MK000E

multinomial_exists

Every finite natural part list, including the empty list, has a witnessed total and multinomial coefficient.

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. ∀ l. ∃ n. ∃ z. Multinomial(b,c,l,n,z)

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

Definition DAG

Multinomial(b,c,l,n,z) · 2Sum(b,c,l,z) · 1

Actual proof prerequisites

beta_sum_exists · checked external prerequisitemultinomial_exists_of_sum
Original expanded first-order statement 
forall b c l. exists n z. (exists mkm_sum_code_total_multinomial mkm_sum_scale_total_multinomial mkm_factor_code_total_multinomial mkm_factor_scale_total_multinomial. (((((exists fs_h_mkm_total_multinomial_trace_start. fs_h_mkm_total_multinomial_trace_start + S (0) = S ((S (0)) * mkm_sum_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_trace_start. mkm_sum_code_total_multinomial = fs_q_mkm_total_multinomial_trace_start * S ((S (0)) * mkm_sum_scale_total_multinomial) + (0))) /\ ((((exists fs_h_mkm_total_multinomial_trace_terminal. fs_h_mkm_total_multinomial_trace_terminal + S (n) = S ((S (l)) * mkm_sum_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_trace_terminal. mkm_sum_code_total_multinomial = fs_q_mkm_total_multinomial_trace_terminal * S ((S (l)) * mkm_sum_scale_total_multinomial) + (n))) /\ forall fs_i_mkm_total_multinomial_trace_steps. (exists fs_lt_mkm_total_multinomial_trace_steps_bound. fs_lt_mkm_total_multinomial_trace_steps_bound + S fs_i_mkm_total_multinomial_trace_steps = l) -> exists fs_a_mkm_total_multinomial_trace_steps fs_r_mkm_total_multinomial_trace_steps fs_s_mkm_total_multinomial_trace_steps. ((((exists fs_h_mkm_total_multinomial_trace_steps_summand. fs_h_mkm_total_multinomial_trace_steps_summand + S (fs_a_mkm_total_multinomial_trace_steps) = S ((S (fs_i_mkm_total_multinomial_trace_steps)) * c)) /\ exists fs_q_mkm_total_multinomial_trace_steps_summand. b = fs_q_mkm_total_multinomial_trace_steps_summand * S ((S (fs_i_mkm_total_multinomial_trace_steps)) * c) + (fs_a_mkm_total_multinomial_trace_steps))) /\ ((((exists fs_h_mkm_total_multinomial_trace_steps_partial. fs_h_mkm_total_multinomial_trace_steps_partial + S (fs_r_mkm_total_multinomial_trace_steps) = S ((S (fs_i_mkm_total_multinomial_trace_steps)) * mkm_sum_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_trace_steps_partial. mkm_sum_code_total_multinomial = fs_q_mkm_total_multinomial_trace_steps_partial * S ((S (fs_i_mkm_total_multinomial_trace_steps)) * mkm_sum_scale_total_multinomial) + (fs_r_mkm_total_multinomial_trace_steps))) /\ ((((exists fs_h_mkm_total_multinomial_trace_steps_successor. fs_h_mkm_total_multinomial_trace_steps_successor + S (fs_s_mkm_total_multinomial_trace_steps) = S ((S (S fs_i_mkm_total_multinomial_trace_steps)) * mkm_sum_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_trace_steps_successor. mkm_sum_code_total_multinomial = fs_q_mkm_total_multinomial_trace_steps_successor * S ((S (S fs_i_mkm_total_multinomial_trace_steps)) * mkm_sum_scale_total_multinomial) + (fs_s_mkm_total_multinomial_trace_steps))) /\ fs_s_mkm_total_multinomial_trace_steps = fs_r_mkm_total_multinomial_trace_steps + fs_a_mkm_total_multinomial_trace_steps)))))) /\ ((forall mkm_index_total_multinomial_factors. (exists mkm_lt_total_multinomial_factors_bound. mkm_lt_total_multinomial_factors_bound + S (mkm_index_total_multinomial_factors) = (l)) -> (exists mkm_value_total_multinomial_factors_point mkm_partial_total_multinomial_factors_point mkm_factor_total_multinomial_factors_point. (((exists fs_h_mkm_total_multinomial_factors_point_source. fs_h_mkm_total_multinomial_factors_point_source + S (mkm_value_total_multinomial_factors_point) = S ((S (mkm_index_total_multinomial_factors)) * c)) /\ exists fs_q_mkm_total_multinomial_factors_point_source. b = fs_q_mkm_total_multinomial_factors_point_source * S ((S (mkm_index_total_multinomial_factors)) * c) + (mkm_value_total_multinomial_factors_point))) /\ ((((exists fs_h_mkm_total_multinomial_factors_point_partial. fs_h_mkm_total_multinomial_factors_point_partial + S (mkm_partial_total_multinomial_factors_point) = S ((S (mkm_index_total_multinomial_factors)) * mkm_sum_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_factors_point_partial. mkm_sum_code_total_multinomial = fs_q_mkm_total_multinomial_factors_point_partial * S ((S (mkm_index_total_multinomial_factors)) * mkm_sum_scale_total_multinomial) + (mkm_partial_total_multinomial_factors_point))) /\ ((((exists bcf_lt_gap_mkm_total_multinomial_factors_point_choose_out_of_range. bcf_lt_gap_mkm_total_multinomial_factors_point_choose_out_of_range + S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point) = mkm_partial_total_multinomial_factors_point) /\ mkm_factor_total_multinomial_factors_point = 0) \/ ((exists bcf_le_gap_mkm_total_multinomial_factors_point_choose_in_range. bcf_le_gap_mkm_total_multinomial_factors_point_choose_in_range + (mkm_partial_total_multinomial_factors_point) = mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point) /\ (exists bcf_row_code_code_mkm_total_multinomial_factors_point_choose bcf_row_code_scale_mkm_total_multinomial_factors_point_choose bcf_row_scale_code_mkm_total_multinomial_factors_point_choose bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose bcf_row_code_mkm_total_multinomial_factors_point_choose bcf_row_scale_mkm_total_multinomial_factors_point_choose. ((forall bcf_row_index_mkm_total_multinomial_factors_point_choose_table. (exists bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_row_bound. bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_row_bound + S (bcf_row_index_mkm_total_multinomial_factors_point_choose_table) = S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) -> exists bcf_row_code_mkm_total_multinomial_factors_point_choose_table bcf_row_scale_mkm_total_multinomial_factors_point_choose_table. ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_row_code. bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_row_code + S (bcf_row_code_mkm_total_multinomial_factors_point_choose_table) = S ((S (bcf_row_index_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_row_code. bcf_row_code_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_row_code * S ((S (bcf_row_index_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose) + (bcf_row_code_mkm_total_multinomial_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_row_scale. bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_row_scale + S (bcf_row_scale_mkm_total_multinomial_factors_point_choose_table) = S ((S (bcf_row_index_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_row_scale. bcf_row_scale_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_row_scale * S ((S (bcf_row_index_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose) + (bcf_row_scale_mkm_total_multinomial_factors_point_choose_table))) /\ ((bcf_row_index_mkm_total_multinomial_factors_point_choose_table = 0 /\ (forall bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row. (exists bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_zero_row_bound. bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_zero_row_bound + S (bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row) = S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) -> exists bcf_value_mkm_total_multinomial_factors_point_choose_table_zero_row. ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_zero_row_entry. bcf_height_mkm_total_multinomial_factors_point_choose_table_zero_row_entry + S (bcf_value_mkm_total_multinomial_factors_point_choose_table_zero_row) = S ((S (bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_zero_row_entry. bcf_row_code_mkm_total_multinomial_factors_point_choose_table = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_zero_row_entry * S ((S (bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose_table) + (bcf_value_mkm_total_multinomial_factors_point_choose_table_zero_row))) /\ ((bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row = 0 /\ bcf_value_mkm_total_multinomial_factors_point_choose_table_zero_row = 1) \/ exists bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_zero_row. bcf_index_mkm_total_multinomial_factors_point_choose_table_zero_row = S bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_zero_row /\ bcf_value_mkm_total_multinomial_factors_point_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_mkm_total_multinomial_factors_point_choose_table bcf_previous_code_mkm_total_multinomial_factors_point_choose_table bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table. bcf_row_index_mkm_total_multinomial_factors_point_choose_table = S bcf_predecessor_mkm_total_multinomial_factors_point_choose_table /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_previous_code. bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_previous_code + S (bcf_previous_code_mkm_total_multinomial_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_previous_code. bcf_row_code_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_previous_code * S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose) + (bcf_previous_code_mkm_total_multinomial_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_previous_scale. bcf_height_mkm_total_multinomial_factors_point_choose_table_decoded_previous_scale + S (bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_previous_scale. bcf_row_scale_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose) + (bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table))) /\ (forall bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step. (exists bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_row_step_bound. bcf_lt_gap_mkm_total_multinomial_factors_point_choose_table_row_step_bound + S (bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step) = S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) -> exists bcf_value_mkm_total_multinomial_factors_point_choose_table_row_step. ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_entry. bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_entry + S (bcf_value_mkm_total_multinomial_factors_point_choose_table_row_step) = S ((S (bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_entry. bcf_row_code_mkm_total_multinomial_factors_point_choose_table = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_entry * S ((S (bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose_table) + (bcf_value_mkm_total_multinomial_factors_point_choose_table_row_step))) /\ ((bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step = 0 /\ bcf_value_mkm_total_multinomial_factors_point_choose_table_row_step = 1) \/ exists bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step bcf_left_mkm_total_multinomial_factors_point_choose_table_row_step bcf_right_mkm_total_multinomial_factors_point_choose_table_row_step. bcf_index_mkm_total_multinomial_factors_point_choose_table_row_step = S bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_previous_left. bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_previous_left + S (bcf_left_mkm_total_multinomial_factors_point_choose_table_row_step) = S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_previous_left. bcf_previous_code_mkm_total_multinomial_factors_point_choose_table = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_previous_left * S ((S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table) + (bcf_left_mkm_total_multinomial_factors_point_choose_table_row_step))) /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_previous_right. bcf_height_mkm_total_multinomial_factors_point_choose_table_row_step_previous_right + S (bcf_right_mkm_total_multinomial_factors_point_choose_table_row_step) = S ((S (S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_previous_right. bcf_previous_code_mkm_total_multinomial_factors_point_choose_table = bcf_quotient_mkm_total_multinomial_factors_point_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_mkm_total_multinomial_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_total_multinomial_factors_point_choose_table) + (bcf_right_mkm_total_multinomial_factors_point_choose_table_row_step))) /\ bcf_value_mkm_total_multinomial_factors_point_choose_table_row_step = bcf_left_mkm_total_multinomial_factors_point_choose_table_row_step + bcf_right_mkm_total_multinomial_factors_point_choose_table_row_step))))))))))) /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_decoded_row_code. bcf_height_mkm_total_multinomial_factors_point_choose_decoded_row_code + S (bcf_row_code_mkm_total_multinomial_factors_point_choose) = S ((S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_row_code. bcf_row_code_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_row_code * S ((S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) * bcf_row_code_scale_mkm_total_multinomial_factors_point_choose) + (bcf_row_code_mkm_total_multinomial_factors_point_choose))) /\ ((((exists bcf_height_mkm_total_multinomial_factors_point_choose_decoded_row_scale. bcf_height_mkm_total_multinomial_factors_point_choose_decoded_row_scale + S (bcf_row_scale_mkm_total_multinomial_factors_point_choose) = S ((S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_row_scale. bcf_row_scale_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_row_scale * S ((S (mkm_partial_total_multinomial_factors_point + mkm_value_total_multinomial_factors_point)) * bcf_row_scale_scale_mkm_total_multinomial_factors_point_choose) + (bcf_row_scale_mkm_total_multinomial_factors_point_choose))) /\ (((exists bcf_height_mkm_total_multinomial_factors_point_choose_decoded_value. bcf_height_mkm_total_multinomial_factors_point_choose_decoded_value + S (mkm_factor_total_multinomial_factors_point) = S ((S (mkm_partial_total_multinomial_factors_point)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_value. bcf_row_code_mkm_total_multinomial_factors_point_choose = bcf_quotient_mkm_total_multinomial_factors_point_choose_decoded_value * S ((S (mkm_partial_total_multinomial_factors_point)) * bcf_row_scale_mkm_total_multinomial_factors_point_choose) + (mkm_factor_total_multinomial_factors_point))))))))) /\ (((exists fs_h_mkm_total_multinomial_factors_point_factor. fs_h_mkm_total_multinomial_factors_point_factor + S (mkm_factor_total_multinomial_factors_point) = S ((S (mkm_index_total_multinomial_factors)) * mkm_factor_scale_total_multinomial)) /\ exists fs_q_mkm_total_multinomial_factors_point_factor. mkm_factor_code_total_multinomial = fs_q_mkm_total_multinomial_factors_point_factor * S ((S (mkm_index_total_multinomial_factors)) * mkm_factor_scale_total_multinomial) + (mkm_factor_total_multinomial_factors_point))))))) /\ (exists ff_u_mkm_total_multinomial_product ff_v_mkm_total_multinomial_product. ((((exists ff_h_mkm_total_multinomial_product_start. ff_h_mkm_total_multinomial_product_start + S (1) = S ((S (0)) * ff_v_mkm_total_multinomial_product)) /\ exists ff_q_mkm_total_multinomial_product_start. ff_u_mkm_total_multinomial_product = ff_q_mkm_total_multinomial_product_start * S ((S (0)) * ff_v_mkm_total_multinomial_product) + (1))) /\ ((((exists ff_h_mkm_total_multinomial_product_terminal. ff_h_mkm_total_multinomial_product_terminal + S (z) = S ((S (l)) * ff_v_mkm_total_multinomial_product)) /\ exists ff_q_mkm_total_multinomial_product_terminal. ff_u_mkm_total_multinomial_product = ff_q_mkm_total_multinomial_product_terminal * S ((S (l)) * ff_v_mkm_total_multinomial_product) + (z))) /\ forall ff_i_mkm_total_multinomial_product. (exists ff_lt_mkm_total_multinomial_product_bound. ff_lt_mkm_total_multinomial_product_bound + S ff_i_mkm_total_multinomial_product = l) -> exists ff_p_mkm_total_multinomial_product ff_r_mkm_total_multinomial_product ff_s_mkm_total_multinomial_product. ((((exists ff_h_mkm_total_multinomial_product_factor. ff_h_mkm_total_multinomial_product_factor + S (ff_p_mkm_total_multinomial_product) = S ((S (ff_i_mkm_total_multinomial_product)) * mkm_factor_scale_total_multinomial)) /\ exists ff_q_mkm_total_multinomial_product_factor. mkm_factor_code_total_multinomial = ff_q_mkm_total_multinomial_product_factor * S ((S (ff_i_mkm_total_multinomial_product)) * mkm_factor_scale_total_multinomial) + (ff_p_mkm_total_multinomial_product))) /\ ((((exists ff_h_mkm_total_multinomial_product_partial. ff_h_mkm_total_multinomial_product_partial + S (ff_r_mkm_total_multinomial_product) = S ((S (ff_i_mkm_total_multinomial_product)) * ff_v_mkm_total_multinomial_product)) /\ exists ff_q_mkm_total_multinomial_product_partial. ff_u_mkm_total_multinomial_product = ff_q_mkm_total_multinomial_product_partial * S ((S (ff_i_mkm_total_multinomial_product)) * ff_v_mkm_total_multinomial_product) + (ff_r_mkm_total_multinomial_product))) /\ ((((exists ff_h_mkm_total_multinomial_product_successor. ff_h_mkm_total_multinomial_product_successor + S (ff_s_mkm_total_multinomial_product) = S ((S (S ff_i_mkm_total_multinomial_product)) * ff_v_mkm_total_multinomial_product)) /\ exists ff_q_mkm_total_multinomial_product_successor. ff_u_mkm_total_multinomial_product = ff_q_mkm_total_multinomial_product_successor * S ((S (S ff_i_mkm_total_multinomial_product)) * ff_v_mkm_total_multinomial_product) + (ff_s_mkm_total_multinomial_product))) /\ ff_s_mkm_total_multinomial_product = ff_r_mkm_total_multinomial_product * ff_p_mkm_total_multinomial_product))))))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 7 reading checkpoints · 2 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.

Named ingredients (1)
multinomial exists of sum
01Fix variables and assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
02Establish hsumL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta sum exists.

  1. L4
    have hsum : ∃ n. Sum(b,c,l,n)Definitions: Sum(b,c,l,n)Original native command in the exact edition
  2. L5
    specialize beta_sum_exists b
  3. L6
    specialize beta_sum_exists c
  4. L7
    specialize beta_sum_exists l
  5. L8
    apply beta_sum_exists
03Separate the logical casesL9–9

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

  1. L9
    cases hsum
04Establish hcoefficientL10–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply multinomial exists of sum.

  1. L10
    have hcoefficient : ∃ z. Multinomial(b,c,l,x,z)Definitions: Multinomial(b,c,l,x,z)Original native command in the exact edition
  2. L11
    specialize multinomial_exists_of_sum b
  3. L12
    specialize multinomial_exists_of_sum c
  4. L13
    specialize multinomial_exists_of_sum l
  5. L14
    specialize multinomial_exists_of_sum x
  6. L15
    apply multinomial_exists_of_sum
  7. L16
    exact hsum_witness
05Separate the logical casesL17–17

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

  1. L17
    cases hcoefficient
06Construct an explicit witnessL18–19

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

  1. L18
    exists x
  2. L19
    exists x1
07Use earlier factsL20–20

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

  1. L20
    exact hcoefficient_witness

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have hsum : ∃ n. Sum(b,c,l,n)
  5. 0005specialize beta_sum_exists b
  6. 0006specialize beta_sum_exists c
  7. 0007specialize beta_sum_exists l
  8. 0008apply beta_sum_exists
  9. 0009cases hsum
  10. 0010have hcoefficient : ∃ z. Multinomial(b,c,l,x,z)
  11. 0011specialize multinomial_exists_of_sum b
  12. 0012specialize multinomial_exists_of_sum c
  13. 0013specialize multinomial_exists_of_sum l
  14. 0014specialize multinomial_exists_of_sum x
  15. 0015apply multinomial_exists_of_sum
  16. 0016exact hsum_witness
  17. 0017cases hcoefficient
  18. 0018exists x
  19. 0019exists x1
  20. 0020exact hcoefficient_witness