MD0001

beta_matrix_cell_exists

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

Every requested finite matrix coordinate has a witnessed beta-decoded entry.

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 b c w i j. exists n. (((exists fs_h_matrix_entry. fs_h_matrix_entry + S (n) = S ((S (i * w + j)) * c)) /\ exists fs_q_matrix_entry. b = fs_q_matrix_entry * S ((S (i * w + j)) * c) + (n)))

Constructive proof overview

Generated structural guide

Every requested finite matrix coordinate has a witnessed beta-decoded entry.

The unchanged tactic script uses 1 declared prerequisite and contains 9 exact native proof lines.

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_exists Stable theorem; checked-use authorized

Direct dependents

MD0003 beta_matrix_cell_exists_unique

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

9 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 assumptionsL1–5

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro w
  4. L4
    intro i
  5. L5
    intro j
02Use earlier factsL6–9

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

  1. L6
    specialize beta_at_exists b
  2. L7
    specialize beta_at_exists c
  3. L8
    specialize beta_at_exists (i * w + j)
  4. L9
    exact beta_at_exists

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro i
  5. 0005intro j
  6. 0006specialize beta_at_exists b
  7. 0007specialize beta_at_exists c
  8. 0008specialize beta_at_exists (i * w + j)
  9. 0009exact beta_at_exists

Separate complete second-wave branches: Full T13 proof · Alpha v27.