EI000A

eisenstein_pair_natural_value_transport

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

An equal signed-pair representative preserves the same 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 p n P N v. p = n + v -> p + N = P + n -> P = N + v

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites 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_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

EI000B eisenstein_coordinate_norm_transport EI0027 eisenstein_coordinate_norm_conjugate EI0028 eisenstein_conjugate_product_is_norm

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 · 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 assumptionsL1–7

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 exact 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]