FS005V · theorem body

four_square_signed_dot_negative_zero

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

A negatively oriented dot-product contribution and its centered square cancel modulo the multiplier.

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,0) → ModEq(k,a · e + e · e,0)

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

PD0008 ModEq

In local proof propositions

none
Exact expanded first-order statement
forall k a e. (exists ftcn_left_fssq_dot_negative_source ftcn_right_fssq_dot_negative_source. (a + e) + (k) * ftcn_left_fssq_dot_negative_source = (0) + (k) * ftcn_right_fssq_dot_negative_source) -> (exists ftcn_left_fssq_dot_negative_result ftcn_right_fssq_dot_negative_result. (a * e + e * e) + (k) * ftcn_left_fssq_dot_negative_result = (0) + (k) * ftcn_right_fssq_dot_negative_result)

Proof neighborhood

Direct theorem prerequisites

FS005K four_square_signed_negative_scale_zero

Direct theorem dependents

FS004N four_square_signed_conjugate_negative_blocks FS004O four_square_signed_natural_positive_first_blocks FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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.

Named ingredients (1)
four square signed negative scale zero
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 four_square_signed_negative_scale_zero k
  2. L6
    specialize four_square_signed_negative_scale_zero a
  3. L7
    specialize four_square_signed_negative_scale_zero e
  4. L8
    specialize four_square_signed_negative_scale_zero e
  5. L9
    apply four_square_signed_negative_scale_zero
  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 four_square_signed_negative_scale_zero k
  6. 0006specialize four_square_signed_negative_scale_zero a
  7. 0007specialize four_square_signed_negative_scale_zero e
  8. 0008specialize four_square_signed_negative_scale_zero e
  9. 0009apply four_square_signed_negative_scale_zero
  10. 0010exact ha