PF0026 · theorem body

fermat_four_coprime_squares

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ a. ∀ b. Coprime(a,b) → Coprime(a · a,b · b)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0005 Coprime

In local proof propositions

PD0005 Coprime
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

coprime_mul_left · Stable closed coprime_mul_right · Stable closed

Direct theorem dependents

PF002I fermat_four_primitive_square_triangle

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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 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 : Coprime(a,b · b)Definitions: Coprime(a,b · b)Original native command in the exact edition
  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 defined command ledger · 16 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hcoprime
  4. 0004have hright : Coprime(a,b · b)
    Exact native replay linehave 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