PA006G

odd_add_even

Stable checked-use theorem · independently closed

An odd natural plus 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 m n. (exists a. m = 2 * a + 1) -> (exists b. n = 2 * b) -> exists c. m + n = 2 * c + 1

Structural proof guide

Generated structural guide

An odd natural plus an even natural is odd.

Use the direct prerequisites mul_add, add_succ_left as previously established PA formulas.

The proof proceeds by case analysis (2), equality transport (2), certified simplification (1).

Referenced ingredients

PA000A mul_add PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA000A mul_add PA000E add_succ_left

Direct dependents

PA00CA even_sum_parity_cases PA00CE odd_sum_iff_opposite_parity

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

10 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 assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro hm
  4. L4
    intro hn
02Separate the logical casesL5–6

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

  1. L5
    cases hm
  2. L6
    cases hn
03Construct an explicit witnessL7–7

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

  1. L7
    exists x + x1
04Calculate and transport equalitiesL8–10

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

  1. L8
    rewrite hm_witness
  2. L9
    rewrite hn_witness
  3. L10
    simp [mul_add, add_succ_left]

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro hm
  4. 0004intro hn
  5. 0005cases hm
  6. 0006cases hn
  7. 0007exists x + x1
  8. 0008rewrite hm_witness
  9. 0009rewrite hn_witness
  10. 0010simp [mul_add, add_succ_left]