Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Definition in prerequisite notation
CRTPrefixLCM(b,c,l,M) ∧ ((M = 0 ∨ Lt(x,M)) ∧ CRTPrefixSolution(r,s,b,c,l,x))
Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
((((forall gcrt_common_index_gfull_secondwave_lcm_own gcrt_common_modulus_gfull_secondwave_lcm_own. (exists ff_lt_gcrt_gfull_secondwave_lcm_own_bound. ff_lt_gcrt_gfull_secondwave_lcm_own_bound + S gcrt_common_index_gfull_secondwave_lcm_own = l) -> (((exists ff_h_gcrt_gfull_secondwave_lcm_own_entry. ff_h_gcrt_gfull_secondwave_lcm_own_entry + S (gcrt_common_modulus_gfull_secondwave_lcm_own) = S ((S (gcrt_common_index_gfull_secondwave_lcm_own)) * c)) /\ exists ff_q_gcrt_gfull_secondwave_lcm_own_entry. b = ff_q_gcrt_gfull_secondwave_lcm_own_entry * S ((S (gcrt_common_index_gfull_secondwave_lcm_own)) * c) + (gcrt_common_modulus_gfull_secondwave_lcm_own))) -> exists gcrt_common_quotient_gfull_secondwave_lcm_own. M = gcrt_common_modulus_gfull_secondwave_lcm_own * gcrt_common_quotient_gfull_secondwave_lcm_own) /\ forall gcrt_lcm_common_gfull_secondwave_lcm. (forall gcrt_common_index_gfull_secondwave_lcm_other gcrt_common_modulus_gfull_secondwave_lcm_other. (exists ff_lt_gcrt_gfull_secondwave_lcm_other_bound. ff_lt_gcrt_gfull_secondwave_lcm_other_bound + S gcrt_common_index_gfull_secondwave_lcm_other = l) -> (((exists ff_h_gcrt_gfull_secondwave_lcm_other_entry. ff_h_gcrt_gfull_secondwave_lcm_other_entry + S (gcrt_common_modulus_gfull_secondwave_lcm_other) = S ((S (gcrt_common_index_gfull_secondwave_lcm_other)) * c)) /\ exists ff_q_gcrt_gfull_secondwave_lcm_other_entry. b = ff_q_gcrt_gfull_secondwave_lcm_other_entry * S ((S (gcrt_common_index_gfull_secondwave_lcm_other)) * c) + (gcrt_common_modulus_gfull_secondwave_lcm_other))) -> exists gcrt_common_quotient_gfull_secondwave_lcm_other. gcrt_lcm_common_gfull_secondwave_lcm = gcrt_common_modulus_gfull_secondwave_lcm_other * gcrt_common_quotient_gfull_secondwave_lcm_other) -> exists gcrt_lcm_quotient_gfull_secondwave_lcm. gcrt_lcm_common_gfull_secondwave_lcm = M * gcrt_lcm_quotient_gfull_secondwave_lcm)) /\ ((M = 0 \/ (exists ff_lt_gcrt_gfull_secondwave_bound. ff_lt_gcrt_gfull_secondwave_bound + S x = M)) /\ (forall gcrt_solution_index_gfull_secondwave_solution gcrt_solution_residue_gfull_secondwave_solution gcrt_solution_modulus_gfull_secondwave_solution. (exists ff_lt_gcrt_gfull_secondwave_solution_bound. ff_lt_gcrt_gfull_secondwave_solution_bound + S gcrt_solution_index_gfull_secondwave_solution = l) -> (((exists ff_h_gcrt_gfull_secondwave_solution_residue. ff_h_gcrt_gfull_secondwave_solution_residue + S (gcrt_solution_residue_gfull_secondwave_solution) = S ((S (gcrt_solution_index_gfull_secondwave_solution)) * s)) /\ exists ff_q_gcrt_gfull_secondwave_solution_residue. r = ff_q_gcrt_gfull_secondwave_solution_residue * S ((S (gcrt_solution_index_gfull_secondwave_solution)) * s) + (gcrt_solution_residue_gfull_secondwave_solution))) -> (((exists ff_h_gcrt_gfull_secondwave_solution_modulus. ff_h_gcrt_gfull_secondwave_solution_modulus + S (gcrt_solution_modulus_gfull_secondwave_solution) = S ((S (gcrt_solution_index_gfull_secondwave_solution)) * c)) /\ exists ff_q_gcrt_gfull_secondwave_solution_modulus. b = ff_q_gcrt_gfull_secondwave_solution_modulus * S ((S (gcrt_solution_index_gfull_secondwave_solution)) * c) + (gcrt_solution_modulus_gfull_secondwave_solution))) -> (exists hgcrt_mod_left_gcrt_gfull_secondwave_solution_congruence hgcrt_mod_right_gcrt_gfull_secondwave_solution_congruence. x + gcrt_solution_modulus_gfull_secondwave_solution * hgcrt_mod_left_gcrt_gfull_secondwave_solution_congruence = gcrt_solution_residue_gfull_secondwave_solution + gcrt_solution_modulus_gfull_secondwave_solution * hgcrt_mod_right_gcrt_gfull_secondwave_solution_congruence))))
The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
none
Checked theorems using this definition
FC0010 · crt_canonical_prefix_solution_implies_normalizedFC0011 · crt_normalized_prefix_solution_uniqueFC0012 · crt_prefix_solution_normalized_existsFC0013 · crt_pairwise_compatible_prefix_normalized_exists_uniqueFC0016 · crt_normalized_prefix_solution_class_iff_lcmFC0017 · crt_normalized_zero_lcm_all_solutions_uniqueFC0018 · crt_positive_normalized_prefix_iff_canonical