BE000D

binary_modular_execution_empty

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The terminal accumulator of any actual empty binary execution is exactly one.

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.

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

The terminal accumulator of any actual empty binary execution is exactly one.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

BE0010 binary_modular_execution_power_correct

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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