GI0011

gaussian_signed_square_difference_compensation

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

Exact signed cross-term compensation for a squared difference; sign reversal preserves the same actual square.

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 z. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p))))))) -> (((((q) * (q))) + (((m) * (m)))) = ((t) + (((((q) * (m))) + (((m) * (q))))))) -> (((((((p) + (m))) * (((p) + (m))))) + (((((n) + (q))) * (((n) + (q)))))) = ((z) + (((((((p) + (m))) * (((n) + (q))))) + (((((n) + (q))) * (((p) + (m))))))))) -> ((z) + (((((((p) * (q))) + (((n) * (m))))) + (((((p) * (q))) + (((n) * (m)))))))) = ((s + t) + (((((((p) * (m))) + (((n) * (q))))) + (((((p) * (m))) + (((n) * (q))))))))

Constructive proof overview

Generated structural guide

Exact signed cross-term compensation for a squared difference; sign reversal preserves the same actual square.

The unchanged tactic script uses 2 declared prerequisites and contains 25 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GI0010 gaussian_signed_square_sum_compensation GI0008 gaussian_signed_square_negated

Direct dependents

GI0019 gaussian_signed_square_lagrange

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

25 script commands · 3 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 (2)
gaussian signed square sum compensationgaussian signed square negated
01Fix variables and assumptionsL1–10

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 z
  8. L8
    intro hfirst
  9. L9
    intro hsecond
  10. L10
    intro hdifference
02Use earlier factsL11–20

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

  1. L11
    specialize gaussian_signed_square_sum_compensation p
  2. L12
    specialize gaussian_signed_square_sum_compensation n
  3. L13
    specialize gaussian_signed_square_sum_compensation m
  4. L14
    specialize gaussian_signed_square_sum_compensation q
  5. L15
    specialize gaussian_signed_square_sum_compensation s
  6. L16
    specialize gaussian_signed_square_sum_compensation t
  7. L17
    specialize gaussian_signed_square_sum_compensation z
  8. L18
    apply gaussian_signed_square_sum_compensation
  9. L19
    exact hfirst
  10. L20
    specialize gaussian_signed_square_negated q
03Use earlier factsL21–25

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

  1. L21
    specialize gaussian_signed_square_negated m
  2. L22
    specialize gaussian_signed_square_negated t
  3. L23
    apply gaussian_signed_square_negated
  4. L24
    exact hsecond
  5. L25
    exact hdifference

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro m
  5. 0005intro s
  6. 0006intro t
  7. 0007intro z
  8. 0008intro hfirst
  9. 0009intro hsecond
  10. 0010intro hdifference
  11. 0011specialize gaussian_signed_square_sum_compensation p
  12. 0012specialize gaussian_signed_square_sum_compensation n
  13. 0013specialize gaussian_signed_square_sum_compensation m
  14. 0014specialize gaussian_signed_square_sum_compensation q
  15. 0015specialize gaussian_signed_square_sum_compensation s
  16. 0016specialize gaussian_signed_square_sum_compensation t
  17. 0017specialize gaussian_signed_square_sum_compensation z
  18. 0018apply gaussian_signed_square_sum_compensation
  19. 0019exact hfirst
  20. 0020specialize gaussian_signed_square_negated q
  21. 0021specialize gaussian_signed_square_negated m
  22. 0022specialize gaussian_signed_square_negated t
  23. 0023apply gaussian_signed_square_negated
  24. 0024exact hsecond
  25. 0025exact hdifference