MX0013

divisor_factor_pair_quotient_product

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

For any actual positive divisor pair, a supplied product quotient equals the product of the supplied cofactors.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall m n d a b u v q. (((~((a)=0)) /\ (((~((b)=0)) /\ (((exists pvs_factor_cdp_quotient_pairleft. (m) = (a) * pvs_factor_cdp_quotient_pairleft) /\ (((exists pvs_factor_cdp_quotient_pairright. (n) = (b) * pvs_factor_cdp_quotient_pairright) /\ ((d)=(a)*(b)))))))))) -> m=a*u -> n=b*v -> m*n=d*q -> q=u*v

Constructive proof overview

Generated structural guide

For any actual positive divisor pair, a supplied product quotient equals the product of the supplied cofactors.

The unchanged tactic script uses 4 declared prerequisites and contains 59 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mul_ne_zero Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized mul_left_cancel_nonzero Stable theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

59 script commands · 15 reading checkpoints · 2 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro d
  4. L4
    intro a
  5. L5
    intro b
  6. L6
    intro u
  7. L7
    intro v
  8. L8
    intro q
  9. L9
    intro hp
  10. L10
    intro hm
02Fix variables and assumptionsL11–12

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro hn
  2. L12
    intro hq
03Separate the logical casesL13–16

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    cases hp
  2. L14
    cases hp_right
  3. L15
    cases hp_right_right
  4. L16
    cases hp_right_right_right
04Establish hdL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul ne zero.

  1. L17
    have hd : ~(d=0)
  2. L18
    intro hzero
  3. L19
    specialize mul_ne_zero (a)
  4. L20
    specialize mul_ne_zero (b)
  5. L21
    apply mul_ne_zero
  6. L22
    exact hp_left
  7. L23
    exact hp_right_left
  8. L24
    trans d
  9. L25
    symm
  10. L26
    exact hp_right_right_right_right
05Use earlier factsL27–27

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L27
    exact hzero
06Establish hprodL28–37

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul assoc.

  1. L28
    have hprod : m*n=d*(u*v)
  2. L29
    rewrite hm
  3. L30
    rewrite hn
  4. L31
    rewrite hp_right_right_right_right
  5. L32
    trans a*(u*(b*v))
  6. L33
    apply mul_assoc
  7. L34
    trans a*((u*b)*v)
  8. L35
    congr
  9. L36
    refl
  10. L37
    symm
07Use earlier factsL38–38

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L38
    apply mul_assoc
08Calculate and transport equalitiesL39–42

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L39
    trans a*((b*u)*v)
  2. L40
    congr
  3. L41
    refl
  4. L42
    congr
09Use earlier factsL43–43

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L43
    apply mul_comm
10Calculate and transport equalitiesL44–47

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L44
    refl
  2. L45
    trans a*(b*(u*v))
  3. L46
    congr
  4. L47
    refl
11Use earlier factsL48–48

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L48
    apply mul_assoc
12Calculate and transport equalitiesL49–49

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L49
    symm
13Use earlier factsL50–55

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L50
    apply mul_assoc
  2. L51
    specialize mul_left_cancel_nonzero (d)
  3. L52
    specialize mul_left_cancel_nonzero (q)
  4. L53
    specialize mul_left_cancel_nonzero (u*v)
  5. L54
    apply mul_left_cancel_nonzero
  6. L55
    exact hd
14Calculate and transport equalitiesL56–57

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L56
    trans m*n
  2. L57
    symm
15Use earlier factsL58–59

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L58
    exact hq
  2. L59
    exact hprod

Library-wide reading audit

Original exact command ledger · 59 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro a
  5. 0005intro b
  6. 0006intro u
  7. 0007intro v
  8. 0008intro q
  9. 0009intro hp
  10. 0010intro hm
  11. 0011intro hn
  12. 0012intro hq
  13. 0013cases hp
  14. 0014cases hp_right
  15. 0015cases hp_right_right
  16. 0016cases hp_right_right_right
  17. 0017have hd : ~(d=0)
  18. 0018intro hzero
  19. 0019specialize mul_ne_zero (a)
  20. 0020specialize mul_ne_zero (b)
  21. 0021apply mul_ne_zero
  22. 0022exact hp_left
  23. 0023exact hp_right_left
  24. 0024trans d
  25. 0025symm
  26. 0026exact hp_right_right_right_right
  27. 0027exact hzero
  28. 0028have hprod : m*n=d*(u*v)
  29. 0029rewrite hm
  30. 0030rewrite hn
  31. 0031rewrite hp_right_right_right_right
  32. 0032trans a*(u*(b*v))
  33. 0033apply mul_assoc
  34. 0034trans a*((u*b)*v)
  35. 0035congr
  36. 0036refl
  37. 0037symm
  38. 0038apply mul_assoc
  39. 0039trans a*((b*u)*v)
  40. 0040congr
  41. 0041refl
  42. 0042congr
  43. 0043apply mul_comm
  44. 0044refl
  45. 0045trans a*(b*(u*v))
  46. 0046congr
  47. 0047refl
  48. 0048apply mul_assoc
  49. 0049symm
  50. 0050apply mul_assoc
  51. 0051specialize mul_left_cancel_nonzero (d)
  52. 0052specialize mul_left_cancel_nonzero (q)
  53. 0053specialize mul_left_cancel_nonzero (u*v)
  54. 0054apply mul_left_cancel_nonzero
  55. 0055exact hd
  56. 0056trans m*n
  57. 0057symm
  58. 0058exact hq
  59. 0059exact hprod