EL0024

odd_prime_lifting_the_exponent

Full guarded odd-prime LTE: for p>2, x>y>0, n>0, p|(x-y), and p not dividing xy, construct x^n,y^n and their positive difference of exact valuation v_p(x-y)+v_p(n).

Alpha v34 checked-use · first admitted v29 · 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.

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

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

  1. L1
    intro p
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro d
  5. L5
    intro n
  6. L6
    intro a
  7. L7
    intro b
  8. L8
    intro hp
  9. L9
    intro hpgt
  10. L10
    intro hxy
02Fix variables and assumptionsL11–17

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

  1. L11
    intro hyzero
  2. L12
    intro hnzero
  3. L13
    intro hbalance
  4. L14
    intro hdiv
  5. L15
    intro hunits
  6. L16
    intro hvd
  7. L17
    intro hvn
03Use earlier factsL18–26

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

  1. L18
    specialize lte_positive_exponent_exact (p)
  2. L19
    specialize lte_positive_exponent_exact (x)
  3. L20
    specialize lte_positive_exponent_exact (y)
  4. L21
    specialize lte_positive_exponent_exact (d)
  5. L22
    specialize lte_positive_exponent_exact (n)
  6. L23
    specialize lte_positive_exponent_exact (a)
  7. L24
    specialize lte_positive_exponent_exact (b)
  8. L25
    apply lte_positive_exponent_exact
  9. L26
    exact hp
04Fix variables and assumptionsL27–27

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

  1. L27
    intro hptwo
05Use earlier factsL28–32

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

  1. L28
    specialize lte_exceeds_two_not_two (p)
  2. L29
    apply lte_exceeds_two_not_two
  3. L30
    exact hpgt
  4. L31
    exact hptwo
  5. L32
    exact hbalance
06Fix variables and assumptionsL33–33

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

  1. L33
    intro hdzero
07Use earlier factsL34–41

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

  1. L34
    specialize lte_strict_difference_nonzero (x)
  2. L35
    specialize lte_strict_difference_nonzero (y)
  3. L36
    specialize lte_strict_difference_nonzero (d)
  4. L37
    apply lte_strict_difference_nonzero
  5. L38
    exact hxy
  6. L39
    exact hbalance
  7. L40
    exact hdzero
  8. L41
    exact hdiv
08Fix variables and assumptionsL42–42

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

  1. L42
    intro hydiv
09Use earlier factsL43–51

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

  1. L43
    specialize lte_nondivisor_product_right (p)
  2. L44
    specialize lte_nondivisor_product_right (x)
  3. L45
    specialize lte_nondivisor_product_right (y)
  4. L46
    apply lte_nondivisor_product_right
  5. L47
    exact hunits
  6. L48
    exact hydiv
  7. L49
    exact hnzero
  8. L50
    exact hvd
  9. L51
    exact hvn

Library-wide reading audit

Original defined command ledger · 51 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro y
  4. 0004intro d
  5. 0005intro n
  6. 0006intro a
  7. 0007intro b
  8. 0008intro hp
  9. 0009intro hpgt
  10. 0010intro hxy
  11. 0011intro hyzero
  12. 0012intro hnzero
  13. 0013intro hbalance
  14. 0014intro hdiv
  15. 0015intro hunits
  16. 0016intro hvd
  17. 0017intro hvn
  18. 0018specialize lte_positive_exponent_exact (p)
  19. 0019specialize lte_positive_exponent_exact (x)
  20. 0020specialize lte_positive_exponent_exact (y)
  21. 0021specialize lte_positive_exponent_exact (d)
  22. 0022specialize lte_positive_exponent_exact (n)
  23. 0023specialize lte_positive_exponent_exact (a)
  24. 0024specialize lte_positive_exponent_exact (b)
  25. 0025apply lte_positive_exponent_exact
  26. 0026exact hp
  27. 0027intro hptwo
  28. 0028specialize lte_exceeds_two_not_two (p)
  29. 0029apply lte_exceeds_two_not_two
  30. 0030exact hpgt
  31. 0031exact hptwo
  32. 0032exact hbalance
  33. 0033intro hdzero
  34. 0034specialize lte_strict_difference_nonzero (x)
  35. 0035specialize lte_strict_difference_nonzero (y)
  36. 0036specialize lte_strict_difference_nonzero (d)
  37. 0037apply lte_strict_difference_nonzero
  38. 0038exact hxy
  39. 0039exact hbalance
  40. 0040exact hdzero
  41. 0041exact hdiv
  42. 0042intro hydiv
  43. 0043specialize lte_nondivisor_product_right (p)
  44. 0044specialize lte_nondivisor_product_right (x)
  45. 0045specialize lte_nondivisor_product_right (y)
  46. 0046apply lte_nondivisor_product_right
  47. 0047exact hunits
  48. 0048exact hydiv
  49. 0049exact hnzero
  50. 0050exact hvd
  51. 0051exact hvn