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
∀ d. ∀ n. Le(d,n) → Lt(n,d + d) → ∃ x. DivRem(n,d,1,x)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 d n. (exists bcf_le_gap_bdqob_lower. bcf_le_gap_bdqob_lower + (d) = n) -> (exists bcf_lt_gap_bdqob_upper. bcf_lt_gap_bdqob_upper + S (n) = d + d) -> exists r. (((n) = (d) * (1) + (r) /\ (exists bcf_lt_gap_bdqob_result_bound. bcf_lt_gap_bdqob_result_bound + S (r) = d)))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 (3)
01Fix variables and assumptionsL1–4
02Separate the logical casesL5–5
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L5
cases hlower
03Establish hsumL6–9
04Establish hremainderL10–16
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add lt cancel left.
05Construct an explicit witnessL17–17
Supply the displayed value, then prove that it has the required property.
- L17
exists x
06Separate the logical casesL18–18
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L18
split
07Calculate and transport equalitiesL19–20
08Use earlier factsL21–22
09Calculate and transport equalitiesL23–24
10Use earlier factsL25–25
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L25
exact hremainder
Original defined command ledger · 25 lines
- 0001
intro d - 0002
intro n - 0003
intro hlower - 0004
intro hupper - 0005
cases hlower - 0006
have hsum : d + x = n - 0007
trans x + d - 0008
apply add_comm - 0009
exact hlower_witness - 0010
have hremainder : Lt(x,d)Exact native replay line
have hremainder : exists bcf_lt_gap_bdqob_remainder. bcf_lt_gap_bdqob_remainder + S (x) = d - 0011
rewrite <- hsum at hupper - 0012
specialize add_lt_cancel_left d - 0013
specialize add_lt_cancel_left x - 0014
specialize add_lt_cancel_left d - 0015
apply add_lt_cancel_left - 0016
exact hupper - 0017
exists x - 0018
split - 0019
trans d + x - 0020
symm - 0021
exact hsum - 0022
specialize mul_one d - 0023
rewrite mul_one - 0024
refl - 0025
exact hremainder