FactorialValuation

Readable signature

FactorialValuation(p, n, e)

Exact expansion

exists bfv_factorial_bertrand_defined_factorial_valuation. ((exists ff_b_bertrand_defined_factorial_valuation_factorial ff_c_bertrand_defined_factorial_valuation_factorial. ((forall ff_i_bertrand_defined_factorial_valuation_factorial_range. (exists ff_lt_bertrand_defined_factorial_valuation_factorial_range_bound. ff_lt_bertrand_defined_factorial_valuation_factorial_range_bound + S ff_i_bertrand_defined_factorial_valuation_factorial_range = n) -> (((exists ff_h_bertrand_defined_factorial_valuation_factorial_range_decoded. ff_h_bertrand_defined_factorial_valuation_factorial_range_decoded + S (1 + ff_i_bertrand_defined_factorial_valuation_factorial_range) = S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_range)) * ff_c_bertrand_defined_factorial_valuation_factorial)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_range_decoded. ff_b_bertrand_defined_factorial_valuation_factorial = ff_q_bertrand_defined_factorial_valuation_factorial_range_decoded * S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_range)) * ff_c_bertrand_defined_factorial_valuation_factorial) + (1 + ff_i_bertrand_defined_factorial_valuation_factorial_range)))) /\ (exists ff_u_bertrand_defined_factorial_valuation_factorial_product ff_v_bertrand_defined_factorial_valuation_factorial_product. ((((exists ff_h_bertrand_defined_factorial_valuation_factorial_product_start. ff_h_bertrand_defined_factorial_valuation_factorial_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_factorial_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_product_start. ff_u_bertrand_defined_factorial_valuation_factorial_product = ff_q_bertrand_defined_factorial_valuation_factorial_product_start * S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_factorial_product) + (1))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_factorial_product_terminal. ff_h_bertrand_defined_factorial_valuation_factorial_product_terminal + S (bfv_factorial_bertrand_defined_factorial_valuation) = S ((S (n)) * ff_v_bertrand_defined_factorial_valuation_factorial_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_product_terminal. ff_u_bertrand_defined_factorial_valuation_factorial_product = ff_q_bertrand_defined_factorial_valuation_factorial_product_terminal * S ((S (n)) * ff_v_bertrand_defined_factorial_valuation_factorial_product) + (bfv_factorial_bertrand_defined_factorial_valuation))) /\ forall ff_i_bertrand_defined_factorial_valuation_factorial_product. (exists ff_lt_bertrand_defined_factorial_valuation_factorial_product_bound. ff_lt_bertrand_defined_factorial_valuation_factorial_product_bound + S ff_i_bertrand_defined_factorial_valuation_factorial_product = n) -> exists ff_p_bertrand_defined_factorial_valuation_factorial_product ff_r_bertrand_defined_factorial_valuation_factorial_product ff_s_bertrand_defined_factorial_valuation_factorial_product. ((((exists ff_h_bertrand_defined_factorial_valuation_factorial_product_factor. ff_h_bertrand_defined_factorial_valuation_factorial_product_factor + S (ff_p_bertrand_defined_factorial_valuation_factorial_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_c_bertrand_defined_factorial_valuation_factorial)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_product_factor. ff_b_bertrand_defined_factorial_valuation_factorial = ff_q_bertrand_defined_factorial_valuation_factorial_product_factor * S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_c_bertrand_defined_factorial_valuation_factorial) + (ff_p_bertrand_defined_factorial_valuation_factorial_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_factorial_product_partial. ff_h_bertrand_defined_factorial_valuation_factorial_product_partial + S (ff_r_bertrand_defined_factorial_valuation_factorial_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_v_bertrand_defined_factorial_valuation_factorial_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_product_partial. ff_u_bertrand_defined_factorial_valuation_factorial_product = ff_q_bertrand_defined_factorial_valuation_factorial_product_partial * S ((S (ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_v_bertrand_defined_factorial_valuation_factorial_product) + (ff_r_bertrand_defined_factorial_valuation_factorial_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_factorial_product_successor. ff_h_bertrand_defined_factorial_valuation_factorial_product_successor + S (ff_s_bertrand_defined_factorial_valuation_factorial_product) = S ((S (S ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_v_bertrand_defined_factorial_valuation_factorial_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_factorial_product_successor. ff_u_bertrand_defined_factorial_valuation_factorial_product = ff_q_bertrand_defined_factorial_valuation_factorial_product_successor * S ((S (S ff_i_bertrand_defined_factorial_valuation_factorial_product)) * ff_v_bertrand_defined_factorial_valuation_factorial_product) + (ff_s_bertrand_defined_factorial_valuation_factorial_product))) /\ ff_s_bertrand_defined_factorial_valuation_factorial_product = ff_r_bertrand_defined_factorial_valuation_factorial_product * ff_p_bertrand_defined_factorial_valuation_factorial_product)))))))) /\ (((exists bpv_gap_bertrand_defined_factorial_valuation_valuation_exponent_bound. bpv_gap_bertrand_defined_factorial_valuation_valuation_exponent_bound + e = bfv_factorial_bertrand_defined_factorial_valuation) /\ (exists bpv_result_bertrand_defined_factorial_valuation_valuation_selected. ((exists ff_b_bertrand_defined_factorial_valuation_valuation_selected_power ff_c_bertrand_defined_factorial_valuation_valuation_selected_power. ((forall ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_repeat. (exists ff_lt_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_bound. ff_lt_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_bound + S ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_repeat = e) -> (((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_decoded. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_repeat)) * ff_c_bertrand_defined_factorial_valuation_valuation_selected_power)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_decoded. ff_b_bertrand_defined_factorial_valuation_valuation_selected_power = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_repeat_decoded * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_repeat)) * ff_c_bertrand_defined_factorial_valuation_valuation_selected_power) + (p)))) /\ (exists ff_u_bertrand_defined_factorial_valuation_valuation_selected_power_product ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product. ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_start. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_start. ff_u_bertrand_defined_factorial_valuation_valuation_selected_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_start * S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product) + (1))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_terminal. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_terminal + S (bpv_result_bertrand_defined_factorial_valuation_valuation_selected) = S ((S (e)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_terminal. ff_u_bertrand_defined_factorial_valuation_valuation_selected_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product) + (bpv_result_bertrand_defined_factorial_valuation_valuation_selected))) /\ forall ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product. (exists ff_lt_bertrand_defined_factorial_valuation_valuation_selected_power_product_bound. ff_lt_bertrand_defined_factorial_valuation_valuation_selected_power_product_bound + S ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product = e) -> exists ff_p_bertrand_defined_factorial_valuation_valuation_selected_power_product ff_r_bertrand_defined_factorial_valuation_valuation_selected_power_product ff_s_bertrand_defined_factorial_valuation_valuation_selected_power_product. ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_factor. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_factor + S (ff_p_bertrand_defined_factorial_valuation_valuation_selected_power_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_c_bertrand_defined_factorial_valuation_valuation_selected_power)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_factor. ff_b_bertrand_defined_factorial_valuation_valuation_selected_power = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_factor * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_c_bertrand_defined_factorial_valuation_valuation_selected_power) + (ff_p_bertrand_defined_factorial_valuation_valuation_selected_power_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_partial. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_partial + S (ff_r_bertrand_defined_factorial_valuation_valuation_selected_power_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_partial. ff_u_bertrand_defined_factorial_valuation_valuation_selected_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_partial * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product) + (ff_r_bertrand_defined_factorial_valuation_valuation_selected_power_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_successor. ff_h_bertrand_defined_factorial_valuation_valuation_selected_power_product_successor + S (ff_s_bertrand_defined_factorial_valuation_valuation_selected_power_product) = S ((S (S ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_successor. ff_u_bertrand_defined_factorial_valuation_valuation_selected_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_selected_power_product_successor * S ((S (S ff_i_bertrand_defined_factorial_valuation_valuation_selected_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_selected_power_product) + (ff_s_bertrand_defined_factorial_valuation_valuation_selected_power_product))) /\ ff_s_bertrand_defined_factorial_valuation_valuation_selected_power_product = ff_r_bertrand_defined_factorial_valuation_valuation_selected_power_product * ff_p_bertrand_defined_factorial_valuation_valuation_selected_power_product)))))))) /\ (exists bpv_factor_bertrand_defined_factorial_valuation_valuation_selected_divides. bfv_factorial_bertrand_defined_factorial_valuation = bpv_result_bertrand_defined_factorial_valuation_valuation_selected * bpv_factor_bertrand_defined_factorial_valuation_valuation_selected_divides)))) /\ forall bpv_candidate_bertrand_defined_factorial_valuation_valuation. (exists bpv_gap_bertrand_defined_factorial_valuation_valuation_candidate_bound. bpv_gap_bertrand_defined_factorial_valuation_valuation_candidate_bound + bpv_candidate_bertrand_defined_factorial_valuation_valuation = bfv_factorial_bertrand_defined_factorial_valuation) -> (exists bpv_result_bertrand_defined_factorial_valuation_valuation_candidate. ((exists ff_b_bertrand_defined_factorial_valuation_valuation_candidate_power ff_c_bertrand_defined_factorial_valuation_valuation_candidate_power. ((forall ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat. (exists ff_lt_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_bound. ff_lt_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_bound + S ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat = bpv_candidate_bertrand_defined_factorial_valuation_valuation) -> (((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_decoded. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat)) * ff_c_bertrand_defined_factorial_valuation_valuation_candidate_power)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_decoded. ff_b_bertrand_defined_factorial_valuation_valuation_candidate_power = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_repeat)) * ff_c_bertrand_defined_factorial_valuation_valuation_candidate_power) + (p)))) /\ (exists ff_u_bertrand_defined_factorial_valuation_valuation_candidate_power_product ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product. ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_start. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_start. ff_u_bertrand_defined_factorial_valuation_valuation_candidate_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product) + (1))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_terminal. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_terminal + S (bpv_result_bertrand_defined_factorial_valuation_valuation_candidate) = S ((S (bpv_candidate_bertrand_defined_factorial_valuation_valuation)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_terminal. ff_u_bertrand_defined_factorial_valuation_valuation_candidate_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bertrand_defined_factorial_valuation_valuation)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product) + (bpv_result_bertrand_defined_factorial_valuation_valuation_candidate))) /\ forall ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product. (exists ff_lt_bertrand_defined_factorial_valuation_valuation_candidate_power_product_bound. ff_lt_bertrand_defined_factorial_valuation_valuation_candidate_power_product_bound + S ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product = bpv_candidate_bertrand_defined_factorial_valuation_valuation) -> exists ff_p_bertrand_defined_factorial_valuation_valuation_candidate_power_product ff_r_bertrand_defined_factorial_valuation_valuation_candidate_power_product ff_s_bertrand_defined_factorial_valuation_valuation_candidate_power_product. ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_factor. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_factor + S (ff_p_bertrand_defined_factorial_valuation_valuation_candidate_power_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_c_bertrand_defined_factorial_valuation_valuation_candidate_power)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_factor. ff_b_bertrand_defined_factorial_valuation_valuation_candidate_power = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_factor * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_c_bertrand_defined_factorial_valuation_valuation_candidate_power) + (ff_p_bertrand_defined_factorial_valuation_valuation_candidate_power_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_partial. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_partial + S (ff_r_bertrand_defined_factorial_valuation_valuation_candidate_power_product) = S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_partial. ff_u_bertrand_defined_factorial_valuation_valuation_candidate_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_partial * S ((S (ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product) + (ff_r_bertrand_defined_factorial_valuation_valuation_candidate_power_product))) /\ ((((exists ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_successor. ff_h_bertrand_defined_factorial_valuation_valuation_candidate_power_product_successor + S (ff_s_bertrand_defined_factorial_valuation_valuation_candidate_power_product) = S ((S (S ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) /\ exists ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_successor. ff_u_bertrand_defined_factorial_valuation_valuation_candidate_power_product = ff_q_bertrand_defined_factorial_valuation_valuation_candidate_power_product_successor * S ((S (S ff_i_bertrand_defined_factorial_valuation_valuation_candidate_power_product)) * ff_v_bertrand_defined_factorial_valuation_valuation_candidate_power_product) + (ff_s_bertrand_defined_factorial_valuation_valuation_candidate_power_product))) /\ ff_s_bertrand_defined_factorial_valuation_valuation_candidate_power_product = ff_r_bertrand_defined_factorial_valuation_valuation_candidate_power_product * ff_p_bertrand_defined_factorial_valuation_valuation_candidate_power_product)))))))) /\ (exists bpv_factor_bertrand_defined_factorial_valuation_valuation_candidate_divides. bfv_factorial_bertrand_defined_factorial_valuation = bpv_result_bertrand_defined_factorial_valuation_valuation_candidate * bpv_factor_bertrand_defined_factorial_valuation_valuation_candidate_divides))) -> (exists bpv_gap_bertrand_defined_factorial_valuation_valuation_maximal. bpv_gap_bertrand_defined_factorial_valuation_valuation_maximal + bpv_candidate_bertrand_defined_factorial_valuation_valuation = e)))

This node is conservative notation, not a theorem, axiom, predicate constant, or kernel rule. Its expansion remains in the unchanged first-order language.

Definition neighborhood

Depends on conservative definitions

Used by conservative definitions

All transitive conservative prerequisites

Used by theorem statements or local proof propositions

Grand-campaign planning vocabulary

Locate FactorialValuation in the global campaign vocabulary →

Reviewed FactorialValuation corresponds to blueprint FactorialValuation with checked argument positions [0, 1, 2].

The global atlas describes planning vocabulary and does not itself certify a definition or theorem. The reviewed expansion and conservative dependency DAG on this page are the actual family-local reading definitions.