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
∀ p. ∀ n. ∀ h. p = S n → Prime(p) → n = h + h → ¬ModEq(p,1,n)Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.
Definitions used by this theorem
In the theorem statement
2 occurrences
In local proof propositions
2 occurrences
Exact expanded native-PA statement
forall p n h. p = S n -> ((~(p = 1) /\ forall esi_prime_left_ecb_prime esi_prime_right_ecb_prime. p = esi_prime_left_ecb_prime * esi_prime_right_ecb_prime -> esi_prime_left_ecb_prime = 1 \/ esi_prime_right_ecb_prime = 1)) -> n = h + h -> ~(exists wpp_mod_left_ecb_one_mod_predecessor wpp_mod_right_ecb_one_mod_predecessor. (1) + p * wpp_mod_left_ecb_one_mod_predecessor = (n) + p * wpp_mod_right_ecb_one_mod_predecessor)Proof neighborhood
Direct theorem prerequisites
PA00BO double_predecessor_ne_one PA0061 prime_is_succ_succ PA002U mod_eq_bounded_unique PA0001 zero_addDirect theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.
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 (4)
01Fix variables and assumptionsL1–7
02Establish hp_shapeL8–11
03Separate the logical casesL12–12
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L12
cases hp_shape
04Establish h1pL13–13
Establish this local claim before using it. It is not an additional assumption.
05Construct an explicit witnessL14–14
Supply the displayed value, then prove that it has the required property.
- L14
exists x
06Calculate and transport equalitiesL15–16
07Establish hnpL17–17
Establish this local claim before using it. It is not an additional assumption.
08Construct an explicit witnessL18–18
Supply the displayed value, then prove that it has the required property.
- L18
exists 0
09Calculate and transport equalitiesL19–19
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L19
trans S n
10Use earlier factsL20–20
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L20
apply zero_add
11Calculate and transport equalitiesL21–21
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L21
symm
12Use earlier factsL22–22
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L22
exact hpn
13Establish h1nL23–32
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq bounded unique.
14Use earlier factsL33–34
15Calculate and transport equalitiesL35–35
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L35
symm
16Use earlier factsL36–36
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L36
exact h1n
Original defined command ledger · 36 lines
- 0001
intro p - 0002
intro n - 0003
intro h - 0004
intro hpn - 0005
intro hp - 0006
intro heven - 0007
intro hone_mod - 0008
have hp_shape : exists k. p = S (S k) - 0009
specialize prime_is_succ_succ p - 0010
apply prime_is_succ_succ - 0011
exact hp - 0012
cases hp_shape - 0013
have h1p : Lt(1,p)Exact native replay line
have h1p : exists gap. gap + S 1 = p - 0014
exists x - 0015
rewrite hp_shape_witness - 0016
simp - 0017
have hnp : Lt(n,p)Exact native replay line
have hnp : exists gap. gap + S n = p - 0018
exists 0 - 0019
trans S n - 0020
apply zero_add - 0021
symm - 0022
exact hpn - 0023
have h1n : 1 = n - 0024
specialize mod_eq_bounded_unique p - 0025
specialize mod_eq_bounded_unique 1 - 0026
specialize mod_eq_bounded_unique n - 0027
apply mod_eq_bounded_unique - 0028
exact h1p - 0029
exact hnp - 0030
exact hone_mod - 0031
specialize double_predecessor_ne_one n - 0032
specialize double_predecessor_ne_one h - 0033
apply double_predecessor_ne_one - 0034
exact heven - 0035
symm - 0036
exact h1n