MK000E

multinomial_exists

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

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

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 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))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 20 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_sum_exists Stable theorem; checked-use authorized MK000D multinomial_exists_of_sum

Direct dependents

none

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

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.

Named ingredients (1)
multinomial exists of sum

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
  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
  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 exact command ledger · 20 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004have hsum : exists n. exists fs_u_mkm_total_sum fs_v_mkm_total_sum. ((((exists fs_h_mkm_total_sum_body_start. fs_h_mkm_total_sum_body_start + S (0) = S ((S (0)) * fs_v_mkm_total_sum)) /\ exists fs_q_mkm_total_sum_body_start. fs_u_mkm_total_sum = fs_q_mkm_total_sum_body_start * S ((S (0)) * fs_v_mkm_total_sum) + (0))) /\ ((((exists fs_h_mkm_total_sum_body_terminal. fs_h_mkm_total_sum_body_terminal + S (n) = S ((S (l)) * fs_v_mkm_total_sum)) /\ exists fs_q_mkm_total_sum_body_terminal. fs_u_mkm_total_sum = fs_q_mkm_total_sum_body_terminal * S ((S (l)) * fs_v_mkm_total_sum) + (n))) /\ forall fs_i_mkm_total_sum_body_steps. (exists fs_lt_mkm_total_sum_body_steps_bound. fs_lt_mkm_total_sum_body_steps_bound + S fs_i_mkm_total_sum_body_steps = l) -> exists fs_a_mkm_total_sum_body_steps fs_r_mkm_total_sum_body_steps fs_s_mkm_total_sum_body_steps. ((((exists fs_h_mkm_total_sum_body_steps_summand. fs_h_mkm_total_sum_body_steps_summand + S (fs_a_mkm_total_sum_body_steps) = S ((S (fs_i_mkm_total_sum_body_steps)) * c)) /\ exists fs_q_mkm_total_sum_body_steps_summand. b = fs_q_mkm_total_sum_body_steps_summand * S ((S (fs_i_mkm_total_sum_body_steps)) * c) + (fs_a_mkm_total_sum_body_steps))) /\ ((((exists fs_h_mkm_total_sum_body_steps_partial. fs_h_mkm_total_sum_body_steps_partial + S (fs_r_mkm_total_sum_body_steps) = S ((S (fs_i_mkm_total_sum_body_steps)) * fs_v_mkm_total_sum)) /\ exists fs_q_mkm_total_sum_body_steps_partial. fs_u_mkm_total_sum = fs_q_mkm_total_sum_body_steps_partial * S ((S (fs_i_mkm_total_sum_body_steps)) * fs_v_mkm_total_sum) + (fs_r_mkm_total_sum_body_steps))) /\ ((((exists fs_h_mkm_total_sum_body_steps_successor. fs_h_mkm_total_sum_body_steps_successor + S (fs_s_mkm_total_sum_body_steps) = S ((S (S fs_i_mkm_total_sum_body_steps)) * fs_v_mkm_total_sum)) /\ exists fs_q_mkm_total_sum_body_steps_successor. fs_u_mkm_total_sum = fs_q_mkm_total_sum_body_steps_successor * S ((S (S fs_i_mkm_total_sum_body_steps)) * fs_v_mkm_total_sum) + (fs_s_mkm_total_sum_body_steps))) /\ fs_s_mkm_total_sum_body_steps = fs_r_mkm_total_sum_body_steps + fs_a_mkm_total_sum_body_steps)))))
  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 : exists z. exists mkm_sum_code_total_coefficient mkm_sum_scale_total_coefficient mkm_factor_code_total_coefficient mkm_factor_scale_total_coefficient. (((((exists fs_h_mkm_total_coefficient_trace_start. fs_h_mkm_total_coefficient_trace_start + S (0) = S ((S (0)) * mkm_sum_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_trace_start. mkm_sum_code_total_coefficient = fs_q_mkm_total_coefficient_trace_start * S ((S (0)) * mkm_sum_scale_total_coefficient) + (0))) /\ ((((exists fs_h_mkm_total_coefficient_trace_terminal. fs_h_mkm_total_coefficient_trace_terminal + S (x) = S ((S (l)) * mkm_sum_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_trace_terminal. mkm_sum_code_total_coefficient = fs_q_mkm_total_coefficient_trace_terminal * S ((S (l)) * mkm_sum_scale_total_coefficient) + (x))) /\ forall fs_i_mkm_total_coefficient_trace_steps. (exists fs_lt_mkm_total_coefficient_trace_steps_bound. fs_lt_mkm_total_coefficient_trace_steps_bound + S fs_i_mkm_total_coefficient_trace_steps = l) -> exists fs_a_mkm_total_coefficient_trace_steps fs_r_mkm_total_coefficient_trace_steps fs_s_mkm_total_coefficient_trace_steps. ((((exists fs_h_mkm_total_coefficient_trace_steps_summand. fs_h_mkm_total_coefficient_trace_steps_summand + S (fs_a_mkm_total_coefficient_trace_steps) = S ((S (fs_i_mkm_total_coefficient_trace_steps)) * c)) /\ exists fs_q_mkm_total_coefficient_trace_steps_summand. b = fs_q_mkm_total_coefficient_trace_steps_summand * S ((S (fs_i_mkm_total_coefficient_trace_steps)) * c) + (fs_a_mkm_total_coefficient_trace_steps))) /\ ((((exists fs_h_mkm_total_coefficient_trace_steps_partial. fs_h_mkm_total_coefficient_trace_steps_partial + S (fs_r_mkm_total_coefficient_trace_steps) = S ((S (fs_i_mkm_total_coefficient_trace_steps)) * mkm_sum_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_trace_steps_partial. mkm_sum_code_total_coefficient = fs_q_mkm_total_coefficient_trace_steps_partial * S ((S (fs_i_mkm_total_coefficient_trace_steps)) * mkm_sum_scale_total_coefficient) + (fs_r_mkm_total_coefficient_trace_steps))) /\ ((((exists fs_h_mkm_total_coefficient_trace_steps_successor. fs_h_mkm_total_coefficient_trace_steps_successor + S (fs_s_mkm_total_coefficient_trace_steps) = S ((S (S fs_i_mkm_total_coefficient_trace_steps)) * mkm_sum_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_trace_steps_successor. mkm_sum_code_total_coefficient = fs_q_mkm_total_coefficient_trace_steps_successor * S ((S (S fs_i_mkm_total_coefficient_trace_steps)) * mkm_sum_scale_total_coefficient) + (fs_s_mkm_total_coefficient_trace_steps))) /\ fs_s_mkm_total_coefficient_trace_steps = fs_r_mkm_total_coefficient_trace_steps + fs_a_mkm_total_coefficient_trace_steps)))))) /\ ((forall mkm_index_total_coefficient_factors. (exists mkm_lt_total_coefficient_factors_bound. mkm_lt_total_coefficient_factors_bound + S (mkm_index_total_coefficient_factors) = (l)) -> (exists mkm_value_total_coefficient_factors_point mkm_partial_total_coefficient_factors_point mkm_factor_total_coefficient_factors_point. (((exists fs_h_mkm_total_coefficient_factors_point_source. fs_h_mkm_total_coefficient_factors_point_source + S (mkm_value_total_coefficient_factors_point) = S ((S (mkm_index_total_coefficient_factors)) * c)) /\ exists fs_q_mkm_total_coefficient_factors_point_source. b = fs_q_mkm_total_coefficient_factors_point_source * S ((S (mkm_index_total_coefficient_factors)) * c) + (mkm_value_total_coefficient_factors_point))) /\ ((((exists fs_h_mkm_total_coefficient_factors_point_partial. fs_h_mkm_total_coefficient_factors_point_partial + S (mkm_partial_total_coefficient_factors_point) = S ((S (mkm_index_total_coefficient_factors)) * mkm_sum_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_factors_point_partial. mkm_sum_code_total_coefficient = fs_q_mkm_total_coefficient_factors_point_partial * S ((S (mkm_index_total_coefficient_factors)) * mkm_sum_scale_total_coefficient) + (mkm_partial_total_coefficient_factors_point))) /\ ((((exists bcf_lt_gap_mkm_total_coefficient_factors_point_choose_out_of_range. bcf_lt_gap_mkm_total_coefficient_factors_point_choose_out_of_range + S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point) = mkm_partial_total_coefficient_factors_point) /\ mkm_factor_total_coefficient_factors_point = 0) \/ ((exists bcf_le_gap_mkm_total_coefficient_factors_point_choose_in_range. bcf_le_gap_mkm_total_coefficient_factors_point_choose_in_range + (mkm_partial_total_coefficient_factors_point) = mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point) /\ (exists bcf_row_code_code_mkm_total_coefficient_factors_point_choose bcf_row_code_scale_mkm_total_coefficient_factors_point_choose bcf_row_scale_code_mkm_total_coefficient_factors_point_choose bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose bcf_row_code_mkm_total_coefficient_factors_point_choose bcf_row_scale_mkm_total_coefficient_factors_point_choose. ((forall bcf_row_index_mkm_total_coefficient_factors_point_choose_table. (exists bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_row_bound. bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_row_bound + S (bcf_row_index_mkm_total_coefficient_factors_point_choose_table) = S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) -> exists bcf_row_code_mkm_total_coefficient_factors_point_choose_table bcf_row_scale_mkm_total_coefficient_factors_point_choose_table. ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_row_code. bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_row_code + S (bcf_row_code_mkm_total_coefficient_factors_point_choose_table) = S ((S (bcf_row_index_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_row_code. bcf_row_code_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_row_code * S ((S (bcf_row_index_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose) + (bcf_row_code_mkm_total_coefficient_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_row_scale. bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_row_scale + S (bcf_row_scale_mkm_total_coefficient_factors_point_choose_table) = S ((S (bcf_row_index_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_row_scale. bcf_row_scale_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_row_scale * S ((S (bcf_row_index_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose) + (bcf_row_scale_mkm_total_coefficient_factors_point_choose_table))) /\ ((bcf_row_index_mkm_total_coefficient_factors_point_choose_table = 0 /\ (forall bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row. (exists bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_zero_row_bound. bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_zero_row_bound + S (bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row) = S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) -> exists bcf_value_mkm_total_coefficient_factors_point_choose_table_zero_row. ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_zero_row_entry. bcf_height_mkm_total_coefficient_factors_point_choose_table_zero_row_entry + S (bcf_value_mkm_total_coefficient_factors_point_choose_table_zero_row) = S ((S (bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_zero_row_entry. bcf_row_code_mkm_total_coefficient_factors_point_choose_table = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_zero_row_entry * S ((S (bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose_table) + (bcf_value_mkm_total_coefficient_factors_point_choose_table_zero_row))) /\ ((bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row = 0 /\ bcf_value_mkm_total_coefficient_factors_point_choose_table_zero_row = 1) \/ exists bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_zero_row. bcf_index_mkm_total_coefficient_factors_point_choose_table_zero_row = S bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_zero_row /\ bcf_value_mkm_total_coefficient_factors_point_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_mkm_total_coefficient_factors_point_choose_table bcf_previous_code_mkm_total_coefficient_factors_point_choose_table bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table. bcf_row_index_mkm_total_coefficient_factors_point_choose_table = S bcf_predecessor_mkm_total_coefficient_factors_point_choose_table /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_previous_code. bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_previous_code + S (bcf_previous_code_mkm_total_coefficient_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_previous_code. bcf_row_code_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_previous_code * S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose) + (bcf_previous_code_mkm_total_coefficient_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_previous_scale. bcf_height_mkm_total_coefficient_factors_point_choose_table_decoded_previous_scale + S (bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_previous_scale. bcf_row_scale_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose) + (bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table))) /\ (forall bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step. (exists bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_row_step_bound. bcf_lt_gap_mkm_total_coefficient_factors_point_choose_table_row_step_bound + S (bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step) = S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) -> exists bcf_value_mkm_total_coefficient_factors_point_choose_table_row_step. ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_entry. bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_entry + S (bcf_value_mkm_total_coefficient_factors_point_choose_table_row_step) = S ((S (bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_entry. bcf_row_code_mkm_total_coefficient_factors_point_choose_table = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_entry * S ((S (bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose_table) + (bcf_value_mkm_total_coefficient_factors_point_choose_table_row_step))) /\ ((bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step = 0 /\ bcf_value_mkm_total_coefficient_factors_point_choose_table_row_step = 1) \/ exists bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step bcf_left_mkm_total_coefficient_factors_point_choose_table_row_step bcf_right_mkm_total_coefficient_factors_point_choose_table_row_step. bcf_index_mkm_total_coefficient_factors_point_choose_table_row_step = S bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_previous_left. bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_previous_left + S (bcf_left_mkm_total_coefficient_factors_point_choose_table_row_step) = S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_previous_left. bcf_previous_code_mkm_total_coefficient_factors_point_choose_table = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_previous_left * S ((S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table) + (bcf_left_mkm_total_coefficient_factors_point_choose_table_row_step))) /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_previous_right. bcf_height_mkm_total_coefficient_factors_point_choose_table_row_step_previous_right + S (bcf_right_mkm_total_coefficient_factors_point_choose_table_row_step) = S ((S (S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_previous_right. bcf_previous_code_mkm_total_coefficient_factors_point_choose_table = bcf_quotient_mkm_total_coefficient_factors_point_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_mkm_total_coefficient_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_total_coefficient_factors_point_choose_table) + (bcf_right_mkm_total_coefficient_factors_point_choose_table_row_step))) /\ bcf_value_mkm_total_coefficient_factors_point_choose_table_row_step = bcf_left_mkm_total_coefficient_factors_point_choose_table_row_step + bcf_right_mkm_total_coefficient_factors_point_choose_table_row_step))))))))))) /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_decoded_row_code. bcf_height_mkm_total_coefficient_factors_point_choose_decoded_row_code + S (bcf_row_code_mkm_total_coefficient_factors_point_choose) = S ((S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_row_code. bcf_row_code_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_row_code * S ((S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) * bcf_row_code_scale_mkm_total_coefficient_factors_point_choose) + (bcf_row_code_mkm_total_coefficient_factors_point_choose))) /\ ((((exists bcf_height_mkm_total_coefficient_factors_point_choose_decoded_row_scale. bcf_height_mkm_total_coefficient_factors_point_choose_decoded_row_scale + S (bcf_row_scale_mkm_total_coefficient_factors_point_choose) = S ((S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_row_scale. bcf_row_scale_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_row_scale * S ((S (mkm_partial_total_coefficient_factors_point + mkm_value_total_coefficient_factors_point)) * bcf_row_scale_scale_mkm_total_coefficient_factors_point_choose) + (bcf_row_scale_mkm_total_coefficient_factors_point_choose))) /\ (((exists bcf_height_mkm_total_coefficient_factors_point_choose_decoded_value. bcf_height_mkm_total_coefficient_factors_point_choose_decoded_value + S (mkm_factor_total_coefficient_factors_point) = S ((S (mkm_partial_total_coefficient_factors_point)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose)) /\ exists bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_value. bcf_row_code_mkm_total_coefficient_factors_point_choose = bcf_quotient_mkm_total_coefficient_factors_point_choose_decoded_value * S ((S (mkm_partial_total_coefficient_factors_point)) * bcf_row_scale_mkm_total_coefficient_factors_point_choose) + (mkm_factor_total_coefficient_factors_point))))))))) /\ (((exists fs_h_mkm_total_coefficient_factors_point_factor. fs_h_mkm_total_coefficient_factors_point_factor + S (mkm_factor_total_coefficient_factors_point) = S ((S (mkm_index_total_coefficient_factors)) * mkm_factor_scale_total_coefficient)) /\ exists fs_q_mkm_total_coefficient_factors_point_factor. mkm_factor_code_total_coefficient = fs_q_mkm_total_coefficient_factors_point_factor * S ((S (mkm_index_total_coefficient_factors)) * mkm_factor_scale_total_coefficient) + (mkm_factor_total_coefficient_factors_point))))))) /\ (exists ff_u_mkm_total_coefficient_product ff_v_mkm_total_coefficient_product. ((((exists ff_h_mkm_total_coefficient_product_start. ff_h_mkm_total_coefficient_product_start + S (1) = S ((S (0)) * ff_v_mkm_total_coefficient_product)) /\ exists ff_q_mkm_total_coefficient_product_start. ff_u_mkm_total_coefficient_product = ff_q_mkm_total_coefficient_product_start * S ((S (0)) * ff_v_mkm_total_coefficient_product) + (1))) /\ ((((exists ff_h_mkm_total_coefficient_product_terminal. ff_h_mkm_total_coefficient_product_terminal + S (z) = S ((S (l)) * ff_v_mkm_total_coefficient_product)) /\ exists ff_q_mkm_total_coefficient_product_terminal. ff_u_mkm_total_coefficient_product = ff_q_mkm_total_coefficient_product_terminal * S ((S (l)) * ff_v_mkm_total_coefficient_product) + (z))) /\ forall ff_i_mkm_total_coefficient_product. (exists ff_lt_mkm_total_coefficient_product_bound. ff_lt_mkm_total_coefficient_product_bound + S ff_i_mkm_total_coefficient_product = l) -> exists ff_p_mkm_total_coefficient_product ff_r_mkm_total_coefficient_product ff_s_mkm_total_coefficient_product. ((((exists ff_h_mkm_total_coefficient_product_factor. ff_h_mkm_total_coefficient_product_factor + S (ff_p_mkm_total_coefficient_product) = S ((S (ff_i_mkm_total_coefficient_product)) * mkm_factor_scale_total_coefficient)) /\ exists ff_q_mkm_total_coefficient_product_factor. mkm_factor_code_total_coefficient = ff_q_mkm_total_coefficient_product_factor * S ((S (ff_i_mkm_total_coefficient_product)) * mkm_factor_scale_total_coefficient) + (ff_p_mkm_total_coefficient_product))) /\ ((((exists ff_h_mkm_total_coefficient_product_partial. ff_h_mkm_total_coefficient_product_partial + S (ff_r_mkm_total_coefficient_product) = S ((S (ff_i_mkm_total_coefficient_product)) * ff_v_mkm_total_coefficient_product)) /\ exists ff_q_mkm_total_coefficient_product_partial. ff_u_mkm_total_coefficient_product = ff_q_mkm_total_coefficient_product_partial * S ((S (ff_i_mkm_total_coefficient_product)) * ff_v_mkm_total_coefficient_product) + (ff_r_mkm_total_coefficient_product))) /\ ((((exists ff_h_mkm_total_coefficient_product_successor. ff_h_mkm_total_coefficient_product_successor + S (ff_s_mkm_total_coefficient_product) = S ((S (S ff_i_mkm_total_coefficient_product)) * ff_v_mkm_total_coefficient_product)) /\ exists ff_q_mkm_total_coefficient_product_successor. ff_u_mkm_total_coefficient_product = ff_q_mkm_total_coefficient_product_successor * S ((S (S ff_i_mkm_total_coefficient_product)) * ff_v_mkm_total_coefficient_product) + (ff_s_mkm_total_coefficient_product))) /\ ff_s_mkm_total_coefficient_product = ff_r_mkm_total_coefficient_product * ff_p_mkm_total_coefficient_product)))))))
  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