BX000B

binary_modular_multiply_residue_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every multiplication step has an actual canonical modular residue.

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 first-order arithmetic statement

forall m x y. (exists ff_modulus_gap_binary_guard. ff_modulus_gap_binary_guard + S 1 = m) -> exists r. (((exists ff_gap_binary_product. ff_gap_binary_product + S (r) = m) /\ (exists ff_left_binary_product_congruence ff_right_binary_product_congruence. (x * y) + m * ff_left_binary_product_congruence = (r) + m * ff_right_binary_product_congruence)))

Constructive proof overview

Generated structural guide

Every multiplication step has an actual canonical modular residue.

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v21 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

BX0002 binary_canonical_residue_exists

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

8 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)
binary canonical residue exists
01Fix variables and assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro hmodulus
02Use earlier factsL5–8

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

  1. L5
    specialize binary_canonical_residue_exists m
  2. L6
    specialize binary_canonical_residue_exists (x * y)
  3. L7
    apply binary_canonical_residue_exists
  4. L8
    exact hmodulus

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro m
  2. 0002intro x
  3. 0003intro y
  4. 0004intro hmodulus
  5. 0005specialize binary_canonical_residue_exists m
  6. 0006specialize binary_canonical_residue_exists (x * y)
  7. 0007apply binary_canonical_residue_exists
  8. 0008exact hmodulus