PE0004

least_prime_above_unique

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

The least prime above any natural is uniquely determined by its minimality, not by its code.

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 a p q. (((~(p = 1) /\ forall bpr_left_pc_pen_unique_left_prime bpr_right_pc_pen_unique_left_prime. p = bpr_left_pc_pen_unique_left_prime * bpr_right_pc_pen_unique_left_prime -> bpr_left_pc_pen_unique_left_prime = 1 \/ bpr_right_pc_pen_unique_left_prime = 1)) /\ ((exists pc_lt_pen_unique_left_greater. pc_lt_pen_unique_left_greater + S (a) = (p)) /\ forall pen_comparison_unique_left. ((~(pen_comparison_unique_left = 1) /\ forall bpr_left_pc_pen_unique_left_comparison bpr_right_pc_pen_unique_left_comparison. pen_comparison_unique_left = bpr_left_pc_pen_unique_left_comparison * bpr_right_pc_pen_unique_left_comparison -> bpr_left_pc_pen_unique_left_comparison = 1 \/ bpr_right_pc_pen_unique_left_comparison = 1)) -> (exists pc_lt_pen_unique_left_above. pc_lt_pen_unique_left_above + S (a) = (pen_comparison_unique_left)) -> (exists pc_le_pen_unique_left_minimal. pc_le_pen_unique_left_minimal + (p) = (pen_comparison_unique_left)))) -> (((~(q = 1) /\ forall bpr_left_pc_pen_unique_right_prime bpr_right_pc_pen_unique_right_prime. q = bpr_left_pc_pen_unique_right_prime * bpr_right_pc_pen_unique_right_prime -> bpr_left_pc_pen_unique_right_prime = 1 \/ bpr_right_pc_pen_unique_right_prime = 1)) /\ ((exists pc_lt_pen_unique_right_greater. pc_lt_pen_unique_right_greater + S (a) = (q)) /\ forall pen_comparison_unique_right. ((~(pen_comparison_unique_right = 1) /\ forall bpr_left_pc_pen_unique_right_comparison bpr_right_pc_pen_unique_right_comparison. pen_comparison_unique_right = bpr_left_pc_pen_unique_right_comparison * bpr_right_pc_pen_unique_right_comparison -> bpr_left_pc_pen_unique_right_comparison = 1 \/ bpr_right_pc_pen_unique_right_comparison = 1)) -> (exists pc_lt_pen_unique_right_above. pc_lt_pen_unique_right_above + S (a) = (pen_comparison_unique_right)) -> (exists pc_le_pen_unique_right_minimal. pc_le_pen_unique_right_minimal + (q) = (pen_comparison_unique_right)))) -> p = q

Constructive proof overview

Generated structural guide

The least prime above any natural is uniquely determined by its minimality, not by its code.

The unchanged tactic script uses 1 declared prerequisite and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_antisymm Stable theorem; checked-use authorized

Direct dependents

PE0006 least_prime_above_exists_unique

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

20 script commands · 4 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro p
  3. L3
    intro q
  4. L4
    intro hp
  5. L5
    intro hq
02Separate the logical casesL6–9

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

  1. L6
    cases hp
  2. L7
    cases hp_right
  3. L8
    cases hq
  4. L9
    cases hq_right
03Use earlier factsL10–19

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

  1. L10
    specialize le_antisymm p
  2. L11
    specialize le_antisymm q
  3. L12
    apply le_antisymm
  4. L13
    specialize hp_right_right q
  5. L14
    apply hp_right_right
  6. L15
    exact hq_left
  7. L16
    exact hq_right_left
  8. L17
    specialize hq_right_right p
  9. L18
    apply hq_right_right
  10. L19
    exact hp_left
04Use earlier factsL20–20

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

  1. L20
    exact hp_right_left

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro a
  2. 0002intro p
  3. 0003intro q
  4. 0004intro hp
  5. 0005intro hq
  6. 0006cases hp
  7. 0007cases hp_right
  8. 0008cases hq
  9. 0009cases hq_right
  10. 0010specialize le_antisymm p
  11. 0011specialize le_antisymm q
  12. 0012apply le_antisymm
  13. 0013specialize hp_right_right q
  14. 0014apply hp_right_right
  15. 0015exact hq_left
  16. 0016exact hq_right_left
  17. 0017specialize hq_right_right p
  18. 0018apply hq_right_right
  19. 0019exact hp_left
  20. 0020exact hp_right_left