EL0005

lte_adjacent_coefficient_identity

Consecutive triangular coefficients advance by exactly twice the next index.

Alpha v34 checked-use · first admitted v29 · 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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ n. S n · n + 2 · S n = S S n · S n

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

Definition DAG

none

Actual proof prerequisites

mul_comm · checked external prerequisitemul_add · checked external prerequisite
Original expanded first-order statement 
forall n. S n * n + 2 * S n = S (S n) * S n

Complete tactic proof in conservative notation

All 13 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

13 script commands · 7 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
02Calculate and transport equalitiesL2–4

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

  1. L2
    trans S n * n + S n * 2
  2. L3
    congr
  3. L4
    refl
03Use earlier factsL5–5

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

  1. L5
    apply mul_comm
04Calculate and transport equalitiesL6–7

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

  1. L6
    trans S n * (n + 2)
  2. L7
    symm
05Use earlier factsL8–8

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

  1. L8
    apply mul_add
06Calculate and transport equalitiesL9–12

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

  1. L9
    trans S n * S (S n)
  2. L10
    congr
  3. L11
    refl
  4. L12
    simp
07Use earlier factsL13–13

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

  1. L13
    apply mul_comm

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro n
  2. 0002trans S n * n + S n * 2
  3. 0003congr
  4. 0004refl
  5. 0005apply mul_comm
  6. 0006trans S n * (n + 2)
  7. 0007symm
  8. 0008apply mul_add
  9. 0009trans S n * S (S n)
  10. 0010congr
  11. 0011refl
  12. 0012simp
  13. 0013apply mul_comm