GF0032

gaussian_ring_multiply_add_real_positive

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

Guided ordinary-HA distributivity for the actual Gaussian real positive component; no AC search or new tactic.

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 ap an bp bn cp cn dp dn ep en fp fn. ((((((ap) * (((cp) + (ep))))) + (((an) * (((cn) + (en))))))) + (((((bp) * (((dn) + (fn))))) + (((bn) * (((dp) + (fp))))))))=((((((((ap) * (cp))) + (((an) * (cn))))) + (((((bp) * (dn))) + (((bn) * (dp))))))) + (((((((ap) * (ep))) + (((an) * (en))))) + (((((bp) * (fn))) + (((bn) * (fp))))))))

Constructive proof overview

Generated structural guide

Guided ordinary-HA distributivity for the actual Gaussian real positive component; no AC search or new tactic.

The unchanged tactic script uses 4 declared prerequisites and contains 44 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mul_add 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

GF0036 gaussian_ring_raw_multiply_add_distributive

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

44 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro cp
  6. L6
    intro cn
  7. L7
    intro dp
  8. L8
    intro dn
  9. L9
    intro ep
  10. L10
    intro en
02Fix variables and assumptionsL11–12

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

  1. L11
    intro fp
  2. L12
    intro fn
03Calculate and transport equalitiesL13–18

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

  1. L13
    trans ((((ap) * (cp))) + ((((ap) * (ep))) + ((((an) * (cn))) + ((((an) * (en))) + ((((bp) * (dn))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))))
  2. L14
    simp [mul_add, add_assoc]
  3. L15
    trans ((((ap) * (cp))) + ((((an) * (cn))) + ((((bp) * (dn))) + ((((bn) * (dp))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp))))))))))
  4. L16
    congr
  5. L17
    refl
  6. L18
    trans ((((an) * (cn))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (dn))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp)))))))))
04Use earlier factsL19–19

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

  1. L19
    apply four_square_add_swap_right_tail
05Calculate and transport equalitiesL20–25

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

  1. L20
    congr
  2. L21
    refl
  3. L22
    trans ((((bp) * (dn))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))
  4. L23
    trans ((((ap) * (ep))) + ((((bp) * (dn))) + ((((an) * (en))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))
  5. L24
    congr
  6. L25
    refl
06Use earlier factsL26–27

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

  1. L26
    apply four_square_add_swap_right_tail
  2. L27
    apply four_square_add_swap_right_tail
07Calculate and transport equalitiesL28–36

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

  1. L28
    congr
  2. L29
    refl
  3. L30
    trans ((((bn) * (dp))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp)))))))
  4. L31
    trans ((((ap) * (ep))) + ((((bn) * (dp))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp)))))))
  5. L32
    congr
  6. L33
    refl
  7. L34
    trans ((((an) * (en))) + ((((bn) * (dp))) + ((((bp) * (fn))) + (((bn) * (fp))))))
  8. L35
    congr
  9. L36
    refl
08Use earlier factsL37–39

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

  1. L37
    apply four_square_add_swap_right_tail
  2. L38
    apply four_square_add_swap_right_tail
  3. L39
    apply four_square_add_swap_right_tail
09Calculate and transport equalitiesL40–44

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

  1. L40
    congr
  2. L41
    refl
  3. L42
    refl
  4. L43
    symm
  5. L44
    simp [mul_add, add_assoc]

Library-wide reading audit

Original exact command ledger · 44 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro cp
  6. 0006intro cn
  7. 0007intro dp
  8. 0008intro dn
  9. 0009intro ep
  10. 0010intro en
  11. 0011intro fp
  12. 0012intro fn
  13. 0013trans ((((ap) * (cp))) + ((((ap) * (ep))) + ((((an) * (cn))) + ((((an) * (en))) + ((((bp) * (dn))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))))
  14. 0014simp [mul_add, add_assoc]
  15. 0015trans ((((ap) * (cp))) + ((((an) * (cn))) + ((((bp) * (dn))) + ((((bn) * (dp))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp))))))))))
  16. 0016congr
  17. 0017refl
  18. 0018trans ((((an) * (cn))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (dn))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp)))))))))
  19. 0019apply four_square_add_swap_right_tail
  20. 0020congr
  21. 0021refl
  22. 0022trans ((((bp) * (dn))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))
  23. 0023trans ((((ap) * (ep))) + ((((bp) * (dn))) + ((((an) * (en))) + ((((bp) * (fn))) + ((((bn) * (dp))) + (((bn) * (fp))))))))
  24. 0024congr
  25. 0025refl
  26. 0026apply four_square_add_swap_right_tail
  27. 0027apply four_square_add_swap_right_tail
  28. 0028congr
  29. 0029refl
  30. 0030trans ((((bn) * (dp))) + ((((ap) * (ep))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp)))))))
  31. 0031trans ((((ap) * (ep))) + ((((bn) * (dp))) + ((((an) * (en))) + ((((bp) * (fn))) + (((bn) * (fp)))))))
  32. 0032congr
  33. 0033refl
  34. 0034trans ((((an) * (en))) + ((((bn) * (dp))) + ((((bp) * (fn))) + (((bn) * (fp))))))
  35. 0035congr
  36. 0036refl
  37. 0037apply four_square_add_swap_right_tail
  38. 0038apply four_square_add_swap_right_tail
  39. 0039apply four_square_add_swap_right_tail
  40. 0040congr
  41. 0041refl
  42. 0042refl
  43. 0043symm
  44. 0044simp [mul_add, add_assoc]