FS0046

four_square_bounded_multiple_is_zero

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

A natural strictly below a modulus can be divisible by that modulus only when it is zero.

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 a. (exists fsri_gap_bounded_multiple. fsri_gap_bounded_multiple + S (a) = (p)) -> (exists fsri_factor_bounded_multiple. (a) = (p) * fsri_factor_bounded_multiple) -> a = 0

Constructive proof overview

Generated structural guide

A natural strictly below a modulus can be divisible by that modulus only when it is zero.

The unchanged tactic script uses 2 declared prerequisites and contains 17 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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

eq_decidable Stable theorem; checked-use authorized bounded_nonzero_not_divides Alpha theorem; checked-use authorized

Direct dependents

FS0049 four_square_ordered_half_square_injective

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

17 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hbound
  4. L4
    intro hmultiple
02Establish hcaseL5–8

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hcase : a = 0 \/ ~(a = 0)
  2. L6
    specialize eq_decidable a
  3. L7
    specialize eq_decidable 0
  4. L8
    exact eq_decidable
03Separate the logical casesL9–9

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

  1. L9
    cases hcase
04Use earlier factsL10–10

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

  1. L10
    exact hcase_left
05Separate the logical casesL11–11

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

  1. L11
    exfalso
06Use earlier factsL12–17

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

  1. L12
    specialize bounded_nonzero_not_divides p
  2. L13
    specialize bounded_nonzero_not_divides a
  3. L14
    apply bounded_nonzero_not_divides
  4. L15
    exact hcase_right
  5. L16
    exact hbound
  6. L17
    exact hmultiple

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hbound
  4. 0004intro hmultiple
  5. 0005have hcase : a = 0 \/ ~(a = 0)
  6. 0006specialize eq_decidable a
  7. 0007specialize eq_decidable 0
  8. 0008exact eq_decidable
  9. 0009cases hcase
  10. 0010exact hcase_left
  11. 0011exfalso
  12. 0012specialize bounded_nonzero_not_divides p
  13. 0013specialize bounded_nonzero_not_divides a
  14. 0014apply bounded_nonzero_not_divides
  15. 0015exact hcase_right
  16. 0016exact hbound
  17. 0017exact hmultiple