PA00C9

odd_multiplier_parity_iff

Alpha v34 checked-use theorem · independently closed; not Stable

An odd multiplier preserves both parity classes exactly.

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 p q. (exists pod_odd_multiplier. p = 2 * pod_odd_multiplier + 1) -> (((((exists pod_even_product_even. p * q = 2 * pod_even_product_even) -> (exists pod_even_factor_even. q = 2 * pod_even_factor_even)) /\ ((exists pod_even_factor_even. q = 2 * pod_even_factor_even) -> (exists pod_even_product_even. p * q = 2 * pod_even_product_even)))) /\ ((((exists pod_odd_product_odd. p * q = 2 * pod_odd_product_odd + 1) -> (exists pod_odd_factor_odd. q = 2 * pod_odd_factor_odd + 1)) /\ ((exists pod_odd_factor_odd. q = 2 * pod_odd_factor_odd + 1) -> (exists pod_odd_product_odd. p * q = 2 * pod_odd_product_odd + 1)))))

Structural proof guide

Generated structural guide

An odd multiplier preserves both parity classes exactly.

Use the direct prerequisites odd_multiplier_even_product_iff, odd_multiplier_odd_product_iff as previously established PA formulas.

The proof proceeds by direct introduction and elimination.

Referenced ingredients

PA00C7 odd_multiplier_even_product_iff PA00C8 odd_multiplier_odd_product_iff

Proof neighborhood

Direct dependencies

PA00C7 odd_multiplier_even_product_iff PA00C8 odd_multiplier_odd_product_iff

Direct dependents

PA00CK odd_product_division_mod_two

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

12 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.

Named ingredients (2)
odd multiplier even product iffodd multiplier odd product iff
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro hp
02Separate the logical casesL4–4

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

  1. L4
    split
03Use earlier factsL5–12

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

  1. L5
    specialize odd_multiplier_even_product_iff p
  2. L6
    specialize odd_multiplier_even_product_iff q
  3. L7
    apply odd_multiplier_even_product_iff
  4. L8
    exact hp
  5. L9
    specialize odd_multiplier_odd_product_iff p
  6. L10
    specialize odd_multiplier_odd_product_iff q
  7. L11
    apply odd_multiplier_odd_product_iff
  8. L12
    exact hp

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro hp
  4. 0004split
  5. 0005specialize odd_multiplier_even_product_iff p
  6. 0006specialize odd_multiplier_even_product_iff q
  7. 0007apply odd_multiplier_even_product_iff
  8. 0008exact hp
  9. 0009specialize odd_multiplier_odd_product_iff p
  10. 0010specialize odd_multiplier_odd_product_iff q
  11. 0011apply odd_multiplier_odd_product_iff
  12. 0012exact hp