PE000C

first_primes_double_exponential_bound

For every positive k, construct exactly the first k primes and both power witnesses proving p_k < 2^(2^k).

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

∀ k. ¬k = 0 → ∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ i. k = S z ∧ (InitialPrimeList(x,y,k) ∧ (BetaAt(x,y,z,n) ∧ (PowTwo(k,m) ∧ (PowTwo(m,i) ∧ Lt(n,i)))))

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

Definition DAG

PowTwo(e,p) · 5InitialPrimeChain(b,c,k) · 1InitialPrimeList(b,c,k) · 1Lt(a,b) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

nonzero_is_succ · checked external prerequisiteinitial_prime_chain_bounded_existsbinary_power_two_exists · checked external prerequisitebinary_power_two_dominates_successor · checked external prerequisitebinary_power_two_exponent_monotone · checked external prerequisitelt_of_lt_of_le · checked external prerequisite
Original expanded first-order statement 
forall k. ~(k = 0) -> exists b c j p e B. k = S j /\ (((k = 0 \/ exists pen_last_index_effective_list. k = S pen_last_index_effective_list /\ ((((exists fs_h_pen_effective_list_chain_initial. fs_h_pen_effective_list_chain_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_effective_list_chain_initial. b = fs_q_pen_effective_list_chain_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_effective_list_chain. (exists pc_lt_pen_effective_list_chain_bound. pc_lt_pen_effective_list_chain_bound + S (pen_index_effective_list_chain) = (pen_last_index_effective_list)) -> exists pen_previous_effective_list_chain pen_following_effective_list_chain. (((exists fs_h_pen_effective_list_chain_previous. fs_h_pen_effective_list_chain_previous + S (pen_previous_effective_list_chain) = S ((S (pen_index_effective_list_chain)) * c)) /\ exists fs_q_pen_effective_list_chain_previous. b = fs_q_pen_effective_list_chain_previous * S ((S (pen_index_effective_list_chain)) * c) + (pen_previous_effective_list_chain))) /\ ((((exists fs_h_pen_effective_list_chain_following. fs_h_pen_effective_list_chain_following + S (pen_following_effective_list_chain) = S ((S (S pen_index_effective_list_chain)) * c)) /\ exists fs_q_pen_effective_list_chain_following. b = fs_q_pen_effective_list_chain_following * S ((S (S pen_index_effective_list_chain)) * c) + (pen_following_effective_list_chain))) /\ (((~(pen_following_effective_list_chain = 1) /\ forall bpr_left_pc_pen_effective_list_chain_next_prime bpr_right_pc_pen_effective_list_chain_next_prime. pen_following_effective_list_chain = bpr_left_pc_pen_effective_list_chain_next_prime * bpr_right_pc_pen_effective_list_chain_next_prime -> bpr_left_pc_pen_effective_list_chain_next_prime = 1 \/ bpr_right_pc_pen_effective_list_chain_next_prime = 1)) /\ ((exists pc_lt_pen_effective_list_chain_next_greater. pc_lt_pen_effective_list_chain_next_greater + S (pen_previous_effective_list_chain) = (pen_following_effective_list_chain)) /\ forall pen_comparison_effective_list_chain_next. ((~(pen_comparison_effective_list_chain_next = 1) /\ forall bpr_left_pc_pen_effective_list_chain_next_comparison bpr_right_pc_pen_effective_list_chain_next_comparison. pen_comparison_effective_list_chain_next = bpr_left_pc_pen_effective_list_chain_next_comparison * bpr_right_pc_pen_effective_list_chain_next_comparison -> bpr_left_pc_pen_effective_list_chain_next_comparison = 1 \/ bpr_right_pc_pen_effective_list_chain_next_comparison = 1)) -> (exists pc_lt_pen_effective_list_chain_next_above. pc_lt_pen_effective_list_chain_next_above + S (pen_previous_effective_list_chain) = (pen_comparison_effective_list_chain_next)) -> (exists pc_le_pen_effective_list_chain_next_minimal. pc_le_pen_effective_list_chain_next_minimal + (pen_following_effective_list_chain) = (pen_comparison_effective_list_chain_next)))))))) /\ ((((exists fs_h_pen_effective_last. fs_h_pen_effective_last + S (p) = S ((S (j)) * c)) /\ exists fs_q_pen_effective_last. b = fs_q_pen_effective_last * S ((S (j)) * c) + (p))) /\ ((exists pa_b_bl_pen_effective_exponent pa_c_bl_pen_effective_exponent. ((forall pa_i_bl_pen_effective_exponent_repeat. (exists pa_lt_bl_pen_effective_exponent_repeat_bound. pa_lt_bl_pen_effective_exponent_repeat_bound + S pa_i_bl_pen_effective_exponent_repeat = k) -> (((exists pa_h_bl_pen_effective_exponent_repeat_decoded. pa_h_bl_pen_effective_exponent_repeat_decoded + S (2) = S ((S (pa_i_bl_pen_effective_exponent_repeat)) * pa_c_bl_pen_effective_exponent)) /\ exists pa_q_bl_pen_effective_exponent_repeat_decoded. pa_b_bl_pen_effective_exponent = pa_q_bl_pen_effective_exponent_repeat_decoded * S ((S (pa_i_bl_pen_effective_exponent_repeat)) * pa_c_bl_pen_effective_exponent) + (2)))) /\ (exists pa_u_bl_pen_effective_exponent_product pa_v_bl_pen_effective_exponent_product. ((((exists pa_h_bl_pen_effective_exponent_product_start. pa_h_bl_pen_effective_exponent_product_start + S (1) = S ((S (0)) * pa_v_bl_pen_effective_exponent_product)) /\ exists pa_q_bl_pen_effective_exponent_product_start. pa_u_bl_pen_effective_exponent_product = pa_q_bl_pen_effective_exponent_product_start * S ((S (0)) * pa_v_bl_pen_effective_exponent_product) + (1))) /\ ((((exists pa_h_bl_pen_effective_exponent_product_terminal. pa_h_bl_pen_effective_exponent_product_terminal + S (e) = S ((S (k)) * pa_v_bl_pen_effective_exponent_product)) /\ exists pa_q_bl_pen_effective_exponent_product_terminal. pa_u_bl_pen_effective_exponent_product = pa_q_bl_pen_effective_exponent_product_terminal * S ((S (k)) * pa_v_bl_pen_effective_exponent_product) + (e))) /\ forall pa_i_bl_pen_effective_exponent_product. (exists pa_lt_bl_pen_effective_exponent_product_bound. pa_lt_bl_pen_effective_exponent_product_bound + S pa_i_bl_pen_effective_exponent_product = k) -> exists pa_p_bl_pen_effective_exponent_product pa_r_bl_pen_effective_exponent_product pa_s_bl_pen_effective_exponent_product. ((((exists pa_h_bl_pen_effective_exponent_product_factor. pa_h_bl_pen_effective_exponent_product_factor + S (pa_p_bl_pen_effective_exponent_product) = S ((S (pa_i_bl_pen_effective_exponent_product)) * pa_c_bl_pen_effective_exponent)) /\ exists pa_q_bl_pen_effective_exponent_product_factor. pa_b_bl_pen_effective_exponent = pa_q_bl_pen_effective_exponent_product_factor * S ((S (pa_i_bl_pen_effective_exponent_product)) * pa_c_bl_pen_effective_exponent) + (pa_p_bl_pen_effective_exponent_product))) /\ ((((exists pa_h_bl_pen_effective_exponent_product_partial. pa_h_bl_pen_effective_exponent_product_partial + S (pa_r_bl_pen_effective_exponent_product) = S ((S (pa_i_bl_pen_effective_exponent_product)) * pa_v_bl_pen_effective_exponent_product)) /\ exists pa_q_bl_pen_effective_exponent_product_partial. pa_u_bl_pen_effective_exponent_product = pa_q_bl_pen_effective_exponent_product_partial * S ((S (pa_i_bl_pen_effective_exponent_product)) * pa_v_bl_pen_effective_exponent_product) + (pa_r_bl_pen_effective_exponent_product))) /\ ((((exists pa_h_bl_pen_effective_exponent_product_successor. pa_h_bl_pen_effective_exponent_product_successor + S (pa_s_bl_pen_effective_exponent_product) = S ((S (S pa_i_bl_pen_effective_exponent_product)) * pa_v_bl_pen_effective_exponent_product)) /\ exists pa_q_bl_pen_effective_exponent_product_successor. pa_u_bl_pen_effective_exponent_product = pa_q_bl_pen_effective_exponent_product_successor * S ((S (S pa_i_bl_pen_effective_exponent_product)) * pa_v_bl_pen_effective_exponent_product) + (pa_s_bl_pen_effective_exponent_product))) /\ pa_s_bl_pen_effective_exponent_product = pa_r_bl_pen_effective_exponent_product * pa_p_bl_pen_effective_exponent_product)))))))) /\ ((exists pa_b_bl_pen_effective_bound pa_c_bl_pen_effective_bound. ((forall pa_i_bl_pen_effective_bound_repeat. (exists pa_lt_bl_pen_effective_bound_repeat_bound. pa_lt_bl_pen_effective_bound_repeat_bound + S pa_i_bl_pen_effective_bound_repeat = e) -> (((exists pa_h_bl_pen_effective_bound_repeat_decoded. pa_h_bl_pen_effective_bound_repeat_decoded + S (2) = S ((S (pa_i_bl_pen_effective_bound_repeat)) * pa_c_bl_pen_effective_bound)) /\ exists pa_q_bl_pen_effective_bound_repeat_decoded. pa_b_bl_pen_effective_bound = pa_q_bl_pen_effective_bound_repeat_decoded * S ((S (pa_i_bl_pen_effective_bound_repeat)) * pa_c_bl_pen_effective_bound) + (2)))) /\ (exists pa_u_bl_pen_effective_bound_product pa_v_bl_pen_effective_bound_product. ((((exists pa_h_bl_pen_effective_bound_product_start. pa_h_bl_pen_effective_bound_product_start + S (1) = S ((S (0)) * pa_v_bl_pen_effective_bound_product)) /\ exists pa_q_bl_pen_effective_bound_product_start. pa_u_bl_pen_effective_bound_product = pa_q_bl_pen_effective_bound_product_start * S ((S (0)) * pa_v_bl_pen_effective_bound_product) + (1))) /\ ((((exists pa_h_bl_pen_effective_bound_product_terminal. pa_h_bl_pen_effective_bound_product_terminal + S (B) = S ((S (e)) * pa_v_bl_pen_effective_bound_product)) /\ exists pa_q_bl_pen_effective_bound_product_terminal. pa_u_bl_pen_effective_bound_product = pa_q_bl_pen_effective_bound_product_terminal * S ((S (e)) * pa_v_bl_pen_effective_bound_product) + (B))) /\ forall pa_i_bl_pen_effective_bound_product. (exists pa_lt_bl_pen_effective_bound_product_bound. pa_lt_bl_pen_effective_bound_product_bound + S pa_i_bl_pen_effective_bound_product = e) -> exists pa_p_bl_pen_effective_bound_product pa_r_bl_pen_effective_bound_product pa_s_bl_pen_effective_bound_product. ((((exists pa_h_bl_pen_effective_bound_product_factor. pa_h_bl_pen_effective_bound_product_factor + S (pa_p_bl_pen_effective_bound_product) = S ((S (pa_i_bl_pen_effective_bound_product)) * pa_c_bl_pen_effective_bound)) /\ exists pa_q_bl_pen_effective_bound_product_factor. pa_b_bl_pen_effective_bound = pa_q_bl_pen_effective_bound_product_factor * S ((S (pa_i_bl_pen_effective_bound_product)) * pa_c_bl_pen_effective_bound) + (pa_p_bl_pen_effective_bound_product))) /\ ((((exists pa_h_bl_pen_effective_bound_product_partial. pa_h_bl_pen_effective_bound_product_partial + S (pa_r_bl_pen_effective_bound_product) = S ((S (pa_i_bl_pen_effective_bound_product)) * pa_v_bl_pen_effective_bound_product)) /\ exists pa_q_bl_pen_effective_bound_product_partial. pa_u_bl_pen_effective_bound_product = pa_q_bl_pen_effective_bound_product_partial * S ((S (pa_i_bl_pen_effective_bound_product)) * pa_v_bl_pen_effective_bound_product) + (pa_r_bl_pen_effective_bound_product))) /\ ((((exists pa_h_bl_pen_effective_bound_product_successor. pa_h_bl_pen_effective_bound_product_successor + S (pa_s_bl_pen_effective_bound_product) = S ((S (S pa_i_bl_pen_effective_bound_product)) * pa_v_bl_pen_effective_bound_product)) /\ exists pa_q_bl_pen_effective_bound_product_successor. pa_u_bl_pen_effective_bound_product = pa_q_bl_pen_effective_bound_product_successor * S ((S (S pa_i_bl_pen_effective_bound_product)) * pa_v_bl_pen_effective_bound_product) + (pa_s_bl_pen_effective_bound_product))) /\ pa_s_bl_pen_effective_bound_product = pa_r_bl_pen_effective_bound_product * pa_p_bl_pen_effective_bound_product)))))))) /\ (exists pc_lt_pen_effective_strict. pc_lt_pen_effective_strict + S (p) = (B))))))

Complete tactic proof in conservative notation

All 62 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

62 script commands · 25 reading checkpoints · 4 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.

Named ingredients (1)
initial prime chain bounded exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro k
  2. L2
    intro hk
02Establish hjL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L3
    have hj : exists j. k = S j
  2. L4
    specialize nonzero_is_succ k
  3. L5
    apply nonzero_is_succ
  4. L6
    exact hk
03Separate the logical casesL7–7

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

  1. L7
    cases hj
04Establish hcL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply initial prime chain bounded exists.

  1. L8
    have hc : ∃ b. ∃ c. ∃ p. ∃ P. InitialPrimeChain(b,c,x) ∧ (BetaAt(b,c,x,p) ∧ (PowTwo(S S x,P) ∧ Lt(p,P)))Definitions: InitialPrimeChain(b,c,x)BetaAt(b,c,x,p)PowTwo(S S x,P)Lt(p,P)Original native command in the exact edition
  2. L9
    specialize initial_prime_chain_bounded_exists x
  3. L10
    apply initial_prime_chain_bounded_exists
05Separate the logical casesL11–17

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

  1. L11
    cases hc
  2. L12
    cases hc_witness
  3. L13
    cases hc_witness_witness
  4. L14
    cases hc_witness_witness_witness
  5. L15
    cases hc_witness_witness_witness_witness
  6. L16
    cases hc_witness_witness_witness_witness_right
  7. L17
    cases hc_witness_witness_witness_witness_right_right
06Establish heL18–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply binary power two exists.

  1. L18
    have he : ∃ e. PowTwo(k,e)Definitions: PowTwo(k,e)Original native command in the exact edition
  2. L19
    specialize binary_power_two_exists k
  3. L20
    apply binary_power_two_exists
07Separate the logical casesL21–21

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

  1. L21
    cases he
08Establish hBL22–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply binary power two exists.

  1. L22
    have hB : ∃ B. PowTwo(x5,B)Definitions: PowTwo(x5,B)Original native command in the exact edition
  2. L23
    specialize binary_power_two_exists x5
  3. L24
    apply binary_power_two_exists
09Separate the logical casesL25–25

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

  1. L25
    cases hB
10Construct an explicit witnessL26–31

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

  1. L26
    exists x1
  2. L27
    exists x2
  3. L28
    exists x
  4. L29
    exists x3
  5. L30
    exists x5
  6. L31
    exists x6
11Separate the logical casesL32–32

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

  1. L32
    split
12Use earlier factsL33–33

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

  1. L33
    exact hj_witness
13Separate the logical casesL34–35

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

  1. L34
    split
  2. L35
    right
14Construct an explicit witnessL36–36

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

  1. L36
    exists x
15Separate the logical casesL37–37

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

  1. L37
    split
16Use earlier factsL38–39

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

  1. L38
    exact hj_witness
  2. L39
    exact hc_witness_witness_witness_witness_left
17Separate the logical casesL40–40

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

  1. L40
    split
18Use earlier factsL41–41

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

  1. L41
    exact hc_witness_witness_witness_witness_right_left
19Separate the logical casesL42–42

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

  1. L42
    split
20Use earlier factsL43–43

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

  1. L43
    exact he_witness
21Separate the logical casesL44–44

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

  1. L44
    split
22Use earlier factsL45–54

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

  1. L45
    exact hB_witness
  2. L46
    specialize lt_of_lt_of_le x3
  3. L47
    specialize lt_of_lt_of_le x4
  4. L48
    specialize lt_of_lt_of_le x6
  5. L49
    apply lt_of_lt_of_le
  6. L50
    exact hc_witness_witness_witness_witness_right_right_right
  7. L51
    specialize binary_power_two_exponent_monotone (S (S x))
  8. L52
    specialize binary_power_two_exponent_monotone x5
  9. L53
    specialize binary_power_two_exponent_monotone x4
  10. L54
    specialize binary_power_two_exponent_monotone x6
23Use earlier factsL55–55

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

  1. L55
    apply binary_power_two_exponent_monotone
24Calculate and transport equalitiesL56–56

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L56
    rewrite <- hj_witness
25Use earlier factsL57–62

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

  1. L57
    specialize binary_power_two_dominates_successor k
  2. L58
    specialize binary_power_two_dominates_successor x5
  3. L59
    apply binary_power_two_dominates_successor
  4. L60
    exact he_witness
  5. L61
    exact hc_witness_witness_witness_witness_right_right_left
  6. L62
    exact hB_witness

Library-wide reading audit

Original defined command ledger · 62 lines
  1. 0001intro k
  2. 0002intro hk
  3. 0003have hj : exists j. k = S j
  4. 0004specialize nonzero_is_succ k
  5. 0005apply nonzero_is_succ
  6. 0006exact hk
  7. 0007cases hj
  8. 0008have hc : ∃ b. ∃ c. ∃ p. ∃ P. InitialPrimeChain(b,c,x) ∧ (BetaAt(b,c,x,p) ∧ (PowTwo(S S x,P) ∧ Lt(p,P)))
  9. 0009specialize initial_prime_chain_bounded_exists x
  10. 0010apply initial_prime_chain_bounded_exists
  11. 0011cases hc
  12. 0012cases hc_witness
  13. 0013cases hc_witness_witness
  14. 0014cases hc_witness_witness_witness
  15. 0015cases hc_witness_witness_witness_witness
  16. 0016cases hc_witness_witness_witness_witness_right
  17. 0017cases hc_witness_witness_witness_witness_right_right
  18. 0018have he : ∃ e. PowTwo(k,e)
  19. 0019specialize binary_power_two_exists k
  20. 0020apply binary_power_two_exists
  21. 0021cases he
  22. 0022have hB : ∃ B. PowTwo(x5,B)
  23. 0023specialize binary_power_two_exists x5
  24. 0024apply binary_power_two_exists
  25. 0025cases hB
  26. 0026exists x1
  27. 0027exists x2
  28. 0028exists x
  29. 0029exists x3
  30. 0030exists x5
  31. 0031exists x6
  32. 0032split
  33. 0033exact hj_witness
  34. 0034split
  35. 0035right
  36. 0036exists x
  37. 0037split
  38. 0038exact hj_witness
  39. 0039exact hc_witness_witness_witness_witness_left
  40. 0040split
  41. 0041exact hc_witness_witness_witness_witness_right_left
  42. 0042split
  43. 0043exact he_witness
  44. 0044split
  45. 0045exact hB_witness
  46. 0046specialize lt_of_lt_of_le x3
  47. 0047specialize lt_of_lt_of_le x4
  48. 0048specialize lt_of_lt_of_le x6
  49. 0049apply lt_of_lt_of_le
  50. 0050exact hc_witness_witness_witness_witness_right_right_right
  51. 0051specialize binary_power_two_exponent_monotone (S (S x))
  52. 0052specialize binary_power_two_exponent_monotone x5
  53. 0053specialize binary_power_two_exponent_monotone x4
  54. 0054specialize binary_power_two_exponent_monotone x6
  55. 0055apply binary_power_two_exponent_monotone
  56. 0056rewrite <- hj_witness
  57. 0057specialize binary_power_two_dominates_successor k
  58. 0058specialize binary_power_two_dominates_successor x5
  59. 0059apply binary_power_two_dominates_successor
  60. 0060exact he_witness
  61. 0061exact hc_witness_witness_witness_witness_right_right_left
  62. 0062exact hB_witness