PF000I

pythagorean_parameter_odd_square

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

The square of a witnessed odd Euclidean parameter has an explicit odd half.

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 m. (exists q. m = 2 * q + 1) -> exists q. m * m = 2 * q + 1

Constructive proof overview

Generated structural guide

The square of a witnessed odd Euclidean parameter has an explicit odd half.

The unchanged tactic script uses 1 declared prerequisite and contains 7 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

odd_mul_odd Stable theorem; checked-use authorized

Direct dependents

PF000J pythagorean_even_odd_square_gap_odd PF000K pythagorean_odd_even_square_gap_odd PF000M pythagorean_opposite_parity_hypotenuse_odd PF001P pythagorean_even_square_has_even_root PF002C fermat_four_even_square_root

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

7 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 assumptionsL1–2

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

  1. L1
    intro m
  2. L2
    intro hodd
02Use earlier factsL3–7

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

  1. L3
    specialize odd_mul_odd m
  2. L4
    specialize odd_mul_odd m
  3. L5
    apply odd_mul_odd
  4. L6
    exact hodd
  5. L7
    exact hodd

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro m
  2. 0002intro hodd
  3. 0003specialize odd_mul_odd m
  4. 0004specialize odd_mul_odd m
  5. 0005apply odd_mul_odd
  6. 0006exact hodd
  7. 0007exact hodd