EI0002

eisenstein_natural_norm_gap_value

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

Ordering the coordinates gives the explicit subtraction-free norm a²+ad+d².

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 a d n. ((((a) * (a)) + ((a + d) * (a + d))) = (((a) * (a + d)) + (n))) -> n = a * a + a * d + d * d

Constructive proof overview

Generated structural guide

Ordering the coordinates gives the explicit subtraction-free norm a²+ad+d².

The unchanged tactic script uses 9 declared prerequisites and contains 12 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_left_cancel Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized natural_mul_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

EI0004 eisenstein_natural_norm_zero

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

12 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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro d
  3. L3
    intro n
  4. L4
    intro hnorm
02Use earlier factsL5–8

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

  1. L5
    specialize add_left_cancel (a * (a + d))
  2. L6
    specialize add_left_cancel n
  3. L7
    specialize add_left_cancel (a * a + a * d + d * d)
  4. L8
    apply add_left_cancel
03Calculate and transport equalitiesL9–10

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

  1. L9
    trans a * a + (a + d) * (a + d)
  2. L10
    symm
04Use earlier factsL11–11

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

  1. L11
    exact hnorm
05Calculate and transport equalitiesL12–12

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

  1. L12
    simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro a
  2. 0002intro d
  3. 0003intro n
  4. 0004intro hnorm
  5. 0005specialize add_left_cancel (a * (a + d))
  6. 0006specialize add_left_cancel n
  7. 0007specialize add_left_cancel (a * a + a * d + d * d)
  8. 0008apply add_left_cancel
  9. 0009trans a * a + (a + d) * (a + d)
  10. 0010symm
  11. 0011exact hnorm
  12. 0012simp [add_mul, mul_add, mul_assoc, add_assoc, add_comm, mul_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]