PA0016 — add

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 k. (n + m) * k = n * k + m * k

Structural proof guide

Generated structural guide

Multiplication distributes over addition on the left.

Use the direct prerequisites mul_comm, mul_add as previously established PA formulas.

The proof proceeds by certified simplification (1).

Referenced ingredients

PA000H mul_comm PA000A mul_add

Proof neighborhood

Direct dependencies

PA000H mul_comm PA000A mul_add

Direct dependents

PA0017 common_divisor_beta_moduli_divides_gap_times_c PA001J balanced_bezout_euclid_step PA001N balanced_combination_scale_right PA001Y mod_eq_mul_right PA002M mul_le_mul_right PA005N square_decomp PA006D odd_mul_odd PA00DQ odd_half_cross_product_gap

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

4 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.

01Fix variables and assumptionsL1–3

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

L1
intro n
L2
intro m
L3
intro k

02Calculate and transport equalitiesL4–4

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

L4
simp [mul_comm, mul_add]

Library-wide reading audit

Original exact command ledger · 4 lines