DL0023

matrix_recursive_matrix_equality_refl

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

Every actual finite signed matrix is pointwise equal to itself.

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 pb pc nb nc d. (((forall mdr_i_matrix_reflp mdr_a_matrix_reflp. (exists mdr_gap_matrix_reflpb. mdr_gap_matrix_reflpb + S (mdr_i_matrix_reflp) = ((d) * (d))) -> (((exists ff_h_mdr_matrix_reflpo. ff_h_mdr_matrix_reflpo + S (mdr_a_matrix_reflp) = S ((S (mdr_i_matrix_reflp)) * pc)) /\ exists ff_q_mdr_matrix_reflpo. pb = ff_q_mdr_matrix_reflpo * S ((S (mdr_i_matrix_reflp)) * pc) + (mdr_a_matrix_reflp))) -> (((exists ff_h_mdr_matrix_reflpn. ff_h_mdr_matrix_reflpn + S (mdr_a_matrix_reflp) = S ((S (mdr_i_matrix_reflp)) * pc)) /\ exists ff_q_mdr_matrix_reflpn. pb = ff_q_mdr_matrix_reflpn * S ((S (mdr_i_matrix_reflp)) * pc) + (mdr_a_matrix_reflp)))) /\ (forall mdr_i_matrix_refln mdr_a_matrix_refln. (exists mdr_gap_matrix_reflnb. mdr_gap_matrix_reflnb + S (mdr_i_matrix_refln) = ((d) * (d))) -> (((exists ff_h_mdr_matrix_reflno. ff_h_mdr_matrix_reflno + S (mdr_a_matrix_refln) = S ((S (mdr_i_matrix_refln)) * nc)) /\ exists ff_q_mdr_matrix_reflno. nb = ff_q_mdr_matrix_reflno * S ((S (mdr_i_matrix_refln)) * nc) + (mdr_a_matrix_refln))) -> (((exists ff_h_mdr_matrix_reflnn. ff_h_mdr_matrix_reflnn + S (mdr_a_matrix_refln) = S ((S (mdr_i_matrix_refln)) * nc)) /\ exists ff_q_mdr_matrix_reflnn. nb = ff_q_mdr_matrix_reflnn * S ((S (mdr_i_matrix_refln)) * nc) + (mdr_a_matrix_refln))))))

Constructive proof overview

Generated structural guide

Every actual finite signed matrix is pointwise equal to itself.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

DL0002 matrix_recursive_prefix_refl

Direct dependents

DL0027 signed_recursive_determinant_functional DL0029 signed_recursive_determinant_from_evaluated_cofactors

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

14 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)
matrix recursive prefix refl
01Fix variables and assumptionsL1–5

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

  1. L1
    intro pb
  2. L2
    intro pc
  3. L3
    intro nb
  4. L4
    intro nc
  5. L5
    intro d
02Separate the logical casesL6–6

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

  1. L6
    split
03Use earlier factsL7–14

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

  1. L7
    specialize matrix_recursive_prefix_refl (pb)
  2. L8
    specialize matrix_recursive_prefix_refl (pc)
  3. L9
    specialize matrix_recursive_prefix_refl (d * d)
  4. L10
    apply matrix_recursive_prefix_refl
  5. L11
    specialize matrix_recursive_prefix_refl (nb)
  6. L12
    specialize matrix_recursive_prefix_refl (nc)
  7. L13
    specialize matrix_recursive_prefix_refl (d * d)
  8. L14
    apply matrix_recursive_prefix_refl

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003intro nb
  4. 0004intro nc
  5. 0005intro d
  6. 0006split
  7. 0007specialize matrix_recursive_prefix_refl (pb)
  8. 0008specialize matrix_recursive_prefix_refl (pc)
  9. 0009specialize matrix_recursive_prefix_refl (d * d)
  10. 0010apply matrix_recursive_prefix_refl
  11. 0011specialize matrix_recursive_prefix_refl (nb)
  12. 0012specialize matrix_recursive_prefix_refl (nc)
  13. 0013specialize matrix_recursive_prefix_refl (d * d)
  14. 0014apply matrix_recursive_prefix_refl