PA00DG

distinct_odd_prime_half_row_count_choice

Alpha v16 checked-use theorem · independently closed; not Stable

Each bounded row has one semantic row-count witness.

Exact expanded PA statement

forall p q h k i. p = 2 * h + 1 -> q = 2 * k + 1 -> ((~(p = 1) /\ forall frp_prime_left_rectangle_count_prime_p frp_prime_right_rectangle_count_prime_p. p = frp_prime_left_rectangle_count_prime_p * frp_prime_right_rectangle_count_prime_p -> frp_prime_left_rectangle_count_prime_p = 1 \/ frp_prime_right_rectangle_count_prime_p = 1)) -> ((~(q = 1) /\ forall frp_prime_left_rectangle_count_prime_q frp_prime_right_rectangle_count_prime_q. q = frp_prime_left_rectangle_count_prime_q * frp_prime_right_rectangle_count_prime_q -> frp_prime_left_rectangle_count_prime_q = 1 \/ frp_prime_right_rectangle_count_prime_q = 1)) -> ~(p = q) -> (exists erc_lt_gap_rectangle_count_row_bound. erc_lt_gap_rectangle_count_row_bound + S (i) = h) -> exists n. (exists erc_row_code_rectangle_count_point_witness erc_row_scale_rectangle_count_point_witness. ((forall eri_column_erc_rectangle_count_point_witness_row. (exists eri_gap_erc_rectangle_count_point_witness_row_bound. eri_gap_erc_rectangle_count_point_witness_row_bound + S (eri_column_erc_rectangle_count_point_witness_row) = k) -> exists eri_bit_erc_rectangle_count_point_witness_row. ((((exists ff_h_eri_erc_rectangle_count_point_witness_row_decoded. ff_h_eri_erc_rectangle_count_point_witness_row_decoded + S (eri_bit_erc_rectangle_count_point_witness_row) = S ((S (eri_column_erc_rectangle_count_point_witness_row)) * erc_row_scale_rectangle_count_point_witness)) /\ exists ff_q_eri_erc_rectangle_count_point_witness_row_decoded. erc_row_code_rectangle_count_point_witness = ff_q_eri_erc_rectangle_count_point_witness_row_decoded * S ((S (eri_column_erc_rectangle_count_point_witness_row)) * erc_row_scale_rectangle_count_point_witness) + (eri_bit_erc_rectangle_count_point_witness_row))) /\ (((eri_bit_erc_rectangle_count_point_witness_row = 0 /\ ((exists eri_gap_erc_rectangle_count_point_witness_row_choice_left. eri_gap_erc_rectangle_count_point_witness_row_choice_left + S (q * S i) = p * S eri_column_erc_rectangle_count_point_witness_row) /\ ~(exists eri_gap_erc_rectangle_count_point_witness_row_choice_right. eri_gap_erc_rectangle_count_point_witness_row_choice_right + S (p * S eri_column_erc_rectangle_count_point_witness_row) = q * S i))) \/ (eri_bit_erc_rectangle_count_point_witness_row = 1 /\ ((exists eri_gap_erc_rectangle_count_point_witness_row_choice_right. eri_gap_erc_rectangle_count_point_witness_row_choice_right + S (p * S eri_column_erc_rectangle_count_point_witness_row) = q * S i) /\ ~(exists eri_gap_erc_rectangle_count_point_witness_row_choice_left. eri_gap_erc_rectangle_count_point_witness_row_choice_left + S (q * S i) = p * S eri_column_erc_rectangle_count_point_witness_row))))))) /\ (((exists ff_u_erc_rectangle_count_point_witness_count_sum ff_v_erc_rectangle_count_point_witness_count_sum. ((((exists ff_h_erc_rectangle_count_point_witness_count_sum_start. ff_h_erc_rectangle_count_point_witness_count_sum_start + S (0) = S ((S (0)) * ff_v_erc_rectangle_count_point_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_point_witness_count_sum_start. ff_u_erc_rectangle_count_point_witness_count_sum = ff_q_erc_rectangle_count_point_witness_count_sum_start * S ((S (0)) * ff_v_erc_rectangle_count_point_witness_count_sum) + (0))) /\ ((((exists ff_h_erc_rectangle_count_point_witness_count_sum_terminal. ff_h_erc_rectangle_count_point_witness_count_sum_terminal + S (n) = S ((S (k)) * ff_v_erc_rectangle_count_point_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_point_witness_count_sum_terminal. ff_u_erc_rectangle_count_point_witness_count_sum = ff_q_erc_rectangle_count_point_witness_count_sum_terminal * S ((S (k)) * ff_v_erc_rectangle_count_point_witness_count_sum) + (n))) /\ forall ff_i_erc_rectangle_count_point_witness_count_sum. (exists ff_lt_erc_rectangle_count_point_witness_count_sum_bound. ff_lt_erc_rectangle_count_point_witness_count_sum_bound + S ff_i_erc_rectangle_count_point_witness_count_sum = k) -> exists ff_a_erc_rectangle_count_point_witness_count_sum ff_r_erc_rectangle_count_point_witness_count_sum ff_s_erc_rectangle_count_point_witness_count_sum. ((((exists ff_h_erc_rectangle_count_point_witness_count_sum_summand. ff_h_erc_rectangle_count_point_witness_count_sum_summand + S (ff_a_erc_rectangle_count_point_witness_count_sum) = S ((S (ff_i_erc_rectangle_count_point_witness_count_sum)) * erc_row_scale_rectangle_count_point_witness)) /\ exists ff_q_erc_rectangle_count_point_witness_count_sum_summand. erc_row_code_rectangle_count_point_witness = ff_q_erc_rectangle_count_point_witness_count_sum_summand * S ((S (ff_i_erc_rectangle_count_point_witness_count_sum)) * erc_row_scale_rectangle_count_point_witness) + (ff_a_erc_rectangle_count_point_witness_count_sum))) /\ ((((exists ff_h_erc_rectangle_count_point_witness_count_sum_partial. ff_h_erc_rectangle_count_point_witness_count_sum_partial + S (ff_r_erc_rectangle_count_point_witness_count_sum) = S ((S (ff_i_erc_rectangle_count_point_witness_count_sum)) * ff_v_erc_rectangle_count_point_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_point_witness_count_sum_partial. ff_u_erc_rectangle_count_point_witness_count_sum = ff_q_erc_rectangle_count_point_witness_count_sum_partial * S ((S (ff_i_erc_rectangle_count_point_witness_count_sum)) * ff_v_erc_rectangle_count_point_witness_count_sum) + (ff_r_erc_rectangle_count_point_witness_count_sum))) /\ ((((exists ff_h_erc_rectangle_count_point_witness_count_sum_successor. ff_h_erc_rectangle_count_point_witness_count_sum_successor + S (ff_s_erc_rectangle_count_point_witness_count_sum) = S ((S (S ff_i_erc_rectangle_count_point_witness_count_sum)) * ff_v_erc_rectangle_count_point_witness_count_sum)) /\ exists ff_q_erc_rectangle_count_point_witness_count_sum_successor. ff_u_erc_rectangle_count_point_witness_count_sum = ff_q_erc_rectangle_count_point_witness_count_sum_successor * S ((S (S ff_i_erc_rectangle_count_point_witness_count_sum)) * ff_v_erc_rectangle_count_point_witness_count_sum) + (ff_s_erc_rectangle_count_point_witness_count_sum))) /\ ff_s_erc_rectangle_count_point_witness_count_sum = ff_r_erc_rectangle_count_point_witness_count_sum + ff_a_erc_rectangle_count_point_witness_count_sum)))))) /\ (forall ff_i_erc_rectangle_count_point_witness_count_bits. (exists ff_lt_erc_rectangle_count_point_witness_count_bits_bound. ff_lt_erc_rectangle_count_point_witness_count_bits_bound + S ff_i_erc_rectangle_count_point_witness_count_bits = k) -> exists ff_bit_erc_rectangle_count_point_witness_count_bits. ((((exists ff_h_erc_rectangle_count_point_witness_count_bits_decoded. ff_h_erc_rectangle_count_point_witness_count_bits_decoded + S (ff_bit_erc_rectangle_count_point_witness_count_bits) = S ((S (ff_i_erc_rectangle_count_point_witness_count_bits)) * erc_row_scale_rectangle_count_point_witness)) /\ exists ff_q_erc_rectangle_count_point_witness_count_bits_decoded. erc_row_code_rectangle_count_point_witness = ff_q_erc_rectangle_count_point_witness_count_bits_decoded * S ((S (ff_i_erc_rectangle_count_point_witness_count_bits)) * erc_row_scale_rectangle_count_point_witness) + (ff_bit_erc_rectangle_count_point_witness_count_bits))) /\ (ff_bit_erc_rectangle_count_point_witness_count_bits = 0 \/ ff_bit_erc_rectangle_count_point_witness_count_bits = 1)))))))

Structural proof guide

Generated structural guide

Each bounded row has one semantic row-count witness.

Use the direct prerequisites distinct_odd_prime_half_row_count_exists as previously established PA formulas.

The proof proceeds by case analysis (3), intermediate claims (1).

Referenced ingredients

PA00DF distinct_odd_prime_half_row_count_exists

Proof neighborhood

Direct dependencies

PA00DF distinct_odd_prime_half_row_count_exists

Direct dependents

PA00DH distinct_odd_prime_half_row_count_choices_bounded

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v16 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

  1. 0001intro p
  2. 0002intro q
  3. 0003intro h
  4. 0004intro k
  5. 0005intro i
  6. 0006intro hpodd
  7. 0007intro hqodd
  8. 0008intro hp
  9. 0009intro hq
  10. 0010intro hpq
  11. 0011intro hi
  12. 0012have hpackage : exists rb rc n. ((forall eri_column_rectangle_count_point_row. (exists eri_gap_rectangle_count_point_row_bound. eri_gap_rectangle_count_point_row_bound + S (eri_column_rectangle_count_point_row) = k) -> exists eri_bit_rectangle_count_point_row. ((((exists ff_h_eri_rectangle_count_point_row_decoded. ff_h_eri_rectangle_count_point_row_decoded + S (eri_bit_rectangle_count_point_row) = S ((S (eri_column_rectangle_count_point_row)) * rc)) /\ exists ff_q_eri_rectangle_count_point_row_decoded. rb = ff_q_eri_rectangle_count_point_row_decoded * S ((S (eri_column_rectangle_count_point_row)) * rc) + (eri_bit_rectangle_count_point_row))) /\ (((eri_bit_rectangle_count_point_row = 0 /\ ((exists eri_gap_rectangle_count_point_row_choice_left. eri_gap_rectangle_count_point_row_choice_left + S (q * S i) = p * S eri_column_rectangle_count_point_row) /\ ~(exists eri_gap_rectangle_count_point_row_choice_right. eri_gap_rectangle_count_point_row_choice_right + S (p * S eri_column_rectangle_count_point_row) = q * S i))) \/ (eri_bit_rectangle_count_point_row = 1 /\ ((exists eri_gap_rectangle_count_point_row_choice_right. eri_gap_rectangle_count_point_row_choice_right + S (p * S eri_column_rectangle_count_point_row) = q * S i) /\ ~(exists eri_gap_rectangle_count_point_row_choice_left. eri_gap_rectangle_count_point_row_choice_left + S (q * S i) = p * S eri_column_rectangle_count_point_row))))))) /\ (((exists ff_u_rectangle_count_point_count_sum ff_v_rectangle_count_point_count_sum. ((((exists ff_h_rectangle_count_point_count_sum_start. ff_h_rectangle_count_point_count_sum_start + S (0) = S ((S (0)) * ff_v_rectangle_count_point_count_sum)) /\ exists ff_q_rectangle_count_point_count_sum_start. ff_u_rectangle_count_point_count_sum = ff_q_rectangle_count_point_count_sum_start * S ((S (0)) * ff_v_rectangle_count_point_count_sum) + (0))) /\ ((((exists ff_h_rectangle_count_point_count_sum_terminal. ff_h_rectangle_count_point_count_sum_terminal + S (n) = S ((S (k)) * ff_v_rectangle_count_point_count_sum)) /\ exists ff_q_rectangle_count_point_count_sum_terminal. ff_u_rectangle_count_point_count_sum = ff_q_rectangle_count_point_count_sum_terminal * S ((S (k)) * ff_v_rectangle_count_point_count_sum) + (n))) /\ forall ff_i_rectangle_count_point_count_sum. (exists ff_lt_rectangle_count_point_count_sum_bound. ff_lt_rectangle_count_point_count_sum_bound + S ff_i_rectangle_count_point_count_sum = k) -> exists ff_a_rectangle_count_point_count_sum ff_r_rectangle_count_point_count_sum ff_s_rectangle_count_point_count_sum. ((((exists ff_h_rectangle_count_point_count_sum_summand. ff_h_rectangle_count_point_count_sum_summand + S (ff_a_rectangle_count_point_count_sum) = S ((S (ff_i_rectangle_count_point_count_sum)) * rc)) /\ exists ff_q_rectangle_count_point_count_sum_summand. rb = ff_q_rectangle_count_point_count_sum_summand * S ((S (ff_i_rectangle_count_point_count_sum)) * rc) + (ff_a_rectangle_count_point_count_sum))) /\ ((((exists ff_h_rectangle_count_point_count_sum_partial. ff_h_rectangle_count_point_count_sum_partial + S (ff_r_rectangle_count_point_count_sum) = S ((S (ff_i_rectangle_count_point_count_sum)) * ff_v_rectangle_count_point_count_sum)) /\ exists ff_q_rectangle_count_point_count_sum_partial. ff_u_rectangle_count_point_count_sum = ff_q_rectangle_count_point_count_sum_partial * S ((S (ff_i_rectangle_count_point_count_sum)) * ff_v_rectangle_count_point_count_sum) + (ff_r_rectangle_count_point_count_sum))) /\ ((((exists ff_h_rectangle_count_point_count_sum_successor. ff_h_rectangle_count_point_count_sum_successor + S (ff_s_rectangle_count_point_count_sum) = S ((S (S ff_i_rectangle_count_point_count_sum)) * ff_v_rectangle_count_point_count_sum)) /\ exists ff_q_rectangle_count_point_count_sum_successor. ff_u_rectangle_count_point_count_sum = ff_q_rectangle_count_point_count_sum_successor * S ((S (S ff_i_rectangle_count_point_count_sum)) * ff_v_rectangle_count_point_count_sum) + (ff_s_rectangle_count_point_count_sum))) /\ ff_s_rectangle_count_point_count_sum = ff_r_rectangle_count_point_count_sum + ff_a_rectangle_count_point_count_sum)))))) /\ (forall ff_i_rectangle_count_point_count_bits. (exists ff_lt_rectangle_count_point_count_bits_bound. ff_lt_rectangle_count_point_count_bits_bound + S ff_i_rectangle_count_point_count_bits = k) -> exists ff_bit_rectangle_count_point_count_bits. ((((exists ff_h_rectangle_count_point_count_bits_decoded. ff_h_rectangle_count_point_count_bits_decoded + S (ff_bit_rectangle_count_point_count_bits) = S ((S (ff_i_rectangle_count_point_count_bits)) * rc)) /\ exists ff_q_rectangle_count_point_count_bits_decoded. rb = ff_q_rectangle_count_point_count_bits_decoded * S ((S (ff_i_rectangle_count_point_count_bits)) * rc) + (ff_bit_rectangle_count_point_count_bits))) /\ (ff_bit_rectangle_count_point_count_bits = 0 \/ ff_bit_rectangle_count_point_count_bits = 1))))))
  13. 0013specialize distinct_odd_prime_half_row_count_exists p
  14. 0014specialize distinct_odd_prime_half_row_count_exists q
  15. 0015specialize distinct_odd_prime_half_row_count_exists h
  16. 0016specialize distinct_odd_prime_half_row_count_exists k
  17. 0017specialize distinct_odd_prime_half_row_count_exists i
  18. 0018apply distinct_odd_prime_half_row_count_exists
  19. 0019exact hpodd
  20. 0020exact hqodd
  21. 0021exact hp
  22. 0022exact hq
  23. 0023exact hpq
  24. 0024exact hi
  25. 0025cases hpackage
  26. 0026cases hpackage_witness
  27. 0027cases hpackage_witness_witness
  28. 0028exists x2
  29. 0029exists x
  30. 0030exists x1
  31. 0031exact hpackage_witness_witness_witness