EI000D

eisenstein_norm_square_balance

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

Actual natural coordinate squares give the exact signed cross-term balance sa+sb=ENorm+ab.

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 sa sb N. (((((ap) * (ap))) + (((an) * (an)))) = ((sa) + (((((ap) * (an))) + (((an) * (ap))))))) -> (((((bp) * (bp))) + (((bn) * (bn)))) = ((sb) + (((((bp) * (bn))) + (((bn) * (bp))))))) -> (((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp)))))) = ((((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))) + (N))) -> (sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))) = (((((ap) * (bp))) + (((an) * (bn))))) + N

Constructive proof overview

Generated structural guide

Actual natural coordinate squares give the exact signed cross-term balance sa+sb=ENorm+ab.

The unchanged tactic script uses 4 declared prerequisites and contains 19 exact native proof lines.

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

Proof neighborhood

Direct dependencies

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

EI000F eisenstein_norm_weighted_square_identity

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

19 script commands · 5 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 sa
  6. L6
    intro sb
  7. L7
    intro N
  8. L8
    intro ha
  9. L9
    intro hb
  10. L10
    intro hnorm
02Use earlier factsL11–14

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

  1. L11
    specialize add_right_cancel ((sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L12
    specialize add_right_cancel ((((((ap) * (bp))) + (((an) * (bn))))) + N)
  3. L13
    specialize add_right_cancel ((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))
  4. L14
    apply add_right_cancel
03Calculate and transport equalitiesL15–17

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

  1. L15
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L16
    simp [ha, hb, add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L17
    trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + N
04Use earlier factsL18–18

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

  1. L18
    exact hnorm
05Calculate and transport equalitiesL19–19

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

  1. L19
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro sa
  6. 0006intro sb
  7. 0007intro N
  8. 0008intro ha
  9. 0009intro hb
  10. 0010intro hnorm
  11. 0011specialize add_right_cancel ((sa + sb) + (((((ap) * (bn))) + (((an) * (bp))))))
  12. 0012specialize add_right_cancel ((((((ap) * (bp))) + (((an) * (bn))))) + N)
  13. 0013specialize add_right_cancel ((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))
  14. 0014apply add_right_cancel
  15. 0015trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  16. 0016simp [ha, hb, add_assoc, add_comm, four_square_add_swap_right_tail]
  17. 0017trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + N
  18. 0018exact hnorm
  19. 0019simp [add_assoc, add_comm, four_square_add_swap_right_tail]