SK0005

coprime_squared_pair

The squares of two coprime naturals are genuinely coprime.

Alpha v34 checked-use · first admitted v29 · 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.

For n>1 the exponent gcd and a real beta table classify and witness all positive root degrees. The unit n=1 has a separate uniform certificate for every positive degree. Zero is excluded. NaturalSquarefreeDecomposition is deliberately distinct from the unrelated polynomial definition.

Exact theorem in conservative defined notation

∀ a. ∀ b. (∀ x. Dvd(x,a) → Dvd(x,b) → x = 1) → ∀ x. Dvd(x,a · a) → Dvd(x,b · b) → x = 1

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

Definition DAG

Dvd(d,n) · 6

Actual proof prerequisites

coprime_mul_left · checked external prerequisitecoprime_mul_right · checked external prerequisite
Original expanded first-order statement 
forall a b. (forall sfd_common_divisor_square_coprime_source. (exists pvs_factor_square_coprime_sourceleft. (a) = (sfd_common_divisor_square_coprime_source) * pvs_factor_square_coprime_sourceleft) -> (exists pvs_factor_square_coprime_sourceright. (b) = (sfd_common_divisor_square_coprime_source) * pvs_factor_square_coprime_sourceright) -> sfd_common_divisor_square_coprime_source = 1) -> (forall sfd_common_divisor_square_coprime_result. (exists pvs_factor_square_coprime_resultleft. (a * a) = (sfd_common_divisor_square_coprime_result) * pvs_factor_square_coprime_resultleft) -> (exists pvs_factor_square_coprime_resultright. (b * b) = (sfd_common_divisor_square_coprime_result) * pvs_factor_square_coprime_resultright) -> sfd_common_divisor_square_coprime_result = 1)

Complete tactic proof in conservative notation

All 16 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

16 script commands · 3 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hcop
02Establish hleftL4–13

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

  1. L4
    have hleft : ∀ sfd_common_divisor_square_coprime_left. Dvd(sfd_common_divisor_square_coprime_left,a · a) → Dvd(sfd_common_divisor_square_coprime_left,b) → sfd_common_divisor_square_coprime_left = 1Definitions: Dvd(sfd_common_divisor_square_coprime_left,a · a)Dvd(sfd_common_divisor_square_coprime_left,b)Original native command in the exact edition
  2. L5
    specialize coprime_mul_left (a)
  3. L6
    specialize coprime_mul_left (a)
  4. L7
    specialize coprime_mul_left (b)
  5. L8
    apply coprime_mul_left
  6. L9
    exact hcop
  7. L10
    exact hcop
  8. L11
    specialize coprime_mul_right (a * a)
  9. L12
    specialize coprime_mul_right (b)
  10. L13
    specialize coprime_mul_right (b)
03Use earlier factsL14–16

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

  1. L14
    apply coprime_mul_right
  2. L15
    exact hleft
  3. L16
    exact hleft

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hcop
  4. 0004have hleft : ∀ sfd_common_divisor_square_coprime_left. Dvd(sfd_common_divisor_square_coprime_left,a · a) → Dvd(sfd_common_divisor_square_coprime_left,b) → sfd_common_divisor_square_coprime_left = 1
  5. 0005specialize coprime_mul_left (a)
  6. 0006specialize coprime_mul_left (a)
  7. 0007specialize coprime_mul_left (b)
  8. 0008apply coprime_mul_left
  9. 0009exact hcop
  10. 0010exact hcop
  11. 0011specialize coprime_mul_right (a * a)
  12. 0012specialize coprime_mul_right (b)
  13. 0013specialize coprime_mul_right (b)
  14. 0014apply coprime_mul_right
  15. 0015exact hleft
  16. 0016exact hleft