PF000O

pythagorean_parameter_divisor_divides_square

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

Every explicitly witnessed divisor of a parameter also divides its natural square.

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 c a. (exists q. a = c * q) -> exists q. a * a = c * q

Constructive proof overview

Generated structural guide

Every explicitly witnessed divisor of a parameter also divides its natural square.

The unchanged tactic script uses 1 declared prerequisite and contains 8 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

multiple_mul_right Stable theorem; checked-use authorized

Direct dependents

PF000P pythagorean_square_gap_coprime_first_parameter PF000Q pythagorean_square_gap_coprime_second_parameter PF000W pythagorean_primitive_hypotenuse_coprime_first_leg PF001O pythagorean_coprime_square_roots

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

8 script commands · 2 reading checkpoints · 0 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 c
  2. L2
    intro a
  3. L3
    intro hfactor
02Use earlier factsL4–8

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

  1. L4
    specialize multiple_mul_right c
  2. L5
    specialize multiple_mul_right a
  3. L6
    specialize multiple_mul_right a
  4. L7
    apply multiple_mul_right
  5. L8
    exact hfactor

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro c
  2. 0002intro a
  3. 0003intro hfactor
  4. 0004specialize multiple_mul_right c
  5. 0005specialize multiple_mul_right a
  6. 0006specialize multiple_mul_right a
  7. 0007apply multiple_mul_right
  8. 0008exact hfactor