PC001B

primorial_factor_choice_one_le

Every dense primorial factor is at least one, including nonprime positions.

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

∀ i. ∀ a. Prime(S i) ∧ a = S i ∨ ¬Prime(S i) ∧ a = 1 → Lt(0,a)

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

Definition DAG

Lt(a,b) · 1Prime(p) · 2

Actual proof prerequisites

le_refl · checked external prerequisite
Original expanded first-order statement 
forall i a. (((((~(S (i) = 1) /\ forall bpr_left_pc_prim_positive_source_prime bpr_right_pc_prim_positive_source_prime. S (i) = bpr_left_pc_prim_positive_source_prime * bpr_right_pc_prim_positive_source_prime -> bpr_left_pc_prim_positive_source_prime = 1 \/ bpr_right_pc_prim_positive_source_prime = 1)) /\ a = S (i)) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_prim_positive_source_prime bpr_right_pc_prim_positive_source_prime. S (i) = bpr_left_pc_prim_positive_source_prime * bpr_right_pc_prim_positive_source_prime -> bpr_left_pc_prim_positive_source_prime = 1 \/ bpr_right_pc_prim_positive_source_prime = 1)) /\ a = 1))) -> (exists pc_le_prim_positive_result. pc_le_prim_positive_result + (1) = (a))

Complete tactic proof in conservative notation

All 12 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

12 script commands · 8 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 assumptionsL1–3

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

  1. L1
    intro i
  2. L2
    intro a
  3. L3
    intro h
02Separate the logical casesL4–5

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

  1. L4
    cases h
  2. L5
    cases h_left
03Calculate and transport equalitiesL6–6

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

  1. L6
    rewrite h_left_right
04Construct an explicit witnessL7–7

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

  1. L7
    exists i
05Calculate and transport equalitiesL8–8

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

  1. L8
    simp
06Separate the logical casesL9–9

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

  1. L9
    cases h_right
07Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite h_right_right
08Use earlier factsL11–12

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

  1. L11
    specialize le_refl 1
  2. L12
    apply le_refl

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro i
  2. 0002intro a
  3. 0003intro h
  4. 0004cases h
  5. 0005cases h_left
  6. 0006rewrite h_left_right
  7. 0007exists i
  8. 0008simp
  9. 0009cases h_right
  10. 0010rewrite h_right_right
  11. 0011specialize le_refl 1
  12. 0012apply le_refl