BD0006

binary_digit_half_below_double

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

If a binary quotient/digit value is below twice a bound, its exact quotient is below that bound.

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.

Exact expanded first-order arithmetic statement

forall n h d p. ((((d = 0) \/ (d = 1)) /\ n = (h + h) + d)) -> (exists gap. gap + S n = p + p) -> exists gap. gap + S h = p

Constructive proof overview

Generated structural guide

If a binary quotient/digit value is below twice a bound, its exact quotient is below that bound.

The unchanged tactic script uses 5 declared prerequisites and contains 37 exact native proof lines.

Alpha v34 checked-use · first admitted v23 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

le_or_lt Stable theorem; checked-use authorized four_square_descent_add_le_add Alpha theorem; checked-use authorized le_add_right Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

BD0007 binary_digit_bounded_prefix_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

37 script commands · 9 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro n
  2. L2
    intro h
  3. L3
    intro d
  4. L4
    intro p
  5. L5
    intro hsplit
  6. L6
    intro hbound
02Separate the logical casesL7–7

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

  1. L7
    cases hsplit
03Use earlier factsL8–9

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

  1. L8
    specialize le_or_lt p
  2. L9
    specialize le_or_lt h
04Separate the logical casesL10–10

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

  1. L10
    cases le_or_lt
05Establish hdoubleL11–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square descent add le add.

  1. L11
    have hdouble : exists gap. gap + (p + p) = h + h
  2. L12
    specialize four_square_descent_add_le_add p
  3. L13
    specialize four_square_descent_add_le_add h
  4. L14
    specialize four_square_descent_add_le_add p
  5. L15
    specialize four_square_descent_add_le_add h
  6. L16
    apply four_square_descent_add_le_add
  7. L17
    exact le_or_lt_left
  8. L18
    exact le_or_lt_left
06Establish hsumL19–23

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

  1. L19
    have hsum : exists gap. gap + (h + h) = n
  2. L20
    rewrite hsplit_right
  3. L21
    specialize le_add_right (h + h)
  4. L22
    specialize le_add_right d
  5. L23
    exact le_add_right
07Establish hreverseL24–30

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

  1. L24
    have hreverse : exists gap. gap + (p + p) = n
  2. L25
    specialize le_trans (p + p)
  3. L26
    specialize le_trans (h + h)
  4. L27
    specialize le_trans n
  5. L28
    apply le_trans
  6. L29
    exact hdouble
  7. L30
    exact hsum
08Separate the logical casesL31–31

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

  1. L31
    exfalso
09Use earlier factsL32–37

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

  1. L32
    specialize lt_not_le n
  2. L33
    specialize lt_not_le (p + p)
  3. L34
    apply lt_not_le
  4. L35
    exact hbound
  5. L36
    exact hreverse
  6. L37
    exact le_or_lt_right

Library-wide reading audit

Original exact command ledger · 37 lines
  1. 0001intro n
  2. 0002intro h
  3. 0003intro d
  4. 0004intro p
  5. 0005intro hsplit
  6. 0006intro hbound
  7. 0007cases hsplit
  8. 0008specialize le_or_lt p
  9. 0009specialize le_or_lt h
  10. 0010cases le_or_lt
  11. 0011have hdouble : exists gap. gap + (p + p) = h + h
  12. 0012specialize four_square_descent_add_le_add p
  13. 0013specialize four_square_descent_add_le_add h
  14. 0014specialize four_square_descent_add_le_add p
  15. 0015specialize four_square_descent_add_le_add h
  16. 0016apply four_square_descent_add_le_add
  17. 0017exact le_or_lt_left
  18. 0018exact le_or_lt_left
  19. 0019have hsum : exists gap. gap + (h + h) = n
  20. 0020rewrite hsplit_right
  21. 0021specialize le_add_right (h + h)
  22. 0022specialize le_add_right d
  23. 0023exact le_add_right
  24. 0024have hreverse : exists gap. gap + (p + p) = n
  25. 0025specialize le_trans (p + p)
  26. 0026specialize le_trans (h + h)
  27. 0027specialize le_trans n
  28. 0028apply le_trans
  29. 0029exact hdouble
  30. 0030exact hsum
  31. 0031exfalso
  32. 0032specialize lt_not_le n
  33. 0033specialize lt_not_le (p + p)
  34. 0034apply lt_not_le
  35. 0035exact hbound
  36. 0036exact hreverse
  37. 0037exact le_or_lt_right

Separate complete second-wave branches: Full T13 proof · Alpha v27.