BT0005

mul_succ_left

Stable checked-use theorem · independently kernel verified

A successor can move through multiplication on the left.

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 PA statement

forall n m. S n * m = n * m + m

Structural proof guide

A successor can move through multiplication on the left.

Direct prerequisites: add_comm, add_assoc. The authored body proceeds by structural induction (1).

Proof neighborhood

Direct dependencies

BT0002 add_comm BT0003 add_assoc

Direct dependents

BT0006 mul_comm BT0024 two_large_factors_impossible BT004E mod_eq_predecessor_cancel BT0054 base_le_beta_modulus BT006N prime_three BT00TT choose_weighted_vertical BT00UE primorial_positive BT00YF division_three_scaled_upper_of_quotient_lt BT011K prime_eighty_three

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

6 script commands · 2 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.

Named ingredients (1)
add comm
01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Induction on mL2–6

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction m
  2. L3
    simp
  3. L4
    specialize add_comm n
  4. L5
    specialize add_comm m
  5. L6
    simp [IH, add_comm, add_assoc]

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro n
  2. 0002induction m
  3. 0003simp
  4. 0004specialize add_comm n
  5. 0005specialize add_comm m
  6. 0006simp [IH, add_comm, add_assoc]