FS005V

four_square_signed_dot_negative_zero

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 10 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

Proof neighborhood

Direct dependencies

FS005K four_square_signed_negative_scale_zero

Direct 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

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

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.

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 exact 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