BT00QD

prime_two_le

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every prime is at least two in witness-defined order.

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 PA statement

forall p. ((~(p = 1) /\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \/ frm_prime_right_bpvl_prime = 1)) -> (exists bpvl_gap_prime_two. bpvl_gap_prime_two + (2) = (p))

Structural proof guide

Every prime is at least two in witness-defined order.

Direct prerequisites: prime_is_succ_succ. The authored body proceeds by case analysis (1), intermediate claims (1), equality transport (3).

Proof neighborhood

Direct dependencies

BT00AW prime_is_succ_succ

Direct dependents

BT00QF prime_power_exponent_le

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

15 script commands · 6 reading checkpoints · 1 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.

Named ingredients (1)
prime is succ succ
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hp
02Establish hshapeL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime is succ succ.

  1. L3
    have hshape : exists k. p = S (S k)
  2. L4
    specialize prime_is_succ_succ p
  3. L5
    apply prime_is_succ_succ
  4. L6
    exact hp
03Separate the logical casesL7–7

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

  1. L7
    cases hshape
04Construct an explicit witnessL8–8

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

  1. L8
    exists x
05Calculate and transport equalitiesL9–14

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    trans S (S x)
  2. L10
    rewrite PA4
  3. L11
    rewrite PA4
  4. L12
    rewrite PA3
  5. L13
    refl
  6. L14
    symm
06Use earlier factsL15–15

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

  1. L15
    exact hshape_witness

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003have hshape : exists k. p = S (S k)
  4. 0004specialize prime_is_succ_succ p
  5. 0005apply prime_is_succ_succ
  6. 0006exact hp
  7. 0007cases hshape
  8. 0008exists x
  9. 0009trans S (S x)
  10. 0010rewrite PA4
  11. 0011rewrite PA4
  12. 0012rewrite PA3
  13. 0013refl
  14. 0014symm
  15. 0015exact hshape_witness