PC000F

beta_cutoff_prefix_empty

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

The empty cutoff prefix is valid for every source and threshold.

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 u b c d f. forall pc_index_cut_empty. (exists pc_lt_cut_empty_bound. pc_lt_cut_empty_bound + S (pc_index_cut_empty) = (0)) -> exists pc_bit_cut_empty. (((exists fs_h_pc_cut_empty_entry. fs_h_pc_cut_empty_entry + S (pc_bit_cut_empty) = S ((S (pc_index_cut_empty)) * f)) /\ exists fs_q_pc_cut_empty_entry. d = fs_q_pc_cut_empty_entry * S ((S (pc_index_cut_empty)) * f) + (pc_bit_cut_empty))) /\ ((((exists pc_lt_cut_empty_choice_below. pc_lt_cut_empty_choice_below + S (pc_index_cut_empty) = (u)) /\ pc_bit_cut_empty = 0) \/ ((exists pc_le_cut_empty_choice_above. pc_le_cut_empty_choice_above + (u) = (pc_index_cut_empty)) /\ (((exists fs_h_pc_cut_empty_choice_source. fs_h_pc_cut_empty_choice_source + S (pc_bit_cut_empty) = S ((S (pc_index_cut_empty)) * c)) /\ exists fs_q_pc_cut_empty_choice_source. b = fs_q_pc_cut_empty_choice_source * S ((S (pc_index_cut_empty)) * c) + (pc_bit_cut_empty))))))

Constructive proof overview

Generated structural guide

The empty cutoff prefix is valid for every source and threshold.

The unchanged tactic script uses 2 declared prerequisites and contains 14 exact native proof lines.

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

Proof neighborhood

Direct dependencies

lt_not_le Stable theorem; checked-use authorized zero_le Stable theorem; checked-use authorized

Direct dependents

PC0013 beta_cutoff_prefix_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

14 script commands · 3 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 assumptionsL1–7

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

  1. L1
    intro u
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro f
  6. L6
    intro i
  7. L7
    intro hi
02Separate the logical casesL8–8

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

  1. L8
    exfalso
03Use earlier factsL9–14

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

  1. L9
    specialize lt_not_le i
  2. L10
    specialize lt_not_le 0
  3. L11
    apply lt_not_le
  4. L12
    exact hi
  5. L13
    specialize zero_le i
  6. L14
    apply zero_le

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro u
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro f
  6. 0006intro i
  7. 0007intro hi
  8. 0008exfalso
  9. 0009specialize lt_not_le i
  10. 0010specialize lt_not_le 0
  11. 0011apply lt_not_le
  12. 0012exact hi
  13. 0013specialize zero_le i
  14. 0014apply zero_le