DL0034

matrix_rank_prefix_equality_symmetric

Finite beta-prefix equality is symmetric because beta decoding is total and functional.

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. ∀ u. ∀ v. ∀ l. (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → BetaAt(u,v,x,y)) → ∀ x. ∀ y. Lt(x,l) → BetaAt(u,v,x,y) → BetaAt(b,c,x,y)

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

Definition DAG

Lt(a,b) · 2BetaAt(b,c,i,x) · 5

Actual proof prerequisites

beta_at_exists · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c u v l. (forall mdr_i_symmetric_source mdr_a_symmetric_source. (exists mdr_gap_symmetric_sourceb. mdr_gap_symmetric_sourceb + S (mdr_i_symmetric_source) = (l)) -> (((exists ff_h_mdr_symmetric_sourceo. ff_h_mdr_symmetric_sourceo + S (mdr_a_symmetric_source) = S ((S (mdr_i_symmetric_source)) * c)) /\ exists ff_q_mdr_symmetric_sourceo. b = ff_q_mdr_symmetric_sourceo * S ((S (mdr_i_symmetric_source)) * c) + (mdr_a_symmetric_source))) -> (((exists ff_h_mdr_symmetric_sourcen. ff_h_mdr_symmetric_sourcen + S (mdr_a_symmetric_source) = S ((S (mdr_i_symmetric_source)) * v)) /\ exists ff_q_mdr_symmetric_sourcen. u = ff_q_mdr_symmetric_sourcen * S ((S (mdr_i_symmetric_source)) * v) + (mdr_a_symmetric_source)))) -> (forall mdr_i_symmetric_result mdr_a_symmetric_result. (exists mdr_gap_symmetric_resultb. mdr_gap_symmetric_resultb + S (mdr_i_symmetric_result) = (l)) -> (((exists ff_h_mdr_symmetric_resulto. ff_h_mdr_symmetric_resulto + S (mdr_a_symmetric_result) = S ((S (mdr_i_symmetric_result)) * v)) /\ exists ff_q_mdr_symmetric_resulto. u = ff_q_mdr_symmetric_resulto * S ((S (mdr_i_symmetric_result)) * v) + (mdr_a_symmetric_result))) -> (((exists ff_h_mdr_symmetric_resultn. ff_h_mdr_symmetric_resultn + S (mdr_a_symmetric_result) = S ((S (mdr_i_symmetric_result)) * c)) /\ exists ff_q_mdr_symmetric_resultn. b = ff_q_mdr_symmetric_resultn * S ((S (mdr_i_symmetric_result)) * c) + (mdr_a_symmetric_result))))

Complete tactic proof in conservative notation

All 32 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

32 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 u
  4. L4
    intro v
  5. L5
    intro l
  6. L6
    intro hprefix
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro hi
  10. L10
    intro ha
02Establish hvalueL11–15

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

  1. L11
    have hvalue : ∃ z. BetaAt(b,c,i,z)Definitions: BetaAt(b,c,i,z)Original native command in the exact edition
  2. L12
    specialize beta_at_exists (b)
  3. L13
    specialize beta_at_exists (c)
  4. L14
    specialize beta_at_exists (i)
  5. L15
    apply beta_at_exists
03Separate the logical casesL16–16

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

  1. L16
    cases hvalue
04Establish heqL17–26

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

  1. L17
    have heq : a = x
  2. L18
    specialize beta_at_unique (u)
  3. L19
    specialize beta_at_unique (v)
  4. L20
    specialize beta_at_unique (i)
  5. L21
    specialize beta_at_unique (a)
  6. L22
    specialize beta_at_unique (x)
  7. L23
    apply beta_at_unique
  8. L24
    exact ha
  9. L25
    specialize hprefix (i)
  10. L26
    specialize hprefix (x)
05Use earlier factsL27–29

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

  1. L27
    apply hprefix
  2. L28
    exact hi
  3. L29
    exact hvalue_witness
06Calculate and transport equalitiesL30–31

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

  1. L30
    rewrite heq
  2. L31
    rewrite heq
07Use earlier factsL32–32

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

  1. L32
    exact hvalue_witness

Library-wide reading audit

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