PA00FQ

odd_half_of_mod4_three_exact

Alpha v16 checked-use theorem · independently closed; not Stable

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

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-v16 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

  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