BE0005

binary_execution_step_digit

Every actual modular square-and-multiply transition explicitly carries a binary digit.

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

∀ m. ∀ r. ∀ a. ∀ d. ∀ s. BinaryModularStep(m,r,a,d,s) → d = 0 ∨ d = 1

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

Definition DAG

BinaryModularStep(m,u,a,b,r) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall m r a d s. ((((d = 0) /\ (((exists ff_gap_binary_step_square. ff_gap_binary_step_square + S (s) = m) /\ (exists ff_left_binary_step_square_congruence ff_right_binary_step_square_congruence. (r * r) + m * ff_left_binary_step_square_congruence = (s) + m * ff_right_binary_step_square_congruence)))) \/ ((d = 1) /\ (((exists ff_gap_binary_step_multiply. ff_gap_binary_step_multiply + S (s) = m) /\ (exists ff_left_binary_step_multiply_congruence ff_right_binary_step_multiply_congruence. ((r * r) * a) + m * ff_left_binary_step_multiply_congruence = (s) + m * ff_right_binary_step_multiply_congruence)))))) -> (d = 0 \/ d = 1)

Complete unchanged native tactic proof

All 13 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

13 script commands · 5 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 assumptionsL1–6

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

  1. L1
    intro m
  2. L2
    intro r
  3. L3
    intro a
  4. L4
    intro d
  5. L5
    intro s
  6. L6
    intro hstep
02Separate the logical casesL7–9

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

  1. L7
    cases hstep
  2. L8
    cases hstep_left
  3. L9
    left
03Use earlier factsL10–10

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

  1. L10
    exact hstep_left_left
04Separate the logical casesL11–12

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

  1. L11
    cases hstep_right
  2. L12
    right
05Use earlier factsL13–13

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

  1. L13
    exact hstep_right_left

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro m
  2. 0002intro r
  3. 0003intro a
  4. 0004intro d
  5. 0005intro s
  6. 0006intro hstep
  7. 0007cases hstep
  8. 0008cases hstep_left
  9. 0009left
  10. 0010exact hstep_left_left
  11. 0011cases hstep_right
  12. 0012right
  13. 0013exact hstep_right_left