FC0004

crt_gcd_coprime_cofactors

A nonzero gcd yields both exact natural cofactors and their constructive coprimality.

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. ∀ g. ¬g = 0 → IsGCD(g,a,b) → ∃ x. ∃ y. a = g · x ∧ (b = g · y ∧ (∀ z. Dvd(z,x) → Dvd(z,y) → z = 1))

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

Definition DAG

Dvd(d,n) · 4IsGCD(g,a,b) · 1

Actual proof prerequisites

is_gcd_dvd_left · checked external prerequisiteis_gcd_dvd_right · checked external prerequisiteis_gcd_quotients_coprime_nonzero · checked external prerequisite
Original expanded first-order statement 
forall a b g. ~(g = 0) -> ((((exists ec_gcd_left_gfull_cofactors. a = g * ec_gcd_left_gfull_cofactors) /\ (exists ec_gcd_right_gfull_cofactors. b = g * ec_gcd_right_gfull_cofactors)) /\ forall ec_gcd_common_gfull_cofactors. (exists ec_gcd_common_left_gfull_cofactors. a = ec_gcd_common_gfull_cofactors * ec_gcd_common_left_gfull_cofactors) -> (exists ec_gcd_common_right_gfull_cofactors. b = ec_gcd_common_gfull_cofactors * ec_gcd_common_right_gfull_cofactors) -> exists ec_gcd_greatest_gfull_cofactors. g = ec_gcd_common_gfull_cofactors * ec_gcd_greatest_gfull_cofactors)) -> exists A B. (a = g * A /\ (b = g * B /\ (forall frp_divisor_gfull_cofactors. (exists frp_left_factor_gfull_cofactors. A = frp_divisor_gfull_cofactors * frp_left_factor_gfull_cofactors) -> (exists frp_right_factor_gfull_cofactors. B = frp_divisor_gfull_cofactors * frp_right_factor_gfull_cofactors) -> frp_divisor_gfull_cofactors = 1)))

Complete tactic proof in conservative notation

All 35 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

35 script commands · 11 reading checkpoints · 2 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro g
  4. L4
    intro hn
  5. L5
    intro hg
02Establish hleftL6–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply is gcd dvd left.

  1. L6
  2. L7
    specialize is_gcd_dvd_left g
  3. L8
    specialize is_gcd_dvd_left a
  4. L9
    specialize is_gcd_dvd_left b
  5. L10
    apply is_gcd_dvd_left
  6. L11
    exact hg
03Separate the logical casesL12–12

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

  1. L12
    cases hleft
04Establish hrightL13–18

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

  1. L13
    have hright : Dvd(g,b)Definitions: Dvd(g,b)Original native command in the exact edition
  2. L14
    specialize is_gcd_dvd_right g
  3. L15
    specialize is_gcd_dvd_right a
  4. L16
    specialize is_gcd_dvd_right b
  5. L17
    apply is_gcd_dvd_right
  6. L18
    exact hg
05Separate the logical casesL19–19

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

  1. L19
    cases hright
06Construct an explicit witnessL20–21

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

  1. L20
    exists x
  2. L21
    exists x1
07Separate the logical casesL22–22

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

  1. L22
    split
08Use earlier factsL23–23

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

  1. L23
    exact hleft_witness
09Separate the logical casesL24–24

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

  1. L24
    split
10Use earlier factsL25–34

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

  1. L25
    exact hright_witness
  2. L26
    specialize is_gcd_quotients_coprime_nonzero g
  3. L27
    specialize is_gcd_quotients_coprime_nonzero a
  4. L28
    specialize is_gcd_quotients_coprime_nonzero b
  5. L29
    specialize is_gcd_quotients_coprime_nonzero x
  6. L30
    specialize is_gcd_quotients_coprime_nonzero x1
  7. L31
    apply is_gcd_quotients_coprime_nonzero
  8. L32
    exact hg
  9. L33
    exact hn
  10. L34
    exact hleft_witness
11Use earlier factsL35–35

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

  1. L35
    exact hright_witness

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro g
  4. 0004intro hn
  5. 0005intro hg
  6. 0006have hleft : Dvd(g,a)
  7. 0007specialize is_gcd_dvd_left g
  8. 0008specialize is_gcd_dvd_left a
  9. 0009specialize is_gcd_dvd_left b
  10. 0010apply is_gcd_dvd_left
  11. 0011exact hg
  12. 0012cases hleft
  13. 0013have hright : Dvd(g,b)
  14. 0014specialize is_gcd_dvd_right g
  15. 0015specialize is_gcd_dvd_right a
  16. 0016specialize is_gcd_dvd_right b
  17. 0017apply is_gcd_dvd_right
  18. 0018exact hg
  19. 0019cases hright
  20. 0020exists x
  21. 0021exists x1
  22. 0022split
  23. 0023exact hleft_witness
  24. 0024split
  25. 0025exact hright_witness
  26. 0026specialize is_gcd_quotients_coprime_nonzero g
  27. 0027specialize is_gcd_quotients_coprime_nonzero a
  28. 0028specialize is_gcd_quotients_coprime_nonzero b
  29. 0029specialize is_gcd_quotients_coprime_nonzero x
  30. 0030specialize is_gcd_quotients_coprime_nonzero x1
  31. 0031apply is_gcd_quotients_coprime_nonzero
  32. 0032exact hg
  33. 0033exact hn
  34. 0034exact hleft_witness
  35. 0035exact hright_witness