GI000D

gaussian_signed_square_product

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

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

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 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)))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GI000A gaussian_signed_product_square_positive GI000B gaussian_signed_product_square_negative GI000C gaussian_signed_product_square_compensation

Direct dependents

GI001A gaussian_signed_norm_product GI001F gaussian_signed_square_scaled

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

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.

Named ingredients (3)
gaussian signed product square positivegaussian signed product square negativegaussian signed product square compensation
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 exact 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