CD0018

finite_beta_value_one_iff

For an actual decoded entry, being one is exactly characteristic membership.

Alpha v34 checked-use · first admitted v27 · 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.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ i. ∀ a. BetaAt(b,c,i,a) → (a = 1 → BetaAt(b,c,i,1)) ∧ (BetaAt(b,c,i,1) → a = 1)

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

Definition DAG

BetaAt(b,c,i,x) · 3

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c i a. (((exists fs_h_fms_value_one. fs_h_fms_value_one + S (a) = S ((S (i)) * c)) /\ exists fs_q_fms_value_one. b = fs_q_fms_value_one * S ((S (i)) * c) + (a))) -> ((((a=1) -> (((exists fs_h_fms_value_one_result. fs_h_fms_value_one_result + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_value_one_result. b = fs_q_fms_value_one_result * S ((S (i)) * c) + (1)))) /\ ((((exists fs_h_fms_value_one_result. fs_h_fms_value_one_result + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_value_one_result. b = fs_q_fms_value_one_result * S ((S (i)) * c) + (1))) -> (a=1))))

Complete tactic proof in conservative notation

All 19 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

19 script commands · 7 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 assumptionsL1–5

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
  4. L4
    intro a
  5. L5
    intro ha
02Separate the logical casesL6–6

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

  1. L6
    split
03Fix variables and assumptionsL7–7

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

  1. L7
    intro he
04Calculate and transport equalitiesL8–9

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

  1. L8
    rewrite he at ha
  2. L9
    rewrite he at ha
05Use earlier factsL10–10

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

  1. L10
    exact ha
06Fix variables and assumptionsL11–11

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

  1. L11
    intro hone
07Use earlier factsL12–19

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

  1. L12
    specialize beta_at_unique b
  2. L13
    specialize beta_at_unique c
  3. L14
    specialize beta_at_unique i
  4. L15
    specialize beta_at_unique a
  5. L16
    specialize beta_at_unique 1
  6. L17
    apply beta_at_unique
  7. L18
    exact ha
  8. L19
    exact hone

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004intro a
  5. 0005intro ha
  6. 0006split
  7. 0007intro he
  8. 0008rewrite he at ha
  9. 0009rewrite he at ha
  10. 0010exact ha
  11. 0011intro hone
  12. 0012specialize beta_at_unique b
  13. 0013specialize beta_at_unique c
  14. 0014specialize beta_at_unique i
  15. 0015specialize beta_at_unique a
  16. 0016specialize beta_at_unique 1
  17. 0017apply beta_at_unique
  18. 0018exact ha
  19. 0019exact hone