GI000D

gaussian_signed_square_product

The actual square of a signed product is the product of its two actual natural squares.

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. ∀ q. ∀ m. ∀ s. ∀ t. SignedDifferenceSquare(p,n,s) → SignedDifferenceSquare(q,m,t) → SignedDifferenceSquare(p · q + n · m,p · m + n · q,s · t)

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

Definition DAG

SignedDifferenceSquare(p,n,s) · 3

Actual proof prerequisites

gaussian_signed_product_square_positivegaussian_signed_product_square_negativegaussian_signed_product_square_compensation
Original expanded first-order statement 
forall p n q m s t. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p))))))) -> (((((q) * (q))) + (((m) * (m)))) = ((t) + (((((q) * (m))) + (((m) * (q))))))) -> (((((((((p) * (q))) + (((n) * (m))))) * (((((p) * (q))) + (((n) * (m))))))) + (((((((p) * (m))) + (((n) * (q))))) * (((((p) * (m))) + (((n) * (q)))))))) = ((s * t) + (((((((((p) * (q))) + (((n) * (m))))) * (((((p) * (m))) + (((n) * (q))))))) + (((((((p) * (m))) + (((n) * (q))))) * (((((p) * (q))) + (((n) * (m)))))))))))

Complete tactic proof in conservative notation

All 18 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

18 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.

Named ingredients (3)
gaussian signed product square positivegaussian signed product square compensationgaussian signed product square negative
01Fix variables and assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro m
  5. L5
    intro s
  6. L6
    intro t
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Calculate and transport equalitiesL9–9

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

  1. L9
    trans ((((((((p) * (p))) + (((n) * (n))))) * (((((q) * (q))) + (((m) * (m))))))) + (((((((p) * (n))) + (((n) * (p))))) * (((((q) * (m))) + (((m) * (q))))))))
03Use earlier factsL10–10

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

  1. L10
    apply gaussian_signed_product_square_positive
04Calculate and transport equalitiesL11–11

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

  1. L11
    trans ((s * t) + (((((((((p) * (p))) + (((n) * (n))))) * (((((q) * (m))) + (((m) * (q))))))) + (((((((p) * (n))) + (((n) * (p))))) * (((((q) * (q))) + (((m) * (m))))))))))
05Use earlier factsL12–14

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

  1. L12
    apply gaussian_signed_product_square_compensation
  2. L13
    exact hfirst
  3. L14
    exact hsecond
06Calculate and transport equalitiesL15–17

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

  1. L15
    congr
  2. L16
    refl
  3. L17
    symm
07Use earlier factsL18–18

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

  1. L18
    apply gaussian_signed_product_square_negative

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro m
  5. 0005intro s
  6. 0006intro t
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009trans ((((((((p) * (p))) + (((n) * (n))))) * (((((q) * (q))) + (((m) * (m))))))) + (((((((p) * (n))) + (((n) * (p))))) * (((((q) * (m))) + (((m) * (q))))))))
  10. 0010apply gaussian_signed_product_square_positive
  11. 0011trans ((s * t) + (((((((((p) * (p))) + (((n) * (n))))) * (((((q) * (m))) + (((m) * (q))))))) + (((((((p) * (n))) + (((n) * (p))))) * (((((q) * (q))) + (((m) * (m))))))))))
  12. 0012apply gaussian_signed_product_square_compensation
  13. 0013exact hfirst
  14. 0014exact hsecond
  15. 0015congr
  16. 0016refl
  17. 0017symm
  18. 0018apply gaussian_signed_product_square_negative