PA003H

beta_sum_exists

Stable checked-use theorem · independently closed

Every decoded beta prefix has a relational finite sum.

Exact expanded PA statement

forall b c l. exists n. (exists ff_u_x ff_v_x. ((((exists ff_h_x_start. ff_h_x_start + S (0) = S ((S (0)) * ff_v_x)) /\ exists ff_q_x_start. ff_u_x = ff_q_x_start * S ((S (0)) * ff_v_x) + (0))) /\ ((((exists ff_h_x_terminal. ff_h_x_terminal + S (n) = S ((S (l)) * ff_v_x)) /\ exists ff_q_x_terminal. ff_u_x = ff_q_x_terminal * S ((S (l)) * ff_v_x) + (n))) /\ forall ff_i_x. (exists ff_lt_x_bound. ff_lt_x_bound + S ff_i_x = l) -> exists ff_a_x ff_r_x ff_s_x. ((((exists ff_h_x_summand. ff_h_x_summand + S (ff_a_x) = S ((S (ff_i_x)) * c)) /\ exists ff_q_x_summand. b = ff_q_x_summand * S ((S (ff_i_x)) * c) + (ff_a_x))) /\ ((((exists ff_h_x_partial. ff_h_x_partial + S (ff_r_x) = S ((S (ff_i_x)) * ff_v_x)) /\ exists ff_q_x_partial. ff_u_x = ff_q_x_partial * S ((S (ff_i_x)) * ff_v_x) + (ff_r_x))) /\ ((((exists ff_h_x_successor. ff_h_x_successor + S (ff_s_x) = S ((S (S ff_i_x)) * ff_v_x)) /\ exists ff_q_x_successor. ff_u_x = ff_q_x_successor * S ((S (S ff_i_x)) * ff_v_x) + (ff_s_x))) /\ ff_s_x = ff_r_x + ff_a_x))))))

Structural proof guide

Generated structural guide

Every decoded beta prefix has a relational finite sum.

Use the direct prerequisites beta_prefix_sum_trace_exists, beta_at_exists as previously established PA formulas.

The proof proceeds by case analysis (4), intermediate claims (2).

Referenced ingredients

PA003G beta_prefix_sum_trace_exists PA0029 beta_at_exists

Proof neighborhood

Direct dependencies

PA003G beta_prefix_sum_trace_exists PA0029 beta_at_exists

Direct dependents

PA003I bit_count_exists PA00C1 prime_scaled_half_quotient_sum_exists PA00CX beta_sum_permutation_invariant PA00D6 gauss_eisenstein_sign_count_mod_quotient_sum PA00DM distinct_odd_prime_half_rectangle_total_exists PA00EJ eisenstein_transposed_column_count_total_exists PA00EL beta_repeat_sum_exists_exact PA00FC eisenstein_fubini_universal PA00FF distinct_odd_prime_eisenstein_quotient_sum_identity

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. 0004have htrace : exists fs_u_exists_trace fs_v_exists_trace. ((((exists fs_h_exists_trace_start. fs_h_exists_trace_start + S (0) = S ((S (0)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_start. fs_u_exists_trace = fs_q_exists_trace_start * S ((S (0)) * fs_v_exists_trace) + (0))) /\ forall fs_i_exists_trace_steps. (exists fs_lt_exists_trace_steps_bound. fs_lt_exists_trace_steps_bound + S fs_i_exists_trace_steps = l) -> exists fs_a_exists_trace_steps fs_r_exists_trace_steps fs_s_exists_trace_steps. ((((exists fs_h_exists_trace_steps_summand. fs_h_exists_trace_steps_summand + S (fs_a_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * c)) /\ exists fs_q_exists_trace_steps_summand. b = fs_q_exists_trace_steps_summand * S ((S (fs_i_exists_trace_steps)) * c) + (fs_a_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_partial. fs_h_exists_trace_steps_partial + S (fs_r_exists_trace_steps) = S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_partial. fs_u_exists_trace = fs_q_exists_trace_steps_partial * S ((S (fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_r_exists_trace_steps))) /\ ((((exists fs_h_exists_trace_steps_successor. fs_h_exists_trace_steps_successor + S (fs_s_exists_trace_steps) = S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace)) /\ exists fs_q_exists_trace_steps_successor. fs_u_exists_trace = fs_q_exists_trace_steps_successor * S ((S (S fs_i_exists_trace_steps)) * fs_v_exists_trace) + (fs_s_exists_trace_steps))) /\ fs_s_exists_trace_steps = fs_r_exists_trace_steps + fs_a_exists_trace_steps))))
  5. 0005specialize beta_prefix_sum_trace_exists b
  6. 0006specialize beta_prefix_sum_trace_exists c
  7. 0007specialize beta_prefix_sum_trace_exists l
  8. 0008exact beta_prefix_sum_trace_exists
  9. 0009cases htrace
  10. 0010cases htrace_witness
  11. 0011cases htrace_witness_witness
  12. 0012have hterminal : exists n. ((exists fs_h_sum_terminal. fs_h_sum_terminal + S (n) = S ((S (l)) * x1)) /\ exists fs_q_sum_terminal. x = fs_q_sum_terminal * S ((S (l)) * x1) + (n))
  13. 0013specialize beta_at_exists x
  14. 0014specialize beta_at_exists x1
  15. 0015specialize beta_at_exists l
  16. 0016exact beta_at_exists
  17. 0017cases hterminal
  18. 0018exists x2
  19. 0019exists x
  20. 0020exists x1
  21. 0021split
  22. 0022exact htrace_witness_witness_left
  23. 0023split
  24. 0024exact hterminal_witness
  25. 0025exact htrace_witness_witness_right