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
∀ i. ∃ a. Prime(S i) ∧ a = S i ∨ ¬Prime(S i) ∧ a = 1Every 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
2 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall i. exists a. (((((~(S (i) = 1) /\ forall bpr_left_bpfc_exists_prime bpr_right_bpfc_exists_prime. S (i) = bpr_left_bpfc_exists_prime * bpr_right_bpfc_exists_prime -> bpr_left_bpfc_exists_prime = 1 \/ bpr_right_bpfc_exists_prime = 1)) /\ a = S (i)) \/ (~((~(S (i) = 1) /\ forall bpr_left_bpfc_exists_prime bpr_right_bpfc_exists_prime. S (i) = bpr_left_bpfc_exists_prime * bpr_right_bpfc_exists_prime -> bpr_left_bpfc_exists_prime = 1 \/ bpr_right_bpfc_exists_prime = 1)) /\ a = 1)))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 (1)
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro i
02Use earlier factsL2–2
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L2
specialize prime_decidable (S i)
03Separate the logical casesL3–3
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L3
cases prime_decidable
04Construct an explicit witnessL4–4
Supply the displayed value, then prove that it has the required property.
- L4
exists S i
05Separate the logical casesL5–6
06Use earlier factsL7–7
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L7
exact prime_decidable_left
07Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
refl
08Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists 1
09Separate the logical casesL10–11
10Use earlier factsL12–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L12
exact prime_decidable_right
11Calculate and transport equalitiesL13–13
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L13
refl