BX0005

binary_exponent_split_exists

Every exponent has an exact half and a witnessed zero-or-one binary digit.

Alpha v34 checked-use · first admitted v21 · 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.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ e. ∃ h. ∃ b. BinaryExponentSplit(e,h,b)

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

Definition DAG

BinaryExponentSplit(e,h,b) · 1

Actual proof prerequisites

parity_cases · checked external prerequisitetwo_mul_eq_add_self · checked external prerequisite
Original expanded first-order statement 
forall e. exists h b. ((((b = 0) \/ (b = 1)) /\ e = (h + h) + b))

Complete unchanged native tactic proof

All 24 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

24 script commands · 17 reading checkpoints · 0 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 assumptionsL1–1

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

  1. L1
    intro e
02Use earlier factsL2–2

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

  1. L2
    specialize parity_cases e
03Separate the logical casesL3–4

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

  1. L3
    cases parity_cases
  2. L4
    cases parity_cases_witness
04Construct an explicit witnessL5–6

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

  1. L5
    exists x
  2. L6
    exists 0
05Separate the logical casesL7–8

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

  1. L7
    split
  2. L8
    left
06Calculate and transport equalitiesL9–10

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

  1. L9
    refl
  2. L10
    trans 2 * x
07Use earlier factsL11–11

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

  1. L11
    exact parity_cases_witness_left
08Calculate and transport equalitiesL12–12

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

  1. L12
    trans x + x
09Use earlier factsL13–13

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

  1. L13
    apply two_mul_eq_add_self
10Calculate and transport equalitiesL14–14

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

  1. L14
    simp
11Construct an explicit witnessL15–16

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

  1. L15
    exists x
  2. L16
    exists 1
12Separate the logical casesL17–18

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

  1. L17
    split
  2. L18
    right
13Calculate and transport equalitiesL19–20

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

  1. L19
    refl
  2. L20
    trans 2 * x + 1
14Use earlier factsL21–21

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

  1. L21
    exact parity_cases_witness_right
15Calculate and transport equalitiesL22–22

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

  1. L22
    congr
16Use earlier factsL23–23

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

  1. L23
    apply two_mul_eq_add_self
17Calculate and transport equalitiesL24–24

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

  1. L24
    refl

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro e
  2. 0002specialize parity_cases e
  3. 0003cases parity_cases
  4. 0004cases parity_cases_witness
  5. 0005exists x
  6. 0006exists 0
  7. 0007split
  8. 0008left
  9. 0009refl
  10. 0010trans 2 * x
  11. 0011exact parity_cases_witness_left
  12. 0012trans x + x
  13. 0013apply two_mul_eq_add_self
  14. 0014simp
  15. 0015exists x
  16. 0016exists 1
  17. 0017split
  18. 0018right
  19. 0019refl
  20. 0020trans 2 * x + 1
  21. 0021exact parity_cases_witness_right
  22. 0022congr
  23. 0023apply two_mul_eq_add_self
  24. 0024refl