PA00EL

beta_repeat_sum_exists_exact

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

Every value and length admit a constant prefix with its exact sum.

Exact expanded PA statement

forall a l. exists b c n. (forall ff_i_repeatsum_exists_repeat. (exists ff_lt_repeatsum_exists_repeat_bound. ff_lt_repeatsum_exists_repeat_bound + S ff_i_repeatsum_exists_repeat = l) -> (((exists ff_h_repeatsum_exists_repeat_decoded. ff_h_repeatsum_exists_repeat_decoded + S (a) = S ((S (ff_i_repeatsum_exists_repeat)) * c)) /\ exists ff_q_repeatsum_exists_repeat_decoded. b = ff_q_repeatsum_exists_repeat_decoded * S ((S (ff_i_repeatsum_exists_repeat)) * c) + (a)))) /\ ((exists ff_u_repeatsum_exists_sum ff_v_repeatsum_exists_sum. ((((exists ff_h_repeatsum_exists_sum_start. ff_h_repeatsum_exists_sum_start + S (0) = S ((S (0)) * ff_v_repeatsum_exists_sum)) /\ exists ff_q_repeatsum_exists_sum_start. ff_u_repeatsum_exists_sum = ff_q_repeatsum_exists_sum_start * S ((S (0)) * ff_v_repeatsum_exists_sum) + (0))) /\ ((((exists ff_h_repeatsum_exists_sum_terminal. ff_h_repeatsum_exists_sum_terminal + S (n) = S ((S (l)) * ff_v_repeatsum_exists_sum)) /\ exists ff_q_repeatsum_exists_sum_terminal. ff_u_repeatsum_exists_sum = ff_q_repeatsum_exists_sum_terminal * S ((S (l)) * ff_v_repeatsum_exists_sum) + (n))) /\ forall ff_i_repeatsum_exists_sum. (exists ff_lt_repeatsum_exists_sum_bound. ff_lt_repeatsum_exists_sum_bound + S ff_i_repeatsum_exists_sum = l) -> exists ff_a_repeatsum_exists_sum ff_r_repeatsum_exists_sum ff_s_repeatsum_exists_sum. ((((exists ff_h_repeatsum_exists_sum_summand. ff_h_repeatsum_exists_sum_summand + S (ff_a_repeatsum_exists_sum) = S ((S (ff_i_repeatsum_exists_sum)) * c)) /\ exists ff_q_repeatsum_exists_sum_summand. b = ff_q_repeatsum_exists_sum_summand * S ((S (ff_i_repeatsum_exists_sum)) * c) + (ff_a_repeatsum_exists_sum))) /\ ((((exists ff_h_repeatsum_exists_sum_partial. ff_h_repeatsum_exists_sum_partial + S (ff_r_repeatsum_exists_sum) = S ((S (ff_i_repeatsum_exists_sum)) * ff_v_repeatsum_exists_sum)) /\ exists ff_q_repeatsum_exists_sum_partial. ff_u_repeatsum_exists_sum = ff_q_repeatsum_exists_sum_partial * S ((S (ff_i_repeatsum_exists_sum)) * ff_v_repeatsum_exists_sum) + (ff_r_repeatsum_exists_sum))) /\ ((((exists ff_h_repeatsum_exists_sum_successor. ff_h_repeatsum_exists_sum_successor + S (ff_s_repeatsum_exists_sum) = S ((S (S ff_i_repeatsum_exists_sum)) * ff_v_repeatsum_exists_sum)) /\ exists ff_q_repeatsum_exists_sum_successor. ff_u_repeatsum_exists_sum = ff_q_repeatsum_exists_sum_successor * S ((S (S ff_i_repeatsum_exists_sum)) * ff_v_repeatsum_exists_sum) + (ff_s_repeatsum_exists_sum))) /\ ff_s_repeatsum_exists_sum = ff_r_repeatsum_exists_sum + ff_a_repeatsum_exists_sum)))))) /\ n = l * a)

Structural proof guide

Generated structural guide

Every value and length admit a constant prefix with its exact sum.

Use the direct prerequisites beta_repeat_exists, beta_sum_exists, beta_repeat_sum_exact as previously established PA formulas.

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

Referenced ingredients

PA0045 beta_repeat_exists PA003H beta_sum_exists PA00EK beta_repeat_sum_exact

Proof neighborhood

Direct dependencies

PA0045 beta_repeat_exists PA003H beta_sum_exists PA00EK beta_repeat_sum_exact

Direct dependents

PA00EQ eisenstein_rectangle_plus_column_count_total

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 a
  2. 0002intro l
  3. 0003have hrepeat : exists b c. (forall ff_i_repeatsum_exists_repeat. (exists ff_lt_repeatsum_exists_repeat_bound. ff_lt_repeatsum_exists_repeat_bound + S ff_i_repeatsum_exists_repeat = l) -> (((exists ff_h_repeatsum_exists_repeat_decoded. ff_h_repeatsum_exists_repeat_decoded + S (a) = S ((S (ff_i_repeatsum_exists_repeat)) * c)) /\ exists ff_q_repeatsum_exists_repeat_decoded. b = ff_q_repeatsum_exists_repeat_decoded * S ((S (ff_i_repeatsum_exists_repeat)) * c) + (a))))
  4. 0004specialize beta_repeat_exists a
  5. 0005specialize beta_repeat_exists l
  6. 0006exact beta_repeat_exists
  7. 0007cases hrepeat
  8. 0008cases hrepeat_witness
  9. 0009have hsum : exists n. (exists ff_u_repeatsum_exists_trace ff_v_repeatsum_exists_trace. ((((exists ff_h_repeatsum_exists_trace_start. ff_h_repeatsum_exists_trace_start + S (0) = S ((S (0)) * ff_v_repeatsum_exists_trace)) /\ exists ff_q_repeatsum_exists_trace_start. ff_u_repeatsum_exists_trace = ff_q_repeatsum_exists_trace_start * S ((S (0)) * ff_v_repeatsum_exists_trace) + (0))) /\ ((((exists ff_h_repeatsum_exists_trace_terminal. ff_h_repeatsum_exists_trace_terminal + S (n) = S ((S (l)) * ff_v_repeatsum_exists_trace)) /\ exists ff_q_repeatsum_exists_trace_terminal. ff_u_repeatsum_exists_trace = ff_q_repeatsum_exists_trace_terminal * S ((S (l)) * ff_v_repeatsum_exists_trace) + (n))) /\ forall ff_i_repeatsum_exists_trace. (exists ff_lt_repeatsum_exists_trace_bound. ff_lt_repeatsum_exists_trace_bound + S ff_i_repeatsum_exists_trace = l) -> exists ff_a_repeatsum_exists_trace ff_r_repeatsum_exists_trace ff_s_repeatsum_exists_trace. ((((exists ff_h_repeatsum_exists_trace_summand. ff_h_repeatsum_exists_trace_summand + S (ff_a_repeatsum_exists_trace) = S ((S (ff_i_repeatsum_exists_trace)) * x1)) /\ exists ff_q_repeatsum_exists_trace_summand. x = ff_q_repeatsum_exists_trace_summand * S ((S (ff_i_repeatsum_exists_trace)) * x1) + (ff_a_repeatsum_exists_trace))) /\ ((((exists ff_h_repeatsum_exists_trace_partial. ff_h_repeatsum_exists_trace_partial + S (ff_r_repeatsum_exists_trace) = S ((S (ff_i_repeatsum_exists_trace)) * ff_v_repeatsum_exists_trace)) /\ exists ff_q_repeatsum_exists_trace_partial. ff_u_repeatsum_exists_trace = ff_q_repeatsum_exists_trace_partial * S ((S (ff_i_repeatsum_exists_trace)) * ff_v_repeatsum_exists_trace) + (ff_r_repeatsum_exists_trace))) /\ ((((exists ff_h_repeatsum_exists_trace_successor. ff_h_repeatsum_exists_trace_successor + S (ff_s_repeatsum_exists_trace) = S ((S (S ff_i_repeatsum_exists_trace)) * ff_v_repeatsum_exists_trace)) /\ exists ff_q_repeatsum_exists_trace_successor. ff_u_repeatsum_exists_trace = ff_q_repeatsum_exists_trace_successor * S ((S (S ff_i_repeatsum_exists_trace)) * ff_v_repeatsum_exists_trace) + (ff_s_repeatsum_exists_trace))) /\ ff_s_repeatsum_exists_trace = ff_r_repeatsum_exists_trace + ff_a_repeatsum_exists_trace))))))
  10. 0010specialize beta_sum_exists x
  11. 0011specialize beta_sum_exists x1
  12. 0012specialize beta_sum_exists l
  13. 0013exact beta_sum_exists
  14. 0014cases hsum
  15. 0015have hexact : x2 = l * a
  16. 0016specialize beta_repeat_sum_exact x
  17. 0017specialize beta_repeat_sum_exact x1
  18. 0018specialize beta_repeat_sum_exact a
  19. 0019specialize beta_repeat_sum_exact l
  20. 0020specialize beta_repeat_sum_exact x2
  21. 0021apply beta_repeat_sum_exact
  22. 0022exact hrepeat_witness_witness
  23. 0023exact hsum_witness
  24. 0024exists x
  25. 0025exists x1
  26. 0026exists x2
  27. 0027split
  28. 0028exact hrepeat_witness_witness
  29. 0029split
  30. 0030exact hsum_witness
  31. 0031exact hexact