FS005U · theorem body

four_square_signed_dot_positive

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

A positively oriented coordinate contributes its centered square modulo the multiplier.

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. ∀ e. ModEq(k,a,e)ModEq(k,a · e,e · e)

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

In local proof propositions

none
Exact expanded first-order statement
forall k a e. (exists ftcn_left_fssq_dot_positive_source ftcn_right_fssq_dot_positive_source. (a) + (k) * ftcn_left_fssq_dot_positive_source = (e) + (k) * ftcn_right_fssq_dot_positive_source) -> (exists ftcn_left_fssq_dot_positive_result ftcn_right_fssq_dot_positive_result. (a * e) + (k) * ftcn_left_fssq_dot_positive_result = (e * e) + (k) * ftcn_right_fssq_dot_positive_result)

Proof neighborhood

Direct theorem prerequisites

mod_eq_mul_right · Stable closed

Direct theorem dependents

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

10 script commands · 2 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 assumptionsL1–4

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro e
  4. L4
    intro ha
02Use earlier factsL5–10

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

  1. L5
    specialize mod_eq_mul_right k
  2. L6
    specialize mod_eq_mul_right a
  3. L7
    specialize mod_eq_mul_right e
  4. L8
    specialize mod_eq_mul_right e
  5. L9
    apply mod_eq_mul_right
  6. L10
    exact ha

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro e
  4. 0004intro ha
  5. 0005specialize mod_eq_mul_right k
  6. 0006specialize mod_eq_mul_right a
  7. 0007specialize mod_eq_mul_right e
  8. 0008specialize mod_eq_mul_right e
  9. 0009apply mod_eq_mul_right
  10. 0010exact ha