GI0008

gaussian_signed_square_negated

Negating an arbitrary signed representative leaves its actual square unchanged.

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

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.

The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ p. ∀ n. ∀ s. SignedDifferenceSquare(p,n,s) → SignedDifferenceSquare(n,p,s)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

SignedDifferenceSquare(p,n,s) · 2

Actual proof prerequisites

add_comm · checked external prerequisite
Original expanded first-order statement 
forall p n s. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p))))))) -> (((((n) * (n))) + (((p) * (p)))) = ((s) + (((((n) * (p))) + (((p) * (n)))))))

Complete tactic proof in conservative notation

All 11 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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 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 defined 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