EI0007

eisenstein_coordinate_norm_functional

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

The subtraction-free signed-coordinate norm has a unique natural value.

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

Constructive proof overview

Generated structural guide

The subtraction-free signed-coordinate norm has a unique natural value.

The unchanged tactic script uses 1 declared prerequisite and contains 16 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

Direct dependents

EI0033 eisenstein_signed_euclidean_division_exists EI0037 eisenstein_norm_functional

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

16 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 assumptionsL1–8

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 m
  7. L7
    intro hn
  8. L8
    intro hm
02Use earlier factsL9–12

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

  1. L9
    specialize add_left_cancel ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))
  2. L10
    specialize add_left_cancel n
  3. L11
    specialize add_left_cancel m
  4. L12
    apply add_left_cancel
03Calculate and transport equalitiesL13–14

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

  1. L13
    trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  2. L14
    symm
04Use earlier factsL15–16

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

  1. L15
    exact hn
  2. L16
    exact hm

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro n
  6. 0006intro m
  7. 0007intro hn
  8. 0008intro hm
  9. 0009specialize add_left_cancel ((((((((ap) * (an))) + (((an) * (ap))))) + (((((bp) * (bn))) + (((bn) * (bp))))))) + (((((ap) * (bp))) + (((an) * (bn))))))
  10. 0010specialize add_left_cancel n
  11. 0011specialize add_left_cancel m
  12. 0012apply add_left_cancel
  13. 0013trans ((((((((ap) * (ap))) + (((an) * (an))))) + (((((bp) * (bp))) + (((bn) * (bn))))))) + (((((ap) * (bn))) + (((an) * (bp))))))
  14. 0014symm
  15. 0015exact hn
  16. 0016exact hm