FS0020

four_square_descent_zero_centered_remainder_divides

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

A coordinate with centered magnitude zero is an actual natural multiple of the modulus in either signed-remainder 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 k a. (((exists fsd_center_bound_zero. fsd_center_bound_zero + (0 + 0) = k) /\ ((exists fsd_center_lower_zero. a = k * fsd_center_lower_zero + 0) \/ (exists fsd_center_upper_zero. a + 0 = k * fsd_center_upper_zero)))) -> exists q. a = k * q

Constructive proof overview

Generated structural guide

A coordinate with centered magnitude zero is an actual natural multiple of the modulus in either signed-remainder branch.

The unchanged tactic script uses 0 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

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

none

Direct dependents

FS0023 four_square_descent_bounded_centered_quotient_nonzero

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 · 9 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro hcenter
02Separate the logical casesL4–6

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

  1. L4
    cases hcenter
  2. L5
    cases hcenter_right
  3. L6
    cases hcenter_right_left
03Construct an explicit witnessL7–7

Supply the displayed value, then prove that it has the required property.

  1. L7
    exists x
04Calculate and transport equalitiesL8–8

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L8
    rewrite PA3 at hcenter_right_left_witness
05Use earlier factsL9–9

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

  1. L9
    exact hcenter_right_left_witness
06Separate the logical casesL10–10

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

  1. L10
    cases hcenter_right_right
07Construct an explicit witnessL11–11

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists x
08Calculate and transport equalitiesL12–12

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L12
    rewrite PA3 at hcenter_right_right_witness
09Use earlier factsL13–13

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

  1. L13
    exact hcenter_right_right_witness

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro hcenter
  4. 0004cases hcenter
  5. 0005cases hcenter_right
  6. 0006cases hcenter_right_left
  7. 0007exists x
  8. 0008rewrite PA3 at hcenter_right_left_witness
  9. 0009exact hcenter_right_left_witness
  10. 0010cases hcenter_right_right
  11. 0011exists x
  12. 0012rewrite PA3 at hcenter_right_right_witness
  13. 0013exact hcenter_right_right_witness