File size: 3,936 Bytes
180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | 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 _ _
/-- A reference always carries the intended label. -/
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
/-- Perfect occurrences are preferred whenever the class has one. -/
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
/-- An unrepresented class refers to its first occurrence, which is
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]
/-- In the absence of a perfect representative, the selected occurrence
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
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