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. ∀ s. ∀ q. ∀ r. Lt(2,n) → FloorSqrt(n + n,s) → DivRem(n + n,3,q,r) → ∃ x. s + x = qEvery 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 s q r. (exists bcf_lt_gap_b5rbfsltq_positive. bcf_lt_gap_b5rbfsltq_positive + S (2) = n) -> (((exists bcs_sqrt_lower_gap_b5rbfsltq_floor. bcs_sqrt_lower_gap_b5rbfsltq_floor + (s) * (s) = (n + n)) /\ exists bcs_sqrt_upper_gap_b5rbfsltq_floor. bcs_sqrt_upper_gap_b5rbfsltq_floor + S (n + n) = S (s) * S (s))) -> (((n + n) = (3) * (q) + (r) /\ (exists bcf_lt_gap_b5rbfsltq_division_bound. bcf_lt_gap_b5rbfsltq_division_bound + S (r) = 3))) -> exists g. s + g = qProof 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 (2)
01Fix variables and assumptionsL1–7
02Establish hboundL8–16
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply floor sqrt le third quotient.
03Separate the logical casesL17–17
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L17
cases hbound
04Construct an explicit witnessL18–18
Supply the displayed value, then prove that it has the required property.
- L18
exists x
05Use earlier factsL19–20
06Calculate and transport equalitiesL21–21
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L21
rewrite add_comm
07Use earlier factsL22–22
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L22
exact hbound_witness
Original defined command ledger · 22 lines
- 0001
intro n - 0002
intro s - 0003
intro q - 0004
intro r - 0005
intro hpositive - 0006
intro hfloor - 0007
intro hdivision - 0008
have hbound : Le(s,q)Exact native replay line
have hbound : exists bcf_le_gap_b5rbfsltq_result. bcf_le_gap_b5rbfsltq_result + (s) = q - 0009
specialize floor_sqrt_le_third_quotient n - 0010
specialize floor_sqrt_le_third_quotient s - 0011
specialize floor_sqrt_le_third_quotient q - 0012
specialize floor_sqrt_le_third_quotient r - 0013
apply floor_sqrt_le_third_quotient - 0014
exact hpositive - 0015
exact hfloor - 0016
exact hdivision - 0017
cases hbound - 0018
exists x - 0019
specialize add_comm s - 0020
specialize add_comm x - 0021
rewrite add_comm - 0022
exact hbound_witness