BT00Q6

bounded_power_valuation_exists

Alpha body-checked · checked-use disabled

Every explicit exponent bound has a greatest power-divisor exponent.

Exact expanded PA statement

forall p a B. exists e. (((exists bpv_gap_bounded_exponent_bound. bpv_gap_bounded_exponent_bound + e = B) /\ (exists bpv_result_bounded_selected. ((exists ff_b_bounded_selected_power ff_c_bounded_selected_power. ((forall ff_i_bounded_selected_power_repeat. (exists ff_lt_bounded_selected_power_repeat_bound. ff_lt_bounded_selected_power_repeat_bound + S ff_i_bounded_selected_power_repeat = e) -> (((exists ff_h_bounded_selected_power_repeat_decoded. ff_h_bounded_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power)) /\ exists ff_q_bounded_selected_power_repeat_decoded. ff_b_bounded_selected_power = ff_q_bounded_selected_power_repeat_decoded * S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power) + (p)))) /\ (exists ff_u_bounded_selected_power_product ff_v_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_start. ff_h_bounded_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_selected_power_product)) /\ exists ff_q_bounded_selected_power_product_start. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_start * S ((S (0)) * ff_v_bounded_selected_power_product) + (1))) /\ ((((exists ff_h_bounded_selected_power_product_terminal. ff_h_bounded_selected_power_product_terminal + S (bpv_result_bounded_selected) = S ((S (e)) * ff_v_bounded_selected_power_product)) /\ exists ff_q_bounded_selected_power_product_terminal. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_terminal * S ((S (e)) * ff_v_bounded_selected_power_product) + (bpv_result_bounded_selected))) /\ forall ff_i_bounded_selected_power_product. (exists ff_lt_bounded_selected_power_product_bound. ff_lt_bounded_selected_power_product_bound + S ff_i_bounded_selected_power_product = e) -> exists ff_p_bounded_selected_power_product ff_r_bounded_selected_power_product ff_s_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_factor. ff_h_bounded_selected_power_product_factor + S (ff_p_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power)) /\ exists ff_q_bounded_selected_power_product_factor. ff_b_bounded_selected_power = ff_q_bounded_selected_power_product_factor * S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power) + (ff_p_bounded_selected_power_product))) /\ ((((exists ff_h_bounded_selected_power_product_partial. ff_h_bounded_selected_power_product_partial + S (ff_r_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\ exists ff_q_bounded_selected_power_product_partial. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_partial * S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_r_bounded_selected_power_product))) /\ ((((exists ff_h_bounded_selected_power_product_successor. ff_h_bounded_selected_power_product_successor + S (ff_s_bounded_selected_power_product) = S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\ exists ff_q_bounded_selected_power_product_successor. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_successor * S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_s_bounded_selected_power_product))) /\ ff_s_bounded_selected_power_product = ff_r_bounded_selected_power_product * ff_p_bounded_selected_power_product)))))))) /\ (exists bpv_factor_bounded_selected_divides. a = bpv_result_bounded_selected * bpv_factor_bounded_selected_divides)))) /\ forall bpv_candidate_bounded. (exists bpv_gap_bounded_candidate_bound. bpv_gap_bounded_candidate_bound + bpv_candidate_bounded = B) -> (exists bpv_result_bounded_candidate. ((exists ff_b_bounded_candidate_power ff_c_bounded_candidate_power. ((forall ff_i_bounded_candidate_power_repeat. (exists ff_lt_bounded_candidate_power_repeat_bound. ff_lt_bounded_candidate_power_repeat_bound + S ff_i_bounded_candidate_power_repeat = bpv_candidate_bounded) -> (((exists ff_h_bounded_candidate_power_repeat_decoded. ff_h_bounded_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power)) /\ exists ff_q_bounded_candidate_power_repeat_decoded. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_repeat_decoded * S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power) + (p)))) /\ (exists ff_u_bounded_candidate_power_product ff_v_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_start. ff_h_bounded_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_candidate_power_product)) /\ exists ff_q_bounded_candidate_power_product_start. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_start * S ((S (0)) * ff_v_bounded_candidate_power_product) + (1))) /\ ((((exists ff_h_bounded_candidate_power_product_terminal. ff_h_bounded_candidate_power_product_terminal + S (bpv_result_bounded_candidate) = S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product)) /\ exists ff_q_bounded_candidate_power_product_terminal. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_terminal * S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product) + (bpv_result_bounded_candidate))) /\ forall ff_i_bounded_candidate_power_product. (exists ff_lt_bounded_candidate_power_product_bound. ff_lt_bounded_candidate_power_product_bound + S ff_i_bounded_candidate_power_product = bpv_candidate_bounded) -> exists ff_p_bounded_candidate_power_product ff_r_bounded_candidate_power_product ff_s_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_factor. ff_h_bounded_candidate_power_product_factor + S (ff_p_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power)) /\ exists ff_q_bounded_candidate_power_product_factor. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_product_factor * S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power) + (ff_p_bounded_candidate_power_product))) /\ ((((exists ff_h_bounded_candidate_power_product_partial. ff_h_bounded_candidate_power_product_partial + S (ff_r_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\ exists ff_q_bounded_candidate_power_product_partial. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_partial * S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_r_bounded_candidate_power_product))) /\ ((((exists ff_h_bounded_candidate_power_product_successor. ff_h_bounded_candidate_power_product_successor + S (ff_s_bounded_candidate_power_product) = S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\ exists ff_q_bounded_candidate_power_product_successor. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_successor * S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_s_bounded_candidate_power_product))) /\ ff_s_bounded_candidate_power_product = ff_r_bounded_candidate_power_product * ff_p_bounded_candidate_power_product)))))))) /\ (exists bpv_factor_bounded_candidate_divides. a = bpv_result_bounded_candidate * bpv_factor_bounded_candidate_divides))) -> (exists bpv_gap_bounded_maximal. bpv_gap_bounded_maximal + bpv_candidate_bounded = e))

Structural proof guide

Every explicit exponent bound has a greatest power-divisor exponent.

Direct prerequisites: bounded_power_valuation_search, power_divides_zero, zero_le. The authored body proceeds by case analysis (2), intermediate claims (2).

Proof neighborhood

Direct dependencies

BT00Q5 bounded_power_valuation_search BT00Q4 power_divides_zero BT000W zero_le

Direct dependents

BT00Q7 power_valuation_exists

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

  1. 0001intro p
  2. 0002intro a
  3. 0003intro B
  4. 0004have hsearch : (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))
  5. 0005specialize bounded_power_valuation_search B
  6. 0006specialize bounded_power_valuation_search p
  7. 0007specialize bounded_power_valuation_search a
  8. 0008exact bounded_power_valuation_search
  9. 0009cases hsearch
  10. 0010have hzero : (exists bpvi_result_exists_zero. ((exists bpvi_b_exists_zero_power bpvi_c_exists_zero_power. ((forall bpvi_i_exists_zero_power. (exists bpvi_repeat_gap_exists_zero_power. bpvi_repeat_gap_exists_zero_power + S bpvi_i_exists_zero_power = 0) -> (((exists bpvi_h_exists_zero_power_repeat. bpvi_h_exists_zero_power_repeat + S (p) = S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_repeat. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_repeat * S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power) + (p)))) /\ (exists bpvi_u_exists_zero_power bpvi_v_exists_zero_power. ((((exists bpvi_h_exists_zero_power_start. bpvi_h_exists_zero_power_start + S (1) = S ((S (0)) * bpvi_v_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_start. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_start * S ((S (0)) * bpvi_v_exists_zero_power) + (1))) /\ ((((exists bpvi_h_exists_zero_power_terminal. bpvi_h_exists_zero_power_terminal + S (bpvi_result_exists_zero) = S ((S (0)) * bpvi_v_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_terminal. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_terminal * S ((S (0)) * bpvi_v_exists_zero_power) + (bpvi_result_exists_zero))) /\ forall bpvi_j_exists_zero_power. (exists bpvi_product_gap_exists_zero_power. bpvi_product_gap_exists_zero_power + S bpvi_j_exists_zero_power = 0) -> exists bpvi_factor_exists_zero_power bpvi_partial_exists_zero_power bpvi_successor_exists_zero_power. ((((exists bpvi_h_exists_zero_power_factor. bpvi_h_exists_zero_power_factor + S (bpvi_factor_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_factor. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_factor * S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power) + (bpvi_factor_exists_zero_power))) /\ ((((exists bpvi_h_exists_zero_power_partial. bpvi_h_exists_zero_power_partial + S (bpvi_partial_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_partial. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_partial * S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_partial_exists_zero_power))) /\ ((((exists bpvi_h_exists_zero_power_successor. bpvi_h_exists_zero_power_successor + S (bpvi_successor_exists_zero_power) = S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\ exists bpvi_q_exists_zero_power_successor. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_successor * S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_successor_exists_zero_power))) /\ bpvi_successor_exists_zero_power = bpvi_partial_exists_zero_power * bpvi_factor_exists_zero_power)))))))) /\ exists bpvi_divisor_factor_exists_zero. a = bpvi_result_exists_zero * bpvi_divisor_factor_exists_zero))
  11. 0011specialize power_divides_zero p
  12. 0012specialize power_divides_zero a
  13. 0013specialize power_divides_zero 0
  14. 0014apply power_divides_zero
  15. 0015refl
  16. 0016specialize hsearch_left 0
  17. 0017exfalso
  18. 0018apply hsearch_left
  19. 0019specialize zero_le B
  20. 0020exact zero_le
  21. 0021exact hzero
  22. 0022cases hsearch_right
  23. 0023exists x
  24. 0024exact hsearch_right_witness