PE0001

foundation_primes_above_every_bound

G021: every supplied natural bound has an actual larger prime; no nonempty list or prime-search witness is supplied.

Alpha v34 checked-use · first admitted v28 · 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.

Every successor is the globally least prime above its predecessor. This is not a sparse Bertrand chain. The bound theorem constructs the list and both power witnesses from k≠0 alone; the separate total-list theorem includes k=0.

Exact theorem in conservative defined notation

∀ B. ∃ p. Prime(p) ∧ Lt(B,p)

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

Definition DAG

Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_unbounded · checked external prerequisite
Original expanded first-order statement 
forall B. exists p. (((~(p = 1) /\ forall frm_prime_left_fsat_unbounded frm_prime_right_fsat_unbounded. p = frm_prime_left_fsat_unbounded * frm_prime_right_fsat_unbounded -> frm_prime_left_fsat_unbounded = 1 \/ frm_prime_right_fsat_unbounded = 1)) /\ (exists fsat_gap_unbounded. fsat_gap_unbounded + S (B) = (p)))

Complete tactic proof in conservative notation

All 8 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

8 script commands · 6 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 assumptionsL1–1

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

  1. L1
    intro B
02Use earlier factsL2–2

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

  1. L2
    specialize prime_unbounded B
03Separate the logical casesL3–4

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

  1. L3
    cases prime_unbounded
  2. L4
    cases prime_unbounded_witness
04Construct an explicit witnessL5–5

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists x
05Separate the logical casesL6–6

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

  1. L6
    split
06Use earlier factsL7–8

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

  1. L7
    exact prime_unbounded_witness_right
  2. L8
    exact prime_unbounded_witness_left

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro B
  2. 0002specialize prime_unbounded B
  3. 0003cases prime_unbounded
  4. 0004cases prime_unbounded_witness
  5. 0005exists x
  6. 0006split
  7. 0007exact prime_unbounded_witness_right
  8. 0008exact prime_unbounded_witness_left