MX0010

coprime_divisor_factor_pair_bounds

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

For positive inputs each coordinate lies in its actual divisor window, and the coordinates are coprime.

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. ~(m=0) -> ~(n=0) -> (forall sfd_common_divisor_cdp_bounds_coprime. (exists pvs_factor_cdp_bounds_coprimeleft. (m) = (sfd_common_divisor_cdp_bounds_coprime) * pvs_factor_cdp_bounds_coprimeleft) -> (exists pvs_factor_cdp_bounds_coprimeright. (n) = (sfd_common_divisor_cdp_bounds_coprime) * pvs_factor_cdp_bounds_coprimeright) -> sfd_common_divisor_cdp_bounds_coprime = 1) -> (((~((a)=0)) /\ (((~((b)=0)) /\ (((exists pvs_factor_cdp_bounds_pairleft. (m) = (a) * pvs_factor_cdp_bounds_pairleft) /\ (((exists pvs_factor_cdp_bounds_pairright. (n) = (b) * pvs_factor_cdp_bounds_pairright) /\ ((d)=(a)*(b)))))))))) -> (((exists pvs_le_gap_cdp_bounds_resultleft. pvs_le_gap_cdp_bounds_resultleft + (a) = (m)) /\ (((exists pvs_le_gap_cdp_bounds_resultright. pvs_le_gap_cdp_bounds_resultright + (b) = (n)) /\ (forall sfd_common_divisor_cdp_bounds_resultcoprime. (exists pvs_factor_cdp_bounds_resultcoprimeleft. (a) = (sfd_common_divisor_cdp_bounds_resultcoprime) * pvs_factor_cdp_bounds_resultcoprimeleft) -> (exists pvs_factor_cdp_bounds_resultcoprimeright. (b) = (sfd_common_divisor_cdp_bounds_resultcoprime) * pvs_factor_cdp_bounds_resultcoprimeright) -> sfd_common_divisor_cdp_bounds_resultcoprime = 1)))))

Constructive proof overview

Generated structural guide

For positive inputs each coordinate lies in its actual divisor window, and the coordinates are coprime.

The unchanged tactic script uses 2 declared prerequisites and contains 33 exact native proof lines.

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

Proof neighborhood

Direct dependencies

divisor_le_nonzero Stable theorem; checked-use authorized crt_coprime_divisor_pair Alpha theorem; checked-use authorized

Direct dependents

MX0011 coprime_divisor_factor_pair_exists_unique MX0054 dirichlet_coprime_grid_support_covering

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

33 script commands · 6 reading checkpoints · 0 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 assumptionsL1–9

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 hm
  7. L7
    intro hn
  8. L8
    intro hc
  9. L9
    intro hp
02Separate the logical casesL10–14

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

  1. L10
    cases hp
  2. L11
    cases hp_right
  3. L12
    cases hp_right_right
  4. L13
    cases hp_right_right_right
  5. L14
    split
03Use earlier factsL15–19

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

  1. L15
    specialize divisor_le_nonzero (a)
  2. L16
    specialize divisor_le_nonzero (m)
  3. L17
    apply divisor_le_nonzero
  4. L18
    exact hm
  5. L19
    exact hp_right_right_left
04Separate the logical casesL20–20

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

  1. L20
    split
05Use earlier factsL21–30

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

  1. L21
    specialize divisor_le_nonzero (b)
  2. L22
    specialize divisor_le_nonzero (n)
  3. L23
    apply divisor_le_nonzero
  4. L24
    exact hn
  5. L25
    exact hp_right_right_right_left
  6. L26
    specialize crt_coprime_divisor_pair (m)
  7. L27
    specialize crt_coprime_divisor_pair (n)
  8. L28
    specialize crt_coprime_divisor_pair (a)
  9. L29
    specialize crt_coprime_divisor_pair (b)
  10. L30
    apply crt_coprime_divisor_pair
06Use earlier factsL31–33

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

  1. L31
    exact hc
  2. L32
    exact hp_right_right_left
  3. L33
    exact hp_right_right_right_left

Library-wide reading audit

Original exact command ledger · 33 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro a
  5. 0005intro b
  6. 0006intro hm
  7. 0007intro hn
  8. 0008intro hc
  9. 0009intro hp
  10. 0010cases hp
  11. 0011cases hp_right
  12. 0012cases hp_right_right
  13. 0013cases hp_right_right_right
  14. 0014split
  15. 0015specialize divisor_le_nonzero (a)
  16. 0016specialize divisor_le_nonzero (m)
  17. 0017apply divisor_le_nonzero
  18. 0018exact hm
  19. 0019exact hp_right_right_left
  20. 0020split
  21. 0021specialize divisor_le_nonzero (b)
  22. 0022specialize divisor_le_nonzero (n)
  23. 0023apply divisor_le_nonzero
  24. 0024exact hn
  25. 0025exact hp_right_right_right_left
  26. 0026specialize crt_coprime_divisor_pair (m)
  27. 0027specialize crt_coprime_divisor_pair (n)
  28. 0028specialize crt_coprime_divisor_pair (a)
  29. 0029specialize crt_coprime_divisor_pair (b)
  30. 0030apply crt_coprime_divisor_pair
  31. 0031exact hc
  32. 0032exact hp_right_right_left
  33. 0033exact hp_right_right_right_left