PA006R — four_mul_eq_double

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 a. 4 * a = 2 * (2 * a)

Structural proof guide

Generated structural guide

Multiplication by four is iterated doubling.

Use the direct prerequisites mul_assoc as previously established PA formulas.

The proof proceeds by closed numeral normalization (1).

Referenced ingredients

PA000B mul_assoc

Proof neighborhood

Direct dependencies

PA000B mul_assoc

Direct dependents

PA00FI odd_half_of_mod4_one_exact PA00FJ odd_half_even_iff_mod4_one PA00FQ odd_half_of_mod4_three_exact PA00FR odd_half_odd_iff_mod4_three

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

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.

Named ingredients (1)

mul assoc

01Fix variables and assumptionsL1–1

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

L1
intro a

02Calculate and transport equalitiesL2–5

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

L2
trans (2 * 2) * a
L3
congr
L4
norm_num
L5
refl

03Use earlier factsL6–6

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

L6
apply mul_assoc

Library-wide reading audit

Original exact command ledger · 6 lines