PA0056 · theorem

odd_not_even

Stable checked-use theorem · independently closed

No odd natural is even.

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

∀ n. Odd(n) → ¬Even(n)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n. (exists b. n = 2 * b + 1) -> ~(exists a. n = 2 * a)

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

11 script commands · 3 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.

Named ingredients (1)
01Fix variables and assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro ho
  3. L3
    intro he
02Separate the logical casesL4–5

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

  1. L4
    cases ho
  2. L5
    cases he
03Use earlier factsL6–11

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

  1. L6
    specialize even_odd_exclusive_pointwise n
  2. L7
    specialize even_odd_exclusive_pointwise x1
  3. L8
    specialize even_odd_exclusive_pointwise x
  4. L9
    apply even_odd_exclusive_pointwise
  5. L10
    exact he_witness
  6. L11
    exact ho_witness

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro n
  2. 0002intro ho
  3. 0003intro he
  4. 0004cases ho
  5. 0005cases he
  6. 0006specialize even_odd_exclusive_pointwise n
  7. 0007specialize even_odd_exclusive_pointwise x1
  8. 0008specialize even_odd_exclusive_pointwise x
  9. 0009apply even_odd_exclusive_pointwise
  10. 0010exact he_witness
  11. 0011exact ho_witness