BA0020

cf_approximation_zero_current_sum_as_difference

The only small-denominator nonnegative combination is an actual previous-column difference with zero current coefficient.

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

∀ u. ∀ U. ∀ v. ∀ V. ∀ rp. ∀ rn. ∀ t. ∀ c. ∀ d. rp = rn + (c · u + d · U) ∧ t = 0 + (c · v + d · V) → c = 0 → rp + 0 · u = rn + d · U ∧ t + 0 · v = 0 + d · V

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

Definition DAG

none

Actual proof prerequisites

mul_zero_left · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall u U v V rp rn t c d. ((((rp) = (rn) + ((c) * (u) + (d) * (U))) /\ ((t) = (0) + ((c) * (v) + (d) * (V))))) -> c = 0 -> ((((rp) + (0) * (u) = (rn) + (d) * (U)) /\ ((t) + (0) * (v) = (0) + (d) * (V))))

Complete tactic proof in conservative notation

All 25 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

25 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 assumptionsL1–10

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

  1. L1
    intro u
  2. L2
    intro U
  3. L3
    intro v
  4. L4
    intro V
  5. L5
    intro rp
  6. L6
    intro rn
  7. L7
    intro t
  8. L8
    intro c
  9. L9
    intro d
  10. L10
    intro h
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hc
03Separate the logical casesL12–13

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

  1. L12
    cases h
  2. L13
    split
04Calculate and transport equalitiesL14–16

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

  1. L14
    trans rp
  2. L15
    simp [mul_zero_left]
  3. L16
    trans rn + (c * u + d * U)
05Use earlier factsL17–17

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

  1. L17
    exact h_left
06Calculate and transport equalitiesL18–22

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

  1. L18
    rewrite hc
  2. L19
    simp [mul_zero_left, zero_add]
  3. L20
    trans t
  4. L21
    simp [mul_zero_left]
  5. L22
    trans 0 + (c * v + d * V)
07Use earlier factsL23–23

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

  1. L23
    exact h_right
08Calculate and transport equalitiesL24–25

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

  1. L24
    rewrite hc
  2. L25
    simp [mul_zero_left, zero_add]

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro rp
  6. 0006intro rn
  7. 0007intro t
  8. 0008intro c
  9. 0009intro d
  10. 0010intro h
  11. 0011intro hc
  12. 0012cases h
  13. 0013split
  14. 0014trans rp
  15. 0015simp [mul_zero_left]
  16. 0016trans rn + (c * u + d * U)
  17. 0017exact h_left
  18. 0018rewrite hc
  19. 0019simp [mul_zero_left, zero_add]
  20. 0020trans t
  21. 0021simp [mul_zero_left]
  22. 0022trans 0 + (c * v + d * V)
  23. 0023exact h_right
  24. 0024rewrite hc
  25. 0025simp [mul_zero_left, zero_add]