SK0005

coprime_squared_pair

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The squares of two coprime naturals are genuinely coprime.

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

The squares of two coprime naturals are genuinely coprime.

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

coprime_mul_left Stable theorem; checked-use authorized coprime_mul_right Stable theorem; checked-use authorized

Direct dependents

SK0006 squarefree_coprime_square_factor_is_one

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : forall sfd_common_divisor_square_coprime_left. (exists pvs_factor_square_coprime_leftleft. (a * a) = (sfd_common_divisor_square_coprime_left) * pvs_factor_square_coprime_leftleft) -> (exists pvs_factor_square_coprime_leftright. (b) = (sfd_common_divisor_square_coprime_left) * pvs_factor_square_coprime_leftright) -> sfd_common_divisor_square_coprime_left = 1
  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 exact command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hcop
  4. 0004have hleft : forall sfd_common_divisor_square_coprime_left. (exists pvs_factor_square_coprime_leftleft. (a * a) = (sfd_common_divisor_square_coprime_left) * pvs_factor_square_coprime_leftleft) -> (exists pvs_factor_square_coprime_leftright. (b) = (sfd_common_divisor_square_coprime_left) * pvs_factor_square_coprime_leftright) -> 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