| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Band.OutputBands |
| import OAI.RepresentationTheory.UniversalSquare.Capacity.RefinedExhaustion |
|
|
| namespace OAI |
|
|
| /-! Numeric Bands. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Saxl.Balance |
|
|
| def BandsValid (es : List (ℕ × ℕ)) : Prop := |
| (∀ e ∈ es, e.1 ≤ e.2) ∧ ∀ e ∈ es, ∀ f ∈ es, |
| (e.1 = f.1 → e.2 = f.2) ∧ (e.1 < f.1 → e.2 < f.1) |
| |
| instance (es : List (ℕ × ℕ)) : Decidable (BandsValid es) := by |
| unfold BandsValid |
| infer_instance |
| |
| def bandStarts (es : List (ℕ × ℕ)) : Finset ℕ := (es.map Prod.fst).toFinset |
| |
| def bandUpper (es : List (ℕ × ℕ)) (k : ℕ) : ℕ := |
| es.toFinset.sup (fun e => if e.1 = k then e.2 else 0) |
| |
| lemma bandUpper_eq {es : List (ℕ × ℕ)} (h : BandsValid es) {e : ℕ × ℕ} |
| (he : e ∈ es) : bandUpper es e.1 = e.2 := by |
| apply le_antisymm |
| · apply Finset.sup_le |
| intro f hf |
| by_cases hf' : f.1 = e.1 |
| · simp only [ite_eq_left hf'] |
| exact ((h.2 f (List.mem_toFinset.mp hf) e he).1 hf').le |
| · simp only [ite_eq_right hf']; exact Nat.zero_le _ |
| · have hh := Finset.le_sup (f := fun f : ℕ × ℕ => if f.1 = e.1 then f.2 else 0) |
| (List.mem_toFinset.mpr he) |
| simpa only [bandUpper, ite_true] using hh |
| |
| lemma mem_bandStarts {es : List (ℕ × ℕ)} {k : ℕ} : |
| k ∈ bandStarts es ↔ ∃ e ∈ es, e.1 = k := by simp [bandStarts] |
| |
| def numericBands (es : List (ℕ × ℕ)) (h : BandsValid es) : OutputBands where |
| starts := bandStarts es |
| upper := bandUpper es |
| nonempty := by |
| intro k hk |
| obtain ⟨e,he,rfl⟩ := mem_bandStarts.mp hk |
| rw [bandUpper_eq h he] |
| exact h.1 e he |
| separated := by |
| intro k hk l hl hkl |
| obtain ⟨e,he,rfl⟩ := mem_bandStarts.mp hk |
| obtain ⟨f,hf,rfl⟩ := mem_bandStarts.mp hl |
| rw [bandUpper_eq h he] |
| exact (h.2 e he f hf).2 hkl |
|
|
| lemma numericBands_iff {es : List (ℕ × ℕ)} (h : BandsValid es) {e : ℕ × ℕ} |
| (he : e ∈ es) {d : ℕ} (z : Fin (d*d)) : |
| ((numericBands es h).alphabet d z ∧ (numericBands es h).letterLabel d z = e.1) ↔ |
| output z ∈ Set.Icc e.1 e.2 := by |
| rw [(numericBands es h).letter_iff (mem_bandStarts.mpr ⟨e,he,rfl⟩)] |
| rw [show (numericBands es h).upper e.1 = e.2 from bandUpper_eq h he] |
|
|
| end Saxl.Balance |
| end |
| end |
|
|
| end OAI |
|
|
