CD0018

finite_beta_value_one_iff

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

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

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 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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 19 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

Direct dependents

CD001B finite_bit_union_intersection_count_balance

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

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.

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 exact 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