continued_fraction_terminal_convergent_is_exact

Exact expanded first-order arithmetic statement

forall a b s h e k u v. (exists cf_gcd_terminal_history. ((((exists ff_h_cf_terminal_history_initial_state. ff_h_cf_terminal_history_initial_state + S (((cf_gcd_terminal_history) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_terminal_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_terminal_history_initial_state. h = ff_q_cf_terminal_history_initial_state * S ((S (0)) * e) + (((cf_gcd_terminal_history) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_terminal_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_terminal_history_terminal_state. ff_h_cf_terminal_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 (S k)) * e)) /\ exists ff_q_cf_terminal_history_terminal_state. h = ff_q_cf_terminal_history_terminal_state * S ((S (S k)) * 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_terminal_history. (exists ff_lt_cf_terminal_history_index. ff_lt_cf_terminal_history_index + S cf_index_terminal_history = S k) -> exists cf_old_a_terminal_history cf_old_b_terminal_history cf_tail_terminal_history cf_new_a_terminal_history cf_new_b_terminal_history cf_head_terminal_history cf_quotient_terminal_history. ((((exists ff_h_cf_terminal_history_previous_state. ff_h_cf_terminal_history_previous_state + S (((cf_old_a_terminal_history) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history)))) * S ((cf_old_a_terminal_history) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history)))) + ((((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history))) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history))))) = S ((S (cf_index_terminal_history)) * e)) /\ exists ff_q_cf_terminal_history_previous_state. h = ff_q_cf_terminal_history_previous_state * S ((S (cf_index_terminal_history)) * e) + (((cf_old_a_terminal_history) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history)))) * S ((cf_old_a_terminal_history) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history)))) + ((((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history))) + (((cf_old_b_terminal_history) + (cf_tail_terminal_history)) * S ((cf_old_b_terminal_history) + (cf_tail_terminal_history)) + ((cf_tail_terminal_history) + (cf_tail_terminal_history))))))) /\ ((((exists ff_h_cf_terminal_history_following_state. ff_h_cf_terminal_history_following_state + S (((cf_new_a_terminal_history) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history)))) * S ((cf_new_a_terminal_history) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history)))) + ((((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history))) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history))))) = S ((S (S cf_index_terminal_history)) * e)) /\ exists ff_q_cf_terminal_history_following_state. h = ff_q_cf_terminal_history_following_state * S ((S (S cf_index_terminal_history)) * e) + (((cf_new_a_terminal_history) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history)))) * S ((cf_new_a_terminal_history) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history)))) + ((((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history))) + (((cf_new_b_terminal_history) + (cf_head_terminal_history)) * S ((cf_new_b_terminal_history) + (cf_head_terminal_history)) + ((cf_head_terminal_history) + (cf_head_terminal_history))))))) /\ (cf_new_b_terminal_history = cf_old_a_terminal_history /\ (cf_new_a_terminal_history = cf_new_b_terminal_history * cf_quotient_terminal_history + cf_old_b_terminal_history /\ ((exists ff_lt_cf_terminal_history_remainder. ff_lt_cf_terminal_history_remainder + S cf_old_b_terminal_history = cf_new_b_terminal_history) /\ (cf_head_terminal_history = S ((cf_quotient_terminal_history + cf_tail_terminal_history) * S (cf_quotient_terminal_history + cf_tail_terminal_history) + (cf_tail_terminal_history + cf_tail_terminal_history))))))))))) -> (exists cfc_previous_numerator_terminal_convergent cfc_previous_denominator_terminal_convergent cfc_code_terminal_convergent cfc_scale_terminal_convergent. ((~(v = 0)) /\ (exists cfc_tail_terminal_convergentcomputation. ((exists cfc_state_terminal_convergentcomputationinitial. ((exists cfc_left_terminal_convergentcomputationinitialcode cfc_right_terminal_convergentcomputationinitialcode cfc_matrix_terminal_convergentcomputationinitialcode. ((cfc_left_terminal_convergentcomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_terminal_convergentcomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_terminal_convergentcomputationinitialcode = ((cfc_left_terminal_convergentcomputationinitialcode) + (cfc_right_terminal_convergentcomputationinitialcode)) * S ((cfc_left_terminal_convergentcomputationinitialcode) + (cfc_right_terminal_convergentcomputationinitialcode)) + ((cfc_right_terminal_convergentcomputationinitialcode) + (cfc_right_terminal_convergentcomputationinitialcode))) /\ ((cfc_state_terminal_convergentcomputationinitial) = ((cfc_tail_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationinitialcode)) * S ((cfc_tail_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationinitialcode)) + ((cfc_matrix_terminal_convergentcomputationinitialcode) + (cfc_matrix_terminal_convergentcomputationinitialcode))))))) /\ (((exists ff_h_terminal_convergentcomputationinitialentry. ff_h_terminal_convergentcomputationinitialentry + S (cfc_state_terminal_convergentcomputationinitial) = S ((S (0)) * cfc_scale_terminal_convergent)) /\ exists ff_q_terminal_convergentcomputationinitialentry. cfc_code_terminal_convergent = ff_q_terminal_convergentcomputationinitialentry * S ((S (0)) * cfc_scale_terminal_convergent) + (cfc_state_terminal_convergentcomputationinitial))))) /\ ((exists cfc_state_terminal_convergentcomputationterminal. ((exists cfc_left_terminal_convergentcomputationterminalcode cfc_right_terminal_convergentcomputationterminalcode cfc_matrix_terminal_convergentcomputationterminalcode. ((cfc_left_terminal_convergentcomputationterminalcode = ((u) + (cfc_previous_numerator_terminal_convergent)) * S ((u) + (cfc_previous_numerator_terminal_convergent)) + ((cfc_previous_numerator_terminal_convergent) + (cfc_previous_numerator_terminal_convergent))) /\ ((cfc_right_terminal_convergentcomputationterminalcode = ((v) + (cfc_previous_denominator_terminal_convergent)) * S ((v) + (cfc_previous_denominator_terminal_convergent)) + ((cfc_previous_denominator_terminal_convergent) + (cfc_previous_denominator_terminal_convergent))) /\ ((cfc_matrix_terminal_convergentcomputationterminalcode = ((cfc_left_terminal_convergentcomputationterminalcode) + (cfc_right_terminal_convergentcomputationterminalcode)) * S ((cfc_left_terminal_convergentcomputationterminalcode) + (cfc_right_terminal_convergentcomputationterminalcode)) + ((cfc_right_terminal_convergentcomputationterminalcode) + (cfc_right_terminal_convergentcomputationterminalcode))) /\ ((cfc_state_terminal_convergentcomputationterminal) = ((s) + (cfc_matrix_terminal_convergentcomputationterminalcode)) * S ((s) + (cfc_matrix_terminal_convergentcomputationterminalcode)) + ((cfc_matrix_terminal_convergentcomputationterminalcode) + (cfc_matrix_terminal_convergentcomputationterminalcode))))))) /\ (((exists ff_h_terminal_convergentcomputationterminalentry. ff_h_terminal_convergentcomputationterminalentry + S (cfc_state_terminal_convergentcomputationterminal) = S ((S (S (k))) * cfc_scale_terminal_convergent)) /\ exists ff_q_terminal_convergentcomputationterminalentry. cfc_code_terminal_convergent = ff_q_terminal_convergentcomputationterminalentry * S ((S (S (k))) * cfc_scale_terminal_convergent) + (cfc_state_terminal_convergentcomputationterminal))))) /\ (forall cfc_index_terminal_convergentcomputation. (exists cfba_gap_terminal_convergentcomputationbound. cfba_gap_terminal_convergentcomputationbound + S (cfc_index_terminal_convergentcomputation) = (S (k))) -> exists cfc_old_terminal_convergentcomputation cfc_a_terminal_convergentcomputation cfc_b_terminal_convergentcomputation cfc_c_terminal_convergentcomputation cfc_d_terminal_convergentcomputation cfc_new_terminal_convergentcomputation cfc_quotient_terminal_convergentcomputation. ((exists cfc_state_terminal_convergentcomputationprevious. ((exists cfc_left_terminal_convergentcomputationpreviouscode cfc_right_terminal_convergentcomputationpreviouscode cfc_matrix_terminal_convergentcomputationpreviouscode. ((cfc_left_terminal_convergentcomputationpreviouscode = ((cfc_a_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation)) * S ((cfc_a_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation)) + ((cfc_b_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation))) /\ ((cfc_right_terminal_convergentcomputationpreviouscode = ((cfc_c_terminal_convergentcomputation) + (cfc_d_terminal_convergentcomputation)) * S ((cfc_c_terminal_convergentcomputation) + (cfc_d_terminal_convergentcomputation)) + ((cfc_d_terminal_convergentcomputation) + (cfc_d_terminal_convergentcomputation))) /\ ((cfc_matrix_terminal_convergentcomputationpreviouscode = ((cfc_left_terminal_convergentcomputationpreviouscode) + (cfc_right_terminal_convergentcomputationpreviouscode)) * S ((cfc_left_terminal_convergentcomputationpreviouscode) + (cfc_right_terminal_convergentcomputationpreviouscode)) + ((cfc_right_terminal_convergentcomputationpreviouscode) + (cfc_right_terminal_convergentcomputationpreviouscode))) /\ ((cfc_state_terminal_convergentcomputationprevious) = ((cfc_old_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationpreviouscode)) * S ((cfc_old_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationpreviouscode)) + ((cfc_matrix_terminal_convergentcomputationpreviouscode) + (cfc_matrix_terminal_convergentcomputationpreviouscode))))))) /\ (((exists ff_h_terminal_convergentcomputationpreviousentry. ff_h_terminal_convergentcomputationpreviousentry + S (cfc_state_terminal_convergentcomputationprevious) = S ((S (cfc_index_terminal_convergentcomputation)) * cfc_scale_terminal_convergent)) /\ exists ff_q_terminal_convergentcomputationpreviousentry. cfc_code_terminal_convergent = ff_q_terminal_convergentcomputationpreviousentry * S ((S (cfc_index_terminal_convergentcomputation)) * cfc_scale_terminal_convergent) + (cfc_state_terminal_convergentcomputationprevious))))) /\ ((exists cfc_state_terminal_convergentcomputationfollowing. ((exists cfc_left_terminal_convergentcomputationfollowingcode cfc_right_terminal_convergentcomputationfollowingcode cfc_matrix_terminal_convergentcomputationfollowingcode. ((cfc_left_terminal_convergentcomputationfollowingcode = (((cfc_quotient_terminal_convergentcomputation * cfc_a_terminal_convergentcomputation + cfc_c_terminal_convergentcomputation)) + ((cfc_quotient_terminal_convergentcomputation * cfc_b_terminal_convergentcomputation + cfc_d_terminal_convergentcomputation))) * S (((cfc_quotient_terminal_convergentcomputation * cfc_a_terminal_convergentcomputation + cfc_c_terminal_convergentcomputation)) + ((cfc_quotient_terminal_convergentcomputation * cfc_b_terminal_convergentcomputation + cfc_d_terminal_convergentcomputation))) + (((cfc_quotient_terminal_convergentcomputation * cfc_b_terminal_convergentcomputation + cfc_d_terminal_convergentcomputation)) + ((cfc_quotient_terminal_convergentcomputation * cfc_b_terminal_convergentcomputation + cfc_d_terminal_convergentcomputation)))) /\ ((cfc_right_terminal_convergentcomputationfollowingcode = ((cfc_a_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation)) * S ((cfc_a_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation)) + ((cfc_b_terminal_convergentcomputation) + (cfc_b_terminal_convergentcomputation))) /\ ((cfc_matrix_terminal_convergentcomputationfollowingcode = ((cfc_left_terminal_convergentcomputationfollowingcode) + (cfc_right_terminal_convergentcomputationfollowingcode)) * S ((cfc_left_terminal_convergentcomputationfollowingcode) + (cfc_right_terminal_convergentcomputationfollowingcode)) + ((cfc_right_terminal_convergentcomputationfollowingcode) + (cfc_right_terminal_convergentcomputationfollowingcode))) /\ ((cfc_state_terminal_convergentcomputationfollowing) = ((cfc_new_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationfollowingcode)) * S ((cfc_new_terminal_convergentcomputation) + (cfc_matrix_terminal_convergentcomputationfollowingcode)) + ((cfc_matrix_terminal_convergentcomputationfollowingcode) + (cfc_matrix_terminal_convergentcomputationfollowingcode))))))) /\ (((exists ff_h_terminal_convergentcomputationfollowingentry. ff_h_terminal_convergentcomputationfollowingentry + S (cfc_state_terminal_convergentcomputationfollowing) = S ((S (S cfc_index_terminal_convergentcomputation)) * cfc_scale_terminal_convergent)) /\ exists ff_q_terminal_convergentcomputationfollowingentry. cfc_code_terminal_convergent = ff_q_terminal_convergentcomputationfollowingentry * S ((S (S cfc_index_terminal_convergentcomputation)) * cfc_scale_terminal_convergent) + (cfc_state_terminal_convergentcomputationfollowing))))) /\ (cfc_new_terminal_convergentcomputation = S ((cfc_quotient_terminal_convergentcomputation + cfc_old_terminal_convergentcomputation) * S (cfc_quotient_terminal_convergentcomputation + cfc_old_terminal_convergentcomputation) + (cfc_old_terminal_convergentcomputation + cfc_old_terminal_convergentcomputation))))))))))) -> a * v = b * u