PA004V

finite_no_top_successor_gate

Stable checked-use theorem · independently closed

The no-top branch of the constructive successor induction is complete.

Exact expanded PA statement

forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) -> ((forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n)))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))

Structural proof guide

Generated structural guide

The no-top branch of the constructive successor induction is complete.

Use the direct prerequisites finite_bounded_prefix_without_top, finite_injective_prefix_succ, finite_surjective_succ_from_prefix as previously established PA formulas.

The proof proceeds by intermediate claims (4).

Referenced ingredients

PA004P finite_bounded_prefix_without_top PA004Q finite_injective_prefix_succ PA004T finite_surjective_succ_from_prefix

Proof neighborhood

Direct dependencies

PA004P finite_bounded_prefix_without_top PA004Q finite_injective_prefix_succ PA004T finite_surjective_succ_from_prefix

Direct dependents

PA004W finite_bounded_injective_surjective

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 n
  4. 0004intro sn
  5. 0005intro hsn
  6. 0006intro hbounded
  7. 0007intro hinj
  8. 0008intro hmissing
  9. 0009intro hinduction
  10. 0010have hnotop : forall i. (exists h. h + S i = n) -> ~((exists h. h + S n = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + n)
  11. 0011intro i
  12. 0012intro hi
  13. 0013intro hentry
  14. 0014apply hmissing
  15. 0015exists i
  16. 0016split
  17. 0017exact hi
  18. 0018exact hentry
  19. 0019have hprefix_bounded : forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))
  20. 0020specialize finite_bounded_prefix_without_top b
  21. 0021specialize finite_bounded_prefix_without_top c
  22. 0022specialize finite_bounded_prefix_without_top n
  23. 0023specialize finite_bounded_prefix_without_top sn
  24. 0024apply finite_bounded_prefix_without_top
  25. 0025exact hsn
  26. 0026exact hbounded
  27. 0027exact hnotop
  28. 0028have hprefix_injective : forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix
  29. 0029specialize finite_injective_prefix_succ b
  30. 0030specialize finite_injective_prefix_succ c
  31. 0031specialize finite_injective_prefix_succ n
  32. 0032specialize finite_injective_prefix_succ sn
  33. 0033apply finite_injective_prefix_succ
  34. 0034exact hsn
  35. 0035exact hinj
  36. 0036have hprefix_surjective : forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))
  37. 0037apply hinduction
  38. 0038exact hprefix_bounded
  39. 0039exact hprefix_injective
  40. 0040specialize finite_surjective_succ_from_prefix b
  41. 0041specialize finite_surjective_succ_from_prefix c
  42. 0042specialize finite_surjective_succ_from_prefix n
  43. 0043specialize finite_surjective_succ_from_prefix sn
  44. 0044apply finite_surjective_succ_from_prefix
  45. 0045exact hsn
  46. 0046exact hbounded
  47. 0047exact hinj
  48. 0048exact hprefix_surjective