FS0020 · theorem body

four_square_descent_zero_centered_remainder_divides

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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

∀ k. ∀ a. Le(0 + 0,k) ∧ ((∃ x. a = k · x + 0) ∨ Dvd(k,a + 0)) → Dvd(k,a)

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

PD0001 Le PD0003 Dvd

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

FS0023 four_square_descent_bounded_centered_quotient_nonzero

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

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.

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

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 defined 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