BT00W0

power_valuation_nonzero_exponent_divides_base

Alpha body-checked · checked-use disabled

A nonzero valuation exponent exposes the base as a divisor.

Exact expanded PA statement

forall p c e. (((exists bpv_gap_bpvnedb_source_exponent_bound. bpv_gap_bpvnedb_source_exponent_bound + e = c) /\ (exists bpv_result_bpvnedb_source_selected. ((exists ff_b_bpvnedb_source_selected_power ff_c_bpvnedb_source_selected_power. ((forall ff_i_bpvnedb_source_selected_power_repeat. (exists ff_lt_bpvnedb_source_selected_power_repeat_bound. ff_lt_bpvnedb_source_selected_power_repeat_bound + S ff_i_bpvnedb_source_selected_power_repeat = e) -> (((exists ff_h_bpvnedb_source_selected_power_repeat_decoded. ff_h_bpvnedb_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power)) /\ exists ff_q_bpvnedb_source_selected_power_repeat_decoded. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power) + (p)))) /\ (exists ff_u_bpvnedb_source_selected_power_product ff_v_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_start. ff_h_bpvnedb_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_selected_power_product)) /\ exists ff_q_bpvnedb_source_selected_power_product_start. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_selected_power_product) + (1))) /\ ((((exists ff_h_bpvnedb_source_selected_power_product_terminal. ff_h_bpvnedb_source_selected_power_product_terminal + S (bpv_result_bpvnedb_source_selected) = S ((S (e)) * ff_v_bpvnedb_source_selected_power_product)) /\ exists ff_q_bpvnedb_source_selected_power_product_terminal. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_terminal * S ((S (e)) * ff_v_bpvnedb_source_selected_power_product) + (bpv_result_bpvnedb_source_selected))) /\ forall ff_i_bpvnedb_source_selected_power_product. (exists ff_lt_bpvnedb_source_selected_power_product_bound. ff_lt_bpvnedb_source_selected_power_product_bound + S ff_i_bpvnedb_source_selected_power_product = e) -> exists ff_p_bpvnedb_source_selected_power_product ff_r_bpvnedb_source_selected_power_product ff_s_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_factor. ff_h_bpvnedb_source_selected_power_product_factor + S (ff_p_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power)) /\ exists ff_q_bpvnedb_source_selected_power_product_factor. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_product_factor * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power) + (ff_p_bpvnedb_source_selected_power_product))) /\ ((((exists ff_h_bpvnedb_source_selected_power_product_partial. ff_h_bpvnedb_source_selected_power_product_partial + S (ff_r_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\ exists ff_q_bpvnedb_source_selected_power_product_partial. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_partial * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_r_bpvnedb_source_selected_power_product))) /\ ((((exists ff_h_bpvnedb_source_selected_power_product_successor. ff_h_bpvnedb_source_selected_power_product_successor + S (ff_s_bpvnedb_source_selected_power_product) = S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\ exists ff_q_bpvnedb_source_selected_power_product_successor. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_successor * S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_s_bpvnedb_source_selected_power_product))) /\ ff_s_bpvnedb_source_selected_power_product = ff_r_bpvnedb_source_selected_power_product * ff_p_bpvnedb_source_selected_power_product)))))))) /\ (exists bpv_factor_bpvnedb_source_selected_divides. c = bpv_result_bpvnedb_source_selected * bpv_factor_bpvnedb_source_selected_divides)))) /\ forall bpv_candidate_bpvnedb_source. (exists bpv_gap_bpvnedb_source_candidate_bound. bpv_gap_bpvnedb_source_candidate_bound + bpv_candidate_bpvnedb_source = c) -> (exists bpv_result_bpvnedb_source_candidate. ((exists ff_b_bpvnedb_source_candidate_power ff_c_bpvnedb_source_candidate_power. ((forall ff_i_bpvnedb_source_candidate_power_repeat. (exists ff_lt_bpvnedb_source_candidate_power_repeat_bound. ff_lt_bpvnedb_source_candidate_power_repeat_bound + S ff_i_bpvnedb_source_candidate_power_repeat = bpv_candidate_bpvnedb_source) -> (((exists ff_h_bpvnedb_source_candidate_power_repeat_decoded. ff_h_bpvnedb_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power)) /\ exists ff_q_bpvnedb_source_candidate_power_repeat_decoded. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power) + (p)))) /\ (exists ff_u_bpvnedb_source_candidate_power_product ff_v_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_start. ff_h_bpvnedb_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product)) /\ exists ff_q_bpvnedb_source_candidate_power_product_start. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product) + (1))) /\ ((((exists ff_h_bpvnedb_source_candidate_power_product_terminal. ff_h_bpvnedb_source_candidate_power_product_terminal + S (bpv_result_bpvnedb_source_candidate) = S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product)) /\ exists ff_q_bpvnedb_source_candidate_power_product_terminal. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_terminal * S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product) + (bpv_result_bpvnedb_source_candidate))) /\ forall ff_i_bpvnedb_source_candidate_power_product. (exists ff_lt_bpvnedb_source_candidate_power_product_bound. ff_lt_bpvnedb_source_candidate_power_product_bound + S ff_i_bpvnedb_source_candidate_power_product = bpv_candidate_bpvnedb_source) -> exists ff_p_bpvnedb_source_candidate_power_product ff_r_bpvnedb_source_candidate_power_product ff_s_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_factor. ff_h_bpvnedb_source_candidate_power_product_factor + S (ff_p_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power)) /\ exists ff_q_bpvnedb_source_candidate_power_product_factor. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_product_factor * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power) + (ff_p_bpvnedb_source_candidate_power_product))) /\ ((((exists ff_h_bpvnedb_source_candidate_power_product_partial. ff_h_bpvnedb_source_candidate_power_product_partial + S (ff_r_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\ exists ff_q_bpvnedb_source_candidate_power_product_partial. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_partial * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_r_bpvnedb_source_candidate_power_product))) /\ ((((exists ff_h_bpvnedb_source_candidate_power_product_successor. ff_h_bpvnedb_source_candidate_power_product_successor + S (ff_s_bpvnedb_source_candidate_power_product) = S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\ exists ff_q_bpvnedb_source_candidate_power_product_successor. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_successor * S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_s_bpvnedb_source_candidate_power_product))) /\ ff_s_bpvnedb_source_candidate_power_product = ff_r_bpvnedb_source_candidate_power_product * ff_p_bpvnedb_source_candidate_power_product)))))))) /\ (exists bpv_factor_bpvnedb_source_candidate_divides. c = bpv_result_bpvnedb_source_candidate * bpv_factor_bpvnedb_source_candidate_divides))) -> (exists bpv_gap_bpvnedb_source_maximal. bpv_gap_bpvnedb_source_maximal + bpv_candidate_bpvnedb_source = e)) -> ~(e = 0) -> (exists bpr_quotient_bpvnedb_result. c = (p) * bpr_quotient_bpvnedb_result)

Structural proof guide

A nonzero valuation exponent exposes the base as a divisor.

Direct prerequisites: one_le_of_ne_zero, power_valuation_power_divides, power_divides_exponent_antitone, pow_one. The authored body proceeds by case analysis (2), intermediate claims (4), equality transport (1).

Proof neighborhood

Direct dependencies

BT0010 one_le_of_ne_zero BT00Q9 power_valuation_power_divides BT00QK power_divides_exponent_antitone BT0094 pow_one

Direct dependents

BT00YJ no_bertrand_central_nonzero_valuation_live_ranges

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 c
  3. 0003intro e
  4. 0004intro hvaluation
  5. 0005intro hexponent
  6. 0006have hone : exists bpr_le_gap_bpvnedb_one_bound. bpr_le_gap_bpvnedb_one_bound + (1) = (e)
  7. 0007specialize one_le_of_ne_zero e
  8. 0008apply one_le_of_ne_zero
  9. 0009exact hexponent
  10. 0010have hselected : exists bpvi_result_bpvnedb_selected. ((exists bpvi_b_bpvnedb_selected_power bpvi_c_bpvnedb_selected_power. ((forall bpvi_i_bpvnedb_selected_power. (exists bpvi_repeat_gap_bpvnedb_selected_power. bpvi_repeat_gap_bpvnedb_selected_power + S bpvi_i_bpvnedb_selected_power = e) -> (((exists bpvi_h_bpvnedb_selected_power_repeat. bpvi_h_bpvnedb_selected_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_repeat. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_repeat * S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (p)))) /\ (exists bpvi_u_bpvnedb_selected_power bpvi_v_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_start. bpvi_h_bpvnedb_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_start. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_start * S ((S (0)) * bpvi_v_bpvnedb_selected_power) + (1))) /\ ((((exists bpvi_h_bpvnedb_selected_power_terminal. bpvi_h_bpvnedb_selected_power_terminal + S (bpvi_result_bpvnedb_selected) = S ((S (e)) * bpvi_v_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_terminal. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_terminal * S ((S (e)) * bpvi_v_bpvnedb_selected_power) + (bpvi_result_bpvnedb_selected))) /\ forall bpvi_j_bpvnedb_selected_power. (exists bpvi_product_gap_bpvnedb_selected_power. bpvi_product_gap_bpvnedb_selected_power + S bpvi_j_bpvnedb_selected_power = e) -> exists bpvi_factor_bpvnedb_selected_power bpvi_partial_bpvnedb_selected_power bpvi_successor_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_factor. bpvi_h_bpvnedb_selected_power_factor + S (bpvi_factor_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_factor. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_factor * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (bpvi_factor_bpvnedb_selected_power))) /\ ((((exists bpvi_h_bpvnedb_selected_power_partial. bpvi_h_bpvnedb_selected_power_partial + S (bpvi_partial_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_partial. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_partial * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_partial_bpvnedb_selected_power))) /\ ((((exists bpvi_h_bpvnedb_selected_power_successor. bpvi_h_bpvnedb_selected_power_successor + S (bpvi_successor_bpvnedb_selected_power) = S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\ exists bpvi_q_bpvnedb_selected_power_successor. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_successor * S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_successor_bpvnedb_selected_power))) /\ bpvi_successor_bpvnedb_selected_power = bpvi_partial_bpvnedb_selected_power * bpvi_factor_bpvnedb_selected_power)))))))) /\ exists bpvi_divisor_factor_bpvnedb_selected. c = bpvi_result_bpvnedb_selected * bpvi_divisor_factor_bpvnedb_selected)
  11. 0011specialize power_valuation_power_divides p
  12. 0012specialize power_valuation_power_divides c
  13. 0013specialize power_valuation_power_divides e
  14. 0014apply power_valuation_power_divides
  15. 0015exact hvaluation
  16. 0016have hunit : exists bpvi_result_bpvnedb_unit. ((exists bpvi_b_bpvnedb_unit_power bpvi_c_bpvnedb_unit_power. ((forall bpvi_i_bpvnedb_unit_power. (exists bpvi_repeat_gap_bpvnedb_unit_power. bpvi_repeat_gap_bpvnedb_unit_power + S bpvi_i_bpvnedb_unit_power = 1) -> (((exists bpvi_h_bpvnedb_unit_power_repeat. bpvi_h_bpvnedb_unit_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_repeat. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_repeat * S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (p)))) /\ (exists bpvi_u_bpvnedb_unit_power bpvi_v_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_start. bpvi_h_bpvnedb_unit_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_start. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_start * S ((S (0)) * bpvi_v_bpvnedb_unit_power) + (1))) /\ ((((exists bpvi_h_bpvnedb_unit_power_terminal. bpvi_h_bpvnedb_unit_power_terminal + S (bpvi_result_bpvnedb_unit) = S ((S (1)) * bpvi_v_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_terminal. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_terminal * S ((S (1)) * bpvi_v_bpvnedb_unit_power) + (bpvi_result_bpvnedb_unit))) /\ forall bpvi_j_bpvnedb_unit_power. (exists bpvi_product_gap_bpvnedb_unit_power. bpvi_product_gap_bpvnedb_unit_power + S bpvi_j_bpvnedb_unit_power = 1) -> exists bpvi_factor_bpvnedb_unit_power bpvi_partial_bpvnedb_unit_power bpvi_successor_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_factor. bpvi_h_bpvnedb_unit_power_factor + S (bpvi_factor_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_factor. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_factor * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (bpvi_factor_bpvnedb_unit_power))) /\ ((((exists bpvi_h_bpvnedb_unit_power_partial. bpvi_h_bpvnedb_unit_power_partial + S (bpvi_partial_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_partial. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_partial * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_partial_bpvnedb_unit_power))) /\ ((((exists bpvi_h_bpvnedb_unit_power_successor. bpvi_h_bpvnedb_unit_power_successor + S (bpvi_successor_bpvnedb_unit_power) = S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\ exists bpvi_q_bpvnedb_unit_power_successor. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_successor * S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_successor_bpvnedb_unit_power))) /\ bpvi_successor_bpvnedb_unit_power = bpvi_partial_bpvnedb_unit_power * bpvi_factor_bpvnedb_unit_power)))))))) /\ exists bpvi_divisor_factor_bpvnedb_unit. c = bpvi_result_bpvnedb_unit * bpvi_divisor_factor_bpvnedb_unit)
  17. 0017specialize power_divides_exponent_antitone p
  18. 0018specialize power_divides_exponent_antitone 1
  19. 0019specialize power_divides_exponent_antitone e
  20. 0020specialize power_divides_exponent_antitone c
  21. 0021apply power_divides_exponent_antitone
  22. 0022exact hone
  23. 0023exact hselected
  24. 0024cases hunit
  25. 0025cases hunit_witness
  26. 0026have hvalue : x = p
  27. 0027specialize pow_one p
  28. 0028specialize pow_one 1
  29. 0029specialize pow_one x
  30. 0030apply pow_one
  31. 0031refl
  32. 0032exact hunit_witness_left
  33. 0033rewrite hvalue at hunit_witness_right
  34. 0034exact hunit_witness_right