FC0008

crt_gcd_lcm_distributes_zero_comparison

Comparison with zero turns all three gcds into their other inputs and preserves the exact original LCM.

Alpha v34 checked-use · first admitted v27 · 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.

All finite lists are included, even the empty list and zero moduli. A positive LCM gives x<M; at zero LCM congruence is exact equality and normalization deliberately does not require the impossible x<0.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ n. ∀ L. ∀ ga. ∀ gb. ∀ g. n = 0 → Dvd(a,L) ∧ Dvd(b,L) ∧ (∀ x. Dvd(a,x) → Dvd(b,x) → Dvd(L,x)) → IsGCD(ga,a,n) → IsGCD(gb,b,n) → IsGCD(g,L,n) → Dvd(ga,g) ∧ Dvd(gb,g) ∧ (∀ x. Dvd(ga,x) → Dvd(gb,x) → Dvd(g,x))

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

Definition DAG

Dvd(d,n) · 10IsGCD(g,a,b) · 3

Actual proof prerequisites

crt_gcd_zero_right_value
Original expanded first-order statement 
forall a b n L ga gb g. n = 0 -> ((((exists hscale_left_factor_gfull_zero_n_L. L = a * hscale_left_factor_gfull_zero_n_L) /\ (exists hscale_right_factor_gfull_zero_n_L. L = b * hscale_right_factor_gfull_zero_n_L)) /\ forall hscale_common_gfull_zero_n_L. (exists hscale_left_common_gfull_zero_n_L. hscale_common_gfull_zero_n_L = a * hscale_left_common_gfull_zero_n_L) -> (exists hscale_right_common_gfull_zero_n_L. hscale_common_gfull_zero_n_L = b * hscale_right_common_gfull_zero_n_L) -> exists hscale_least_factor_gfull_zero_n_L. hscale_common_gfull_zero_n_L = L * hscale_least_factor_gfull_zero_n_L)) -> ((((exists ec_gcd_left_gfull_zero_n_ga. a = ga * ec_gcd_left_gfull_zero_n_ga) /\ (exists ec_gcd_right_gfull_zero_n_ga. n = ga * ec_gcd_right_gfull_zero_n_ga)) /\ forall ec_gcd_common_gfull_zero_n_ga. (exists ec_gcd_common_left_gfull_zero_n_ga. a = ec_gcd_common_gfull_zero_n_ga * ec_gcd_common_left_gfull_zero_n_ga) -> (exists ec_gcd_common_right_gfull_zero_n_ga. n = ec_gcd_common_gfull_zero_n_ga * ec_gcd_common_right_gfull_zero_n_ga) -> exists ec_gcd_greatest_gfull_zero_n_ga. ga = ec_gcd_common_gfull_zero_n_ga * ec_gcd_greatest_gfull_zero_n_ga)) -> ((((exists ec_gcd_left_gfull_zero_n_gb. b = gb * ec_gcd_left_gfull_zero_n_gb) /\ (exists ec_gcd_right_gfull_zero_n_gb. n = gb * ec_gcd_right_gfull_zero_n_gb)) /\ forall ec_gcd_common_gfull_zero_n_gb. (exists ec_gcd_common_left_gfull_zero_n_gb. b = ec_gcd_common_gfull_zero_n_gb * ec_gcd_common_left_gfull_zero_n_gb) -> (exists ec_gcd_common_right_gfull_zero_n_gb. n = ec_gcd_common_gfull_zero_n_gb * ec_gcd_common_right_gfull_zero_n_gb) -> exists ec_gcd_greatest_gfull_zero_n_gb. gb = ec_gcd_common_gfull_zero_n_gb * ec_gcd_greatest_gfull_zero_n_gb)) -> ((((exists ec_gcd_left_gfull_zero_n_g. L = g * ec_gcd_left_gfull_zero_n_g) /\ (exists ec_gcd_right_gfull_zero_n_g. n = g * ec_gcd_right_gfull_zero_n_g)) /\ forall ec_gcd_common_gfull_zero_n_g. (exists ec_gcd_common_left_gfull_zero_n_g. L = ec_gcd_common_gfull_zero_n_g * ec_gcd_common_left_gfull_zero_n_g) -> (exists ec_gcd_common_right_gfull_zero_n_g. n = ec_gcd_common_gfull_zero_n_g * ec_gcd_common_right_gfull_zero_n_g) -> exists ec_gcd_greatest_gfull_zero_n_g. g = ec_gcd_common_gfull_zero_n_g * ec_gcd_greatest_gfull_zero_n_g)) -> ((((exists hscale_left_factor_gfull_zero_n_result. g = ga * hscale_left_factor_gfull_zero_n_result) /\ (exists hscale_right_factor_gfull_zero_n_result. g = gb * hscale_right_factor_gfull_zero_n_result)) /\ forall hscale_common_gfull_zero_n_result. (exists hscale_left_common_gfull_zero_n_result. hscale_common_gfull_zero_n_result = ga * hscale_left_common_gfull_zero_n_result) -> (exists hscale_right_common_gfull_zero_n_result. hscale_common_gfull_zero_n_result = gb * hscale_right_common_gfull_zero_n_result) -> exists hscale_least_factor_gfull_zero_n_result. hscale_common_gfull_zero_n_result = g * hscale_least_factor_gfull_zero_n_result))

Complete tactic proof in conservative notation

All 41 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

41 script commands · 7 reading checkpoints · 3 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.

Named ingredients (1)
crt gcd zero right value
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 n
  4. L4
    intro L
  5. L5
    intro ga
  6. L6
    intro gb
  7. L7
    intro g
  8. L8
    intro hn
  9. L9
    intro hL
  10. L10
    intro hga
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hgb
  2. L12
    intro hg
03Establish heqaL13–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply crt gcd zero right value.

  1. L13
    have heqa : ga = a
  2. L14
    specialize crt_gcd_zero_right_value a
  3. L15
    specialize crt_gcd_zero_right_value ga
  4. L16
    apply crt_gcd_zero_right_value
  5. L17
    rewrite <- hn
  6. L18
    rewrite <- hn
  7. L19
    exact hga
04Establish heqbL20–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply crt gcd zero right value.

  1. L20
    have heqb : gb = b
  2. L21
    specialize crt_gcd_zero_right_value b
  3. L22
    specialize crt_gcd_zero_right_value gb
  4. L23
    apply crt_gcd_zero_right_value
  5. L24
    rewrite <- hn
  6. L25
    rewrite <- hn
  7. L26
    exact hgb
05Establish heqgL27–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply crt gcd zero right value.

  1. L27
    have heqg : g = L
  2. L28
    specialize crt_gcd_zero_right_value L
  3. L29
    specialize crt_gcd_zero_right_value g
  4. L30
    apply crt_gcd_zero_right_value
  5. L31
    rewrite <- hn
  6. L32
    rewrite <- hn
  7. L33
    exact hg
  8. L34
    rewrite heqa
  9. L35
    rewrite heqa
  10. L36
    rewrite heqb
06Calculate and transport equalitiesL37–40

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

  1. L37
    rewrite heqb
  2. L38
    rewrite heqg
  3. L39
    rewrite heqg
  4. L40
    rewrite heqg
07Use earlier factsL41–41

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

  1. L41
    exact hL

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro n
  4. 0004intro L
  5. 0005intro ga
  6. 0006intro gb
  7. 0007intro g
  8. 0008intro hn
  9. 0009intro hL
  10. 0010intro hga
  11. 0011intro hgb
  12. 0012intro hg
  13. 0013have heqa : ga = a
  14. 0014specialize crt_gcd_zero_right_value a
  15. 0015specialize crt_gcd_zero_right_value ga
  16. 0016apply crt_gcd_zero_right_value
  17. 0017rewrite <- hn
  18. 0018rewrite <- hn
  19. 0019exact hga
  20. 0020have heqb : gb = b
  21. 0021specialize crt_gcd_zero_right_value b
  22. 0022specialize crt_gcd_zero_right_value gb
  23. 0023apply crt_gcd_zero_right_value
  24. 0024rewrite <- hn
  25. 0025rewrite <- hn
  26. 0026exact hgb
  27. 0027have heqg : g = L
  28. 0028specialize crt_gcd_zero_right_value L
  29. 0029specialize crt_gcd_zero_right_value g
  30. 0030apply crt_gcd_zero_right_value
  31. 0031rewrite <- hn
  32. 0032rewrite <- hn
  33. 0033exact hg
  34. 0034rewrite heqa
  35. 0035rewrite heqa
  36. 0036rewrite heqb
  37. 0037rewrite heqb
  38. 0038rewrite heqg
  39. 0039rewrite heqg
  40. 0040rewrite heqg
  41. 0041exact hL