TS0021

prime_mod_four_trichotomy

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

Every prime lies in exactly the constructive residue branches two, one modulo four, or three modulo four.

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_ftsc_prime frm_prime_right_ftsc_prime. p = frm_prime_left_ftsc_prime * frm_prime_right_ftsc_prime -> frm_prime_left_ftsc_prime = 1 \/ frm_prime_right_ftsc_prime = 1)) -> (p = 2 \/ ((exists ftsc_four_one_ftsc_prime. (p) = 4 * ftsc_four_one_ftsc_prime + 1) \/ (exists ftsc_four_three_ftsc_prime. (p) = 4 * ftsc_four_three_ftsc_prime + 3)))

Constructive proof overview

Generated structural guide

Every prime lies in exactly the constructive residue branches two, one modulo four, or three modulo four.

The unchanged tactic script uses 2 declared prerequisites and contains 13 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

TS0020 prime_is_two_or_odd odd_mod4_cases Stable theorem; checked-use authorized

Direct dependents

TS0028 prime_is_two_squares_iff_two_or_one_mod_four TS003B prime_mod_four_good_or_three

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

13 script commands · 6 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 is two or odd
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 hparityL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime is two or odd.

  1. L3
    have hparity : p = 2 \/ (exists ftsc_odd_ftsc_prime. (p) = 2 * ftsc_odd_ftsc_prime + 1)
  2. L4
    specialize prime_is_two_or_odd p
  3. L5
    apply prime_is_two_or_odd
  4. L6
    exact hprime
03Separate the logical casesL7–8

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

  1. L7
    cases hparity
  2. L8
    left
04Use earlier factsL9–9

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

  1. L9
    exact hparity_left
05Separate the logical casesL10–10

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

  1. L10
    right
06Use earlier factsL11–13

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

  1. L11
    specialize odd_mod4_cases p
  2. L12
    apply odd_mod4_cases
  3. L13
    exact hparity_right

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003have hparity : p = 2 \/ (exists ftsc_odd_ftsc_prime. (p) = 2 * ftsc_odd_ftsc_prime + 1)
  4. 0004specialize prime_is_two_or_odd p
  5. 0005apply prime_is_two_or_odd
  6. 0006exact hprime
  7. 0007cases hparity
  8. 0008left
  9. 0009exact hparity_left
  10. 0010right
  11. 0011specialize odd_mod4_cases p
  12. 0012apply odd_mod4_cases
  13. 0013exact hparity_right