FS001O · theorem body

four_square_descent_norm_bound_forces_smaller_multiplier

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

A centered quaternion norm k·r strictly below k² forces its quotient r to be strictly smaller than k.

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. ∀ r. ∀ z. k · r = z → Lt(z,k · k) → Lt(r,k)

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

PD0002 Lt

In local proof propositions

PD0001 Le
Exact expanded first-order statement
forall k r z. k * r = z -> (exists gap. gap + S z = k * k) -> exists gap. gap + S r = k

Proof neighborhood

Direct theorem prerequisites

le_or_lt · Stable closed mul_le_mul_left · Stable closed lt_not_le · Stable closed

Direct theorem dependents

FS000N four_square_odd_prime_bounded_modular_seed FS0024 four_square_descent_odd_centered_strict_step

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

22 script commands · 5 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.

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 assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro r
  3. L3
    intro z
  4. L4
    intro hnorm
  5. L5
    intro hbound
02Use earlier factsL6–7

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

  1. L6
    specialize le_or_lt k
  2. L7
    specialize le_or_lt r
03Separate the logical casesL8–9

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

  1. L8
    cases le_or_lt
  2. L9
    exfalso
04Establish hscaledL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L10
    have hscaled : Le(k · k,k · r)Definitions: Le(k · k,k · r)Original native command in the exact edition
  2. L11
    specialize mul_le_mul_left k
  3. L12
    specialize mul_le_mul_left r
  4. L13
    specialize mul_le_mul_left k
  5. L14
    apply mul_le_mul_left
  6. L15
    exact le_or_lt_left
  7. L16
    rewrite hnorm at hscaled
  8. L17
    specialize lt_not_le z
  9. L18
    specialize lt_not_le (k * k)
  10. L19
    apply lt_not_le
05Use earlier factsL20–22

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

  1. L20
    exact hbound
  2. L21
    exact hscaled
  3. L22
    exact le_or_lt_right

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro k
  2. 0002intro r
  3. 0003intro z
  4. 0004intro hnorm
  5. 0005intro hbound
  6. 0006specialize le_or_lt k
  7. 0007specialize le_or_lt r
  8. 0008cases le_or_lt
  9. 0009exfalso
  10. 0010have hscaled : Le(k · k,k · r)
    Exact native replay linehave hscaled : exists t. t + k * k = k * r
  11. 0011specialize mul_le_mul_left k
  12. 0012specialize mul_le_mul_left r
  13. 0013specialize mul_le_mul_left k
  14. 0014apply mul_le_mul_left
  15. 0015exact le_or_lt_left
  16. 0016rewrite hnorm at hscaled
  17. 0017specialize lt_not_le z
  18. 0018specialize lt_not_le (k * k)
  19. 0019apply lt_not_le
  20. 0020exact hbound
  21. 0021exact hscaled
  22. 0022exact le_or_lt_right