PA006P

beta_sum_functional

Stable checked-use theorem · independently closed

The relational finite sum has a unique natural value.

Exact expanded PA statement

forall b c l n m. (exists ff_u_l ff_v_l. ((((exists ff_h_l_start. ff_h_l_start + S (0) = S ((S (0)) * ff_v_l)) /\ exists ff_q_l_start. ff_u_l = ff_q_l_start * S ((S (0)) * ff_v_l) + (0))) /\ ((((exists ff_h_l_terminal. ff_h_l_terminal + S (n) = S ((S (l)) * ff_v_l)) /\ exists ff_q_l_terminal. ff_u_l = ff_q_l_terminal * S ((S (l)) * ff_v_l) + (n))) /\ forall ff_i_l. (exists ff_lt_l_bound. ff_lt_l_bound + S ff_i_l = l) -> exists ff_a_l ff_r_l ff_s_l. ((((exists ff_h_l_summand. ff_h_l_summand + S (ff_a_l) = S ((S (ff_i_l)) * c)) /\ exists ff_q_l_summand. b = ff_q_l_summand * S ((S (ff_i_l)) * c) + (ff_a_l))) /\ ((((exists ff_h_l_partial. ff_h_l_partial + S (ff_r_l) = S ((S (ff_i_l)) * ff_v_l)) /\ exists ff_q_l_partial. ff_u_l = ff_q_l_partial * S ((S (ff_i_l)) * ff_v_l) + (ff_r_l))) /\ ((((exists ff_h_l_successor. ff_h_l_successor + S (ff_s_l) = S ((S (S ff_i_l)) * ff_v_l)) /\ exists ff_q_l_successor. ff_u_l = ff_q_l_successor * S ((S (S ff_i_l)) * ff_v_l) + (ff_s_l))) /\ ff_s_l = ff_r_l + ff_a_l)))))) -> (exists ff_u_r ff_v_r. ((((exists ff_h_r_start. ff_h_r_start + S (0) = S ((S (0)) * ff_v_r)) /\ exists ff_q_r_start. ff_u_r = ff_q_r_start * S ((S (0)) * ff_v_r) + (0))) /\ ((((exists ff_h_r_terminal. ff_h_r_terminal + S (m) = S ((S (l)) * ff_v_r)) /\ exists ff_q_r_terminal. ff_u_r = ff_q_r_terminal * S ((S (l)) * ff_v_r) + (m))) /\ forall ff_i_r. (exists ff_lt_r_bound. ff_lt_r_bound + S ff_i_r = l) -> exists ff_a_r ff_r_r ff_s_r. ((((exists ff_h_r_summand. ff_h_r_summand + S (ff_a_r) = S ((S (ff_i_r)) * c)) /\ exists ff_q_r_summand. b = ff_q_r_summand * S ((S (ff_i_r)) * c) + (ff_a_r))) /\ ((((exists ff_h_r_partial. ff_h_r_partial + S (ff_r_r) = S ((S (ff_i_r)) * ff_v_r)) /\ exists ff_q_r_partial. ff_u_r = ff_q_r_partial * S ((S (ff_i_r)) * ff_v_r) + (ff_r_r))) /\ ((((exists ff_h_r_successor. ff_h_r_successor + S (ff_s_r) = S ((S (S ff_i_r)) * ff_v_r)) /\ exists ff_q_r_successor. ff_u_r = ff_q_r_successor * S ((S (S ff_i_r)) * ff_v_r) + (ff_s_r))) /\ ff_s_r = ff_r_r + ff_a_r)))))) -> n = m

Structural proof guide

Generated structural guide

The relational finite sum has a unique natural value.

Use the direct prerequisites beta_sum_trace_functional as previously established PA formulas.

The proof proceeds by case analysis (4).

Referenced ingredients

PA006K beta_sum_trace_functional

Proof neighborhood

Direct dependencies

PA006K beta_sum_trace_functional

Direct dependents

PA006Q bit_count_functional PA00E5 distinct_odd_prime_quotient_sum_equals_rectangle_total PA00F6 eisenstein_transposed_column_counts_extensional PA00FB eisenstein_successor_terminal_sum_matches_last_column

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro n
  5. 0005intro m
  6. 0006intro hn
  7. 0007intro hm
  8. 0008cases hn
  9. 0009cases hn_witness
  10. 0010cases hm
  11. 0011cases hm_witness
  12. 0012specialize beta_sum_trace_functional b
  13. 0013specialize beta_sum_trace_functional c
  14. 0014specialize beta_sum_trace_functional l
  15. 0015specialize beta_sum_trace_functional n
  16. 0016specialize beta_sum_trace_functional x
  17. 0017specialize beta_sum_trace_functional x1
  18. 0018specialize beta_sum_trace_functional m
  19. 0019specialize beta_sum_trace_functional x2
  20. 0020specialize beta_sum_trace_functional x3
  21. 0021apply beta_sum_trace_functional
  22. 0022exact hn_witness_witness
  23. 0023exact hm_witness_witness