BA0027

cf_approximation_natural_error_as_signed

Every natural numerator, including zero, is an actual signed numerator with negative component zero.

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. ∀ r. ∀ t. ∀ D. NaturalAbsDifference(a · t,b · r,D) → RationalApproximationError(a,b,r,0,t,D)

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

Definition DAG

NaturalAbsDifference(p,n,D) · 1RationalApproximationError(a,b,rp,rn,t,E) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall a b r t D. (((a * t) = (b * r) + (D)) \/ ((b * r) = (a * t) + (D))) -> (((a * t + b * 0) = (b * r) + (D)) \/ ((b * r) = (a * t + b * 0) + (D)))

Complete tactic proof in conservative notation

All 15 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

15 script commands · 8 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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro r
  4. L4
    intro t
  5. L5
    intro D
  6. L6
    intro h
02Separate the logical casesL7–8

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

  1. L7
    cases h
  2. L8
    left
03Calculate and transport equalitiesL9–10

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

  1. L9
    trans a * t
  2. L10
    simp
04Use earlier factsL11–11

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

  1. L11
    exact h_left
05Separate the logical casesL12–12

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

  1. L12
    right
06Calculate and transport equalitiesL13–13

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

  1. L13
    trans a * t + D
07Use earlier factsL14–14

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

  1. L14
    exact h_right
08Calculate and transport equalitiesL15–15

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

  1. L15
    simp

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro r
  4. 0004intro t
  5. 0005intro D
  6. 0006intro h
  7. 0007cases h
  8. 0008left
  9. 0009trans a * t
  10. 0010simp
  11. 0011exact h_left
  12. 0012right
  13. 0013trans a * t + D
  14. 0014exact h_right
  15. 0015simp