PA00FR · theorem

odd_half_odd_iff_mod4_three

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

For a fixed odd decomposition, a modulo-four-three modulus is equivalent to an odd 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 → (Odd(h) → Mod4Three(p)) ∧ (Mod4Three(p) → Odd(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

PD0010 Odd PD0012 Mod4Three

4 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p h. p = 2 * h + 1 -> ((((exists poh_odd_half. h = 2 * poh_odd_half + 1) -> (exists poh_three_modulus. p = 4 * poh_three_modulus + 3)) /\ ((exists poh_three_modulus. p = 4 * poh_three_modulus + 3) -> (exists poh_odd_half. h = 2 * poh_odd_half + 1))))

Proof neighborhood

Direct theorem prerequisites

PA000A mul_add PA006R four_mul_eq_double_double PA00FQ odd_half_of_mod4_three_exact

Direct theorem dependents

PA00FU qres_opposite_status_from_mod_four_three

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

24 script commands · 11 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 three 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 hodd
04Separate the logical casesL6–6

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

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

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

  1. L7
    exists x
06Calculate and transport equalitiesL8–14

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

  1. L8
    rewrite hp
  2. L9
    rewrite hodd_witness
  3. L10
    simp [mul_add]
  4. L11
    congr
  5. L12
    congr
  6. L13
    congr
  7. L14
    symm
07Use earlier factsL15–15

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

  1. L15
    apply four_mul_eq_double_double
08Fix variables and assumptionsL16–16

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

  1. L16
    intro hthree
09Separate the logical casesL17–17

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

  1. L17
    cases hthree
10Construct an explicit witnessL18–18

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

  1. L18
    exists x
11Use earlier factsL19–24

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

  1. L19
    specialize odd_half_of_mod4_three_exact p
  2. L20
    specialize odd_half_of_mod4_three_exact h
  3. L21
    specialize odd_half_of_mod4_three_exact x
  4. L22
    apply odd_half_of_mod4_three_exact
  5. L23
    exact hp
  6. L24
    exact hthree_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hp
  4. 0004split
  5. 0005intro hodd
  6. 0006cases hodd
  7. 0007exists x
  8. 0008rewrite hp
  9. 0009rewrite hodd_witness
  10. 0010simp [mul_add]
  11. 0011congr
  12. 0012congr
  13. 0013congr
  14. 0014symm
  15. 0015apply four_mul_eq_double_double
  16. 0016intro hthree
  17. 0017cases hthree
  18. 0018exists x
  19. 0019specialize odd_half_of_mod4_three_exact p
  20. 0020specialize odd_half_of_mod4_three_exact h
  21. 0021specialize odd_half_of_mod4_three_exact x
  22. 0022apply odd_half_of_mod4_three_exact
  23. 0023exact hp
  24. 0024exact hthree_witness