BT0046

dvd_to_mod_zero

Stable checked-use theorem · independently kernel verified

A multiple is balanced-congruent to zero.

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 m a. (exists k. a = m * k) -> exists u v. a + m * u = 0 + m * v

Structural proof guide

A multiple is balanced-congruent to zero.

Direct prerequisites: zero_add. The authored body proceeds by case analysis (1), equality transport (1).

Proof neighborhood

Direct dependencies

BT0000 zero_add

Direct dependents

BT004F binary_crt

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

8 script commands · 4 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.

  1. L1
    intro m
  2. L2
    intro a
  3. L3
    intro h
02Separate the logical casesL4–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases h
03Construct an explicit witnessL5–6

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists 0
  2. L6
    exists x
04Calculate and transport equalitiesL7–8

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

  1. L7
    rewrite h_witness
  2. L8
    simp [zero_add]

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro m
  2. 0002intro a
  3. 0003intro h
  4. 0004cases h
  5. 0005exists 0
  6. 0006exists x
  7. 0007rewrite h_witness
  8. 0008simp [zero_add]