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. BetaAt(qb,qc,0,n) → BetaAt(qb,qc,0,n) ∧ (∀ x. Lt(x,0) → ∃ y. ∃ z. ∃ m. BetaAt(qb,qc,x,y) ∧ (BetaAt(qb,qc,S x,z) ∧ (BetaAt(db,dc,x,m) ∧ DivRem(y,p,z,m))))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. (((exists ff_h_lmd_initial_source. ff_h_lmd_initial_source + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_initial_source. qb = ff_q_lmd_initial_source * S ((S (0)) * qc) + (n))) -> (((((exists ff_h_lmd_empty_initial. ff_h_lmd_empty_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_empty_initial. qb = ff_q_lmd_empty_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_empty. (exists lmd_gap_empty_index. lmd_gap_empty_index + S (lmd_index_empty) = (0)) -> exists lmd_current_empty lmd_successor_empty lmd_digit_empty. ((((exists ff_h_lmd_empty_current. ff_h_lmd_empty_current + S (lmd_current_empty) = S ((S (lmd_index_empty)) * qc)) /\ exists ff_q_lmd_empty_current. qb = ff_q_lmd_empty_current * S ((S (lmd_index_empty)) * qc) + (lmd_current_empty))) /\ ((((exists ff_h_lmd_empty_successor. ff_h_lmd_empty_successor + S (lmd_successor_empty) = S ((S (S lmd_index_empty)) * qc)) /\ exists ff_q_lmd_empty_successor. qb = ff_q_lmd_empty_successor * S ((S (S lmd_index_empty)) * qc) + (lmd_successor_empty))) /\ ((((exists ff_h_lmd_empty_digit. ff_h_lmd_empty_digit + S (lmd_digit_empty) = S ((S (lmd_index_empty)) * dc)) /\ exists ff_q_lmd_empty_digit. db = ff_q_lmd_empty_digit * S ((S (lmd_index_empty)) * dc) + (lmd_digit_empty))) /\ ((lmd_current_empty = (p) * (lmd_successor_empty) + (lmd_digit_empty)) /\ (exists lmd_gap_empty_digit_bound. lmd_gap_empty_digit_bound + S (lmd_digit_empty) = (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 assumptionsL1–7
02Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
split
03Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
exact hinitial
04Fix variables and assumptionsL10–11
05Separate the logical casesL12–13
06Calculate and transport equalitiesL14–14
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L14
rewrite PA4 at hi_witness