PE0004

least_prime_above_unique

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

Alpha v34 checked-use · first admitted v28 · 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.

Every successor is the globally least prime above its predecessor. This is not a sparse Bertrand chain. The bound theorem constructs the list and both power witnesses from k≠0 alone; the separate total-list theorem includes k=0.

Exact theorem in conservative defined notation

∀ a. ∀ p. ∀ q. NextPrime(a,p) → NextPrime(a,q) → p = q

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

Definition DAG

NextPrime(a,p) · 2

Actual proof prerequisites

le_antisymm · checked external prerequisite
Original expanded first-order 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

Complete tactic proof in conservative notation

All 20 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

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.

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 assumptionsL1–5

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 defined 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