BA0009

cf_approximation_identity_current_absolute_error

The invariant's current error is exactly the absolute cross-product error, including the zero terminal value.

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. ∀ E. ∀ F. AlternatingConvergentIdentity(a,b,u,U,v,V,E,F) → NaturalAbsDifference(a · v,b · u,E)

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

Definition DAG

NaturalAbsDifference(p,n,D) · 1AlternatingConvergentIdentity(a,b,u,U,v,V,E,F) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall a b u U v V E 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))))) -> (((a * v) = (b * u) + (E)) \/ ((b * u) = (a * v) + (E)))

Complete tactic proof in conservative notation

All 18 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

18 script commands · 5 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 assumptionsL1–9

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 E
  8. L8
    intro F
  9. L9
    intro hi
02Separate the logical casesL10–13

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

  1. L10
    cases hi
  2. L11
    cases hi_left
  3. L12
    cases hi_left_right
  4. L13
    left
03Use earlier factsL14–14

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

  1. L14
    exact hi_left_right_left
04Separate the logical casesL15–17

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

  1. L15
    cases hi_right
  2. L16
    cases hi_right_right
  3. L17
    right
05Use earlier factsL18–18

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

  1. L18
    exact hi_right_right_left

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro u
  4. 0004intro U
  5. 0005intro v
  6. 0006intro V
  7. 0007intro E
  8. 0008intro F
  9. 0009intro hi
  10. 0010cases hi
  11. 0011cases hi_left
  12. 0012cases hi_left_right
  13. 0013left
  14. 0014exact hi_left_right_left
  15. 0015cases hi_right
  16. 0016cases hi_right_right
  17. 0017right
  18. 0018exact hi_right_right_left