| import Mathlib.Data.Finset.Max |
| import Mathlib.Data.Fintype.Fin |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! Canonical occurrence representatives for the imperfect-position code. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| variable {α : Type*} [DecidableEq α] {N : ℕ} |
|
|
| def columnRepresentativeCandidates (label : Fin N → α) (perfect : Finset (Fin N)) |
| (i : Fin N) : Finset (Fin N) := |
| let p := perfect.filter (fun j => label j = label i) |
| if p.Nonempty then p else univ.filter (fun j => label j = label i) |
|
|
| lemma columnRepresentativeCandidates_nonempty (label : Fin N → α) |
| (perfect : Finset (Fin N)) (i : Fin N) : |
| (columnRepresentativeCandidates label perfect i).Nonempty := by |
| dsimp only [columnRepresentativeCandidates] |
| split_ifs with h |
| · exact h |
| · exact ⟨i, mem_filter.mpr ⟨mem_univ _, rfl⟩⟩ |
|
|
| def columnRepresentative (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) : Fin N := |
| (columnRepresentativeCandidates label perfect i).min' |
| (columnRepresentativeCandidates_nonempty label perfect i) |
|
|
| lemma columnRepresentative_mem (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) : |
| columnRepresentative label perfect i ∈ columnRepresentativeCandidates label perfect i := |
| min'_mem _ _ |
|
|
| / |
| theorem columnRepresentative_label (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) : |
| label (columnRepresentative label perfect i) = label i := by |
| have hm := columnRepresentative_mem label perfect i |
| dsimp only [columnRepresentativeCandidates] at hm |
| split_ifs at hm <;> exact (mem_filter.mp hm).2 |
|
|
| / |
| theorem columnRepresentative_perfect (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) |
| (h : ∃ j ∈ perfect, label j = label i) : columnRepresentative label perfect i ∈ perfect := by |
| have hp : (perfect.filter (fun j => label j = label i)).Nonempty := by |
| obtain ⟨j, hj, heq⟩ := h |
| exact ⟨j, mem_filter.mpr ⟨hj, heq⟩⟩ |
| have hm := columnRepresentative_mem label perfect i |
| simp only [columnRepresentativeCandidates, hp, ite_true] at hm |
| exact (mem_filter.mp hm).1 |
|
|
| / |
| necessarily imperfect; two such classes cannot share a reference. -/ |
| theorem columnRepresentative_imperfect (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) |
| (h : ¬∃ j ∈ perfect, label j = label i) : columnRepresentative label perfect i ∉ perfect := by |
| intro hm |
| exact h ⟨_, hm, columnRepresentative_label label perfect i⟩ |
|
|
| theorem columnRepresentative_eq_iff (label : Fin N → α) (perfect : Finset (Fin N)) |
| (i j : Fin N) : columnRepresentative label perfect i = columnRepresentative label perfect j ↔ |
| label i = label j := by |
| constructor |
| · intro h |
| exact (columnRepresentative_label label perfect i).symm.trans |
| ((congrArg label h).trans (columnRepresentative_label label perfect j)) |
| · intro h |
| have hc : columnRepresentativeCandidates label perfect i = |
| columnRepresentativeCandidates label perfect j := by |
| simp only [columnRepresentativeCandidates, h] |
| unfold columnRepresentative |
| simp only [hc] |
|
|
| / |
| precedes every other occurrence in its class. -/ |
| theorem columnRepresentative_first (label : Fin N → α) (perfect : Finset (Fin N)) (i : Fin N) |
| (h : ¬∃ j ∈ perfect, label j = label i) (j : Fin N) (hj : label j = label i) : |
| columnRepresentative label perfect i ≤ j := by |
| apply min'_le |
| have hp : ¬(perfect.filter (fun j => label j = label i)).Nonempty := by |
| rintro ⟨k, hk⟩ |
| exact h ⟨k, (mem_filter.mp hk).1, (mem_filter.mp hk).2⟩ |
| simp only [columnRepresentativeCandidates, hp, ite_false] |
| exact mem_filter.mpr ⟨mem_univ _, hj⟩ |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
