BP000A

bertrand_chain_successor_preserves_guard

Every strict Bertrand successor preserves the initial 1<n domain guard.

Alpha v34 checked-use · first admitted v20 · 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.

Exact theorem in conservative defined notation

∀ a. ∀ p. Lt(1,a) → BertrandWindow(a,p) → Lt(1,p)

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

Definition DAG

BertrandWindow(n,p) · 1Lt(a,b) · 2

Actual proof prerequisites

lt_trans · checked external prerequisite
Original expanded first-order statement 
forall a p. (exists bcf_lt_gap_bpc_chain_terminal. bcf_lt_gap_bpc_chain_terminal + S (1) = a) -> ((((~(p = 1) /\ forall frm_prime_left_bpc_next_prime frm_prime_right_bpc_next_prime. p = frm_prime_left_bpc_next_prime * frm_prime_right_bpc_next_prime -> frm_prime_left_bpc_next_prime = 1 \/ frm_prime_right_bpc_next_prime = 1)) /\ ((exists bcf_lt_gap_bpc_next_lower. bcf_lt_gap_bpc_next_lower + S (a) = p) /\ (exists bcf_lt_gap_bpc_next_upper. bcf_lt_gap_bpc_next_upper + S (p) = a + a)))) -> (exists bcf_lt_gap_bpc_chain_next. bcf_lt_gap_bpc_chain_next + S (1) = p)

Complete unchanged native tactic proof

All 12 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

12 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.

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 assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro p
  3. L3
    intro hguard
  4. L4
    intro hwindow
02Separate the logical casesL5–6

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

  1. L5
    cases hwindow
  2. L6
    cases hwindow_right
03Use earlier factsL7–12

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

  1. L7
    specialize lt_trans 1
  2. L8
    specialize lt_trans a
  3. L9
    specialize lt_trans p
  4. L10
    apply lt_trans
  5. L11
    exact hguard
  6. L12
    exact hwindow_right_left

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro p
  3. 0003intro hguard
  4. 0004intro hwindow
  5. 0005cases hwindow
  6. 0006cases hwindow_right
  7. 0007specialize lt_trans 1
  8. 0008specialize lt_trans a
  9. 0009specialize lt_trans p
  10. 0010apply lt_trans
  11. 0011exact hguard
  12. 0012exact hwindow_right_left