CD0023

finite_bit_zero_nonmember

A decoded zero bit excludes characteristic membership without any additional decidability assumption.

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. BetaAt(b,c,i,0) → ¬BetaAt(b,c,i,1)

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

Definition DAG

BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisitefinite_bit_zero_one_conflict
Original expanded first-order statement 
forall b c i. (((exists fs_h_fms_zero_not_one. fs_h_fms_zero_not_one + S (0) = S ((S (i)) * c)) /\ exists fs_q_fms_zero_not_one. b = fs_q_fms_zero_not_one * S ((S (i)) * c) + (0))) -> ~(((exists fs_h_fms_zero_not_one_result. fs_h_fms_zero_not_one_result + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_zero_not_one_result. b = fs_q_fms_zero_not_one_result * S ((S (i)) * c) + (1)))

Complete tactic proof in conservative notation

All 16 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

16 script commands · 4 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.

Named ingredients (1)
finite bit zero one conflict
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 hz
  5. L5
    intro ho
02Use earlier factsL6–7

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

  1. L6
    specialize finite_bit_zero_one_conflict 0
  2. L7
    apply finite_bit_zero_one_conflict
03Calculate and transport equalitiesL8–8

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

  1. L8
    refl
04Use earlier factsL9–16

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

  1. L9
    specialize beta_at_unique b
  2. L10
    specialize beta_at_unique c
  3. L11
    specialize beta_at_unique i
  4. L12
    specialize beta_at_unique 0
  5. L13
    specialize beta_at_unique 1
  6. L14
    apply beta_at_unique
  7. L15
    exact hz
  8. L16
    exact ho

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004intro hz
  5. 0005intro ho
  6. 0006specialize finite_bit_zero_one_conflict 0
  7. 0007apply finite_bit_zero_one_conflict
  8. 0008refl
  9. 0009specialize beta_at_unique b
  10. 0010specialize beta_at_unique c
  11. 0011specialize beta_at_unique i
  12. 0012specialize beta_at_unique 0
  13. 0013specialize beta_at_unique 1
  14. 0014apply beta_at_unique
  15. 0015exact hz
  16. 0016exact ho