TS0002 · theorem body

square_mod_four_zero_or_one

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ z. (∃ x. z · z = 4 · x + 0) ∨ Mod4One(z · z)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0011 Mod4One

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

parity_cases · Stable closed TS0000 even_square_is_four_multiple TS0001 odd_square_is_four_multiple_plus_one

Direct theorem dependents

TS0003 sum_two_squares_mod_four_cases

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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