EI000A

eisenstein_pair_natural_value_transport

An equal signed-pair representative preserves the same natural value.

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

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.

A floor quotient in the fundamental parallelogram already gives the required strict norm decrease; global nearest-point optimality is not asserted. The shared carrier is identical to the Gaussian carrier, but the multiplication law and norm are different. Eisenstein gcd, factorization, and prime classification remain separate targets.

Exact theorem in conservative defined notation

∀ p. ∀ n. ∀ P. ∀ N. ∀ v. p = n + v → p + N = P + n → P = N + v

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

none

Actual proof prerequisites

add_right_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall p n P N v. p = n + v -> p + N = P + n -> P = N + v

Complete tactic proof in conservative notation

All 16 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

16 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro P
  4. L4
    intro N
  5. L5
    intro v
  6. L6
    intro hvalue
  7. L7
    intro hcross
02Use earlier factsL8–11

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

  1. L8
    specialize add_right_cancel P
  2. L9
    specialize add_right_cancel (N + v)
  3. L10
    specialize add_right_cancel n
  4. L11
    apply add_right_cancel
03Calculate and transport equalitiesL12–13

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

  1. L12
    trans p + N
  2. L13
    symm
04Use earlier factsL14–14

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

  1. L14
    exact hcross
05Calculate and transport equalitiesL15–16

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

  1. L15
    rewrite hvalue
  2. L16
    simp [add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro P
  4. 0004intro N
  5. 0005intro v
  6. 0006intro hvalue
  7. 0007intro hcross
  8. 0008specialize add_right_cancel P
  9. 0009specialize add_right_cancel (N + v)
  10. 0010specialize add_right_cancel n
  11. 0011apply add_right_cancel
  12. 0012trans p + N
  13. 0013symm
  14. 0014exact hcross
  15. 0015rewrite hvalue
  16. 0016simp [add_assoc, add_comm, four_square_add_swap_right_tail]