FC0007

crt_gcd_lcm_distributes_zero_left

The zero-left-modulus boundary of gcd--LCM distributivity follows from exact divisibility, not a positivity assumption.

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. a = 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) · 15IsGCD(g,a,b) · 3

Actual proof prerequisites

is_lcm_symm · checked external prerequisitecrt_gcd_lcm_distributes_divisibility · checked external prerequisite
Original expanded first-order statement 
forall a b n L ga gb g. a = 0 -> ((((exists hscale_left_factor_gfull_zero_left_L. L = a * hscale_left_factor_gfull_zero_left_L) /\ (exists hscale_right_factor_gfull_zero_left_L. L = b * hscale_right_factor_gfull_zero_left_L)) /\ forall hscale_common_gfull_zero_left_L. (exists hscale_left_common_gfull_zero_left_L. hscale_common_gfull_zero_left_L = a * hscale_left_common_gfull_zero_left_L) -> (exists hscale_right_common_gfull_zero_left_L. hscale_common_gfull_zero_left_L = b * hscale_right_common_gfull_zero_left_L) -> exists hscale_least_factor_gfull_zero_left_L. hscale_common_gfull_zero_left_L = L * hscale_least_factor_gfull_zero_left_L)) -> ((((exists ec_gcd_left_gfull_zero_left_ga. a = ga * ec_gcd_left_gfull_zero_left_ga) /\ (exists ec_gcd_right_gfull_zero_left_ga. n = ga * ec_gcd_right_gfull_zero_left_ga)) /\ forall ec_gcd_common_gfull_zero_left_ga. (exists ec_gcd_common_left_gfull_zero_left_ga. a = ec_gcd_common_gfull_zero_left_ga * ec_gcd_common_left_gfull_zero_left_ga) -> (exists ec_gcd_common_right_gfull_zero_left_ga. n = ec_gcd_common_gfull_zero_left_ga * ec_gcd_common_right_gfull_zero_left_ga) -> exists ec_gcd_greatest_gfull_zero_left_ga. ga = ec_gcd_common_gfull_zero_left_ga * ec_gcd_greatest_gfull_zero_left_ga)) -> ((((exists ec_gcd_left_gfull_zero_left_gb. b = gb * ec_gcd_left_gfull_zero_left_gb) /\ (exists ec_gcd_right_gfull_zero_left_gb. n = gb * ec_gcd_right_gfull_zero_left_gb)) /\ forall ec_gcd_common_gfull_zero_left_gb. (exists ec_gcd_common_left_gfull_zero_left_gb. b = ec_gcd_common_gfull_zero_left_gb * ec_gcd_common_left_gfull_zero_left_gb) -> (exists ec_gcd_common_right_gfull_zero_left_gb. n = ec_gcd_common_gfull_zero_left_gb * ec_gcd_common_right_gfull_zero_left_gb) -> exists ec_gcd_greatest_gfull_zero_left_gb. gb = ec_gcd_common_gfull_zero_left_gb * ec_gcd_greatest_gfull_zero_left_gb)) -> ((((exists ec_gcd_left_gfull_zero_left_g. L = g * ec_gcd_left_gfull_zero_left_g) /\ (exists ec_gcd_right_gfull_zero_left_g. n = g * ec_gcd_right_gfull_zero_left_g)) /\ forall ec_gcd_common_gfull_zero_left_g. (exists ec_gcd_common_left_gfull_zero_left_g. L = ec_gcd_common_gfull_zero_left_g * ec_gcd_common_left_gfull_zero_left_g) -> (exists ec_gcd_common_right_gfull_zero_left_g. n = ec_gcd_common_gfull_zero_left_g * ec_gcd_common_right_gfull_zero_left_g) -> exists ec_gcd_greatest_gfull_zero_left_g. g = ec_gcd_common_gfull_zero_left_g * ec_gcd_greatest_gfull_zero_left_g)) -> ((((exists hscale_left_factor_gfull_zero_left_result. g = ga * hscale_left_factor_gfull_zero_left_result) /\ (exists hscale_right_factor_gfull_zero_left_result. g = gb * hscale_right_factor_gfull_zero_left_result)) /\ forall hscale_common_gfull_zero_left_result. (exists hscale_left_common_gfull_zero_left_result. hscale_common_gfull_zero_left_result = ga * hscale_left_common_gfull_zero_left_result) -> (exists hscale_right_common_gfull_zero_left_result. hscale_common_gfull_zero_left_result = gb * hscale_right_common_gfull_zero_left_result) -> exists hscale_least_factor_gfull_zero_left_result. hscale_common_gfull_zero_left_result = g * hscale_least_factor_gfull_zero_left_result))

Complete tactic proof in conservative notation

All 38 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

38 script commands · 7 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.

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 assumptionsL1–10

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 ha
  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 hswapL13–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply is lcm symm.

  1. L13
    have hswap : Dvd(b,L) ∧ Dvd(a,L) ∧ (∀ x. Dvd(b,x) → Dvd(a,x) → Dvd(L,x))Definitions: Dvd(b,L)Dvd(a,L)Dvd(b,x)Dvd(a,x)Dvd(L,x)Original native command in the exact edition
  2. L14
    specialize is_lcm_symm L
  3. L15
    specialize is_lcm_symm a
  4. L16
    specialize is_lcm_symm b
  5. L17
    apply is_lcm_symm
  6. L18
    exact hL
  7. L19
    specialize is_lcm_symm g
  8. L20
    specialize is_lcm_symm gb
  9. L21
    specialize is_lcm_symm ga
  10. L22
    apply is_lcm_symm
04Use earlier factsL23–30

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

  1. L23
    specialize crt_gcd_lcm_distributes_divisibility b
  2. L24
    specialize crt_gcd_lcm_distributes_divisibility a
  3. L25
    specialize crt_gcd_lcm_distributes_divisibility n
  4. L26
    specialize crt_gcd_lcm_distributes_divisibility L
  5. L27
    specialize crt_gcd_lcm_distributes_divisibility gb
  6. L28
    specialize crt_gcd_lcm_distributes_divisibility ga
  7. L29
    specialize crt_gcd_lcm_distributes_divisibility g
  8. L30
    apply crt_gcd_lcm_distributes_divisibility
05Construct an explicit witnessL31–31

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

  1. L31
    exists 0
06Calculate and transport equalitiesL32–33

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

  1. L32
    rewrite ha
  2. L33
    symm
07Use earlier factsL34–38

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

  1. L34
    apply PA5
  2. L35
    exact hswap
  3. L36
    exact hgb
  4. L37
    exact hga
  5. L38
    exact hg

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro n
  4. 0004intro L
  5. 0005intro ga
  6. 0006intro gb
  7. 0007intro g
  8. 0008intro ha
  9. 0009intro hL
  10. 0010intro hga
  11. 0011intro hgb
  12. 0012intro hg
  13. 0013have hswap : Dvd(b,L) ∧ Dvd(a,L) ∧ (∀ x. Dvd(b,x) → Dvd(a,x) → Dvd(L,x))
  14. 0014specialize is_lcm_symm L
  15. 0015specialize is_lcm_symm a
  16. 0016specialize is_lcm_symm b
  17. 0017apply is_lcm_symm
  18. 0018exact hL
  19. 0019specialize is_lcm_symm g
  20. 0020specialize is_lcm_symm gb
  21. 0021specialize is_lcm_symm ga
  22. 0022apply is_lcm_symm
  23. 0023specialize crt_gcd_lcm_distributes_divisibility b
  24. 0024specialize crt_gcd_lcm_distributes_divisibility a
  25. 0025specialize crt_gcd_lcm_distributes_divisibility n
  26. 0026specialize crt_gcd_lcm_distributes_divisibility L
  27. 0027specialize crt_gcd_lcm_distributes_divisibility gb
  28. 0028specialize crt_gcd_lcm_distributes_divisibility ga
  29. 0029specialize crt_gcd_lcm_distributes_divisibility g
  30. 0030apply crt_gcd_lcm_distributes_divisibility
  31. 0031exists 0
  32. 0032rewrite ha
  33. 0033symm
  34. 0034apply PA5
  35. 0035exact hswap
  36. 0036exact hgb
  37. 0037exact hga
  38. 0038exact hg