PE0001

foundation_primes_above_every_bound

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

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

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 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)))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_unbounded Stable theorem; checked-use authorized

Direct dependents

none

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

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.

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 exact 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