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.
All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.
Exact theorem in conservative defined notation
∀ p. ∀ x. ∀ y. ∀ d. ∀ n. ∀ a. ∀ b. ¬p = 1 ∧ (∀ z. ∀ m. p = z · m → z = 1 ∨ m = 1) → Lt(2,p) → Lt(y,x) → ¬y = 0 → ¬n = 0 → x = y + d → Dvd(p,d) → ¬Dvd(p,x · y) → BoundedPowerValuation(p,d,d,a) → BoundedPowerValuation(p,n,n,b) → ∃ z. ∃ m. ∃ k. LiftedPowerDifference(p,x,y,n,a + b,z,m,k)
Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall p x y d n a b. (~((p) = 1) /\ forall pvs_left_public_lte_prime pvs_right_public_lte_prime. (p) = pvs_left_public_lte_prime * pvs_right_public_lte_prime -> pvs_left_public_lte_prime = 1 \/ pvs_right_public_lte_prime = 1) -> (exists olte_gap_public_lte_odd. olte_gap_public_lte_odd + S (2) = (p)) -> (exists olte_gap_public_lte_order. olte_gap_public_lte_order + S (y) = (x)) -> ~(y = 0) -> ~(n = 0) -> x = y + d -> (exists olte_factor_public_lte_divisor. (d) = (p) * olte_factor_public_lte_divisor) -> ~(exists olte_factor_public_lte_units. (x * y) = (p) * olte_factor_public_lte_units) -> (((exists bpd_gap_pvs_public_lte_difference_valuation_selected_bound. bpd_gap_pvs_public_lte_difference_valuation_selected_bound + (a) = (d)) /\ (exists bpvi_result_pvs_public_lte_difference_valuation_selected. ((exists bpvi_b_pvs_public_lte_difference_valuation_selected_power bpvi_c_pvs_public_lte_difference_valuation_selected_power. ((forall bpvi_i_pvs_public_lte_difference_valuation_selected_power. (exists bpvi_repeat_gap_pvs_public_lte_difference_valuation_selected_power. bpvi_repeat_gap_pvs_public_lte_difference_valuation_selected_power + S bpvi_i_pvs_public_lte_difference_valuation_selected_power = a) -> (((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_repeat. bpvi_h_pvs_public_lte_difference_valuation_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_public_lte_difference_valuation_selected_power)) * bpvi_c_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_repeat. bpvi_b_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_repeat * S ((S (bpvi_i_pvs_public_lte_difference_valuation_selected_power)) * bpvi_c_pvs_public_lte_difference_valuation_selected_power) + (p)))) /\ (exists bpvi_u_pvs_public_lte_difference_valuation_selected_power bpvi_v_pvs_public_lte_difference_valuation_selected_power. ((((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_start. bpvi_h_pvs_public_lte_difference_valuation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_start. bpvi_u_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_start * S ((S (0)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_terminal. bpvi_h_pvs_public_lte_difference_valuation_selected_power_terminal + S (bpvi_result_pvs_public_lte_difference_valuation_selected) = S ((S (a)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_terminal. bpvi_u_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_terminal * S ((S (a)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power) + (bpvi_result_pvs_public_lte_difference_valuation_selected))) /\ forall bpvi_j_pvs_public_lte_difference_valuation_selected_power. (exists bpvi_product_gap_pvs_public_lte_difference_valuation_selected_power. bpvi_product_gap_pvs_public_lte_difference_valuation_selected_power + S bpvi_j_pvs_public_lte_difference_valuation_selected_power = a) -> exists bpvi_factor_pvs_public_lte_difference_valuation_selected_power bpvi_partial_pvs_public_lte_difference_valuation_selected_power bpvi_successor_pvs_public_lte_difference_valuation_selected_power. ((((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_factor. bpvi_h_pvs_public_lte_difference_valuation_selected_power_factor + S (bpvi_factor_pvs_public_lte_difference_valuation_selected_power) = S ((S (bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_c_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_factor. bpvi_b_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_factor * S ((S (bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_c_pvs_public_lte_difference_valuation_selected_power) + (bpvi_factor_pvs_public_lte_difference_valuation_selected_power))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_partial. bpvi_h_pvs_public_lte_difference_valuation_selected_power_partial + S (bpvi_partial_pvs_public_lte_difference_valuation_selected_power) = S ((S (bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_partial. bpvi_u_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_partial * S ((S (bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power) + (bpvi_partial_pvs_public_lte_difference_valuation_selected_power))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_selected_power_successor. bpvi_h_pvs_public_lte_difference_valuation_selected_power_successor + S (bpvi_successor_pvs_public_lte_difference_valuation_selected_power) = S ((S (S bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_selected_power_successor. bpvi_u_pvs_public_lte_difference_valuation_selected_power = bpvi_q_pvs_public_lte_difference_valuation_selected_power_successor * S ((S (S bpvi_j_pvs_public_lte_difference_valuation_selected_power)) * bpvi_v_pvs_public_lte_difference_valuation_selected_power) + (bpvi_successor_pvs_public_lte_difference_valuation_selected_power))) /\ bpvi_successor_pvs_public_lte_difference_valuation_selected_power = bpvi_partial_pvs_public_lte_difference_valuation_selected_power * bpvi_factor_pvs_public_lte_difference_valuation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_public_lte_difference_valuation_selected. d = bpvi_result_pvs_public_lte_difference_valuation_selected * bpvi_divisor_factor_pvs_public_lte_difference_valuation_selected))) /\ forall bpd_candidate_pvs_public_lte_difference_valuation. (exists bpd_gap_pvs_public_lte_difference_valuation_candidate_bound. bpd_gap_pvs_public_lte_difference_valuation_candidate_bound + (bpd_candidate_pvs_public_lte_difference_valuation) = (d)) -> (exists bpvi_result_pvs_public_lte_difference_valuation_candidate. ((exists bpvi_b_pvs_public_lte_difference_valuation_candidate_power bpvi_c_pvs_public_lte_difference_valuation_candidate_power. ((forall bpvi_i_pvs_public_lte_difference_valuation_candidate_power. (exists bpvi_repeat_gap_pvs_public_lte_difference_valuation_candidate_power. bpvi_repeat_gap_pvs_public_lte_difference_valuation_candidate_power + S bpvi_i_pvs_public_lte_difference_valuation_candidate_power = bpd_candidate_pvs_public_lte_difference_valuation) -> (((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_repeat. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_c_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_repeat. bpvi_b_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_repeat * S ((S (bpvi_i_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_c_pvs_public_lte_difference_valuation_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_public_lte_difference_valuation_candidate_power bpvi_v_pvs_public_lte_difference_valuation_candidate_power. ((((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_start. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_start. bpvi_u_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_terminal. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_terminal + S (bpvi_result_pvs_public_lte_difference_valuation_candidate) = S ((S (bpd_candidate_pvs_public_lte_difference_valuation)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_terminal. bpvi_u_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_terminal * S ((S (bpd_candidate_pvs_public_lte_difference_valuation)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power) + (bpvi_result_pvs_public_lte_difference_valuation_candidate))) /\ forall bpvi_j_pvs_public_lte_difference_valuation_candidate_power. (exists bpvi_product_gap_pvs_public_lte_difference_valuation_candidate_power. bpvi_product_gap_pvs_public_lte_difference_valuation_candidate_power + S bpvi_j_pvs_public_lte_difference_valuation_candidate_power = bpd_candidate_pvs_public_lte_difference_valuation) -> exists bpvi_factor_pvs_public_lte_difference_valuation_candidate_power bpvi_partial_pvs_public_lte_difference_valuation_candidate_power bpvi_successor_pvs_public_lte_difference_valuation_candidate_power. ((((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_factor. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_factor + S (bpvi_factor_pvs_public_lte_difference_valuation_candidate_power) = S ((S (bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_c_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_factor. bpvi_b_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_factor * S ((S (bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_c_pvs_public_lte_difference_valuation_candidate_power) + (bpvi_factor_pvs_public_lte_difference_valuation_candidate_power))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_partial. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_partial + S (bpvi_partial_pvs_public_lte_difference_valuation_candidate_power) = S ((S (bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_partial. bpvi_u_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_partial * S ((S (bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power) + (bpvi_partial_pvs_public_lte_difference_valuation_candidate_power))) /\ ((((exists bpvi_h_pvs_public_lte_difference_valuation_candidate_power_successor. bpvi_h_pvs_public_lte_difference_valuation_candidate_power_successor + S (bpvi_successor_pvs_public_lte_difference_valuation_candidate_power) = S ((S (S bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_difference_valuation_candidate_power_successor. bpvi_u_pvs_public_lte_difference_valuation_candidate_power = bpvi_q_pvs_public_lte_difference_valuation_candidate_power_successor * S ((S (S bpvi_j_pvs_public_lte_difference_valuation_candidate_power)) * bpvi_v_pvs_public_lte_difference_valuation_candidate_power) + (bpvi_successor_pvs_public_lte_difference_valuation_candidate_power))) /\ bpvi_successor_pvs_public_lte_difference_valuation_candidate_power = bpvi_partial_pvs_public_lte_difference_valuation_candidate_power * bpvi_factor_pvs_public_lte_difference_valuation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_public_lte_difference_valuation_candidate. d = bpvi_result_pvs_public_lte_difference_valuation_candidate * bpvi_divisor_factor_pvs_public_lte_difference_valuation_candidate)) -> (exists bpd_gap_pvs_public_lte_difference_valuation_maximal. bpd_gap_pvs_public_lte_difference_valuation_maximal + (bpd_candidate_pvs_public_lte_difference_valuation) = (a))) -> (((exists bpd_gap_pvs_public_lte_exponent_valuation_selected_bound. bpd_gap_pvs_public_lte_exponent_valuation_selected_bound + (b) = (n)) /\ (exists bpvi_result_pvs_public_lte_exponent_valuation_selected. ((exists bpvi_b_pvs_public_lte_exponent_valuation_selected_power bpvi_c_pvs_public_lte_exponent_valuation_selected_power. ((forall bpvi_i_pvs_public_lte_exponent_valuation_selected_power. (exists bpvi_repeat_gap_pvs_public_lte_exponent_valuation_selected_power. bpvi_repeat_gap_pvs_public_lte_exponent_valuation_selected_power + S bpvi_i_pvs_public_lte_exponent_valuation_selected_power = b) -> (((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_repeat. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_c_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_repeat. bpvi_b_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_repeat * S ((S (bpvi_i_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_c_pvs_public_lte_exponent_valuation_selected_power) + (p)))) /\ (exists bpvi_u_pvs_public_lte_exponent_valuation_selected_power bpvi_v_pvs_public_lte_exponent_valuation_selected_power. ((((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_start. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_start. bpvi_u_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_start * S ((S (0)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_terminal. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_terminal + S (bpvi_result_pvs_public_lte_exponent_valuation_selected) = S ((S (b)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_terminal. bpvi_u_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_terminal * S ((S (b)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power) + (bpvi_result_pvs_public_lte_exponent_valuation_selected))) /\ forall bpvi_j_pvs_public_lte_exponent_valuation_selected_power. (exists bpvi_product_gap_pvs_public_lte_exponent_valuation_selected_power. bpvi_product_gap_pvs_public_lte_exponent_valuation_selected_power + S bpvi_j_pvs_public_lte_exponent_valuation_selected_power = b) -> exists bpvi_factor_pvs_public_lte_exponent_valuation_selected_power bpvi_partial_pvs_public_lte_exponent_valuation_selected_power bpvi_successor_pvs_public_lte_exponent_valuation_selected_power. ((((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_factor. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_factor + S (bpvi_factor_pvs_public_lte_exponent_valuation_selected_power) = S ((S (bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_c_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_factor. bpvi_b_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_factor * S ((S (bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_c_pvs_public_lte_exponent_valuation_selected_power) + (bpvi_factor_pvs_public_lte_exponent_valuation_selected_power))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_partial. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_partial + S (bpvi_partial_pvs_public_lte_exponent_valuation_selected_power) = S ((S (bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_partial. bpvi_u_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_partial * S ((S (bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power) + (bpvi_partial_pvs_public_lte_exponent_valuation_selected_power))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_selected_power_successor. bpvi_h_pvs_public_lte_exponent_valuation_selected_power_successor + S (bpvi_successor_pvs_public_lte_exponent_valuation_selected_power) = S ((S (S bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_selected_power_successor. bpvi_u_pvs_public_lte_exponent_valuation_selected_power = bpvi_q_pvs_public_lte_exponent_valuation_selected_power_successor * S ((S (S bpvi_j_pvs_public_lte_exponent_valuation_selected_power)) * bpvi_v_pvs_public_lte_exponent_valuation_selected_power) + (bpvi_successor_pvs_public_lte_exponent_valuation_selected_power))) /\ bpvi_successor_pvs_public_lte_exponent_valuation_selected_power = bpvi_partial_pvs_public_lte_exponent_valuation_selected_power * bpvi_factor_pvs_public_lte_exponent_valuation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_public_lte_exponent_valuation_selected. n = bpvi_result_pvs_public_lte_exponent_valuation_selected * bpvi_divisor_factor_pvs_public_lte_exponent_valuation_selected))) /\ forall bpd_candidate_pvs_public_lte_exponent_valuation. (exists bpd_gap_pvs_public_lte_exponent_valuation_candidate_bound. bpd_gap_pvs_public_lte_exponent_valuation_candidate_bound + (bpd_candidate_pvs_public_lte_exponent_valuation) = (n)) -> (exists bpvi_result_pvs_public_lte_exponent_valuation_candidate. ((exists bpvi_b_pvs_public_lte_exponent_valuation_candidate_power bpvi_c_pvs_public_lte_exponent_valuation_candidate_power. ((forall bpvi_i_pvs_public_lte_exponent_valuation_candidate_power. (exists bpvi_repeat_gap_pvs_public_lte_exponent_valuation_candidate_power. bpvi_repeat_gap_pvs_public_lte_exponent_valuation_candidate_power + S bpvi_i_pvs_public_lte_exponent_valuation_candidate_power = bpd_candidate_pvs_public_lte_exponent_valuation) -> (((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_repeat. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_c_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_repeat. bpvi_b_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_repeat * S ((S (bpvi_i_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_c_pvs_public_lte_exponent_valuation_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_public_lte_exponent_valuation_candidate_power bpvi_v_pvs_public_lte_exponent_valuation_candidate_power. ((((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_start. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_start. bpvi_u_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_terminal. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_terminal + S (bpvi_result_pvs_public_lte_exponent_valuation_candidate) = S ((S (bpd_candidate_pvs_public_lte_exponent_valuation)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_terminal. bpvi_u_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_terminal * S ((S (bpd_candidate_pvs_public_lte_exponent_valuation)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power) + (bpvi_result_pvs_public_lte_exponent_valuation_candidate))) /\ forall bpvi_j_pvs_public_lte_exponent_valuation_candidate_power. (exists bpvi_product_gap_pvs_public_lte_exponent_valuation_candidate_power. bpvi_product_gap_pvs_public_lte_exponent_valuation_candidate_power + S bpvi_j_pvs_public_lte_exponent_valuation_candidate_power = bpd_candidate_pvs_public_lte_exponent_valuation) -> exists bpvi_factor_pvs_public_lte_exponent_valuation_candidate_power bpvi_partial_pvs_public_lte_exponent_valuation_candidate_power bpvi_successor_pvs_public_lte_exponent_valuation_candidate_power. ((((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_factor. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_factor + S (bpvi_factor_pvs_public_lte_exponent_valuation_candidate_power) = S ((S (bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_c_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_factor. bpvi_b_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_factor * S ((S (bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_c_pvs_public_lte_exponent_valuation_candidate_power) + (bpvi_factor_pvs_public_lte_exponent_valuation_candidate_power))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_partial. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_partial + S (bpvi_partial_pvs_public_lte_exponent_valuation_candidate_power) = S ((S (bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_partial. bpvi_u_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_partial * S ((S (bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power) + (bpvi_partial_pvs_public_lte_exponent_valuation_candidate_power))) /\ ((((exists bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_successor. bpvi_h_pvs_public_lte_exponent_valuation_candidate_power_successor + S (bpvi_successor_pvs_public_lte_exponent_valuation_candidate_power) = S ((S (S bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power)) /\ exists bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_successor. bpvi_u_pvs_public_lte_exponent_valuation_candidate_power = bpvi_q_pvs_public_lte_exponent_valuation_candidate_power_successor * S ((S (S bpvi_j_pvs_public_lte_exponent_valuation_candidate_power)) * bpvi_v_pvs_public_lte_exponent_valuation_candidate_power) + (bpvi_successor_pvs_public_lte_exponent_valuation_candidate_power))) /\ bpvi_successor_pvs_public_lte_exponent_valuation_candidate_power = bpvi_partial_pvs_public_lte_exponent_valuation_candidate_power * bpvi_factor_pvs_public_lte_exponent_valuation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_public_lte_exponent_valuation_candidate. n = bpvi_result_pvs_public_lte_exponent_valuation_candidate * bpvi_divisor_factor_pvs_public_lte_exponent_valuation_candidate)) -> (exists bpd_gap_pvs_public_lte_exponent_valuation_maximal. bpd_gap_pvs_public_lte_exponent_valuation_maximal + (bpd_candidate_pvs_public_lte_exponent_valuation) = (b))) -> exists X Y D. (((exists pa_b_olte_public_lte_resultA pa_c_olte_public_lte_resultA. ((forall pa_i_olte_public_lte_resultA_repeat. (exists pa_lt_olte_public_lte_resultA_repeat_bound. pa_lt_olte_public_lte_resultA_repeat_bound + S pa_i_olte_public_lte_resultA_repeat = n) -> (((exists pa_h_olte_public_lte_resultA_repeat_decoded. pa_h_olte_public_lte_resultA_repeat_decoded + S (x) = S ((S (pa_i_olte_public_lte_resultA_repeat)) * pa_c_olte_public_lte_resultA)) /\ exists pa_q_olte_public_lte_resultA_repeat_decoded. pa_b_olte_public_lte_resultA = pa_q_olte_public_lte_resultA_repeat_decoded * S ((S (pa_i_olte_public_lte_resultA_repeat)) * pa_c_olte_public_lte_resultA) + (x)))) /\ (exists pa_u_olte_public_lte_resultA_product pa_v_olte_public_lte_resultA_product. ((((exists pa_h_olte_public_lte_resultA_product_start. pa_h_olte_public_lte_resultA_product_start + S (1) = S ((S (0)) * pa_v_olte_public_lte_resultA_product)) /\ exists pa_q_olte_public_lte_resultA_product_start. pa_u_olte_public_lte_resultA_product = pa_q_olte_public_lte_resultA_product_start * S ((S (0)) * pa_v_olte_public_lte_resultA_product) + (1))) /\ ((((exists pa_h_olte_public_lte_resultA_product_terminal. pa_h_olte_public_lte_resultA_product_terminal + S (X) = S ((S (n)) * pa_v_olte_public_lte_resultA_product)) /\ exists pa_q_olte_public_lte_resultA_product_terminal. pa_u_olte_public_lte_resultA_product = pa_q_olte_public_lte_resultA_product_terminal * S ((S (n)) * pa_v_olte_public_lte_resultA_product) + (X))) /\ forall pa_i_olte_public_lte_resultA_product. (exists pa_lt_olte_public_lte_resultA_product_bound. pa_lt_olte_public_lte_resultA_product_bound + S pa_i_olte_public_lte_resultA_product = n) -> exists pa_p_olte_public_lte_resultA_product pa_r_olte_public_lte_resultA_product pa_s_olte_public_lte_resultA_product. ((((exists pa_h_olte_public_lte_resultA_product_factor. pa_h_olte_public_lte_resultA_product_factor + S (pa_p_olte_public_lte_resultA_product) = S ((S (pa_i_olte_public_lte_resultA_product)) * pa_c_olte_public_lte_resultA)) /\ exists pa_q_olte_public_lte_resultA_product_factor. pa_b_olte_public_lte_resultA = pa_q_olte_public_lte_resultA_product_factor * S ((S (pa_i_olte_public_lte_resultA_product)) * pa_c_olte_public_lte_resultA) + (pa_p_olte_public_lte_resultA_product))) /\ ((((exists pa_h_olte_public_lte_resultA_product_partial. pa_h_olte_public_lte_resultA_product_partial + S (pa_r_olte_public_lte_resultA_product) = S ((S (pa_i_olte_public_lte_resultA_product)) * pa_v_olte_public_lte_resultA_product)) /\ exists pa_q_olte_public_lte_resultA_product_partial. pa_u_olte_public_lte_resultA_product = pa_q_olte_public_lte_resultA_product_partial * S ((S (pa_i_olte_public_lte_resultA_product)) * pa_v_olte_public_lte_resultA_product) + (pa_r_olte_public_lte_resultA_product))) /\ ((((exists pa_h_olte_public_lte_resultA_product_successor. pa_h_olte_public_lte_resultA_product_successor + S (pa_s_olte_public_lte_resultA_product) = S ((S (S pa_i_olte_public_lte_resultA_product)) * pa_v_olte_public_lte_resultA_product)) /\ exists pa_q_olte_public_lte_resultA_product_successor. pa_u_olte_public_lte_resultA_product = pa_q_olte_public_lte_resultA_product_successor * S ((S (S pa_i_olte_public_lte_resultA_product)) * pa_v_olte_public_lte_resultA_product) + (pa_s_olte_public_lte_resultA_product))) /\ pa_s_olte_public_lte_resultA_product = pa_r_olte_public_lte_resultA_product * pa_p_olte_public_lte_resultA_product)))))))) /\ (((exists pa_b_olte_public_lte_resultB pa_c_olte_public_lte_resultB. ((forall pa_i_olte_public_lte_resultB_repeat. (exists pa_lt_olte_public_lte_resultB_repeat_bound. pa_lt_olte_public_lte_resultB_repeat_bound + S pa_i_olte_public_lte_resultB_repeat = n) -> (((exists pa_h_olte_public_lte_resultB_repeat_decoded. pa_h_olte_public_lte_resultB_repeat_decoded + S (y) = S ((S (pa_i_olte_public_lte_resultB_repeat)) * pa_c_olte_public_lte_resultB)) /\ exists pa_q_olte_public_lte_resultB_repeat_decoded. pa_b_olte_public_lte_resultB = pa_q_olte_public_lte_resultB_repeat_decoded * S ((S (pa_i_olte_public_lte_resultB_repeat)) * pa_c_olte_public_lte_resultB) + (y)))) /\ (exists pa_u_olte_public_lte_resultB_product pa_v_olte_public_lte_resultB_product. ((((exists pa_h_olte_public_lte_resultB_product_start. pa_h_olte_public_lte_resultB_product_start + S (1) = S ((S (0)) * pa_v_olte_public_lte_resultB_product)) /\ exists pa_q_olte_public_lte_resultB_product_start. pa_u_olte_public_lte_resultB_product = pa_q_olte_public_lte_resultB_product_start * S ((S (0)) * pa_v_olte_public_lte_resultB_product) + (1))) /\ ((((exists pa_h_olte_public_lte_resultB_product_terminal. pa_h_olte_public_lte_resultB_product_terminal + S (Y) = S ((S (n)) * pa_v_olte_public_lte_resultB_product)) /\ exists pa_q_olte_public_lte_resultB_product_terminal. pa_u_olte_public_lte_resultB_product = pa_q_olte_public_lte_resultB_product_terminal * S ((S (n)) * pa_v_olte_public_lte_resultB_product) + (Y))) /\ forall pa_i_olte_public_lte_resultB_product. (exists pa_lt_olte_public_lte_resultB_product_bound. pa_lt_olte_public_lte_resultB_product_bound + S pa_i_olte_public_lte_resultB_product = n) -> exists pa_p_olte_public_lte_resultB_product pa_r_olte_public_lte_resultB_product pa_s_olte_public_lte_resultB_product. ((((exists pa_h_olte_public_lte_resultB_product_factor. pa_h_olte_public_lte_resultB_product_factor + S (pa_p_olte_public_lte_resultB_product) = S ((S (pa_i_olte_public_lte_resultB_product)) * pa_c_olte_public_lte_resultB)) /\ exists pa_q_olte_public_lte_resultB_product_factor. pa_b_olte_public_lte_resultB = pa_q_olte_public_lte_resultB_product_factor * S ((S (pa_i_olte_public_lte_resultB_product)) * pa_c_olte_public_lte_resultB) + (pa_p_olte_public_lte_resultB_product))) /\ ((((exists pa_h_olte_public_lte_resultB_product_partial. pa_h_olte_public_lte_resultB_product_partial + S (pa_r_olte_public_lte_resultB_product) = S ((S (pa_i_olte_public_lte_resultB_product)) * pa_v_olte_public_lte_resultB_product)) /\ exists pa_q_olte_public_lte_resultB_product_partial. pa_u_olte_public_lte_resultB_product = pa_q_olte_public_lte_resultB_product_partial * S ((S (pa_i_olte_public_lte_resultB_product)) * pa_v_olte_public_lte_resultB_product) + (pa_r_olte_public_lte_resultB_product))) /\ ((((exists pa_h_olte_public_lte_resultB_product_successor. pa_h_olte_public_lte_resultB_product_successor + S (pa_s_olte_public_lte_resultB_product) = S ((S (S pa_i_olte_public_lte_resultB_product)) * pa_v_olte_public_lte_resultB_product)) /\ exists pa_q_olte_public_lte_resultB_product_successor. pa_u_olte_public_lte_resultB_product = pa_q_olte_public_lte_resultB_product_successor * S ((S (S pa_i_olte_public_lte_resultB_product)) * pa_v_olte_public_lte_resultB_product) + (pa_s_olte_public_lte_resultB_product))) /\ pa_s_olte_public_lte_resultB_product = pa_r_olte_public_lte_resultB_product * pa_p_olte_public_lte_resultB_product)))))))) /\ ((((X) = (Y) + (D)) /\ (((~((D) = 0)) /\ (((exists olte_factor_public_lte_resultdivides. (D) = (p) * olte_factor_public_lte_resultdivides) /\ (((~(exists olte_factor_public_lte_resultunit. (Y) = (p) * olte_factor_public_lte_resultunit)) /\ (((exists bpd_gap_pvs_olte_public_lte_resultvaluation_selected_bound. bpd_gap_pvs_olte_public_lte_resultvaluation_selected_bound + (a + b) = (D)) /\ (exists bpvi_result_pvs_olte_public_lte_resultvaluation_selected. ((exists bpvi_b_pvs_olte_public_lte_resultvaluation_selected_power bpvi_c_pvs_olte_public_lte_resultvaluation_selected_power. ((forall bpvi_i_pvs_olte_public_lte_resultvaluation_selected_power. (exists bpvi_repeat_gap_pvs_olte_public_lte_resultvaluation_selected_power. bpvi_repeat_gap_pvs_olte_public_lte_resultvaluation_selected_power + S bpvi_i_pvs_olte_public_lte_resultvaluation_selected_power = a + b) -> (((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_repeat. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_repeat. bpvi_b_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_repeat * S ((S (bpvi_i_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_selected_power) + (p)))) /\ (exists bpvi_u_pvs_olte_public_lte_resultvaluation_selected_power bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power. ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_start. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_start. bpvi_u_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_start * S ((S (0)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_terminal. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_terminal + S (bpvi_result_pvs_olte_public_lte_resultvaluation_selected) = S ((S (a + b)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_terminal. bpvi_u_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_terminal * S ((S (a + b)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power) + (bpvi_result_pvs_olte_public_lte_resultvaluation_selected))) /\ forall bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power. (exists bpvi_product_gap_pvs_olte_public_lte_resultvaluation_selected_power. bpvi_product_gap_pvs_olte_public_lte_resultvaluation_selected_power + S bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power = a + b) -> exists bpvi_factor_pvs_olte_public_lte_resultvaluation_selected_power bpvi_partial_pvs_olte_public_lte_resultvaluation_selected_power bpvi_successor_pvs_olte_public_lte_resultvaluation_selected_power. ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_factor. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_factor + S (bpvi_factor_pvs_olte_public_lte_resultvaluation_selected_power) = S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_factor. bpvi_b_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_factor * S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_selected_power) + (bpvi_factor_pvs_olte_public_lte_resultvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_partial. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_partial + S (bpvi_partial_pvs_olte_public_lte_resultvaluation_selected_power) = S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_partial. bpvi_u_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_partial * S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power) + (bpvi_partial_pvs_olte_public_lte_resultvaluation_selected_power))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_successor. bpvi_h_pvs_olte_public_lte_resultvaluation_selected_power_successor + S (bpvi_successor_pvs_olte_public_lte_resultvaluation_selected_power) = S ((S (S bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_successor. bpvi_u_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_q_pvs_olte_public_lte_resultvaluation_selected_power_successor * S ((S (S bpvi_j_pvs_olte_public_lte_resultvaluation_selected_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_selected_power) + (bpvi_successor_pvs_olte_public_lte_resultvaluation_selected_power))) /\ bpvi_successor_pvs_olte_public_lte_resultvaluation_selected_power = bpvi_partial_pvs_olte_public_lte_resultvaluation_selected_power * bpvi_factor_pvs_olte_public_lte_resultvaluation_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_olte_public_lte_resultvaluation_selected. D = bpvi_result_pvs_olte_public_lte_resultvaluation_selected * bpvi_divisor_factor_pvs_olte_public_lte_resultvaluation_selected))) /\ forall bpd_candidate_pvs_olte_public_lte_resultvaluation. (exists bpd_gap_pvs_olte_public_lte_resultvaluation_candidate_bound. bpd_gap_pvs_olte_public_lte_resultvaluation_candidate_bound + (bpd_candidate_pvs_olte_public_lte_resultvaluation) = (D)) -> (exists bpvi_result_pvs_olte_public_lte_resultvaluation_candidate. ((exists bpvi_b_pvs_olte_public_lte_resultvaluation_candidate_power bpvi_c_pvs_olte_public_lte_resultvaluation_candidate_power. ((forall bpvi_i_pvs_olte_public_lte_resultvaluation_candidate_power. (exists bpvi_repeat_gap_pvs_olte_public_lte_resultvaluation_candidate_power. bpvi_repeat_gap_pvs_olte_public_lte_resultvaluation_candidate_power + S bpvi_i_pvs_olte_public_lte_resultvaluation_candidate_power = bpd_candidate_pvs_olte_public_lte_resultvaluation) -> (((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_repeat. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_repeat. bpvi_b_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_repeat * S ((S (bpvi_i_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_olte_public_lte_resultvaluation_candidate_power bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power. ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_start. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_start. bpvi_u_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_start * S ((S (0)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_terminal. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_terminal + S (bpvi_result_pvs_olte_public_lte_resultvaluation_candidate) = S ((S (bpd_candidate_pvs_olte_public_lte_resultvaluation)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_terminal. bpvi_u_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_terminal * S ((S (bpd_candidate_pvs_olte_public_lte_resultvaluation)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power) + (bpvi_result_pvs_olte_public_lte_resultvaluation_candidate))) /\ forall bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power. (exists bpvi_product_gap_pvs_olte_public_lte_resultvaluation_candidate_power. bpvi_product_gap_pvs_olte_public_lte_resultvaluation_candidate_power + S bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power = bpd_candidate_pvs_olte_public_lte_resultvaluation) -> exists bpvi_factor_pvs_olte_public_lte_resultvaluation_candidate_power bpvi_partial_pvs_olte_public_lte_resultvaluation_candidate_power bpvi_successor_pvs_olte_public_lte_resultvaluation_candidate_power. ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_factor. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_factor + S (bpvi_factor_pvs_olte_public_lte_resultvaluation_candidate_power) = S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_factor. bpvi_b_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_factor * S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_c_pvs_olte_public_lte_resultvaluation_candidate_power) + (bpvi_factor_pvs_olte_public_lte_resultvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_partial. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_partial + S (bpvi_partial_pvs_olte_public_lte_resultvaluation_candidate_power) = S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_partial. bpvi_u_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_partial * S ((S (bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power) + (bpvi_partial_pvs_olte_public_lte_resultvaluation_candidate_power))) /\ ((((exists bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_successor. bpvi_h_pvs_olte_public_lte_resultvaluation_candidate_power_successor + S (bpvi_successor_pvs_olte_public_lte_resultvaluation_candidate_power) = S ((S (S bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power)) /\ exists bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_successor. bpvi_u_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_q_pvs_olte_public_lte_resultvaluation_candidate_power_successor * S ((S (S bpvi_j_pvs_olte_public_lte_resultvaluation_candidate_power)) * bpvi_v_pvs_olte_public_lte_resultvaluation_candidate_power) + (bpvi_successor_pvs_olte_public_lte_resultvaluation_candidate_power))) /\ bpvi_successor_pvs_olte_public_lte_resultvaluation_candidate_power = bpvi_partial_pvs_olte_public_lte_resultvaluation_candidate_power * bpvi_factor_pvs_olte_public_lte_resultvaluation_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_olte_public_lte_resultvaluation_candidate. D = bpvi_result_pvs_olte_public_lte_resultvaluation_candidate * bpvi_divisor_factor_pvs_olte_public_lte_resultvaluation_candidate)) -> (exists bpd_gap_pvs_olte_public_lte_resultvaluation_maximal. bpd_gap_pvs_olte_public_lte_resultvaluation_maximal + (bpd_candidate_pvs_olte_public_lte_resultvaluation) = (a + b)))))))))))))))Complete tactic proof in conservative notation
All 51 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
51 script commands · 9 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 (4)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–17
03Use earlier factsL18–26
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L18
specialize lte_positive_exponent_exact (p) - L19
specialize lte_positive_exponent_exact (x) - L20
specialize lte_positive_exponent_exact (y) - L21
specialize lte_positive_exponent_exact (d) - L22
specialize lte_positive_exponent_exact (n) - L23
specialize lte_positive_exponent_exact (a) - L24
specialize lte_positive_exponent_exact (b) - L25
apply lte_positive_exponent_exact - L26
exact hp
04Fix variables and assumptionsL27–27
Work with arbitrary variables or the premises of the current implication.
- L27
intro hptwo
05Use earlier factsL28–32
06Fix variables and assumptionsL33–33
Work with arbitrary variables or the premises of the current implication.
- L33
intro hdzero
07Use earlier factsL34–41
Instantiate or apply named facts and discharge the corresponding proof obligations.
08Fix variables and assumptionsL42–42
Work with arbitrary variables or the premises of the current implication.
- L42
intro hydiv
09Use earlier factsL43–51
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original defined command ledger · 51 lines
- 0001
intro p - 0002
intro x - 0003
intro y - 0004
intro d - 0005
intro n - 0006
intro a - 0007
intro b - 0008
intro hp - 0009
intro hpgt - 0010
intro hxy - 0011
intro hyzero - 0012
intro hnzero - 0013
intro hbalance - 0014
intro hdiv - 0015
intro hunits - 0016
intro hvd - 0017
intro hvn - 0018
specialize lte_positive_exponent_exact (p) - 0019
specialize lte_positive_exponent_exact (x) - 0020
specialize lte_positive_exponent_exact (y) - 0021
specialize lte_positive_exponent_exact (d) - 0022
specialize lte_positive_exponent_exact (n) - 0023
specialize lte_positive_exponent_exact (a) - 0024
specialize lte_positive_exponent_exact (b) - 0025
apply lte_positive_exponent_exact - 0026
exact hp - 0027
intro hptwo - 0028
specialize lte_exceeds_two_not_two (p) - 0029
apply lte_exceeds_two_not_two - 0030
exact hpgt - 0031
exact hptwo - 0032
exact hbalance - 0033
intro hdzero - 0034
specialize lte_strict_difference_nonzero (x) - 0035
specialize lte_strict_difference_nonzero (y) - 0036
specialize lte_strict_difference_nonzero (d) - 0037
apply lte_strict_difference_nonzero - 0038
exact hxy - 0039
exact hbalance - 0040
exact hdzero - 0041
exact hdiv - 0042
intro hydiv - 0043
specialize lte_nondivisor_product_right (p) - 0044
specialize lte_nondivisor_product_right (x) - 0045
specialize lte_nondivisor_product_right (y) - 0046
apply lte_nondivisor_product_right - 0047
exact hunits - 0048
exact hydiv - 0049
exact hnzero - 0050
exact hvd - 0051
exact hvn