MN0001

matrix_skip_index_exists

Every minor coordinate has a constructive source coordinate that skips an arbitrary deleted index.

Alpha v34 checked-use · first admitted v24 · independently kernel and Lean verified; not Stable

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.

Historical partial components only: this chapter proves arbitrary signed cofactor minors and exact signed determinants through dimension four. T13 is now closed by the separate Alpha-v27 integer-linear-algebra branch: arbitrary determinant data, rank, and integer column spans, without a claim of lattice index or normal forms. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ i. ∀ r. ∃ s. MatrixSkipIndex(i,r,s)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

MatrixSkipIndex(i,r,s) · 1

Actual proof prerequisites

le_or_lt · checked external prerequisite
Original expanded first-order statement 
forall i r. exists s. ((((exists ff_gap_mdm_lt_result_before. ff_gap_mdm_lt_result_before + S (i) = (r)) /\ s = i) \/ ((exists ff_gap_mdm_le_result_after. ff_gap_mdm_le_result_after + (r) = (i)) /\ s = S i)))

Complete unchanged native tactic proof

All 15 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

15 script commands · 11 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro i
  2. L2
    intro r
02Use earlier factsL3–4

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

  1. L3
    specialize le_or_lt r
  2. L4
    specialize le_or_lt i
03Separate the logical casesL5–5

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

  1. L5
    cases le_or_lt
04Construct an explicit witnessL6–6

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

  1. L6
    exists S i
05Separate the logical casesL7–8

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

  1. L7
    right
  2. L8
    split
06Use earlier factsL9–9

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

  1. L9
    exact le_or_lt_left
07Calculate and transport equalitiesL10–10

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

  1. L10
    refl
08Construct an explicit witnessL11–11

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

  1. L11
    exists i
09Separate the logical casesL12–13

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

  1. L12
    left
  2. L13
    split
10Use earlier factsL14–14

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

  1. L14
    exact le_or_lt_right
11Calculate and transport equalitiesL15–15

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

  1. L15
    refl

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro i
  2. 0002intro r
  3. 0003specialize le_or_lt r
  4. 0004specialize le_or_lt i
  5. 0005cases le_or_lt
  6. 0006exists S i
  7. 0007right
  8. 0008split
  9. 0009exact le_or_lt_left
  10. 0010refl
  11. 0011exists i
  12. 0012left
  13. 0013split
  14. 0014exact le_or_lt_right
  15. 0015refl