TS0002

square_mod_four_zero_or_one

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

Every natural square is constructively either zero or one 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 z. (exists fts_four_square_zero. z * z = 4 * fts_four_square_zero + 0) \/ (exists fts_four_square_one. z * z = 4 * fts_four_square_one + 1)

Constructive proof overview

Generated structural guide

Every natural square is constructively either zero or one modulo four.

The unchanged tactic script uses 3 declared prerequisites and contains 14 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

parity_cases Stable theorem; checked-use authorized TS0000 even_square_is_four_multiple TS0001 odd_square_is_four_multiple_plus_one

Direct dependents

TS0003 sum_two_squares_mod_four_cases

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

14 script commands · 6 reading checkpoints · 0 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)
even square is four multipleodd square is four multiple plus one
01Fix variables and assumptionsL1–1

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

  1. L1
    intro z
02Use earlier factsL2–2

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

  1. L2
    specialize parity_cases z
03Separate the logical casesL3–5

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

  1. L3
    cases parity_cases
  2. L4
    cases parity_cases_witness
  3. L5
    left
04Use earlier factsL6–9

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

  1. L6
    specialize even_square_is_four_multiple z
  2. L7
    specialize even_square_is_four_multiple x
  3. L8
    apply even_square_is_four_multiple
  4. L9
    exact parity_cases_witness_left
05Separate the logical casesL10–10

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

  1. L10
    right
06Use earlier factsL11–14

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

  1. L11
    specialize odd_square_is_four_multiple_plus_one z
  2. L12
    specialize odd_square_is_four_multiple_plus_one x
  3. L13
    apply odd_square_is_four_multiple_plus_one
  4. L14
    exact parity_cases_witness_right

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro z
  2. 0002specialize parity_cases z
  3. 0003cases parity_cases
  4. 0004cases parity_cases_witness
  5. 0005left
  6. 0006specialize even_square_is_four_multiple z
  7. 0007specialize even_square_is_four_multiple x
  8. 0008apply even_square_is_four_multiple
  9. 0009exact parity_cases_witness_left
  10. 0010right
  11. 0011specialize odd_square_is_four_multiple_plus_one z
  12. 0012specialize odd_square_is_four_multiple_plus_one x
  13. 0013apply odd_square_is_four_multiple_plus_one
  14. 0014exact parity_cases_witness_right