EI0008

eisenstein_coordinate_norm_negation

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

Negating both genuine signed coordinates leaves the Eisenstein norm unchanged.

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 n. (((((((((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))) -> (((((((((an) * (an))) + (((ap) * (ap))))) + (((((bn) * (bn))) + (((bp) * (bp))))))) + (((((an) * (bp))) + (((ap) * (bn)))))) = ((((((((((an) * (ap))) + (((ap) * (an))))) + (((((bn) * (bp))) + (((bp) * (bn))))))) + (((((an) * (bn))) + (((ap) * (bp))))))) + (n)))

Constructive proof overview

Generated structural guide

Negating both genuine signed coordinates leaves the Eisenstein norm unchanged.

The unchanged tactic script uses 4 declared prerequisites and contains 11 exact native proof lines.

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

Proof neighborhood

Direct dependencies

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 mul_comm Stable theorem; checked-use authorized

Direct dependents

none

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

11 script commands · 4 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 assumptionsL1–6

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 n
  6. L6
    intro hnorm
02Calculate and transport equalitiesL7–9

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

  1. L7
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L8
    simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]
  3. L9
    trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + n
03Use earlier factsL10–10

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

  1. L10
    exact hnorm
04Calculate and transport equalitiesL11–11

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

  1. L11
    simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro n
  6. 0006intro hnorm
  7. 0007trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  8. 0008simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]
  9. 0009trans ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn)))))) + n
  10. 0010exact hnorm
  11. 0011simp [add_assoc, add_comm, four_square_add_swap_right_tail, mul_comm]