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.
Exact expanded 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)))Structural proof guide
Every index has its exact prime-or-one selector factor.
Direct prerequisites: prime_decidable. The authored body proceeds by case analysis (1).
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.
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