PF0013 · theorem body

pythagorean_coordinate_parity_choice

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ a. Even(a) ∨ Odd(a)

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

Proof neighborhood

Direct theorem prerequisites

parity_cases · Stable closed

Direct theorem dependents

PF0014 pythagorean_primitive_legs_opposite_parity PF001P pythagorean_even_square_has_even_root

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

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.

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 assumptionsL1–1

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 defined 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