PA00FQ

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.

Exact expanded PA statement

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

Structural proof guide

Generated structural guide

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

Use the direct prerequisites mul_add, four_mul_eq_double_double, odd_half_unique as previously established PA formulas.

The proof proceeds by intermediate claims (1), certified simplification (1).

Referenced ingredients

PA000A mul_add PA006R four_mul_eq_double_double PA006T odd_half_unique

Proof neighborhood

Direct dependencies

PA000A mul_add PA006R four_mul_eq_double_double PA006T odd_half_unique

Direct dependents

PA00FR odd_half_odd_iff_mod4_three

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

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.

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 exact 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