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. p = S n → Prime(p) → ∃ x. ∃ y. InversePrefix(p,n,x,y,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
0 occurrences
Exact expanded native-PA statement
forall p n. p = S n -> ((~(p = 1) /\ forall wip_prime_left_full_prime wip_prime_right_full_prime. p = wip_prime_left_full_prime * wip_prime_right_full_prime -> wip_prime_left_full_prime = 1 \/ wip_prime_right_full_prime = 1)) -> exists b c. (forall wip_index_full_result. (exists wip_gap_full_result_prefix_bound. wip_gap_full_result_prefix_bound + S wip_index_full_result = n) -> exists wip_mate_full_result. ((((exists wip_beta_height_full_result_decoded. wip_beta_height_full_result_decoded + S (wip_mate_full_result) = S ((S (wip_index_full_result)) * c)) /\ exists wip_beta_quotient_full_result_decoded. b = wip_beta_quotient_full_result_decoded * S ((S (wip_index_full_result)) * c) + (wip_mate_full_result))) /\ ((exists wip_gap_full_result_inverse_index_bound. wip_gap_full_result_inverse_index_bound + S wip_index_full_result = n) /\ ((exists wip_gap_full_result_inverse_mate_bound. wip_gap_full_result_inverse_mate_bound + S wip_mate_full_result = n) /\ (exists wip_mod_left_full_result_inverse_mod wip_mod_right_full_result_inverse_mod. ((S wip_index_full_result) * S wip_mate_full_result) + p * wip_mod_left_full_result_inverse_mod = 1 + p * wip_mod_right_full_result_inverse_mod)))))Proof neighborhood
Direct theorem prerequisites
Direct 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.
Instantiate the bounded construction at its own length
This short wrapper reuses prime_inverse_prefix_exists_bounded with n as both the requested prefix length and its bound. The original hypotheses discharge the prime and range conditions; le_refl supplies the remaining self-bound. The construction itself belongs to the linked bounded theorem.
Named ingredients (2)
01Fix variables and assumptionsL1–4
02Use earlier factsL5–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original defined command ledger · 12 lines
- 0001
intro p - 0002
intro n - 0003
intro hpn - 0004
intro hp - 0005
specialize prime_inverse_prefix_exists_bounded p - 0006
specialize prime_inverse_prefix_exists_bounded n - 0007
specialize prime_inverse_prefix_exists_bounded n - 0008
apply prime_inverse_prefix_exists_bounded - 0009
exact hpn - 0010
exact hp - 0011
specialize le_refl n - 0012
exact le_refl