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
∀ a. ∀ b. ∀ n. Prime(b) → Le(a,n) → Lt(n,b) → Le(b,a + a) → ∃ x. Prime(x) ∧ (Lt(n,x) ∧ Le(x,n + n))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
7 occurrences
In local proof propositions
4 occurrences
Exact expanded native-PA statement
forall a b n. ((~(b = 1) /\ forall bpr_left_bb8ci_prime bpr_right_bb8ci_prime. b = bpr_left_bb8ci_prime * bpr_right_bb8ci_prime -> bpr_left_bb8ci_prime = 1 \/ bpr_right_bb8ci_prime = 1)) -> (exists bpr_le_gap_bb8ci_lower. bpr_le_gap_bb8ci_lower + (a) = (n)) -> (exists bpr_gap_bb8ci_strict. bpr_gap_bb8ci_strict + S (n) = b) -> (exists bpr_le_gap_bb8ci_cover. bpr_le_gap_bb8ci_cover + (b) = (a + a)) -> exists p. ((~(p = 1) /\ forall bpr_left_bb8ci_result_prime bpr_right_bb8ci_result_prime. p = bpr_left_bb8ci_result_prime * bpr_right_bb8ci_result_prime -> bpr_left_bb8ci_result_prime = 1 \/ bpr_right_bb8ci_result_prime = 1)) /\ ((exists bpr_gap_bb8ci_result_strict. bpr_gap_bb8ci_result_strict + S (n) = p) /\ (exists bpr_le_gap_bb8ci_result_upper. bpr_le_gap_bb8ci_result_upper + (p) = (n + n)))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 assumptionsL1–7
02Establish hfirstL8–13
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add le add right.
- L8
have hfirst : Le(a + a,n + a)Definitions: Le(a + a,n + a)Original native command in the exact edition - L9
specialize add_le_add_right a - L10
specialize add_le_add_right n - L11
specialize add_le_add_right a - L12
apply add_le_add_right - L13
exact hlower
03Establish hsecondL14–19
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add le add left.
- L14
have hsecond : Le(n + a,n + n)Definitions: Le(n + a,n + n)Original native command in the exact edition - L15
specialize add_le_add_left a - L16
specialize add_le_add_left n - L17
specialize add_le_add_left n - L18
apply add_le_add_left - L19
exact hlower
04Establish hdoubleL20–26
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le trans.
- L20
have hdouble : Le(a + a,n + n)Definitions: Le(a + a,n + n)Original native command in the exact edition - L21
specialize le_trans (a + a) - L22
specialize le_trans (n + a) - L23
specialize le_trans (n + n) - L24
apply le_trans - L25
exact hfirst - L26
exact hsecond
05Establish hupperL27–33
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le trans.
06Construct an explicit witnessL34–34
Supply the displayed value, then prove that it has the required property.
- L34
exists b
07Separate the logical casesL35–35
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L35
split
08Use earlier factsL36–36
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L36
exact hprime
09Separate the logical casesL37–37
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L37
split
Original defined command ledger · 39 lines
- 0001
intro a - 0002
intro b - 0003
intro n - 0004
intro hprime - 0005
intro hlower - 0006
intro hstrict - 0007
intro hcover - 0008
have hfirst : Le(a + a,n + a)Exact native replay line
have hfirst : exists k. k + (a + a) = n + a - 0009
specialize add_le_add_right a - 0010
specialize add_le_add_right n - 0011
specialize add_le_add_right a - 0012
apply add_le_add_right - 0013
exact hlower - 0014
have hsecond : Le(n + a,n + n)Exact native replay line
have hsecond : exists k. k + (n + a) = n + n - 0015
specialize add_le_add_left a - 0016
specialize add_le_add_left n - 0017
specialize add_le_add_left n - 0018
apply add_le_add_left - 0019
exact hlower - 0020
have hdouble : Le(a + a,n + n)Exact native replay line
have hdouble : exists k. k + (a + a) = n + n - 0021
specialize le_trans (a + a) - 0022
specialize le_trans (n + a) - 0023
specialize le_trans (n + n) - 0024
apply le_trans - 0025
exact hfirst - 0026
exact hsecond - 0027
have hupper : Le(b,n + n)Exact native replay line
have hupper : exists k. k + b = n + n - 0028
specialize le_trans b - 0029
specialize le_trans (a + a) - 0030
specialize le_trans (n + n) - 0031
apply le_trans - 0032
exact hcover - 0033
exact hdouble - 0034
exists b - 0035
split - 0036
exact hprime - 0037
split - 0038
exact hstrict - 0039
exact hupper