DL0098

matrix_integer_selected_prefix_point_at

Every output decoded from a complete selected-submatrix prefix has its actual source-point certificate, independently of the existential value witness used by the prefix.

Alpha v34 checked-use · first admitted v27 · 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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ w. ∀ rb. ∀ rc. ∀ cb. ∀ cc. ∀ q. ∀ u. ∀ v. ∀ i. ∀ a. (∀ x. Lt(x,q · q) → ∃ y. (∃ z. ∃ n. ∃ m. ∃ k. x = q · z + n ∧ (Lt(n,q) ∧ (BetaAt(rb,rc,z,m) ∧ (BetaAt(cb,cc,n,k) ∧ BetaAt(b,c,m · w + k,y))))) ∧ BetaAt(u,v,x,y)) → Lt(i,q · q) → BetaAt(u,v,i,a) → ∃ x. ∃ y. ∃ z. ∃ n. i = q · x + y ∧ (Lt(y,q) ∧ (BetaAt(rb,rc,x,z) ∧ (BetaAt(cb,cc,y,n) ∧ BetaAt(b,c,z · w + n,a))))

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

Definition DAG

Lt(a,b) · 5BetaAt(b,c,i,x) · 12

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c w rb rc cb cc q u v i a. (forall mdr_i_selected_prefix. (exists mdr_gap_selected_prefixbound. mdr_gap_selected_prefixbound + S (mdr_i_selected_prefix) = (q * q)) -> exists mdr_a_selected_prefix. (((exists mdr_r_selected_prefixpoint mdr_s_selected_prefixpoint mdr_u_selected_prefixpoint mdr_v_selected_prefixpoint. ((mdr_i_selected_prefix = (q) * mdr_r_selected_prefixpoint + mdr_s_selected_prefixpoint) /\ ((exists mdr_gap_selected_prefixpointcolumn. mdr_gap_selected_prefixpointcolumn + S (mdr_s_selected_prefixpoint) = (q)) /\ ((((exists ff_h_mdr_selected_prefixpointrow_index. ff_h_mdr_selected_prefixpointrow_index + S (mdr_u_selected_prefixpoint) = S ((S (mdr_r_selected_prefixpoint)) * rc)) /\ exists ff_q_mdr_selected_prefixpointrow_index. rb = ff_q_mdr_selected_prefixpointrow_index * S ((S (mdr_r_selected_prefixpoint)) * rc) + (mdr_u_selected_prefixpoint))) /\ ((((exists ff_h_mdr_selected_prefixpointcolumn_index. ff_h_mdr_selected_prefixpointcolumn_index + S (mdr_v_selected_prefixpoint) = S ((S (mdr_s_selected_prefixpoint)) * cc)) /\ exists ff_q_mdr_selected_prefixpointcolumn_index. cb = ff_q_mdr_selected_prefixpointcolumn_index * S ((S (mdr_s_selected_prefixpoint)) * cc) + (mdr_v_selected_prefixpoint))) /\ (((exists ff_h_mdr_selected_prefixpointsource. ff_h_mdr_selected_prefixpointsource + S (mdr_a_selected_prefix) = S ((S ((mdr_u_selected_prefixpoint) * (w) + (mdr_v_selected_prefixpoint))) * c)) /\ exists ff_q_mdr_selected_prefixpointsource. b = ff_q_mdr_selected_prefixpointsource * S ((S ((mdr_u_selected_prefixpoint) * (w) + (mdr_v_selected_prefixpoint))) * c) + (mdr_a_selected_prefix)))))))) /\ (((exists ff_h_mdr_selected_prefixoutput. ff_h_mdr_selected_prefixoutput + S (mdr_a_selected_prefix) = S ((S (mdr_i_selected_prefix)) * v)) /\ exists ff_q_mdr_selected_prefixoutput. u = ff_q_mdr_selected_prefixoutput * S ((S (mdr_i_selected_prefix)) * v) + (mdr_a_selected_prefix)))))) -> (exists mdr_gap_selected_index. mdr_gap_selected_index + S (i) = (q * q)) -> (((exists ff_h_mdr_selected_value. ff_h_mdr_selected_value + S (a) = S ((S (i)) * v)) /\ exists ff_q_mdr_selected_value. u = ff_q_mdr_selected_value * S ((S (i)) * v) + (a))) -> (exists mdr_r_actual_selected_point mdr_s_actual_selected_point mdr_u_actual_selected_point mdr_v_actual_selected_point. ((i = (q) * mdr_r_actual_selected_point + mdr_s_actual_selected_point) /\ ((exists mdr_gap_actual_selected_pointcolumn. mdr_gap_actual_selected_pointcolumn + S (mdr_s_actual_selected_point) = (q)) /\ ((((exists ff_h_mdr_actual_selected_pointrow_index. ff_h_mdr_actual_selected_pointrow_index + S (mdr_u_actual_selected_point) = S ((S (mdr_r_actual_selected_point)) * rc)) /\ exists ff_q_mdr_actual_selected_pointrow_index. rb = ff_q_mdr_actual_selected_pointrow_index * S ((S (mdr_r_actual_selected_point)) * rc) + (mdr_u_actual_selected_point))) /\ ((((exists ff_h_mdr_actual_selected_pointcolumn_index. ff_h_mdr_actual_selected_pointcolumn_index + S (mdr_v_actual_selected_point) = S ((S (mdr_s_actual_selected_point)) * cc)) /\ exists ff_q_mdr_actual_selected_pointcolumn_index. cb = ff_q_mdr_actual_selected_pointcolumn_index * S ((S (mdr_s_actual_selected_point)) * cc) + (mdr_v_actual_selected_point))) /\ (((exists ff_h_mdr_actual_selected_pointsource. ff_h_mdr_actual_selected_pointsource + S (a) = S ((S ((mdr_u_actual_selected_point) * (w) + (mdr_v_actual_selected_point))) * c)) /\ exists ff_q_mdr_actual_selected_pointsource. b = ff_q_mdr_actual_selected_pointsource * S ((S ((mdr_u_actual_selected_point) * (w) + (mdr_v_actual_selected_point))) * c) + (a))))))))

Complete tactic proof in conservative notation

All 33 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

33 script commands · 7 reading checkpoints · 2 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 assumptionsL1–10

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 rb
  5. L5
    intro rc
  6. L6
    intro cb
  7. L7
    intro cc
  8. L8
    intro q
  9. L9
    intro u
  10. L10
    intro v
02Fix variables and assumptionsL11–15

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

  1. L11
    intro i
  2. L12
    intro a
  3. L13
    intro hprefix
  4. L14
    intro hi
  5. L15
    intro ha
03Establish hentryL16–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hprefix.

  1. L16
    have hentry : ∃ z. (∃ x. ∃ y. ∃ n. ∃ m. i = q · x + y ∧ (Lt(y,q) ∧ (BetaAt(rb,rc,x,n) ∧ (BetaAt(cb,cc,y,m) ∧ BetaAt(b,c,n · w + m,z))))) ∧ BetaAt(u,v,i,z)Definitions: Lt(y,q)BetaAt(rb,rc,x,n)BetaAt(cb,cc,y,m)BetaAt(b,c,n · w + m,z)BetaAt(u,v,i,z)Original native command in the exact edition
  2. L17
    specialize hprefix (i)
  3. L18
    apply hprefix
  4. L19
    exact hi
04Separate the logical casesL20–21

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

  1. L20
    cases hentry
  2. L21
    cases hentry_witness
05Establish hvalueL22–31

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L22
    have hvalue : a = x
  2. L23
    specialize beta_at_unique (u)
  3. L24
    specialize beta_at_unique (v)
  4. L25
    specialize beta_at_unique (i)
  5. L26
    specialize beta_at_unique (a)
  6. L27
    specialize beta_at_unique (x)
  7. L28
    apply beta_at_unique
  8. L29
    exact ha
  9. L30
    exact hentry_witness_right
  10. L31
    rewrite hvalue
06Calculate and transport equalitiesL32–32

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

  1. L32
    rewrite hvalue
07Use earlier factsL33–33

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

  1. L33
    exact hentry_witness_left

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro w
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro cb
  7. 0007intro cc
  8. 0008intro q
  9. 0009intro u
  10. 0010intro v
  11. 0011intro i
  12. 0012intro a
  13. 0013intro hprefix
  14. 0014intro hi
  15. 0015intro ha
  16. 0016have hentry : ∃ z. (∃ x. ∃ y. ∃ n. ∃ m. i = q · x + y ∧ (Lt(y,q) ∧ (BetaAt(rb,rc,x,n) ∧ (BetaAt(cb,cc,y,m) ∧ BetaAt(b,c,n · w + m,z))))) ∧ BetaAt(u,v,i,z)
  17. 0017specialize hprefix (i)
  18. 0018apply hprefix
  19. 0019exact hi
  20. 0020cases hentry
  21. 0021cases hentry_witness
  22. 0022have hvalue : a = x
  23. 0023specialize beta_at_unique (u)
  24. 0024specialize beta_at_unique (v)
  25. 0025specialize beta_at_unique (i)
  26. 0026specialize beta_at_unique (a)
  27. 0027specialize beta_at_unique (x)
  28. 0028apply beta_at_unique
  29. 0029exact ha
  30. 0030exact hentry_witness_right
  31. 0031rewrite hvalue
  32. 0032rewrite hvalue
  33. 0033exact hentry_witness_left