BD0006

binary_digit_half_below_double

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

Alpha v34 checked-use · first admitted v23 · 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.

The exact G102 milestone is fully proved for every natural exponent and every modulus greater than one, including actual canonical digits, a beta-coded accumulator execution, modular-power correctness, and the formal bound k≤3·BitLen(e)+2. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ n. ∀ h. ∀ d. ∀ p. (d = 0 ∨ d = 1) ∧ n = h + h + d → Lt(n,p + p) → Lt(h,p)

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

Definition DAG

Lt(a,b) · 2

Actual proof prerequisites

le_or_lt · checked external prerequisitefour_square_descent_add_le_add · checked external prerequisitele_add_right · checked external prerequisitele_trans · checked external prerequisitelt_not_le · checked external prerequisite
Original expanded first-order 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

Complete unchanged native tactic proof

All 37 lines are the exact independently kernel-checked original script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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