crt_pairwise_compatible_dominating_last_canonical_exists_unique

Exact expanded first-order arithmetic statement

forall r s b c l a M. (forall gcrt_positive_index_dominating_canonical_positive gcrt_positive_value_dominating_canonical_positive. (exists ff_lt_gcrt_dominating_canonical_positive_bound. ff_lt_gcrt_dominating_canonical_positive_bound + S gcrt_positive_index_dominating_canonical_positive = S l) -> (((exists ff_h_gcrt_dominating_canonical_positive_entry. ff_h_gcrt_dominating_canonical_positive_entry + S (gcrt_positive_value_dominating_canonical_positive) = S ((S (gcrt_positive_index_dominating_canonical_positive)) * c)) /\ exists ff_q_gcrt_dominating_canonical_positive_entry. b = ff_q_gcrt_dominating_canonical_positive_entry * S ((S (gcrt_positive_index_dominating_canonical_positive)) * c) + (gcrt_positive_value_dominating_canonical_positive))) -> ~(gcrt_positive_value_dominating_canonical_positive = 0)) -> (forall gcomp_left_index_dominating_canonical_pairs gcomp_right_index_dominating_canonical_pairs gcomp_left_residue_dominating_canonical_pairs gcomp_right_residue_dominating_canonical_pairs gcomp_left_modulus_dominating_canonical_pairs gcomp_right_modulus_dominating_canonical_pairs gcomp_pair_gcd_dominating_canonical_pairs. (exists ff_lt_gcrt_dominating_canonical_pairs_left_bound. ff_lt_gcrt_dominating_canonical_pairs_left_bound + S gcomp_left_index_dominating_canonical_pairs = S l) -> (exists ff_lt_gcrt_dominating_canonical_pairs_right_bound. ff_lt_gcrt_dominating_canonical_pairs_right_bound + S gcomp_right_index_dominating_canonical_pairs = S l) -> (((exists ff_h_gcrt_dominating_canonical_pairs_left_residue. ff_h_gcrt_dominating_canonical_pairs_left_residue + S (gcomp_left_residue_dominating_canonical_pairs) = S ((S (gcomp_left_index_dominating_canonical_pairs)) * s)) /\ exists ff_q_gcrt_dominating_canonical_pairs_left_residue. r = ff_q_gcrt_dominating_canonical_pairs_left_residue * S ((S (gcomp_left_index_dominating_canonical_pairs)) * s) + (gcomp_left_residue_dominating_canonical_pairs))) -> (((exists ff_h_gcrt_dominating_canonical_pairs_right_residue. ff_h_gcrt_dominating_canonical_pairs_right_residue + S (gcomp_right_residue_dominating_canonical_pairs) = S ((S (gcomp_right_index_dominating_canonical_pairs)) * s)) /\ exists ff_q_gcrt_dominating_canonical_pairs_right_residue. r = ff_q_gcrt_dominating_canonical_pairs_right_residue * S ((S (gcomp_right_index_dominating_canonical_pairs)) * s) + (gcomp_right_residue_dominating_canonical_pairs))) -> (((exists ff_h_gcrt_dominating_canonical_pairs_left_modulus. ff_h_gcrt_dominating_canonical_pairs_left_modulus + S (gcomp_left_modulus_dominating_canonical_pairs) = S ((S (gcomp_left_index_dominating_canonical_pairs)) * c)) /\ exists ff_q_gcrt_dominating_canonical_pairs_left_modulus. b = ff_q_gcrt_dominating_canonical_pairs_left_modulus * S ((S (gcomp_left_index_dominating_canonical_pairs)) * c) + (gcomp_left_modulus_dominating_canonical_pairs))) -> (((exists ff_h_gcrt_dominating_canonical_pairs_right_modulus. ff_h_gcrt_dominating_canonical_pairs_right_modulus + S (gcomp_right_modulus_dominating_canonical_pairs) = S ((S (gcomp_right_index_dominating_canonical_pairs)) * c)) /\ exists ff_q_gcrt_dominating_canonical_pairs_right_modulus. b = ff_q_gcrt_dominating_canonical_pairs_right_modulus * S ((S (gcomp_right_index_dominating_canonical_pairs)) * c) + (gcomp_right_modulus_dominating_canonical_pairs))) -> ((((exists hag_left_factor_gcomp_dominating_canonical_pairs_gcd. gcomp_left_modulus_dominating_canonical_pairs = gcomp_pair_gcd_dominating_canonical_pairs * hag_left_factor_gcomp_dominating_canonical_pairs_gcd) /\ (exists hag_right_factor_gcomp_dominating_canonical_pairs_gcd. gcomp_right_modulus_dominating_canonical_pairs = gcomp_pair_gcd_dominating_canonical_pairs * hag_right_factor_gcomp_dominating_canonical_pairs_gcd)) /\ forall hag_divisor_gcomp_dominating_canonical_pairs_gcd. (exists hag_common_left_gcomp_dominating_canonical_pairs_gcd. gcomp_left_modulus_dominating_canonical_pairs = hag_divisor_gcomp_dominating_canonical_pairs_gcd * hag_common_left_gcomp_dominating_canonical_pairs_gcd) -> (exists hag_common_right_gcomp_dominating_canonical_pairs_gcd. gcomp_right_modulus_dominating_canonical_pairs = hag_divisor_gcomp_dominating_canonical_pairs_gcd * hag_common_right_gcomp_dominating_canonical_pairs_gcd) -> exists hag_greatest_factor_gcomp_dominating_canonical_pairs_gcd. gcomp_pair_gcd_dominating_canonical_pairs = hag_divisor_gcomp_dominating_canonical_pairs_gcd * hag_greatest_factor_gcomp_dominating_canonical_pairs_gcd)) -> (exists hgcrt_mod_left_gcrt_dominating_canonical_pairs_result hgcrt_mod_right_gcrt_dominating_canonical_pairs_result. gcomp_left_residue_dominating_canonical_pairs + gcomp_pair_gcd_dominating_canonical_pairs * hgcrt_mod_left_gcrt_dominating_canonical_pairs_result = gcomp_right_residue_dominating_canonical_pairs + gcomp_pair_gcd_dominating_canonical_pairs * hgcrt_mod_right_gcrt_dominating_canonical_pairs_result)) -> (((exists ff_h_gcrt_dominating_canonical_residue. ff_h_gcrt_dominating_canonical_residue + S (a) = S ((S (l)) * s)) /\ exists ff_q_gcrt_dominating_canonical_residue. r = ff_q_gcrt_dominating_canonical_residue * S ((S (l)) * s) + (a))) -> (((exists ff_h_gcrt_dominating_canonical_modulus. ff_h_gcrt_dominating_canonical_modulus + S (M) = S ((S (l)) * c)) /\ exists ff_q_gcrt_dominating_canonical_modulus. b = ff_q_gcrt_dominating_canonical_modulus * S ((S (l)) * c) + (M))) -> (forall gcrt_common_index_dominating_canonical_common gcrt_common_modulus_dominating_canonical_common. (exists ff_lt_gcrt_dominating_canonical_common_bound. ff_lt_gcrt_dominating_canonical_common_bound + S gcrt_common_index_dominating_canonical_common = l) -> (((exists ff_h_gcrt_dominating_canonical_common_entry. ff_h_gcrt_dominating_canonical_common_entry + S (gcrt_common_modulus_dominating_canonical_common) = S ((S (gcrt_common_index_dominating_canonical_common)) * c)) /\ exists ff_q_gcrt_dominating_canonical_common_entry. b = ff_q_gcrt_dominating_canonical_common_entry * S ((S (gcrt_common_index_dominating_canonical_common)) * c) + (gcrt_common_modulus_dominating_canonical_common))) -> exists gcrt_common_quotient_dominating_canonical_common. M = gcrt_common_modulus_dominating_canonical_common * gcrt_common_quotient_dominating_canonical_common) -> exists x L. ((((((forall gcrt_common_index_dominating_canonical_chosen_lcm_own gcrt_common_modulus_dominating_canonical_chosen_lcm_own. (exists ff_lt_gcrt_dominating_canonical_chosen_lcm_own_bound. ff_lt_gcrt_dominating_canonical_chosen_lcm_own_bound + S gcrt_common_index_dominating_canonical_chosen_lcm_own = S l) -> (((exists ff_h_gcrt_dominating_canonical_chosen_lcm_own_entry. ff_h_gcrt_dominating_canonical_chosen_lcm_own_entry + S (gcrt_common_modulus_dominating_canonical_chosen_lcm_own) = S ((S (gcrt_common_index_dominating_canonical_chosen_lcm_own)) * c)) /\ exists ff_q_gcrt_dominating_canonical_chosen_lcm_own_entry. b = ff_q_gcrt_dominating_canonical_chosen_lcm_own_entry * S ((S (gcrt_common_index_dominating_canonical_chosen_lcm_own)) * c) + (gcrt_common_modulus_dominating_canonical_chosen_lcm_own))) -> exists gcrt_common_quotient_dominating_canonical_chosen_lcm_own. L = gcrt_common_modulus_dominating_canonical_chosen_lcm_own * gcrt_common_quotient_dominating_canonical_chosen_lcm_own) /\ forall gcrt_lcm_common_dominating_canonical_chosen_lcm. (forall gcrt_common_index_dominating_canonical_chosen_lcm_other gcrt_common_modulus_dominating_canonical_chosen_lcm_other. (exists ff_lt_gcrt_dominating_canonical_chosen_lcm_other_bound. ff_lt_gcrt_dominating_canonical_chosen_lcm_other_bound + S gcrt_common_index_dominating_canonical_chosen_lcm_other = S l) -> (((exists ff_h_gcrt_dominating_canonical_chosen_lcm_other_entry. ff_h_gcrt_dominating_canonical_chosen_lcm_other_entry + S (gcrt_common_modulus_dominating_canonical_chosen_lcm_other) = S ((S (gcrt_common_index_dominating_canonical_chosen_lcm_other)) * c)) /\ exists ff_q_gcrt_dominating_canonical_chosen_lcm_other_entry. b = ff_q_gcrt_dominating_canonical_chosen_lcm_other_entry * S ((S (gcrt_common_index_dominating_canonical_chosen_lcm_other)) * c) + (gcrt_common_modulus_dominating_canonical_chosen_lcm_other))) -> exists gcrt_common_quotient_dominating_canonical_chosen_lcm_other. gcrt_lcm_common_dominating_canonical_chosen_lcm = gcrt_common_modulus_dominating_canonical_chosen_lcm_other * gcrt_common_quotient_dominating_canonical_chosen_lcm_other) -> exists gcrt_lcm_quotient_dominating_canonical_chosen_lcm. gcrt_lcm_common_dominating_canonical_chosen_lcm = L * gcrt_lcm_quotient_dominating_canonical_chosen_lcm)) /\ ((exists ff_lt_gcrt_dominating_canonical_chosen_bounded. ff_lt_gcrt_dominating_canonical_chosen_bounded + S x = L) /\ (forall gcrt_solution_index_dominating_canonical_chosen_solution gcrt_solution_residue_dominating_canonical_chosen_solution gcrt_solution_modulus_dominating_canonical_chosen_solution. (exists ff_lt_gcrt_dominating_canonical_chosen_solution_bound. ff_lt_gcrt_dominating_canonical_chosen_solution_bound + S gcrt_solution_index_dominating_canonical_chosen_solution = S l) -> (((exists ff_h_gcrt_dominating_canonical_chosen_solution_residue. ff_h_gcrt_dominating_canonical_chosen_solution_residue + S (gcrt_solution_residue_dominating_canonical_chosen_solution) = S ((S (gcrt_solution_index_dominating_canonical_chosen_solution)) * s)) /\ exists ff_q_gcrt_dominating_canonical_chosen_solution_residue. r = ff_q_gcrt_dominating_canonical_chosen_solution_residue * S ((S (gcrt_solution_index_dominating_canonical_chosen_solution)) * s) + (gcrt_solution_residue_dominating_canonical_chosen_solution))) -> (((exists ff_h_gcrt_dominating_canonical_chosen_solution_modulus. ff_h_gcrt_dominating_canonical_chosen_solution_modulus + S (gcrt_solution_modulus_dominating_canonical_chosen_solution) = S ((S (gcrt_solution_index_dominating_canonical_chosen_solution)) * c)) /\ exists ff_q_gcrt_dominating_canonical_chosen_solution_modulus. b = ff_q_gcrt_dominating_canonical_chosen_solution_modulus * S ((S (gcrt_solution_index_dominating_canonical_chosen_solution)) * c) + (gcrt_solution_modulus_dominating_canonical_chosen_solution))) -> (exists hgcrt_mod_left_gcrt_dominating_canonical_chosen_solution_congruence hgcrt_mod_right_gcrt_dominating_canonical_chosen_solution_congruence. x + gcrt_solution_modulus_dominating_canonical_chosen_solution * hgcrt_mod_left_gcrt_dominating_canonical_chosen_solution_congruence = gcrt_solution_residue_dominating_canonical_chosen_solution + gcrt_solution_modulus_dominating_canonical_chosen_solution * hgcrt_mod_right_gcrt_dominating_canonical_chosen_solution_congruence))))) /\ forall y. (((((forall gcrt_common_index_dominating_canonical_compared_lcm_own gcrt_common_modulus_dominating_canonical_compared_lcm_own. (exists ff_lt_gcrt_dominating_canonical_compared_lcm_own_bound. ff_lt_gcrt_dominating_canonical_compared_lcm_own_bound + S gcrt_common_index_dominating_canonical_compared_lcm_own = S l) -> (((exists ff_h_gcrt_dominating_canonical_compared_lcm_own_entry. ff_h_gcrt_dominating_canonical_compared_lcm_own_entry + S (gcrt_common_modulus_dominating_canonical_compared_lcm_own) = S ((S (gcrt_common_index_dominating_canonical_compared_lcm_own)) * c)) /\ exists ff_q_gcrt_dominating_canonical_compared_lcm_own_entry. b = ff_q_gcrt_dominating_canonical_compared_lcm_own_entry * S ((S (gcrt_common_index_dominating_canonical_compared_lcm_own)) * c) + (gcrt_common_modulus_dominating_canonical_compared_lcm_own))) -> exists gcrt_common_quotient_dominating_canonical_compared_lcm_own. L = gcrt_common_modulus_dominating_canonical_compared_lcm_own * gcrt_common_quotient_dominating_canonical_compared_lcm_own) /\ forall gcrt_lcm_common_dominating_canonical_compared_lcm. (forall gcrt_common_index_dominating_canonical_compared_lcm_other gcrt_common_modulus_dominating_canonical_compared_lcm_other. (exists ff_lt_gcrt_dominating_canonical_compared_lcm_other_bound. ff_lt_gcrt_dominating_canonical_compared_lcm_other_bound + S gcrt_common_index_dominating_canonical_compared_lcm_other = S l) -> (((exists ff_h_gcrt_dominating_canonical_compared_lcm_other_entry. ff_h_gcrt_dominating_canonical_compared_lcm_other_entry + S (gcrt_common_modulus_dominating_canonical_compared_lcm_other) = S ((S (gcrt_common_index_dominating_canonical_compared_lcm_other)) * c)) /\ exists ff_q_gcrt_dominating_canonical_compared_lcm_other_entry. b = ff_q_gcrt_dominating_canonical_compared_lcm_other_entry * S ((S (gcrt_common_index_dominating_canonical_compared_lcm_other)) * c) + (gcrt_common_modulus_dominating_canonical_compared_lcm_other))) -> exists gcrt_common_quotient_dominating_canonical_compared_lcm_other. gcrt_lcm_common_dominating_canonical_compared_lcm = gcrt_common_modulus_dominating_canonical_compared_lcm_other * gcrt_common_quotient_dominating_canonical_compared_lcm_other) -> exists gcrt_lcm_quotient_dominating_canonical_compared_lcm. gcrt_lcm_common_dominating_canonical_compared_lcm = L * gcrt_lcm_quotient_dominating_canonical_compared_lcm)) /\ ((exists ff_lt_gcrt_dominating_canonical_compared_bounded. ff_lt_gcrt_dominating_canonical_compared_bounded + S y = L) /\ (forall gcrt_solution_index_dominating_canonical_compared_solution gcrt_solution_residue_dominating_canonical_compared_solution gcrt_solution_modulus_dominating_canonical_compared_solution. (exists ff_lt_gcrt_dominating_canonical_compared_solution_bound. ff_lt_gcrt_dominating_canonical_compared_solution_bound + S gcrt_solution_index_dominating_canonical_compared_solution = S l) -> (((exists ff_h_gcrt_dominating_canonical_compared_solution_residue. ff_h_gcrt_dominating_canonical_compared_solution_residue + S (gcrt_solution_residue_dominating_canonical_compared_solution) = S ((S (gcrt_solution_index_dominating_canonical_compared_solution)) * s)) /\ exists ff_q_gcrt_dominating_canonical_compared_solution_residue. r = ff_q_gcrt_dominating_canonical_compared_solution_residue * S ((S (gcrt_solution_index_dominating_canonical_compared_solution)) * s) + (gcrt_solution_residue_dominating_canonical_compared_solution))) -> (((exists ff_h_gcrt_dominating_canonical_compared_solution_modulus. ff_h_gcrt_dominating_canonical_compared_solution_modulus + S (gcrt_solution_modulus_dominating_canonical_compared_solution) = S ((S (gcrt_solution_index_dominating_canonical_compared_solution)) * c)) /\ exists ff_q_gcrt_dominating_canonical_compared_solution_modulus. b = ff_q_gcrt_dominating_canonical_compared_solution_modulus * S ((S (gcrt_solution_index_dominating_canonical_compared_solution)) * c) + (gcrt_solution_modulus_dominating_canonical_compared_solution))) -> (exists hgcrt_mod_left_gcrt_dominating_canonical_compared_solution_congruence hgcrt_mod_right_gcrt_dominating_canonical_compared_solution_congruence. y + gcrt_solution_modulus_dominating_canonical_compared_solution * hgcrt_mod_left_gcrt_dominating_canonical_compared_solution_congruence = gcrt_solution_residue_dominating_canonical_compared_solution + gcrt_solution_modulus_dominating_canonical_compared_solution * hgcrt_mod_right_gcrt_dominating_canonical_compared_solution_congruence))))) -> y = x)