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.
Exact expanded first-order arithmetic statement
forall r s b c l x M. (((((forall gcrt_common_index_gfull_canonical_to_normal_source_lcm_own gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_own. (exists ff_lt_gcrt_gfull_canonical_to_normal_source_lcm_own_bound. ff_lt_gcrt_gfull_canonical_to_normal_source_lcm_own_bound + S gcrt_common_index_gfull_canonical_to_normal_source_lcm_own = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_source_lcm_own_entry. ff_h_gcrt_gfull_canonical_to_normal_source_lcm_own_entry + S (gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_own) = S ((S (gcrt_common_index_gfull_canonical_to_normal_source_lcm_own)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_source_lcm_own_entry. b = ff_q_gcrt_gfull_canonical_to_normal_source_lcm_own_entry * S ((S (gcrt_common_index_gfull_canonical_to_normal_source_lcm_own)) * c) + (gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_own))) -> exists gcrt_common_quotient_gfull_canonical_to_normal_source_lcm_own. M = gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_own * gcrt_common_quotient_gfull_canonical_to_normal_source_lcm_own) /\ forall gcrt_lcm_common_gfull_canonical_to_normal_source_lcm. (forall gcrt_common_index_gfull_canonical_to_normal_source_lcm_other gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_other. (exists ff_lt_gcrt_gfull_canonical_to_normal_source_lcm_other_bound. ff_lt_gcrt_gfull_canonical_to_normal_source_lcm_other_bound + S gcrt_common_index_gfull_canonical_to_normal_source_lcm_other = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_source_lcm_other_entry. ff_h_gcrt_gfull_canonical_to_normal_source_lcm_other_entry + S (gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_other) = S ((S (gcrt_common_index_gfull_canonical_to_normal_source_lcm_other)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_source_lcm_other_entry. b = ff_q_gcrt_gfull_canonical_to_normal_source_lcm_other_entry * S ((S (gcrt_common_index_gfull_canonical_to_normal_source_lcm_other)) * c) + (gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_other))) -> exists gcrt_common_quotient_gfull_canonical_to_normal_source_lcm_other. gcrt_lcm_common_gfull_canonical_to_normal_source_lcm = gcrt_common_modulus_gfull_canonical_to_normal_source_lcm_other * gcrt_common_quotient_gfull_canonical_to_normal_source_lcm_other) -> exists gcrt_lcm_quotient_gfull_canonical_to_normal_source_lcm. gcrt_lcm_common_gfull_canonical_to_normal_source_lcm = M * gcrt_lcm_quotient_gfull_canonical_to_normal_source_lcm)) /\ ((exists ff_lt_gcrt_gfull_canonical_to_normal_source_bounded. ff_lt_gcrt_gfull_canonical_to_normal_source_bounded + S x = M) /\ (forall gcrt_solution_index_gfull_canonical_to_normal_source_solution gcrt_solution_residue_gfull_canonical_to_normal_source_solution gcrt_solution_modulus_gfull_canonical_to_normal_source_solution. (exists ff_lt_gcrt_gfull_canonical_to_normal_source_solution_bound. ff_lt_gcrt_gfull_canonical_to_normal_source_solution_bound + S gcrt_solution_index_gfull_canonical_to_normal_source_solution = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_source_solution_residue. ff_h_gcrt_gfull_canonical_to_normal_source_solution_residue + S (gcrt_solution_residue_gfull_canonical_to_normal_source_solution) = S ((S (gcrt_solution_index_gfull_canonical_to_normal_source_solution)) * s)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_source_solution_residue. r = ff_q_gcrt_gfull_canonical_to_normal_source_solution_residue * S ((S (gcrt_solution_index_gfull_canonical_to_normal_source_solution)) * s) + (gcrt_solution_residue_gfull_canonical_to_normal_source_solution))) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_source_solution_modulus. ff_h_gcrt_gfull_canonical_to_normal_source_solution_modulus + S (gcrt_solution_modulus_gfull_canonical_to_normal_source_solution) = S ((S (gcrt_solution_index_gfull_canonical_to_normal_source_solution)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_source_solution_modulus. b = ff_q_gcrt_gfull_canonical_to_normal_source_solution_modulus * S ((S (gcrt_solution_index_gfull_canonical_to_normal_source_solution)) * c) + (gcrt_solution_modulus_gfull_canonical_to_normal_source_solution))) -> (exists hgcrt_mod_left_gcrt_gfull_canonical_to_normal_source_solution_congruence hgcrt_mod_right_gcrt_gfull_canonical_to_normal_source_solution_congruence. x + gcrt_solution_modulus_gfull_canonical_to_normal_source_solution * hgcrt_mod_left_gcrt_gfull_canonical_to_normal_source_solution_congruence = gcrt_solution_residue_gfull_canonical_to_normal_source_solution + gcrt_solution_modulus_gfull_canonical_to_normal_source_solution * hgcrt_mod_right_gcrt_gfull_canonical_to_normal_source_solution_congruence))))) -> (((((forall gcrt_common_index_gfull_canonical_to_normal_result_lcm_own gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_own. (exists ff_lt_gcrt_gfull_canonical_to_normal_result_lcm_own_bound. ff_lt_gcrt_gfull_canonical_to_normal_result_lcm_own_bound + S gcrt_common_index_gfull_canonical_to_normal_result_lcm_own = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_result_lcm_own_entry. ff_h_gcrt_gfull_canonical_to_normal_result_lcm_own_entry + S (gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_own) = S ((S (gcrt_common_index_gfull_canonical_to_normal_result_lcm_own)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_result_lcm_own_entry. b = ff_q_gcrt_gfull_canonical_to_normal_result_lcm_own_entry * S ((S (gcrt_common_index_gfull_canonical_to_normal_result_lcm_own)) * c) + (gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_own))) -> exists gcrt_common_quotient_gfull_canonical_to_normal_result_lcm_own. M = gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_own * gcrt_common_quotient_gfull_canonical_to_normal_result_lcm_own) /\ forall gcrt_lcm_common_gfull_canonical_to_normal_result_lcm. (forall gcrt_common_index_gfull_canonical_to_normal_result_lcm_other gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_other. (exists ff_lt_gcrt_gfull_canonical_to_normal_result_lcm_other_bound. ff_lt_gcrt_gfull_canonical_to_normal_result_lcm_other_bound + S gcrt_common_index_gfull_canonical_to_normal_result_lcm_other = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_result_lcm_other_entry. ff_h_gcrt_gfull_canonical_to_normal_result_lcm_other_entry + S (gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_other) = S ((S (gcrt_common_index_gfull_canonical_to_normal_result_lcm_other)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_result_lcm_other_entry. b = ff_q_gcrt_gfull_canonical_to_normal_result_lcm_other_entry * S ((S (gcrt_common_index_gfull_canonical_to_normal_result_lcm_other)) * c) + (gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_other))) -> exists gcrt_common_quotient_gfull_canonical_to_normal_result_lcm_other. gcrt_lcm_common_gfull_canonical_to_normal_result_lcm = gcrt_common_modulus_gfull_canonical_to_normal_result_lcm_other * gcrt_common_quotient_gfull_canonical_to_normal_result_lcm_other) -> exists gcrt_lcm_quotient_gfull_canonical_to_normal_result_lcm. gcrt_lcm_common_gfull_canonical_to_normal_result_lcm = M * gcrt_lcm_quotient_gfull_canonical_to_normal_result_lcm)) /\ ((M = 0 \/ (exists ff_lt_gcrt_gfull_canonical_to_normal_result_bound. ff_lt_gcrt_gfull_canonical_to_normal_result_bound + S x = M)) /\ (forall gcrt_solution_index_gfull_canonical_to_normal_result_solution gcrt_solution_residue_gfull_canonical_to_normal_result_solution gcrt_solution_modulus_gfull_canonical_to_normal_result_solution. (exists ff_lt_gcrt_gfull_canonical_to_normal_result_solution_bound. ff_lt_gcrt_gfull_canonical_to_normal_result_solution_bound + S gcrt_solution_index_gfull_canonical_to_normal_result_solution = l) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_result_solution_residue. ff_h_gcrt_gfull_canonical_to_normal_result_solution_residue + S (gcrt_solution_residue_gfull_canonical_to_normal_result_solution) = S ((S (gcrt_solution_index_gfull_canonical_to_normal_result_solution)) * s)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_result_solution_residue. r = ff_q_gcrt_gfull_canonical_to_normal_result_solution_residue * S ((S (gcrt_solution_index_gfull_canonical_to_normal_result_solution)) * s) + (gcrt_solution_residue_gfull_canonical_to_normal_result_solution))) -> (((exists ff_h_gcrt_gfull_canonical_to_normal_result_solution_modulus. ff_h_gcrt_gfull_canonical_to_normal_result_solution_modulus + S (gcrt_solution_modulus_gfull_canonical_to_normal_result_solution) = S ((S (gcrt_solution_index_gfull_canonical_to_normal_result_solution)) * c)) /\ exists ff_q_gcrt_gfull_canonical_to_normal_result_solution_modulus. b = ff_q_gcrt_gfull_canonical_to_normal_result_solution_modulus * S ((S (gcrt_solution_index_gfull_canonical_to_normal_result_solution)) * c) + (gcrt_solution_modulus_gfull_canonical_to_normal_result_solution))) -> (exists hgcrt_mod_left_gcrt_gfull_canonical_to_normal_result_solution_congruence hgcrt_mod_right_gcrt_gfull_canonical_to_normal_result_solution_congruence. x + gcrt_solution_modulus_gfull_canonical_to_normal_result_solution * hgcrt_mod_left_gcrt_gfull_canonical_to_normal_result_solution_congruence = gcrt_solution_residue_gfull_canonical_to_normal_result_solution + gcrt_solution_modulus_gfull_canonical_to_normal_result_solution * hgcrt_mod_right_gcrt_gfull_canonical_to_normal_result_solution_congruence)))))Constructive proof overview
Generated structural guide
Every historical strictly bounded canonical solution is also a zero-safe normalized solution, without changing either definition.
The unchanged tactic script uses 0 declared prerequisites and contains 16 exact native proof lines.
Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
Direct dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
01Fix variables and assumptionsL1–8
02Separate the logical casesL9–11
03Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
exact hc_left