PA005A · theorem

even_successor_to_odd

Stable checked-use theorem · independently closed

If a successor is even, its predecessor 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.

Statement with defined notation

∀ n. Even(S n) → 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

2 occurrences

In local proof propositions

PD0010 Odd

1 occurrences

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

Proof neighborhood

Direct theorem prerequisites

PA0057 parity_cases PA0058 successor_odd_of_even PA0059 even_not_odd

Direct theorem dependents

PA005E pow_predecessor_parity_mod

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

17 script commands · 8 reading checkpoints · 1 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 (3)
parity casessuccessor odd of eveneven not odd
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hse
02Use earlier factsL3–3

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

  1. L3
    specialize parity_cases n
03Separate the logical casesL4–6

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

  1. L4
    cases parity_cases
  2. L5
    cases parity_cases_witness
  3. L6
    exfalso
04Establish hsoL7–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply successor odd of even.

  1. L7
  2. L8
    specialize successor_odd_of_even n
  3. L9
    apply successor_odd_of_even
05Construct an explicit witnessL10–10

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

  1. L10
    exists x
06Use earlier factsL11–15

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

  1. L11
    exact parity_cases_witness_left
  2. L12
    specialize even_not_odd (S n)
  3. L13
    apply even_not_odd
  4. L14
    exact hse
  5. L15
    exact hso
07Construct an explicit witnessL16–16

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

  1. L16
    exists x
08Use earlier factsL17–17

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

  1. L17
    exact parity_cases_witness_right

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro n
  2. 0002intro hse
  3. 0003specialize parity_cases n
  4. 0004cases parity_cases
  5. 0005cases parity_cases_witness
  6. 0006exfalso
  7. 0007have hso : Odd(S n)
    Exact native replay linehave hso : exists b. S n = 2 * b + 1
  8. 0008specialize successor_odd_of_even n
  9. 0009apply successor_odd_of_even
  10. 0010exists x
  11. 0011exact parity_cases_witness_left
  12. 0012specialize even_not_odd (S n)
  13. 0013apply even_not_odd
  14. 0014exact hse
  15. 0015exact hso
  16. 0016exists x
  17. 0017exact parity_cases_witness_right