MK0008

multinomial_binomial_prefix_empty

The empty multinomial factor prefix requires no binomial entries.

Alpha v34 checked-use · first admitted v27 · 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.

The carry relation contains actual quotient columns and carry bits, not the desired valuation. The theorem includes all finite lists and zero parts. It uses sequential binary-column carries; a separate simultaneous-grid or permutation-invariance theorem is not asserted.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ sb. ∀ sc. ∀ cb. ∀ cc. MultinomialBinomialPrefix(b,c,sb,sc,cb,cc,0)

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

Definition DAG

MultinomialBinomialPrefix(b,c,sb,sc,cb,cc,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall b c sb sc cb cc. forall mkm_index_binempty. (exists mkm_lt_binempty_bound. mkm_lt_binempty_bound + S (mkm_index_binempty) = (0)) -> (exists mkm_value_binempty_point mkm_partial_binempty_point mkm_factor_binempty_point. (((exists fs_h_mkm_binempty_point_source. fs_h_mkm_binempty_point_source + S (mkm_value_binempty_point) = S ((S (mkm_index_binempty)) * c)) /\ exists fs_q_mkm_binempty_point_source. b = fs_q_mkm_binempty_point_source * S ((S (mkm_index_binempty)) * c) + (mkm_value_binempty_point))) /\ ((((exists fs_h_mkm_binempty_point_partial. fs_h_mkm_binempty_point_partial + S (mkm_partial_binempty_point) = S ((S (mkm_index_binempty)) * sc)) /\ exists fs_q_mkm_binempty_point_partial. sb = fs_q_mkm_binempty_point_partial * S ((S (mkm_index_binempty)) * sc) + (mkm_partial_binempty_point))) /\ ((((exists bcf_lt_gap_mkm_binempty_point_choose_out_of_range. bcf_lt_gap_mkm_binempty_point_choose_out_of_range + S (mkm_partial_binempty_point + mkm_value_binempty_point) = mkm_partial_binempty_point) /\ mkm_factor_binempty_point = 0) \/ ((exists bcf_le_gap_mkm_binempty_point_choose_in_range. bcf_le_gap_mkm_binempty_point_choose_in_range + (mkm_partial_binempty_point) = mkm_partial_binempty_point + mkm_value_binempty_point) /\ (exists bcf_row_code_code_mkm_binempty_point_choose bcf_row_code_scale_mkm_binempty_point_choose bcf_row_scale_code_mkm_binempty_point_choose bcf_row_scale_scale_mkm_binempty_point_choose bcf_row_code_mkm_binempty_point_choose bcf_row_scale_mkm_binempty_point_choose. ((forall bcf_row_index_mkm_binempty_point_choose_table. (exists bcf_lt_gap_mkm_binempty_point_choose_table_row_bound. bcf_lt_gap_mkm_binempty_point_choose_table_row_bound + S (bcf_row_index_mkm_binempty_point_choose_table) = S (mkm_partial_binempty_point + mkm_value_binempty_point)) -> exists bcf_row_code_mkm_binempty_point_choose_table bcf_row_scale_mkm_binempty_point_choose_table. ((((exists bcf_height_mkm_binempty_point_choose_table_decoded_row_code. bcf_height_mkm_binempty_point_choose_table_decoded_row_code + S (bcf_row_code_mkm_binempty_point_choose_table) = S ((S (bcf_row_index_mkm_binempty_point_choose_table)) * bcf_row_code_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_decoded_row_code. bcf_row_code_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_table_decoded_row_code * S ((S (bcf_row_index_mkm_binempty_point_choose_table)) * bcf_row_code_scale_mkm_binempty_point_choose) + (bcf_row_code_mkm_binempty_point_choose_table))) /\ ((((exists bcf_height_mkm_binempty_point_choose_table_decoded_row_scale. bcf_height_mkm_binempty_point_choose_table_decoded_row_scale + S (bcf_row_scale_mkm_binempty_point_choose_table) = S ((S (bcf_row_index_mkm_binempty_point_choose_table)) * bcf_row_scale_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_decoded_row_scale. bcf_row_scale_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_table_decoded_row_scale * S ((S (bcf_row_index_mkm_binempty_point_choose_table)) * bcf_row_scale_scale_mkm_binempty_point_choose) + (bcf_row_scale_mkm_binempty_point_choose_table))) /\ ((bcf_row_index_mkm_binempty_point_choose_table = 0 /\ (forall bcf_index_mkm_binempty_point_choose_table_zero_row. (exists bcf_lt_gap_mkm_binempty_point_choose_table_zero_row_bound. bcf_lt_gap_mkm_binempty_point_choose_table_zero_row_bound + S (bcf_index_mkm_binempty_point_choose_table_zero_row) = S (mkm_partial_binempty_point + mkm_value_binempty_point)) -> exists bcf_value_mkm_binempty_point_choose_table_zero_row. ((((exists bcf_height_mkm_binempty_point_choose_table_zero_row_entry. bcf_height_mkm_binempty_point_choose_table_zero_row_entry + S (bcf_value_mkm_binempty_point_choose_table_zero_row) = S ((S (bcf_index_mkm_binempty_point_choose_table_zero_row)) * bcf_row_scale_mkm_binempty_point_choose_table)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_zero_row_entry. bcf_row_code_mkm_binempty_point_choose_table = bcf_quotient_mkm_binempty_point_choose_table_zero_row_entry * S ((S (bcf_index_mkm_binempty_point_choose_table_zero_row)) * bcf_row_scale_mkm_binempty_point_choose_table) + (bcf_value_mkm_binempty_point_choose_table_zero_row))) /\ ((bcf_index_mkm_binempty_point_choose_table_zero_row = 0 /\ bcf_value_mkm_binempty_point_choose_table_zero_row = 1) \/ exists bcf_predecessor_mkm_binempty_point_choose_table_zero_row. bcf_index_mkm_binempty_point_choose_table_zero_row = S bcf_predecessor_mkm_binempty_point_choose_table_zero_row /\ bcf_value_mkm_binempty_point_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_mkm_binempty_point_choose_table bcf_previous_code_mkm_binempty_point_choose_table bcf_previous_scale_mkm_binempty_point_choose_table. bcf_row_index_mkm_binempty_point_choose_table = S bcf_predecessor_mkm_binempty_point_choose_table /\ ((((exists bcf_height_mkm_binempty_point_choose_table_decoded_previous_code. bcf_height_mkm_binempty_point_choose_table_decoded_previous_code + S (bcf_previous_code_mkm_binempty_point_choose_table) = S ((S (bcf_predecessor_mkm_binempty_point_choose_table)) * bcf_row_code_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_decoded_previous_code. bcf_row_code_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_table_decoded_previous_code * S ((S (bcf_predecessor_mkm_binempty_point_choose_table)) * bcf_row_code_scale_mkm_binempty_point_choose) + (bcf_previous_code_mkm_binempty_point_choose_table))) /\ ((((exists bcf_height_mkm_binempty_point_choose_table_decoded_previous_scale. bcf_height_mkm_binempty_point_choose_table_decoded_previous_scale + S (bcf_previous_scale_mkm_binempty_point_choose_table) = S ((S (bcf_predecessor_mkm_binempty_point_choose_table)) * bcf_row_scale_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_decoded_previous_scale. bcf_row_scale_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_mkm_binempty_point_choose_table)) * bcf_row_scale_scale_mkm_binempty_point_choose) + (bcf_previous_scale_mkm_binempty_point_choose_table))) /\ (forall bcf_index_mkm_binempty_point_choose_table_row_step. (exists bcf_lt_gap_mkm_binempty_point_choose_table_row_step_bound. bcf_lt_gap_mkm_binempty_point_choose_table_row_step_bound + S (bcf_index_mkm_binempty_point_choose_table_row_step) = S (mkm_partial_binempty_point + mkm_value_binempty_point)) -> exists bcf_value_mkm_binempty_point_choose_table_row_step. ((((exists bcf_height_mkm_binempty_point_choose_table_row_step_entry. bcf_height_mkm_binempty_point_choose_table_row_step_entry + S (bcf_value_mkm_binempty_point_choose_table_row_step) = S ((S (bcf_index_mkm_binempty_point_choose_table_row_step)) * bcf_row_scale_mkm_binempty_point_choose_table)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_row_step_entry. bcf_row_code_mkm_binempty_point_choose_table = bcf_quotient_mkm_binempty_point_choose_table_row_step_entry * S ((S (bcf_index_mkm_binempty_point_choose_table_row_step)) * bcf_row_scale_mkm_binempty_point_choose_table) + (bcf_value_mkm_binempty_point_choose_table_row_step))) /\ ((bcf_index_mkm_binempty_point_choose_table_row_step = 0 /\ bcf_value_mkm_binempty_point_choose_table_row_step = 1) \/ exists bcf_predecessor_mkm_binempty_point_choose_table_row_step bcf_left_mkm_binempty_point_choose_table_row_step bcf_right_mkm_binempty_point_choose_table_row_step. bcf_index_mkm_binempty_point_choose_table_row_step = S bcf_predecessor_mkm_binempty_point_choose_table_row_step /\ ((((exists bcf_height_mkm_binempty_point_choose_table_row_step_previous_left. bcf_height_mkm_binempty_point_choose_table_row_step_previous_left + S (bcf_left_mkm_binempty_point_choose_table_row_step) = S ((S (bcf_predecessor_mkm_binempty_point_choose_table_row_step)) * bcf_previous_scale_mkm_binempty_point_choose_table)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_row_step_previous_left. bcf_previous_code_mkm_binempty_point_choose_table = bcf_quotient_mkm_binempty_point_choose_table_row_step_previous_left * S ((S (bcf_predecessor_mkm_binempty_point_choose_table_row_step)) * bcf_previous_scale_mkm_binempty_point_choose_table) + (bcf_left_mkm_binempty_point_choose_table_row_step))) /\ ((((exists bcf_height_mkm_binempty_point_choose_table_row_step_previous_right. bcf_height_mkm_binempty_point_choose_table_row_step_previous_right + S (bcf_right_mkm_binempty_point_choose_table_row_step) = S ((S (S (bcf_predecessor_mkm_binempty_point_choose_table_row_step))) * bcf_previous_scale_mkm_binempty_point_choose_table)) /\ exists bcf_quotient_mkm_binempty_point_choose_table_row_step_previous_right. bcf_previous_code_mkm_binempty_point_choose_table = bcf_quotient_mkm_binempty_point_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_mkm_binempty_point_choose_table_row_step))) * bcf_previous_scale_mkm_binempty_point_choose_table) + (bcf_right_mkm_binempty_point_choose_table_row_step))) /\ bcf_value_mkm_binempty_point_choose_table_row_step = bcf_left_mkm_binempty_point_choose_table_row_step + bcf_right_mkm_binempty_point_choose_table_row_step))))))))))) /\ ((((exists bcf_height_mkm_binempty_point_choose_decoded_row_code. bcf_height_mkm_binempty_point_choose_decoded_row_code + S (bcf_row_code_mkm_binempty_point_choose) = S ((S (mkm_partial_binempty_point + mkm_value_binempty_point)) * bcf_row_code_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_decoded_row_code. bcf_row_code_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_decoded_row_code * S ((S (mkm_partial_binempty_point + mkm_value_binempty_point)) * bcf_row_code_scale_mkm_binempty_point_choose) + (bcf_row_code_mkm_binempty_point_choose))) /\ ((((exists bcf_height_mkm_binempty_point_choose_decoded_row_scale. bcf_height_mkm_binempty_point_choose_decoded_row_scale + S (bcf_row_scale_mkm_binempty_point_choose) = S ((S (mkm_partial_binempty_point + mkm_value_binempty_point)) * bcf_row_scale_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_decoded_row_scale. bcf_row_scale_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_decoded_row_scale * S ((S (mkm_partial_binempty_point + mkm_value_binempty_point)) * bcf_row_scale_scale_mkm_binempty_point_choose) + (bcf_row_scale_mkm_binempty_point_choose))) /\ (((exists bcf_height_mkm_binempty_point_choose_decoded_value. bcf_height_mkm_binempty_point_choose_decoded_value + S (mkm_factor_binempty_point) = S ((S (mkm_partial_binempty_point)) * bcf_row_scale_mkm_binempty_point_choose)) /\ exists bcf_quotient_mkm_binempty_point_choose_decoded_value. bcf_row_code_mkm_binempty_point_choose = bcf_quotient_mkm_binempty_point_choose_decoded_value * S ((S (mkm_partial_binempty_point)) * bcf_row_scale_mkm_binempty_point_choose) + (mkm_factor_binempty_point))))))))) /\ (((exists fs_h_mkm_binempty_point_factor. fs_h_mkm_binempty_point_factor + S (mkm_factor_binempty_point) = S ((S (mkm_index_binempty)) * cc)) /\ exists fs_q_mkm_binempty_point_factor. cb = fs_q_mkm_binempty_point_factor * S ((S (mkm_index_binempty)) * cc) + (mkm_factor_binempty_point))))))

Complete tactic proof in conservative notation

All 15 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

15 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 assumptionsL1–8

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro sb
  4. L4
    intro sc
  5. L5
    intro cb
  6. L6
    intro cc
  7. L7
    intro i
  8. L8
    intro hi
02Separate the logical casesL9–9

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

  1. L9
    exfalso
03Use earlier factsL10–15

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

  1. L10
    specialize lt_not_le i
  2. L11
    specialize lt_not_le 0
  3. L12
    apply lt_not_le
  4. L13
    exact hi
  5. L14
    specialize zero_le i
  6. L15
    apply zero_le

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro sb
  4. 0004intro sc
  5. 0005intro cb
  6. 0006intro cc
  7. 0007intro i
  8. 0008intro hi
  9. 0009exfalso
  10. 0010specialize lt_not_le i
  11. 0011specialize lt_not_le 0
  12. 0012apply lt_not_le
  13. 0013exact hi
  14. 0014specialize zero_le i
  15. 0015apply zero_le