continued_fraction_has_exact_terminal_convergent

forall a b s. (exists cf_a_pred_zero_initial_fraction cf_b_pred_zero_initial_fraction cf_code_zero_initial_fraction cf_scale_zero_initial_fraction cf_length_pred_zero_initial_fraction. (a = S cf_a_pred_zero_initial_fraction /\ (b = S cf_b_pred_zero_initial_fraction /\ (exists cf_gcd_zero_initial_fraction_trace. ((((exists ff_h_cf_zero_initial_fraction_trace_initial_state. ff_h_cf_zero_initial_fraction_trace_initial_state + S (((cf_gcd_zero_initial_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_zero_initial_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_zero_initial_fraction)) /\ exists ff_q_cf_zero_initial_fraction_trace_initial_state. cf_code_zero_initial_fraction = ff_q_cf_zero_initial_fraction_trace_initial_state * S ((S (0)) * cf_scale_zero_initial_fraction) + (((cf_gcd_zero_initial_fraction_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_zero_initial_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_zero_initial_fraction_trace_terminal_state. ff_h_cf_zero_initial_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_zero_initial_fraction)) * cf_scale_zero_initial_fraction)) /\ exists ff_q_cf_zero_initial_fraction_trace_terminal_state. cf_code_zero_initial_fraction = ff_q_cf_zero_initial_fraction_trace_terminal_state * S ((S (S cf_length_pred_zero_initial_fraction)) * cf_scale_zero_initial_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_zero_initial_fraction_trace. (exists ff_lt_cf_zero_initial_fraction_trace_index. ff_lt_cf_zero_initial_fraction_trace_index + S cf_index_zero_initial_fraction_trace = S cf_length_pred_zero_initial_fraction) -> exists cf_old_a_zero_initial_fraction_trace cf_old_b_zero_initial_fraction_trace cf_tail_zero_initial_fraction_trace cf_new_a_zero_initial_fraction_trace cf_new_b_zero_initial_fraction_trace cf_head_zero_initial_fraction_trace cf_quotient_zero_initial_fraction_trace. ((((exists ff_h_cf_zero_initial_fraction_trace_previous_state. ff_h_cf_zero_initial_fraction_trace_previous_state + S (((cf_old_a_zero_initial_fraction_trace) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)))) * S ((cf_old_a_zero_initial_fraction_trace) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)))) + ((((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace))) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace))))) = S ((S (cf_index_zero_initial_fraction_trace)) * cf_scale_zero_initial_fraction)) /\ exists ff_q_cf_zero_initial_fraction_trace_previous_state. cf_code_zero_initial_fraction = ff_q_cf_zero_initial_fraction_trace_previous_state * S ((S (cf_index_zero_initial_fraction_trace)) * cf_scale_zero_initial_fraction) + (((cf_old_a_zero_initial_fraction_trace) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)))) * S ((cf_old_a_zero_initial_fraction_trace) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)))) + ((((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace))) + (((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) * S ((cf_old_b_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace)) + ((cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace))))))) /\ ((((exists ff_h_cf_zero_initial_fraction_trace_following_state. ff_h_cf_zero_initial_fraction_trace_following_state + S (((cf_new_a_zero_initial_fraction_trace) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)))) * S ((cf_new_a_zero_initial_fraction_trace) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)))) + ((((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace))) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace))))) = S ((S (S cf_index_zero_initial_fraction_trace)) * cf_scale_zero_initial_fraction)) /\ exists ff_q_cf_zero_initial_fraction_trace_following_state. cf_code_zero_initial_fraction = ff_q_cf_zero_initial_fraction_trace_following_state * S ((S (S cf_index_zero_initial_fraction_trace)) * cf_scale_zero_initial_fraction) + (((cf_new_a_zero_initial_fraction_trace) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)))) * S ((cf_new_a_zero_initial_fraction_trace) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)))) + ((((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace))) + (((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) * S ((cf_new_b_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace)) + ((cf_head_zero_initial_fraction_trace) + (cf_head_zero_initial_fraction_trace))))))) /\ (cf_new_b_zero_initial_fraction_trace = cf_old_a_zero_initial_fraction_trace /\ (cf_new_a_zero_initial_fraction_trace = cf_new_b_zero_initial_fraction_trace * cf_quotient_zero_initial_fraction_trace + cf_old_b_zero_initial_fraction_trace /\ ((exists ff_lt_cf_zero_initial_fraction_trace_remainder. ff_lt_cf_zero_initial_fraction_trace_remainder + S cf_old_b_zero_initial_fraction_trace = cf_new_b_zero_initial_fraction_trace) /\ (cf_head_zero_initial_fraction_trace = S ((cf_quotient_zero_initial_fraction_trace + cf_tail_zero_initial_fraction_trace) * S (cf_quotient_zero_initial_fraction_trace + cf_tail_zero_initial_fraction_trace) + (cf_tail_zero_initial_fraction_trace + cf_tail_zero_initial_fraction_trace)))))))))))))) -> exists i u v. ((exists cfc_previous_numerator_public_terminal cfc_previous_denominator_public_terminal cfc_code_public_terminal cfc_scale_public_terminal. ((~(v = 0)) /\ (exists cfc_tail_public_terminalcomputation. ((exists cfc_state_public_terminalcomputationinitial. ((exists cfc_left_public_terminalcomputationinitialcode cfc_right_public_terminalcomputationinitialcode cfc_matrix_public_terminalcomputationinitialcode. ((cfc_left_public_terminalcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_public_terminalcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_public_terminalcomputationinitialcode = ((cfc_left_public_terminalcomputationinitialcode) + (cfc_right_public_terminalcomputationinitialcode)) * S ((cfc_left_public_terminalcomputationinitialcode) + (cfc_right_public_terminalcomputationinitialcode)) + ((cfc_right_public_terminalcomputationinitialcode) + (cfc_right_public_terminalcomputationinitialcode))) /\ ((cfc_state_public_terminalcomputationinitial) = ((cfc_tail_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationinitialcode)) * S ((cfc_tail_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationinitialcode)) + ((cfc_matrix_public_terminalcomputationinitialcode) + (cfc_matrix_public_terminalcomputationinitialcode))))))) /\ (((exists ff_h_public_terminalcomputationinitialentry. ff_h_public_terminalcomputationinitialentry + S (cfc_state_public_terminalcomputationinitial) = S ((S (0)) * cfc_scale_public_terminal)) /\ exists ff_q_public_terminalcomputationinitialentry. cfc_code_public_terminal = ff_q_public_terminalcomputationinitialentry * S ((S (0)) * cfc_scale_public_terminal) + (cfc_state_public_terminalcomputationinitial))))) /\ ((exists cfc_state_public_terminalcomputationterminal. ((exists cfc_left_public_terminalcomputationterminalcode cfc_right_public_terminalcomputationterminalcode cfc_matrix_public_terminalcomputationterminalcode. ((cfc_left_public_terminalcomputationterminalcode = ((u) + (cfc_previous_numerator_public_terminal)) * S ((u) + (cfc_previous_numerator_public_terminal)) + ((cfc_previous_numerator_public_terminal) + (cfc_previous_numerator_public_terminal))) /\ ((cfc_right_public_terminalcomputationterminalcode = ((v) + (cfc_previous_denominator_public_terminal)) * S ((v) + (cfc_previous_denominator_public_terminal)) + ((cfc_previous_denominator_public_terminal) + (cfc_previous_denominator_public_terminal))) /\ ((cfc_matrix_public_terminalcomputationterminalcode = ((cfc_left_public_terminalcomputationterminalcode) + (cfc_right_public_terminalcomputationterminalcode)) * S ((cfc_left_public_terminalcomputationterminalcode) + (cfc_right_public_terminalcomputationterminalcode)) + ((cfc_right_public_terminalcomputationterminalcode) + (cfc_right_public_terminalcomputationterminalcode))) /\ ((cfc_state_public_terminalcomputationterminal) = ((s) + (cfc_matrix_public_terminalcomputationterminalcode)) * S ((s) + (cfc_matrix_public_terminalcomputationterminalcode)) + ((cfc_matrix_public_terminalcomputationterminalcode) + (cfc_matrix_public_terminalcomputationterminalcode))))))) /\ (((exists ff_h_public_terminalcomputationterminalentry. ff_h_public_terminalcomputationterminalentry + S (cfc_state_public_terminalcomputationterminal) = S ((S (S (i))) * cfc_scale_public_terminal)) /\ exists ff_q_public_terminalcomputationterminalentry. cfc_code_public_terminal = ff_q_public_terminalcomputationterminalentry * S ((S (S (i))) * cfc_scale_public_terminal) + (cfc_state_public_terminalcomputationterminal))))) /\ (forall cfc_index_public_terminalcomputation. (exists cfba_gap_public_terminalcomputationbound. cfba_gap_public_terminalcomputationbound + S (cfc_index_public_terminalcomputation) = (S (i))) -> exists cfc_old_public_terminalcomputation cfc_a_public_terminalcomputation cfc_b_public_terminalcomputation cfc_c_public_terminalcomputation cfc_d_public_terminalcomputation cfc_new_public_terminalcomputation cfc_quotient_public_terminalcomputation. ((exists cfc_state_public_terminalcomputationprevious. ((exists cfc_left_public_terminalcomputationpreviouscode cfc_right_public_terminalcomputationpreviouscode cfc_matrix_public_terminalcomputationpreviouscode. ((cfc_left_public_terminalcomputationpreviouscode = ((cfc_a_public_terminalcomputation) + (cfc_b_public_terminalcomputation)) * S ((cfc_a_public_terminalcomputation) + (cfc_b_public_terminalcomputation)) + ((cfc_b_public_terminalcomputation) + (cfc_b_public_terminalcomputation))) /\ ((cfc_right_public_terminalcomputationpreviouscode = ((cfc_c_public_terminalcomputation) + (cfc_d_public_terminalcomputation)) * S ((cfc_c_public_terminalcomputation) + (cfc_d_public_terminalcomputation)) + ((cfc_d_public_terminalcomputation) + (cfc_d_public_terminalcomputation))) /\ ((cfc_matrix_public_terminalcomputationpreviouscode = ((cfc_left_public_terminalcomputationpreviouscode) + (cfc_right_public_terminalcomputationpreviouscode)) * S ((cfc_left_public_terminalcomputationpreviouscode) + (cfc_right_public_terminalcomputationpreviouscode)) + ((cfc_right_public_terminalcomputationpreviouscode) + (cfc_right_public_terminalcomputationpreviouscode))) /\ ((cfc_state_public_terminalcomputationprevious) = ((cfc_old_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationpreviouscode)) * S ((cfc_old_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationpreviouscode)) + ((cfc_matrix_public_terminalcomputationpreviouscode) + (cfc_matrix_public_terminalcomputationpreviouscode))))))) /\ (((exists ff_h_public_terminalcomputationpreviousentry. ff_h_public_terminalcomputationpreviousentry + S (cfc_state_public_terminalcomputationprevious) = S ((S (cfc_index_public_terminalcomputation)) * cfc_scale_public_terminal)) /\ exists ff_q_public_terminalcomputationpreviousentry. cfc_code_public_terminal = ff_q_public_terminalcomputationpreviousentry * S ((S (cfc_index_public_terminalcomputation)) * cfc_scale_public_terminal) + (cfc_state_public_terminalcomputationprevious))))) /\ ((exists cfc_state_public_terminalcomputationfollowing. ((exists cfc_left_public_terminalcomputationfollowingcode cfc_right_public_terminalcomputationfollowingcode cfc_matrix_public_terminalcomputationfollowingcode. ((cfc_left_public_terminalcomputationfollowingcode = (((cfc_quotient_public_terminalcomputation * cfc_a_public_terminalcomputation + cfc_c_public_terminalcomputation)) + ((cfc_quotient_public_terminalcomputation * cfc_b_public_terminalcomputation + cfc_d_public_terminalcomputation))) * S (((cfc_quotient_public_terminalcomputation * cfc_a_public_terminalcomputation + cfc_c_public_terminalcomputation)) + ((cfc_quotient_public_terminalcomputation * cfc_b_public_terminalcomputation + cfc_d_public_terminalcomputation))) + (((cfc_quotient_public_terminalcomputation * cfc_b_public_terminalcomputation + cfc_d_public_terminalcomputation)) + ((cfc_quotient_public_terminalcomputation * cfc_b_public_terminalcomputation + cfc_d_public_terminalcomputation)))) /\ ((cfc_right_public_terminalcomputationfollowingcode = ((cfc_a_public_terminalcomputation) + (cfc_b_public_terminalcomputation)) * S ((cfc_a_public_terminalcomputation) + (cfc_b_public_terminalcomputation)) + ((cfc_b_public_terminalcomputation) + (cfc_b_public_terminalcomputation))) /\ ((cfc_matrix_public_terminalcomputationfollowingcode = ((cfc_left_public_terminalcomputationfollowingcode) + (cfc_right_public_terminalcomputationfollowingcode)) * S ((cfc_left_public_terminalcomputationfollowingcode) + (cfc_right_public_terminalcomputationfollowingcode)) + ((cfc_right_public_terminalcomputationfollowingcode) + (cfc_right_public_terminalcomputationfollowingcode))) /\ ((cfc_state_public_terminalcomputationfollowing) = ((cfc_new_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationfollowingcode)) * S ((cfc_new_public_terminalcomputation) + (cfc_matrix_public_terminalcomputationfollowingcode)) + ((cfc_matrix_public_terminalcomputationfollowingcode) + (cfc_matrix_public_terminalcomputationfollowingcode))))))) /\ (((exists ff_h_public_terminalcomputationfollowingentry. ff_h_public_terminalcomputationfollowingentry + S (cfc_state_public_terminalcomputationfollowing) = S ((S (S cfc_index_public_terminalcomputation)) * cfc_scale_public_terminal)) /\ exists ff_q_public_terminalcomputationfollowingentry. cfc_code_public_terminal = ff_q_public_terminalcomputationfollowingentry * S ((S (S cfc_index_public_terminalcomputation)) * cfc_scale_public_terminal) + (cfc_state_public_terminalcomputationfollowing))))) /\ (cfc_new_public_terminalcomputation = S ((cfc_quotient_public_terminalcomputation + cfc_old_public_terminalcomputation) * S (cfc_quotient_public_terminalcomputation + cfc_old_public_terminalcomputation) + (cfc_old_public_terminalcomputation + cfc_old_public_terminalcomputation))))))))))) /\ (a * v = b * u))