EL0004

euclidean_log_double_successor

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

The exact two-step budget for a successor exponent grows by two.

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 n. S n + S n = S (S (n + n))

Constructive proof overview

Generated structural guide

The exact two-step budget for a successor exponent grows by two.

The unchanged tactic script uses 1 declared prerequisite and contains 6 exact native proof lines.

Alpha v34 checked-use · first admitted v23 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_succ_left Stable theorem; checked-use authorized

Direct dependents

EL0009 euclidean_log_budget_successor_power

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

6 script commands · 3 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 assumptionsL1–1

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

  1. L1
    intro n
02Use earlier factsL2–3

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

  1. L2
    specialize add_succ_left n
  2. L3
    specialize add_succ_left (S n)
03Calculate and transport equalitiesL4–6

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

  1. L4
    rewrite add_succ_left
  2. L5
    rewrite PA4
  3. L6
    refl

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro n
  2. 0002specialize add_succ_left n
  3. 0003specialize add_succ_left (S n)
  4. 0004rewrite add_succ_left
  5. 0005rewrite PA4
  6. 0006refl

Separate complete second-wave branches: Full T13 proof · Alpha v27.