PC0010

beta_cutoff_prefix_drop_last

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

Restrict an actual cutoff table to its preceding prefix.

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 l. (forall pc_index_cut_drop_source. (exists pc_lt_cut_drop_source_bound. pc_lt_cut_drop_source_bound + S (pc_index_cut_drop_source) = (S l)) -> exists pc_bit_cut_drop_source. (((exists fs_h_pc_cut_drop_source_entry. fs_h_pc_cut_drop_source_entry + S (pc_bit_cut_drop_source) = S ((S (pc_index_cut_drop_source)) * f)) /\ exists fs_q_pc_cut_drop_source_entry. d = fs_q_pc_cut_drop_source_entry * S ((S (pc_index_cut_drop_source)) * f) + (pc_bit_cut_drop_source))) /\ ((((exists pc_lt_cut_drop_source_choice_below. pc_lt_cut_drop_source_choice_below + S (pc_index_cut_drop_source) = (u)) /\ pc_bit_cut_drop_source = 0) \/ ((exists pc_le_cut_drop_source_choice_above. pc_le_cut_drop_source_choice_above + (u) = (pc_index_cut_drop_source)) /\ (((exists fs_h_pc_cut_drop_source_choice_source. fs_h_pc_cut_drop_source_choice_source + S (pc_bit_cut_drop_source) = S ((S (pc_index_cut_drop_source)) * c)) /\ exists fs_q_pc_cut_drop_source_choice_source. b = fs_q_pc_cut_drop_source_choice_source * S ((S (pc_index_cut_drop_source)) * c) + (pc_bit_cut_drop_source))))))) -> (forall pc_index_cut_drop_target. (exists pc_lt_cut_drop_target_bound. pc_lt_cut_drop_target_bound + S (pc_index_cut_drop_target) = (l)) -> exists pc_bit_cut_drop_target. (((exists fs_h_pc_cut_drop_target_entry. fs_h_pc_cut_drop_target_entry + S (pc_bit_cut_drop_target) = S ((S (pc_index_cut_drop_target)) * f)) /\ exists fs_q_pc_cut_drop_target_entry. d = fs_q_pc_cut_drop_target_entry * S ((S (pc_index_cut_drop_target)) * f) + (pc_bit_cut_drop_target))) /\ ((((exists pc_lt_cut_drop_target_choice_below. pc_lt_cut_drop_target_choice_below + S (pc_index_cut_drop_target) = (u)) /\ pc_bit_cut_drop_target = 0) \/ ((exists pc_le_cut_drop_target_choice_above. pc_le_cut_drop_target_choice_above + (u) = (pc_index_cut_drop_target)) /\ (((exists fs_h_pc_cut_drop_target_choice_source. fs_h_pc_cut_drop_target_choice_source + S (pc_bit_cut_drop_target) = S ((S (pc_index_cut_drop_target)) * c)) /\ exists fs_q_pc_cut_drop_target_choice_source. b = fs_q_pc_cut_drop_target_choice_source * S ((S (pc_index_cut_drop_target)) * c) + (pc_bit_cut_drop_target)))))))

Constructive proof overview

Generated structural guide

Restrict an actual cutoff table to its preceding prefix.

The unchanged tactic script uses 1 declared prerequisite and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_succ Stable theorem; checked-use authorized

Direct dependents

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

15 script commands · 2 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 assumptionsL1–9

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 l
  7. L7
    intro h
  8. L8
    intro i
  9. L9
    intro hi
02Use earlier factsL10–15

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

  1. L10
    specialize h i
  2. L11
    apply h
  3. L12
    specialize le_succ (S i)
  4. L13
    specialize le_succ l
  5. L14
    apply le_succ
  6. L15
    exact hi

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro u
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro f
  6. 0006intro l
  7. 0007intro h
  8. 0008intro i
  9. 0009intro hi
  10. 0010specialize h i
  11. 0011apply h
  12. 0012specialize le_succ (S i)
  13. 0013specialize le_succ l
  14. 0014apply le_succ
  15. 0015exact hi