BA0008

cf_approximation_identity_entry_transport

Substitution along the actual recurrence equations transports the determinant/error invariant without assuming it as part of a convergent certificate.

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

∀ a. ∀ b. ∀ u. ∀ U. ∀ v. ∀ V. ∀ p. ∀ P. ∀ q. ∀ Q. ∀ E. ∀ F. u = p → U = P → v = q → V = Q → AlternatingConvergentIdentity(a,b,p,P,q,Q,E,F) → AlternatingConvergentIdentity(a,b,u,U,v,V,E,F)

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

Definition DAG

AlternatingConvergentIdentity(a,b,u,U,v,V,E,F) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall a b u U v V p P q Q E F. u = p -> U = P -> v = q -> V = Q -> ((((p * Q + 1 = P * q) /\ ((a * q = b * p + E) /\ (b * P = a * Q + F)))) \/ (((P * q + 1 = p * Q) /\ ((b * p = a * q + E) /\ (a * Q = b * P + F))))) -> ((((u * V + 1 = U * v) /\ ((a * v = b * u + E) /\ (b * U = a * V + F)))) \/ (((U * v + 1 = u * V) /\ ((b * u = a * v + E) /\ (a * V = b * U + F)))))

Complete tactic proof in conservative notation

All 50 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

50 script commands · 18 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 a
  2. L2
    intro b
  3. L3
    intro u
  4. L4
    intro U
  5. L5
    intro v
  6. L6
    intro V
  7. L7
    intro p
  8. L8
    intro P
  9. L9
    intro q
  10. L10
    intro Q
02Fix variables and assumptionsL11–17

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

  1. L11
    intro E
  2. L12
    intro F
  3. L13
    intro hu
  4. L14
    intro hU
  5. L15
    intro hv
  6. L16
    intro hV
  7. L17
    intro hi
03Separate the logical casesL18–22

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

  1. L18
    cases hi
  2. L19
    cases hi_left
  3. L20
    cases hi_left_right
  4. L21
    left
  5. L22
    split
04Calculate and transport equalitiesL23–26

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

  1. L23
    rewrite hu
  2. L24
    rewrite hU
  3. L25
    rewrite hv
  4. L26
    rewrite hV
05Use earlier factsL27–27

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

  1. L27
    exact hi_left_left
06Separate the logical casesL28–28

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

  1. L28
    split
07Calculate and transport equalitiesL29–30

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

  1. L29
    rewrite hu
  2. L30
    rewrite hv
08Use earlier factsL31–31

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

  1. L31
    exact hi_left_right_left
09Calculate and transport equalitiesL32–33

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

  1. L32
    rewrite hU
  2. L33
    rewrite hV
10Use earlier factsL34–34

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

  1. L34
    exact hi_left_right_right
11Separate the logical casesL35–38

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

  1. L35
    cases hi_right
  2. L36
    cases hi_right_right
  3. L37
    right
  4. L38
    split
12Calculate and transport equalitiesL39–42

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

  1. L39
    rewrite hu
  2. L40
    rewrite hU
  3. L41
    rewrite hv
  4. L42
    rewrite hV
13Use earlier factsL43–43

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

  1. L43
    exact hi_right_left
14Separate the logical casesL44–44

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

  1. L44
    split
15Calculate and transport equalitiesL45–46

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

  1. L45
    rewrite hu
  2. L46
    rewrite hv
16Use earlier factsL47–47

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

  1. L47
    exact hi_right_right_left
17Calculate and transport equalitiesL48–49

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

  1. L48
    rewrite hU
  2. L49
    rewrite hV
18Use earlier factsL50–50

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

  1. L50
    exact hi_right_right_right

Library-wide reading audit

Original defined command ledger · 50 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro u
  4. 0004intro U
  5. 0005intro v
  6. 0006intro V
  7. 0007intro p
  8. 0008intro P
  9. 0009intro q
  10. 0010intro Q
  11. 0011intro E
  12. 0012intro F
  13. 0013intro hu
  14. 0014intro hU
  15. 0015intro hv
  16. 0016intro hV
  17. 0017intro hi
  18. 0018cases hi
  19. 0019cases hi_left
  20. 0020cases hi_left_right
  21. 0021left
  22. 0022split
  23. 0023rewrite hu
  24. 0024rewrite hU
  25. 0025rewrite hv
  26. 0026rewrite hV
  27. 0027exact hi_left_left
  28. 0028split
  29. 0029rewrite hu
  30. 0030rewrite hv
  31. 0031exact hi_left_right_left
  32. 0032rewrite hU
  33. 0033rewrite hV
  34. 0034exact hi_left_right_right
  35. 0035cases hi_right
  36. 0036cases hi_right_right
  37. 0037right
  38. 0038split
  39. 0039rewrite hu
  40. 0040rewrite hU
  41. 0041rewrite hv
  42. 0042rewrite hV
  43. 0043exact hi_right_left
  44. 0044split
  45. 0045rewrite hu
  46. 0046rewrite hv
  47. 0047exact hi_right_right_left
  48. 0048rewrite hU
  49. 0049rewrite hV
  50. 0050exact hi_right_right_right