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dd14c26 | 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 90 91 92 93 94 95 | 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
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