EL000C

lte_prime_nondivisor_one

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

An actual prime does not divide one.

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 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)

Constructive proof overview

Generated structural guide

An actual prime does not divide one.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mul_eq_one_components Stable theorem; checked-use authorized

Direct dependents

EL0010 lte_nondivisor_power EL0016 lte_prime_self_valuation_value EL001C lte_coprime_power_difference_quotient

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

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.

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 exact 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