PE0007

initial_prime_chain_singleton_exists

The one-entry initial prime chain is an actual beta code of the prime two.

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

∃ b. ∃ c. InitialPrimeChain(b,c,0)

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

Definition DAG

InitialPrimeChain(b,c,k) · 1

Actual proof prerequisites

bertrand_chain_singleton_code_exists · checked external prerequisitelt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
exists b c. (((exists fs_h_pen_singleton_initial. fs_h_pen_singleton_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_singleton_initial. b = fs_q_pen_singleton_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_singleton. (exists pc_lt_pen_singleton_bound. pc_lt_pen_singleton_bound + S (pen_index_singleton) = (0)) -> exists pen_previous_singleton pen_following_singleton. (((exists fs_h_pen_singleton_previous. fs_h_pen_singleton_previous + S (pen_previous_singleton) = S ((S (pen_index_singleton)) * c)) /\ exists fs_q_pen_singleton_previous. b = fs_q_pen_singleton_previous * S ((S (pen_index_singleton)) * c) + (pen_previous_singleton))) /\ ((((exists fs_h_pen_singleton_following. fs_h_pen_singleton_following + S (pen_following_singleton) = S ((S (S pen_index_singleton)) * c)) /\ exists fs_q_pen_singleton_following. b = fs_q_pen_singleton_following * S ((S (S pen_index_singleton)) * c) + (pen_following_singleton))) /\ (((~(pen_following_singleton = 1) /\ forall bpr_left_pc_pen_singleton_next_prime bpr_right_pc_pen_singleton_next_prime. pen_following_singleton = bpr_left_pc_pen_singleton_next_prime * bpr_right_pc_pen_singleton_next_prime -> bpr_left_pc_pen_singleton_next_prime = 1 \/ bpr_right_pc_pen_singleton_next_prime = 1)) /\ ((exists pc_lt_pen_singleton_next_greater. pc_lt_pen_singleton_next_greater + S (pen_previous_singleton) = (pen_following_singleton)) /\ forall pen_comparison_singleton_next. ((~(pen_comparison_singleton_next = 1) /\ forall bpr_left_pc_pen_singleton_next_comparison bpr_right_pc_pen_singleton_next_comparison. pen_comparison_singleton_next = bpr_left_pc_pen_singleton_next_comparison * bpr_right_pc_pen_singleton_next_comparison -> bpr_left_pc_pen_singleton_next_comparison = 1 \/ bpr_right_pc_pen_singleton_next_comparison = 1)) -> (exists pc_lt_pen_singleton_next_above. pc_lt_pen_singleton_next_above + S (pen_previous_singleton) = (pen_comparison_singleton_next)) -> (exists pc_le_pen_singleton_next_minimal. pc_le_pen_singleton_next_minimal + (pen_following_singleton) = (pen_comparison_singleton_next)))))

Complete tactic proof in conservative notation

All 16 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

16 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.

01Use earlier factsL1–1

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

  1. L1
    specialize bertrand_chain_singleton_code_exists 2
02Separate the logical casesL2–3

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

  1. L2
    cases bertrand_chain_singleton_code_exists
  2. L3
    cases bertrand_chain_singleton_code_exists_witness
03Construct an explicit witnessL4–5

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

  1. L4
    exists x
  2. L5
    exists x1
04Separate the logical casesL6–6

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

  1. L6
    split
05Use earlier factsL7–7

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

  1. L7
    exact bertrand_chain_singleton_code_exists_witness_witness
06Fix variables and assumptionsL8–9

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

  1. L8
    intro i
  2. L9
    intro hi
07Separate the logical casesL10–10

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

  1. L10
    exfalso
08Use earlier factsL11–16

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

  1. L11
    specialize lt_not_le i
  2. L12
    specialize lt_not_le 0
  3. L13
    apply lt_not_le
  4. L14
    exact hi
  5. L15
    specialize zero_le i
  6. L16
    apply zero_le

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001specialize bertrand_chain_singleton_code_exists 2
  2. 0002cases bertrand_chain_singleton_code_exists
  3. 0003cases bertrand_chain_singleton_code_exists_witness
  4. 0004exists x
  5. 0005exists x1
  6. 0006split
  7. 0007exact bertrand_chain_singleton_code_exists_witness_witness
  8. 0008intro i
  9. 0009intro hi
  10. 0010exfalso
  11. 0011specialize lt_not_le i
  12. 0012specialize lt_not_le 0
  13. 0013apply lt_not_le
  14. 0014exact hi
  15. 0015specialize zero_le i
  16. 0016apply zero_le