GI0008

gaussian_signed_square_negated

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Negating an arbitrary signed representative leaves its actual square unchanged.

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 p n s. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p))))))) -> (((((n) * (n))) + (((p) * (p)))) = ((s) + (((((n) * (p))) + (((p) * (n)))))))

Constructive proof overview

Generated structural guide

Negating an arbitrary signed representative leaves its actual square unchanged.

The unchanged tactic script uses 1 declared prerequisite and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_comm Stable theorem; checked-use authorized

Direct dependents

GI0011 gaussian_signed_square_difference_compensation GI001C gaussian_signed_norm_conjugate

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

11 script commands · 7 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro s
  4. L4
    intro hsquare
02Calculate and transport equalitiesL5–5

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L5
    trans ((((p) * (p))) + (((n) * (n))))
03Use earlier factsL6–6

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

  1. L6
    apply add_comm
04Calculate and transport equalitiesL7–7

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    trans ((s) + (((((p) * (n))) + (((n) * (p))))))
05Use earlier factsL8–8

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

  1. L8
    exact hsquare
06Calculate and transport equalitiesL9–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    congr
  2. L10
    refl
07Use earlier factsL11–11

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

  1. L11
    apply add_comm

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro s
  4. 0004intro hsquare
  5. 0005trans ((((p) * (p))) + (((n) * (n))))
  6. 0006apply add_comm
  7. 0007trans ((s) + (((((p) * (n))) + (((n) * (p))))))
  8. 0008exact hsquare
  9. 0009congr
  10. 0010refl
  11. 0011apply add_comm