GI0010

gaussian_signed_square_sum_compensation

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

Exact signed cross-term compensation for a squared sum, with no unproved norm premise.

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

Constructive proof overview

Generated structural guide

Exact signed cross-term compensation for a squared sum, with no unproved norm premise.

The unchanged tactic script uses 6 declared prerequisites and contains 29 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GI000E gaussian_signed_sum_square_positive GI000F gaussian_signed_sum_square_negative add_right_cancel Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

GI0011 gaussian_signed_square_difference_compensation 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

29 script commands · 9 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 sum square positivegaussian signed sum square negative
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 hsum
02Use earlier factsL11–14

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

  1. L11
    specialize add_right_cancel ((z) + (((((((p) * (m))) + (((n) * (q))))) + (((((p) * (m))) + (((n) * (q))))))))
  2. L12
    specialize add_right_cancel ((s + t) + (((((((p) * (q))) + (((n) * (m))))) + (((((p) * (q))) + (((n) * (m))))))))
  3. L13
    specialize add_right_cancel ((((((p) * (n))) + (((n) * (p))))) + (((((q) * (m))) + (((m) * (q))))))
  4. L14
    apply add_right_cancel
03Calculate and transport equalitiesL15–20

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

  1. L15
    trans ((z) + (((((((((p) * (n))) + (((n) * (p))))) + (((((q) * (m))) + (((m) * (q))))))) + (((((((p) * (m))) + (((n) * (q))))) + (((((p) * (m))) + (((n) * (q))))))))))
  2. L16
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L17
    trans ((z) + (((((((p) + (q))) * (((n) + (m))))) + (((((n) + (m))) * (((p) + (q))))))))
  4. L18
    congr
  5. L19
    refl
  6. L20
    symm
04Use earlier factsL21–21

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

  1. L21
    apply gaussian_signed_sum_square_negative
05Calculate and transport equalitiesL22–23

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

  1. L22
    trans ((((((p) + (q))) * (((p) + (q))))) + (((((n) + (m))) * (((n) + (m))))))
  2. L23
    symm
06Use earlier factsL24–24

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

  1. L24
    exact hsum
07Calculate and transport equalitiesL25–25

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

  1. L25
    trans ((((((((p) * (p))) + (((n) * (n))))) + (((((q) * (q))) + (((m) * (m))))))) + (((((((p) * (q))) + (((n) * (m))))) + (((((p) * (q))) + (((n) * (m))))))))
08Use earlier factsL26–26

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

  1. L26
    apply gaussian_signed_sum_square_positive
09Calculate and transport equalitiesL27–29

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

  1. L27
    rewrite hfirst
  2. L28
    rewrite hsecond
  3. L29
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 29 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 hsum
  11. 0011specialize add_right_cancel ((z) + (((((((p) * (m))) + (((n) * (q))))) + (((((p) * (m))) + (((n) * (q))))))))
  12. 0012specialize add_right_cancel ((s + t) + (((((((p) * (q))) + (((n) * (m))))) + (((((p) * (q))) + (((n) * (m))))))))
  13. 0013specialize add_right_cancel ((((((p) * (n))) + (((n) * (p))))) + (((((q) * (m))) + (((m) * (q))))))
  14. 0014apply add_right_cancel
  15. 0015trans ((z) + (((((((((p) * (n))) + (((n) * (p))))) + (((((q) * (m))) + (((m) * (q))))))) + (((((((p) * (m))) + (((n) * (q))))) + (((((p) * (m))) + (((n) * (q))))))))))
  16. 0016simp [add_assoc, add_comm, four_square_add_swap_right_tail]
  17. 0017trans ((z) + (((((((p) + (q))) * (((n) + (m))))) + (((((n) + (m))) * (((p) + (q))))))))
  18. 0018congr
  19. 0019refl
  20. 0020symm
  21. 0021apply gaussian_signed_sum_square_negative
  22. 0022trans ((((((p) + (q))) * (((p) + (q))))) + (((((n) + (m))) * (((n) + (m))))))
  23. 0023symm
  24. 0024exact hsum
  25. 0025trans ((((((((p) * (p))) + (((n) * (n))))) + (((((q) * (q))) + (((m) * (m))))))) + (((((((p) * (q))) + (((n) * (m))))) + (((((p) * (q))) + (((n) * (m))))))))
  26. 0026apply gaussian_signed_sum_square_positive
  27. 0027rewrite hfirst
  28. 0028rewrite hsecond
  29. 0029simp [add_assoc, add_comm, four_square_add_swap_right_tail]