Actual divisor sums and Möbius tables — exact evidence

Exact theorem nodes and inherited prerequisites

• DV0001 arithmetic_signed_table_component_prefix_preserved · actual bundle node 277
• DV0002 arithmetic_signed_table_equal_entry_transport · actual bundle node 278
• DV0003 arithmetic_signed_table_extend_at · actual bundle node 279
• DV0004 arithmetic_signed_table_append · actual bundle node 280
• DV0005 arithmetic_signed_table_singleton · actual bundle node 281
• DV0006 arithmetic_signed_sum_exists · actual bundle node 282
• DV0007 arithmetic_signed_sum_append_transport · actual bundle node 283
• DV0008 mobius_table_zero_constructor · actual bundle node 284
• DV0009 mobius_table_append · actual bundle node 285
• DV000A mobius_table_exists · actual bundle node 286
• DV000B mobius_table_lookup · actual bundle node 287
• DV000C mobius_table_entry_iff · actual bundle node 288
• DV000D mobius_table_one_entry · actual bundle node 289
• DV000E mobius_table_extensional · actual bundle node 290
• DV000F mobius_table_restrict · actual bundle node 291
• DV0010 divisor_mask_entry_zero · actual bundle node 292
• DV0011 divisor_mask_entry_from_quotient · actual bundle node 293
• DV0012 divisor_mask_entry_from_nondivisor · actual bundle node 294
• DV0013 divisor_mask_entry_exists · actual bundle node 295
• DV0014 divisor_mask_entry_functional · actual bundle node 296
• DV0015 divisor_mask_entry_quotient_input · actual bundle node 297
• DV0016 divisor_mask_entry_omitted_value · actual bundle node 298
• DV0017 divisor_mask_prefix_zero_constructor · actual bundle node 299
• DV0018 divisor_mask_prefix_append · actual bundle node 300
• DV0019 divisor_mask_prefix_exists · actual bundle node 301
• DV001A divisor_mask_prefix_extensional · actual bundle node 302
• DV001B divisor_mask_prefix_restrict · actual bundle node 303
• DV001C divisor_mask_positive_quotient_entry · actual bundle node 304
• DV001D divisor_mask_omitted_entry · actual bundle node 305
• DV001E divisor_mask_entry_positive_source_extensional · actual bundle node 306
• DV001F divisor_mask_positive_source_extensional · actual bundle node 307
• DV0020 signed_divisor_sum_exists · actual bundle node 308
• DV0021 signed_divisor_sum_functional · actual bundle node 309
• DV0022 signed_divisor_sum_exists_unique · actual bundle node 310
• DV0023 signed_divisor_sum_zero_excluded · actual bundle node 311
• DV0024 signed_divisor_sum_one · actual bundle node 312
• DV0025 signed_divisor_sum_positive_source_extensional · actual bundle node 313
• add_comm · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n m. n + m = m + n

• beta_prefix_extend · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\ exists q. z = q * S ((S k) * c) + s) /\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\ exists q. z = q * S ((S i) * c) + a))

• divisor_signed_sum_empty_value · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F z. (exists dst_positive_code_empty_sum dst_positive_scale_empty_sum dst_negative_code_empty_sum dst_negative_scale_empty_sum dst_positive_sum_empty_sum dst_negative_sum_empty_sum. (((F) = (((((dst_positive_code_empty_sum) + (dst_positive_scale_empty_sum)) * S ((dst_positive_code_empty_sum) + (dst_positive_scale_empty_sum)) + ((dst_positive_scale_empty_sum) + (dst_positive_scale_empty_sum))) + (((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) * S ((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) + ((dst_negative_scale_empty_sum) + (dst_negative_scale_empty_sum)))) * S ((((dst_positive_code_empty_sum) + (dst_positive_scale_empty_sum)) * S ((dst_positive_code_empty_sum) + (dst_positive_scale_empty_sum)) + ((dst_positive_scale_empty_sum) + (dst_positive_scale_empty_sum))) + (((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) * S ((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) + ((dst_negative_scale_empty_sum) + (dst_negative_scale_empty_sum)))) + ((((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) * S ((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) + ((dst_negative_scale_empty_sum) + (dst_negative_scale_empty_sum))) + (((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) * S ((dst_negative_code_empty_sum) + (dst_negative_scale_empty_sum)) + ((dst_negative_scale_empty_sum) + (dst_negative_scale_empty_sum)))))) /\ (((exists fs_u_dst_empty_sumpositive fs_v_dst_empty_sumpositive. ((((exists fs_h_dst_empty_sumpositive_body_start. fs_h_dst_empty_sumpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_empty_sumpositive)) /\ exists fs_q_dst_empty_sumpositive_body_start. fs_u_dst_empty_sumpositive = fs_q_dst_empty_sumpositive_body_start * S ((S (0)) * fs_v_dst_empty_sumpositive) + (0))) /\ ((((exists fs_h_dst_empty_sumpositive_body_terminal. fs_h_dst_empty_sumpositive_body_terminal + S (dst_positive_sum_empty_sum) = S ((S (0)) * fs_v_dst_empty_sumpositive)) /\ exists fs_q_dst_empty_sumpositive_body_terminal. fs_u_dst_empty_sumpositive = fs_q_dst_empty_sumpositive_body_terminal * S ((S (0)) * fs_v_dst_empty_sumpositive) + (dst_positive_sum_empty_sum))) /\ forall fs_i_dst_empty_sumpositive_body_steps. (exists fs_lt_dst_empty_sumpositive_body_steps_bound. fs_lt_dst_empty_sumpositive_body_steps_bound + S fs_i_dst_empty_sumpositive_body_steps = 0) -> exists fs_a_dst_empty_sumpositive_body_steps fs_r_dst_empty_sumpositive_body_steps fs_s_dst_empty_sumpositive_body_steps. ((((exists fs_h_dst_empty_sumpositive_body_steps_summand. fs_h_dst_empty_sumpositive_body_steps_summand + S (fs_a_dst_empty_sumpositive_body_steps) = S ((S (fs_i_dst_empty_sumpositive_body_steps)) * dst_positive_scale_empty_sum)) /\ exists fs_q_dst_empty_sumpositive_body_steps_summand. dst_positive_code_empty_sum = fs_q_dst_empty_sumpositive_body_steps_summand * S ((S (fs_i_dst_empty_sumpositive_body_steps)) * dst_positive_scale_empty_sum) + (fs_a_dst_empty_sumpositive_body_steps))) /\ ((((exists fs_h_dst_empty_sumpositive_body_steps_partial. fs_h_dst_empty_sumpositive_body_steps_partial + S (fs_r_dst_empty_sumpositive_body_steps) = S ((S (fs_i_dst_empty_sumpositive_body_steps)) * fs_v_dst_empty_sumpositive)) /\ exists fs_q_dst_empty_sumpositive_body_steps_partial. fs_u_dst_empty_sumpositive = fs_q_dst_empty_sumpositive_body_steps_partial * S ((S (fs_i_dst_empty_sumpositive_body_steps)) * fs_v_dst_empty_sumpositive) + (fs_r_dst_empty_sumpositive_body_steps))) /\ ((((exists fs_h_dst_empty_sumpositive_body_steps_successor. fs_h_dst_empty_sumpositive_body_steps_successor + S (fs_s_dst_empty_sumpositive_body_steps) = S ((S (S fs_i_dst_empty_sumpositive_body_steps)) * fs_v_dst_empty_sumpositive)) /\ exists fs_q_dst_empty_sumpositive_body_steps_successor. fs_u_dst_empty_sumpositive = fs_q_dst_empty_sumpositive_body_steps_successor * S ((S (S fs_i_dst_empty_sumpositive_body_steps)) * fs_v_dst_empty_sumpositive) + (fs_s_dst_empty_sumpositive_body_steps))) /\ fs_s_dst_empty_sumpositive_body_steps = fs_r_dst_empty_sumpositive_body_steps + fs_a_dst_empty_sumpositive_body_steps)))))) /\ (((exists fs_u_dst_empty_sumnegative fs_v_dst_empty_sumnegative. ((((exists fs_h_dst_empty_sumnegative_body_start. fs_h_dst_empty_sumnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_empty_sumnegative)) /\ exists fs_q_dst_empty_sumnegative_body_start. fs_u_dst_empty_sumnegative = fs_q_dst_empty_sumnegative_body_start * S ((S (0)) * fs_v_dst_empty_sumnegative) + (0))) /\ ((((exists fs_h_dst_empty_sumnegative_body_terminal. fs_h_dst_empty_sumnegative_body_terminal + S (dst_negative_sum_empty_sum) = S ((S (0)) * fs_v_dst_empty_sumnegative)) /\ exists fs_q_dst_empty_sumnegative_body_terminal. fs_u_dst_empty_sumnegative = fs_q_dst_empty_sumnegative_body_terminal * S ((S (0)) * fs_v_dst_empty_sumnegative) + (dst_negative_sum_empty_sum))) /\ forall fs_i_dst_empty_sumnegative_body_steps. (exists fs_lt_dst_empty_sumnegative_body_steps_bound. fs_lt_dst_empty_sumnegative_body_steps_bound + S fs_i_dst_empty_sumnegative_body_steps = 0) -> exists fs_a_dst_empty_sumnegative_body_steps fs_r_dst_empty_sumnegative_body_steps fs_s_dst_empty_sumnegative_body_steps. ((((exists fs_h_dst_empty_sumnegative_body_steps_summand. fs_h_dst_empty_sumnegative_body_steps_summand + S (fs_a_dst_empty_sumnegative_body_steps) = S ((S (fs_i_dst_empty_sumnegative_body_steps)) * dst_negative_scale_empty_sum)) /\ exists fs_q_dst_empty_sumnegative_body_steps_summand. dst_negative_code_empty_sum = fs_q_dst_empty_sumnegative_body_steps_summand * S ((S (fs_i_dst_empty_sumnegative_body_steps)) * dst_negative_scale_empty_sum) + (fs_a_dst_empty_sumnegative_body_steps))) /\ ((((exists fs_h_dst_empty_sumnegative_body_steps_partial. fs_h_dst_empty_sumnegative_body_steps_partial + S (fs_r_dst_empty_sumnegative_body_steps) = S ((S (fs_i_dst_empty_sumnegative_body_steps)) * fs_v_dst_empty_sumnegative)) /\ exists fs_q_dst_empty_sumnegative_body_steps_partial. fs_u_dst_empty_sumnegative = fs_q_dst_empty_sumnegative_body_steps_partial * S ((S (fs_i_dst_empty_sumnegative_body_steps)) * fs_v_dst_empty_sumnegative) + (fs_r_dst_empty_sumnegative_body_steps))) /\ ((((exists fs_h_dst_empty_sumnegative_body_steps_successor. fs_h_dst_empty_sumnegative_body_steps_successor + S (fs_s_dst_empty_sumnegative_body_steps) = S ((S (S fs_i_dst_empty_sumnegative_body_steps)) * fs_v_dst_empty_sumnegative)) /\ exists fs_q_dst_empty_sumnegative_body_steps_successor. fs_u_dst_empty_sumnegative = fs_q_dst_empty_sumnegative_body_steps_successor * S ((S (S fs_i_dst_empty_sumnegative_body_steps)) * fs_v_dst_empty_sumnegative) + (fs_s_dst_empty_sumnegative_body_steps))) /\ fs_s_dst_empty_sumnegative_body_steps = fs_r_dst_empty_sumnegative_body_steps + fs_a_dst_empty_sumnegative_body_steps)))))) /\ (exists ge_balance_positive_empty_sumresult ge_balance_negative_empty_sumresult. (((((z) = 2 * (ge_balance_positive_empty_sumresult) /\ (ge_balance_negative_empty_sumresult) = 0) \/ exists ge_signed_half_empty_sumresultdecode. (((z) = 2 * ge_signed_half_empty_sumresultdecode + 1 /\ (ge_balance_positive_empty_sumresult) = 0) /\ (ge_balance_negative_empty_sumresult) = S ge_signed_half_empty_sumresultdecode))) /\ ((dst_positive_sum_empty_sum) + ge_balance_negative_empty_sumresult = (dst_negative_sum_empty_sum) + ge_balance_positive_empty_sumresult))))))))) -> z = 0

• divisor_signed_sum_exists_from_components · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F pb pc nb nc l. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> exists z. (exists dst_positive_code_sum_exists_result dst_positive_scale_sum_exists_result dst_negative_code_sum_exists_result dst_negative_scale_sum_exists_result dst_positive_sum_sum_exists_result dst_negative_sum_sum_exists_result. (((F) = (((((dst_positive_code_sum_exists_result) + (dst_positive_scale_sum_exists_result)) * S ((dst_positive_code_sum_exists_result) + (dst_positive_scale_sum_exists_result)) + ((dst_positive_scale_sum_exists_result) + (dst_positive_scale_sum_exists_result))) + (((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) * S ((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) + ((dst_negative_scale_sum_exists_result) + (dst_negative_scale_sum_exists_result)))) * S ((((dst_positive_code_sum_exists_result) + (dst_positive_scale_sum_exists_result)) * S ((dst_positive_code_sum_exists_result) + (dst_positive_scale_sum_exists_result)) + ((dst_positive_scale_sum_exists_result) + (dst_positive_scale_sum_exists_result))) + (((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) * S ((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) + ((dst_negative_scale_sum_exists_result) + (dst_negative_scale_sum_exists_result)))) + ((((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) * S ((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) + ((dst_negative_scale_sum_exists_result) + (dst_negative_scale_sum_exists_result))) + (((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) * S ((dst_negative_code_sum_exists_result) + (dst_negative_scale_sum_exists_result)) + ((dst_negative_scale_sum_exists_result) + (dst_negative_scale_sum_exists_result)))))) /\ (((exists fs_u_dst_sum_exists_resultpositive fs_v_dst_sum_exists_resultpositive. ((((exists fs_h_dst_sum_exists_resultpositive_body_start. fs_h_dst_sum_exists_resultpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_exists_resultpositive)) /\ exists fs_q_dst_sum_exists_resultpositive_body_start. fs_u_dst_sum_exists_resultpositive = fs_q_dst_sum_exists_resultpositive_body_start * S ((S (0)) * fs_v_dst_sum_exists_resultpositive) + (0))) /\ ((((exists fs_h_dst_sum_exists_resultpositive_body_terminal. fs_h_dst_sum_exists_resultpositive_body_terminal + S (dst_positive_sum_sum_exists_result) = S ((S (l)) * fs_v_dst_sum_exists_resultpositive)) /\ exists fs_q_dst_sum_exists_resultpositive_body_terminal. fs_u_dst_sum_exists_resultpositive = fs_q_dst_sum_exists_resultpositive_body_terminal * S ((S (l)) * fs_v_dst_sum_exists_resultpositive) + (dst_positive_sum_sum_exists_result))) /\ forall fs_i_dst_sum_exists_resultpositive_body_steps. (exists fs_lt_dst_sum_exists_resultpositive_body_steps_bound. fs_lt_dst_sum_exists_resultpositive_body_steps_bound + S fs_i_dst_sum_exists_resultpositive_body_steps = l) -> exists fs_a_dst_sum_exists_resultpositive_body_steps fs_r_dst_sum_exists_resultpositive_body_steps fs_s_dst_sum_exists_resultpositive_body_steps. ((((exists fs_h_dst_sum_exists_resultpositive_body_steps_summand. fs_h_dst_sum_exists_resultpositive_body_steps_summand + S (fs_a_dst_sum_exists_resultpositive_body_steps) = S ((S (fs_i_dst_sum_exists_resultpositive_body_steps)) * dst_positive_scale_sum_exists_result)) /\ exists fs_q_dst_sum_exists_resultpositive_body_steps_summand. dst_positive_code_sum_exists_result = fs_q_dst_sum_exists_resultpositive_body_steps_summand * S ((S (fs_i_dst_sum_exists_resultpositive_body_steps)) * dst_positive_scale_sum_exists_result) + (fs_a_dst_sum_exists_resultpositive_body_steps))) /\ ((((exists fs_h_dst_sum_exists_resultpositive_body_steps_partial. fs_h_dst_sum_exists_resultpositive_body_steps_partial + S (fs_r_dst_sum_exists_resultpositive_body_steps) = S ((S (fs_i_dst_sum_exists_resultpositive_body_steps)) * fs_v_dst_sum_exists_resultpositive)) /\ exists fs_q_dst_sum_exists_resultpositive_body_steps_partial. fs_u_dst_sum_exists_resultpositive = fs_q_dst_sum_exists_resultpositive_body_steps_partial * S ((S (fs_i_dst_sum_exists_resultpositive_body_steps)) * fs_v_dst_sum_exists_resultpositive) + (fs_r_dst_sum_exists_resultpositive_body_steps))) /\ ((((exists fs_h_dst_sum_exists_resultpositive_body_steps_successor. fs_h_dst_sum_exists_resultpositive_body_steps_successor + S (fs_s_dst_sum_exists_resultpositive_body_steps) = S ((S (S fs_i_dst_sum_exists_resultpositive_body_steps)) * fs_v_dst_sum_exists_resultpositive)) /\ exists fs_q_dst_sum_exists_resultpositive_body_steps_successor. fs_u_dst_sum_exists_resultpositive = fs_q_dst_sum_exists_resultpositive_body_steps_successor * S ((S (S fs_i_dst_sum_exists_resultpositive_body_steps)) * fs_v_dst_sum_exists_resultpositive) + (fs_s_dst_sum_exists_resultpositive_body_steps))) /\ fs_s_dst_sum_exists_resultpositive_body_steps = fs_r_dst_sum_exists_resultpositive_body_steps + fs_a_dst_sum_exists_resultpositive_body_steps)))))) /\ (((exists fs_u_dst_sum_exists_resultnegative fs_v_dst_sum_exists_resultnegative. ((((exists fs_h_dst_sum_exists_resultnegative_body_start. fs_h_dst_sum_exists_resultnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_exists_resultnegative)) /\ exists fs_q_dst_sum_exists_resultnegative_body_start. fs_u_dst_sum_exists_resultnegative = fs_q_dst_sum_exists_resultnegative_body_start * S ((S (0)) * fs_v_dst_sum_exists_resultnegative) + (0))) /\ ((((exists fs_h_dst_sum_exists_resultnegative_body_terminal. fs_h_dst_sum_exists_resultnegative_body_terminal + S (dst_negative_sum_sum_exists_result) = S ((S (l)) * fs_v_dst_sum_exists_resultnegative)) /\ exists fs_q_dst_sum_exists_resultnegative_body_terminal. fs_u_dst_sum_exists_resultnegative = fs_q_dst_sum_exists_resultnegative_body_terminal * S ((S (l)) * fs_v_dst_sum_exists_resultnegative) + (dst_negative_sum_sum_exists_result))) /\ forall fs_i_dst_sum_exists_resultnegative_body_steps. (exists fs_lt_dst_sum_exists_resultnegative_body_steps_bound. fs_lt_dst_sum_exists_resultnegative_body_steps_bound + S fs_i_dst_sum_exists_resultnegative_body_steps = l) -> exists fs_a_dst_sum_exists_resultnegative_body_steps fs_r_dst_sum_exists_resultnegative_body_steps fs_s_dst_sum_exists_resultnegative_body_steps. ((((exists fs_h_dst_sum_exists_resultnegative_body_steps_summand. fs_h_dst_sum_exists_resultnegative_body_steps_summand + S (fs_a_dst_sum_exists_resultnegative_body_steps) = S ((S (fs_i_dst_sum_exists_resultnegative_body_steps)) * dst_negative_scale_sum_exists_result)) /\ exists fs_q_dst_sum_exists_resultnegative_body_steps_summand. dst_negative_code_sum_exists_result = fs_q_dst_sum_exists_resultnegative_body_steps_summand * S ((S (fs_i_dst_sum_exists_resultnegative_body_steps)) * dst_negative_scale_sum_exists_result) + (fs_a_dst_sum_exists_resultnegative_body_steps))) /\ ((((exists fs_h_dst_sum_exists_resultnegative_body_steps_partial. fs_h_dst_sum_exists_resultnegative_body_steps_partial + S (fs_r_dst_sum_exists_resultnegative_body_steps) = S ((S (fs_i_dst_sum_exists_resultnegative_body_steps)) * fs_v_dst_sum_exists_resultnegative)) /\ exists fs_q_dst_sum_exists_resultnegative_body_steps_partial. fs_u_dst_sum_exists_resultnegative = fs_q_dst_sum_exists_resultnegative_body_steps_partial * S ((S (fs_i_dst_sum_exists_resultnegative_body_steps)) * fs_v_dst_sum_exists_resultnegative) + (fs_r_dst_sum_exists_resultnegative_body_steps))) /\ ((((exists fs_h_dst_sum_exists_resultnegative_body_steps_successor. fs_h_dst_sum_exists_resultnegative_body_steps_successor + S (fs_s_dst_sum_exists_resultnegative_body_steps) = S ((S (S fs_i_dst_sum_exists_resultnegative_body_steps)) * fs_v_dst_sum_exists_resultnegative)) /\ exists fs_q_dst_sum_exists_resultnegative_body_steps_successor. fs_u_dst_sum_exists_resultnegative = fs_q_dst_sum_exists_resultnegative_body_steps_successor * S ((S (S fs_i_dst_sum_exists_resultnegative_body_steps)) * fs_v_dst_sum_exists_resultnegative) + (fs_s_dst_sum_exists_resultnegative_body_steps))) /\ fs_s_dst_sum_exists_resultnegative_body_steps = fs_r_dst_sum_exists_resultnegative_body_steps + fs_a_dst_sum_exists_resultnegative_body_steps)))))) /\ (exists ge_balance_positive_sum_exists_resultresult ge_balance_negative_sum_exists_resultresult. (((((z) = 2 * (ge_balance_positive_sum_exists_resultresult) /\ (ge_balance_negative_sum_exists_resultresult) = 0) \/ exists ge_signed_half_sum_exists_resultresultdecode. (((z) = 2 * ge_signed_half_sum_exists_resultresultdecode + 1 /\ (ge_balance_positive_sum_exists_resultresult) = 0) /\ (ge_balance_negative_sum_exists_resultresult) = S ge_signed_half_sum_exists_resultresultdecode))) /\ ((dst_positive_sum_sum_exists_result) + ge_balance_negative_sum_exists_resultresult = (dst_negative_sum_sum_exists_result) + ge_balance_positive_sum_exists_resultresult)))))))))

• divisor_signed_sum_extensional · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F G l a b. (forall dst_index_sum_ext_entries dst_first_sum_ext_entries dst_second_sum_ext_entries. (exists pvs_gap_sum_ext_entriesbound. pvs_gap_sum_ext_entriesbound + S (dst_index_sum_ext_entries) = (l)) -> (exists dst_positive_code_sum_ext_entriesfirst dst_positive_scale_sum_ext_entriesfirst dst_negative_code_sum_ext_entriesfirst dst_negative_scale_sum_ext_entriesfirst dst_positive_sum_ext_entriesfirst dst_negative_sum_ext_entriesfirst. (((F) = (((((dst_positive_code_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst)) * S ((dst_positive_code_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst)) + ((dst_positive_scale_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst))) + (((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) * S ((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) + ((dst_negative_scale_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)))) * S ((((dst_positive_code_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst)) * S ((dst_positive_code_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst)) + ((dst_positive_scale_sum_ext_entriesfirst) + (dst_positive_scale_sum_ext_entriesfirst))) + (((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) * S ((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) + ((dst_negative_scale_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)))) + ((((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) * S ((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) + ((dst_negative_scale_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst))) + (((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) * S ((dst_negative_code_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)) + ((dst_negative_scale_sum_ext_entriesfirst) + (dst_negative_scale_sum_ext_entriesfirst)))))) /\ (((((exists ff_h_pvs_sum_ext_entriesfirstpositive. ff_h_pvs_sum_ext_entriesfirstpositive + S (dst_positive_sum_ext_entriesfirst) = S ((S (dst_index_sum_ext_entries)) * dst_positive_scale_sum_ext_entriesfirst)) /\ exists ff_q_pvs_sum_ext_entriesfirstpositive. dst_positive_code_sum_ext_entriesfirst = ff_q_pvs_sum_ext_entriesfirstpositive * S ((S (dst_index_sum_ext_entries)) * dst_positive_scale_sum_ext_entriesfirst) + (dst_positive_sum_ext_entriesfirst))) /\ (((((exists ff_h_pvs_sum_ext_entriesfirstnegative. ff_h_pvs_sum_ext_entriesfirstnegative + S (dst_negative_sum_ext_entriesfirst) = S ((S (dst_index_sum_ext_entries)) * dst_negative_scale_sum_ext_entriesfirst)) /\ exists ff_q_pvs_sum_ext_entriesfirstnegative. dst_negative_code_sum_ext_entriesfirst = ff_q_pvs_sum_ext_entriesfirstnegative * S ((S (dst_index_sum_ext_entries)) * dst_negative_scale_sum_ext_entriesfirst) + (dst_negative_sum_ext_entriesfirst))) /\ (exists ge_balance_positive_sum_ext_entriesfirstvalue ge_balance_negative_sum_ext_entriesfirstvalue. (((((dst_first_sum_ext_entries) = 2 * (ge_balance_positive_sum_ext_entriesfirstvalue) /\ (ge_balance_negative_sum_ext_entriesfirstvalue) = 0) \/ exists ge_signed_half_sum_ext_entriesfirstvaluedecode. (((dst_first_sum_ext_entries) = 2 * ge_signed_half_sum_ext_entriesfirstvaluedecode + 1 /\ (ge_balance_positive_sum_ext_entriesfirstvalue) = 0) /\ (ge_balance_negative_sum_ext_entriesfirstvalue) = S ge_signed_half_sum_ext_entriesfirstvaluedecode))) /\ ((dst_positive_sum_ext_entriesfirst) + ge_balance_negative_sum_ext_entriesfirstvalue = (dst_negative_sum_ext_entriesfirst) + ge_balance_positive_sum_ext_entriesfirstvalue))))))))) -> (exists dst_positive_code_sum_ext_entriessecond dst_positive_scale_sum_ext_entriessecond dst_negative_code_sum_ext_entriessecond dst_negative_scale_sum_ext_entriessecond dst_positive_sum_ext_entriessecond dst_negative_sum_ext_entriessecond. (((G) = (((((dst_positive_code_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond)) * S ((dst_positive_code_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond)) + ((dst_positive_scale_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond))) + (((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) * S ((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) + ((dst_negative_scale_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)))) * S ((((dst_positive_code_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond)) * S ((dst_positive_code_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond)) + ((dst_positive_scale_sum_ext_entriessecond) + (dst_positive_scale_sum_ext_entriessecond))) + (((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) * S ((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) + ((dst_negative_scale_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)))) + ((((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) * S ((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) + ((dst_negative_scale_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond))) + (((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) * S ((dst_negative_code_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)) + ((dst_negative_scale_sum_ext_entriessecond) + (dst_negative_scale_sum_ext_entriessecond)))))) /\ (((((exists ff_h_pvs_sum_ext_entriessecondpositive. ff_h_pvs_sum_ext_entriessecondpositive + S (dst_positive_sum_ext_entriessecond) = S ((S (dst_index_sum_ext_entries)) * dst_positive_scale_sum_ext_entriessecond)) /\ exists ff_q_pvs_sum_ext_entriessecondpositive. dst_positive_code_sum_ext_entriessecond = ff_q_pvs_sum_ext_entriessecondpositive * S ((S (dst_index_sum_ext_entries)) * dst_positive_scale_sum_ext_entriessecond) + (dst_positive_sum_ext_entriessecond))) /\ (((((exists ff_h_pvs_sum_ext_entriessecondnegative. ff_h_pvs_sum_ext_entriessecondnegative + S (dst_negative_sum_ext_entriessecond) = S ((S (dst_index_sum_ext_entries)) * dst_negative_scale_sum_ext_entriessecond)) /\ exists ff_q_pvs_sum_ext_entriessecondnegative. dst_negative_code_sum_ext_entriessecond = ff_q_pvs_sum_ext_entriessecondnegative * S ((S (dst_index_sum_ext_entries)) * dst_negative_scale_sum_ext_entriessecond) + (dst_negative_sum_ext_entriessecond))) /\ (exists ge_balance_positive_sum_ext_entriessecondvalue ge_balance_negative_sum_ext_entriessecondvalue. (((((dst_second_sum_ext_entries) = 2 * (ge_balance_positive_sum_ext_entriessecondvalue) /\ (ge_balance_negative_sum_ext_entriessecondvalue) = 0) \/ exists ge_signed_half_sum_ext_entriessecondvaluedecode. (((dst_second_sum_ext_entries) = 2 * ge_signed_half_sum_ext_entriessecondvaluedecode + 1 /\ (ge_balance_positive_sum_ext_entriessecondvalue) = 0) /\ (ge_balance_negative_sum_ext_entriessecondvalue) = S ge_signed_half_sum_ext_entriessecondvaluedecode))) /\ ((dst_positive_sum_ext_entriessecond) + ge_balance_negative_sum_ext_entriessecondvalue = (dst_negative_sum_ext_entriessecond) + ge_balance_positive_sum_ext_entriessecondvalue))))))))) -> dst_first_sum_ext_entries = dst_second_sum_ext_entries) -> (exists dst_positive_code_sum_ext_first dst_positive_scale_sum_ext_first dst_negative_code_sum_ext_first dst_negative_scale_sum_ext_first dst_positive_sum_sum_ext_first dst_negative_sum_sum_ext_first. (((F) = (((((dst_positive_code_sum_ext_first) + (dst_positive_scale_sum_ext_first)) * S ((dst_positive_code_sum_ext_first) + (dst_positive_scale_sum_ext_first)) + ((dst_positive_scale_sum_ext_first) + (dst_positive_scale_sum_ext_first))) + (((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) * S ((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) + ((dst_negative_scale_sum_ext_first) + (dst_negative_scale_sum_ext_first)))) * S ((((dst_positive_code_sum_ext_first) + (dst_positive_scale_sum_ext_first)) * S ((dst_positive_code_sum_ext_first) + (dst_positive_scale_sum_ext_first)) + ((dst_positive_scale_sum_ext_first) + (dst_positive_scale_sum_ext_first))) + (((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) * S ((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) + ((dst_negative_scale_sum_ext_first) + (dst_negative_scale_sum_ext_first)))) + ((((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) * S ((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) + ((dst_negative_scale_sum_ext_first) + (dst_negative_scale_sum_ext_first))) + (((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) * S ((dst_negative_code_sum_ext_first) + (dst_negative_scale_sum_ext_first)) + ((dst_negative_scale_sum_ext_first) + (dst_negative_scale_sum_ext_first)))))) /\ (((exists fs_u_dst_sum_ext_firstpositive fs_v_dst_sum_ext_firstpositive. ((((exists fs_h_dst_sum_ext_firstpositive_body_start. fs_h_dst_sum_ext_firstpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_ext_firstpositive)) /\ exists fs_q_dst_sum_ext_firstpositive_body_start. fs_u_dst_sum_ext_firstpositive = fs_q_dst_sum_ext_firstpositive_body_start * S ((S (0)) * fs_v_dst_sum_ext_firstpositive) + (0))) /\ ((((exists fs_h_dst_sum_ext_firstpositive_body_terminal. fs_h_dst_sum_ext_firstpositive_body_terminal + S (dst_positive_sum_sum_ext_first) = S ((S (l)) * fs_v_dst_sum_ext_firstpositive)) /\ exists fs_q_dst_sum_ext_firstpositive_body_terminal. fs_u_dst_sum_ext_firstpositive = fs_q_dst_sum_ext_firstpositive_body_terminal * S ((S (l)) * fs_v_dst_sum_ext_firstpositive) + (dst_positive_sum_sum_ext_first))) /\ forall fs_i_dst_sum_ext_firstpositive_body_steps. (exists fs_lt_dst_sum_ext_firstpositive_body_steps_bound. fs_lt_dst_sum_ext_firstpositive_body_steps_bound + S fs_i_dst_sum_ext_firstpositive_body_steps = l) -> exists fs_a_dst_sum_ext_firstpositive_body_steps fs_r_dst_sum_ext_firstpositive_body_steps fs_s_dst_sum_ext_firstpositive_body_steps. ((((exists fs_h_dst_sum_ext_firstpositive_body_steps_summand. fs_h_dst_sum_ext_firstpositive_body_steps_summand + S (fs_a_dst_sum_ext_firstpositive_body_steps) = S ((S (fs_i_dst_sum_ext_firstpositive_body_steps)) * dst_positive_scale_sum_ext_first)) /\ exists fs_q_dst_sum_ext_firstpositive_body_steps_summand. dst_positive_code_sum_ext_first = fs_q_dst_sum_ext_firstpositive_body_steps_summand * S ((S (fs_i_dst_sum_ext_firstpositive_body_steps)) * dst_positive_scale_sum_ext_first) + (fs_a_dst_sum_ext_firstpositive_body_steps))) /\ ((((exists fs_h_dst_sum_ext_firstpositive_body_steps_partial. fs_h_dst_sum_ext_firstpositive_body_steps_partial + S (fs_r_dst_sum_ext_firstpositive_body_steps) = S ((S (fs_i_dst_sum_ext_firstpositive_body_steps)) * fs_v_dst_sum_ext_firstpositive)) /\ exists fs_q_dst_sum_ext_firstpositive_body_steps_partial. fs_u_dst_sum_ext_firstpositive = fs_q_dst_sum_ext_firstpositive_body_steps_partial * S ((S (fs_i_dst_sum_ext_firstpositive_body_steps)) * fs_v_dst_sum_ext_firstpositive) + (fs_r_dst_sum_ext_firstpositive_body_steps))) /\ ((((exists fs_h_dst_sum_ext_firstpositive_body_steps_successor. fs_h_dst_sum_ext_firstpositive_body_steps_successor + S (fs_s_dst_sum_ext_firstpositive_body_steps) = S ((S (S fs_i_dst_sum_ext_firstpositive_body_steps)) * fs_v_dst_sum_ext_firstpositive)) /\ exists fs_q_dst_sum_ext_firstpositive_body_steps_successor. fs_u_dst_sum_ext_firstpositive = fs_q_dst_sum_ext_firstpositive_body_steps_successor * S ((S (S fs_i_dst_sum_ext_firstpositive_body_steps)) * fs_v_dst_sum_ext_firstpositive) + (fs_s_dst_sum_ext_firstpositive_body_steps))) /\ fs_s_dst_sum_ext_firstpositive_body_steps = fs_r_dst_sum_ext_firstpositive_body_steps + fs_a_dst_sum_ext_firstpositive_body_steps)))))) /\ (((exists fs_u_dst_sum_ext_firstnegative fs_v_dst_sum_ext_firstnegative. ((((exists fs_h_dst_sum_ext_firstnegative_body_start. fs_h_dst_sum_ext_firstnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_ext_firstnegative)) /\ exists fs_q_dst_sum_ext_firstnegative_body_start. fs_u_dst_sum_ext_firstnegative = fs_q_dst_sum_ext_firstnegative_body_start * S ((S (0)) * fs_v_dst_sum_ext_firstnegative) + (0))) /\ ((((exists fs_h_dst_sum_ext_firstnegative_body_terminal. fs_h_dst_sum_ext_firstnegative_body_terminal + S (dst_negative_sum_sum_ext_first) = S ((S (l)) * fs_v_dst_sum_ext_firstnegative)) /\ exists fs_q_dst_sum_ext_firstnegative_body_terminal. fs_u_dst_sum_ext_firstnegative = fs_q_dst_sum_ext_firstnegative_body_terminal * S ((S (l)) * fs_v_dst_sum_ext_firstnegative) + (dst_negative_sum_sum_ext_first))) /\ forall fs_i_dst_sum_ext_firstnegative_body_steps. (exists fs_lt_dst_sum_ext_firstnegative_body_steps_bound. fs_lt_dst_sum_ext_firstnegative_body_steps_bound + S fs_i_dst_sum_ext_firstnegative_body_steps = l) -> exists fs_a_dst_sum_ext_firstnegative_body_steps fs_r_dst_sum_ext_firstnegative_body_steps fs_s_dst_sum_ext_firstnegative_body_steps. ((((exists fs_h_dst_sum_ext_firstnegative_body_steps_summand. fs_h_dst_sum_ext_firstnegative_body_steps_summand + S (fs_a_dst_sum_ext_firstnegative_body_steps) = S ((S (fs_i_dst_sum_ext_firstnegative_body_steps)) * dst_negative_scale_sum_ext_first)) /\ exists fs_q_dst_sum_ext_firstnegative_body_steps_summand. dst_negative_code_sum_ext_first = fs_q_dst_sum_ext_firstnegative_body_steps_summand * S ((S (fs_i_dst_sum_ext_firstnegative_body_steps)) * dst_negative_scale_sum_ext_first) + (fs_a_dst_sum_ext_firstnegative_body_steps))) /\ ((((exists fs_h_dst_sum_ext_firstnegative_body_steps_partial. fs_h_dst_sum_ext_firstnegative_body_steps_partial + S (fs_r_dst_sum_ext_firstnegative_body_steps) = S ((S (fs_i_dst_sum_ext_firstnegative_body_steps)) * fs_v_dst_sum_ext_firstnegative)) /\ exists fs_q_dst_sum_ext_firstnegative_body_steps_partial. fs_u_dst_sum_ext_firstnegative = fs_q_dst_sum_ext_firstnegative_body_steps_partial * S ((S (fs_i_dst_sum_ext_firstnegative_body_steps)) * fs_v_dst_sum_ext_firstnegative) + (fs_r_dst_sum_ext_firstnegative_body_steps))) /\ ((((exists fs_h_dst_sum_ext_firstnegative_body_steps_successor. fs_h_dst_sum_ext_firstnegative_body_steps_successor + S (fs_s_dst_sum_ext_firstnegative_body_steps) = S ((S (S fs_i_dst_sum_ext_firstnegative_body_steps)) * fs_v_dst_sum_ext_firstnegative)) /\ exists fs_q_dst_sum_ext_firstnegative_body_steps_successor. fs_u_dst_sum_ext_firstnegative = fs_q_dst_sum_ext_firstnegative_body_steps_successor * S ((S (S fs_i_dst_sum_ext_firstnegative_body_steps)) * fs_v_dst_sum_ext_firstnegative) + (fs_s_dst_sum_ext_firstnegative_body_steps))) /\ fs_s_dst_sum_ext_firstnegative_body_steps = fs_r_dst_sum_ext_firstnegative_body_steps + fs_a_dst_sum_ext_firstnegative_body_steps)))))) /\ (exists ge_balance_positive_sum_ext_firstresult ge_balance_negative_sum_ext_firstresult. (((((a) = 2 * (ge_balance_positive_sum_ext_firstresult) /\ (ge_balance_negative_sum_ext_firstresult) = 0) \/ exists ge_signed_half_sum_ext_firstresultdecode. (((a) = 2 * ge_signed_half_sum_ext_firstresultdecode + 1 /\ (ge_balance_positive_sum_ext_firstresult) = 0) /\ (ge_balance_negative_sum_ext_firstresult) = S ge_signed_half_sum_ext_firstresultdecode))) /\ ((dst_positive_sum_sum_ext_first) + ge_balance_negative_sum_ext_firstresult = (dst_negative_sum_sum_ext_first) + ge_balance_positive_sum_ext_firstresult))))))))) -> (exists dst_positive_code_sum_ext_second dst_positive_scale_sum_ext_second dst_negative_code_sum_ext_second dst_negative_scale_sum_ext_second dst_positive_sum_sum_ext_second dst_negative_sum_sum_ext_second. (((G) = (((((dst_positive_code_sum_ext_second) + (dst_positive_scale_sum_ext_second)) * S ((dst_positive_code_sum_ext_second) + (dst_positive_scale_sum_ext_second)) + ((dst_positive_scale_sum_ext_second) + (dst_positive_scale_sum_ext_second))) + (((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) * S ((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) + ((dst_negative_scale_sum_ext_second) + (dst_negative_scale_sum_ext_second)))) * S ((((dst_positive_code_sum_ext_second) + (dst_positive_scale_sum_ext_second)) * S ((dst_positive_code_sum_ext_second) + (dst_positive_scale_sum_ext_second)) + ((dst_positive_scale_sum_ext_second) + (dst_positive_scale_sum_ext_second))) + (((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) * S ((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) + ((dst_negative_scale_sum_ext_second) + (dst_negative_scale_sum_ext_second)))) + ((((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) * S ((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) + ((dst_negative_scale_sum_ext_second) + (dst_negative_scale_sum_ext_second))) + (((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) * S ((dst_negative_code_sum_ext_second) + (dst_negative_scale_sum_ext_second)) + ((dst_negative_scale_sum_ext_second) + (dst_negative_scale_sum_ext_second)))))) /\ (((exists fs_u_dst_sum_ext_secondpositive fs_v_dst_sum_ext_secondpositive. ((((exists fs_h_dst_sum_ext_secondpositive_body_start. fs_h_dst_sum_ext_secondpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_ext_secondpositive)) /\ exists fs_q_dst_sum_ext_secondpositive_body_start. fs_u_dst_sum_ext_secondpositive = fs_q_dst_sum_ext_secondpositive_body_start * S ((S (0)) * fs_v_dst_sum_ext_secondpositive) + (0))) /\ ((((exists fs_h_dst_sum_ext_secondpositive_body_terminal. fs_h_dst_sum_ext_secondpositive_body_terminal + S (dst_positive_sum_sum_ext_second) = S ((S (l)) * fs_v_dst_sum_ext_secondpositive)) /\ exists fs_q_dst_sum_ext_secondpositive_body_terminal. fs_u_dst_sum_ext_secondpositive = fs_q_dst_sum_ext_secondpositive_body_terminal * S ((S (l)) * fs_v_dst_sum_ext_secondpositive) + (dst_positive_sum_sum_ext_second))) /\ forall fs_i_dst_sum_ext_secondpositive_body_steps. (exists fs_lt_dst_sum_ext_secondpositive_body_steps_bound. fs_lt_dst_sum_ext_secondpositive_body_steps_bound + S fs_i_dst_sum_ext_secondpositive_body_steps = l) -> exists fs_a_dst_sum_ext_secondpositive_body_steps fs_r_dst_sum_ext_secondpositive_body_steps fs_s_dst_sum_ext_secondpositive_body_steps. ((((exists fs_h_dst_sum_ext_secondpositive_body_steps_summand. fs_h_dst_sum_ext_secondpositive_body_steps_summand + S (fs_a_dst_sum_ext_secondpositive_body_steps) = S ((S (fs_i_dst_sum_ext_secondpositive_body_steps)) * dst_positive_scale_sum_ext_second)) /\ exists fs_q_dst_sum_ext_secondpositive_body_steps_summand. dst_positive_code_sum_ext_second = fs_q_dst_sum_ext_secondpositive_body_steps_summand * S ((S (fs_i_dst_sum_ext_secondpositive_body_steps)) * dst_positive_scale_sum_ext_second) + (fs_a_dst_sum_ext_secondpositive_body_steps))) /\ ((((exists fs_h_dst_sum_ext_secondpositive_body_steps_partial. fs_h_dst_sum_ext_secondpositive_body_steps_partial + S (fs_r_dst_sum_ext_secondpositive_body_steps) = S ((S (fs_i_dst_sum_ext_secondpositive_body_steps)) * fs_v_dst_sum_ext_secondpositive)) /\ exists fs_q_dst_sum_ext_secondpositive_body_steps_partial. fs_u_dst_sum_ext_secondpositive = fs_q_dst_sum_ext_secondpositive_body_steps_partial * S ((S (fs_i_dst_sum_ext_secondpositive_body_steps)) * fs_v_dst_sum_ext_secondpositive) + (fs_r_dst_sum_ext_secondpositive_body_steps))) /\ ((((exists fs_h_dst_sum_ext_secondpositive_body_steps_successor. fs_h_dst_sum_ext_secondpositive_body_steps_successor + S (fs_s_dst_sum_ext_secondpositive_body_steps) = S ((S (S fs_i_dst_sum_ext_secondpositive_body_steps)) * fs_v_dst_sum_ext_secondpositive)) /\ exists fs_q_dst_sum_ext_secondpositive_body_steps_successor. fs_u_dst_sum_ext_secondpositive = fs_q_dst_sum_ext_secondpositive_body_steps_successor * S ((S (S fs_i_dst_sum_ext_secondpositive_body_steps)) * fs_v_dst_sum_ext_secondpositive) + (fs_s_dst_sum_ext_secondpositive_body_steps))) /\ fs_s_dst_sum_ext_secondpositive_body_steps = fs_r_dst_sum_ext_secondpositive_body_steps + fs_a_dst_sum_ext_secondpositive_body_steps)))))) /\ (((exists fs_u_dst_sum_ext_secondnegative fs_v_dst_sum_ext_secondnegative. ((((exists fs_h_dst_sum_ext_secondnegative_body_start. fs_h_dst_sum_ext_secondnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_sum_ext_secondnegative)) /\ exists fs_q_dst_sum_ext_secondnegative_body_start. fs_u_dst_sum_ext_secondnegative = fs_q_dst_sum_ext_secondnegative_body_start * S ((S (0)) * fs_v_dst_sum_ext_secondnegative) + (0))) /\ ((((exists fs_h_dst_sum_ext_secondnegative_body_terminal. fs_h_dst_sum_ext_secondnegative_body_terminal + S (dst_negative_sum_sum_ext_second) = S ((S (l)) * fs_v_dst_sum_ext_secondnegative)) /\ exists fs_q_dst_sum_ext_secondnegative_body_terminal. fs_u_dst_sum_ext_secondnegative = fs_q_dst_sum_ext_secondnegative_body_terminal * S ((S (l)) * fs_v_dst_sum_ext_secondnegative) + (dst_negative_sum_sum_ext_second))) /\ forall fs_i_dst_sum_ext_secondnegative_body_steps. (exists fs_lt_dst_sum_ext_secondnegative_body_steps_bound. fs_lt_dst_sum_ext_secondnegative_body_steps_bound + S fs_i_dst_sum_ext_secondnegative_body_steps = l) -> exists fs_a_dst_sum_ext_secondnegative_body_steps fs_r_dst_sum_ext_secondnegative_body_steps fs_s_dst_sum_ext_secondnegative_body_steps. ((((exists fs_h_dst_sum_ext_secondnegative_body_steps_summand. fs_h_dst_sum_ext_secondnegative_body_steps_summand + S (fs_a_dst_sum_ext_secondnegative_body_steps) = S ((S (fs_i_dst_sum_ext_secondnegative_body_steps)) * dst_negative_scale_sum_ext_second)) /\ exists fs_q_dst_sum_ext_secondnegative_body_steps_summand. dst_negative_code_sum_ext_second = fs_q_dst_sum_ext_secondnegative_body_steps_summand * S ((S (fs_i_dst_sum_ext_secondnegative_body_steps)) * dst_negative_scale_sum_ext_second) + (fs_a_dst_sum_ext_secondnegative_body_steps))) /\ ((((exists fs_h_dst_sum_ext_secondnegative_body_steps_partial. fs_h_dst_sum_ext_secondnegative_body_steps_partial + S (fs_r_dst_sum_ext_secondnegative_body_steps) = S ((S (fs_i_dst_sum_ext_secondnegative_body_steps)) * fs_v_dst_sum_ext_secondnegative)) /\ exists fs_q_dst_sum_ext_secondnegative_body_steps_partial. fs_u_dst_sum_ext_secondnegative = fs_q_dst_sum_ext_secondnegative_body_steps_partial * S ((S (fs_i_dst_sum_ext_secondnegative_body_steps)) * fs_v_dst_sum_ext_secondnegative) + (fs_r_dst_sum_ext_secondnegative_body_steps))) /\ ((((exists fs_h_dst_sum_ext_secondnegative_body_steps_successor. fs_h_dst_sum_ext_secondnegative_body_steps_successor + S (fs_s_dst_sum_ext_secondnegative_body_steps) = S ((S (S fs_i_dst_sum_ext_secondnegative_body_steps)) * fs_v_dst_sum_ext_secondnegative)) /\ exists fs_q_dst_sum_ext_secondnegative_body_steps_successor. fs_u_dst_sum_ext_secondnegative = fs_q_dst_sum_ext_secondnegative_body_steps_successor * S ((S (S fs_i_dst_sum_ext_secondnegative_body_steps)) * fs_v_dst_sum_ext_secondnegative) + (fs_s_dst_sum_ext_secondnegative_body_steps))) /\ fs_s_dst_sum_ext_secondnegative_body_steps = fs_r_dst_sum_ext_secondnegative_body_steps + fs_a_dst_sum_ext_secondnegative_body_steps)))))) /\ (exists ge_balance_positive_sum_ext_secondresult ge_balance_negative_sum_ext_secondresult. (((((b) = 2 * (ge_balance_positive_sum_ext_secondresult) /\ (ge_balance_negative_sum_ext_secondresult) = 0) \/ exists ge_signed_half_sum_ext_secondresultdecode. (((b) = 2 * ge_signed_half_sum_ext_secondresultdecode + 1 /\ (ge_balance_positive_sum_ext_secondresult) = 0) /\ (ge_balance_negative_sum_ext_secondresult) = S ge_signed_half_sum_ext_secondresultdecode))) /\ ((dst_positive_sum_sum_ext_second) + ge_balance_negative_sum_ext_secondresult = (dst_negative_sum_sum_ext_second) + ge_balance_positive_sum_ext_secondresult))))))))) -> a = b

• divisor_signed_sum_successor_intro · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F l a b c. (exists dst_positive_code_step_sum dst_positive_scale_step_sum dst_negative_code_step_sum dst_negative_scale_step_sum dst_positive_sum_step_sum dst_negative_sum_step_sum. (((F) = (((((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) * S ((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) + ((dst_positive_scale_step_sum) + (dst_positive_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))) * S ((((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) * S ((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) + ((dst_positive_scale_step_sum) + (dst_positive_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))) + ((((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))))) /\ (((exists fs_u_dst_step_sumpositive fs_v_dst_step_sumpositive. ((((exists fs_h_dst_step_sumpositive_body_start. fs_h_dst_step_sumpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_start. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_start * S ((S (0)) * fs_v_dst_step_sumpositive) + (0))) /\ ((((exists fs_h_dst_step_sumpositive_body_terminal. fs_h_dst_step_sumpositive_body_terminal + S (dst_positive_sum_step_sum) = S ((S (l)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_terminal. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_terminal * S ((S (l)) * fs_v_dst_step_sumpositive) + (dst_positive_sum_step_sum))) /\ forall fs_i_dst_step_sumpositive_body_steps. (exists fs_lt_dst_step_sumpositive_body_steps_bound. fs_lt_dst_step_sumpositive_body_steps_bound + S fs_i_dst_step_sumpositive_body_steps = l) -> exists fs_a_dst_step_sumpositive_body_steps fs_r_dst_step_sumpositive_body_steps fs_s_dst_step_sumpositive_body_steps. ((((exists fs_h_dst_step_sumpositive_body_steps_summand. fs_h_dst_step_sumpositive_body_steps_summand + S (fs_a_dst_step_sumpositive_body_steps) = S ((S (fs_i_dst_step_sumpositive_body_steps)) * dst_positive_scale_step_sum)) /\ exists fs_q_dst_step_sumpositive_body_steps_summand. dst_positive_code_step_sum = fs_q_dst_step_sumpositive_body_steps_summand * S ((S (fs_i_dst_step_sumpositive_body_steps)) * dst_positive_scale_step_sum) + (fs_a_dst_step_sumpositive_body_steps))) /\ ((((exists fs_h_dst_step_sumpositive_body_steps_partial. fs_h_dst_step_sumpositive_body_steps_partial + S (fs_r_dst_step_sumpositive_body_steps) = S ((S (fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_steps_partial. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_steps_partial * S ((S (fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive) + (fs_r_dst_step_sumpositive_body_steps))) /\ ((((exists fs_h_dst_step_sumpositive_body_steps_successor. fs_h_dst_step_sumpositive_body_steps_successor + S (fs_s_dst_step_sumpositive_body_steps) = S ((S (S fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_steps_successor. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_steps_successor * S ((S (S fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive) + (fs_s_dst_step_sumpositive_body_steps))) /\ fs_s_dst_step_sumpositive_body_steps = fs_r_dst_step_sumpositive_body_steps + fs_a_dst_step_sumpositive_body_steps)))))) /\ (((exists fs_u_dst_step_sumnegative fs_v_dst_step_sumnegative. ((((exists fs_h_dst_step_sumnegative_body_start. fs_h_dst_step_sumnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_start. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_start * S ((S (0)) * fs_v_dst_step_sumnegative) + (0))) /\ ((((exists fs_h_dst_step_sumnegative_body_terminal. fs_h_dst_step_sumnegative_body_terminal + S (dst_negative_sum_step_sum) = S ((S (l)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_terminal. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_terminal * S ((S (l)) * fs_v_dst_step_sumnegative) + (dst_negative_sum_step_sum))) /\ forall fs_i_dst_step_sumnegative_body_steps. (exists fs_lt_dst_step_sumnegative_body_steps_bound. fs_lt_dst_step_sumnegative_body_steps_bound + S fs_i_dst_step_sumnegative_body_steps = l) -> exists fs_a_dst_step_sumnegative_body_steps fs_r_dst_step_sumnegative_body_steps fs_s_dst_step_sumnegative_body_steps. ((((exists fs_h_dst_step_sumnegative_body_steps_summand. fs_h_dst_step_sumnegative_body_steps_summand + S (fs_a_dst_step_sumnegative_body_steps) = S ((S (fs_i_dst_step_sumnegative_body_steps)) * dst_negative_scale_step_sum)) /\ exists fs_q_dst_step_sumnegative_body_steps_summand. dst_negative_code_step_sum = fs_q_dst_step_sumnegative_body_steps_summand * S ((S (fs_i_dst_step_sumnegative_body_steps)) * dst_negative_scale_step_sum) + (fs_a_dst_step_sumnegative_body_steps))) /\ ((((exists fs_h_dst_step_sumnegative_body_steps_partial. fs_h_dst_step_sumnegative_body_steps_partial + S (fs_r_dst_step_sumnegative_body_steps) = S ((S (fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_steps_partial. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_steps_partial * S ((S (fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative) + (fs_r_dst_step_sumnegative_body_steps))) /\ ((((exists fs_h_dst_step_sumnegative_body_steps_successor. fs_h_dst_step_sumnegative_body_steps_successor + S (fs_s_dst_step_sumnegative_body_steps) = S ((S (S fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_steps_successor. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_steps_successor * S ((S (S fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative) + (fs_s_dst_step_sumnegative_body_steps))) /\ fs_s_dst_step_sumnegative_body_steps = fs_r_dst_step_sumnegative_body_steps + fs_a_dst_step_sumnegative_body_steps)))))) /\ (exists ge_balance_positive_step_sumresult ge_balance_negative_step_sumresult. (((((a) = 2 * (ge_balance_positive_step_sumresult) /\ (ge_balance_negative_step_sumresult) = 0) \/ exists ge_signed_half_step_sumresultdecode. (((a) = 2 * ge_signed_half_step_sumresultdecode + 1 /\ (ge_balance_positive_step_sumresult) = 0) /\ (ge_balance_negative_step_sumresult) = S ge_signed_half_step_sumresultdecode))) /\ ((dst_positive_sum_step_sum) + ge_balance_negative_step_sumresult = (dst_negative_sum_step_sum) + ge_balance_positive_step_sumresult))))))))) -> (exists dst_positive_code_step_entry dst_positive_scale_step_entry dst_negative_code_step_entry dst_negative_scale_step_entry dst_positive_step_entry dst_negative_step_entry. (((F) = (((((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) * S ((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) + ((dst_positive_scale_step_entry) + (dst_positive_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))) * S ((((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) * S ((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) + ((dst_positive_scale_step_entry) + (dst_positive_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))) + ((((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))))) /\ (((((exists ff_h_pvs_step_entrypositive. ff_h_pvs_step_entrypositive + S (dst_positive_step_entry) = S ((S (l)) * dst_positive_scale_step_entry)) /\ exists ff_q_pvs_step_entrypositive. dst_positive_code_step_entry = ff_q_pvs_step_entrypositive * S ((S (l)) * dst_positive_scale_step_entry) + (dst_positive_step_entry))) /\ (((((exists ff_h_pvs_step_entrynegative. ff_h_pvs_step_entrynegative + S (dst_negative_step_entry) = S ((S (l)) * dst_negative_scale_step_entry)) /\ exists ff_q_pvs_step_entrynegative. dst_negative_code_step_entry = ff_q_pvs_step_entrynegative * S ((S (l)) * dst_negative_scale_step_entry) + (dst_negative_step_entry))) /\ (exists ge_balance_positive_step_entryvalue ge_balance_negative_step_entryvalue. (((((b) = 2 * (ge_balance_positive_step_entryvalue) /\ (ge_balance_negative_step_entryvalue) = 0) \/ exists ge_signed_half_step_entryvaluedecode. (((b) = 2 * ge_signed_half_step_entryvaluedecode + 1 /\ (ge_balance_positive_step_entryvalue) = 0) /\ (ge_balance_negative_step_entryvalue) = S ge_signed_half_step_entryvaluedecode))) /\ ((dst_positive_step_entry) + ge_balance_negative_step_entryvalue = (dst_negative_step_entry) + ge_balance_positive_step_entryvalue))))))))) -> (exists dsa_ap_step_add dsa_an_step_add dsa_bp_step_add dsa_bn_step_add dsa_cp_step_add dsa_cn_step_add. (((((a) = 2 * (dsa_ap_step_add) /\ (dsa_an_step_add) = 0) \/ exists ge_signed_half_step_addleft. (((a) = 2 * ge_signed_half_step_addleft + 1 /\ (dsa_ap_step_add) = 0) /\ (dsa_an_step_add) = S ge_signed_half_step_addleft))) /\ ((((((b) = 2 * (dsa_bp_step_add) /\ (dsa_bn_step_add) = 0) \/ exists ge_signed_half_step_addright. (((b) = 2 * ge_signed_half_step_addright + 1 /\ (dsa_bp_step_add) = 0) /\ (dsa_bn_step_add) = S ge_signed_half_step_addright))) /\ ((((((c) = 2 * (dsa_cp_step_add) /\ (dsa_cn_step_add) = 0) \/ exists ge_signed_half_step_addoutput. (((c) = 2 * ge_signed_half_step_addoutput + 1 /\ (dsa_cp_step_add) = 0) /\ (dsa_cn_step_add) = S ge_signed_half_step_addoutput))) /\ ((dsa_ap_step_add + dsa_bp_step_add) + dsa_cn_step_add = (dsa_an_step_add + dsa_bn_step_add) + dsa_cp_step_add))))))) -> (exists dst_positive_code_step_result dst_positive_scale_step_result dst_negative_code_step_result dst_negative_scale_step_result dst_positive_sum_step_result dst_negative_sum_step_result. (((F) = (((((dst_positive_code_step_result) + (dst_positive_scale_step_result)) * S ((dst_positive_code_step_result) + (dst_positive_scale_step_result)) + ((dst_positive_scale_step_result) + (dst_positive_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))) * S ((((dst_positive_code_step_result) + (dst_positive_scale_step_result)) * S ((dst_positive_code_step_result) + (dst_positive_scale_step_result)) + ((dst_positive_scale_step_result) + (dst_positive_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))) + ((((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))))) /\ (((exists fs_u_dst_step_resultpositive fs_v_dst_step_resultpositive. ((((exists fs_h_dst_step_resultpositive_body_start. fs_h_dst_step_resultpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_start. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_start * S ((S (0)) * fs_v_dst_step_resultpositive) + (0))) /\ ((((exists fs_h_dst_step_resultpositive_body_terminal. fs_h_dst_step_resultpositive_body_terminal + S (dst_positive_sum_step_result) = S ((S (S l)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_terminal. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_terminal * S ((S (S l)) * fs_v_dst_step_resultpositive) + (dst_positive_sum_step_result))) /\ forall fs_i_dst_step_resultpositive_body_steps. (exists fs_lt_dst_step_resultpositive_body_steps_bound. fs_lt_dst_step_resultpositive_body_steps_bound + S fs_i_dst_step_resultpositive_body_steps = S l) -> exists fs_a_dst_step_resultpositive_body_steps fs_r_dst_step_resultpositive_body_steps fs_s_dst_step_resultpositive_body_steps. ((((exists fs_h_dst_step_resultpositive_body_steps_summand. fs_h_dst_step_resultpositive_body_steps_summand + S (fs_a_dst_step_resultpositive_body_steps) = S ((S (fs_i_dst_step_resultpositive_body_steps)) * dst_positive_scale_step_result)) /\ exists fs_q_dst_step_resultpositive_body_steps_summand. dst_positive_code_step_result = fs_q_dst_step_resultpositive_body_steps_summand * S ((S (fs_i_dst_step_resultpositive_body_steps)) * dst_positive_scale_step_result) + (fs_a_dst_step_resultpositive_body_steps))) /\ ((((exists fs_h_dst_step_resultpositive_body_steps_partial. fs_h_dst_step_resultpositive_body_steps_partial + S (fs_r_dst_step_resultpositive_body_steps) = S ((S (fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_steps_partial. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_steps_partial * S ((S (fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive) + (fs_r_dst_step_resultpositive_body_steps))) /\ ((((exists fs_h_dst_step_resultpositive_body_steps_successor. fs_h_dst_step_resultpositive_body_steps_successor + S (fs_s_dst_step_resultpositive_body_steps) = S ((S (S fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_steps_successor. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_steps_successor * S ((S (S fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive) + (fs_s_dst_step_resultpositive_body_steps))) /\ fs_s_dst_step_resultpositive_body_steps = fs_r_dst_step_resultpositive_body_steps + fs_a_dst_step_resultpositive_body_steps)))))) /\ (((exists fs_u_dst_step_resultnegative fs_v_dst_step_resultnegative. ((((exists fs_h_dst_step_resultnegative_body_start. fs_h_dst_step_resultnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_start. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_start * S ((S (0)) * fs_v_dst_step_resultnegative) + (0))) /\ ((((exists fs_h_dst_step_resultnegative_body_terminal. fs_h_dst_step_resultnegative_body_terminal + S (dst_negative_sum_step_result) = S ((S (S l)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_terminal. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_terminal * S ((S (S l)) * fs_v_dst_step_resultnegative) + (dst_negative_sum_step_result))) /\ forall fs_i_dst_step_resultnegative_body_steps. (exists fs_lt_dst_step_resultnegative_body_steps_bound. fs_lt_dst_step_resultnegative_body_steps_bound + S fs_i_dst_step_resultnegative_body_steps = S l) -> exists fs_a_dst_step_resultnegative_body_steps fs_r_dst_step_resultnegative_body_steps fs_s_dst_step_resultnegative_body_steps. ((((exists fs_h_dst_step_resultnegative_body_steps_summand. fs_h_dst_step_resultnegative_body_steps_summand + S (fs_a_dst_step_resultnegative_body_steps) = S ((S (fs_i_dst_step_resultnegative_body_steps)) * dst_negative_scale_step_result)) /\ exists fs_q_dst_step_resultnegative_body_steps_summand. dst_negative_code_step_result = fs_q_dst_step_resultnegative_body_steps_summand * S ((S (fs_i_dst_step_resultnegative_body_steps)) * dst_negative_scale_step_result) + (fs_a_dst_step_resultnegative_body_steps))) /\ ((((exists fs_h_dst_step_resultnegative_body_steps_partial. fs_h_dst_step_resultnegative_body_steps_partial + S (fs_r_dst_step_resultnegative_body_steps) = S ((S (fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_steps_partial. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_steps_partial * S ((S (fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative) + (fs_r_dst_step_resultnegative_body_steps))) /\ ((((exists fs_h_dst_step_resultnegative_body_steps_successor. fs_h_dst_step_resultnegative_body_steps_successor + S (fs_s_dst_step_resultnegative_body_steps) = S ((S (S fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_steps_successor. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_steps_successor * S ((S (S fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative) + (fs_s_dst_step_resultnegative_body_steps))) /\ fs_s_dst_step_resultnegative_body_steps = fs_r_dst_step_resultnegative_body_steps + fs_a_dst_step_resultnegative_body_steps)))))) /\ (exists ge_balance_positive_step_resultresult ge_balance_negative_step_resultresult. (((((c) = 2 * (ge_balance_positive_step_resultresult) /\ (ge_balance_negative_step_resultresult) = 0) \/ exists ge_signed_half_step_resultresultdecode. (((c) = 2 * ge_signed_half_step_resultresultdecode + 1 /\ (ge_balance_positive_step_resultresult) = 0) /\ (ge_balance_negative_step_resultresult) = S ge_signed_half_step_resultresultdecode))) /\ ((dst_positive_sum_step_result) + ge_balance_negative_step_resultresult = (dst_negative_sum_step_result) + ge_balance_positive_step_resultresult)))))))))

• divisor_signed_table_at_from_components · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F pb pc nb nc i p n z. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> (((exists ff_h_pvs_entry_positive. ff_h_pvs_entry_positive + S (p) = S ((S (i)) * pc)) /\ exists ff_q_pvs_entry_positive. pb = ff_q_pvs_entry_positive * S ((S (i)) * pc) + (p))) -> (((exists ff_h_pvs_entry_negative. ff_h_pvs_entry_negative + S (n) = S ((S (i)) * nc)) /\ exists ff_q_pvs_entry_negative. nb = ff_q_pvs_entry_negative * S ((S (i)) * nc) + (n))) -> (exists ge_balance_positive_entry_balance ge_balance_negative_entry_balance. (((((z) = 2 * (ge_balance_positive_entry_balance) /\ (ge_balance_negative_entry_balance) = 0) \/ exists ge_signed_half_entry_balancedecode. (((z) = 2 * ge_signed_half_entry_balancedecode + 1 /\ (ge_balance_positive_entry_balance) = 0) /\ (ge_balance_negative_entry_balance) = S ge_signed_half_entry_balancedecode))) /\ ((p) + ge_balance_negative_entry_balance = (n) + ge_balance_positive_entry_balance))) -> (exists dst_positive_code_entry_result dst_positive_scale_entry_result dst_negative_code_entry_result dst_negative_scale_entry_result dst_positive_entry_result dst_negative_entry_result. (((F) = (((((dst_positive_code_entry_result) + (dst_positive_scale_entry_result)) * S ((dst_positive_code_entry_result) + (dst_positive_scale_entry_result)) + ((dst_positive_scale_entry_result) + (dst_positive_scale_entry_result))) + (((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) * S ((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) + ((dst_negative_scale_entry_result) + (dst_negative_scale_entry_result)))) * S ((((dst_positive_code_entry_result) + (dst_positive_scale_entry_result)) * S ((dst_positive_code_entry_result) + (dst_positive_scale_entry_result)) + ((dst_positive_scale_entry_result) + (dst_positive_scale_entry_result))) + (((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) * S ((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) + ((dst_negative_scale_entry_result) + (dst_negative_scale_entry_result)))) + ((((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) * S ((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) + ((dst_negative_scale_entry_result) + (dst_negative_scale_entry_result))) + (((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) * S ((dst_negative_code_entry_result) + (dst_negative_scale_entry_result)) + ((dst_negative_scale_entry_result) + (dst_negative_scale_entry_result)))))) /\ (((((exists ff_h_pvs_entry_resultpositive. ff_h_pvs_entry_resultpositive + S (dst_positive_entry_result) = S ((S (i)) * dst_positive_scale_entry_result)) /\ exists ff_q_pvs_entry_resultpositive. dst_positive_code_entry_result = ff_q_pvs_entry_resultpositive * S ((S (i)) * dst_positive_scale_entry_result) + (dst_positive_entry_result))) /\ (((((exists ff_h_pvs_entry_resultnegative. ff_h_pvs_entry_resultnegative + S (dst_negative_entry_result) = S ((S (i)) * dst_negative_scale_entry_result)) /\ exists ff_q_pvs_entry_resultnegative. dst_negative_code_entry_result = ff_q_pvs_entry_resultnegative * S ((S (i)) * dst_negative_scale_entry_result) + (dst_negative_entry_result))) /\ (exists ge_balance_positive_entry_resultvalue ge_balance_negative_entry_resultvalue. (((((z) = 2 * (ge_balance_positive_entry_resultvalue) /\ (ge_balance_negative_entry_resultvalue) = 0) \/ exists ge_signed_half_entry_resultvaluedecode. (((z) = 2 * ge_signed_half_entry_resultvaluedecode + 1 /\ (ge_balance_positive_entry_resultvalue) = 0) /\ (ge_balance_negative_entry_resultvalue) = S ge_signed_half_entry_resultvaluedecode))) /\ ((dst_positive_entry_result) + ge_balance_negative_entry_resultvalue = (dst_negative_entry_result) + ge_balance_positive_entry_resultvalue)))))))))

• divisor_signed_table_at_functional · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F i a b. (exists dst_positive_code_functional_first dst_positive_scale_functional_first dst_negative_code_functional_first dst_negative_scale_functional_first dst_positive_functional_first dst_negative_functional_first. (((F) = (((((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) * S ((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) + ((dst_positive_scale_functional_first) + (dst_positive_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))) * S ((((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) * S ((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) + ((dst_positive_scale_functional_first) + (dst_positive_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))) + ((((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))))) /\ (((((exists ff_h_pvs_functional_firstpositive. ff_h_pvs_functional_firstpositive + S (dst_positive_functional_first) = S ((S (i)) * dst_positive_scale_functional_first)) /\ exists ff_q_pvs_functional_firstpositive. dst_positive_code_functional_first = ff_q_pvs_functional_firstpositive * S ((S (i)) * dst_positive_scale_functional_first) + (dst_positive_functional_first))) /\ (((((exists ff_h_pvs_functional_firstnegative. ff_h_pvs_functional_firstnegative + S (dst_negative_functional_first) = S ((S (i)) * dst_negative_scale_functional_first)) /\ exists ff_q_pvs_functional_firstnegative. dst_negative_code_functional_first = ff_q_pvs_functional_firstnegative * S ((S (i)) * dst_negative_scale_functional_first) + (dst_negative_functional_first))) /\ (exists ge_balance_positive_functional_firstvalue ge_balance_negative_functional_firstvalue. (((((a) = 2 * (ge_balance_positive_functional_firstvalue) /\ (ge_balance_negative_functional_firstvalue) = 0) \/ exists ge_signed_half_functional_firstvaluedecode. (((a) = 2 * ge_signed_half_functional_firstvaluedecode + 1 /\ (ge_balance_positive_functional_firstvalue) = 0) /\ (ge_balance_negative_functional_firstvalue) = S ge_signed_half_functional_firstvaluedecode))) /\ ((dst_positive_functional_first) + ge_balance_negative_functional_firstvalue = (dst_negative_functional_first) + ge_balance_positive_functional_firstvalue))))))))) -> (exists dst_positive_code_functional_second dst_positive_scale_functional_second dst_negative_code_functional_second dst_negative_scale_functional_second dst_positive_functional_second dst_negative_functional_second. (((F) = (((((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) * S ((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) + ((dst_positive_scale_functional_second) + (dst_positive_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))) * S ((((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) * S ((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) + ((dst_positive_scale_functional_second) + (dst_positive_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))) + ((((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))))) /\ (((((exists ff_h_pvs_functional_secondpositive. ff_h_pvs_functional_secondpositive + S (dst_positive_functional_second) = S ((S (i)) * dst_positive_scale_functional_second)) /\ exists ff_q_pvs_functional_secondpositive. dst_positive_code_functional_second = ff_q_pvs_functional_secondpositive * S ((S (i)) * dst_positive_scale_functional_second) + (dst_positive_functional_second))) /\ (((((exists ff_h_pvs_functional_secondnegative. ff_h_pvs_functional_secondnegative + S (dst_negative_functional_second) = S ((S (i)) * dst_negative_scale_functional_second)) /\ exists ff_q_pvs_functional_secondnegative. dst_negative_code_functional_second = ff_q_pvs_functional_secondnegative * S ((S (i)) * dst_negative_scale_functional_second) + (dst_negative_functional_second))) /\ (exists ge_balance_positive_functional_secondvalue ge_balance_negative_functional_secondvalue. (((((b) = 2 * (ge_balance_positive_functional_secondvalue) /\ (ge_balance_negative_functional_secondvalue) = 0) \/ exists ge_signed_half_functional_secondvaluedecode. (((b) = 2 * ge_signed_half_functional_secondvaluedecode + 1 /\ (ge_balance_positive_functional_secondvalue) = 0) /\ (ge_balance_negative_functional_secondvalue) = S ge_signed_half_functional_secondvaluedecode))) /\ ((dst_positive_functional_second) + ge_balance_negative_functional_secondvalue = (dst_negative_functional_second) + ge_balance_positive_functional_secondvalue))))))))) -> a = b

• divisor_signed_table_at_to_components · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall F pb pc nb nc i z. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> (exists dst_positive_code_unpack_entry dst_positive_scale_unpack_entry dst_negative_code_unpack_entry dst_negative_scale_unpack_entry dst_positive_unpack_entry dst_negative_unpack_entry. (((F) = (((((dst_positive_code_unpack_entry) + (dst_positive_scale_unpack_entry)) * S ((dst_positive_code_unpack_entry) + (dst_positive_scale_unpack_entry)) + ((dst_positive_scale_unpack_entry) + (dst_positive_scale_unpack_entry))) + (((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) * S ((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) + ((dst_negative_scale_unpack_entry) + (dst_negative_scale_unpack_entry)))) * S ((((dst_positive_code_unpack_entry) + (dst_positive_scale_unpack_entry)) * S ((dst_positive_code_unpack_entry) + (dst_positive_scale_unpack_entry)) + ((dst_positive_scale_unpack_entry) + (dst_positive_scale_unpack_entry))) + (((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) * S ((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) + ((dst_negative_scale_unpack_entry) + (dst_negative_scale_unpack_entry)))) + ((((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) * S ((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) + ((dst_negative_scale_unpack_entry) + (dst_negative_scale_unpack_entry))) + (((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) * S ((dst_negative_code_unpack_entry) + (dst_negative_scale_unpack_entry)) + ((dst_negative_scale_unpack_entry) + (dst_negative_scale_unpack_entry)))))) /\ (((((exists ff_h_pvs_unpack_entrypositive. ff_h_pvs_unpack_entrypositive + S (dst_positive_unpack_entry) = S ((S (i)) * dst_positive_scale_unpack_entry)) /\ exists ff_q_pvs_unpack_entrypositive. dst_positive_code_unpack_entry = ff_q_pvs_unpack_entrypositive * S ((S (i)) * dst_positive_scale_unpack_entry) + (dst_positive_unpack_entry))) /\ (((((exists ff_h_pvs_unpack_entrynegative. ff_h_pvs_unpack_entrynegative + S (dst_negative_unpack_entry) = S ((S (i)) * dst_negative_scale_unpack_entry)) /\ exists ff_q_pvs_unpack_entrynegative. dst_negative_code_unpack_entry = ff_q_pvs_unpack_entrynegative * S ((S (i)) * dst_negative_scale_unpack_entry) + (dst_negative_unpack_entry))) /\ (exists ge_balance_positive_unpack_entryvalue ge_balance_negative_unpack_entryvalue. (((((z) = 2 * (ge_balance_positive_unpack_entryvalue) /\ (ge_balance_negative_unpack_entryvalue) = 0) \/ exists ge_signed_half_unpack_entryvaluedecode. (((z) = 2 * ge_signed_half_unpack_entryvaluedecode + 1 /\ (ge_balance_positive_unpack_entryvalue) = 0) /\ (ge_balance_negative_unpack_entryvalue) = S ge_signed_half_unpack_entryvaluedecode))) /\ ((dst_positive_unpack_entry) + ge_balance_negative_unpack_entryvalue = (dst_negative_unpack_entry) + ge_balance_positive_unpack_entryvalue))))))))) -> exists p n. (((((exists ff_h_pvs_unpack_resultpositive. ff_h_pvs_unpack_resultpositive + S (p) = S ((S (i)) * pc)) /\ exists ff_q_pvs_unpack_resultpositive. pb = ff_q_pvs_unpack_resultpositive * S ((S (i)) * pc) + (p))) /\ (((((exists ff_h_pvs_unpack_resultnegative. ff_h_pvs_unpack_resultnegative + S (n) = S ((S (i)) * nc)) /\ exists ff_q_pvs_unpack_resultnegative. nb = ff_q_pvs_unpack_resultnegative * S ((S (i)) * nc) + (n))) /\ (exists ge_balance_positive_unpack_resultvalue ge_balance_negative_unpack_resultvalue. (((((z) = 2 * (ge_balance_positive_unpack_resultvalue) /\ (ge_balance_negative_unpack_resultvalue) = 0) \/ exists ge_signed_half_unpack_resultvaluedecode. (((z) = 2 * ge_signed_half_unpack_resultvaluedecode + 1 /\ (ge_balance_positive_unpack_resultvalue) = 0) /\ (ge_balance_negative_unpack_resultvalue) = S ge_signed_half_unpack_resultvaluedecode))) /\ ((p) + ge_balance_negative_unpack_resultvalue = (n) + ge_balance_positive_unpack_resultvalue)))))))

• divisor_signed_table_components · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall N F. (exists dst_positive_code_extract_table dst_positive_scale_extract_table dst_negative_code_extract_table dst_negative_scale_extract_table. (((F) = (((((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) * S ((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) + ((dst_positive_scale_extract_table) + (dst_positive_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))) * S ((((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) * S ((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) + ((dst_positive_scale_extract_table) + (dst_positive_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))) + ((((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))))) /\ (forall dst_index_extract_table. (exists pvs_le_gap_extract_tabledomain. pvs_le_gap_extract_tabledomain + (dst_index_extract_table) = (N)) -> exists dst_positive_extract_table dst_negative_extract_table dst_value_extract_table. ((((exists ff_h_pvs_extract_tableentrypositive. ff_h_pvs_extract_tableentrypositive + S (dst_positive_extract_table) = S ((S (dst_index_extract_table)) * dst_positive_scale_extract_table)) /\ exists ff_q_pvs_extract_tableentrypositive. dst_positive_code_extract_table = ff_q_pvs_extract_tableentrypositive * S ((S (dst_index_extract_table)) * dst_positive_scale_extract_table) + (dst_positive_extract_table))) /\ (((((exists ff_h_pvs_extract_tableentrynegative. ff_h_pvs_extract_tableentrynegative + S (dst_negative_extract_table) = S ((S (dst_index_extract_table)) * dst_negative_scale_extract_table)) /\ exists ff_q_pvs_extract_tableentrynegative. dst_negative_code_extract_table = ff_q_pvs_extract_tableentrynegative * S ((S (dst_index_extract_table)) * dst_negative_scale_extract_table) + (dst_negative_extract_table))) /\ (exists ge_balance_positive_extract_tableentryvalue ge_balance_negative_extract_tableentryvalue. (((((dst_value_extract_table) = 2 * (ge_balance_positive_extract_tableentryvalue) /\ (ge_balance_negative_extract_tableentryvalue) = 0) \/ exists ge_signed_half_extract_tableentryvaluedecode. (((dst_value_extract_table) = 2 * ge_signed_half_extract_tableentryvaluedecode + 1 /\ (ge_balance_positive_extract_tableentryvalue) = 0) /\ (ge_balance_negative_extract_tableentryvalue) = S ge_signed_half_extract_tableentryvaluedecode))) /\ ((dst_positive_extract_table) + ge_balance_negative_extract_tableentryvalue = (dst_negative_extract_table) + ge_balance_positive_extract_tableentryvalue))))))))) -> exists pb pc nb nc. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))))))

• divisor_signed_table_from_components · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall N F pb pc nb nc. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> (exists dst_positive_code_constructor_table dst_positive_scale_constructor_table dst_negative_code_constructor_table dst_negative_scale_constructor_table. (((F) = (((((dst_positive_code_constructor_table) + (dst_positive_scale_constructor_table)) * S ((dst_positive_code_constructor_table) + (dst_positive_scale_constructor_table)) + ((dst_positive_scale_constructor_table) + (dst_positive_scale_constructor_table))) + (((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) * S ((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) + ((dst_negative_scale_constructor_table) + (dst_negative_scale_constructor_table)))) * S ((((dst_positive_code_constructor_table) + (dst_positive_scale_constructor_table)) * S ((dst_positive_code_constructor_table) + (dst_positive_scale_constructor_table)) + ((dst_positive_scale_constructor_table) + (dst_positive_scale_constructor_table))) + (((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) * S ((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) + ((dst_negative_scale_constructor_table) + (dst_negative_scale_constructor_table)))) + ((((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) * S ((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) + ((dst_negative_scale_constructor_table) + (dst_negative_scale_constructor_table))) + (((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) * S ((dst_negative_code_constructor_table) + (dst_negative_scale_constructor_table)) + ((dst_negative_scale_constructor_table) + (dst_negative_scale_constructor_table)))))) /\ (forall dst_index_constructor_table. (exists pvs_le_gap_constructor_tabledomain. pvs_le_gap_constructor_tabledomain + (dst_index_constructor_table) = (N)) -> exists dst_positive_constructor_table dst_negative_constructor_table dst_value_constructor_table. ((((exists ff_h_pvs_constructor_tableentrypositive. ff_h_pvs_constructor_tableentrypositive + S (dst_positive_constructor_table) = S ((S (dst_index_constructor_table)) * dst_positive_scale_constructor_table)) /\ exists ff_q_pvs_constructor_tableentrypositive. dst_positive_code_constructor_table = ff_q_pvs_constructor_tableentrypositive * S ((S (dst_index_constructor_table)) * dst_positive_scale_constructor_table) + (dst_positive_constructor_table))) /\ (((((exists ff_h_pvs_constructor_tableentrynegative. ff_h_pvs_constructor_tableentrynegative + S (dst_negative_constructor_table) = S ((S (dst_index_constructor_table)) * dst_negative_scale_constructor_table)) /\ exists ff_q_pvs_constructor_tableentrynegative. dst_negative_code_constructor_table = ff_q_pvs_constructor_tableentrynegative * S ((S (dst_index_constructor_table)) * dst_negative_scale_constructor_table) + (dst_negative_constructor_table))) /\ (exists ge_balance_positive_constructor_tableentryvalue ge_balance_negative_constructor_tableentryvalue. (((((dst_value_constructor_table) = 2 * (ge_balance_positive_constructor_tableentryvalue) /\ (ge_balance_negative_constructor_tableentryvalue) = 0) \/ exists ge_signed_half_constructor_tableentryvaluedecode. (((dst_value_constructor_table) = 2 * ge_signed_half_constructor_tableentryvaluedecode + 1 /\ (ge_balance_positive_constructor_tableentryvalue) = 0) /\ (ge_balance_negative_constructor_tableentryvalue) = S ge_signed_half_constructor_tableentryvaluedecode))) /\ ((dst_positive_constructor_table) + ge_balance_negative_constructor_tableentryvalue = (dst_negative_constructor_table) + ge_balance_positive_constructor_tableentryvalue)))))))))

• divisor_signed_table_lookup · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall N F i. (exists dst_positive_code_lookup_table dst_positive_scale_lookup_table dst_negative_code_lookup_table dst_negative_scale_lookup_table. (((F) = (((((dst_positive_code_lookup_table) + (dst_positive_scale_lookup_table)) * S ((dst_positive_code_lookup_table) + (dst_positive_scale_lookup_table)) + ((dst_positive_scale_lookup_table) + (dst_positive_scale_lookup_table))) + (((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) * S ((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) + ((dst_negative_scale_lookup_table) + (dst_negative_scale_lookup_table)))) * S ((((dst_positive_code_lookup_table) + (dst_positive_scale_lookup_table)) * S ((dst_positive_code_lookup_table) + (dst_positive_scale_lookup_table)) + ((dst_positive_scale_lookup_table) + (dst_positive_scale_lookup_table))) + (((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) * S ((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) + ((dst_negative_scale_lookup_table) + (dst_negative_scale_lookup_table)))) + ((((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) * S ((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) + ((dst_negative_scale_lookup_table) + (dst_negative_scale_lookup_table))) + (((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) * S ((dst_negative_code_lookup_table) + (dst_negative_scale_lookup_table)) + ((dst_negative_scale_lookup_table) + (dst_negative_scale_lookup_table)))))) /\ (forall dst_index_lookup_table. (exists pvs_le_gap_lookup_tabledomain. pvs_le_gap_lookup_tabledomain + (dst_index_lookup_table) = (N)) -> exists dst_positive_lookup_table dst_negative_lookup_table dst_value_lookup_table. ((((exists ff_h_pvs_lookup_tableentrypositive. ff_h_pvs_lookup_tableentrypositive + S (dst_positive_lookup_table) = S ((S (dst_index_lookup_table)) * dst_positive_scale_lookup_table)) /\ exists ff_q_pvs_lookup_tableentrypositive. dst_positive_code_lookup_table = ff_q_pvs_lookup_tableentrypositive * S ((S (dst_index_lookup_table)) * dst_positive_scale_lookup_table) + (dst_positive_lookup_table))) /\ (((((exists ff_h_pvs_lookup_tableentrynegative. ff_h_pvs_lookup_tableentrynegative + S (dst_negative_lookup_table) = S ((S (dst_index_lookup_table)) * dst_negative_scale_lookup_table)) /\ exists ff_q_pvs_lookup_tableentrynegative. dst_negative_code_lookup_table = ff_q_pvs_lookup_tableentrynegative * S ((S (dst_index_lookup_table)) * dst_negative_scale_lookup_table) + (dst_negative_lookup_table))) /\ (exists ge_balance_positive_lookup_tableentryvalue ge_balance_negative_lookup_tableentryvalue. (((((dst_value_lookup_table) = 2 * (ge_balance_positive_lookup_tableentryvalue) /\ (ge_balance_negative_lookup_tableentryvalue) = 0) \/ exists ge_signed_half_lookup_tableentryvaluedecode. (((dst_value_lookup_table) = 2 * ge_signed_half_lookup_tableentryvaluedecode + 1 /\ (ge_balance_positive_lookup_tableentryvalue) = 0) /\ (ge_balance_negative_lookup_tableentryvalue) = S ge_signed_half_lookup_tableentryvaluedecode))) /\ ((dst_positive_lookup_table) + ge_balance_negative_lookup_tableentryvalue = (dst_negative_lookup_table) + ge_balance_positive_lookup_tableentryvalue))))))))) -> (exists pvs_le_gap_lookup_domain. pvs_le_gap_lookup_domain + (i) = (N)) -> exists z. (exists dst_positive_code_lookup_result dst_positive_scale_lookup_result dst_negative_code_lookup_result dst_negative_scale_lookup_result dst_positive_lookup_result dst_negative_lookup_result. (((F) = (((((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) * S ((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) + ((dst_positive_scale_lookup_result) + (dst_positive_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))) * S ((((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) * S ((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) + ((dst_positive_scale_lookup_result) + (dst_positive_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))) + ((((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))))) /\ (((((exists ff_h_pvs_lookup_resultpositive. ff_h_pvs_lookup_resultpositive + S (dst_positive_lookup_result) = S ((S (i)) * dst_positive_scale_lookup_result)) /\ exists ff_q_pvs_lookup_resultpositive. dst_positive_code_lookup_result = ff_q_pvs_lookup_resultpositive * S ((S (i)) * dst_positive_scale_lookup_result) + (dst_positive_lookup_result))) /\ (((((exists ff_h_pvs_lookup_resultnegative. ff_h_pvs_lookup_resultnegative + S (dst_negative_lookup_result) = S ((S (i)) * dst_negative_scale_lookup_result)) /\ exists ff_q_pvs_lookup_resultnegative. dst_negative_code_lookup_result = ff_q_pvs_lookup_resultnegative * S ((S (i)) * dst_negative_scale_lookup_result) + (dst_negative_lookup_result))) /\ (exists ge_balance_positive_lookup_resultvalue ge_balance_negative_lookup_resultvalue. (((((z) = 2 * (ge_balance_positive_lookup_resultvalue) /\ (ge_balance_negative_lookup_resultvalue) = 0) \/ exists ge_signed_half_lookup_resultvaluedecode. (((z) = 2 * ge_signed_half_lookup_resultvaluedecode + 1 /\ (ge_balance_positive_lookup_resultvalue) = 0) /\ (ge_balance_negative_lookup_resultvalue) = S ge_signed_half_lookup_resultvaluedecode))) /\ ((dst_positive_lookup_result) + ge_balance_negative_lookup_resultvalue = (dst_negative_lookup_result) + ge_balance_positive_lookup_resultvalue)))))))))

• divisor_signed_table_restrict · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall N K F. (exists dst_positive_code_restriction_source dst_positive_scale_restriction_source dst_negative_code_restriction_source dst_negative_scale_restriction_source. (((F) = (((((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) * S ((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) + ((dst_positive_scale_restriction_source) + (dst_positive_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))) * S ((((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) * S ((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) + ((dst_positive_scale_restriction_source) + (dst_positive_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))) + ((((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))))) /\ (forall dst_index_restriction_source. (exists pvs_le_gap_restriction_sourcedomain. pvs_le_gap_restriction_sourcedomain + (dst_index_restriction_source) = (N)) -> exists dst_positive_restriction_source dst_negative_restriction_source dst_value_restriction_source. ((((exists ff_h_pvs_restriction_sourceentrypositive. ff_h_pvs_restriction_sourceentrypositive + S (dst_positive_restriction_source) = S ((S (dst_index_restriction_source)) * dst_positive_scale_restriction_source)) /\ exists ff_q_pvs_restriction_sourceentrypositive. dst_positive_code_restriction_source = ff_q_pvs_restriction_sourceentrypositive * S ((S (dst_index_restriction_source)) * dst_positive_scale_restriction_source) + (dst_positive_restriction_source))) /\ (((((exists ff_h_pvs_restriction_sourceentrynegative. ff_h_pvs_restriction_sourceentrynegative + S (dst_negative_restriction_source) = S ((S (dst_index_restriction_source)) * dst_negative_scale_restriction_source)) /\ exists ff_q_pvs_restriction_sourceentrynegative. dst_negative_code_restriction_source = ff_q_pvs_restriction_sourceentrynegative * S ((S (dst_index_restriction_source)) * dst_negative_scale_restriction_source) + (dst_negative_restriction_source))) /\ (exists ge_balance_positive_restriction_sourceentryvalue ge_balance_negative_restriction_sourceentryvalue. (((((dst_value_restriction_source) = 2 * (ge_balance_positive_restriction_sourceentryvalue) /\ (ge_balance_negative_restriction_sourceentryvalue) = 0) \/ exists ge_signed_half_restriction_sourceentryvaluedecode. (((dst_value_restriction_source) = 2 * ge_signed_half_restriction_sourceentryvaluedecode + 1 /\ (ge_balance_positive_restriction_sourceentryvalue) = 0) /\ (ge_balance_negative_restriction_sourceentryvalue) = S ge_signed_half_restriction_sourceentryvaluedecode))) /\ ((dst_positive_restriction_source) + ge_balance_negative_restriction_sourceentryvalue = (dst_negative_restriction_source) + ge_balance_positive_restriction_sourceentryvalue))))))))) -> (exists pvs_le_gap_restriction_bound. pvs_le_gap_restriction_bound + (K) = (N)) -> (exists dst_positive_code_restriction_target dst_positive_scale_restriction_target dst_negative_code_restriction_target dst_negative_scale_restriction_target. (((F) = (((((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) * S ((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) + ((dst_positive_scale_restriction_target) + (dst_positive_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))) * S ((((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) * S ((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) + ((dst_positive_scale_restriction_target) + (dst_positive_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))) + ((((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))))) /\ (forall dst_index_restriction_target. (exists pvs_le_gap_restriction_targetdomain. pvs_le_gap_restriction_targetdomain + (dst_index_restriction_target) = (K)) -> exists dst_positive_restriction_target dst_negative_restriction_target dst_value_restriction_target. ((((exists ff_h_pvs_restriction_targetentrypositive. ff_h_pvs_restriction_targetentrypositive + S (dst_positive_restriction_target) = S ((S (dst_index_restriction_target)) * dst_positive_scale_restriction_target)) /\ exists ff_q_pvs_restriction_targetentrypositive. dst_positive_code_restriction_target = ff_q_pvs_restriction_targetentrypositive * S ((S (dst_index_restriction_target)) * dst_positive_scale_restriction_target) + (dst_positive_restriction_target))) /\ (((((exists ff_h_pvs_restriction_targetentrynegative. ff_h_pvs_restriction_targetentrynegative + S (dst_negative_restriction_target) = S ((S (dst_index_restriction_target)) * dst_negative_scale_restriction_target)) /\ exists ff_q_pvs_restriction_targetentrynegative. dst_negative_code_restriction_target = ff_q_pvs_restriction_targetentrynegative * S ((S (dst_index_restriction_target)) * dst_negative_scale_restriction_target) + (dst_negative_restriction_target))) /\ (exists ge_balance_positive_restriction_targetentryvalue ge_balance_negative_restriction_targetentryvalue. (((((dst_value_restriction_target) = 2 * (ge_balance_positive_restriction_targetentryvalue) /\ (ge_balance_negative_restriction_targetentryvalue) = 0) \/ exists ge_signed_half_restriction_targetentryvaluedecode. (((dst_value_restriction_target) = 2 * ge_signed_half_restriction_targetentryvaluedecode + 1 /\ (ge_balance_positive_restriction_targetentryvalue) = 0) /\ (ge_balance_negative_restriction_targetentryvalue) = S ge_signed_half_restriction_targetentryvaluedecode))) /\ ((dst_positive_restriction_target) + ge_balance_negative_restriction_targetentryvalue = (dst_negative_restriction_target) + ge_balance_positive_restriction_targetentryvalue)))))))))

• eq_decidable · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall a b. a = b \/ ~(a = b)

• le_eq_or_lt · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall a b. (exists k. k + a = b) -> a = b \/ exists k. k + S a = b

• le_of_succ_le_succ · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall a b. (exists k. k + S a = S b) -> exists r. r + a = b

• le_succ_self · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. n <= S n

• le_trans · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n m k. n <= m -> m <= k -> n <= k

• le_zero · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. n <= 0 -> n = 0

• mobius_one · checked inherited prerequisite
  Exact statement in the checked dependency cone

  ((~((1) = 0)) /\ ((((exists mv_square_prime_one_valuesquare. ((~((mv_square_prime_one_valuesquare) = 1) /\ forall pvs_left_one_valuesquareprime pvs_right_one_valuesquareprime. (mv_square_prime_one_valuesquare) = pvs_left_one_valuesquareprime * pvs_right_one_valuesquareprime -> pvs_left_one_valuesquareprime = 1 \/ pvs_right_one_valuesquareprime = 1) /\ (exists pvs_factor_one_valuesquaredivisor. (1) = (mv_square_prime_one_valuesquare * mv_square_prime_one_valuesquare) * pvs_factor_one_valuesquaredivisor))) /\ ((2) = 0))) \/ (((((~((1) = 0)) /\ (forall sfd_prime_one_valuesquarefree. (~((sfd_prime_one_valuesquarefree) = 1) /\ forall pvs_left_one_valuesquarefreedomain pvs_right_one_valuesquarefreedomain. (sfd_prime_one_valuesquarefree) = pvs_left_one_valuesquarefreedomain * pvs_right_one_valuesquarefreedomain -> pvs_left_one_valuesquarefreedomain = 1 \/ pvs_right_one_valuesquarefreedomain = 1) -> (exists pvs_le_gap_one_valuesquarefreebound. pvs_le_gap_one_valuesquarefreebound + (sfd_prime_one_valuesquarefree) = (1)) -> ~(exists pvs_factor_one_valuesquarefreesquare. (1) = (sfd_prime_one_valuesquarefree * sfd_prime_one_valuesquarefree) * pvs_factor_one_valuesquarefreesquare)))) /\ (exists mv_factor_code_one_valuefactors mv_factor_scale_one_valuefactors mv_factor_count_one_valuefactors. (((~(1 = 0) /\ ((exists ff_u_fsat_one_valuefactorsfactorization_product ff_v_fsat_one_valuefactorsfactorization_product. ((((exists ff_h_fsat_one_valuefactorsfactorization_product_start. ff_h_fsat_one_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_start. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_one_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_terminal. ff_h_fsat_one_valuefactorsfactorization_product_terminal + S (1) = S ((S (mv_factor_count_one_valuefactors)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_terminal. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_one_valuefactors)) * ff_v_fsat_one_valuefactorsfactorization_product) + (1))) /\ forall ff_i_fsat_one_valuefactorsfactorization_product. (exists ff_lt_fsat_one_valuefactorsfactorization_product_bound. ff_lt_fsat_one_valuefactorsfactorization_product_bound + S ff_i_fsat_one_valuefactorsfactorization_product = mv_factor_count_one_valuefactors) -> exists ff_p_fsat_one_valuefactorsfactorization_product ff_r_fsat_one_valuefactorsfactorization_product ff_s_fsat_one_valuefactorsfactorization_product. ((((exists ff_h_fsat_one_valuefactorsfactorization_product_factor. ff_h_fsat_one_valuefactorsfactorization_product_factor + S (ff_p_fsat_one_valuefactorsfactorization_product) = S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * mv_factor_scale_one_valuefactors)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_factor. mv_factor_code_one_valuefactors = ff_q_fsat_one_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * mv_factor_scale_one_valuefactors) + (ff_p_fsat_one_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_partial. ff_h_fsat_one_valuefactorsfactorization_product_partial + S (ff_r_fsat_one_valuefactorsfactorization_product) = S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_partial. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product) + (ff_r_fsat_one_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_successor. ff_h_fsat_one_valuefactorsfactorization_product_successor + S (ff_s_fsat_one_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_successor. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product) + (ff_s_fsat_one_valuefactorsfactorization_product))) /\ ff_s_fsat_one_valuefactorsfactorization_product = ff_r_fsat_one_valuefactorsfactorization_product * ff_p_fsat_one_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_one_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_one_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_one_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_one_valuefactorsfactorization_primes = (mv_factor_count_one_valuefactors)) -> exists ftsf_factor_fsat_one_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_one_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_one_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_one_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_one_valuefactorsfactorization_primes)) * mv_factor_scale_one_valuefactors)) /\ exists ff_q_ftsf_fsat_one_valuefactorsfactorization_primes_entry. mv_factor_code_one_valuefactors = ff_q_ftsf_fsat_one_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_one_valuefactorsfactorization_primes)) * mv_factor_scale_one_valuefactors) + (ftsf_factor_fsat_one_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_one_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_one_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_one_valuefactorsparityeven. (mv_factor_count_one_valuefactors) = 2 * mv_even_half_one_valuefactorsparityeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_one_valuefactorsparityodd. (mv_factor_count_one_valuefactors) = 2 * mv_odd_half_one_valuefactorsparityodd + 1) /\ ((2) = 1))))))))))

• mobius_value_exists · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. ~(n = 0) -> exists z. (((~((n) = 0)) /\ ((((exists mv_square_prime_exists_valuesquare. ((~((mv_square_prime_exists_valuesquare) = 1) /\ forall pvs_left_exists_valuesquareprime pvs_right_exists_valuesquareprime. (mv_square_prime_exists_valuesquare) = pvs_left_exists_valuesquareprime * pvs_right_exists_valuesquareprime -> pvs_left_exists_valuesquareprime = 1 \/ pvs_right_exists_valuesquareprime = 1) /\ (exists pvs_factor_exists_valuesquaredivisor. (n) = (mv_square_prime_exists_valuesquare * mv_square_prime_exists_valuesquare) * pvs_factor_exists_valuesquaredivisor))) /\ ((z) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_exists_valuesquarefree. (~((sfd_prime_exists_valuesquarefree) = 1) /\ forall pvs_left_exists_valuesquarefreedomain pvs_right_exists_valuesquarefreedomain. (sfd_prime_exists_valuesquarefree) = pvs_left_exists_valuesquarefreedomain * pvs_right_exists_valuesquarefreedomain -> pvs_left_exists_valuesquarefreedomain = 1 \/ pvs_right_exists_valuesquarefreedomain = 1) -> (exists pvs_le_gap_exists_valuesquarefreebound. pvs_le_gap_exists_valuesquarefreebound + (sfd_prime_exists_valuesquarefree) = (n)) -> ~(exists pvs_factor_exists_valuesquarefreesquare. (n) = (sfd_prime_exists_valuesquarefree * sfd_prime_exists_valuesquarefree) * pvs_factor_exists_valuesquarefreesquare)))) /\ (exists mv_factor_code_exists_valuefactors mv_factor_scale_exists_valuefactors mv_factor_count_exists_valuefactors. (((~(n = 0) /\ ((exists ff_u_fsat_exists_valuefactorsfactorization_product ff_v_fsat_exists_valuefactorsfactorization_product. ((((exists ff_h_fsat_exists_valuefactorsfactorization_product_start. ff_h_fsat_exists_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_exists_valuefactorsfactorization_product)) /\ exists ff_q_fsat_exists_valuefactorsfactorization_product_start. ff_u_fsat_exists_valuefactorsfactorization_product = ff_q_fsat_exists_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_exists_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_exists_valuefactorsfactorization_product_terminal. ff_h_fsat_exists_valuefactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_exists_valuefactors)) * ff_v_fsat_exists_valuefactorsfactorization_product)) /\ exists ff_q_fsat_exists_valuefactorsfactorization_product_terminal. ff_u_fsat_exists_valuefactorsfactorization_product = ff_q_fsat_exists_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_exists_valuefactors)) * ff_v_fsat_exists_valuefactorsfactorization_product) + (n))) /\ forall ff_i_fsat_exists_valuefactorsfactorization_product. (exists ff_lt_fsat_exists_valuefactorsfactorization_product_bound. ff_lt_fsat_exists_valuefactorsfactorization_product_bound + S ff_i_fsat_exists_valuefactorsfactorization_product = mv_factor_count_exists_valuefactors) -> exists ff_p_fsat_exists_valuefactorsfactorization_product ff_r_fsat_exists_valuefactorsfactorization_product ff_s_fsat_exists_valuefactorsfactorization_product. ((((exists ff_h_fsat_exists_valuefactorsfactorization_product_factor. ff_h_fsat_exists_valuefactorsfactorization_product_factor + S (ff_p_fsat_exists_valuefactorsfactorization_product) = S ((S (ff_i_fsat_exists_valuefactorsfactorization_product)) * mv_factor_scale_exists_valuefactors)) /\ exists ff_q_fsat_exists_valuefactorsfactorization_product_factor. mv_factor_code_exists_valuefactors = ff_q_fsat_exists_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_exists_valuefactorsfactorization_product)) * mv_factor_scale_exists_valuefactors) + (ff_p_fsat_exists_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_exists_valuefactorsfactorization_product_partial. ff_h_fsat_exists_valuefactorsfactorization_product_partial + S (ff_r_fsat_exists_valuefactorsfactorization_product) = S ((S (ff_i_fsat_exists_valuefactorsfactorization_product)) * ff_v_fsat_exists_valuefactorsfactorization_product)) /\ exists ff_q_fsat_exists_valuefactorsfactorization_product_partial. ff_u_fsat_exists_valuefactorsfactorization_product = ff_q_fsat_exists_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_exists_valuefactorsfactorization_product)) * ff_v_fsat_exists_valuefactorsfactorization_product) + (ff_r_fsat_exists_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_exists_valuefactorsfactorization_product_successor. ff_h_fsat_exists_valuefactorsfactorization_product_successor + S (ff_s_fsat_exists_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_exists_valuefactorsfactorization_product)) * ff_v_fsat_exists_valuefactorsfactorization_product)) /\ exists ff_q_fsat_exists_valuefactorsfactorization_product_successor. ff_u_fsat_exists_valuefactorsfactorization_product = ff_q_fsat_exists_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_exists_valuefactorsfactorization_product)) * ff_v_fsat_exists_valuefactorsfactorization_product) + (ff_s_fsat_exists_valuefactorsfactorization_product))) /\ ff_s_fsat_exists_valuefactorsfactorization_product = ff_r_fsat_exists_valuefactorsfactorization_product * ff_p_fsat_exists_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_exists_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_exists_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_exists_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_exists_valuefactorsfactorization_primes = (mv_factor_count_exists_valuefactors)) -> exists ftsf_factor_fsat_exists_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_exists_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_exists_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_exists_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_exists_valuefactorsfactorization_primes)) * mv_factor_scale_exists_valuefactors)) /\ exists ff_q_ftsf_fsat_exists_valuefactorsfactorization_primes_entry. mv_factor_code_exists_valuefactors = ff_q_ftsf_fsat_exists_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_exists_valuefactorsfactorization_primes)) * mv_factor_scale_exists_valuefactors) + (ftsf_factor_fsat_exists_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_exists_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_exists_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_exists_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_exists_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_exists_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_exists_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_exists_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_exists_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_exists_valuefactorsparityeven. (mv_factor_count_exists_valuefactors) = 2 * mv_even_half_exists_valuefactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_exists_valuefactorsparityodd. (mv_factor_count_exists_valuefactors) = 2 * mv_odd_half_exists_valuefactorsparityodd + 1) /\ ((z) = 1)))))))))))

• mobius_value_functional · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n a b. (((~((n) = 0)) /\ ((((exists mv_square_prime_functional_firstsquare. ((~((mv_square_prime_functional_firstsquare) = 1) /\ forall pvs_left_functional_firstsquareprime pvs_right_functional_firstsquareprime. (mv_square_prime_functional_firstsquare) = pvs_left_functional_firstsquareprime * pvs_right_functional_firstsquareprime -> pvs_left_functional_firstsquareprime = 1 \/ pvs_right_functional_firstsquareprime = 1) /\ (exists pvs_factor_functional_firstsquaredivisor. (n) = (mv_square_prime_functional_firstsquare * mv_square_prime_functional_firstsquare) * pvs_factor_functional_firstsquaredivisor))) /\ ((a) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_functional_firstsquarefree. (~((sfd_prime_functional_firstsquarefree) = 1) /\ forall pvs_left_functional_firstsquarefreedomain pvs_right_functional_firstsquarefreedomain. (sfd_prime_functional_firstsquarefree) = pvs_left_functional_firstsquarefreedomain * pvs_right_functional_firstsquarefreedomain -> pvs_left_functional_firstsquarefreedomain = 1 \/ pvs_right_functional_firstsquarefreedomain = 1) -> (exists pvs_le_gap_functional_firstsquarefreebound. pvs_le_gap_functional_firstsquarefreebound + (sfd_prime_functional_firstsquarefree) = (n)) -> ~(exists pvs_factor_functional_firstsquarefreesquare. (n) = (sfd_prime_functional_firstsquarefree * sfd_prime_functional_firstsquarefree) * pvs_factor_functional_firstsquarefreesquare)))) /\ (exists mv_factor_code_functional_firstfactors mv_factor_scale_functional_firstfactors mv_factor_count_functional_firstfactors. (((~(n = 0) /\ ((exists ff_u_fsat_functional_firstfactorsfactorization_product ff_v_fsat_functional_firstfactorsfactorization_product. ((((exists ff_h_fsat_functional_firstfactorsfactorization_product_start. ff_h_fsat_functional_firstfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_functional_firstfactorsfactorization_product)) /\ exists ff_q_fsat_functional_firstfactorsfactorization_product_start. ff_u_fsat_functional_firstfactorsfactorization_product = ff_q_fsat_functional_firstfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_functional_firstfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_functional_firstfactorsfactorization_product_terminal. ff_h_fsat_functional_firstfactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_functional_firstfactors)) * ff_v_fsat_functional_firstfactorsfactorization_product)) /\ exists ff_q_fsat_functional_firstfactorsfactorization_product_terminal. ff_u_fsat_functional_firstfactorsfactorization_product = ff_q_fsat_functional_firstfactorsfactorization_product_terminal * S ((S (mv_factor_count_functional_firstfactors)) * ff_v_fsat_functional_firstfactorsfactorization_product) + (n))) /\ forall ff_i_fsat_functional_firstfactorsfactorization_product. (exists ff_lt_fsat_functional_firstfactorsfactorization_product_bound. ff_lt_fsat_functional_firstfactorsfactorization_product_bound + S ff_i_fsat_functional_firstfactorsfactorization_product = mv_factor_count_functional_firstfactors) -> exists ff_p_fsat_functional_firstfactorsfactorization_product ff_r_fsat_functional_firstfactorsfactorization_product ff_s_fsat_functional_firstfactorsfactorization_product. ((((exists ff_h_fsat_functional_firstfactorsfactorization_product_factor. ff_h_fsat_functional_firstfactorsfactorization_product_factor + S (ff_p_fsat_functional_firstfactorsfactorization_product) = S ((S (ff_i_fsat_functional_firstfactorsfactorization_product)) * mv_factor_scale_functional_firstfactors)) /\ exists ff_q_fsat_functional_firstfactorsfactorization_product_factor. mv_factor_code_functional_firstfactors = ff_q_fsat_functional_firstfactorsfactorization_product_factor * S ((S (ff_i_fsat_functional_firstfactorsfactorization_product)) * mv_factor_scale_functional_firstfactors) + (ff_p_fsat_functional_firstfactorsfactorization_product))) /\ ((((exists ff_h_fsat_functional_firstfactorsfactorization_product_partial. ff_h_fsat_functional_firstfactorsfactorization_product_partial + S (ff_r_fsat_functional_firstfactorsfactorization_product) = S ((S (ff_i_fsat_functional_firstfactorsfactorization_product)) * ff_v_fsat_functional_firstfactorsfactorization_product)) /\ exists ff_q_fsat_functional_firstfactorsfactorization_product_partial. ff_u_fsat_functional_firstfactorsfactorization_product = ff_q_fsat_functional_firstfactorsfactorization_product_partial * S ((S (ff_i_fsat_functional_firstfactorsfactorization_product)) * ff_v_fsat_functional_firstfactorsfactorization_product) + (ff_r_fsat_functional_firstfactorsfactorization_product))) /\ ((((exists ff_h_fsat_functional_firstfactorsfactorization_product_successor. ff_h_fsat_functional_firstfactorsfactorization_product_successor + S (ff_s_fsat_functional_firstfactorsfactorization_product) = S ((S (S ff_i_fsat_functional_firstfactorsfactorization_product)) * ff_v_fsat_functional_firstfactorsfactorization_product)) /\ exists ff_q_fsat_functional_firstfactorsfactorization_product_successor. ff_u_fsat_functional_firstfactorsfactorization_product = ff_q_fsat_functional_firstfactorsfactorization_product_successor * S ((S (S ff_i_fsat_functional_firstfactorsfactorization_product)) * ff_v_fsat_functional_firstfactorsfactorization_product) + (ff_s_fsat_functional_firstfactorsfactorization_product))) /\ ff_s_fsat_functional_firstfactorsfactorization_product = ff_r_fsat_functional_firstfactorsfactorization_product * ff_p_fsat_functional_firstfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_functional_firstfactorsfactorization_primes. (exists ftsf_gap_fsat_functional_firstfactorsfactorization_primes_bound. ftsf_gap_fsat_functional_firstfactorsfactorization_primes_bound + S ftsf_index_fsat_functional_firstfactorsfactorization_primes = (mv_factor_count_functional_firstfactors)) -> exists ftsf_factor_fsat_functional_firstfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_functional_firstfactorsfactorization_primes_entry. ff_h_ftsf_fsat_functional_firstfactorsfactorization_primes_entry + S (ftsf_factor_fsat_functional_firstfactorsfactorization_primes) = S ((S (ftsf_index_fsat_functional_firstfactorsfactorization_primes)) * mv_factor_scale_functional_firstfactors)) /\ exists ff_q_ftsf_fsat_functional_firstfactorsfactorization_primes_entry. mv_factor_code_functional_firstfactors = ff_q_ftsf_fsat_functional_firstfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_functional_firstfactorsfactorization_primes)) * mv_factor_scale_functional_firstfactors) + (ftsf_factor_fsat_functional_firstfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_functional_firstfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_functional_firstfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_functional_firstfactorsfactorization_primes_prime. ftsf_factor_fsat_functional_firstfactorsfactorization_primes = frm_prime_left_ftsf_fsat_functional_firstfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_functional_firstfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_functional_firstfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_functional_firstfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_functional_firstfactorsparityeven. (mv_factor_count_functional_firstfactors) = 2 * mv_even_half_functional_firstfactorsparityeven) /\ ((a) = 2))) \/ (((exists mv_odd_half_functional_firstfactorsparityodd. (mv_factor_count_functional_firstfactors) = 2 * mv_odd_half_functional_firstfactorsparityodd + 1) /\ ((a) = 1))))))))))) -> (((~((n) = 0)) /\ ((((exists mv_square_prime_functional_secondsquare. ((~((mv_square_prime_functional_secondsquare) = 1) /\ forall pvs_left_functional_secondsquareprime pvs_right_functional_secondsquareprime. (mv_square_prime_functional_secondsquare) = pvs_left_functional_secondsquareprime * pvs_right_functional_secondsquareprime -> pvs_left_functional_secondsquareprime = 1 \/ pvs_right_functional_secondsquareprime = 1) /\ (exists pvs_factor_functional_secondsquaredivisor. (n) = (mv_square_prime_functional_secondsquare * mv_square_prime_functional_secondsquare) * pvs_factor_functional_secondsquaredivisor))) /\ ((b) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_functional_secondsquarefree. (~((sfd_prime_functional_secondsquarefree) = 1) /\ forall pvs_left_functional_secondsquarefreedomain pvs_right_functional_secondsquarefreedomain. (sfd_prime_functional_secondsquarefree) = pvs_left_functional_secondsquarefreedomain * pvs_right_functional_secondsquarefreedomain -> pvs_left_functional_secondsquarefreedomain = 1 \/ pvs_right_functional_secondsquarefreedomain = 1) -> (exists pvs_le_gap_functional_secondsquarefreebound. pvs_le_gap_functional_secondsquarefreebound + (sfd_prime_functional_secondsquarefree) = (n)) -> ~(exists pvs_factor_functional_secondsquarefreesquare. (n) = (sfd_prime_functional_secondsquarefree * sfd_prime_functional_secondsquarefree) * pvs_factor_functional_secondsquarefreesquare)))) /\ (exists mv_factor_code_functional_secondfactors mv_factor_scale_functional_secondfactors mv_factor_count_functional_secondfactors. (((~(n = 0) /\ ((exists ff_u_fsat_functional_secondfactorsfactorization_product ff_v_fsat_functional_secondfactorsfactorization_product. ((((exists ff_h_fsat_functional_secondfactorsfactorization_product_start. ff_h_fsat_functional_secondfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_functional_secondfactorsfactorization_product)) /\ exists ff_q_fsat_functional_secondfactorsfactorization_product_start. ff_u_fsat_functional_secondfactorsfactorization_product = ff_q_fsat_functional_secondfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_functional_secondfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_functional_secondfactorsfactorization_product_terminal. ff_h_fsat_functional_secondfactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_functional_secondfactors)) * ff_v_fsat_functional_secondfactorsfactorization_product)) /\ exists ff_q_fsat_functional_secondfactorsfactorization_product_terminal. ff_u_fsat_functional_secondfactorsfactorization_product = ff_q_fsat_functional_secondfactorsfactorization_product_terminal * S ((S (mv_factor_count_functional_secondfactors)) * ff_v_fsat_functional_secondfactorsfactorization_product) + (n))) /\ forall ff_i_fsat_functional_secondfactorsfactorization_product. (exists ff_lt_fsat_functional_secondfactorsfactorization_product_bound. ff_lt_fsat_functional_secondfactorsfactorization_product_bound + S ff_i_fsat_functional_secondfactorsfactorization_product = mv_factor_count_functional_secondfactors) -> exists ff_p_fsat_functional_secondfactorsfactorization_product ff_r_fsat_functional_secondfactorsfactorization_product ff_s_fsat_functional_secondfactorsfactorization_product. ((((exists ff_h_fsat_functional_secondfactorsfactorization_product_factor. ff_h_fsat_functional_secondfactorsfactorization_product_factor + S (ff_p_fsat_functional_secondfactorsfactorization_product) = S ((S (ff_i_fsat_functional_secondfactorsfactorization_product)) * mv_factor_scale_functional_secondfactors)) /\ exists ff_q_fsat_functional_secondfactorsfactorization_product_factor. mv_factor_code_functional_secondfactors = ff_q_fsat_functional_secondfactorsfactorization_product_factor * S ((S (ff_i_fsat_functional_secondfactorsfactorization_product)) * mv_factor_scale_functional_secondfactors) + (ff_p_fsat_functional_secondfactorsfactorization_product))) /\ ((((exists ff_h_fsat_functional_secondfactorsfactorization_product_partial. ff_h_fsat_functional_secondfactorsfactorization_product_partial + S (ff_r_fsat_functional_secondfactorsfactorization_product) = S ((S (ff_i_fsat_functional_secondfactorsfactorization_product)) * ff_v_fsat_functional_secondfactorsfactorization_product)) /\ exists ff_q_fsat_functional_secondfactorsfactorization_product_partial. ff_u_fsat_functional_secondfactorsfactorization_product = ff_q_fsat_functional_secondfactorsfactorization_product_partial * S ((S (ff_i_fsat_functional_secondfactorsfactorization_product)) * ff_v_fsat_functional_secondfactorsfactorization_product) + (ff_r_fsat_functional_secondfactorsfactorization_product))) /\ ((((exists ff_h_fsat_functional_secondfactorsfactorization_product_successor. ff_h_fsat_functional_secondfactorsfactorization_product_successor + S (ff_s_fsat_functional_secondfactorsfactorization_product) = S ((S (S ff_i_fsat_functional_secondfactorsfactorization_product)) * ff_v_fsat_functional_secondfactorsfactorization_product)) /\ exists ff_q_fsat_functional_secondfactorsfactorization_product_successor. ff_u_fsat_functional_secondfactorsfactorization_product = ff_q_fsat_functional_secondfactorsfactorization_product_successor * S ((S (S ff_i_fsat_functional_secondfactorsfactorization_product)) * ff_v_fsat_functional_secondfactorsfactorization_product) + (ff_s_fsat_functional_secondfactorsfactorization_product))) /\ ff_s_fsat_functional_secondfactorsfactorization_product = ff_r_fsat_functional_secondfactorsfactorization_product * ff_p_fsat_functional_secondfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_functional_secondfactorsfactorization_primes. (exists ftsf_gap_fsat_functional_secondfactorsfactorization_primes_bound. ftsf_gap_fsat_functional_secondfactorsfactorization_primes_bound + S ftsf_index_fsat_functional_secondfactorsfactorization_primes = (mv_factor_count_functional_secondfactors)) -> exists ftsf_factor_fsat_functional_secondfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_functional_secondfactorsfactorization_primes_entry. ff_h_ftsf_fsat_functional_secondfactorsfactorization_primes_entry + S (ftsf_factor_fsat_functional_secondfactorsfactorization_primes) = S ((S (ftsf_index_fsat_functional_secondfactorsfactorization_primes)) * mv_factor_scale_functional_secondfactors)) /\ exists ff_q_ftsf_fsat_functional_secondfactorsfactorization_primes_entry. mv_factor_code_functional_secondfactors = ff_q_ftsf_fsat_functional_secondfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_functional_secondfactorsfactorization_primes)) * mv_factor_scale_functional_secondfactors) + (ftsf_factor_fsat_functional_secondfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_functional_secondfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_functional_secondfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_functional_secondfactorsfactorization_primes_prime. ftsf_factor_fsat_functional_secondfactorsfactorization_primes = frm_prime_left_ftsf_fsat_functional_secondfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_functional_secondfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_functional_secondfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_functional_secondfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_functional_secondfactorsparityeven. (mv_factor_count_functional_secondfactors) = 2 * mv_even_half_functional_secondfactorsparityeven) /\ ((b) = 2))) \/ (((exists mv_odd_half_functional_secondfactorsparityodd. (mv_factor_count_functional_secondfactors) = 2 * mv_odd_half_functional_secondfactorsparityodd + 1) /\ ((b) = 1))))))))))) -> a = b

• multiple_decidable_nonzero · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall d n. ~(d = 0) -> (exists q. n = d * q) \/ ~(exists q. n = d * q)

• one_mul · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. 1 * n = n

• signed_add_zero_left · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall input. (exists sa_lp_zero_left sa_ln_zero_left sa_rp_zero_left sa_rn_zero_left sa_op_zero_left sa_on_zero_left. (((0 = 2 * sa_lp_zero_left /\ sa_ln_zero_left = 0) \/ exists sd_half_zero_left_left. ((0 = 2 * sd_half_zero_left_left + 1 /\ sa_lp_zero_left = 0) /\ sa_ln_zero_left = S sd_half_zero_left_left)) /\ (((input = 2 * sa_rp_zero_left /\ sa_rn_zero_left = 0) \/ exists sd_half_zero_left_right. ((input = 2 * sd_half_zero_left_right + 1 /\ sa_rp_zero_left = 0) /\ sa_rn_zero_left = S sd_half_zero_left_right)) /\ (((input = 2 * sa_op_zero_left /\ sa_on_zero_left = 0) \/ exists sd_half_zero_left_output. ((input = 2 * sd_half_zero_left_output + 1 /\ sa_op_zero_left = 0) /\ sa_on_zero_left = S sd_half_zero_left_output)) /\ (sa_lp_zero_left + sa_rp_zero_left) + sa_on_zero_left = (sa_ln_zero_left + sa_rn_zero_left) + sa_op_zero_left))))

• signed_decode_total · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall code. exists pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_total. ((code = 2 * sd_half_total + 1 /\ pos = 0) /\ neg = S sd_half_total))

• succ_le_succ · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall a b. (exists k. k + a = b) -> exists r. r + S a = S b

• zero_add · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. 0 + n = n

• zero_le · checked inherited prerequisite
  Exact statement in the checked dependency cone

  forall n. 0 <= n