Equality of lengths transports all actual last-entry indices and the finite preservation bound of a witnessed swap.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
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.
Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.
forall b c d e k l i p q. k=l -> (((((exists ff_h_pfp_swap_length_sourceoldi. ff_h_pfp_swap_length_sourceoldi + S (p) = S ((S (i)) * c)) /\ exists ff_q_pfp_swap_length_sourceoldi. b = ff_q_pfp_swap_length_sourceoldi * S ((S (i)) * c) + (p))) /\ (((((exists ff_h_pfp_swap_length_sourceoldlast. ff_h_pfp_swap_length_sourceoldlast + S (q) = S ((S (k)) * c)) /\ exists ff_q_pfp_swap_length_sourceoldlast. b = ff_q_pfp_swap_length_sourceoldlast * S ((S (k)) * c) + (q))) /\ (((((exists ff_h_pfp_swap_length_sourcenewi. ff_h_pfp_swap_length_sourcenewi + S (q) = S ((S (i)) * e)) /\ exists ff_q_pfp_swap_length_sourcenewi. d = ff_q_pfp_swap_length_sourcenewi * S ((S (i)) * e) + (q))) /\ (((((exists ff_h_pfp_swap_length_sourcenewlast. ff_h_pfp_swap_length_sourcenewlast + S (p) = S ((S (k)) * e)) /\ exists ff_q_pfp_swap_length_sourcenewlast. d = ff_q_pfp_swap_length_sourcenewlast * S ((S (k)) * e) + (p))) /\ (forall pfp_j_swap_length_source pfp_a_swap_length_source. (exists pfp_gap_swap_length_sourcebound. pfp_gap_swap_length_sourcebound + S (pfp_j_swap_length_source) = (S (k))) -> ~(pfp_j_swap_length_source = i) -> ~(pfp_j_swap_length_source = k) -> (((exists ff_h_pfp_swap_length_sourceold. ff_h_pfp_swap_length_sourceold + S (pfp_a_swap_length_source) = S ((S (pfp_j_swap_length_source)) * c)) /\ exists ff_q_pfp_swap_length_sourceold. b = ff_q_pfp_swap_length_sourceold * S ((S (pfp_j_swap_length_source)) * c) + (pfp_a_swap_length_source))) -> (((exists ff_h_pfp_swap_length_sourcenew. ff_h_pfp_swap_length_sourcenew + S (pfp_a_swap_length_source) = S ((S (pfp_j_swap_length_source)) * e)) /\ exists ff_q_pfp_swap_length_sourcenew. d = ff_q_pfp_swap_length_sourcenew * S ((S (pfp_j_swap_length_source)) * e) + (pfp_a_swap_length_source)))))))))))) -> (((((exists ff_h_pfp_swap_length_targetoldi. ff_h_pfp_swap_length_targetoldi + S (p) = S ((S (i)) * c)) /\ exists ff_q_pfp_swap_length_targetoldi. b = ff_q_pfp_swap_length_targetoldi * S ((S (i)) * c) + (p))) /\ (((((exists ff_h_pfp_swap_length_targetoldlast. ff_h_pfp_swap_length_targetoldlast + S (q) = S ((S (l)) * c)) /\ exists ff_q_pfp_swap_length_targetoldlast. b = ff_q_pfp_swap_length_targetoldlast * S ((S (l)) * c) + (q))) /\ (((((exists ff_h_pfp_swap_length_targetnewi. ff_h_pfp_swap_length_targetnewi + S (q) = S ((S (i)) * e)) /\ exists ff_q_pfp_swap_length_targetnewi. d = ff_q_pfp_swap_length_targetnewi * S ((S (i)) * e) + (q))) /\ (((((exists ff_h_pfp_swap_length_targetnewlast. ff_h_pfp_swap_length_targetnewlast + S (p) = S ((S (l)) * e)) /\ exists ff_q_pfp_swap_length_targetnewlast. d = ff_q_pfp_swap_length_targetnewlast * S ((S (l)) * e) + (p))) /\ (forall pfp_j_swap_length_target pfp_a_swap_length_target. (exists pfp_gap_swap_length_targetbound. pfp_gap_swap_length_targetbound + S (pfp_j_swap_length_target) = (S (l))) -> ~(pfp_j_swap_length_target = i) -> ~(pfp_j_swap_length_target = l) -> (((exists ff_h_pfp_swap_length_targetold. ff_h_pfp_swap_length_targetold + S (pfp_a_swap_length_target) = S ((S (pfp_j_swap_length_target)) * c)) /\ exists ff_q_pfp_swap_length_targetold. b = ff_q_pfp_swap_length_targetold * S ((S (pfp_j_swap_length_target)) * c) + (pfp_a_swap_length_target))) -> (((exists ff_h_pfp_swap_length_targetnew. ff_h_pfp_swap_length_targetnew + S (pfp_a_swap_length_target) = S ((S (pfp_j_swap_length_target)) * e)) /\ exists ff_q_pfp_swap_length_targetnew. d = ff_q_pfp_swap_length_targetnew * S ((S (pfp_j_swap_length_target)) * e) + (pfp_a_swap_length_target))))))))))))
Complete tactic proof in conservative notation
All 18 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
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.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.
01Fix variables and assumptionsL1–10
Work with arbitrary variables or the premises of the current implication.