PA00FQ · theorem

odd_half_of_mod4_three_exact

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

The half of a fixed odd number congruent to three modulo four is exactly 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

forall p h a. p = 2 * h + 1 -> p = 4 * a + 3 -> h = 2 * a + 1

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

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p h a. p = 2 * h + 1 -> p = 4 * a + 3 -> h = 2 * a + 1

Proof neighborhood

Direct theorem prerequisites

PA000A mul_add PA006R four_mul_eq_double_double PA006T odd_half_unique

Direct theorem dependents

PA00FR odd_half_odd_iff_mod4_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

19 script commands · 3 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 (2)
four mul eq double doubleodd half unique
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro a
  4. L4
    intro hp
  5. L5
    intro hfour
02Establish hcanonicalL6–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four mul eq double double.

  1. L6
    have hcanonical : p = 2 * (2 * a + 1) + 1
  2. L7
    trans 4 * a + 3
  3. L8
    exact hfour
  4. L9
    simp [mul_add]
  5. L10
    congr
  6. L11
    congr
  7. L12
    congr
  8. L13
    apply four_mul_eq_double_double
  9. L14
    specialize odd_half_unique p
  10. L15
    specialize odd_half_unique h
03Use earlier factsL16–19

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

  1. L16
    specialize odd_half_unique (2 * a + 1)
  2. L17
    apply odd_half_unique
  3. L18
    exact hp
  4. L19
    exact hcanonical

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro hp
  5. 0005intro hfour
  6. 0006have hcanonical : p = 2 * (2 * a + 1) + 1
  7. 0007trans 4 * a + 3
  8. 0008exact hfour
  9. 0009simp [mul_add]
  10. 0010congr
  11. 0011congr
  12. 0012congr
  13. 0013apply four_mul_eq_double_double
  14. 0014specialize odd_half_unique p
  15. 0015specialize odd_half_unique h
  16. 0016specialize odd_half_unique (2 * a + 1)
  17. 0017apply odd_half_unique
  18. 0018exact hp
  19. 0019exact hcanonical