PC002B

binary_split_upper_bound

An actual binary-split integer is at most twice its half plus one.

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

∀ N. ∀ h. ∀ d. d = 0 ∨ d = 1 → N = h + h + d → Le(N,S (h + h))

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

Definition DAG

Le(a,b) · 1

Actual proof prerequisites

le_succ_self · checked external prerequisitele_refl · checked external prerequisite
Original expanded first-order statement 
forall N h d. (d = 0 \/ d = 1) -> N = (h + h) + d -> (exists pc_le_split_upper_result. pc_le_split_upper_result + (N) = (S (h + h)))

Complete tactic proof in conservative notation

All 19 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

19 script commands · 6 reading checkpoints · 2 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 assumptionsL1–5

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 hd
  5. L5
    intro he
02Calculate and transport equalitiesL6–6

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

  1. L6
    rewrite he
03Separate the logical casesL7–7

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

  1. L7
    cases hd
04Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite hd_left
05Establish hzL9–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA3.

  1. L9
    have hz : (h + h) + 0 = h + h
  2. L10
    apply PA3
  3. L11
    rewrite hz
  4. L12
    specialize le_succ_self (h + h)
  5. L13
    apply le_succ_self
  6. L14
    rewrite hd_right
06Establish honeL15–19

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

  1. L15
    have hone : (h + h) + 1 = S (h + h)
  2. L16
    simp
  3. L17
    rewrite hone
  4. L18
    specialize le_refl (S (h + h))
  5. L19
    apply le_refl

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro N
  2. 0002intro h
  3. 0003intro d
  4. 0004intro hd
  5. 0005intro he
  6. 0006rewrite he
  7. 0007cases hd
  8. 0008rewrite hd_left
  9. 0009have hz : (h + h) + 0 = h + h
  10. 0010apply PA3
  11. 0011rewrite hz
  12. 0012specialize le_succ_self (h + h)
  13. 0013apply le_succ_self
  14. 0014rewrite hd_right
  15. 0015have hone : (h + h) + 1 = S (h + h)
  16. 0016simp
  17. 0017rewrite hone
  18. 0018specialize le_refl (S (h + h))
  19. 0019apply le_refl