EL000C

lte_prime_nondivisor_one

An actual prime does not divide one.

Alpha v34 checked-use · first admitted v29 · 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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ p. ¬p = 1 ∧ (∀ x. ∀ y. p = x · y → x = 1 ∨ y = 1) → ¬Dvd(p,1)

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

Definition DAG

Dvd(d,n) · 1

Actual proof prerequisites

mul_eq_one_components · checked external prerequisite
Original expanded first-order statement 
forall p. (~((p) = 1) /\ forall pvs_left_unit_prime pvs_right_unit_prime. (p) = pvs_left_unit_prime * pvs_right_unit_prime -> pvs_left_unit_prime = 1 \/ pvs_right_unit_prime = 1) -> ~(exists olte_factor_unit. (1) = (p) * olte_factor_unit)

Complete tactic proof in conservative notation

All 14 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

14 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.

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 p
  2. L2
    intro hp
  3. L3
    intro hdiv
02Separate the logical casesL4–5

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

  1. L4
    cases hdiv
  2. L5
    cases hp
03Use earlier factsL6–6

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

  1. L6
    apply hp_left
04Establish hcomponentsL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul eq one components.

  1. L7
    have hcomponents : p = 1 /\ x = 1
  2. L8
    specialize mul_eq_one_components (p)
  3. L9
    specialize mul_eq_one_components (x)
  4. L10
    apply mul_eq_one_components
  5. L11
    symm
  6. L12
    exact hdiv_witness
05Separate the logical casesL13–13

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

  1. L13
    cases hcomponents
06Use earlier factsL14–14

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

  1. L14
    exact hcomponents_left

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003intro hdiv
  4. 0004cases hdiv
  5. 0005cases hp
  6. 0006apply hp_left
  7. 0007have hcomponents : p = 1 /\ x = 1
  8. 0008specialize mul_eq_one_components (p)
  9. 0009specialize mul_eq_one_components (x)
  10. 0010apply mul_eq_one_components
  11. 0011symm
  12. 0012exact hdiv_witness
  13. 0013cases hcomponents
  14. 0014exact hcomponents_left