PA00CG · theorem

odd_division_parity_iff

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

An exact quotient-remainder equation with odd coefficient preserves the complete parity classification.

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

∀ p. ∀ q. ∀ r. ∀ n. Odd(p) → n = p · q + r → (Even(n) → Even(q + r)) ∧ (Even(q + r) → Even(n)) ∧ ((Odd(n) → Odd(q + r)) ∧ (Odd(q + r) → Odd(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

PD0009 Even PD0010 Odd

9 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p q r n. (exists pod_odd_multiplier. p = 2 * pod_odd_multiplier + 1) -> n = p * q + r -> (((((exists pod_even_division_n_even. n = 2 * pod_even_division_n_even) -> (exists pod_even_division_qr_even. q + r = 2 * pod_even_division_qr_even)) /\ ((exists pod_even_division_qr_even. q + r = 2 * pod_even_division_qr_even) -> (exists pod_even_division_n_even. n = 2 * pod_even_division_n_even)))) /\ ((((exists pod_odd_division_n_odd. n = 2 * pod_odd_division_n_odd + 1) -> (exists pod_odd_division_qr_odd. q + r = 2 * pod_odd_division_qr_odd + 1)) /\ ((exists pod_odd_division_qr_odd. q + r = 2 * pod_odd_division_qr_odd + 1) -> (exists pod_odd_division_n_odd. n = 2 * pod_odd_division_n_odd + 1)))))

Proof neighborhood

Direct theorem prerequisites

PA00CC odd_division_even_iff PA00CF odd_division_odd_iff

Direct theorem dependents

PA00CK odd_product_division_mod_two

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

21 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.

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 (2)
odd division even iffodd division odd iff
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro r
  4. L4
    intro n
  5. L5
    intro hp
  6. L6
    intro hdivision
02Separate the logical casesL7–7

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

  1. L7
    split
03Use earlier factsL8–17

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

  1. L8
    specialize odd_division_even_iff p
  2. L9
    specialize odd_division_even_iff q
  3. L10
    specialize odd_division_even_iff r
  4. L11
    specialize odd_division_even_iff n
  5. L12
    apply odd_division_even_iff
  6. L13
    exact hp
  7. L14
    exact hdivision
  8. L15
    specialize odd_division_odd_iff p
  9. L16
    specialize odd_division_odd_iff q
  10. L17
    specialize odd_division_odd_iff r
04Use earlier factsL18–21

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

  1. L18
    specialize odd_division_odd_iff n
  2. L19
    apply odd_division_odd_iff
  3. L20
    exact hp
  4. L21
    exact hdivision

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro r
  4. 0004intro n
  5. 0005intro hp
  6. 0006intro hdivision
  7. 0007split
  8. 0008specialize odd_division_even_iff p
  9. 0009specialize odd_division_even_iff q
  10. 0010specialize odd_division_even_iff r
  11. 0011specialize odd_division_even_iff n
  12. 0012apply odd_division_even_iff
  13. 0013exact hp
  14. 0014exact hdivision
  15. 0015specialize odd_division_odd_iff p
  16. 0016specialize odd_division_odd_iff q
  17. 0017specialize odd_division_odd_iff r
  18. 0018specialize odd_division_odd_iff n
  19. 0019apply odd_division_odd_iff
  20. 0020exact hp
  21. 0021exact hdivision