CD0023

finite_bit_zero_nonmember

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized CD0017 finite_bit_zero_one_conflict

Direct dependents

CD002C finite_partial_sumset_empty CD0041 prime_modular_normalized_boundary_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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