crt_pairwise_compatible_prefix_solvable_iff

Exact expanded first-order arithmetic statement

forall r s b c l. (((forall gcomp_left_index_gfull_solvable_pairs_forward gcomp_right_index_gfull_solvable_pairs_forward gcomp_left_residue_gfull_solvable_pairs_forward gcomp_right_residue_gfull_solvable_pairs_forward gcomp_left_modulus_gfull_solvable_pairs_forward gcomp_right_modulus_gfull_solvable_pairs_forward gcomp_pair_gcd_gfull_solvable_pairs_forward. (exists ff_lt_gcrt_gfull_solvable_pairs_forward_left_bound. ff_lt_gcrt_gfull_solvable_pairs_forward_left_bound + S gcomp_left_index_gfull_solvable_pairs_forward = l) -> (exists ff_lt_gcrt_gfull_solvable_pairs_forward_right_bound. ff_lt_gcrt_gfull_solvable_pairs_forward_right_bound + S gcomp_right_index_gfull_solvable_pairs_forward = l) -> (((exists ff_h_gcrt_gfull_solvable_pairs_forward_left_residue. ff_h_gcrt_gfull_solvable_pairs_forward_left_residue + S (gcomp_left_residue_gfull_solvable_pairs_forward) = S ((S (gcomp_left_index_gfull_solvable_pairs_forward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_pairs_forward_left_residue. r = ff_q_gcrt_gfull_solvable_pairs_forward_left_residue * S ((S (gcomp_left_index_gfull_solvable_pairs_forward)) * s) + (gcomp_left_residue_gfull_solvable_pairs_forward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_forward_right_residue. ff_h_gcrt_gfull_solvable_pairs_forward_right_residue + S (gcomp_right_residue_gfull_solvable_pairs_forward) = S ((S (gcomp_right_index_gfull_solvable_pairs_forward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_pairs_forward_right_residue. r = ff_q_gcrt_gfull_solvable_pairs_forward_right_residue * S ((S (gcomp_right_index_gfull_solvable_pairs_forward)) * s) + (gcomp_right_residue_gfull_solvable_pairs_forward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_forward_left_modulus. ff_h_gcrt_gfull_solvable_pairs_forward_left_modulus + S (gcomp_left_modulus_gfull_solvable_pairs_forward) = S ((S (gcomp_left_index_gfull_solvable_pairs_forward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_pairs_forward_left_modulus. b = ff_q_gcrt_gfull_solvable_pairs_forward_left_modulus * S ((S (gcomp_left_index_gfull_solvable_pairs_forward)) * c) + (gcomp_left_modulus_gfull_solvable_pairs_forward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_forward_right_modulus. ff_h_gcrt_gfull_solvable_pairs_forward_right_modulus + S (gcomp_right_modulus_gfull_solvable_pairs_forward) = S ((S (gcomp_right_index_gfull_solvable_pairs_forward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_pairs_forward_right_modulus. b = ff_q_gcrt_gfull_solvable_pairs_forward_right_modulus * S ((S (gcomp_right_index_gfull_solvable_pairs_forward)) * c) + (gcomp_right_modulus_gfull_solvable_pairs_forward))) -> ((((exists hag_left_factor_gcomp_gfull_solvable_pairs_forward_gcd. gcomp_left_modulus_gfull_solvable_pairs_forward = gcomp_pair_gcd_gfull_solvable_pairs_forward * hag_left_factor_gcomp_gfull_solvable_pairs_forward_gcd) /\ (exists hag_right_factor_gcomp_gfull_solvable_pairs_forward_gcd. gcomp_right_modulus_gfull_solvable_pairs_forward = gcomp_pair_gcd_gfull_solvable_pairs_forward * hag_right_factor_gcomp_gfull_solvable_pairs_forward_gcd)) /\ forall hag_divisor_gcomp_gfull_solvable_pairs_forward_gcd. (exists hag_common_left_gcomp_gfull_solvable_pairs_forward_gcd. gcomp_left_modulus_gfull_solvable_pairs_forward = hag_divisor_gcomp_gfull_solvable_pairs_forward_gcd * hag_common_left_gcomp_gfull_solvable_pairs_forward_gcd) -> (exists hag_common_right_gcomp_gfull_solvable_pairs_forward_gcd. gcomp_right_modulus_gfull_solvable_pairs_forward = hag_divisor_gcomp_gfull_solvable_pairs_forward_gcd * hag_common_right_gcomp_gfull_solvable_pairs_forward_gcd) -> exists hag_greatest_factor_gcomp_gfull_solvable_pairs_forward_gcd. gcomp_pair_gcd_gfull_solvable_pairs_forward = hag_divisor_gcomp_gfull_solvable_pairs_forward_gcd * hag_greatest_factor_gcomp_gfull_solvable_pairs_forward_gcd)) -> (exists hgcrt_mod_left_gcrt_gfull_solvable_pairs_forward_result hgcrt_mod_right_gcrt_gfull_solvable_pairs_forward_result. gcomp_left_residue_gfull_solvable_pairs_forward + gcomp_pair_gcd_gfull_solvable_pairs_forward * hgcrt_mod_left_gcrt_gfull_solvable_pairs_forward_result = gcomp_right_residue_gfull_solvable_pairs_forward + gcomp_pair_gcd_gfull_solvable_pairs_forward * hgcrt_mod_right_gcrt_gfull_solvable_pairs_forward_result)) -> exists x. (forall gcrt_solution_index_gfull_solvable_solution_forward gcrt_solution_residue_gfull_solvable_solution_forward gcrt_solution_modulus_gfull_solvable_solution_forward. (exists ff_lt_gcrt_gfull_solvable_solution_forward_bound. ff_lt_gcrt_gfull_solvable_solution_forward_bound + S gcrt_solution_index_gfull_solvable_solution_forward = l) -> (((exists ff_h_gcrt_gfull_solvable_solution_forward_residue. ff_h_gcrt_gfull_solvable_solution_forward_residue + S (gcrt_solution_residue_gfull_solvable_solution_forward) = S ((S (gcrt_solution_index_gfull_solvable_solution_forward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_solution_forward_residue. r = ff_q_gcrt_gfull_solvable_solution_forward_residue * S ((S (gcrt_solution_index_gfull_solvable_solution_forward)) * s) + (gcrt_solution_residue_gfull_solvable_solution_forward))) -> (((exists ff_h_gcrt_gfull_solvable_solution_forward_modulus. ff_h_gcrt_gfull_solvable_solution_forward_modulus + S (gcrt_solution_modulus_gfull_solvable_solution_forward) = S ((S (gcrt_solution_index_gfull_solvable_solution_forward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_solution_forward_modulus. b = ff_q_gcrt_gfull_solvable_solution_forward_modulus * S ((S (gcrt_solution_index_gfull_solvable_solution_forward)) * c) + (gcrt_solution_modulus_gfull_solvable_solution_forward))) -> (exists hgcrt_mod_left_gcrt_gfull_solvable_solution_forward_congruence hgcrt_mod_right_gcrt_gfull_solvable_solution_forward_congruence. x + gcrt_solution_modulus_gfull_solvable_solution_forward * hgcrt_mod_left_gcrt_gfull_solvable_solution_forward_congruence = gcrt_solution_residue_gfull_solvable_solution_forward + gcrt_solution_modulus_gfull_solvable_solution_forward * hgcrt_mod_right_gcrt_gfull_solvable_solution_forward_congruence))) /\ ((exists x. (forall gcrt_solution_index_gfull_solvable_solution_backward gcrt_solution_residue_gfull_solvable_solution_backward gcrt_solution_modulus_gfull_solvable_solution_backward. (exists ff_lt_gcrt_gfull_solvable_solution_backward_bound. ff_lt_gcrt_gfull_solvable_solution_backward_bound + S gcrt_solution_index_gfull_solvable_solution_backward = l) -> (((exists ff_h_gcrt_gfull_solvable_solution_backward_residue. ff_h_gcrt_gfull_solvable_solution_backward_residue + S (gcrt_solution_residue_gfull_solvable_solution_backward) = S ((S (gcrt_solution_index_gfull_solvable_solution_backward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_solution_backward_residue. r = ff_q_gcrt_gfull_solvable_solution_backward_residue * S ((S (gcrt_solution_index_gfull_solvable_solution_backward)) * s) + (gcrt_solution_residue_gfull_solvable_solution_backward))) -> (((exists ff_h_gcrt_gfull_solvable_solution_backward_modulus. ff_h_gcrt_gfull_solvable_solution_backward_modulus + S (gcrt_solution_modulus_gfull_solvable_solution_backward) = S ((S (gcrt_solution_index_gfull_solvable_solution_backward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_solution_backward_modulus. b = ff_q_gcrt_gfull_solvable_solution_backward_modulus * S ((S (gcrt_solution_index_gfull_solvable_solution_backward)) * c) + (gcrt_solution_modulus_gfull_solvable_solution_backward))) -> (exists hgcrt_mod_left_gcrt_gfull_solvable_solution_backward_congruence hgcrt_mod_right_gcrt_gfull_solvable_solution_backward_congruence. x + gcrt_solution_modulus_gfull_solvable_solution_backward * hgcrt_mod_left_gcrt_gfull_solvable_solution_backward_congruence = gcrt_solution_residue_gfull_solvable_solution_backward + gcrt_solution_modulus_gfull_solvable_solution_backward * hgcrt_mod_right_gcrt_gfull_solvable_solution_backward_congruence))) -> (forall gcomp_left_index_gfull_solvable_pairs_backward gcomp_right_index_gfull_solvable_pairs_backward gcomp_left_residue_gfull_solvable_pairs_backward gcomp_right_residue_gfull_solvable_pairs_backward gcomp_left_modulus_gfull_solvable_pairs_backward gcomp_right_modulus_gfull_solvable_pairs_backward gcomp_pair_gcd_gfull_solvable_pairs_backward. (exists ff_lt_gcrt_gfull_solvable_pairs_backward_left_bound. ff_lt_gcrt_gfull_solvable_pairs_backward_left_bound + S gcomp_left_index_gfull_solvable_pairs_backward = l) -> (exists ff_lt_gcrt_gfull_solvable_pairs_backward_right_bound. ff_lt_gcrt_gfull_solvable_pairs_backward_right_bound + S gcomp_right_index_gfull_solvable_pairs_backward = l) -> (((exists ff_h_gcrt_gfull_solvable_pairs_backward_left_residue. ff_h_gcrt_gfull_solvable_pairs_backward_left_residue + S (gcomp_left_residue_gfull_solvable_pairs_backward) = S ((S (gcomp_left_index_gfull_solvable_pairs_backward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_pairs_backward_left_residue. r = ff_q_gcrt_gfull_solvable_pairs_backward_left_residue * S ((S (gcomp_left_index_gfull_solvable_pairs_backward)) * s) + (gcomp_left_residue_gfull_solvable_pairs_backward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_backward_right_residue. ff_h_gcrt_gfull_solvable_pairs_backward_right_residue + S (gcomp_right_residue_gfull_solvable_pairs_backward) = S ((S (gcomp_right_index_gfull_solvable_pairs_backward)) * s)) /\ exists ff_q_gcrt_gfull_solvable_pairs_backward_right_residue. r = ff_q_gcrt_gfull_solvable_pairs_backward_right_residue * S ((S (gcomp_right_index_gfull_solvable_pairs_backward)) * s) + (gcomp_right_residue_gfull_solvable_pairs_backward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_backward_left_modulus. ff_h_gcrt_gfull_solvable_pairs_backward_left_modulus + S (gcomp_left_modulus_gfull_solvable_pairs_backward) = S ((S (gcomp_left_index_gfull_solvable_pairs_backward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_pairs_backward_left_modulus. b = ff_q_gcrt_gfull_solvable_pairs_backward_left_modulus * S ((S (gcomp_left_index_gfull_solvable_pairs_backward)) * c) + (gcomp_left_modulus_gfull_solvable_pairs_backward))) -> (((exists ff_h_gcrt_gfull_solvable_pairs_backward_right_modulus. ff_h_gcrt_gfull_solvable_pairs_backward_right_modulus + S (gcomp_right_modulus_gfull_solvable_pairs_backward) = S ((S (gcomp_right_index_gfull_solvable_pairs_backward)) * c)) /\ exists ff_q_gcrt_gfull_solvable_pairs_backward_right_modulus. b = ff_q_gcrt_gfull_solvable_pairs_backward_right_modulus * S ((S (gcomp_right_index_gfull_solvable_pairs_backward)) * c) + (gcomp_right_modulus_gfull_solvable_pairs_backward))) -> ((((exists hag_left_factor_gcomp_gfull_solvable_pairs_backward_gcd. gcomp_left_modulus_gfull_solvable_pairs_backward = gcomp_pair_gcd_gfull_solvable_pairs_backward * hag_left_factor_gcomp_gfull_solvable_pairs_backward_gcd) /\ (exists hag_right_factor_gcomp_gfull_solvable_pairs_backward_gcd. gcomp_right_modulus_gfull_solvable_pairs_backward = gcomp_pair_gcd_gfull_solvable_pairs_backward * hag_right_factor_gcomp_gfull_solvable_pairs_backward_gcd)) /\ forall hag_divisor_gcomp_gfull_solvable_pairs_backward_gcd. (exists hag_common_left_gcomp_gfull_solvable_pairs_backward_gcd. gcomp_left_modulus_gfull_solvable_pairs_backward = hag_divisor_gcomp_gfull_solvable_pairs_backward_gcd * hag_common_left_gcomp_gfull_solvable_pairs_backward_gcd) -> (exists hag_common_right_gcomp_gfull_solvable_pairs_backward_gcd. gcomp_right_modulus_gfull_solvable_pairs_backward = hag_divisor_gcomp_gfull_solvable_pairs_backward_gcd * hag_common_right_gcomp_gfull_solvable_pairs_backward_gcd) -> exists hag_greatest_factor_gcomp_gfull_solvable_pairs_backward_gcd. gcomp_pair_gcd_gfull_solvable_pairs_backward = hag_divisor_gcomp_gfull_solvable_pairs_backward_gcd * hag_greatest_factor_gcomp_gfull_solvable_pairs_backward_gcd)) -> (exists hgcrt_mod_left_gcrt_gfull_solvable_pairs_backward_result hgcrt_mod_right_gcrt_gfull_solvable_pairs_backward_result. gcomp_left_residue_gfull_solvable_pairs_backward + gcomp_pair_gcd_gfull_solvable_pairs_backward * hgcrt_mod_left_gcrt_gfull_solvable_pairs_backward_result = gcomp_right_residue_gfull_solvable_pairs_backward + gcomp_pair_gcd_gfull_solvable_pairs_backward * hgcrt_mod_right_gcrt_gfull_solvable_pairs_backward_result))))