PA00FJ · theorem

odd_half_even_iff_mod4_one

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

For a fixed odd decomposition, a modulo-four-one modulus is equivalent to an even half.

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. ∀ h. p = 2 · h + 1 → (Even(h) → Mod4One(p)) ∧ (Mod4One(p) → Even(h))

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 PD0011 Mod4One

4 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p h. p = 2 * h + 1 -> ((((exists poh_even_half. h = 2 * poh_even_half) -> (exists poh_one_modulus. p = 4 * poh_one_modulus + 1)) /\ ((exists poh_one_modulus. p = 4 * poh_one_modulus + 1) -> (exists poh_even_half. h = 2 * poh_even_half))))

Proof neighborhood

Direct theorem prerequisites

PA006R four_mul_eq_double_double PA00FI odd_half_of_mod4_one_exact

Direct theorem dependents

PA00FO qres_same_status_from_mod_four_one

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

22 script commands · 12 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)
four mul eq double doubleodd half of mod4 one exact
01Fix variables and assumptionsL1–3

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

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

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

  1. L4
    split
03Fix variables and assumptionsL5–5

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

  1. L5
    intro heven
04Separate the logical casesL6–6

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

  1. L6
    cases heven
05Construct an explicit witnessL7–7

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

  1. L7
    exists x
06Calculate and transport equalitiesL8–11

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

  1. L8
    rewrite hp
  2. L9
    rewrite heven_witness
  3. L10
    congr
  4. L11
    symm
07Use earlier factsL12–12

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

  1. L12
    apply four_mul_eq_double_double
08Calculate and transport equalitiesL13–13

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

  1. L13
    refl
09Fix variables and assumptionsL14–14

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

  1. L14
    intro hone
10Separate the logical casesL15–15

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

  1. L15
    cases hone
11Construct an explicit witnessL16–16

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

  1. L16
    exists x
12Use earlier factsL17–22

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

  1. L17
    specialize odd_half_of_mod4_one_exact p
  2. L18
    specialize odd_half_of_mod4_one_exact h
  3. L19
    specialize odd_half_of_mod4_one_exact x
  4. L20
    apply odd_half_of_mod4_one_exact
  5. L21
    exact hp
  6. L22
    exact hone_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hp
  4. 0004split
  5. 0005intro heven
  6. 0006cases heven
  7. 0007exists x
  8. 0008rewrite hp
  9. 0009rewrite heven_witness
  10. 0010congr
  11. 0011symm
  12. 0012apply four_mul_eq_double_double
  13. 0013refl
  14. 0014intro hone
  15. 0015cases hone
  16. 0016exists x
  17. 0017specialize odd_half_of_mod4_one_exact p
  18. 0018specialize odd_half_of_mod4_one_exact h
  19. 0019specialize odd_half_of_mod4_one_exact x
  20. 0020apply odd_half_of_mod4_one_exact
  21. 0021exact hp
  22. 0022exact hone_witness