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. ¬n = 0 → ¬n = 1 → Prime(n) ∨ (∃ x. ∃ y. ¬x = 1 ∧ ¬y = 1 ∧ n = x · y)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
1 occurrences
In local proof propositions
2 occurrences
Exact expanded native-PA statement
forall n. ~(n = 0) -> ~(n = 1) -> ((~(n = 1) /\ forall a b. n = a * b -> a = 1 \/ b = 1) \/ exists c d. ((~(c = 1) /\ ~(d = 1)) /\ n = c * 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 (2)
01Fix variables and assumptionsL1–3
02Use earlier factsL4–5
03Establish hsearchL6–8
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply factor search up to.
- L6
have hsearch : (∀ x. ∀ y. Le(x,n) → n = x · y → x = 1 ∨ y = 1) ∨ (∃ x. ∃ y. Le(x,n) ∧ ¬x = 1 ∧ ¬y = 1 ∧ n = x · y)Definitions: Le(x,n)Original native command in the exact edition - L7
apply factor_search_up_to - L8
exact hn0
04Separate the logical casesL9–11
05Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
exact hn1
06Fix variables and assumptionsL13–15
07Use earlier factsL16–22
08Construct an explicit witnessL23–23
Supply the displayed value, then prove that it has the required property.
- L23
exists d
09Use earlier factsL24–25
10Separate the logical casesL26–31
11Construct an explicit witnessL32–33
12Separate the logical casesL34–35
Original defined command ledger · 38 lines
- 0001
intro n - 0002
intro hn0 - 0003
intro hn1 - 0004
specialize factor_search_up_to n - 0005
specialize factor_search_up_to n - 0006
have hsearch : (∀ x. ∀ y. Le(x,n) → n = x · y → x = 1 ∨ y = 1) ∨ (∃ x. ∃ y. Le(x,n) ∧ ¬x = 1 ∧ ¬y = 1 ∧ n = x · y)Exact native replay line
have hsearch : (forall c d. (exists k. k + c = n) -> n = c * d -> c = 1 \/ d = 1) \/ exists c d. ((((exists k. k + c = n) /\ ~(c = 1)) /\ ~(d = 1)) /\ n = c * d) - 0007
apply factor_search_up_to - 0008
exact hn0 - 0009
cases hsearch - 0010
left - 0011
split - 0012
exact hn1 - 0013
intro c - 0014
intro d - 0015
intro hfac - 0016
specialize hsearch_left c - 0017
specialize hsearch_left d - 0018
apply hsearch_left - 0019
specialize divisor_le_nonzero c - 0020
specialize divisor_le_nonzero n - 0021
apply divisor_le_nonzero - 0022
exact hn0 - 0023
exists d - 0024
exact hfac - 0025
exact hfac - 0026
right - 0027
cases hsearch_right - 0028
cases hsearch_right_witness - 0029
cases hsearch_right_witness_witness - 0030
cases hsearch_right_witness_witness_left - 0031
cases hsearch_right_witness_witness_left_left - 0032
exists x - 0033
exists x1 - 0034
split - 0035
split - 0036
exact hsearch_right_witness_witness_left_left_right - 0037
exact hsearch_right_witness_witness_left_right - 0038
exact hsearch_right_witness_witness_right