FS001P · theorem body

four_square_descent_matching_parity_sum_even

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. Even(a) ∧ Even(b) ∨ Odd(a) ∧ Odd(b) → Even(a + 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

PD0009 Even PD0010 Odd

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

even_sum_iff_same_parity · Alpha closed

Direct theorem dependents

FS001T four_square_descent_even_multiplier_matching_parity_halving

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

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.

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 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 defined 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