BL000E

binary_length_one

The positive integer one has exactly one binary digit.

Alpha v34 checked-use · first admitted v22 · 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.

G101 and G102 were OPEN when these BitLen foundations were first admitted in Alpha v22. Both are now CLOSED in Alpha v23: complete canonical exponent digits and both exact logarithmic execution bounds are proved.

Exact theorem in conservative defined notation

BitLen(1,1)

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

Definition DAG

BitLen(n,ell) · 1

Actual proof prerequisites

binary_power_two_existsbinary_power_two_zero_valuebinary_power_two_successor_double
Original expanded first-order statement 
(((1) = 0 /\ (1) = 1) \/ exists ff_exponent_bl_one ff_lower_bl_one ff_upper_bl_one. (((1) = S ff_exponent_bl_one) /\ ((exists ff_positive_bl_one. ff_positive_bl_one + 1 = (1)) /\ ((exists pa_b_bl_one_lower pa_c_bl_one_lower. ((forall pa_i_bl_one_lower_repeat. (exists pa_lt_bl_one_lower_repeat_bound. pa_lt_bl_one_lower_repeat_bound + S pa_i_bl_one_lower_repeat = ff_exponent_bl_one) -> (((exists pa_h_bl_one_lower_repeat_decoded. pa_h_bl_one_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_one_lower_repeat)) * pa_c_bl_one_lower)) /\ exists pa_q_bl_one_lower_repeat_decoded. pa_b_bl_one_lower = pa_q_bl_one_lower_repeat_decoded * S ((S (pa_i_bl_one_lower_repeat)) * pa_c_bl_one_lower) + (2)))) /\ (exists pa_u_bl_one_lower_product pa_v_bl_one_lower_product. ((((exists pa_h_bl_one_lower_product_start. pa_h_bl_one_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_start. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_start * S ((S (0)) * pa_v_bl_one_lower_product) + (1))) /\ ((((exists pa_h_bl_one_lower_product_terminal. pa_h_bl_one_lower_product_terminal + S (ff_lower_bl_one) = S ((S (ff_exponent_bl_one)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_terminal. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_terminal * S ((S (ff_exponent_bl_one)) * pa_v_bl_one_lower_product) + (ff_lower_bl_one))) /\ forall pa_i_bl_one_lower_product. (exists pa_lt_bl_one_lower_product_bound. pa_lt_bl_one_lower_product_bound + S pa_i_bl_one_lower_product = ff_exponent_bl_one) -> exists pa_p_bl_one_lower_product pa_r_bl_one_lower_product pa_s_bl_one_lower_product. ((((exists pa_h_bl_one_lower_product_factor. pa_h_bl_one_lower_product_factor + S (pa_p_bl_one_lower_product) = S ((S (pa_i_bl_one_lower_product)) * pa_c_bl_one_lower)) /\ exists pa_q_bl_one_lower_product_factor. pa_b_bl_one_lower = pa_q_bl_one_lower_product_factor * S ((S (pa_i_bl_one_lower_product)) * pa_c_bl_one_lower) + (pa_p_bl_one_lower_product))) /\ ((((exists pa_h_bl_one_lower_product_partial. pa_h_bl_one_lower_product_partial + S (pa_r_bl_one_lower_product) = S ((S (pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_partial. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_partial * S ((S (pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product) + (pa_r_bl_one_lower_product))) /\ ((((exists pa_h_bl_one_lower_product_successor. pa_h_bl_one_lower_product_successor + S (pa_s_bl_one_lower_product) = S ((S (S pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_successor. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_successor * S ((S (S pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product) + (pa_s_bl_one_lower_product))) /\ pa_s_bl_one_lower_product = pa_r_bl_one_lower_product * pa_p_bl_one_lower_product)))))))) /\ ((exists pa_b_bl_one_upper pa_c_bl_one_upper. ((forall pa_i_bl_one_upper_repeat. (exists pa_lt_bl_one_upper_repeat_bound. pa_lt_bl_one_upper_repeat_bound + S pa_i_bl_one_upper_repeat = 1) -> (((exists pa_h_bl_one_upper_repeat_decoded. pa_h_bl_one_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_one_upper_repeat)) * pa_c_bl_one_upper)) /\ exists pa_q_bl_one_upper_repeat_decoded. pa_b_bl_one_upper = pa_q_bl_one_upper_repeat_decoded * S ((S (pa_i_bl_one_upper_repeat)) * pa_c_bl_one_upper) + (2)))) /\ (exists pa_u_bl_one_upper_product pa_v_bl_one_upper_product. ((((exists pa_h_bl_one_upper_product_start. pa_h_bl_one_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_start. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_start * S ((S (0)) * pa_v_bl_one_upper_product) + (1))) /\ ((((exists pa_h_bl_one_upper_product_terminal. pa_h_bl_one_upper_product_terminal + S (ff_upper_bl_one) = S ((S (1)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_terminal. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_terminal * S ((S (1)) * pa_v_bl_one_upper_product) + (ff_upper_bl_one))) /\ forall pa_i_bl_one_upper_product. (exists pa_lt_bl_one_upper_product_bound. pa_lt_bl_one_upper_product_bound + S pa_i_bl_one_upper_product = 1) -> exists pa_p_bl_one_upper_product pa_r_bl_one_upper_product pa_s_bl_one_upper_product. ((((exists pa_h_bl_one_upper_product_factor. pa_h_bl_one_upper_product_factor + S (pa_p_bl_one_upper_product) = S ((S (pa_i_bl_one_upper_product)) * pa_c_bl_one_upper)) /\ exists pa_q_bl_one_upper_product_factor. pa_b_bl_one_upper = pa_q_bl_one_upper_product_factor * S ((S (pa_i_bl_one_upper_product)) * pa_c_bl_one_upper) + (pa_p_bl_one_upper_product))) /\ ((((exists pa_h_bl_one_upper_product_partial. pa_h_bl_one_upper_product_partial + S (pa_r_bl_one_upper_product) = S ((S (pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_partial. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_partial * S ((S (pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product) + (pa_r_bl_one_upper_product))) /\ ((((exists pa_h_bl_one_upper_product_successor. pa_h_bl_one_upper_product_successor + S (pa_s_bl_one_upper_product) = S ((S (S pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_successor. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_successor * S ((S (S pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product) + (pa_s_bl_one_upper_product))) /\ pa_s_bl_one_upper_product = pa_r_bl_one_upper_product * pa_p_bl_one_upper_product)))))))) /\ ((exists ff_lower_gap_bl_one. ff_lower_gap_bl_one + (ff_lower_bl_one) = (1)) /\ (exists ff_upper_gap_bl_one. ff_upper_gap_bl_one + S (1) = (ff_upper_bl_one))))))))

Complete unchanged native tactic proof

All 41 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

41 script commands · 22 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 (3)
binary power two existsbinary power two zero valuebinary power two successor double

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Establish hzeroL1–3

Establish this local claim before using it. It is not an additional assumption.

  1. L1
    have hzero : ∃ p. PowTwo(0,p)Definitions: PowTwoOriginal native command in the exact edition
  2. L2
    specialize binary_power_two_exists 0
  3. L3
    exact binary_power_two_exists
02Separate the logical casesL4–4

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

  1. L4
    cases hzero
03Establish honeL5–7

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hone : ∃ q. PowTwo(1,q)Definitions: PowTwoOriginal native command in the exact edition
  2. L6
    specialize binary_power_two_exists 1
  3. L7
    exact binary_power_two_exists
04Separate the logical casesL8–8

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

  1. L8
    cases hone
05Establish hvalueL9–12

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

  1. L9
    have hvalue : x = 1
  2. L10
    specialize binary_power_two_zero_value x
  3. L11
    apply binary_power_two_zero_value
  4. L12
    exact hzero_witness
06Establish hdoubleL13–19

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

  1. L13
    have hdouble : x1 = x + x
  2. L14
    specialize binary_power_two_successor_double 0
  3. L15
    specialize binary_power_two_successor_double x
  4. L16
    specialize binary_power_two_successor_double x1
  5. L17
    apply binary_power_two_successor_double
  6. L18
    exact hzero_witness
  7. L19
    exact hone_witness
07Separate the logical casesL20–20

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

  1. L20
    right
08Construct an explicit witnessL21–23

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

  1. L21
    exists 0
  2. L22
    exists x
  3. L23
    exists x1
09Separate the logical casesL24–24

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

  1. L24
    split
10Calculate and transport equalitiesL25–25

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

  1. L25
    refl
11Separate the logical casesL26–26

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

  1. L26
    split
12Construct an explicit witnessL27–27

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

  1. L27
    exists 0
13Calculate and transport equalitiesL28–28

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

  1. L28
    simp
14Separate the logical casesL29–29

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

  1. L29
    split
15Use earlier factsL30–30

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

  1. L30
    exact hzero_witness
16Separate the logical casesL31–31

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

  1. L31
    split
17Use earlier factsL32–32

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

  1. L32
    exact hone_witness
18Separate the logical casesL33–33

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

  1. L33
    split
19Construct an explicit witnessL34–34

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

  1. L34
    exists 0
20Calculate and transport equalitiesL35–36

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

  1. L35
    rewrite hvalue
  2. L36
    simp
21Construct an explicit witnessL37–37

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

  1. L37
    exists 0
22Calculate and transport equalitiesL38–41

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

  1. L38
    rewrite hdouble
  2. L39
    rewrite hvalue
  3. L40
    rewrite hvalue
  4. L41
    simp

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001have hzero : exists p. (exists pa_b_bl_one_lower pa_c_bl_one_lower. ((forall pa_i_bl_one_lower_repeat. (exists pa_lt_bl_one_lower_repeat_bound. pa_lt_bl_one_lower_repeat_bound + S pa_i_bl_one_lower_repeat = 0) -> (((exists pa_h_bl_one_lower_repeat_decoded. pa_h_bl_one_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_one_lower_repeat)) * pa_c_bl_one_lower)) /\ exists pa_q_bl_one_lower_repeat_decoded. pa_b_bl_one_lower = pa_q_bl_one_lower_repeat_decoded * S ((S (pa_i_bl_one_lower_repeat)) * pa_c_bl_one_lower) + (2)))) /\ (exists pa_u_bl_one_lower_product pa_v_bl_one_lower_product. ((((exists pa_h_bl_one_lower_product_start. pa_h_bl_one_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_start. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_start * S ((S (0)) * pa_v_bl_one_lower_product) + (1))) /\ ((((exists pa_h_bl_one_lower_product_terminal. pa_h_bl_one_lower_product_terminal + S (p) = S ((S (0)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_terminal. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_terminal * S ((S (0)) * pa_v_bl_one_lower_product) + (p))) /\ forall pa_i_bl_one_lower_product. (exists pa_lt_bl_one_lower_product_bound. pa_lt_bl_one_lower_product_bound + S pa_i_bl_one_lower_product = 0) -> exists pa_p_bl_one_lower_product pa_r_bl_one_lower_product pa_s_bl_one_lower_product. ((((exists pa_h_bl_one_lower_product_factor. pa_h_bl_one_lower_product_factor + S (pa_p_bl_one_lower_product) = S ((S (pa_i_bl_one_lower_product)) * pa_c_bl_one_lower)) /\ exists pa_q_bl_one_lower_product_factor. pa_b_bl_one_lower = pa_q_bl_one_lower_product_factor * S ((S (pa_i_bl_one_lower_product)) * pa_c_bl_one_lower) + (pa_p_bl_one_lower_product))) /\ ((((exists pa_h_bl_one_lower_product_partial. pa_h_bl_one_lower_product_partial + S (pa_r_bl_one_lower_product) = S ((S (pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_partial. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_partial * S ((S (pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product) + (pa_r_bl_one_lower_product))) /\ ((((exists pa_h_bl_one_lower_product_successor. pa_h_bl_one_lower_product_successor + S (pa_s_bl_one_lower_product) = S ((S (S pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product)) /\ exists pa_q_bl_one_lower_product_successor. pa_u_bl_one_lower_product = pa_q_bl_one_lower_product_successor * S ((S (S pa_i_bl_one_lower_product)) * pa_v_bl_one_lower_product) + (pa_s_bl_one_lower_product))) /\ pa_s_bl_one_lower_product = pa_r_bl_one_lower_product * pa_p_bl_one_lower_product))))))))
  2. 0002specialize binary_power_two_exists 0
  3. 0003exact binary_power_two_exists
  4. 0004cases hzero
  5. 0005have hone : exists q. (exists pa_b_bl_one_upper pa_c_bl_one_upper. ((forall pa_i_bl_one_upper_repeat. (exists pa_lt_bl_one_upper_repeat_bound. pa_lt_bl_one_upper_repeat_bound + S pa_i_bl_one_upper_repeat = 1) -> (((exists pa_h_bl_one_upper_repeat_decoded. pa_h_bl_one_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_one_upper_repeat)) * pa_c_bl_one_upper)) /\ exists pa_q_bl_one_upper_repeat_decoded. pa_b_bl_one_upper = pa_q_bl_one_upper_repeat_decoded * S ((S (pa_i_bl_one_upper_repeat)) * pa_c_bl_one_upper) + (2)))) /\ (exists pa_u_bl_one_upper_product pa_v_bl_one_upper_product. ((((exists pa_h_bl_one_upper_product_start. pa_h_bl_one_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_start. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_start * S ((S (0)) * pa_v_bl_one_upper_product) + (1))) /\ ((((exists pa_h_bl_one_upper_product_terminal. pa_h_bl_one_upper_product_terminal + S (q) = S ((S (1)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_terminal. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_terminal * S ((S (1)) * pa_v_bl_one_upper_product) + (q))) /\ forall pa_i_bl_one_upper_product. (exists pa_lt_bl_one_upper_product_bound. pa_lt_bl_one_upper_product_bound + S pa_i_bl_one_upper_product = 1) -> exists pa_p_bl_one_upper_product pa_r_bl_one_upper_product pa_s_bl_one_upper_product. ((((exists pa_h_bl_one_upper_product_factor. pa_h_bl_one_upper_product_factor + S (pa_p_bl_one_upper_product) = S ((S (pa_i_bl_one_upper_product)) * pa_c_bl_one_upper)) /\ exists pa_q_bl_one_upper_product_factor. pa_b_bl_one_upper = pa_q_bl_one_upper_product_factor * S ((S (pa_i_bl_one_upper_product)) * pa_c_bl_one_upper) + (pa_p_bl_one_upper_product))) /\ ((((exists pa_h_bl_one_upper_product_partial. pa_h_bl_one_upper_product_partial + S (pa_r_bl_one_upper_product) = S ((S (pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_partial. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_partial * S ((S (pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product) + (pa_r_bl_one_upper_product))) /\ ((((exists pa_h_bl_one_upper_product_successor. pa_h_bl_one_upper_product_successor + S (pa_s_bl_one_upper_product) = S ((S (S pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product)) /\ exists pa_q_bl_one_upper_product_successor. pa_u_bl_one_upper_product = pa_q_bl_one_upper_product_successor * S ((S (S pa_i_bl_one_upper_product)) * pa_v_bl_one_upper_product) + (pa_s_bl_one_upper_product))) /\ pa_s_bl_one_upper_product = pa_r_bl_one_upper_product * pa_p_bl_one_upper_product))))))))
  6. 0006specialize binary_power_two_exists 1
  7. 0007exact binary_power_two_exists
  8. 0008cases hone
  9. 0009have hvalue : x = 1
  10. 0010specialize binary_power_two_zero_value x
  11. 0011apply binary_power_two_zero_value
  12. 0012exact hzero_witness
  13. 0013have hdouble : x1 = x + x
  14. 0014specialize binary_power_two_successor_double 0
  15. 0015specialize binary_power_two_successor_double x
  16. 0016specialize binary_power_two_successor_double x1
  17. 0017apply binary_power_two_successor_double
  18. 0018exact hzero_witness
  19. 0019exact hone_witness
  20. 0020right
  21. 0021exists 0
  22. 0022exists x
  23. 0023exists x1
  24. 0024split
  25. 0025refl
  26. 0026split
  27. 0027exists 0
  28. 0028simp
  29. 0029split
  30. 0030exact hzero_witness
  31. 0031split
  32. 0032exact hone_witness
  33. 0033split
  34. 0034exists 0
  35. 0035rewrite hvalue
  36. 0036simp
  37. 0037exists 0
  38. 0038rewrite hdouble
  39. 0039rewrite hvalue
  40. 0040rewrite hvalue
  41. 0041simp