EL0004

euclidean_log_double_successor

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

Alpha v34 checked-use · first admitted v23 · 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.

The exact G101 milestone is fully proved, including the stronger checked bound k≤2·BitLen(b), a real beta-coded execution, and its actual terminal gcd. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ n. S n + S n = S S (n + n)

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

Definition DAG

none

Actual proof prerequisites

add_succ_left · checked external prerequisite
Original expanded first-order statement 
forall n. S n + S n = S (S (n + n))

Complete unchanged native tactic proof

All 6 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–1

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 defined 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