PA0076

gauss_signed_half_prefix_all_bits

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

The sign projection of every encoded signed-half prefix is an AllBits prefix.

Exact expanded PA statement

forall p h a b c mb mc sb sc l. (forall gsp_index_bits_source. (exists gsp_lt_gap_bits_source_index_bound. gsp_lt_gap_bits_source_index_bound + S gsp_index_bits_source = l) -> (exists gsp_value_bits_source_entry gsp_magnitude_bits_source_entry gsp_sign_bits_source_entry. (((exists ff_h_gsp_bits_source_entry_source. ff_h_gsp_bits_source_entry_source + S (gsp_value_bits_source_entry) = S ((S (gsp_index_bits_source)) * c)) /\ exists ff_q_gsp_bits_source_entry_source. b = ff_q_gsp_bits_source_entry_source * S ((S (gsp_index_bits_source)) * c) + (gsp_value_bits_source_entry))) /\ ((((exists ff_h_gsp_bits_source_entry_magnitude. ff_h_gsp_bits_source_entry_magnitude + S (gsp_magnitude_bits_source_entry) = S ((S (gsp_index_bits_source)) * mc)) /\ exists ff_q_gsp_bits_source_entry_magnitude. mb = ff_q_gsp_bits_source_entry_magnitude * S ((S (gsp_index_bits_source)) * mc) + (gsp_magnitude_bits_source_entry))) /\ ((((exists ff_h_gsp_bits_source_entry_sign. ff_h_gsp_bits_source_entry_sign + S (gsp_sign_bits_source_entry) = S ((S (gsp_index_bits_source)) * sc)) /\ exists ff_q_gsp_bits_source_entry_sign. sb = ff_q_gsp_bits_source_entry_sign * S ((S (gsp_index_bits_source)) * sc) + (gsp_sign_bits_source_entry))) /\ ((exists gsp_lt_gap_bits_source_entry_positive. gsp_lt_gap_bits_source_entry_positive + S 0 = gsp_magnitude_bits_source_entry) /\ ((exists gsp_le_gap_bits_source_entry_bounded. gsp_le_gap_bits_source_entry_bounded + gsp_magnitude_bits_source_entry = h) /\ ((gsp_sign_bits_source_entry = 0 \/ gsp_sign_bits_source_entry = 1) /\ (((gsp_sign_bits_source_entry = 0 /\ (exists gsp_mod_left_bits_source_entry_lower gsp_mod_right_bits_source_entry_lower. (a * gsp_value_bits_source_entry) + p * gsp_mod_left_bits_source_entry_lower = (gsp_magnitude_bits_source_entry) + p * gsp_mod_right_bits_source_entry_lower)) \/ (gsp_sign_bits_source_entry = 1 /\ (exists gsp_mod_left_bits_source_entry_reflected gsp_mod_right_bits_source_entry_reflected. (a * gsp_value_bits_source_entry) + p * gsp_mod_left_bits_source_entry_reflected = ((2 * h) * gsp_magnitude_bits_source_entry) + p * gsp_mod_right_bits_source_entry_reflected))))))))))) -> (forall ff_i_gsp_signs. (exists ff_lt_gsp_signs_bound. ff_lt_gsp_signs_bound + S ff_i_gsp_signs = l) -> exists ff_bit_gsp_signs. ((((exists ff_h_gsp_signs_decoded. ff_h_gsp_signs_decoded + S (ff_bit_gsp_signs) = S ((S (ff_i_gsp_signs)) * sc)) /\ exists ff_q_gsp_signs_decoded. sb = ff_q_gsp_signs_decoded * S ((S (ff_i_gsp_signs)) * sc) + (ff_bit_gsp_signs))) /\ (ff_bit_gsp_signs = 0 \/ ff_bit_gsp_signs = 1)))

Structural proof guide

Generated structural guide

The sign projection of every encoded signed-half prefix is an AllBits prefix.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (9), intermediate claims (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA0077 gauss_signed_half_bit_count_exists

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 p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro b
  5. 0005intro c
  6. 0006intro mb
  7. 0007intro mc
  8. 0008intro sb
  9. 0009intro sc
  10. 0010intro l
  11. 0011intro hprefix
  12. 0012intro i
  13. 0013intro hi
  14. 0014have hentry : exists gsp_value_bits_entry gsp_magnitude_bits_entry gsp_sign_bits_entry. (((exists ff_h_gsp_bits_entry_source. ff_h_gsp_bits_entry_source + S (gsp_value_bits_entry) = S ((S (i)) * c)) /\ exists ff_q_gsp_bits_entry_source. b = ff_q_gsp_bits_entry_source * S ((S (i)) * c) + (gsp_value_bits_entry))) /\ ((((exists ff_h_gsp_bits_entry_magnitude. ff_h_gsp_bits_entry_magnitude + S (gsp_magnitude_bits_entry) = S ((S (i)) * mc)) /\ exists ff_q_gsp_bits_entry_magnitude. mb = ff_q_gsp_bits_entry_magnitude * S ((S (i)) * mc) + (gsp_magnitude_bits_entry))) /\ ((((exists ff_h_gsp_bits_entry_sign. ff_h_gsp_bits_entry_sign + S (gsp_sign_bits_entry) = S ((S (i)) * sc)) /\ exists ff_q_gsp_bits_entry_sign. sb = ff_q_gsp_bits_entry_sign * S ((S (i)) * sc) + (gsp_sign_bits_entry))) /\ ((exists gsp_lt_gap_bits_entry_positive. gsp_lt_gap_bits_entry_positive + S 0 = gsp_magnitude_bits_entry) /\ ((exists gsp_le_gap_bits_entry_bounded. gsp_le_gap_bits_entry_bounded + gsp_magnitude_bits_entry = h) /\ ((gsp_sign_bits_entry = 0 \/ gsp_sign_bits_entry = 1) /\ (((gsp_sign_bits_entry = 0 /\ (exists gsp_mod_left_bits_entry_lower gsp_mod_right_bits_entry_lower. (a * gsp_value_bits_entry) + p * gsp_mod_left_bits_entry_lower = (gsp_magnitude_bits_entry) + p * gsp_mod_right_bits_entry_lower)) \/ (gsp_sign_bits_entry = 1 /\ (exists gsp_mod_left_bits_entry_reflected gsp_mod_right_bits_entry_reflected. (a * gsp_value_bits_entry) + p * gsp_mod_left_bits_entry_reflected = ((2 * h) * gsp_magnitude_bits_entry) + p * gsp_mod_right_bits_entry_reflected)))))))))
  15. 0015specialize hprefix i
  16. 0016apply hprefix
  17. 0017exact hi
  18. 0018cases hentry
  19. 0019cases hentry_witness
  20. 0020cases hentry_witness_witness
  21. 0021cases hentry_witness_witness_witness
  22. 0022cases hentry_witness_witness_witness_right
  23. 0023cases hentry_witness_witness_witness_right_right
  24. 0024cases hentry_witness_witness_witness_right_right_right
  25. 0025cases hentry_witness_witness_witness_right_right_right_right
  26. 0026cases hentry_witness_witness_witness_right_right_right_right_right
  27. 0027exists x2
  28. 0028split
  29. 0029exact hentry_witness_witness_witness_right_right_left
  30. 0030exact hentry_witness_witness_witness_right_right_right_right_right_left