continued_fraction_convergent_index_is_valid

forall a b s h e L i u v. (exists cf_gcd_convergent_index_history. ((((exists ff_h_cf_convergent_index_history_initial_state. ff_h_cf_convergent_index_history_initial_state + S (((cf_gcd_convergent_index_history) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_convergent_index_history) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * e)) /\ exists ff_q_cf_convergent_index_history_initial_state. h = ff_q_cf_convergent_index_history_initial_state * S ((S (0)) * e) + (((cf_gcd_convergent_index_history) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_convergent_index_history) + (((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_convergent_index_history_terminal_state. ff_h_cf_convergent_index_history_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 (L)) * e)) /\ exists ff_q_cf_convergent_index_history_terminal_state. h = ff_q_cf_convergent_index_history_terminal_state * S ((S (L)) * e) + (((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_convergent_index_history. (exists ff_lt_cf_convergent_index_history_index. ff_lt_cf_convergent_index_history_index + S cf_index_convergent_index_history = L) -> exists cf_old_a_convergent_index_history cf_old_b_convergent_index_history cf_tail_convergent_index_history cf_new_a_convergent_index_history cf_new_b_convergent_index_history cf_head_convergent_index_history cf_quotient_convergent_index_history. ((((exists ff_h_cf_convergent_index_history_previous_state. ff_h_cf_convergent_index_history_previous_state + S (((cf_old_a_convergent_index_history) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history)))) * S ((cf_old_a_convergent_index_history) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history)))) + ((((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history))) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history))))) = S ((S (cf_index_convergent_index_history)) * e)) /\ exists ff_q_cf_convergent_index_history_previous_state. h = ff_q_cf_convergent_index_history_previous_state * S ((S (cf_index_convergent_index_history)) * e) + (((cf_old_a_convergent_index_history) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history)))) * S ((cf_old_a_convergent_index_history) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history)))) + ((((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history))) + (((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) * S ((cf_old_b_convergent_index_history) + (cf_tail_convergent_index_history)) + ((cf_tail_convergent_index_history) + (cf_tail_convergent_index_history))))))) /\ ((((exists ff_h_cf_convergent_index_history_following_state. ff_h_cf_convergent_index_history_following_state + S (((cf_new_a_convergent_index_history) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history)))) * S ((cf_new_a_convergent_index_history) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history)))) + ((((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history))) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history))))) = S ((S (S cf_index_convergent_index_history)) * e)) /\ exists ff_q_cf_convergent_index_history_following_state. h = ff_q_cf_convergent_index_history_following_state * S ((S (S cf_index_convergent_index_history)) * e) + (((cf_new_a_convergent_index_history) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history)))) * S ((cf_new_a_convergent_index_history) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history)))) + ((((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history))) + (((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) * S ((cf_new_b_convergent_index_history) + (cf_head_convergent_index_history)) + ((cf_head_convergent_index_history) + (cf_head_convergent_index_history))))))) /\ (cf_new_b_convergent_index_history = cf_old_a_convergent_index_history /\ (cf_new_a_convergent_index_history = cf_new_b_convergent_index_history * cf_quotient_convergent_index_history + cf_old_b_convergent_index_history /\ ((exists ff_lt_cf_convergent_index_history_remainder. ff_lt_cf_convergent_index_history_remainder + S cf_old_b_convergent_index_history = cf_new_b_convergent_index_history) /\ (cf_head_convergent_index_history = S ((cf_quotient_convergent_index_history + cf_tail_convergent_index_history) * S (cf_quotient_convergent_index_history + cf_tail_convergent_index_history) + (cf_tail_convergent_index_history + cf_tail_convergent_index_history))))))))))) -> (exists cfc_previous_numerator_existing_convergent cfc_previous_denominator_existing_convergent cfc_code_existing_convergent cfc_scale_existing_convergent. ((~(v = 0)) /\ (exists cfc_tail_existing_convergentcomputation. ((exists cfc_state_existing_convergentcomputationinitial. ((exists cfc_left_existing_convergentcomputationinitialcode cfc_right_existing_convergentcomputationinitialcode cfc_matrix_existing_convergentcomputationinitialcode. ((cfc_left_existing_convergentcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_existing_convergentcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_existing_convergentcomputationinitialcode = ((cfc_left_existing_convergentcomputationinitialcode) + (cfc_right_existing_convergentcomputationinitialcode)) * S ((cfc_left_existing_convergentcomputationinitialcode) + (cfc_right_existing_convergentcomputationinitialcode)) + ((cfc_right_existing_convergentcomputationinitialcode) + (cfc_right_existing_convergentcomputationinitialcode))) /\ ((cfc_state_existing_convergentcomputationinitial) = ((cfc_tail_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationinitialcode)) * S ((cfc_tail_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationinitialcode)) + ((cfc_matrix_existing_convergentcomputationinitialcode) + (cfc_matrix_existing_convergentcomputationinitialcode))))))) /\ (((exists ff_h_existing_convergentcomputationinitialentry. ff_h_existing_convergentcomputationinitialentry + S (cfc_state_existing_convergentcomputationinitial) = S ((S (0)) * cfc_scale_existing_convergent)) /\ exists ff_q_existing_convergentcomputationinitialentry. cfc_code_existing_convergent = ff_q_existing_convergentcomputationinitialentry * S ((S (0)) * cfc_scale_existing_convergent) + (cfc_state_existing_convergentcomputationinitial))))) /\ ((exists cfc_state_existing_convergentcomputationterminal. ((exists cfc_left_existing_convergentcomputationterminalcode cfc_right_existing_convergentcomputationterminalcode cfc_matrix_existing_convergentcomputationterminalcode. ((cfc_left_existing_convergentcomputationterminalcode = ((u) + (cfc_previous_numerator_existing_convergent)) * S ((u) + (cfc_previous_numerator_existing_convergent)) + ((cfc_previous_numerator_existing_convergent) + (cfc_previous_numerator_existing_convergent))) /\ ((cfc_right_existing_convergentcomputationterminalcode = ((v) + (cfc_previous_denominator_existing_convergent)) * S ((v) + (cfc_previous_denominator_existing_convergent)) + ((cfc_previous_denominator_existing_convergent) + (cfc_previous_denominator_existing_convergent))) /\ ((cfc_matrix_existing_convergentcomputationterminalcode = ((cfc_left_existing_convergentcomputationterminalcode) + (cfc_right_existing_convergentcomputationterminalcode)) * S ((cfc_left_existing_convergentcomputationterminalcode) + (cfc_right_existing_convergentcomputationterminalcode)) + ((cfc_right_existing_convergentcomputationterminalcode) + (cfc_right_existing_convergentcomputationterminalcode))) /\ ((cfc_state_existing_convergentcomputationterminal) = ((s) + (cfc_matrix_existing_convergentcomputationterminalcode)) * S ((s) + (cfc_matrix_existing_convergentcomputationterminalcode)) + ((cfc_matrix_existing_convergentcomputationterminalcode) + (cfc_matrix_existing_convergentcomputationterminalcode))))))) /\ (((exists ff_h_existing_convergentcomputationterminalentry. ff_h_existing_convergentcomputationterminalentry + S (cfc_state_existing_convergentcomputationterminal) = S ((S (S (i))) * cfc_scale_existing_convergent)) /\ exists ff_q_existing_convergentcomputationterminalentry. cfc_code_existing_convergent = ff_q_existing_convergentcomputationterminalentry * S ((S (S (i))) * cfc_scale_existing_convergent) + (cfc_state_existing_convergentcomputationterminal))))) /\ (forall cfc_index_existing_convergentcomputation. (exists cfba_gap_existing_convergentcomputationbound. cfba_gap_existing_convergentcomputationbound + S (cfc_index_existing_convergentcomputation) = (S (i))) -> exists cfc_old_existing_convergentcomputation cfc_a_existing_convergentcomputation cfc_b_existing_convergentcomputation cfc_c_existing_convergentcomputation cfc_d_existing_convergentcomputation cfc_new_existing_convergentcomputation cfc_quotient_existing_convergentcomputation. ((exists cfc_state_existing_convergentcomputationprevious. ((exists cfc_left_existing_convergentcomputationpreviouscode cfc_right_existing_convergentcomputationpreviouscode cfc_matrix_existing_convergentcomputationpreviouscode. ((cfc_left_existing_convergentcomputationpreviouscode = ((cfc_a_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation)) * S ((cfc_a_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation)) + ((cfc_b_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation))) /\ ((cfc_right_existing_convergentcomputationpreviouscode = ((cfc_c_existing_convergentcomputation) + (cfc_d_existing_convergentcomputation)) * S ((cfc_c_existing_convergentcomputation) + (cfc_d_existing_convergentcomputation)) + ((cfc_d_existing_convergentcomputation) + (cfc_d_existing_convergentcomputation))) /\ ((cfc_matrix_existing_convergentcomputationpreviouscode = ((cfc_left_existing_convergentcomputationpreviouscode) + (cfc_right_existing_convergentcomputationpreviouscode)) * S ((cfc_left_existing_convergentcomputationpreviouscode) + (cfc_right_existing_convergentcomputationpreviouscode)) + ((cfc_right_existing_convergentcomputationpreviouscode) + (cfc_right_existing_convergentcomputationpreviouscode))) /\ ((cfc_state_existing_convergentcomputationprevious) = ((cfc_old_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationpreviouscode)) * S ((cfc_old_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationpreviouscode)) + ((cfc_matrix_existing_convergentcomputationpreviouscode) + (cfc_matrix_existing_convergentcomputationpreviouscode))))))) /\ (((exists ff_h_existing_convergentcomputationpreviousentry. ff_h_existing_convergentcomputationpreviousentry + S (cfc_state_existing_convergentcomputationprevious) = S ((S (cfc_index_existing_convergentcomputation)) * cfc_scale_existing_convergent)) /\ exists ff_q_existing_convergentcomputationpreviousentry. cfc_code_existing_convergent = ff_q_existing_convergentcomputationpreviousentry * S ((S (cfc_index_existing_convergentcomputation)) * cfc_scale_existing_convergent) + (cfc_state_existing_convergentcomputationprevious))))) /\ ((exists cfc_state_existing_convergentcomputationfollowing. ((exists cfc_left_existing_convergentcomputationfollowingcode cfc_right_existing_convergentcomputationfollowingcode cfc_matrix_existing_convergentcomputationfollowingcode. ((cfc_left_existing_convergentcomputationfollowingcode = (((cfc_quotient_existing_convergentcomputation * cfc_a_existing_convergentcomputation + cfc_c_existing_convergentcomputation)) + ((cfc_quotient_existing_convergentcomputation * cfc_b_existing_convergentcomputation + cfc_d_existing_convergentcomputation))) * S (((cfc_quotient_existing_convergentcomputation * cfc_a_existing_convergentcomputation + cfc_c_existing_convergentcomputation)) + ((cfc_quotient_existing_convergentcomputation * cfc_b_existing_convergentcomputation + cfc_d_existing_convergentcomputation))) + (((cfc_quotient_existing_convergentcomputation * cfc_b_existing_convergentcomputation + cfc_d_existing_convergentcomputation)) + ((cfc_quotient_existing_convergentcomputation * cfc_b_existing_convergentcomputation + cfc_d_existing_convergentcomputation)))) /\ ((cfc_right_existing_convergentcomputationfollowingcode = ((cfc_a_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation)) * S ((cfc_a_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation)) + ((cfc_b_existing_convergentcomputation) + (cfc_b_existing_convergentcomputation))) /\ ((cfc_matrix_existing_convergentcomputationfollowingcode = ((cfc_left_existing_convergentcomputationfollowingcode) + (cfc_right_existing_convergentcomputationfollowingcode)) * S ((cfc_left_existing_convergentcomputationfollowingcode) + (cfc_right_existing_convergentcomputationfollowingcode)) + ((cfc_right_existing_convergentcomputationfollowingcode) + (cfc_right_existing_convergentcomputationfollowingcode))) /\ ((cfc_state_existing_convergentcomputationfollowing) = ((cfc_new_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationfollowingcode)) * S ((cfc_new_existing_convergentcomputation) + (cfc_matrix_existing_convergentcomputationfollowingcode)) + ((cfc_matrix_existing_convergentcomputationfollowingcode) + (cfc_matrix_existing_convergentcomputationfollowingcode))))))) /\ (((exists ff_h_existing_convergentcomputationfollowingentry. ff_h_existing_convergentcomputationfollowingentry + S (cfc_state_existing_convergentcomputationfollowing) = S ((S (S cfc_index_existing_convergentcomputation)) * cfc_scale_existing_convergent)) /\ exists ff_q_existing_convergentcomputationfollowingentry. cfc_code_existing_convergent = ff_q_existing_convergentcomputationfollowingentry * S ((S (S cfc_index_existing_convergentcomputation)) * cfc_scale_existing_convergent) + (cfc_state_existing_convergentcomputationfollowing))))) /\ (cfc_new_existing_convergentcomputation = S ((cfc_quotient_existing_convergentcomputation + cfc_old_existing_convergentcomputation) * S (cfc_quotient_existing_convergentcomputation + cfc_old_existing_convergentcomputation) + (cfc_old_existing_convergentcomputation + cfc_old_existing_convergentcomputation))))))))))) -> (exists cfba_gap_convergent_index_valid. cfba_gap_convergent_index_valid + S (i) = (L))