continued_fraction_convergent_best_approximation

Exact expanded first-order arithmetic statement

forall a b s i u v. (((exists cf_a_pred_best_actual_fraction cf_b_pred_best_actual_fraction cf_code_best_actual_fraction cf_scale_best_actual_fraction cf_length_pred_best_actual_fraction. (a = S cf_a_pred_best_actual_fraction /\ (b = S cf_b_pred_best_actual_fraction /\ (exists cf_gcd_best_actual_fraction_trace. ((((exists ff_h_cf_best_actual_fraction_trace_initial_state. ff_h_cf_best_actual_fraction_trace_initial_state + S (((cf_gcd_best_actual_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_best_actual_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * cf_scale_best_actual_fraction)) /\ exists ff_q_cf_best_actual_fraction_trace_initial_state. cf_code_best_actual_fraction = ff_q_cf_best_actual_fraction_trace_initial_state * S ((S (0)) * cf_scale_best_actual_fraction) + (((cf_gcd_best_actual_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_best_actual_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ff_h_cf_best_actual_fraction_trace_terminal_state. ff_h_cf_best_actual_fraction_trace_terminal_state + S (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))) = S ((S (S cf_length_pred_best_actual_fraction)) * cf_scale_best_actual_fraction)) /\ exists ff_q_cf_best_actual_fraction_trace_terminal_state. cf_code_best_actual_fraction = ff_q_cf_best_actual_fraction_trace_terminal_state * S ((S (S cf_length_pred_best_actual_fraction)) * cf_scale_best_actual_fraction) + (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))))) /\ forall cf_index_best_actual_fraction_trace. (exists ff_lt_cf_best_actual_fraction_trace_index. ff_lt_cf_best_actual_fraction_trace_index + S cf_index_best_actual_fraction_trace = S cf_length_pred_best_actual_fraction) -> exists cf_old_a_best_actual_fraction_trace cf_old_b_best_actual_fraction_trace cf_tail_best_actual_fraction_trace cf_new_a_best_actual_fraction_trace cf_new_b_best_actual_fraction_trace cf_head_best_actual_fraction_trace cf_quotient_best_actual_fraction_trace. ((((exists ff_h_cf_best_actual_fraction_trace_previous_state. ff_h_cf_best_actual_fraction_trace_previous_state + S (((cf_old_a_best_actual_fraction_trace) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)))) * S ((cf_old_a_best_actual_fraction_trace) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)))) + ((((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace))) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace))))) = S ((S (cf_index_best_actual_fraction_trace)) * cf_scale_best_actual_fraction)) /\ exists ff_q_cf_best_actual_fraction_trace_previous_state. cf_code_best_actual_fraction = ff_q_cf_best_actual_fraction_trace_previous_state * S ((S (cf_index_best_actual_fraction_trace)) * cf_scale_best_actual_fraction) + (((cf_old_a_best_actual_fraction_trace) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)))) * S ((cf_old_a_best_actual_fraction_trace) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)))) + ((((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace))) + (((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) * S ((cf_old_b_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace)) + ((cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace))))))) /\ ((((exists ff_h_cf_best_actual_fraction_trace_following_state. ff_h_cf_best_actual_fraction_trace_following_state + S (((cf_new_a_best_actual_fraction_trace) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)))) * S ((cf_new_a_best_actual_fraction_trace) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)))) + ((((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace))) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace))))) = S ((S (S cf_index_best_actual_fraction_trace)) * cf_scale_best_actual_fraction)) /\ exists ff_q_cf_best_actual_fraction_trace_following_state. cf_code_best_actual_fraction = ff_q_cf_best_actual_fraction_trace_following_state * S ((S (S cf_index_best_actual_fraction_trace)) * cf_scale_best_actual_fraction) + (((cf_new_a_best_actual_fraction_trace) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)))) * S ((cf_new_a_best_actual_fraction_trace) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)))) + ((((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace))) + (((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) * S ((cf_new_b_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace)) + ((cf_head_best_actual_fraction_trace) + (cf_head_best_actual_fraction_trace))))))) /\ (cf_new_b_best_actual_fraction_trace = cf_old_a_best_actual_fraction_trace /\ (cf_new_a_best_actual_fraction_trace = cf_new_b_best_actual_fraction_trace * cf_quotient_best_actual_fraction_trace + cf_old_b_best_actual_fraction_trace /\ ((exists ff_lt_cf_best_actual_fraction_trace_remainder. ff_lt_cf_best_actual_fraction_trace_remainder + S cf_old_b_best_actual_fraction_trace = cf_new_b_best_actual_fraction_trace) /\ (cf_head_best_actual_fraction_trace = S ((cf_quotient_best_actual_fraction_trace + cf_tail_best_actual_fraction_trace) * S (cf_quotient_best_actual_fraction_trace + cf_tail_best_actual_fraction_trace) + (cf_tail_best_actual_fraction_trace + cf_tail_best_actual_fraction_trace)))))))))))))) /\ (exists cfc_previous_numerator_best_actual_convergent cfc_previous_denominator_best_actual_convergent cfc_code_best_actual_convergent cfc_scale_best_actual_convergent. ((~(v = 0)) /\ (exists cfc_tail_best_actual_convergentcomputation. ((exists cfc_state_best_actual_convergentcomputationinitial. ((exists cfc_left_best_actual_convergentcomputationinitialcode cfc_right_best_actual_convergentcomputationinitialcode cfc_matrix_best_actual_convergentcomputationinitialcode. ((cfc_left_best_actual_convergentcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_best_actual_convergentcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_best_actual_convergentcomputationinitialcode = ((cfc_left_best_actual_convergentcomputationinitialcode) + (cfc_right_best_actual_convergentcomputationinitialcode)) * S ((cfc_left_best_actual_convergentcomputationinitialcode) + (cfc_right_best_actual_convergentcomputationinitialcode)) + ((cfc_right_best_actual_convergentcomputationinitialcode) + (cfc_right_best_actual_convergentcomputationinitialcode))) /\ ((cfc_state_best_actual_convergentcomputationinitial) = ((cfc_tail_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationinitialcode)) * S ((cfc_tail_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationinitialcode)) + ((cfc_matrix_best_actual_convergentcomputationinitialcode) + (cfc_matrix_best_actual_convergentcomputationinitialcode))))))) /\ (((exists ff_h_best_actual_convergentcomputationinitialentry. ff_h_best_actual_convergentcomputationinitialentry + S (cfc_state_best_actual_convergentcomputationinitial) = S ((S (0)) * cfc_scale_best_actual_convergent)) /\ exists ff_q_best_actual_convergentcomputationinitialentry. cfc_code_best_actual_convergent = ff_q_best_actual_convergentcomputationinitialentry * S ((S (0)) * cfc_scale_best_actual_convergent) + (cfc_state_best_actual_convergentcomputationinitial))))) /\ ((exists cfc_state_best_actual_convergentcomputationterminal. ((exists cfc_left_best_actual_convergentcomputationterminalcode cfc_right_best_actual_convergentcomputationterminalcode cfc_matrix_best_actual_convergentcomputationterminalcode. ((cfc_left_best_actual_convergentcomputationterminalcode = ((u) + (cfc_previous_numerator_best_actual_convergent)) * S ((u) + (cfc_previous_numerator_best_actual_convergent)) + ((cfc_previous_numerator_best_actual_convergent) + (cfc_previous_numerator_best_actual_convergent))) /\ ((cfc_right_best_actual_convergentcomputationterminalcode = ((v) + (cfc_previous_denominator_best_actual_convergent)) * S ((v) + (cfc_previous_denominator_best_actual_convergent)) + ((cfc_previous_denominator_best_actual_convergent) + (cfc_previous_denominator_best_actual_convergent))) /\ ((cfc_matrix_best_actual_convergentcomputationterminalcode = ((cfc_left_best_actual_convergentcomputationterminalcode) + (cfc_right_best_actual_convergentcomputationterminalcode)) * S ((cfc_left_best_actual_convergentcomputationterminalcode) + (cfc_right_best_actual_convergentcomputationterminalcode)) + ((cfc_right_best_actual_convergentcomputationterminalcode) + (cfc_right_best_actual_convergentcomputationterminalcode))) /\ ((cfc_state_best_actual_convergentcomputationterminal) = ((s) + (cfc_matrix_best_actual_convergentcomputationterminalcode)) * S ((s) + (cfc_matrix_best_actual_convergentcomputationterminalcode)) + ((cfc_matrix_best_actual_convergentcomputationterminalcode) + (cfc_matrix_best_actual_convergentcomputationterminalcode))))))) /\ (((exists ff_h_best_actual_convergentcomputationterminalentry. ff_h_best_actual_convergentcomputationterminalentry + S (cfc_state_best_actual_convergentcomputationterminal) = S ((S (S (i))) * cfc_scale_best_actual_convergent)) /\ exists ff_q_best_actual_convergentcomputationterminalentry. cfc_code_best_actual_convergent = ff_q_best_actual_convergentcomputationterminalentry * S ((S (S (i))) * cfc_scale_best_actual_convergent) + (cfc_state_best_actual_convergentcomputationterminal))))) /\ (forall cfc_index_best_actual_convergentcomputation. (exists cfba_gap_best_actual_convergentcomputationbound. cfba_gap_best_actual_convergentcomputationbound + S (cfc_index_best_actual_convergentcomputation) = (S (i))) -> exists cfc_old_best_actual_convergentcomputation cfc_a_best_actual_convergentcomputation cfc_b_best_actual_convergentcomputation cfc_c_best_actual_convergentcomputation cfc_d_best_actual_convergentcomputation cfc_new_best_actual_convergentcomputation cfc_quotient_best_actual_convergentcomputation. ((exists cfc_state_best_actual_convergentcomputationprevious. ((exists cfc_left_best_actual_convergentcomputationpreviouscode cfc_right_best_actual_convergentcomputationpreviouscode cfc_matrix_best_actual_convergentcomputationpreviouscode. ((cfc_left_best_actual_convergentcomputationpreviouscode = ((cfc_a_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation)) * S ((cfc_a_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation)) + ((cfc_b_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation))) /\ ((cfc_right_best_actual_convergentcomputationpreviouscode = ((cfc_c_best_actual_convergentcomputation) + (cfc_d_best_actual_convergentcomputation)) * S ((cfc_c_best_actual_convergentcomputation) + (cfc_d_best_actual_convergentcomputation)) + ((cfc_d_best_actual_convergentcomputation) + (cfc_d_best_actual_convergentcomputation))) /\ ((cfc_matrix_best_actual_convergentcomputationpreviouscode = ((cfc_left_best_actual_convergentcomputationpreviouscode) + (cfc_right_best_actual_convergentcomputationpreviouscode)) * S ((cfc_left_best_actual_convergentcomputationpreviouscode) + (cfc_right_best_actual_convergentcomputationpreviouscode)) + ((cfc_right_best_actual_convergentcomputationpreviouscode) + (cfc_right_best_actual_convergentcomputationpreviouscode))) /\ ((cfc_state_best_actual_convergentcomputationprevious) = ((cfc_old_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationpreviouscode)) * S ((cfc_old_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationpreviouscode)) + ((cfc_matrix_best_actual_convergentcomputationpreviouscode) + (cfc_matrix_best_actual_convergentcomputationpreviouscode))))))) /\ (((exists ff_h_best_actual_convergentcomputationpreviousentry. ff_h_best_actual_convergentcomputationpreviousentry + S (cfc_state_best_actual_convergentcomputationprevious) = S ((S (cfc_index_best_actual_convergentcomputation)) * cfc_scale_best_actual_convergent)) /\ exists ff_q_best_actual_convergentcomputationpreviousentry. cfc_code_best_actual_convergent = ff_q_best_actual_convergentcomputationpreviousentry * S ((S (cfc_index_best_actual_convergentcomputation)) * cfc_scale_best_actual_convergent) + (cfc_state_best_actual_convergentcomputationprevious))))) /\ ((exists cfc_state_best_actual_convergentcomputationfollowing. ((exists cfc_left_best_actual_convergentcomputationfollowingcode cfc_right_best_actual_convergentcomputationfollowingcode cfc_matrix_best_actual_convergentcomputationfollowingcode. ((cfc_left_best_actual_convergentcomputationfollowingcode = (((cfc_quotient_best_actual_convergentcomputation * cfc_a_best_actual_convergentcomputation + cfc_c_best_actual_convergentcomputation)) + ((cfc_quotient_best_actual_convergentcomputation * cfc_b_best_actual_convergentcomputation + cfc_d_best_actual_convergentcomputation))) * S (((cfc_quotient_best_actual_convergentcomputation * cfc_a_best_actual_convergentcomputation + cfc_c_best_actual_convergentcomputation)) + ((cfc_quotient_best_actual_convergentcomputation * cfc_b_best_actual_convergentcomputation + cfc_d_best_actual_convergentcomputation))) + (((cfc_quotient_best_actual_convergentcomputation * cfc_b_best_actual_convergentcomputation + cfc_d_best_actual_convergentcomputation)) + ((cfc_quotient_best_actual_convergentcomputation * cfc_b_best_actual_convergentcomputation + cfc_d_best_actual_convergentcomputation)))) /\ ((cfc_right_best_actual_convergentcomputationfollowingcode = ((cfc_a_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation)) * S ((cfc_a_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation)) + ((cfc_b_best_actual_convergentcomputation) + (cfc_b_best_actual_convergentcomputation))) /\ ((cfc_matrix_best_actual_convergentcomputationfollowingcode = ((cfc_left_best_actual_convergentcomputationfollowingcode) + (cfc_right_best_actual_convergentcomputationfollowingcode)) * S ((cfc_left_best_actual_convergentcomputationfollowingcode) + (cfc_right_best_actual_convergentcomputationfollowingcode)) + ((cfc_right_best_actual_convergentcomputationfollowingcode) + (cfc_right_best_actual_convergentcomputationfollowingcode))) /\ ((cfc_state_best_actual_convergentcomputationfollowing) = ((cfc_new_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationfollowingcode)) * S ((cfc_new_best_actual_convergentcomputation) + (cfc_matrix_best_actual_convergentcomputationfollowingcode)) + ((cfc_matrix_best_actual_convergentcomputationfollowingcode) + (cfc_matrix_best_actual_convergentcomputationfollowingcode))))))) /\ (((exists ff_h_best_actual_convergentcomputationfollowingentry. ff_h_best_actual_convergentcomputationfollowingentry + S (cfc_state_best_actual_convergentcomputationfollowing) = S ((S (S cfc_index_best_actual_convergentcomputation)) * cfc_scale_best_actual_convergent)) /\ exists ff_q_best_actual_convergentcomputationfollowingentry. cfc_code_best_actual_convergent = ff_q_best_actual_convergentcomputationfollowingentry * S ((S (S cfc_index_best_actual_convergentcomputation)) * cfc_scale_best_actual_convergent) + (cfc_state_best_actual_convergentcomputationfollowing))))) /\ (cfc_new_best_actual_convergentcomputation = S ((cfc_quotient_best_actual_convergentcomputation + cfc_old_best_actual_convergentcomputation) * S (cfc_quotient_best_actual_convergentcomputation + cfc_old_best_actual_convergentcomputation) + (cfc_old_best_actual_convergentcomputation + cfc_old_best_actual_convergentcomputation))))))))))))) -> (forall cfc_numerator_best_natural cfc_denominator_best_natural cfc_current_error_best_natural cfc_candidate_error_best_natural. ~(cfc_denominator_best_natural = 0) -> (exists cfba_gap_best_naturaldenominator. cfba_gap_best_naturaldenominator + S (cfc_denominator_best_natural) = (v)) -> ((((a) * (v)) = ((b) * (u)) + (cfc_current_error_best_natural)) \/ (((b) * (u)) = ((a) * (v)) + (cfc_current_error_best_natural))) -> ((((a) * cfc_denominator_best_natural) = ((b) * cfc_numerator_best_natural) + (cfc_candidate_error_best_natural)) \/ (((b) * cfc_numerator_best_natural) = ((a) * cfc_denominator_best_natural) + (cfc_candidate_error_best_natural))) -> (exists cfba_bound_best_naturalresult. cfba_bound_best_naturalresult + (cfc_current_error_best_natural) = (cfc_candidate_error_best_natural)))