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.
Statement with defined notation
∀ p. ∀ n. ∀ qb. ∀ qc. ∀ db. ∀ dc. ∀ l. ∀ i. BetaAt(qb,qc,0,n) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ m. BetaAt(qb,qc,x,y) ∧ (BetaAt(qb,qc,S x,z) ∧ (BetaAt(db,dc,x,m) ∧ DivRem(y,p,z,m)))) → Lt(i,l) → ∃ x. ∃ y. ∃ z. BetaAt(qb,qc,i,x) ∧ (BetaAt(qb,qc,S i,y) ∧ (BetaAt(db,dc,i,z) ∧ DivRem(x,p,y,z)))Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.
Definitions used by this theorem
In the theorem statement
In local proof propositions
Exact expanded first-order statement
forall p n qb qc db dc l i. (((((exists ff_h_lmd_total_result_initial. ff_h_lmd_total_result_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_total_result_initial. qb = ff_q_lmd_total_result_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_total_result. (exists lmd_gap_total_result_index. lmd_gap_total_result_index + S (lmd_index_total_result) = (l)) -> exists lmd_current_total_result lmd_successor_total_result lmd_digit_total_result. ((((exists ff_h_lmd_total_result_current. ff_h_lmd_total_result_current + S (lmd_current_total_result) = S ((S (lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_current. qb = ff_q_lmd_total_result_current * S ((S (lmd_index_total_result)) * qc) + (lmd_current_total_result))) /\ ((((exists ff_h_lmd_total_result_successor. ff_h_lmd_total_result_successor + S (lmd_successor_total_result) = S ((S (S lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_successor. qb = ff_q_lmd_total_result_successor * S ((S (S lmd_index_total_result)) * qc) + (lmd_successor_total_result))) /\ ((((exists ff_h_lmd_total_result_digit. ff_h_lmd_total_result_digit + S (lmd_digit_total_result) = S ((S (lmd_index_total_result)) * dc)) /\ exists ff_q_lmd_total_result_digit. db = ff_q_lmd_total_result_digit * S ((S (lmd_index_total_result)) * dc) + (lmd_digit_total_result))) /\ ((lmd_current_total_result = (p) * (lmd_successor_total_result) + (lmd_digit_total_result)) /\ (exists lmd_gap_total_result_digit_bound. lmd_gap_total_result_digit_bound + S (lmd_digit_total_result) = (p)))))))) -> (exists lmd_gap_point_index. lmd_gap_point_index + S (i) = (l)) -> (exists q Q d. ((((exists ff_h_lmd_point_current. ff_h_lmd_point_current + S (q) = S ((S (i)) * qc)) /\ exists ff_q_lmd_point_current. qb = ff_q_lmd_point_current * S ((S (i)) * qc) + (q))) /\ ((((exists ff_h_lmd_point_next. ff_h_lmd_point_next + S (Q) = S ((S (S i)) * qc)) /\ exists ff_q_lmd_point_next. qb = ff_q_lmd_point_next * S ((S (S i)) * qc) + (Q))) /\ ((((exists ff_h_lmd_point_digit. ff_h_lmd_point_digit + S (d) = S ((S (i)) * dc)) /\ exists ff_q_lmd_point_digit. db = ff_q_lmd_point_digit * S ((S (i)) * dc) + (d))) /\ ((q = p * Q + d) /\ (exists lmd_gap_point_digit_bound. lmd_gap_point_digit_bound + S (d) = (p)))))))Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.
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 assumptionsL1–10
02Separate the logical casesL11–11
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L11
cases hchain