BA0005

cf_approximation_empty_matrix_identity

The identity matrix gives the exact two initial signed errors b and a, including zero values.

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. AlternatingConvergentIdentity(a,b,1,0,0,1,b,a)

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

Definition DAG

AlternatingConvergentIdentity(a,b,u,U,v,V,E,F) · 1

Actual proof prerequisites

zero_add · checked external prerequisite
Original expanded first-order statement 
forall a b. ((((1 * 1 + 1 = 0 * 0) /\ ((a * 0 = b * 1 + b) /\ (b * 0 = a * 1 + a)))) \/ (((0 * 0 + 1 = 1 * 1) /\ ((b * 1 = a * 0 + b) /\ (a * 1 = b * 0 + a)))))

Complete tactic proof in conservative notation

All 8 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

8 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Separate the logical casesL3–4

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

  1. L3
    right
  2. L4
    split
03Calculate and transport equalitiesL5–5

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

  1. L5
    simp [zero_add]
04Separate the logical casesL6–6

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

  1. L6
    split
05Calculate and transport equalitiesL7–8

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

  1. L7
    simp [zero_add]
  2. L8
    simp [zero_add]

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003right
  4. 0004split
  5. 0005simp [zero_add]
  6. 0006split
  7. 0007simp [zero_add]
  8. 0008simp [zero_add]