GC0014

crt_gcd_monotone_under_divisibility

A divisibility relation between inputs transports monotonically to their relational gcd values with any fixed natural, including zero.

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

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.

Historical partial components only: this chapter proves canonical solutions under successive-merge compatibility and in the pairwise-compatible dominating-last case. G011 is now closed in the separate Alpha-v27 generalized-crt branch for arbitrary pairwise-compatible finite lists, including noncoprime moduli. Full G011 proof · Alpha v27

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ n. ∀ ga. ∀ gb. Dvd(a,b) → IsGCD(ga,a,n) → IsGCD(gb,b,n) → Dvd(ga,gb)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Dvd(d,n) · 2IsGCD(g,a,b) · 2

Actual proof prerequisites

is_gcd_dvd_left · checked external prerequisiteis_gcd_dvd_right · checked external prerequisitemultiple_trans · checked external prerequisiteis_gcd_greatest · checked external prerequisite
Original expanded first-order statement 
forall a b n ga gb. (exists q. b = a * q) -> ((((exists hag_left_factor_gcomp_monotone_small. a = ga * hag_left_factor_gcomp_monotone_small) /\ (exists hag_right_factor_gcomp_monotone_small. n = ga * hag_right_factor_gcomp_monotone_small)) /\ forall hag_divisor_gcomp_monotone_small. (exists hag_common_left_gcomp_monotone_small. a = hag_divisor_gcomp_monotone_small * hag_common_left_gcomp_monotone_small) -> (exists hag_common_right_gcomp_monotone_small. n = hag_divisor_gcomp_monotone_small * hag_common_right_gcomp_monotone_small) -> exists hag_greatest_factor_gcomp_monotone_small. ga = hag_divisor_gcomp_monotone_small * hag_greatest_factor_gcomp_monotone_small)) -> ((((exists hag_left_factor_gcomp_monotone_large. b = gb * hag_left_factor_gcomp_monotone_large) /\ (exists hag_right_factor_gcomp_monotone_large. n = gb * hag_right_factor_gcomp_monotone_large)) /\ forall hag_divisor_gcomp_monotone_large. (exists hag_common_left_gcomp_monotone_large. b = hag_divisor_gcomp_monotone_large * hag_common_left_gcomp_monotone_large) -> (exists hag_common_right_gcomp_monotone_large. n = hag_divisor_gcomp_monotone_large * hag_common_right_gcomp_monotone_large) -> exists hag_greatest_factor_gcomp_monotone_large. gb = hag_divisor_gcomp_monotone_large * hag_greatest_factor_gcomp_monotone_large)) -> exists q. gb = ga * q

Complete unchanged native tactic proof

All 29 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

29 script commands · 4 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro n
  4. L4
    intro ga
  5. L5
    intro gb
  6. L6
    intro hab
  7. L7
    intro hga
  8. L8
    intro hgb
02Use earlier factsL9–18

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

  1. L9
    specialize is_gcd_greatest gb
  2. L10
    specialize is_gcd_greatest b
  3. L11
    specialize is_gcd_greatest n
  4. L12
    specialize is_gcd_greatest ga
  5. L13
    apply is_gcd_greatest
  6. L14
    exact hgb
  7. L15
    specialize multiple_trans a
  8. L16
    specialize multiple_trans ga
  9. L17
    specialize multiple_trans b
  10. L18
    apply multiple_trans
03Use earlier factsL19–28

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

  1. L19
    exact hab
  2. L20
    specialize is_gcd_dvd_left ga
  3. L21
    specialize is_gcd_dvd_left a
  4. L22
    specialize is_gcd_dvd_left n
  5. L23
    apply is_gcd_dvd_left
  6. L24
    exact hga
  7. L25
    specialize is_gcd_dvd_right ga
  8. L26
    specialize is_gcd_dvd_right a
  9. L27
    specialize is_gcd_dvd_right n
  10. L28
    apply is_gcd_dvd_right
04Use earlier factsL29–29

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

  1. L29
    exact hga

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro n
  4. 0004intro ga
  5. 0005intro gb
  6. 0006intro hab
  7. 0007intro hga
  8. 0008intro hgb
  9. 0009specialize is_gcd_greatest gb
  10. 0010specialize is_gcd_greatest b
  11. 0011specialize is_gcd_greatest n
  12. 0012specialize is_gcd_greatest ga
  13. 0013apply is_gcd_greatest
  14. 0014exact hgb
  15. 0015specialize multiple_trans a
  16. 0016specialize multiple_trans ga
  17. 0017specialize multiple_trans b
  18. 0018apply multiple_trans
  19. 0019exact hab
  20. 0020specialize is_gcd_dvd_left ga
  21. 0021specialize is_gcd_dvd_left a
  22. 0022specialize is_gcd_dvd_left n
  23. 0023apply is_gcd_dvd_left
  24. 0024exact hga
  25. 0025specialize is_gcd_dvd_right ga
  26. 0026specialize is_gcd_dvd_right a
  27. 0027specialize is_gcd_dvd_right n
  28. 0028apply is_gcd_dvd_right
  29. 0029exact hga