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
∀ n. ∀ q. ∀ r. DivRem(2 · n,3,q,r) → ∃ x. q + x = n ∧ Le(2 · n,6 · x) ∧ Lt(r,3)Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
3 occurrences
In local proof propositions
1 occurrences
Exact expanded native-PA statement
forall n q r. (((2 * (n) = 3 * (q) + (r)) /\ exists bqb_remainder_gap_canonical_source. bqb_remainder_gap_canonical_source + S (r) = 3)) -> exists c. (((((q) + (c) = (n)) /\ exists bqb_budget_gap_canonical_result. bqb_budget_gap_canonical_result + 2 * (n) = 6 * (c))) /\ exists bqb_remainder_gap_canonical_preserved. bqb_remainder_gap_canonical_preserved + S r = 3)Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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.
Named ingredients (1)
01Fix variables and assumptionsL1–4
02Separate the logical casesL5–5
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L5
cases hcanonical
03Establish hbaseL6–11
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply double triple remainder complement budget.
- L6
have hbase : ∃ c. q + c = n ∧ Le(2 · n,6 · c)Definitions: Le(2 · n,6 · c)Original native command in the exact edition - L7
specialize double_triple_remainder_complement_budget n - L8
specialize double_triple_remainder_complement_budget q - L9
specialize double_triple_remainder_complement_budget r - L10
apply double_triple_remainder_complement_budget - L11
exact hcanonical_left
04Separate the logical casesL12–12
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L12
cases hbase
05Construct an explicit witnessL13–13
Supply the displayed value, then prove that it has the required property.
- L13
exists x
06Separate the logical casesL14–14
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L14
split
Original defined command ledger · 16 lines
- 0001
intro n - 0002
intro q - 0003
intro r - 0004
intro hcanonical - 0005
cases hcanonical - 0006
have hbase : ∃ c. q + c = n ∧ Le(2 · n,6 · c)Exact native replay line
have hbase : exists c. ((((q) + (c) = (n)) /\ exists bqb_budget_gap_canonical_base. bqb_budget_gap_canonical_base + 2 * (n) = 6 * (c))) - 0007
specialize double_triple_remainder_complement_budget n - 0008
specialize double_triple_remainder_complement_budget q - 0009
specialize double_triple_remainder_complement_budget r - 0010
apply double_triple_remainder_complement_budget - 0011
exact hcanonical_left - 0012
cases hbase - 0013
exists x - 0014
split - 0015
exact hbase_witness - 0016
exact hcanonical_right