GC0002

crt_coprime_divisor_pair

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

Any divisors of two coprime naturals are themselves constructively 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 original first-admission records.

Exact expanded first-order arithmetic statement

forall a b d e. (forall frp_divisor_gcomp_divisor_source. (exists frp_left_factor_gcomp_divisor_source. a = frp_divisor_gcomp_divisor_source * frp_left_factor_gcomp_divisor_source) -> (exists frp_right_factor_gcomp_divisor_source. b = frp_divisor_gcomp_divisor_source * frp_right_factor_gcomp_divisor_source) -> frp_divisor_gcomp_divisor_source = 1) -> (exists u. a = d * u) -> (exists v. b = e * v) -> (forall frp_divisor_gcomp_divisor_result. (exists frp_left_factor_gcomp_divisor_result. d = frp_divisor_gcomp_divisor_result * frp_left_factor_gcomp_divisor_result) -> (exists frp_right_factor_gcomp_divisor_result. e = frp_divisor_gcomp_divisor_result * frp_right_factor_gcomp_divisor_result) -> frp_divisor_gcomp_divisor_result = 1)

Constructive proof overview

Generated structural guide

Any divisors of two coprime naturals are themselves constructively coprime.

The unchanged tactic script uses 1 declared prerequisite and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

multiple_trans Stable theorem; checked-use authorized

Direct dependents

GC000F crt_is_gcd_coprime_factor_remove GC0011 crt_is_gcd_coprime_product

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

24 script commands · 3 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 assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro hcoprime
  6. L6
    intro hd
  7. L7
    intro he
  8. L8
    intro k
  9. L9
    intro hkd
  10. L10
    intro hke
02Use earlier factsL11–20

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

  1. L11
    specialize hcoprime k
  2. L12
    apply hcoprime
  3. L13
    specialize multiple_trans d
  4. L14
    specialize multiple_trans k
  5. L15
    specialize multiple_trans a
  6. L16
    apply multiple_trans
  7. L17
    exact hd
  8. L18
    exact hkd
  9. L19
    specialize multiple_trans e
  10. L20
    specialize multiple_trans k
03Use earlier factsL21–24

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

  1. L21
    specialize multiple_trans b
  2. L22
    apply multiple_trans
  3. L23
    exact he
  4. L24
    exact hke

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro d
  4. 0004intro e
  5. 0005intro hcoprime
  6. 0006intro hd
  7. 0007intro he
  8. 0008intro k
  9. 0009intro hkd
  10. 0010intro hke
  11. 0011specialize hcoprime k
  12. 0012apply hcoprime
  13. 0013specialize multiple_trans d
  14. 0014specialize multiple_trans k
  15. 0015specialize multiple_trans a
  16. 0016apply multiple_trans
  17. 0017exact hd
  18. 0018exact hkd
  19. 0019specialize multiple_trans e
  20. 0020specialize multiple_trans k
  21. 0021specialize multiple_trans b
  22. 0022apply multiple_trans
  23. 0023exact he
  24. 0024exact hke

Separate complete second-wave branches: Full G011 proof · Alpha v27.