MN0004

matrix_skip_index_bounded

Every minor coordinate below width q maps to an original coordinate strictly below S q.

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. ∀ q. MatrixSkipIndex(i,r,s) → Lt(i,q) → Lt(s,S q)

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

Definition DAG

MatrixSkipIndex(i,r,s) · 1Lt(a,b) · 2

Actual proof prerequisites

lt_of_lt_of_le · checked external prerequisitele_succ_self · checked external prerequisitesucc_le_succ · checked external prerequisite
Original expanded first-order statement 
forall i r s q. ((((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))) -> (exists ff_gap_mdm_lt_bounded_input. ff_gap_mdm_lt_bounded_input + S (i) = (q)) -> (exists ff_gap_mdm_lt_bounded_output. ff_gap_mdm_lt_bounded_output + S (s) = (S q))

Complete unchanged native tactic proof

All 22 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

22 script commands · 7 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 assumptionsL1–6

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

  1. L1
    intro i
  2. L2
    intro r
  3. L3
    intro s
  4. L4
    intro q
  5. L5
    intro hskip
  6. L6
    intro hbound
02Separate the logical casesL7–8

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

  1. L7
    cases hskip
  2. L8
    cases hskip_left
03Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite hskip_left_right
04Use earlier factsL10–16

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

  1. L10
    specialize lt_of_lt_of_le i
  2. L11
    specialize lt_of_lt_of_le q
  3. L12
    specialize lt_of_lt_of_le (S q)
  4. L13
    apply lt_of_lt_of_le
  5. L14
    exact hbound
  6. L15
    specialize le_succ_self q
  7. L16
    exact le_succ_self
05Separate the logical casesL17–17

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

  1. L17
    cases hskip_right
06Calculate and transport equalitiesL18–18

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

  1. L18
    rewrite hskip_right_right
07Use earlier factsL19–22

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

  1. L19
    specialize succ_le_succ (S i)
  2. L20
    specialize succ_le_succ q
  3. L21
    apply succ_le_succ
  4. L22
    exact hbound

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro i
  2. 0002intro r
  3. 0003intro s
  4. 0004intro q
  5. 0005intro hskip
  6. 0006intro hbound
  7. 0007cases hskip
  8. 0008cases hskip_left
  9. 0009rewrite hskip_left_right
  10. 0010specialize lt_of_lt_of_le i
  11. 0011specialize lt_of_lt_of_le q
  12. 0012specialize lt_of_lt_of_le (S q)
  13. 0013apply lt_of_lt_of_le
  14. 0014exact hbound
  15. 0015specialize le_succ_self q
  16. 0016exact le_succ_self
  17. 0017cases hskip_right
  18. 0018rewrite hskip_right_right
  19. 0019specialize succ_le_succ (S i)
  20. 0020specialize succ_le_succ q
  21. 0021apply succ_le_succ
  22. 0022exact hbound