B>, A + B) (< 9,0,A>, A + B) (< 9,0,B>, B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: LeafRules + Details: The following transitions are estimated by its predecessors and are removed [5,8,9] * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (?,1) 11. n02(A,B) -> m2(B,A) True (?,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (?,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (?,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(concat) = 1 p(m0) = 0 p(m1) = 0 p(m2) = 0 p(m3) = 0 p(m4) = 1 p(m8) = 0 p(m9) = 0 p(n0) = 0 p(n00) = 0 p(n01) = 0 p(n02) = 0 The following rules are strictly oriented: [A >= 0 && B >= 0] ==> m4(A,B) = 1 > 0 = n0(A,B) The following rules are weakly oriented: [A >= 0 && B >= 0] ==> concat(A,B) = 1 >= 1 = m4(A,B) [A >= 0 && B >= A] ==> m0(A,B) = 0 >= 0 = m1(A,B) [B >= A] ==> m2(A,B) = 0 >= 0 = m3(A,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] ==> m8(A,B) = 0 >= 0 = m0(C,B) True ==> m1(A,B) = 0 >= 0 = m8(A,B) [B >= C && 1 + A >= C && C >= 1 + A] ==> m9(A,B) = 0 >= 0 = m2(C,B) True ==> m3(A,B) = 0 >= 0 = m9(A,B) True ==> n01(A,B) = 0 >= 0 = m0(0,B) True ==> n02(A,B) = 0 >= 0 = m2(B,A) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] ==> n0(A,B) = 0 >= 0 = c3(n00(D,C),n01(D,C),n02(D,C)) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (?,1) 11. n02(A,B) -> m2(B,A) True (?,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (?,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (?,1) 7. m3(A,B) -> m9(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (1,1) 11. n02(A,B) -> m2(B,A) True (1,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (1,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [2,6,7], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(m2) = 1 + -1*x1 + x2 p(m3) = 1 + -1*x1 + x2 p(m9) = 1 + -1*x1 + x2 The following rules are strictly oriented: [B >= C && 1 + A >= C && C >= 1 + A] ==> m9(A,B) = 1 + -1*A + B > 1 + B + -1*C = m2(C,B) The following rules are weakly oriented: [B >= A] ==> m2(A,B) = 1 + -1*A + B >= 1 + -1*A + B = m3(A,B) True ==> m3(A,B) = 1 + -1*A + B >= 1 + -1*A + B = m9(A,B) We use the following global sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) * Step 7: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (?,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (1 + A + 2*B,1) 7. m3(A,B) -> m9(A,B) True (?,1) 10. n01(A,B) -> m0(0,B) True (1,1) 11. n02(A,B) -> m2(B,A) True (1,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (1,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 8: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (2 + A + 2*B,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (?,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (1 + A + 2*B,1) 7. m3(A,B) -> m9(A,B) True (2 + A + 2*B,1) 10. n01(A,B) -> m0(0,B) True (1,1) 11. n02(A,B) -> m2(B,A) True (1,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (1,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [1,3,4], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(m0) = 1 + -1*x1 + x2 p(m1) = 1 + -1*x1 + x2 p(m8) = 1 + -1*x1 + x2 The following rules are strictly oriented: [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] ==> m8(A,B) = 1 + -1*A + B > 1 + B + -1*C = m0(C,B) The following rules are weakly oriented: [A >= 0 && B >= A] ==> m0(A,B) = 1 + -1*A + B >= 1 + -1*A + B = m1(A,B) True ==> m1(A,B) = 1 + -1*A + B >= 1 + -1*A + B = m8(A,B) We use the following global sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) * Step 9: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (?,1) 2. m2(A,B) -> m3(A,B) [B >= A] (2 + A + 2*B,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (1 + B,1) 4. m1(A,B) -> m8(A,B) True (?,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (1 + A + 2*B,1) 7. m3(A,B) -> m9(A,B) True (2 + A + 2*B,1) 10. n01(A,B) -> m0(0,B) True (1,1) 11. n02(A,B) -> m2(B,A) True (1,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (1,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 10: LocalSizeboundsProc WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. concat(A,B) -> m4(A,B) [A >= 0 && B >= 0] (1,1) 1. m0(A,B) -> m1(A,B) [A >= 0 && B >= A] (2 + B,1) 2. m2(A,B) -> m3(A,B) [B >= A] (2 + A + 2*B,1) 3. m8(A,B) -> m0(C,B) [A >= 0 && B >= 1 + A && 1 + A >= C && C >= 1 + A] (1 + B,1) 4. m1(A,B) -> m8(A,B) True (2 + B,1) 6. m9(A,B) -> m2(C,B) [B >= C && 1 + A >= C && C >= 1 + A] (1 + A + 2*B,1) 7. m3(A,B) -> m9(A,B) True (2 + A + 2*B,1) 10. n01(A,B) -> m0(0,B) True (1,1) 11. n02(A,B) -> m2(B,A) True (1,1) 12. n0(A,B) -> c3(n00(D,C),n01(D,C),n02(D,C)) [C >= 0 && D >= C && B >= C && C >= B && A + C >= D && D >= A + C] (1,1) 13. m4(A,B) -> n0(A,B) [A >= 0 && B >= 0] (1,1) Signature: {(concat,2) ;(m0,2) ;(m1,2) ;(m2,2) ;(m3,2) ;(m4,2) ;(m5,2) ;(m6,2) ;(m7,2) ;(m8,2) ;(m9,2) ;(n0,2) ;(n00,2) ;(n01,2) ;(n02,2)} Flow Graph: [0->{13},1->{4},2->{7},3->{1},4->{3},6->{2},7->{6},10->{1},11->{2},12->{10,11},13->{12}] Sizebounds: (< 0,0,A>, A) (< 0,0,B>, B) (< 1,0,A>, B) (< 1,0,B>, B) (< 2,0,A>, A + B) (< 2,0,B>, A + B) (< 3,0,A>, 1) (< 3,0,B>, B) (< 4,0,A>, B) (< 4,0,B>, B) (< 6,0,A>, 1) (< 6,0,B>, A + B) (< 7,0,A>, A + B) (< 7,0,B>, A + B) (<10,0,A>, 0) (<10,0,B>, B) (<11,0,A>, B) (<11,0,B>, A + B) (<12,0,A>, A + B) (<12,0,B>, B) (<12,1,A>, A + B) (<12,1,B>, B) (<12,2,A>, A + B) (<12,2,B>, B) (<13,0,A>, A) (<13,0,B>, B) + Applied Processor: LocalSizeboundsProc + Details: The problem is already solved. WORST_CASE(?,O(n^1))