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Bounding Least Common Multiples with Triangles ITP 2016: Proof Pearl Hing-Lun Chan and Michael Norrish College of Engineering and Computer Science Australian National University August 2016, Nancy, France. Hing-Lun Chan & Michael Norrish

  1. Bounding Least Common Multiples with Triangles ITP 2016: Proof Pearl Hing-Lun Chan and Michael Norrish College of Engineering and Computer Science Australian National University August 2016, Nancy, France. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 1 / 21

  2. Motivation AKS mechanisation AKS mechanisation Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 2 / 21

  3. Motivation AKS mechanisation AKS mechanisation Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 2 / 21

  4. Motivation AKS mechanisation AKS mechanisation ◦ Replacing 2 m by 2 m / 2 makes the lower bound valid for all m > 0. ◦ This change won’t affect the conclusion: AKS algorithm is in class P . Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 2 / 21

  5. Motivation AKS mechanisation Nair’s Paper Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 3 / 21

  6. Motivation AKS mechanisation Nair’s Paper Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 3 / 21

  7. Motivation AKS mechanisation Nair’s Paper ◦ The cryptic “difference operator” means Leibniz’s Harmonic Triangle! Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 3 / 21

  8. LCM of Consecutive Numbers LCM of Consecutive Numbers LCM of a List list_lcm [1] list_lcm [1; 2] list_lcm [1; 2; 3] list_lcm [1; 2; 3; 4] list_lcm [1; 2; 3; 4; 5] list_lcm [1; 2; 3; 4; 5; 6] Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 4 / 21

  9. LCM of Consecutive Numbers LCM of Consecutive Numbers LCM of a List list_lcm [1] = 1 list_lcm [1; 2] = 2 list_lcm [1; 2; 3] = 6 list_lcm [1; 2; 3; 4] = 12 list_lcm [1; 2; 3; 4; 5] = 60 list_lcm [1; 2; 3; 4; 5; 6] = 60 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 4 / 21

  10. LCM of Consecutive Numbers LCM of Consecutive Numbers LCM of a List ≥ 2 0 = 1 list_lcm [1] = 1 ≥ 2 1 = 2 list_lcm [1; 2] = 2 ≥ 2 2 = 4 list_lcm [1; 2; 3] = 6 ≥ 2 3 = 8 list_lcm [1; 2; 3; 4] = 12 ≥ 2 4 = 16 list_lcm [1; 2; 3; 4; 5] = 60 ≥ 2 5 = 32 list_lcm [1; 2; 3; 4; 5; 6] = 60 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 4 / 21

  11. LCM of Consecutive Numbers LCM of Consecutive Numbers LCM of a List ≥ 2 0 = 1 list_lcm [1] = 1 ≥ 2 1 = 2 list_lcm [1; 2] = 2 ≥ 2 2 = 4 list_lcm [1; 2; 3] = 6 ≥ 2 3 = 8 list_lcm [1; 2; 3; 4] = 12 ≥ 2 4 = 16 list_lcm [1; 2; 3; 4; 5] = 60 ≥ 2 5 = 32 list_lcm [1; 2; 3; 4; 5; 6] = 60 Theorem Lower bound for the LCM of consecutive numbers. ⊢ 2 n ≤ list_lcm [ 1 .. n + 1 ] Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 4 / 21

  12. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound Let ℓ = [ a ; b ; c ] . Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 5 / 21

  13. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound Let ℓ = [ a ; b ; c ] . Since LCM is a common multiple of each element (in fact, the least), a ≤ list_lcm [ a ; b ; c ] b ≤ list_lcm [ a ; b ; c ] c ≤ list_lcm [ a ; b ; c ] Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 5 / 21

  14. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound Let ℓ = [ a ; b ; c ] . Since LCM is a common multiple of each element (in fact, the least), a ≤ list_lcm [ a ; b ; c ] b ≤ list_lcm [ a ; b ; c ] c ≤ list_lcm [ a ; b ; c ] Hence a + b + c ≤ 3 × list_lcm [ a ; b ; c ] Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 5 / 21

  15. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound Let ℓ = [ a ; b ; c ] . Since LCM is a common multiple of each element (in fact, the least), a ≤ list_lcm [ a ; b ; c ] b ≤ list_lcm [ a ; b ; c ] c ≤ list_lcm [ a ; b ; c ] Hence a + b + c ≤ 3 × list_lcm [ a ; b ; c ] Theorem For a list ℓ of positive numbers, SUM ℓ ≤ LENGTH ℓ × list_lcm ℓ . Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 5 / 21

  16. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound – Applications Theorem For a list ℓ of positive numbers, SUM ℓ ≤ LENGTH ℓ × list_lcm ℓ . Naïve application: ( n + 1 )( n + 2 ) ≤ ( n + 1 ) × list_lcm [ 1 .. n + 1 ] 2 ( n + 2 ) ≤ list_lcm [ 1 .. n + 1 ] 2 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 6 / 21

  17. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound – Applications Theorem For a list ℓ of positive numbers, SUM ℓ ≤ LENGTH ℓ × list_lcm ℓ . Naïve application: ( n + 1 )( n + 2 ) ≤ ( n + 1 ) × list_lcm [ 1 .. n + 1 ] 2 ( n + 2 ) disappointing! ≤ list_lcm [ 1 .. n + 1 ] 2 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 6 / 21

  18. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Lower Bound – Applications Theorem For a list ℓ of positive numbers, SUM ℓ ≤ LENGTH ℓ × list_lcm ℓ . Naïve application: ( n + 1 )( n + 2 ) ≤ ( n + 1 ) × list_lcm [ 1 .. n + 1 ] 2 ( n + 2 ) disappointing! ≤ list_lcm [ 1 .. n + 1 ] 2 Need a clever idea to obtain this lower bound: 2 n ≤ list_lcm [ 1 .. much better! n + 1 ] Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 6 / 21

  19. LCM of Consecutive Numbers LCM Lower Bound for a List LCM Bound Comparison list_lcm [ 1 .. n + 1 ] 10 3 2 n ( n + 2 ) / 2 10 2 10 1 10 0 0 2 4 6 8 n Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 7 / 21

  20. Triangles Pascal’s Triangle Yang Hui’s Triangle Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 8 / 21

  21. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  22. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 1 1 1 1 2 1 1 1 1 3 3 1 1 1 1 4 6 4 1 1 1 1 5 10 10 5 1 1 1 1 6 15 20 15 6 1 1 Each boundary entry: always 1. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  23. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 1 2 2 1 1 3 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Each boundary entry: always 1. Each inside entry: sum of two immediate parents. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  24. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 4 6 6 4 1 1 5 10 10 10 5 1 1 6 15 20 15 6 1 Each boundary entry: always 1. Each inside entry: sum of two immediate parents. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  25. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 10 5 5 1 1 6 15 20 15 15 6 1 Each boundary entry: always 1. Each inside entry: sum of two immediate parents. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  26. Triangles Pascal’s Triangle Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 1 6 6 15 15 20 20 15 15 6 6 1 1 Each boundary entry: always 1. Each inside entry: sum of two immediate parents. Sum of the n -th row: n � n � = ( 1 + 1 ) n = 2 n � k k = 0 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 9 / 21

  27. Triangles Leibniz’s Triangle Leibniz’s Harmonic Triangle 1 1 1 1 2 2 1 1 1 3 6 3 1 1 1 1 4 12 12 4 1 1 1 1 1 5 20 30 20 5 1 1 1 1 1 1 6 30 60 60 30 6 1 1 1 1 1 1 1 7 42 105 140 105 42 7 Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 10 / 21

  28. Triangles Leibniz’s Triangle Leibniz’s Harmonic Triangle 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 3 3 6 3 3 1 1 1 1 1 1 4 4 12 12 4 4 1 1 1 1 1 1 1 5 5 20 30 20 5 5 1 1 1 1 1 1 1 1 6 6 30 60 60 30 6 6 1 1 1 1 1 1 1 1 1 7 7 42 105 140 105 42 7 7 1 Each boundary entry: for the n -th row, n = 0 , 1 , · · · ( n + 1 ) Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 10 / 21

  29. Triangles Leibniz’s Triangle Leibniz’s Harmonic Triangle 1 1 1 1 1 2 2 2 1 1 1 1 1 3 3 6 6 3 1 1 1 1 4 12 12 4 1 1 1 1 1 5 20 30 20 5 1 1 1 1 1 1 6 30 60 60 30 6 1 1 1 1 1 1 1 7 42 105 140 105 42 7 1 Each boundary entry: for the n -th row, n = 0 , 1 , · · · ( n + 1 ) Each entry (inside or not): sum of two immediate children. Hing-Lun Chan & Michael Norrish (ANU) Bounding LCM with Triangles ITP 2016 10 / 21

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