FS001P

four_square_descent_matching_parity_sum_even

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

A pair of natural coordinates with matching constructive parity has an explicitly even sum.

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. ((((exists fsd_even_first_sum. a = 2 * fsd_even_first_sum) /\ (exists fsd_even_second_sum. b = 2 * fsd_even_second_sum)) \/ ((exists fsd_odd_first_sum. a = 2 * fsd_odd_first_sum + 1) /\ (exists fsd_odd_second_sum. b = 2 * fsd_odd_second_sum + 1)))) -> exists u. a + b = 2 * u

Constructive proof overview

Generated structural guide

A pair of natural coordinates with matching constructive parity has an explicitly even sum.

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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

even_sum_iff_same_parity Alpha theorem; checked-use authorized

Direct dependents

FS001T four_square_descent_even_multiplier_matching_parity_halving

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 · 4 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 a
  2. L2
    intro b
  3. L3
    intro hmatching
02Use earlier factsL4–5

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

  1. L4
    specialize even_sum_iff_same_parity a
  2. L5
    specialize even_sum_iff_same_parity b
03Separate the logical casesL6–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    cases even_sum_iff_same_parity
04Use earlier factsL7–8

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

  1. L7
    apply even_sum_iff_same_parity_right
  2. L8
    exact hmatching

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hmatching
  4. 0004specialize even_sum_iff_same_parity a
  5. 0005specialize even_sum_iff_same_parity b
  6. 0006cases even_sum_iff_same_parity
  7. 0007apply even_sum_iff_same_parity_right
  8. 0008exact hmatching