BE0004

binary_execution_initial_state

There is a real beta code whose initial square-and-multiply accumulator is one.

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.

G102 was OPEN at this family's Alpha-v22 first admission: complete execution was proved only for a supplied valid beta-coded digit prefix. G102 is now CLOSED in Alpha v23 for every arbitrary exponent, with actual canonical digits and operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∃ z. ∃ d. Beta(z,d,0,1)

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

Definition DAG

Beta(b,c,i,x) · 1

Actual proof prerequisites

beta_at_self_of_bound · checked external prerequisite
Original expanded first-order statement 
exists z d. (((exists ff_h_be_initial. ff_h_be_initial + S (1) = S ((S (0)) * d)) /\ exists ff_q_be_initial. z = ff_q_be_initial * S ((S (0)) * d) + (1)))

Complete unchanged native tactic proof

All 8 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

8 script commands · 4 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.

01Construct an explicit witnessL1–2

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

  1. L1
    exists 1
  2. L2
    exists 1
02Use earlier factsL3–6

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

  1. L3
    specialize beta_at_self_of_bound 1
  2. L4
    specialize beta_at_self_of_bound 0
  3. L5
    specialize beta_at_self_of_bound 1
  4. L6
    apply beta_at_self_of_bound
03Construct an explicit witnessL7–7

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

  1. L7
    exists 0
04Calculate and transport equalitiesL8–8

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

  1. L8
    simp

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001exists 1
  2. 0002exists 1
  3. 0003specialize beta_at_self_of_bound 1
  4. 0004specialize beta_at_self_of_bound 0
  5. 0005specialize beta_at_self_of_bound 1
  6. 0006apply beta_at_self_of_bound
  7. 0007exists 0
  8. 0008simp