PF0026

fermat_four_coprime_squares

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

Coprime natural bases have coprime squares, with no prime-factorization assumption.

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 pff_divisor_ffd_square_source. (exists pff_left_ffd_square_source. (a) = pff_divisor_ffd_square_source * pff_left_ffd_square_source) -> (exists pff_right_ffd_square_source. (b) = pff_divisor_ffd_square_source * pff_right_ffd_square_source) -> pff_divisor_ffd_square_source = 1) -> (forall pff_divisor_ffd_square_result. (exists pff_left_ffd_square_result. (a * a) = pff_divisor_ffd_square_result * pff_left_ffd_square_result) -> (exists pff_right_ffd_square_result. (b * b) = pff_divisor_ffd_square_result * pff_right_ffd_square_result) -> pff_divisor_ffd_square_result = 1)

Constructive proof overview

Generated structural guide

Coprime natural bases have coprime squares, with no prime-factorization assumption.

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

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

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

Direct dependents

PF002I fermat_four_primitive_square_triangle

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 hcoprime
02Establish hrightL4–13

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

  1. L4
    have hright : forall pff_divisor_ffd_square_middle. (exists pff_left_ffd_square_middle. (a) = pff_divisor_ffd_square_middle * pff_left_ffd_square_middle) -> (exists pff_right_ffd_square_middle. (b * b) = pff_divisor_ffd_square_middle * pff_right_ffd_square_middle) -> pff_divisor_ffd_square_middle = 1
  2. L5
    specialize coprime_mul_right (a)
  3. L6
    specialize coprime_mul_right (b)
  4. L7
    specialize coprime_mul_right (b)
  5. L8
    apply coprime_mul_right
  6. L9
    exact hcoprime
  7. L10
    exact hcoprime
  8. L11
    specialize coprime_mul_left (a)
  9. L12
    specialize coprime_mul_left (a)
  10. L13
    specialize coprime_mul_left (b * b)
03Use earlier factsL14–16

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

  1. L14
    apply coprime_mul_left
  2. L15
    exact hright
  3. L16
    exact hright

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hcoprime
  4. 0004have hright : forall pff_divisor_ffd_square_middle. (exists pff_left_ffd_square_middle. (a) = pff_divisor_ffd_square_middle * pff_left_ffd_square_middle) -> (exists pff_right_ffd_square_middle. (b * b) = pff_divisor_ffd_square_middle * pff_right_ffd_square_middle) -> pff_divisor_ffd_square_middle = 1
  5. 0005specialize coprime_mul_right (a)
  6. 0006specialize coprime_mul_right (b)
  7. 0007specialize coprime_mul_right (b)
  8. 0008apply coprime_mul_right
  9. 0009exact hcoprime
  10. 0010exact hcoprime
  11. 0011specialize coprime_mul_left (a)
  12. 0012specialize coprime_mul_left (a)
  13. 0013specialize coprime_mul_left (b * b)
  14. 0014apply coprime_mul_left
  15. 0015exact hright
  16. 0016exact hright