BA0028

cf_convergent_old_history_state_unique

The existing G071 beta history has unique actual dividend, divisor, and forward quotient-list coordinates at each index.

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

∀ h. ∀ e. ∀ j. ∀ a. ∀ b. ∀ s. ∀ A. ∀ B. ∀ T. BetaAt(h,e,j,(a + ((b + s) · S (b + s) + (s + s))) · S (a + ((b + s) · S (b + s) + (s + s))) + ((b + s) · S (b + s) + (s + s) + ((b + s) · S (b + s) + (s + s)))) → BetaAt(h,e,j,(A + ((B + T) · S (B + T) + (T + T))) · S (A + ((B + T) · S (B + T) + (T + T))) + ((B + T) · S (B + T) + (T + T) + ((B + T) · S (B + T) + (T + T)))) → a = A ∧ (b = B ∧ s = T)

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

Definition DAG

NaturalPair(z,a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisitepair_code_injective · checked external prerequisite
Original expanded first-order statement 
forall h e j a b s A B T. (((exists ff_h_cf_old_unique_left_state. ff_h_cf_old_unique_left_state + S (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))) = S ((S (j)) * e)) /\ exists ff_q_cf_old_unique_left_state. h = ff_q_cf_old_unique_left_state * S ((S (j)) * e) + (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))))) -> (((exists ff_h_cf_old_unique_right_state. ff_h_cf_old_unique_right_state + S (((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) * S ((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) + ((((B) + (T)) * S ((B) + (T)) + ((T) + (T))) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T))))) = S ((S (j)) * e)) /\ exists ff_q_cf_old_unique_right_state. h = ff_q_cf_old_unique_right_state * S ((S (j)) * e) + (((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) * S ((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) + ((((B) + (T)) * S ((B) + (T)) + ((T) + (T))) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T))))))) -> ((a = A) /\ ((b = B) /\ (s = T)))

Complete tactic proof in conservative notation

All 40 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

40 script commands · 8 reading checkpoints · 2 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 h
  2. L2
    intro e
  3. L3
    intro j
  4. L4
    intro a
  5. L5
    intro b
  6. L6
    intro s
  7. L7
    intro A
  8. L8
    intro B
  9. L9
    intro T
  10. L10
    intro h1
02Fix variables and assumptionsL11–11

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

  1. L11
    intro h2
03Establish hzL12–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L12
    have hz : NaturalPair((a + ((b + s) · S (b + s) + (s + s))) · S (a + ((b + s) · S (b + s) + (s + s))) + ((b + s) · S (b + s) + (s + s) + ((b + s) · S (b + s) + (s + s))),A,(B + T) · S (B + T) + (T + T))Definitions: NaturalPair((a + ((b + s) · S (b + s) + (s + s))) · S (a + ((b + s) · S (b + s) + (s + s))) + ((b + s) · S (b + s) + (s + s) + ((b + s) · S (b + s) + (s + s))),A,(B + T) · S (B + T) + (T + T))Original native command in the exact edition
  2. L13
    specialize beta_at_unique (h)
  3. L14
    specialize beta_at_unique (e)
  4. L15
    specialize beta_at_unique (j)
  5. L16
    specialize beta_at_unique (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  6. L17
    specialize beta_at_unique (((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) * S ((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) + ((((B) + (T)) * S ((B) + (T)) + ((T) + (T))) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))))
  7. L18
    apply beta_at_unique
  8. L19
    exact h1
  9. L20
    exact h2
04Establish hpL21–29

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pair code injective.

  1. L21
    have hp : a = A ∧ NaturalPair((b + s) · S (b + s) + (s + s),B,T)Definitions: NaturalPair((b + s) · S (b + s) + (s + s),B,T)Original native command in the exact edition
  2. L22
    specialize pair_code_injective (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  3. L23
    specialize pair_code_injective (a)
  4. L24
    specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  5. L25
    specialize pair_code_injective (A)
  6. L26
    specialize pair_code_injective (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))
  7. L27
    apply pair_code_injective
  8. L28
    refl
  9. L29
    exact hz
05Separate the logical casesL30–31

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

  1. L30
    cases hp
  2. L31
    split
06Use earlier factsL32–38

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

  1. L32
    exact hp_left
  2. L33
    specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  3. L34
    specialize pair_code_injective (b)
  4. L35
    specialize pair_code_injective (s)
  5. L36
    specialize pair_code_injective (B)
  6. L37
    specialize pair_code_injective (T)
  7. L38
    apply pair_code_injective
07Calculate and transport equalitiesL39–39

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

  1. L39
    refl
08Use earlier factsL40–40

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

  1. L40
    exact hp_right

Library-wide reading audit

Original defined command ledger · 40 lines
  1. 0001intro h
  2. 0002intro e
  3. 0003intro j
  4. 0004intro a
  5. 0005intro b
  6. 0006intro s
  7. 0007intro A
  8. 0008intro B
  9. 0009intro T
  10. 0010intro h1
  11. 0011intro h2
  12. 0012have hz : NaturalPair((a + ((b + s) · S (b + s) + (s + s))) · S (a + ((b + s) · S (b + s) + (s + s))) + ((b + s) · S (b + s) + (s + s) + ((b + s) · S (b + s) + (s + s))),A,(B + T) · S (B + T) + (T + T))
  13. 0013specialize beta_at_unique (h)
  14. 0014specialize beta_at_unique (e)
  15. 0015specialize beta_at_unique (j)
  16. 0016specialize beta_at_unique (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  17. 0017specialize beta_at_unique (((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) * S ((A) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))) + ((((B) + (T)) * S ((B) + (T)) + ((T) + (T))) + (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))))
  18. 0018apply beta_at_unique
  19. 0019exact h1
  20. 0020exact h2
  21. 0021have hp : a = A ∧ NaturalPair((b + s) · S (b + s) + (s + s),B,T)
  22. 0022specialize pair_code_injective (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))))
  23. 0023specialize pair_code_injective (a)
  24. 0024specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  25. 0025specialize pair_code_injective (A)
  26. 0026specialize pair_code_injective (((B) + (T)) * S ((B) + (T)) + ((T) + (T)))
  27. 0027apply pair_code_injective
  28. 0028refl
  29. 0029exact hz
  30. 0030cases hp
  31. 0031split
  32. 0032exact hp_left
  33. 0033specialize pair_code_injective (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))
  34. 0034specialize pair_code_injective (b)
  35. 0035specialize pair_code_injective (s)
  36. 0036specialize pair_code_injective (B)
  37. 0037specialize pair_code_injective (T)
  38. 0038apply pair_code_injective
  39. 0039refl
  40. 0040exact hp_right