MX0011

coprime_divisor_factor_pair_exists_unique

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

Every positive divisor of a positive coprime product has exactly one bounded positive divisor pair.

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. ~(m=0) -> ~(n=0) -> ~(d=0) -> (forall sfd_common_divisor_cdp_bounded_coprime. (exists pvs_factor_cdp_bounded_coprimeleft. (m) = (sfd_common_divisor_cdp_bounded_coprime) * pvs_factor_cdp_bounded_coprimeleft) -> (exists pvs_factor_cdp_bounded_coprimeright. (n) = (sfd_common_divisor_cdp_bounded_coprime) * pvs_factor_cdp_bounded_coprimeright) -> sfd_common_divisor_cdp_bounded_coprime = 1) -> (exists pvs_factor_cdp_bounded_divisor. (m*n) = (d) * pvs_factor_cdp_bounded_divisor) -> exists a b. (((((~((a)=0)) /\ (((~((b)=0)) /\ (((exists pvs_factor_cdp_bounded_exists_pairleft. (m) = (a) * pvs_factor_cdp_bounded_exists_pairleft) /\ (((exists pvs_factor_cdp_bounded_exists_pairright. (n) = (b) * pvs_factor_cdp_bounded_exists_pairright) /\ ((d)=(a)*(b)))))))))) /\ (((((exists pvs_le_gap_cdp_bounded_exists_boundsleft. pvs_le_gap_cdp_bounded_exists_boundsleft + (a) = (m)) /\ (((exists pvs_le_gap_cdp_bounded_exists_boundsright. pvs_le_gap_cdp_bounded_exists_boundsright + (b) = (n)) /\ (forall sfd_common_divisor_cdp_bounded_exists_boundscoprime. (exists pvs_factor_cdp_bounded_exists_boundscoprimeleft. (a) = (sfd_common_divisor_cdp_bounded_exists_boundscoprime) * pvs_factor_cdp_bounded_exists_boundscoprimeleft) -> (exists pvs_factor_cdp_bounded_exists_boundscoprimeright. (b) = (sfd_common_divisor_cdp_bounded_exists_boundscoprime) * pvs_factor_cdp_bounded_exists_boundscoprimeright) -> sfd_common_divisor_cdp_bounded_exists_boundscoprime = 1))))) /\ (forall c e. (((~((c)=0)) /\ (((~((e)=0)) /\ (((exists pvs_factor_cdp_bounded_exists_uniqueleft. (m) = (c) * pvs_factor_cdp_bounded_exists_uniqueleft) /\ (((exists pvs_factor_cdp_bounded_exists_uniqueright. (n) = (e) * pvs_factor_cdp_bounded_exists_uniqueright) /\ ((d)=(c)*(e)))))))))) -> ((a=c) /\ (b=e)))))))

Constructive proof overview

Generated structural guide

Every positive divisor of a positive coprime product has exactly one bounded positive divisor pair.

The unchanged tactic script uses 3 declared prerequisites and contains 47 exact native proof lines.

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

Proof neighborhood

Direct dependencies

MX000F coprime_divisor_factor_pair_exists MX0010 coprime_divisor_factor_pair_bounds MX000E coprime_divisor_factor_pair_unique

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

47 script commands · 11 reading checkpoints · 1 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.

Named ingredients (3)
coprime divisor factor pair existscoprime divisor factor pair boundscoprime divisor factor pair unique
01Fix variables and assumptionsL1–8

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 hm
  5. L5
    intro hn
  6. L6
    intro hd
  7. L7
    intro hc
  8. L8
    intro hdiv
02Establish hpL9–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply coprime divisor factor pair exists.

  1. L9
    have hp : exists a b. (((~((a)=0)) /\ (((~((b)=0)) /\ (((exists pvs_factor_cdp_bounded_constructleft. (m) = (a) * pvs_factor_cdp_bounded_constructleft) /\ (((exists pvs_factor_cdp_bounded_constructright. (n) = (b) * pvs_factor_cdp_bounded_constructright) /\ ((d)=(a)*(b))))))))))
  2. L10
    specialize coprime_divisor_factor_pair_exists (m)
  3. L11
    specialize coprime_divisor_factor_pair_exists (n)
  4. L12
    specialize coprime_divisor_factor_pair_exists (d)
  5. L13
    apply coprime_divisor_factor_pair_exists
  6. L14
    exact hd
  7. L15
    exact hc
  8. L16
    exact hdiv
03Separate the logical casesL17–18

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

  1. L17
    cases hp
  2. L18
    cases hp_witness
04Construct an explicit witnessL19–20

Supply the displayed value, then prove that it has the required property.

  1. L19
    exists x
  2. L20
    exists x1
05Separate the logical casesL21–21

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

  1. L21
    split
06Use earlier factsL22–22

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

  1. L22
    exact hp_witness_witness
07Separate the logical casesL23–23

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

  1. L23
    split
08Use earlier factsL24–33

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

  1. L24
    specialize coprime_divisor_factor_pair_bounds (m)
  2. L25
    specialize coprime_divisor_factor_pair_bounds (n)
  3. L26
    specialize coprime_divisor_factor_pair_bounds (d)
  4. L27
    specialize coprime_divisor_factor_pair_bounds (x)
  5. L28
    specialize coprime_divisor_factor_pair_bounds (x1)
  6. L29
    apply coprime_divisor_factor_pair_bounds
  7. L30
    exact hm
  8. L31
    exact hn
  9. L32
    exact hc
  10. L33
    exact hp_witness_witness
09Fix variables and assumptionsL34–36

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

  1. L34
    intro c
  2. L35
    intro e
  3. L36
    intro hq
10Use earlier factsL37–46

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

  1. L37
    specialize coprime_divisor_factor_pair_unique (m)
  2. L38
    specialize coprime_divisor_factor_pair_unique (n)
  3. L39
    specialize coprime_divisor_factor_pair_unique (d)
  4. L40
    specialize coprime_divisor_factor_pair_unique (x)
  5. L41
    specialize coprime_divisor_factor_pair_unique (x1)
  6. L42
    specialize coprime_divisor_factor_pair_unique (c)
  7. L43
    specialize coprime_divisor_factor_pair_unique (e)
  8. L44
    apply coprime_divisor_factor_pair_unique
  9. L45
    exact hc
  10. L46
    exact hp_witness_witness
11Use earlier factsL47–47

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

  1. L47
    exact hq

Library-wide reading audit

Original exact command ledger · 47 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hm
  5. 0005intro hn
  6. 0006intro hd
  7. 0007intro hc
  8. 0008intro hdiv
  9. 0009have hp : exists a b. (((~((a)=0)) /\ (((~((b)=0)) /\ (((exists pvs_factor_cdp_bounded_constructleft. (m) = (a) * pvs_factor_cdp_bounded_constructleft) /\ (((exists pvs_factor_cdp_bounded_constructright. (n) = (b) * pvs_factor_cdp_bounded_constructright) /\ ((d)=(a)*(b))))))))))
  10. 0010specialize coprime_divisor_factor_pair_exists (m)
  11. 0011specialize coprime_divisor_factor_pair_exists (n)
  12. 0012specialize coprime_divisor_factor_pair_exists (d)
  13. 0013apply coprime_divisor_factor_pair_exists
  14. 0014exact hd
  15. 0015exact hc
  16. 0016exact hdiv
  17. 0017cases hp
  18. 0018cases hp_witness
  19. 0019exists x
  20. 0020exists x1
  21. 0021split
  22. 0022exact hp_witness_witness
  23. 0023split
  24. 0024specialize coprime_divisor_factor_pair_bounds (m)
  25. 0025specialize coprime_divisor_factor_pair_bounds (n)
  26. 0026specialize coprime_divisor_factor_pair_bounds (d)
  27. 0027specialize coprime_divisor_factor_pair_bounds (x)
  28. 0028specialize coprime_divisor_factor_pair_bounds (x1)
  29. 0029apply coprime_divisor_factor_pair_bounds
  30. 0030exact hm
  31. 0031exact hn
  32. 0032exact hc
  33. 0033exact hp_witness_witness
  34. 0034intro c
  35. 0035intro e
  36. 0036intro hq
  37. 0037specialize coprime_divisor_factor_pair_unique (m)
  38. 0038specialize coprime_divisor_factor_pair_unique (n)
  39. 0039specialize coprime_divisor_factor_pair_unique (d)
  40. 0040specialize coprime_divisor_factor_pair_unique (x)
  41. 0041specialize coprime_divisor_factor_pair_unique (x1)
  42. 0042specialize coprime_divisor_factor_pair_unique (c)
  43. 0043specialize coprime_divisor_factor_pair_unique (e)
  44. 0044apply coprime_divisor_factor_pair_unique
  45. 0045exact hc
  46. 0046exact hp_witness_witness
  47. 0047exact hq