PC0010

beta_cutoff_prefix_drop_last

Restrict an actual cutoff table to its preceding prefix.

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.

These are the exact finite integer inequalities with constant 8, for every N≥2. The proof uses constructive binomial and primorial infrastructure; it does not assume logarithms, asymptotic estimates, the prime number theorem, or a factorization oracle.

Exact theorem in conservative defined notation

∀ u. ∀ b. ∀ c. ∀ d. ∀ f. ∀ l. BetaCutoffPrefix(u,b,c,d,f,S l) → BetaCutoffPrefix(u,b,c,d,f,l)

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

Definition DAG

BetaCutoffPrefix(u,b,c,d,f,l) · 2

Actual proof prerequisites

le_succ · checked external prerequisite
Original expanded first-order 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)))))))

Complete tactic proof in conservative notation

All 15 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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