MK000F

multinomial_nonzero

An actual multinomial coefficient is always nonzero; no undefined zero valuation is used in Kummer's theorem.

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) → ¬z = 0

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

Definition DAG

Multinomial(b,c,l,n,z) · 1

Actual proof prerequisites

multinomial_binomial_prefix_nonzerocrt_positive_moduli_prefix_product_nonzero · checked external prerequisite
Original expanded first-order statement 
forall b c l n z. (exists mkm_sum_code_nonzero_multinomial mkm_sum_scale_nonzero_multinomial mkm_factor_code_nonzero_multinomial mkm_factor_scale_nonzero_multinomial. (((((exists fs_h_mkm_nonzero_multinomial_trace_start. fs_h_mkm_nonzero_multinomial_trace_start + S (0) = S ((S (0)) * mkm_sum_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_trace_start. mkm_sum_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_trace_start * S ((S (0)) * mkm_sum_scale_nonzero_multinomial) + (0))) /\ ((((exists fs_h_mkm_nonzero_multinomial_trace_terminal. fs_h_mkm_nonzero_multinomial_trace_terminal + S (n) = S ((S (l)) * mkm_sum_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_trace_terminal. mkm_sum_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_trace_terminal * S ((S (l)) * mkm_sum_scale_nonzero_multinomial) + (n))) /\ forall fs_i_mkm_nonzero_multinomial_trace_steps. (exists fs_lt_mkm_nonzero_multinomial_trace_steps_bound. fs_lt_mkm_nonzero_multinomial_trace_steps_bound + S fs_i_mkm_nonzero_multinomial_trace_steps = l) -> exists fs_a_mkm_nonzero_multinomial_trace_steps fs_r_mkm_nonzero_multinomial_trace_steps fs_s_mkm_nonzero_multinomial_trace_steps. ((((exists fs_h_mkm_nonzero_multinomial_trace_steps_summand. fs_h_mkm_nonzero_multinomial_trace_steps_summand + S (fs_a_mkm_nonzero_multinomial_trace_steps) = S ((S (fs_i_mkm_nonzero_multinomial_trace_steps)) * c)) /\ exists fs_q_mkm_nonzero_multinomial_trace_steps_summand. b = fs_q_mkm_nonzero_multinomial_trace_steps_summand * S ((S (fs_i_mkm_nonzero_multinomial_trace_steps)) * c) + (fs_a_mkm_nonzero_multinomial_trace_steps))) /\ ((((exists fs_h_mkm_nonzero_multinomial_trace_steps_partial. fs_h_mkm_nonzero_multinomial_trace_steps_partial + S (fs_r_mkm_nonzero_multinomial_trace_steps) = S ((S (fs_i_mkm_nonzero_multinomial_trace_steps)) * mkm_sum_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_trace_steps_partial. mkm_sum_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_trace_steps_partial * S ((S (fs_i_mkm_nonzero_multinomial_trace_steps)) * mkm_sum_scale_nonzero_multinomial) + (fs_r_mkm_nonzero_multinomial_trace_steps))) /\ ((((exists fs_h_mkm_nonzero_multinomial_trace_steps_successor. fs_h_mkm_nonzero_multinomial_trace_steps_successor + S (fs_s_mkm_nonzero_multinomial_trace_steps) = S ((S (S fs_i_mkm_nonzero_multinomial_trace_steps)) * mkm_sum_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_trace_steps_successor. mkm_sum_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_trace_steps_successor * S ((S (S fs_i_mkm_nonzero_multinomial_trace_steps)) * mkm_sum_scale_nonzero_multinomial) + (fs_s_mkm_nonzero_multinomial_trace_steps))) /\ fs_s_mkm_nonzero_multinomial_trace_steps = fs_r_mkm_nonzero_multinomial_trace_steps + fs_a_mkm_nonzero_multinomial_trace_steps)))))) /\ ((forall mkm_index_nonzero_multinomial_factors. (exists mkm_lt_nonzero_multinomial_factors_bound. mkm_lt_nonzero_multinomial_factors_bound + S (mkm_index_nonzero_multinomial_factors) = (l)) -> (exists mkm_value_nonzero_multinomial_factors_point mkm_partial_nonzero_multinomial_factors_point mkm_factor_nonzero_multinomial_factors_point. (((exists fs_h_mkm_nonzero_multinomial_factors_point_source. fs_h_mkm_nonzero_multinomial_factors_point_source + S (mkm_value_nonzero_multinomial_factors_point) = S ((S (mkm_index_nonzero_multinomial_factors)) * c)) /\ exists fs_q_mkm_nonzero_multinomial_factors_point_source. b = fs_q_mkm_nonzero_multinomial_factors_point_source * S ((S (mkm_index_nonzero_multinomial_factors)) * c) + (mkm_value_nonzero_multinomial_factors_point))) /\ ((((exists fs_h_mkm_nonzero_multinomial_factors_point_partial. fs_h_mkm_nonzero_multinomial_factors_point_partial + S (mkm_partial_nonzero_multinomial_factors_point) = S ((S (mkm_index_nonzero_multinomial_factors)) * mkm_sum_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_factors_point_partial. mkm_sum_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_factors_point_partial * S ((S (mkm_index_nonzero_multinomial_factors)) * mkm_sum_scale_nonzero_multinomial) + (mkm_partial_nonzero_multinomial_factors_point))) /\ ((((exists bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_out_of_range. bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_out_of_range + S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point) = mkm_partial_nonzero_multinomial_factors_point) /\ mkm_factor_nonzero_multinomial_factors_point = 0) \/ ((exists bcf_le_gap_mkm_nonzero_multinomial_factors_point_choose_in_range. bcf_le_gap_mkm_nonzero_multinomial_factors_point_choose_in_range + (mkm_partial_nonzero_multinomial_factors_point) = mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point) /\ (exists bcf_row_code_code_mkm_nonzero_multinomial_factors_point_choose bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose bcf_row_scale_code_mkm_nonzero_multinomial_factors_point_choose bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose bcf_row_code_mkm_nonzero_multinomial_factors_point_choose bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose. ((forall bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table. (exists bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_row_bound. bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_row_bound + S (bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table) = S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) -> exists bcf_row_code_mkm_nonzero_multinomial_factors_point_choose_table bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table. ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_code. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_code + S (bcf_row_code_mkm_nonzero_multinomial_factors_point_choose_table) = S ((S (bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_code. bcf_row_code_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_code * S ((S (bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_row_code_mkm_nonzero_multinomial_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_scale. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_scale + S (bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table) = S ((S (bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_scale. bcf_row_scale_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_row_scale * S ((S (bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table))) /\ ((bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table = 0 /\ (forall bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row. (exists bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_bound. bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_bound + S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row) = S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) -> exists bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_zero_row. ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_entry. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_entry + S (bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_zero_row) = S ((S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_entry. bcf_row_code_mkm_nonzero_multinomial_factors_point_choose_table = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_zero_row_entry * S ((S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table) + (bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_zero_row))) /\ ((bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row = 0 /\ bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_zero_row = 1) \/ exists bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_zero_row. bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_zero_row = S bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_zero_row /\ bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table bcf_previous_code_mkm_nonzero_multinomial_factors_point_choose_table bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table. bcf_row_index_mkm_nonzero_multinomial_factors_point_choose_table = S bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_code. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_code + S (bcf_previous_code_mkm_nonzero_multinomial_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_code. bcf_row_code_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_code * S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_previous_code_mkm_nonzero_multinomial_factors_point_choose_table))) /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_scale. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_scale + S (bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table) = S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_scale. bcf_row_scale_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table))) /\ (forall bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step. (exists bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_row_step_bound. bcf_lt_gap_mkm_nonzero_multinomial_factors_point_choose_table_row_step_bound + S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step) = S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) -> exists bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_row_step. ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_entry. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_entry + S (bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_row_step) = S ((S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_entry. bcf_row_code_mkm_nonzero_multinomial_factors_point_choose_table = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_entry * S ((S (bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose_table) + (bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_row_step))) /\ ((bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step = 0 /\ bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_row_step = 1) \/ exists bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step bcf_left_mkm_nonzero_multinomial_factors_point_choose_table_row_step bcf_right_mkm_nonzero_multinomial_factors_point_choose_table_row_step. bcf_index_mkm_nonzero_multinomial_factors_point_choose_table_row_step = S bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_left. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_left + S (bcf_left_mkm_nonzero_multinomial_factors_point_choose_table_row_step) = S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_left. bcf_previous_code_mkm_nonzero_multinomial_factors_point_choose_table = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_left * S ((S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step)) * bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table) + (bcf_left_mkm_nonzero_multinomial_factors_point_choose_table_row_step))) /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_right. bcf_height_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_right + S (bcf_right_mkm_nonzero_multinomial_factors_point_choose_table_row_step) = S ((S (S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_right. bcf_previous_code_mkm_nonzero_multinomial_factors_point_choose_table = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_mkm_nonzero_multinomial_factors_point_choose_table_row_step))) * bcf_previous_scale_mkm_nonzero_multinomial_factors_point_choose_table) + (bcf_right_mkm_nonzero_multinomial_factors_point_choose_table_row_step))) /\ bcf_value_mkm_nonzero_multinomial_factors_point_choose_table_row_step = bcf_left_mkm_nonzero_multinomial_factors_point_choose_table_row_step + bcf_right_mkm_nonzero_multinomial_factors_point_choose_table_row_step))))))))))) /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_row_code. bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_row_code + S (bcf_row_code_mkm_nonzero_multinomial_factors_point_choose) = S ((S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_row_code. bcf_row_code_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_row_code * S ((S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) * bcf_row_code_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_row_code_mkm_nonzero_multinomial_factors_point_choose))) /\ ((((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_row_scale. bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_row_scale + S (bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose) = S ((S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_row_scale. bcf_row_scale_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_row_scale * S ((S (mkm_partial_nonzero_multinomial_factors_point + mkm_value_nonzero_multinomial_factors_point)) * bcf_row_scale_scale_mkm_nonzero_multinomial_factors_point_choose) + (bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose))) /\ (((exists bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_value. bcf_height_mkm_nonzero_multinomial_factors_point_choose_decoded_value + S (mkm_factor_nonzero_multinomial_factors_point) = S ((S (mkm_partial_nonzero_multinomial_factors_point)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose)) /\ exists bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_value. bcf_row_code_mkm_nonzero_multinomial_factors_point_choose = bcf_quotient_mkm_nonzero_multinomial_factors_point_choose_decoded_value * S ((S (mkm_partial_nonzero_multinomial_factors_point)) * bcf_row_scale_mkm_nonzero_multinomial_factors_point_choose) + (mkm_factor_nonzero_multinomial_factors_point))))))))) /\ (((exists fs_h_mkm_nonzero_multinomial_factors_point_factor. fs_h_mkm_nonzero_multinomial_factors_point_factor + S (mkm_factor_nonzero_multinomial_factors_point) = S ((S (mkm_index_nonzero_multinomial_factors)) * mkm_factor_scale_nonzero_multinomial)) /\ exists fs_q_mkm_nonzero_multinomial_factors_point_factor. mkm_factor_code_nonzero_multinomial = fs_q_mkm_nonzero_multinomial_factors_point_factor * S ((S (mkm_index_nonzero_multinomial_factors)) * mkm_factor_scale_nonzero_multinomial) + (mkm_factor_nonzero_multinomial_factors_point))))))) /\ (exists ff_u_mkm_nonzero_multinomial_product ff_v_mkm_nonzero_multinomial_product. ((((exists ff_h_mkm_nonzero_multinomial_product_start. ff_h_mkm_nonzero_multinomial_product_start + S (1) = S ((S (0)) * ff_v_mkm_nonzero_multinomial_product)) /\ exists ff_q_mkm_nonzero_multinomial_product_start. ff_u_mkm_nonzero_multinomial_product = ff_q_mkm_nonzero_multinomial_product_start * S ((S (0)) * ff_v_mkm_nonzero_multinomial_product) + (1))) /\ ((((exists ff_h_mkm_nonzero_multinomial_product_terminal. ff_h_mkm_nonzero_multinomial_product_terminal + S (z) = S ((S (l)) * ff_v_mkm_nonzero_multinomial_product)) /\ exists ff_q_mkm_nonzero_multinomial_product_terminal. ff_u_mkm_nonzero_multinomial_product = ff_q_mkm_nonzero_multinomial_product_terminal * S ((S (l)) * ff_v_mkm_nonzero_multinomial_product) + (z))) /\ forall ff_i_mkm_nonzero_multinomial_product. (exists ff_lt_mkm_nonzero_multinomial_product_bound. ff_lt_mkm_nonzero_multinomial_product_bound + S ff_i_mkm_nonzero_multinomial_product = l) -> exists ff_p_mkm_nonzero_multinomial_product ff_r_mkm_nonzero_multinomial_product ff_s_mkm_nonzero_multinomial_product. ((((exists ff_h_mkm_nonzero_multinomial_product_factor. ff_h_mkm_nonzero_multinomial_product_factor + S (ff_p_mkm_nonzero_multinomial_product) = S ((S (ff_i_mkm_nonzero_multinomial_product)) * mkm_factor_scale_nonzero_multinomial)) /\ exists ff_q_mkm_nonzero_multinomial_product_factor. mkm_factor_code_nonzero_multinomial = ff_q_mkm_nonzero_multinomial_product_factor * S ((S (ff_i_mkm_nonzero_multinomial_product)) * mkm_factor_scale_nonzero_multinomial) + (ff_p_mkm_nonzero_multinomial_product))) /\ ((((exists ff_h_mkm_nonzero_multinomial_product_partial. ff_h_mkm_nonzero_multinomial_product_partial + S (ff_r_mkm_nonzero_multinomial_product) = S ((S (ff_i_mkm_nonzero_multinomial_product)) * ff_v_mkm_nonzero_multinomial_product)) /\ exists ff_q_mkm_nonzero_multinomial_product_partial. ff_u_mkm_nonzero_multinomial_product = ff_q_mkm_nonzero_multinomial_product_partial * S ((S (ff_i_mkm_nonzero_multinomial_product)) * ff_v_mkm_nonzero_multinomial_product) + (ff_r_mkm_nonzero_multinomial_product))) /\ ((((exists ff_h_mkm_nonzero_multinomial_product_successor. ff_h_mkm_nonzero_multinomial_product_successor + S (ff_s_mkm_nonzero_multinomial_product) = S ((S (S ff_i_mkm_nonzero_multinomial_product)) * ff_v_mkm_nonzero_multinomial_product)) /\ exists ff_q_mkm_nonzero_multinomial_product_successor. ff_u_mkm_nonzero_multinomial_product = ff_q_mkm_nonzero_multinomial_product_successor * S ((S (S ff_i_mkm_nonzero_multinomial_product)) * ff_v_mkm_nonzero_multinomial_product) + (ff_s_mkm_nonzero_multinomial_product))) /\ ff_s_mkm_nonzero_multinomial_product = ff_r_mkm_nonzero_multinomial_product * ff_p_mkm_nonzero_multinomial_product)))))))) -> ~(z = 0)

Complete tactic proof in conservative notation

All 29 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

29 script commands · 4 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.

Named ingredients (1)
multinomial binomial prefix nonzero
01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro n
  5. L5
    intro z
  6. L6
    intro h
  7. L7
    intro hzero
02Separate the logical casesL8–13

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

  1. L8
    cases h
  2. L9
    cases h_witness
  3. L10
    cases h_witness_witness
  4. L11
    cases h_witness_witness_witness
  5. L12
    cases h_witness_witness_witness_witness
  6. L13
    cases h_witness_witness_witness_witness_right
03Use earlier factsL14–23

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

  1. L14
    specialize crt_positive_moduli_prefix_product_nonzero x2
  2. L15
    specialize crt_positive_moduli_prefix_product_nonzero x3
  3. L16
    specialize crt_positive_moduli_prefix_product_nonzero l
  4. L17
    specialize crt_positive_moduli_prefix_product_nonzero z
  5. L18
    apply crt_positive_moduli_prefix_product_nonzero
  6. L19
    specialize multinomial_binomial_prefix_nonzero b
  7. L20
    specialize multinomial_binomial_prefix_nonzero c
  8. L21
    specialize multinomial_binomial_prefix_nonzero x
  9. L22
    specialize multinomial_binomial_prefix_nonzero x1
  10. L23
    specialize multinomial_binomial_prefix_nonzero x2
04Use earlier factsL24–29

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

  1. L24
    specialize multinomial_binomial_prefix_nonzero x3
  2. L25
    specialize multinomial_binomial_prefix_nonzero l
  3. L26
    apply multinomial_binomial_prefix_nonzero
  4. L27
    exact h_witness_witness_witness_witness_right_left
  5. L28
    exact h_witness_witness_witness_witness_right_right
  6. L29
    exact hzero

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro n
  5. 0005intro z
  6. 0006intro h
  7. 0007intro hzero
  8. 0008cases h
  9. 0009cases h_witness
  10. 0010cases h_witness_witness
  11. 0011cases h_witness_witness_witness
  12. 0012cases h_witness_witness_witness_witness
  13. 0013cases h_witness_witness_witness_witness_right
  14. 0014specialize crt_positive_moduli_prefix_product_nonzero x2
  15. 0015specialize crt_positive_moduli_prefix_product_nonzero x3
  16. 0016specialize crt_positive_moduli_prefix_product_nonzero l
  17. 0017specialize crt_positive_moduli_prefix_product_nonzero z
  18. 0018apply crt_positive_moduli_prefix_product_nonzero
  19. 0019specialize multinomial_binomial_prefix_nonzero b
  20. 0020specialize multinomial_binomial_prefix_nonzero c
  21. 0021specialize multinomial_binomial_prefix_nonzero x
  22. 0022specialize multinomial_binomial_prefix_nonzero x1
  23. 0023specialize multinomial_binomial_prefix_nonzero x2
  24. 0024specialize multinomial_binomial_prefix_nonzero x3
  25. 0025specialize multinomial_binomial_prefix_nonzero l
  26. 0026apply multinomial_binomial_prefix_nonzero
  27. 0027exact h_witness_witness_witness_witness_right_left
  28. 0028exact h_witness_witness_witness_witness_right_right
  29. 0029exact hzero