| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| section |
| namespace LogConcaveSampling.Frontier |
|
|
| variable {E : Type*} (a b : E → ℕ) |
|
|
| abbrev Live (k : ℕ) := {e : E // a e≤k ∧ k<b e} |
| abbrev Incoming (k : ℕ) := {e : E // b e=k+1} |
| abbrev Outgoing (k : ℕ) := {e : E // a e=k+1} |
| abbrev Spectator (k : ℕ) := {e : E // a e≤k ∧ k+1<b e} |
|
|
| def splitIncoming (hab : ∀e,a e<b e) (k : ℕ) : |
| Live a b k ≃ Incoming b k ⊕ Spectator a b k where |
| toFun e := if h : b e.val=k+1 then Sum.inl ⟨e.val,h⟩ else |
| Sum.inr ⟨e.val,e.property.1,by have := e.property.2; omega⟩ |
| invFun e := match e with |
| | Sum.inl e => ⟨e.val,by have := hab e.val; have := e.property; omega, |
| by have := e.property; omega⟩ |
| | Sum.inr e => ⟨e.val,e.property.1,by have := e.property.2; omega⟩ |
| left_inv e := by |
| dsimp only |
| split_ifs <;> rfl |
| right_inv e := by |
| cases e with |
| | inl e => simp only [e.property,dite_eq_left] |
| | inr e => |
| have hn : b e.val≠k+1 := by have := e.property.2; omega |
| simp [hn] |
| |
| def splitOutgoing (hab : ∀e,a e<b e) (k : ℕ) : |
| Live a b (k+1) ≃ Outgoing a k ⊕ Spectator a b k where |
| toFun e := if h : a e.val=k+1 then Sum.inl ⟨e.val,h⟩ else |
| Sum.inr ⟨e.val,by have := e.property.1; omega,e.property.2⟩ |
| invFun e := match e with |
| | Sum.inl e => ⟨e.val,by have := e.property; omega, |
| by have := hab e.val; have := e.property; omega⟩ |
| | Sum.inr e => ⟨e.val,by have := e.property.1; omega,e.property.2⟩ |
| left_inv e := by |
| dsimp only |
| split_ifs <;> rfl |
| right_inv e := by |
| cases e with |
| | inl e => simp only [e.property,dite_eq_left] |
| | inr e => |
| have hn : a e.val≠k+1 := by have := e.property.1; omega |
| simp [hn] |
| |
| def inputCoordinates {C : Type*} (hab : ∀e,a e<b e) (k : ℕ) : |
| (Live a b k → C) ≃ (Incoming b k → C) × (Spectator a b k → C) := |
| (Equiv.arrowCongr (splitIncoming a b hab k) (Equiv.refl C)).trans |
| (Equiv.sumArrowEquivProdArrow _ _ _) |
|
|
| def outputCoordinates {C : Type*} (hab : ∀e,a e<b e) (k : ℕ) : |
| (Live a b (k+1) → C) ≃ (Outgoing a k → C) × (Spectator a b k → C) := |
| (Equiv.arrowCongr (splitOutgoing a b hab k) (Equiv.refl C)).trans |
| (Equiv.sumArrowEquivProdArrow _ _ _) |
|
|
| lemma inputCoordinates_apply {C : Type*} (hab : ∀e,a e<b e) (k : ℕ) |
| (x : Live a b k → C) : |
| (inputCoordinates a b hab k x).1=(fun e => x ⟨e.val,by |
| have := hab e.val; have := e.property; omega,by have := e.property; omega⟩) ∧ |
| (inputCoordinates a b hab k x).2=(fun e => x ⟨e.val,e.property.1,by have := e.property.2; omega⟩) := by |
| constructor <;> rfl |
| |
| lemma outputCoordinates_apply {C : Type*} (hab : ∀e,a e<b e) (k : ℕ) |
| (x : Live a b (k+1) → C) : |
| (outputCoordinates a b hab k x).1=(fun e => x ⟨e.val,by have := e.property; omega, |
| by have := hab e.val; have := e.property; omega⟩) ∧ |
| (outputCoordinates a b hab k x).2=(fun e => x ⟨e.val,by have := e.property.1; omega,e.property.2⟩) := by |
| constructor <;> rfl |
| end LogConcaveSampling.Frontier |
| |
| end |
| end UpperProof |
| end |
| end |
| end |
| |
| end OAI |
| |
