TS003B

prime_mod_four_good_or_three

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The constructive prime residue trichotomy splits into one represented-prime branch and one three-modulo-four branch.

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 first-order arithmetic statement

forall p. ((~(p = 1) /\ forall frm_prime_left_ftsp_p frm_prime_right_ftsp_p. p = frm_prime_left_ftsp_p * frm_prime_right_ftsp_p -> frm_prime_left_ftsp_p = 1 \/ frm_prime_right_ftsp_p = 1)) -> ((p = 2 \/ exists k. p = 4 * k + 1) \/ exists k. p = 4 * k + 3)

Constructive proof overview

Generated structural guide

The constructive prime residue trichotomy splits into one represented-prime branch and one three-modulo-four branch.

The unchanged tactic script uses 1 declared prerequisite and contains 16 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

TS0021 prime_mod_four_trichotomy

Direct dependents

TS003C all_bad_prime_even_two_square_sufficiency_bounded

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

16 script commands · 8 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 (1)
prime mod four trichotomy
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hprime
02Establish hclassesL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime mod four trichotomy.

  1. L3
    have hclasses : (p = 2 \/ ((exists k. p = 4 * k + 1) \/ (exists k. p = 4 * k + 3)))
  2. L4
    specialize prime_mod_four_trichotomy p
  3. L5
    apply prime_mod_four_trichotomy
  4. L6
    exact hprime
03Separate the logical casesL7–9

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

  1. L7
    cases hclasses
  2. L8
    left
  3. L9
    left
04Use earlier factsL10–10

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

  1. L10
    exact hclasses_left
05Separate the logical casesL11–13

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

  1. L11
    cases hclasses_right
  2. L12
    left
  3. L13
    right
06Use earlier factsL14–14

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

  1. L14
    exact hclasses_right_left
07Separate the logical casesL15–15

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

  1. L15
    right
08Use earlier factsL16–16

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

  1. L16
    exact hclasses_right_right

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003have hclasses : (p = 2 \/ ((exists k. p = 4 * k + 1) \/ (exists k. p = 4 * k + 3)))
  4. 0004specialize prime_mod_four_trichotomy p
  5. 0005apply prime_mod_four_trichotomy
  6. 0006exact hprime
  7. 0007cases hclasses
  8. 0008left
  9. 0009left
  10. 0010exact hclasses_left
  11. 0011cases hclasses_right
  12. 0012left
  13. 0013right
  14. 0014exact hclasses_right_left
  15. 0015right
  16. 0016exact hclasses_right_right