PC0036

prime_count_one

One contributes no prime: the exact prime count at one is zero.

Alpha v34 checked-use · first admitted v27 · 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.

These are the exact finite integer inequalities with constant 8, for every N≥2. The proof uses constructive binomial and primorial infrastructure; it does not assume logarithms, asymptotic estimates, the prime number theorem, or a factorization oracle.

Exact theorem in conservative defined notation

∀ k. PrimeCount(1,k) → k = 0

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

Definition DAG

PrimeCount(x,z) · 1Prime(p) · 2BetaAt(b,c,i,x) · 1Sum(b,c,l,z) · 1

Actual proof prerequisites

beta_sum_succ_decompose · checked external prerequisiteprime_bit_prefix_entryle_refl · checked external prerequisitebeta_sum_zero · checked external prerequisite
Original expanded first-order statement 
forall k. (exists pc_code_count_one_source pc_scale_count_one_source. (forall pc_index_count_one_source_mask. (exists pc_lt_count_one_source_mask_bound. pc_lt_count_one_source_mask_bound + S (pc_index_count_one_source_mask) = (1)) -> exists pc_bit_count_one_source_mask. (((exists fs_h_pc_count_one_source_mask_entry. fs_h_pc_count_one_source_mask_entry + S (pc_bit_count_one_source_mask) = S ((S (pc_index_count_one_source_mask)) * pc_scale_count_one_source)) /\ exists fs_q_pc_count_one_source_mask_entry. pc_code_count_one_source = fs_q_pc_count_one_source_mask_entry * S ((S (pc_index_count_one_source_mask)) * pc_scale_count_one_source) + (pc_bit_count_one_source_mask))) /\ (((((~(S (pc_index_count_one_source_mask) = 1) /\ forall bpr_left_pc_count_one_source_mask_choice_prime bpr_right_pc_count_one_source_mask_choice_prime. S (pc_index_count_one_source_mask) = bpr_left_pc_count_one_source_mask_choice_prime * bpr_right_pc_count_one_source_mask_choice_prime -> bpr_left_pc_count_one_source_mask_choice_prime = 1 \/ bpr_right_pc_count_one_source_mask_choice_prime = 1)) /\ pc_bit_count_one_source_mask = 1) \/ (~((~(S (pc_index_count_one_source_mask) = 1) /\ forall bpr_left_pc_count_one_source_mask_choice_prime bpr_right_pc_count_one_source_mask_choice_prime. S (pc_index_count_one_source_mask) = bpr_left_pc_count_one_source_mask_choice_prime * bpr_right_pc_count_one_source_mask_choice_prime -> bpr_left_pc_count_one_source_mask_choice_prime = 1 \/ bpr_right_pc_count_one_source_mask_choice_prime = 1)) /\ pc_bit_count_one_source_mask = 0)))) /\ (exists fs_u_pc_count_one_source_sum fs_v_pc_count_one_source_sum. ((((exists fs_h_pc_count_one_source_sum_body_start. fs_h_pc_count_one_source_sum_body_start + S (0) = S ((S (0)) * fs_v_pc_count_one_source_sum)) /\ exists fs_q_pc_count_one_source_sum_body_start. fs_u_pc_count_one_source_sum = fs_q_pc_count_one_source_sum_body_start * S ((S (0)) * fs_v_pc_count_one_source_sum) + (0))) /\ ((((exists fs_h_pc_count_one_source_sum_body_terminal. fs_h_pc_count_one_source_sum_body_terminal + S (k) = S ((S (1)) * fs_v_pc_count_one_source_sum)) /\ exists fs_q_pc_count_one_source_sum_body_terminal. fs_u_pc_count_one_source_sum = fs_q_pc_count_one_source_sum_body_terminal * S ((S (1)) * fs_v_pc_count_one_source_sum) + (k))) /\ forall fs_i_pc_count_one_source_sum_body_steps. (exists fs_lt_pc_count_one_source_sum_body_steps_bound. fs_lt_pc_count_one_source_sum_body_steps_bound + S fs_i_pc_count_one_source_sum_body_steps = 1) -> exists fs_a_pc_count_one_source_sum_body_steps fs_r_pc_count_one_source_sum_body_steps fs_s_pc_count_one_source_sum_body_steps. ((((exists fs_h_pc_count_one_source_sum_body_steps_summand. fs_h_pc_count_one_source_sum_body_steps_summand + S (fs_a_pc_count_one_source_sum_body_steps) = S ((S (fs_i_pc_count_one_source_sum_body_steps)) * pc_scale_count_one_source)) /\ exists fs_q_pc_count_one_source_sum_body_steps_summand. pc_code_count_one_source = fs_q_pc_count_one_source_sum_body_steps_summand * S ((S (fs_i_pc_count_one_source_sum_body_steps)) * pc_scale_count_one_source) + (fs_a_pc_count_one_source_sum_body_steps))) /\ ((((exists fs_h_pc_count_one_source_sum_body_steps_partial. fs_h_pc_count_one_source_sum_body_steps_partial + S (fs_r_pc_count_one_source_sum_body_steps) = S ((S (fs_i_pc_count_one_source_sum_body_steps)) * fs_v_pc_count_one_source_sum)) /\ exists fs_q_pc_count_one_source_sum_body_steps_partial. fs_u_pc_count_one_source_sum = fs_q_pc_count_one_source_sum_body_steps_partial * S ((S (fs_i_pc_count_one_source_sum_body_steps)) * fs_v_pc_count_one_source_sum) + (fs_r_pc_count_one_source_sum_body_steps))) /\ ((((exists fs_h_pc_count_one_source_sum_body_steps_successor. fs_h_pc_count_one_source_sum_body_steps_successor + S (fs_s_pc_count_one_source_sum_body_steps) = S ((S (S fs_i_pc_count_one_source_sum_body_steps)) * fs_v_pc_count_one_source_sum)) /\ exists fs_q_pc_count_one_source_sum_body_steps_successor. fs_u_pc_count_one_source_sum = fs_q_pc_count_one_source_sum_body_steps_successor * S ((S (S fs_i_pc_count_one_source_sum_body_steps)) * fs_v_pc_count_one_source_sum) + (fs_s_pc_count_one_source_sum_body_steps))) /\ fs_s_pc_count_one_source_sum_body_steps = fs_r_pc_count_one_source_sum_body_steps + fs_a_pc_count_one_source_sum_body_steps))))))) -> k = 0

Complete tactic proof in conservative notation

All 44 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

44 script commands · 11 reading checkpoints · 3 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)
prime bit prefix entry
01Fix variables and assumptionsL1–2

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

  1. L1
    intro k
  2. L2
    intro h
02Separate the logical casesL3–5

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

  1. L3
    cases h
  2. L4
    cases h_witness
  3. L5
    cases h_witness_witness
03Establish hdL6–12

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

  1. L6
    have hd : ∃ a. ∃ w. BetaAt(x,x1,0,a) ∧ (Sum(x,x1,0,w) ∧ k = w + a)Definitions: BetaAt(x,x1,0,a)Sum(x,x1,0,w)Original native command in the exact edition
  2. L7
    specialize beta_sum_succ_decompose x
  3. L8
    specialize beta_sum_succ_decompose x1
  4. L9
    specialize beta_sum_succ_decompose 0
  5. L10
    specialize beta_sum_succ_decompose k
  6. L11
    apply beta_sum_succ_decompose
  7. L12
    exact h_witness_witness_right
04Separate the logical casesL13–16

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

  1. L13
    cases hd
  2. L14
    cases hd_witness
  3. L15
    cases hd_witness_witness
  4. L16
    cases hd_witness_witness_right
05Establish hcL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime bit prefix entry.

  1. L17
    have hc : Prime(1) ∧ x2 = 1 ∨ ¬Prime(1) ∧ x2 = 0Definitions: Prime(1)Original native command in the exact edition
  2. L18
    specialize prime_bit_prefix_entry x
  3. L19
    specialize prime_bit_prefix_entry x1
  4. L20
    specialize prime_bit_prefix_entry 1
  5. L21
    specialize prime_bit_prefix_entry 0
  6. L22
    specialize prime_bit_prefix_entry x2
  7. L23
    apply prime_bit_prefix_entry
  8. L24
    exact h_witness_witness_left
  9. L25
    specialize le_refl 1
  10. L26
    apply le_refl
06Use earlier factsL27–27

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

  1. L27
    exact hd_witness_witness_left
07Separate the logical casesL28–31

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

  1. L28
    cases hc
  2. L29
    cases hc_left
  3. L30
    cases hc_left_left
  4. L31
    exfalso
08Use earlier factsL32–32

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

  1. L32
    apply hc_left_left_left
09Calculate and transport equalitiesL33–33

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

  1. L33
    refl
10Separate the logical casesL34–34

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

  1. L34
    cases hc_right
11Establish hzL35–44

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta sum zero.

  1. L35
    have hz : x3 = 0
  2. L36
    specialize beta_sum_zero x
  3. L37
    specialize beta_sum_zero x1
  4. L38
    specialize beta_sum_zero x3
  5. L39
    apply beta_sum_zero
  6. L40
    exact hd_witness_witness_right_left
  7. L41
    rewrite hd_witness_witness_right_right
  8. L42
    rewrite hz
  9. L43
    rewrite hc_right_right
  10. L44
    norm_num

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro k
  2. 0002intro h
  3. 0003cases h
  4. 0004cases h_witness
  5. 0005cases h_witness_witness
  6. 0006have hd : ∃ a. ∃ w. BetaAt(x,x1,0,a) ∧ (Sum(x,x1,0,w) ∧ k = w + a)
  7. 0007specialize beta_sum_succ_decompose x
  8. 0008specialize beta_sum_succ_decompose x1
  9. 0009specialize beta_sum_succ_decompose 0
  10. 0010specialize beta_sum_succ_decompose k
  11. 0011apply beta_sum_succ_decompose
  12. 0012exact h_witness_witness_right
  13. 0013cases hd
  14. 0014cases hd_witness
  15. 0015cases hd_witness_witness
  16. 0016cases hd_witness_witness_right
  17. 0017have hc : Prime(1) ∧ x2 = 1 ∨ ¬Prime(1) ∧ x2 = 0
  18. 0018specialize prime_bit_prefix_entry x
  19. 0019specialize prime_bit_prefix_entry x1
  20. 0020specialize prime_bit_prefix_entry 1
  21. 0021specialize prime_bit_prefix_entry 0
  22. 0022specialize prime_bit_prefix_entry x2
  23. 0023apply prime_bit_prefix_entry
  24. 0024exact h_witness_witness_left
  25. 0025specialize le_refl 1
  26. 0026apply le_refl
  27. 0027exact hd_witness_witness_left
  28. 0028cases hc
  29. 0029cases hc_left
  30. 0030cases hc_left_left
  31. 0031exfalso
  32. 0032apply hc_left_left_left
  33. 0033refl
  34. 0034cases hc_right
  35. 0035have hz : x3 = 0
  36. 0036specialize beta_sum_zero x
  37. 0037specialize beta_sum_zero x1
  38. 0038specialize beta_sum_zero x3
  39. 0039apply beta_sum_zero
  40. 0040exact hd_witness_witness_right_left
  41. 0041rewrite hd_witness_witness_right_right
  42. 0042rewrite hz
  43. 0043rewrite hc_right_right
  44. 0044norm_num