PF0013

pythagorean_coordinate_parity_choice

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

Every natural has a constructive disjunction of explicit even and odd witnesses.

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. (exists q. a = 2 * q) \/ (exists q. a = 2 * q + 1)

Constructive proof overview

Generated structural guide

Every natural has a constructive disjunction of explicit even and odd witnesses.

The unchanged tactic script uses 1 declared prerequisite and contains 10 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

parity_cases Stable theorem; checked-use authorized

Direct dependents

PF0014 pythagorean_primitive_legs_opposite_parity PF001P pythagorean_even_square_has_even_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

10 script commands · 8 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 assumptionsL1–1

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

  1. L1
    intro a
02Use earlier factsL2–2

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

  1. L2
    specialize parity_cases a
03Separate the logical casesL3–5

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

  1. L3
    cases parity_cases
  2. L4
    cases parity_cases_witness
  3. L5
    left
04Construct an explicit witnessL6–6

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists x
05Use earlier factsL7–7

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

  1. L7
    exact parity_cases_witness_left
06Separate the logical casesL8–8

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

  1. L8
    right
07Construct an explicit witnessL9–9

Supply the displayed value, then prove that it has the required property.

  1. L9
    exists x
08Use earlier factsL10–10

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

  1. L10
    exact parity_cases_witness_right

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002specialize parity_cases a
  3. 0003cases parity_cases
  4. 0004cases parity_cases_witness
  5. 0005left
  6. 0006exists x
  7. 0007exact parity_cases_witness_left
  8. 0008right
  9. 0009exists x
  10. 0010exact parity_cases_witness_right