PA0058

successor_odd_of_even

Stable checked-use theorem · independently closed

The successor of an even natural is odd.

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 PA statement

forall n. (exists a. n = 2 * a) -> exists b. S n = 2 * b + 1

Structural proof guide

Generated structural guide

The successor of an even natural is odd.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (1), equality transport (1), certified simplification (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA005A even_successor_to_odd

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

6 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 assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro he
02Separate the logical casesL3–3

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

  1. L3
    cases he
03Construct an explicit witnessL4–4

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

  1. L4
    exists x
04Calculate and transport equalitiesL5–6

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L5
    rewrite he_witness
  2. L6
    simp

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro n
  2. 0002intro he
  3. 0003cases he
  4. 0004exists x
  5. 0005rewrite he_witness
  6. 0006simp