continued_fraction_convergent_functional

Exact expanded first-order arithmetic statement

forall s i u v p q. (exists cfc_previous_numerator_functional_convergent_first cfc_previous_denominator_functional_convergent_first cfc_code_functional_convergent_first cfc_scale_functional_convergent_first. ((~(v = 0)) /\ (exists cfc_tail_functional_convergent_firstcomputation. ((exists cfc_state_functional_convergent_firstcomputationinitial. ((exists cfc_left_functional_convergent_firstcomputationinitialcode cfc_right_functional_convergent_firstcomputationinitialcode cfc_matrix_functional_convergent_firstcomputationinitialcode. ((cfc_left_functional_convergent_firstcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_functional_convergent_firstcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_functional_convergent_firstcomputationinitialcode = ((cfc_left_functional_convergent_firstcomputationinitialcode) + (cfc_right_functional_convergent_firstcomputationinitialcode)) * S ((cfc_left_functional_convergent_firstcomputationinitialcode) + (cfc_right_functional_convergent_firstcomputationinitialcode)) + ((cfc_right_functional_convergent_firstcomputationinitialcode) + (cfc_right_functional_convergent_firstcomputationinitialcode))) /\ ((cfc_state_functional_convergent_firstcomputationinitial) = ((cfc_tail_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationinitialcode)) * S ((cfc_tail_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationinitialcode)) + ((cfc_matrix_functional_convergent_firstcomputationinitialcode) + (cfc_matrix_functional_convergent_firstcomputationinitialcode))))))) /\ (((exists ff_h_functional_convergent_firstcomputationinitialentry. ff_h_functional_convergent_firstcomputationinitialentry + S (cfc_state_functional_convergent_firstcomputationinitial) = S ((S (0)) * cfc_scale_functional_convergent_first)) /\ exists ff_q_functional_convergent_firstcomputationinitialentry. cfc_code_functional_convergent_first = ff_q_functional_convergent_firstcomputationinitialentry * S ((S (0)) * cfc_scale_functional_convergent_first) + (cfc_state_functional_convergent_firstcomputationinitial))))) /\ ((exists cfc_state_functional_convergent_firstcomputationterminal. ((exists cfc_left_functional_convergent_firstcomputationterminalcode cfc_right_functional_convergent_firstcomputationterminalcode cfc_matrix_functional_convergent_firstcomputationterminalcode. ((cfc_left_functional_convergent_firstcomputationterminalcode = ((u) + (cfc_previous_numerator_functional_convergent_first)) * S ((u) + (cfc_previous_numerator_functional_convergent_first)) + ((cfc_previous_numerator_functional_convergent_first) + (cfc_previous_numerator_functional_convergent_first))) /\ ((cfc_right_functional_convergent_firstcomputationterminalcode = ((v) + (cfc_previous_denominator_functional_convergent_first)) * S ((v) + (cfc_previous_denominator_functional_convergent_first)) + ((cfc_previous_denominator_functional_convergent_first) + (cfc_previous_denominator_functional_convergent_first))) /\ ((cfc_matrix_functional_convergent_firstcomputationterminalcode = ((cfc_left_functional_convergent_firstcomputationterminalcode) + (cfc_right_functional_convergent_firstcomputationterminalcode)) * S ((cfc_left_functional_convergent_firstcomputationterminalcode) + (cfc_right_functional_convergent_firstcomputationterminalcode)) + ((cfc_right_functional_convergent_firstcomputationterminalcode) + (cfc_right_functional_convergent_firstcomputationterminalcode))) /\ ((cfc_state_functional_convergent_firstcomputationterminal) = ((s) + (cfc_matrix_functional_convergent_firstcomputationterminalcode)) * S ((s) + (cfc_matrix_functional_convergent_firstcomputationterminalcode)) + ((cfc_matrix_functional_convergent_firstcomputationterminalcode) + (cfc_matrix_functional_convergent_firstcomputationterminalcode))))))) /\ (((exists ff_h_functional_convergent_firstcomputationterminalentry. ff_h_functional_convergent_firstcomputationterminalentry + S (cfc_state_functional_convergent_firstcomputationterminal) = S ((S (S (i))) * cfc_scale_functional_convergent_first)) /\ exists ff_q_functional_convergent_firstcomputationterminalentry. cfc_code_functional_convergent_first = ff_q_functional_convergent_firstcomputationterminalentry * S ((S (S (i))) * cfc_scale_functional_convergent_first) + (cfc_state_functional_convergent_firstcomputationterminal))))) /\ (forall cfc_index_functional_convergent_firstcomputation. (exists cfba_gap_functional_convergent_firstcomputationbound. cfba_gap_functional_convergent_firstcomputationbound + S (cfc_index_functional_convergent_firstcomputation) = (S (i))) -> exists cfc_old_functional_convergent_firstcomputation cfc_a_functional_convergent_firstcomputation cfc_b_functional_convergent_firstcomputation cfc_c_functional_convergent_firstcomputation cfc_d_functional_convergent_firstcomputation cfc_new_functional_convergent_firstcomputation cfc_quotient_functional_convergent_firstcomputation. ((exists cfc_state_functional_convergent_firstcomputationprevious. ((exists cfc_left_functional_convergent_firstcomputationpreviouscode cfc_right_functional_convergent_firstcomputationpreviouscode cfc_matrix_functional_convergent_firstcomputationpreviouscode. ((cfc_left_functional_convergent_firstcomputationpreviouscode = ((cfc_a_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation)) * S ((cfc_a_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation)) + ((cfc_b_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation))) /\ ((cfc_right_functional_convergent_firstcomputationpreviouscode = ((cfc_c_functional_convergent_firstcomputation) + (cfc_d_functional_convergent_firstcomputation)) * S ((cfc_c_functional_convergent_firstcomputation) + (cfc_d_functional_convergent_firstcomputation)) + ((cfc_d_functional_convergent_firstcomputation) + (cfc_d_functional_convergent_firstcomputation))) /\ ((cfc_matrix_functional_convergent_firstcomputationpreviouscode = ((cfc_left_functional_convergent_firstcomputationpreviouscode) + (cfc_right_functional_convergent_firstcomputationpreviouscode)) * S ((cfc_left_functional_convergent_firstcomputationpreviouscode) + (cfc_right_functional_convergent_firstcomputationpreviouscode)) + ((cfc_right_functional_convergent_firstcomputationpreviouscode) + (cfc_right_functional_convergent_firstcomputationpreviouscode))) /\ ((cfc_state_functional_convergent_firstcomputationprevious) = ((cfc_old_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationpreviouscode)) * S ((cfc_old_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationpreviouscode)) + ((cfc_matrix_functional_convergent_firstcomputationpreviouscode) + (cfc_matrix_functional_convergent_firstcomputationpreviouscode))))))) /\ (((exists ff_h_functional_convergent_firstcomputationpreviousentry. ff_h_functional_convergent_firstcomputationpreviousentry + S (cfc_state_functional_convergent_firstcomputationprevious) = S ((S (cfc_index_functional_convergent_firstcomputation)) * cfc_scale_functional_convergent_first)) /\ exists ff_q_functional_convergent_firstcomputationpreviousentry. cfc_code_functional_convergent_first = ff_q_functional_convergent_firstcomputationpreviousentry * S ((S (cfc_index_functional_convergent_firstcomputation)) * cfc_scale_functional_convergent_first) + (cfc_state_functional_convergent_firstcomputationprevious))))) /\ ((exists cfc_state_functional_convergent_firstcomputationfollowing. ((exists cfc_left_functional_convergent_firstcomputationfollowingcode cfc_right_functional_convergent_firstcomputationfollowingcode cfc_matrix_functional_convergent_firstcomputationfollowingcode. ((cfc_left_functional_convergent_firstcomputationfollowingcode = (((cfc_quotient_functional_convergent_firstcomputation * cfc_a_functional_convergent_firstcomputation + cfc_c_functional_convergent_firstcomputation)) + ((cfc_quotient_functional_convergent_firstcomputation * cfc_b_functional_convergent_firstcomputation + cfc_d_functional_convergent_firstcomputation))) * S (((cfc_quotient_functional_convergent_firstcomputation * cfc_a_functional_convergent_firstcomputation + cfc_c_functional_convergent_firstcomputation)) + ((cfc_quotient_functional_convergent_firstcomputation * cfc_b_functional_convergent_firstcomputation + cfc_d_functional_convergent_firstcomputation))) + (((cfc_quotient_functional_convergent_firstcomputation * cfc_b_functional_convergent_firstcomputation + cfc_d_functional_convergent_firstcomputation)) + ((cfc_quotient_functional_convergent_firstcomputation * cfc_b_functional_convergent_firstcomputation + cfc_d_functional_convergent_firstcomputation)))) /\ ((cfc_right_functional_convergent_firstcomputationfollowingcode = ((cfc_a_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation)) * S ((cfc_a_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation)) + ((cfc_b_functional_convergent_firstcomputation) + (cfc_b_functional_convergent_firstcomputation))) /\ ((cfc_matrix_functional_convergent_firstcomputationfollowingcode = ((cfc_left_functional_convergent_firstcomputationfollowingcode) + (cfc_right_functional_convergent_firstcomputationfollowingcode)) * S ((cfc_left_functional_convergent_firstcomputationfollowingcode) + (cfc_right_functional_convergent_firstcomputationfollowingcode)) + ((cfc_right_functional_convergent_firstcomputationfollowingcode) + (cfc_right_functional_convergent_firstcomputationfollowingcode))) /\ ((cfc_state_functional_convergent_firstcomputationfollowing) = ((cfc_new_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationfollowingcode)) * S ((cfc_new_functional_convergent_firstcomputation) + (cfc_matrix_functional_convergent_firstcomputationfollowingcode)) + ((cfc_matrix_functional_convergent_firstcomputationfollowingcode) + (cfc_matrix_functional_convergent_firstcomputationfollowingcode))))))) /\ (((exists ff_h_functional_convergent_firstcomputationfollowingentry. ff_h_functional_convergent_firstcomputationfollowingentry + S (cfc_state_functional_convergent_firstcomputationfollowing) = S ((S (S cfc_index_functional_convergent_firstcomputation)) * cfc_scale_functional_convergent_first)) /\ exists ff_q_functional_convergent_firstcomputationfollowingentry. cfc_code_functional_convergent_first = ff_q_functional_convergent_firstcomputationfollowingentry * S ((S (S cfc_index_functional_convergent_firstcomputation)) * cfc_scale_functional_convergent_first) + (cfc_state_functional_convergent_firstcomputationfollowing))))) /\ (cfc_new_functional_convergent_firstcomputation = S ((cfc_quotient_functional_convergent_firstcomputation + cfc_old_functional_convergent_firstcomputation) * S (cfc_quotient_functional_convergent_firstcomputation + cfc_old_functional_convergent_firstcomputation) + (cfc_old_functional_convergent_firstcomputation + cfc_old_functional_convergent_firstcomputation))))))))))) -> (exists cfc_previous_numerator_functional_convergent_second cfc_previous_denominator_functional_convergent_second cfc_code_functional_convergent_second cfc_scale_functional_convergent_second. ((~(q = 0)) /\ (exists cfc_tail_functional_convergent_secondcomputation. ((exists cfc_state_functional_convergent_secondcomputationinitial. ((exists cfc_left_functional_convergent_secondcomputationinitialcode cfc_right_functional_convergent_secondcomputationinitialcode cfc_matrix_functional_convergent_secondcomputationinitialcode. ((cfc_left_functional_convergent_secondcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_functional_convergent_secondcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_functional_convergent_secondcomputationinitialcode = ((cfc_left_functional_convergent_secondcomputationinitialcode) + (cfc_right_functional_convergent_secondcomputationinitialcode)) * S ((cfc_left_functional_convergent_secondcomputationinitialcode) + (cfc_right_functional_convergent_secondcomputationinitialcode)) + ((cfc_right_functional_convergent_secondcomputationinitialcode) + (cfc_right_functional_convergent_secondcomputationinitialcode))) /\ ((cfc_state_functional_convergent_secondcomputationinitial) = ((cfc_tail_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationinitialcode)) * S ((cfc_tail_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationinitialcode)) + ((cfc_matrix_functional_convergent_secondcomputationinitialcode) + (cfc_matrix_functional_convergent_secondcomputationinitialcode))))))) /\ (((exists ff_h_functional_convergent_secondcomputationinitialentry. ff_h_functional_convergent_secondcomputationinitialentry + S (cfc_state_functional_convergent_secondcomputationinitial) = S ((S (0)) * cfc_scale_functional_convergent_second)) /\ exists ff_q_functional_convergent_secondcomputationinitialentry. cfc_code_functional_convergent_second = ff_q_functional_convergent_secondcomputationinitialentry * S ((S (0)) * cfc_scale_functional_convergent_second) + (cfc_state_functional_convergent_secondcomputationinitial))))) /\ ((exists cfc_state_functional_convergent_secondcomputationterminal. ((exists cfc_left_functional_convergent_secondcomputationterminalcode cfc_right_functional_convergent_secondcomputationterminalcode cfc_matrix_functional_convergent_secondcomputationterminalcode. ((cfc_left_functional_convergent_secondcomputationterminalcode = ((p) + (cfc_previous_numerator_functional_convergent_second)) * S ((p) + (cfc_previous_numerator_functional_convergent_second)) + ((cfc_previous_numerator_functional_convergent_second) + (cfc_previous_numerator_functional_convergent_second))) /\ ((cfc_right_functional_convergent_secondcomputationterminalcode = ((q) + (cfc_previous_denominator_functional_convergent_second)) * S ((q) + (cfc_previous_denominator_functional_convergent_second)) + ((cfc_previous_denominator_functional_convergent_second) + (cfc_previous_denominator_functional_convergent_second))) /\ ((cfc_matrix_functional_convergent_secondcomputationterminalcode = ((cfc_left_functional_convergent_secondcomputationterminalcode) + (cfc_right_functional_convergent_secondcomputationterminalcode)) * S ((cfc_left_functional_convergent_secondcomputationterminalcode) + (cfc_right_functional_convergent_secondcomputationterminalcode)) + ((cfc_right_functional_convergent_secondcomputationterminalcode) + (cfc_right_functional_convergent_secondcomputationterminalcode))) /\ ((cfc_state_functional_convergent_secondcomputationterminal) = ((s) + (cfc_matrix_functional_convergent_secondcomputationterminalcode)) * S ((s) + (cfc_matrix_functional_convergent_secondcomputationterminalcode)) + ((cfc_matrix_functional_convergent_secondcomputationterminalcode) + (cfc_matrix_functional_convergent_secondcomputationterminalcode))))))) /\ (((exists ff_h_functional_convergent_secondcomputationterminalentry. ff_h_functional_convergent_secondcomputationterminalentry + S (cfc_state_functional_convergent_secondcomputationterminal) = S ((S (S (i))) * cfc_scale_functional_convergent_second)) /\ exists ff_q_functional_convergent_secondcomputationterminalentry. cfc_code_functional_convergent_second = ff_q_functional_convergent_secondcomputationterminalentry * S ((S (S (i))) * cfc_scale_functional_convergent_second) + (cfc_state_functional_convergent_secondcomputationterminal))))) /\ (forall cfc_index_functional_convergent_secondcomputation. (exists cfba_gap_functional_convergent_secondcomputationbound. cfba_gap_functional_convergent_secondcomputationbound + S (cfc_index_functional_convergent_secondcomputation) = (S (i))) -> exists cfc_old_functional_convergent_secondcomputation cfc_a_functional_convergent_secondcomputation cfc_b_functional_convergent_secondcomputation cfc_c_functional_convergent_secondcomputation cfc_d_functional_convergent_secondcomputation cfc_new_functional_convergent_secondcomputation cfc_quotient_functional_convergent_secondcomputation. ((exists cfc_state_functional_convergent_secondcomputationprevious. ((exists cfc_left_functional_convergent_secondcomputationpreviouscode cfc_right_functional_convergent_secondcomputationpreviouscode cfc_matrix_functional_convergent_secondcomputationpreviouscode. ((cfc_left_functional_convergent_secondcomputationpreviouscode = ((cfc_a_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation)) * S ((cfc_a_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation)) + ((cfc_b_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation))) /\ ((cfc_right_functional_convergent_secondcomputationpreviouscode = ((cfc_c_functional_convergent_secondcomputation) + (cfc_d_functional_convergent_secondcomputation)) * S ((cfc_c_functional_convergent_secondcomputation) + (cfc_d_functional_convergent_secondcomputation)) + ((cfc_d_functional_convergent_secondcomputation) + (cfc_d_functional_convergent_secondcomputation))) /\ ((cfc_matrix_functional_convergent_secondcomputationpreviouscode = ((cfc_left_functional_convergent_secondcomputationpreviouscode) + (cfc_right_functional_convergent_secondcomputationpreviouscode)) * S ((cfc_left_functional_convergent_secondcomputationpreviouscode) + (cfc_right_functional_convergent_secondcomputationpreviouscode)) + ((cfc_right_functional_convergent_secondcomputationpreviouscode) + (cfc_right_functional_convergent_secondcomputationpreviouscode))) /\ ((cfc_state_functional_convergent_secondcomputationprevious) = ((cfc_old_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationpreviouscode)) * S ((cfc_old_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationpreviouscode)) + ((cfc_matrix_functional_convergent_secondcomputationpreviouscode) + (cfc_matrix_functional_convergent_secondcomputationpreviouscode))))))) /\ (((exists ff_h_functional_convergent_secondcomputationpreviousentry. ff_h_functional_convergent_secondcomputationpreviousentry + S (cfc_state_functional_convergent_secondcomputationprevious) = S ((S (cfc_index_functional_convergent_secondcomputation)) * cfc_scale_functional_convergent_second)) /\ exists ff_q_functional_convergent_secondcomputationpreviousentry. cfc_code_functional_convergent_second = ff_q_functional_convergent_secondcomputationpreviousentry * S ((S (cfc_index_functional_convergent_secondcomputation)) * cfc_scale_functional_convergent_second) + (cfc_state_functional_convergent_secondcomputationprevious))))) /\ ((exists cfc_state_functional_convergent_secondcomputationfollowing. ((exists cfc_left_functional_convergent_secondcomputationfollowingcode cfc_right_functional_convergent_secondcomputationfollowingcode cfc_matrix_functional_convergent_secondcomputationfollowingcode. ((cfc_left_functional_convergent_secondcomputationfollowingcode = (((cfc_quotient_functional_convergent_secondcomputation * cfc_a_functional_convergent_secondcomputation + cfc_c_functional_convergent_secondcomputation)) + ((cfc_quotient_functional_convergent_secondcomputation * cfc_b_functional_convergent_secondcomputation + cfc_d_functional_convergent_secondcomputation))) * S (((cfc_quotient_functional_convergent_secondcomputation * cfc_a_functional_convergent_secondcomputation + cfc_c_functional_convergent_secondcomputation)) + ((cfc_quotient_functional_convergent_secondcomputation * cfc_b_functional_convergent_secondcomputation + cfc_d_functional_convergent_secondcomputation))) + (((cfc_quotient_functional_convergent_secondcomputation * cfc_b_functional_convergent_secondcomputation + cfc_d_functional_convergent_secondcomputation)) + ((cfc_quotient_functional_convergent_secondcomputation * cfc_b_functional_convergent_secondcomputation + cfc_d_functional_convergent_secondcomputation)))) /\ ((cfc_right_functional_convergent_secondcomputationfollowingcode = ((cfc_a_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation)) * S ((cfc_a_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation)) + ((cfc_b_functional_convergent_secondcomputation) + (cfc_b_functional_convergent_secondcomputation))) /\ ((cfc_matrix_functional_convergent_secondcomputationfollowingcode = ((cfc_left_functional_convergent_secondcomputationfollowingcode) + (cfc_right_functional_convergent_secondcomputationfollowingcode)) * S ((cfc_left_functional_convergent_secondcomputationfollowingcode) + (cfc_right_functional_convergent_secondcomputationfollowingcode)) + ((cfc_right_functional_convergent_secondcomputationfollowingcode) + (cfc_right_functional_convergent_secondcomputationfollowingcode))) /\ ((cfc_state_functional_convergent_secondcomputationfollowing) = ((cfc_new_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationfollowingcode)) * S ((cfc_new_functional_convergent_secondcomputation) + (cfc_matrix_functional_convergent_secondcomputationfollowingcode)) + ((cfc_matrix_functional_convergent_secondcomputationfollowingcode) + (cfc_matrix_functional_convergent_secondcomputationfollowingcode))))))) /\ (((exists ff_h_functional_convergent_secondcomputationfollowingentry. ff_h_functional_convergent_secondcomputationfollowingentry + S (cfc_state_functional_convergent_secondcomputationfollowing) = S ((S (S cfc_index_functional_convergent_secondcomputation)) * cfc_scale_functional_convergent_second)) /\ exists ff_q_functional_convergent_secondcomputationfollowingentry. cfc_code_functional_convergent_second = ff_q_functional_convergent_secondcomputationfollowingentry * S ((S (S cfc_index_functional_convergent_secondcomputation)) * cfc_scale_functional_convergent_second) + (cfc_state_functional_convergent_secondcomputationfollowing))))) /\ (cfc_new_functional_convergent_secondcomputation = S ((cfc_quotient_functional_convergent_secondcomputation + cfc_old_functional_convergent_secondcomputation) * S (cfc_quotient_functional_convergent_secondcomputation + cfc_old_functional_convergent_secondcomputation) + (cfc_old_functional_convergent_secondcomputation + cfc_old_functional_convergent_secondcomputation))))))))))) -> u = p /\ v = q