GI0007

gaussian_signed_square_functional

The actual nonnegative square of a signed pair is unique by cancellative natural arithmetic.

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

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

Definition DAG

SignedDifferenceSquare(p,n,s) · 2

Actual proof prerequisites

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

Complete tactic proof in conservative notation

All 14 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

14 script commands · 4 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 assumptionsL1–6

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 t
  5. L5
    intro hfirst
  6. L6
    intro hsecond
02Use earlier factsL7–10

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

  1. L7
    specialize add_right_cancel s
  2. L8
    specialize add_right_cancel t
  3. L9
    specialize add_right_cancel ((((p) * (n))) + (((n) * (p))))
  4. L10
    apply add_right_cancel
03Calculate and transport equalitiesL11–12

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

  1. L11
    trans ((((p) * (p))) + (((n) * (n))))
  2. L12
    symm
04Use earlier factsL13–14

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

  1. L13
    exact hfirst
  2. L14
    exact hsecond

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro s
  4. 0004intro t
  5. 0005intro hfirst
  6. 0006intro hsecond
  7. 0007specialize add_right_cancel s
  8. 0008specialize add_right_cancel t
  9. 0009specialize add_right_cancel ((((p) * (n))) + (((n) * (p))))
  10. 0010apply add_right_cancel
  11. 0011trans ((((p) * (p))) + (((n) * (n))))
  12. 0012symm
  13. 0013exact hfirst
  14. 0014exact hsecond