BE000D

binary_modular_execution_empty

The terminal accumulator of any actual empty binary execution is exactly 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

∀ b. ∀ c. ∀ a. ∀ m. ∀ r. BinaryModularExecution(b,c,a,m,0,r) → r = 1

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

Definition DAG

BinaryModularExecution(d,s,a,m,n,r) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c a m r. (exists ff_trace_code_be_empty ff_trace_scale_be_empty. ((((((exists ff_h_be_empty_trace_start. ff_h_be_empty_trace_start + S (1) = S ((S (0)) * ff_trace_scale_be_empty)) /\ exists ff_q_be_empty_trace_start. ff_trace_code_be_empty = ff_q_be_empty_trace_start * S ((S (0)) * ff_trace_scale_be_empty) + (1))) /\ forall ff_index_be_empty_trace. (exists ff_lt_be_empty_trace_bound. ff_lt_be_empty_trace_bound + S ff_index_be_empty_trace = 0) -> exists ff_digit_be_empty_trace ff_previous_be_empty_trace ff_current_be_empty_trace. ((((exists ff_h_be_empty_trace_source. ff_h_be_empty_trace_source + S (ff_digit_be_empty_trace) = S ((S (ff_index_be_empty_trace)) * c)) /\ exists ff_q_be_empty_trace_source. b = ff_q_be_empty_trace_source * S ((S (ff_index_be_empty_trace)) * c) + (ff_digit_be_empty_trace))) /\ ((((exists ff_h_be_empty_trace_before. ff_h_be_empty_trace_before + S (ff_previous_be_empty_trace) = S ((S (ff_index_be_empty_trace)) * ff_trace_scale_be_empty)) /\ exists ff_q_be_empty_trace_before. ff_trace_code_be_empty = ff_q_be_empty_trace_before * S ((S (ff_index_be_empty_trace)) * ff_trace_scale_be_empty) + (ff_previous_be_empty_trace))) /\ ((((exists ff_h_be_empty_trace_after. ff_h_be_empty_trace_after + S (ff_current_be_empty_trace) = S ((S (S ff_index_be_empty_trace)) * ff_trace_scale_be_empty)) /\ exists ff_q_be_empty_trace_after. ff_trace_code_be_empty = ff_q_be_empty_trace_after * S ((S (S ff_index_be_empty_trace)) * ff_trace_scale_be_empty) + (ff_current_be_empty_trace))) /\ ((((ff_digit_be_empty_trace = 0) /\ (((exists ff_gap_binary_be_empty_trace_transition_square. ff_gap_binary_be_empty_trace_transition_square + S (ff_current_be_empty_trace) = m) /\ (exists ff_left_binary_be_empty_trace_transition_square_congruence ff_right_binary_be_empty_trace_transition_square_congruence. (ff_previous_be_empty_trace * ff_previous_be_empty_trace) + m * ff_left_binary_be_empty_trace_transition_square_congruence = (ff_current_be_empty_trace) + m * ff_right_binary_be_empty_trace_transition_square_congruence)))) \/ ((ff_digit_be_empty_trace = 1) /\ (((exists ff_gap_binary_be_empty_trace_transition_multiply. ff_gap_binary_be_empty_trace_transition_multiply + S (ff_current_be_empty_trace) = m) /\ (exists ff_left_binary_be_empty_trace_transition_multiply_congruence ff_right_binary_be_empty_trace_transition_multiply_congruence. ((ff_previous_be_empty_trace * ff_previous_be_empty_trace) * a) + m * ff_left_binary_be_empty_trace_transition_multiply_congruence = (ff_current_be_empty_trace) + m * ff_right_binary_be_empty_trace_transition_multiply_congruence))))))))))) /\ (((exists ff_h_be_empty_terminal. ff_h_be_empty_terminal + S (r) = S ((S (0)) * ff_trace_scale_be_empty)) /\ exists ff_q_be_empty_terminal. ff_trace_code_be_empty = ff_q_be_empty_terminal * S ((S (0)) * ff_trace_scale_be_empty) + (r))))) -> r = 1

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

18 script commands · 3 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 b
  2. L2
    intro c
  3. L3
    intro a
  4. L4
    intro m
  5. L5
    intro r
  6. L6
    intro hexecution
02Separate the logical casesL7–10

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

  1. L7
    cases hexecution
  2. L8
    cases hexecution_witness
  3. L9
    cases hexecution_witness_witness
  4. L10
    cases hexecution_witness_witness_left
03Use earlier factsL11–18

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

  1. L11
    specialize beta_at_unique x
  2. L12
    specialize beta_at_unique x1
  3. L13
    specialize beta_at_unique 0
  4. L14
    specialize beta_at_unique r
  5. L15
    specialize beta_at_unique 1
  6. L16
    apply beta_at_unique
  7. L17
    exact hexecution_witness_witness_right
  8. L18
    exact hexecution_witness_witness_left_left

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro m
  5. 0005intro r
  6. 0006intro hexecution
  7. 0007cases hexecution
  8. 0008cases hexecution_witness
  9. 0009cases hexecution_witness_witness
  10. 0010cases hexecution_witness_witness_left
  11. 0011specialize beta_at_unique x
  12. 0012specialize beta_at_unique x1
  13. 0013specialize beta_at_unique 0
  14. 0014specialize beta_at_unique r
  15. 0015specialize beta_at_unique 1
  16. 0016apply beta_at_unique
  17. 0017exact hexecution_witness_witness_right
  18. 0018exact hexecution_witness_witness_left_left