BA0038

cf_convergent_matrix_state_prefix_transport

Ordinary beta-prefix extension preserves every actually decoded matrix coordinate at all retained indices.

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

∀ h. ∀ e. ∀ H. ∀ E. ∀ k. ∀ j. ∀ s. ∀ u. ∀ U. ∀ v. ∀ V. (∀ x. ∀ y. Lt(x,k) → BetaAt(h,e,x,y) → BetaAt(H,E,x,y)) → Lt(j,k) → ConvergentMatrixAt(h,e,j,s,u,U,v,V) → ConvergentMatrixAt(H,E,j,s,u,U,v,V)

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

Definition DAG

ConvergentMatrixAt(h,e,j,s,u,U,v,V) · 2Lt(a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall h e H E k j s u U v V. (forall cfc_index_state_transport cfc_value_state_transport. (exists cfba_gap_state_transportbound. cfba_gap_state_transportbound + S (cfc_index_state_transport) = (k)) -> (((exists ff_h_state_transportsource. ff_h_state_transportsource + S (cfc_value_state_transport) = S ((S (cfc_index_state_transport)) * e)) /\ exists ff_q_state_transportsource. h = ff_q_state_transportsource * S ((S (cfc_index_state_transport)) * e) + (cfc_value_state_transport))) -> (((exists ff_h_state_transporttarget. ff_h_state_transporttarget + S (cfc_value_state_transport) = S ((S (cfc_index_state_transport)) * E)) /\ exists ff_q_state_transporttarget. H = ff_q_state_transporttarget * S ((S (cfc_index_state_transport)) * E) + (cfc_value_state_transport)))) -> (exists cfba_gap_state_transport_index. cfba_gap_state_transport_index + S (j) = (k)) -> (exists cfc_state_state_transport_source. ((exists cfc_left_state_transport_sourcecode cfc_right_state_transport_sourcecode cfc_matrix_state_transport_sourcecode. ((cfc_left_state_transport_sourcecode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_state_transport_sourcecode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_state_transport_sourcecode = ((cfc_left_state_transport_sourcecode) + (cfc_right_state_transport_sourcecode)) * S ((cfc_left_state_transport_sourcecode) + (cfc_right_state_transport_sourcecode)) + ((cfc_right_state_transport_sourcecode) + (cfc_right_state_transport_sourcecode))) /\ ((cfc_state_state_transport_source) = ((s) + (cfc_matrix_state_transport_sourcecode)) * S ((s) + (cfc_matrix_state_transport_sourcecode)) + ((cfc_matrix_state_transport_sourcecode) + (cfc_matrix_state_transport_sourcecode))))))) /\ (((exists ff_h_state_transport_sourceentry. ff_h_state_transport_sourceentry + S (cfc_state_state_transport_source) = S ((S (j)) * e)) /\ exists ff_q_state_transport_sourceentry. h = ff_q_state_transport_sourceentry * S ((S (j)) * e) + (cfc_state_state_transport_source))))) -> (exists cfc_state_state_transport_target. ((exists cfc_left_state_transport_targetcode cfc_right_state_transport_targetcode cfc_matrix_state_transport_targetcode. ((cfc_left_state_transport_targetcode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_state_transport_targetcode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_state_transport_targetcode = ((cfc_left_state_transport_targetcode) + (cfc_right_state_transport_targetcode)) * S ((cfc_left_state_transport_targetcode) + (cfc_right_state_transport_targetcode)) + ((cfc_right_state_transport_targetcode) + (cfc_right_state_transport_targetcode))) /\ ((cfc_state_state_transport_target) = ((s) + (cfc_matrix_state_transport_targetcode)) * S ((s) + (cfc_matrix_state_transport_targetcode)) + ((cfc_matrix_state_transport_targetcode) + (cfc_matrix_state_transport_targetcode))))))) /\ (((exists ff_h_state_transport_targetentry. ff_h_state_transport_targetentry + S (cfc_state_state_transport_target) = S ((S (j)) * E)) /\ exists ff_q_state_transport_targetentry. H = ff_q_state_transport_targetentry * S ((S (j)) * E) + (cfc_state_state_transport_target)))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 6 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 assumptionsL1–10

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

  1. L1
    intro h
  2. L2
    intro e
  3. L3
    intro H
  4. L4
    intro E
  5. L5
    intro k
  6. L6
    intro j
  7. L7
    intro s
  8. L8
    intro u
  9. L9
    intro U
  10. L10
    intro v
02Fix variables and assumptionsL11–14

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

  1. L11
    intro V
  2. L12
    intro hp
  3. L13
    intro hj
  4. L14
    intro ha
03Separate the logical casesL15–16

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

  1. L15
    cases ha
  2. L16
    cases ha_witness
04Construct an explicit witnessL17–17

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

  1. L17
    exists x
05Separate the logical casesL18–18

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

  1. L18
    split
06Use earlier factsL19–24

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

  1. L19
    exact ha_witness_left
  2. L20
    specialize hp (j)
  3. L21
    specialize hp (x)
  4. L22
    apply hp
  5. L23
    exact hj
  6. L24
    exact ha_witness_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro h
  2. 0002intro e
  3. 0003intro H
  4. 0004intro E
  5. 0005intro k
  6. 0006intro j
  7. 0007intro s
  8. 0008intro u
  9. 0009intro U
  10. 0010intro v
  11. 0011intro V
  12. 0012intro hp
  13. 0013intro hj
  14. 0014intro ha
  15. 0015cases ha
  16. 0016cases ha_witness
  17. 0017exists x
  18. 0018split
  19. 0019exact ha_witness_left
  20. 0020specialize hp (j)
  21. 0021specialize hp (x)
  22. 0022apply hp
  23. 0023exact hj
  24. 0024exact ha_witness_right