BA0032

cf_convergent_matrix_empty_elimination

The zero-step actual prefix product is exactly the identity matrix, regardless of its unconsumed quotient tail.

Alpha v34 checked-use · first admitted v29 · 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.

The initial 0/1 convergent is included: u is natural, not necessarily positive. Comparison denominators are strictly smaller and positive. Signed competitors are represented by an arbitrary difference rp−rn. Approximation inequalities are proved from the trace, never stored as assumptions in Convergent.

Exact theorem in conservative defined notation

∀ s. ∀ h. ∀ e. ∀ u. ∀ U. ∀ v. ∀ V. ConvergentMatrixTrace(s,h,e,0,u,U,v,V) → u = 1 ∧ (U = 0 ∧ (v = 0 ∧ V = 1))

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

Definition DAG

ConvergentMatrixTrace(s,h,e,k,u,U,v,V) · 1

Actual proof prerequisites

cf_convergent_matrix_state_unique
Original expanded first-order statement 
forall s h e u U v V. (exists cfc_tail_matrix_empty. ((exists cfc_state_matrix_emptyinitial. ((exists cfc_left_matrix_emptyinitialcode cfc_right_matrix_emptyinitialcode cfc_matrix_matrix_emptyinitialcode. ((cfc_left_matrix_emptyinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_matrix_emptyinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_matrix_emptyinitialcode = ((cfc_left_matrix_emptyinitialcode) + (cfc_right_matrix_emptyinitialcode)) * S ((cfc_left_matrix_emptyinitialcode) + (cfc_right_matrix_emptyinitialcode)) + ((cfc_right_matrix_emptyinitialcode) + (cfc_right_matrix_emptyinitialcode))) /\ ((cfc_state_matrix_emptyinitial) = ((cfc_tail_matrix_empty) + (cfc_matrix_matrix_emptyinitialcode)) * S ((cfc_tail_matrix_empty) + (cfc_matrix_matrix_emptyinitialcode)) + ((cfc_matrix_matrix_emptyinitialcode) + (cfc_matrix_matrix_emptyinitialcode))))))) /\ (((exists ff_h_matrix_emptyinitialentry. ff_h_matrix_emptyinitialentry + S (cfc_state_matrix_emptyinitial) = S ((S (0)) * e)) /\ exists ff_q_matrix_emptyinitialentry. h = ff_q_matrix_emptyinitialentry * S ((S (0)) * e) + (cfc_state_matrix_emptyinitial))))) /\ ((exists cfc_state_matrix_emptyterminal. ((exists cfc_left_matrix_emptyterminalcode cfc_right_matrix_emptyterminalcode cfc_matrix_matrix_emptyterminalcode. ((cfc_left_matrix_emptyterminalcode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_matrix_emptyterminalcode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_matrix_emptyterminalcode = ((cfc_left_matrix_emptyterminalcode) + (cfc_right_matrix_emptyterminalcode)) * S ((cfc_left_matrix_emptyterminalcode) + (cfc_right_matrix_emptyterminalcode)) + ((cfc_right_matrix_emptyterminalcode) + (cfc_right_matrix_emptyterminalcode))) /\ ((cfc_state_matrix_emptyterminal) = ((s) + (cfc_matrix_matrix_emptyterminalcode)) * S ((s) + (cfc_matrix_matrix_emptyterminalcode)) + ((cfc_matrix_matrix_emptyterminalcode) + (cfc_matrix_matrix_emptyterminalcode))))))) /\ (((exists ff_h_matrix_emptyterminalentry. ff_h_matrix_emptyterminalentry + S (cfc_state_matrix_emptyterminal) = S ((S (0)) * e)) /\ exists ff_q_matrix_emptyterminalentry. h = ff_q_matrix_emptyterminalentry * S ((S (0)) * e) + (cfc_state_matrix_emptyterminal))))) /\ (forall cfc_index_matrix_empty. (exists cfba_gap_matrix_emptybound. cfba_gap_matrix_emptybound + S (cfc_index_matrix_empty) = (0)) -> exists cfc_old_matrix_empty cfc_a_matrix_empty cfc_b_matrix_empty cfc_c_matrix_empty cfc_d_matrix_empty cfc_new_matrix_empty cfc_quotient_matrix_empty. ((exists cfc_state_matrix_emptyprevious. ((exists cfc_left_matrix_emptypreviouscode cfc_right_matrix_emptypreviouscode cfc_matrix_matrix_emptypreviouscode. ((cfc_left_matrix_emptypreviouscode = ((cfc_a_matrix_empty) + (cfc_b_matrix_empty)) * S ((cfc_a_matrix_empty) + (cfc_b_matrix_empty)) + ((cfc_b_matrix_empty) + (cfc_b_matrix_empty))) /\ ((cfc_right_matrix_emptypreviouscode = ((cfc_c_matrix_empty) + (cfc_d_matrix_empty)) * S ((cfc_c_matrix_empty) + (cfc_d_matrix_empty)) + ((cfc_d_matrix_empty) + (cfc_d_matrix_empty))) /\ ((cfc_matrix_matrix_emptypreviouscode = ((cfc_left_matrix_emptypreviouscode) + (cfc_right_matrix_emptypreviouscode)) * S ((cfc_left_matrix_emptypreviouscode) + (cfc_right_matrix_emptypreviouscode)) + ((cfc_right_matrix_emptypreviouscode) + (cfc_right_matrix_emptypreviouscode))) /\ ((cfc_state_matrix_emptyprevious) = ((cfc_old_matrix_empty) + (cfc_matrix_matrix_emptypreviouscode)) * S ((cfc_old_matrix_empty) + (cfc_matrix_matrix_emptypreviouscode)) + ((cfc_matrix_matrix_emptypreviouscode) + (cfc_matrix_matrix_emptypreviouscode))))))) /\ (((exists ff_h_matrix_emptypreviousentry. ff_h_matrix_emptypreviousentry + S (cfc_state_matrix_emptyprevious) = S ((S (cfc_index_matrix_empty)) * e)) /\ exists ff_q_matrix_emptypreviousentry. h = ff_q_matrix_emptypreviousentry * S ((S (cfc_index_matrix_empty)) * e) + (cfc_state_matrix_emptyprevious))))) /\ ((exists cfc_state_matrix_emptyfollowing. ((exists cfc_left_matrix_emptyfollowingcode cfc_right_matrix_emptyfollowingcode cfc_matrix_matrix_emptyfollowingcode. ((cfc_left_matrix_emptyfollowingcode = (((cfc_quotient_matrix_empty * cfc_a_matrix_empty + cfc_c_matrix_empty)) + ((cfc_quotient_matrix_empty * cfc_b_matrix_empty + cfc_d_matrix_empty))) * S (((cfc_quotient_matrix_empty * cfc_a_matrix_empty + cfc_c_matrix_empty)) + ((cfc_quotient_matrix_empty * cfc_b_matrix_empty + cfc_d_matrix_empty))) + (((cfc_quotient_matrix_empty * cfc_b_matrix_empty + cfc_d_matrix_empty)) + ((cfc_quotient_matrix_empty * cfc_b_matrix_empty + cfc_d_matrix_empty)))) /\ ((cfc_right_matrix_emptyfollowingcode = ((cfc_a_matrix_empty) + (cfc_b_matrix_empty)) * S ((cfc_a_matrix_empty) + (cfc_b_matrix_empty)) + ((cfc_b_matrix_empty) + (cfc_b_matrix_empty))) /\ ((cfc_matrix_matrix_emptyfollowingcode = ((cfc_left_matrix_emptyfollowingcode) + (cfc_right_matrix_emptyfollowingcode)) * S ((cfc_left_matrix_emptyfollowingcode) + (cfc_right_matrix_emptyfollowingcode)) + ((cfc_right_matrix_emptyfollowingcode) + (cfc_right_matrix_emptyfollowingcode))) /\ ((cfc_state_matrix_emptyfollowing) = ((cfc_new_matrix_empty) + (cfc_matrix_matrix_emptyfollowingcode)) * S ((cfc_new_matrix_empty) + (cfc_matrix_matrix_emptyfollowingcode)) + ((cfc_matrix_matrix_emptyfollowingcode) + (cfc_matrix_matrix_emptyfollowingcode))))))) /\ (((exists ff_h_matrix_emptyfollowingentry. ff_h_matrix_emptyfollowingentry + S (cfc_state_matrix_emptyfollowing) = S ((S (S cfc_index_matrix_empty)) * e)) /\ exists ff_q_matrix_emptyfollowingentry. h = ff_q_matrix_emptyfollowingentry * S ((S (S cfc_index_matrix_empty)) * e) + (cfc_state_matrix_emptyfollowing))))) /\ (cfc_new_matrix_empty = S ((cfc_quotient_matrix_empty + cfc_old_matrix_empty) * S (cfc_quotient_matrix_empty + cfc_old_matrix_empty) + (cfc_old_matrix_empty + cfc_old_matrix_empty))))))))) -> ((u = 1) /\ ((U = 0) /\ ((v = 0) /\ (V = 1))))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 6 reading checkpoints · 1 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.

Named ingredients (1)
cf convergent matrix state unique
01Fix variables and assumptionsL1–8

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

  1. L1
    intro s
  2. L2
    intro h
  3. L3
    intro e
  4. L4
    intro u
  5. L5
    intro U
  6. L6
    intro v
  7. L7
    intro V
  8. L8
    intro ht
02Separate the logical casesL9–11

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

  1. L9
    cases ht
  2. L10
    cases ht_witness
  3. L11
    cases ht_witness_right
03Establish heqL12–21

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have heq : ((s = x) /\ ((u = 1) /\ ((U = 0) /\ ((v = 0) /\ (V = 1)))))
  2. L13
    specialize cf_convergent_matrix_state_unique (h)
  3. L14
    specialize cf_convergent_matrix_state_unique (e)
  4. L15
    specialize cf_convergent_matrix_state_unique (0)
  5. L16
    specialize cf_convergent_matrix_state_unique (s)
  6. L17
    specialize cf_convergent_matrix_state_unique (u)
  7. L18
    specialize cf_convergent_matrix_state_unique (U)
  8. L19
    specialize cf_convergent_matrix_state_unique (v)
  9. L20
    specialize cf_convergent_matrix_state_unique (V)
  10. L21
    specialize cf_convergent_matrix_state_unique (x)
04Use earlier factsL22–28

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

  1. L22
    specialize cf_convergent_matrix_state_unique (1)
  2. L23
    specialize cf_convergent_matrix_state_unique (0)
  3. L24
    specialize cf_convergent_matrix_state_unique (0)
  4. L25
    specialize cf_convergent_matrix_state_unique (1)
  5. L26
    apply cf_convergent_matrix_state_unique
  6. L27
    exact ht_witness_right_left
  7. L28
    exact ht_witness_left
05Separate the logical casesL29–29

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

  1. L29
    cases heq
06Use earlier factsL30–30

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

  1. L30
    exact heq_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro s
  2. 0002intro h
  3. 0003intro e
  4. 0004intro u
  5. 0005intro U
  6. 0006intro v
  7. 0007intro V
  8. 0008intro ht
  9. 0009cases ht
  10. 0010cases ht_witness
  11. 0011cases ht_witness_right
  12. 0012have heq : ((s = x) /\ ((u = 1) /\ ((U = 0) /\ ((v = 0) /\ (V = 1)))))
  13. 0013specialize cf_convergent_matrix_state_unique (h)
  14. 0014specialize cf_convergent_matrix_state_unique (e)
  15. 0015specialize cf_convergent_matrix_state_unique (0)
  16. 0016specialize cf_convergent_matrix_state_unique (s)
  17. 0017specialize cf_convergent_matrix_state_unique (u)
  18. 0018specialize cf_convergent_matrix_state_unique (U)
  19. 0019specialize cf_convergent_matrix_state_unique (v)
  20. 0020specialize cf_convergent_matrix_state_unique (V)
  21. 0021specialize cf_convergent_matrix_state_unique (x)
  22. 0022specialize cf_convergent_matrix_state_unique (1)
  23. 0023specialize cf_convergent_matrix_state_unique (0)
  24. 0024specialize cf_convergent_matrix_state_unique (0)
  25. 0025specialize cf_convergent_matrix_state_unique (1)
  26. 0026apply cf_convergent_matrix_state_unique
  27. 0027exact ht_witness_right_left
  28. 0028exact ht_witness_left
  29. 0029cases heq
  30. 0030exact heq_right