BT00VQ

primorial_one

Alpha body-checked · checked-use disabled

The inclusive Primorial at one is exactly one.

Exact expanded PA statement

forall z. (exists bpr_code_bpo_source bpr_scale_bpo_source. ((forall bpr_index_bpo_source_mask. (exists bpr_gap_bpo_source_mask_bound. bpr_gap_bpo_source_mask_bound + S (bpr_index_bpo_source_mask) = 1) -> exists bpr_value_bpo_source_mask. ((((exists bpr_height_bpo_source_mask_decoded. bpr_height_bpo_source_mask_decoded + S (bpr_value_bpo_source_mask) = S ((S (bpr_index_bpo_source_mask)) * bpr_scale_bpo_source)) /\ exists bpr_quotient_bpo_source_mask_decoded. bpr_code_bpo_source = bpr_quotient_bpo_source_mask_decoded * S ((S (bpr_index_bpo_source_mask)) * bpr_scale_bpo_source) + (bpr_value_bpo_source_mask))) /\ (((((~(S (bpr_index_bpo_source_mask) = 1) /\ forall bpr_left_bpo_source_mask_choice_prime bpr_right_bpo_source_mask_choice_prime. S (bpr_index_bpo_source_mask) = bpr_left_bpo_source_mask_choice_prime * bpr_right_bpo_source_mask_choice_prime -> bpr_left_bpo_source_mask_choice_prime = 1 \/ bpr_right_bpo_source_mask_choice_prime = 1)) /\ bpr_value_bpo_source_mask = S (bpr_index_bpo_source_mask)) \/ (~((~(S (bpr_index_bpo_source_mask) = 1) /\ forall bpr_left_bpo_source_mask_choice_prime bpr_right_bpo_source_mask_choice_prime. S (bpr_index_bpo_source_mask) = bpr_left_bpo_source_mask_choice_prime * bpr_right_bpo_source_mask_choice_prime -> bpr_left_bpo_source_mask_choice_prime = 1 \/ bpr_right_bpo_source_mask_choice_prime = 1)) /\ bpr_value_bpo_source_mask = 1))))) /\ (exists ff_u_bpo_source_product ff_v_bpo_source_product. ((((exists ff_h_bpo_source_product_start. ff_h_bpo_source_product_start + S (1) = S ((S (0)) * ff_v_bpo_source_product)) /\ exists ff_q_bpo_source_product_start. ff_u_bpo_source_product = ff_q_bpo_source_product_start * S ((S (0)) * ff_v_bpo_source_product) + (1))) /\ ((((exists ff_h_bpo_source_product_terminal. ff_h_bpo_source_product_terminal + S (z) = S ((S (1)) * ff_v_bpo_source_product)) /\ exists ff_q_bpo_source_product_terminal. ff_u_bpo_source_product = ff_q_bpo_source_product_terminal * S ((S (1)) * ff_v_bpo_source_product) + (z))) /\ forall ff_i_bpo_source_product. (exists ff_lt_bpo_source_product_bound. ff_lt_bpo_source_product_bound + S ff_i_bpo_source_product = 1) -> exists ff_p_bpo_source_product ff_r_bpo_source_product ff_s_bpo_source_product. ((((exists ff_h_bpo_source_product_factor. ff_h_bpo_source_product_factor + S (ff_p_bpo_source_product) = S ((S (ff_i_bpo_source_product)) * bpr_scale_bpo_source)) /\ exists ff_q_bpo_source_product_factor. bpr_code_bpo_source = ff_q_bpo_source_product_factor * S ((S (ff_i_bpo_source_product)) * bpr_scale_bpo_source) + (ff_p_bpo_source_product))) /\ ((((exists ff_h_bpo_source_product_partial. ff_h_bpo_source_product_partial + S (ff_r_bpo_source_product) = S ((S (ff_i_bpo_source_product)) * ff_v_bpo_source_product)) /\ exists ff_q_bpo_source_product_partial. ff_u_bpo_source_product = ff_q_bpo_source_product_partial * S ((S (ff_i_bpo_source_product)) * ff_v_bpo_source_product) + (ff_r_bpo_source_product))) /\ ((((exists ff_h_bpo_source_product_successor. ff_h_bpo_source_product_successor + S (ff_s_bpo_source_product) = S ((S (S ff_i_bpo_source_product)) * ff_v_bpo_source_product)) /\ exists ff_q_bpo_source_product_successor. ff_u_bpo_source_product = ff_q_bpo_source_product_successor * S ((S (S ff_i_bpo_source_product)) * ff_v_bpo_source_product) + (ff_s_bpo_source_product))) /\ ff_s_bpo_source_product = ff_r_bpo_source_product * ff_p_bpo_source_product)))))))) -> z = 1

Structural proof guide

The inclusive Primorial at one is exactly one.

Direct prerequisites: primorial_zero, primorial_succ_decompose. The authored body proceeds by case analysis (7), intermediate claims (2), equality transport (4), closed numeral normalization (2).

Proof neighborhood

Direct dependencies

BT00UC primorial_zero BT00UD primorial_succ_decompose

Direct dependents

BT00VV primorial_le_four_pow_bounded

Formal native tactic body

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

  1. 0001intro z
  2. 0002intro hprimorial
  3. 0003have hdecomposition : exists p r. (((((~(S (0) = 1) /\ forall bpr_left_bpo_factor_prime bpr_right_bpo_factor_prime. S (0) = bpr_left_bpo_factor_prime * bpr_right_bpo_factor_prime -> bpr_left_bpo_factor_prime = 1 \/ bpr_right_bpo_factor_prime = 1)) /\ p = S (0)) \/ (~((~(S (0) = 1) /\ forall bpr_left_bpo_factor_prime bpr_right_bpo_factor_prime. S (0) = bpr_left_bpo_factor_prime * bpr_right_bpo_factor_prime -> bpr_left_bpo_factor_prime = 1 \/ bpr_right_bpo_factor_prime = 1)) /\ p = 1))) /\ ((exists bpr_code_bpo_previous bpr_scale_bpo_previous. ((forall bpr_index_bpo_previous_mask. (exists bpr_gap_bpo_previous_mask_bound. bpr_gap_bpo_previous_mask_bound + S (bpr_index_bpo_previous_mask) = 0) -> exists bpr_value_bpo_previous_mask. ((((exists bpr_height_bpo_previous_mask_decoded. bpr_height_bpo_previous_mask_decoded + S (bpr_value_bpo_previous_mask) = S ((S (bpr_index_bpo_previous_mask)) * bpr_scale_bpo_previous)) /\ exists bpr_quotient_bpo_previous_mask_decoded. bpr_code_bpo_previous = bpr_quotient_bpo_previous_mask_decoded * S ((S (bpr_index_bpo_previous_mask)) * bpr_scale_bpo_previous) + (bpr_value_bpo_previous_mask))) /\ (((((~(S (bpr_index_bpo_previous_mask) = 1) /\ forall bpr_left_bpo_previous_mask_choice_prime bpr_right_bpo_previous_mask_choice_prime. S (bpr_index_bpo_previous_mask) = bpr_left_bpo_previous_mask_choice_prime * bpr_right_bpo_previous_mask_choice_prime -> bpr_left_bpo_previous_mask_choice_prime = 1 \/ bpr_right_bpo_previous_mask_choice_prime = 1)) /\ bpr_value_bpo_previous_mask = S (bpr_index_bpo_previous_mask)) \/ (~((~(S (bpr_index_bpo_previous_mask) = 1) /\ forall bpr_left_bpo_previous_mask_choice_prime bpr_right_bpo_previous_mask_choice_prime. S (bpr_index_bpo_previous_mask) = bpr_left_bpo_previous_mask_choice_prime * bpr_right_bpo_previous_mask_choice_prime -> bpr_left_bpo_previous_mask_choice_prime = 1 \/ bpr_right_bpo_previous_mask_choice_prime = 1)) /\ bpr_value_bpo_previous_mask = 1))))) /\ (exists ff_u_bpo_previous_product ff_v_bpo_previous_product. ((((exists ff_h_bpo_previous_product_start. ff_h_bpo_previous_product_start + S (1) = S ((S (0)) * ff_v_bpo_previous_product)) /\ exists ff_q_bpo_previous_product_start. ff_u_bpo_previous_product = ff_q_bpo_previous_product_start * S ((S (0)) * ff_v_bpo_previous_product) + (1))) /\ ((((exists ff_h_bpo_previous_product_terminal. ff_h_bpo_previous_product_terminal + S (r) = S ((S (0)) * ff_v_bpo_previous_product)) /\ exists ff_q_bpo_previous_product_terminal. ff_u_bpo_previous_product = ff_q_bpo_previous_product_terminal * S ((S (0)) * ff_v_bpo_previous_product) + (r))) /\ forall ff_i_bpo_previous_product. (exists ff_lt_bpo_previous_product_bound. ff_lt_bpo_previous_product_bound + S ff_i_bpo_previous_product = 0) -> exists ff_p_bpo_previous_product ff_r_bpo_previous_product ff_s_bpo_previous_product. ((((exists ff_h_bpo_previous_product_factor. ff_h_bpo_previous_product_factor + S (ff_p_bpo_previous_product) = S ((S (ff_i_bpo_previous_product)) * bpr_scale_bpo_previous)) /\ exists ff_q_bpo_previous_product_factor. bpr_code_bpo_previous = ff_q_bpo_previous_product_factor * S ((S (ff_i_bpo_previous_product)) * bpr_scale_bpo_previous) + (ff_p_bpo_previous_product))) /\ ((((exists ff_h_bpo_previous_product_partial. ff_h_bpo_previous_product_partial + S (ff_r_bpo_previous_product) = S ((S (ff_i_bpo_previous_product)) * ff_v_bpo_previous_product)) /\ exists ff_q_bpo_previous_product_partial. ff_u_bpo_previous_product = ff_q_bpo_previous_product_partial * S ((S (ff_i_bpo_previous_product)) * ff_v_bpo_previous_product) + (ff_r_bpo_previous_product))) /\ ((((exists ff_h_bpo_previous_product_successor. ff_h_bpo_previous_product_successor + S (ff_s_bpo_previous_product) = S ((S (S ff_i_bpo_previous_product)) * ff_v_bpo_previous_product)) /\ exists ff_q_bpo_previous_product_successor. ff_u_bpo_previous_product = ff_q_bpo_previous_product_successor * S ((S (S ff_i_bpo_previous_product)) * ff_v_bpo_previous_product) + (ff_s_bpo_previous_product))) /\ ff_s_bpo_previous_product = ff_r_bpo_previous_product * ff_p_bpo_previous_product)))))))) /\ z = r * p)
  4. 0004specialize primorial_succ_decompose 0
  5. 0005specialize primorial_succ_decompose z
  6. 0006apply primorial_succ_decompose
  7. 0007exact hprimorial
  8. 0008cases hdecomposition
  9. 0009cases hdecomposition_witness
  10. 0010cases hdecomposition_witness_witness
  11. 0011cases hdecomposition_witness_witness_right
  12. 0012have hr : x1 = 1
  13. 0013apply primorial_zero
  14. 0014exact hdecomposition_witness_witness_right_left
  15. 0015cases hdecomposition_witness_witness_left
  16. 0016cases hdecomposition_witness_witness_left_left
  17. 0017rewrite hr at hdecomposition_witness_witness_right_right
  18. 0018rewrite hdecomposition_witness_witness_left_left_right at hdecomposition_witness_witness_right_right
  19. 0019trans 1 * 1
  20. 0020exact hdecomposition_witness_witness_right_right
  21. 0021norm_num
  22. 0022cases hdecomposition_witness_witness_left_right
  23. 0023rewrite hr at hdecomposition_witness_witness_right_right
  24. 0024rewrite hdecomposition_witness_witness_left_right_right at hdecomposition_witness_witness_right_right
  25. 0025trans 1 * 1
  26. 0026exact hdecomposition_witness_witness_right_right
  27. 0027norm_num