d60ef53d54
This PR changes `Lean.Order.CCPO` and `.CompleteLattice` to carry a Prop. This avoids the `CCPO IO` instance from being `noncomputable`.
136 lines
4.3 KiB
Text
136 lines
4.3 KiB
Text
-- Since we do not have ENNReal in core, we just axiomatize it all for this test
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opaque ENNReal : Type
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axiom ENNReal.sup : ∀ {α}, (α → ENNReal) → ENNReal
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axiom ENNReal.sum : ∀ {α}, (α → ENNReal) → ENNReal
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axiom ENNReal.add : ENNReal → ENNReal → ENNReal
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axiom ENNReal.mul : ENNReal → ENNReal → ENNReal
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noncomputable instance : Add ENNReal where add := .add
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noncomputable instance : Mul ENNReal where mul := .mul
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@[instance] axiom ENNReal.zero : Zero ENNReal
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axiom ENNReal.one : ENNReal
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axiom ENNReal.one_half : ENNReal
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@[simp] axiom ENNReal.mul_one : ∀ x, x * ENNReal.one = x
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@[simp] axiom ENNReal.mul_zero : ∀ (x : ENNReal), x * 0 = 0
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@[simp] axiom ENNReal.add_zero : ∀ (x : ENNReal), x + 0 = x
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@[simp] axiom ENNReal.zero_add : ∀ (x : ENNReal), 0 + x = x
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@[simp] axiom ENNReal.sum_bool : ∀ f, sum f = f true + f false
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@[simp] axiom ENNReal.sum_const_zero : ∀ α, ENNReal.sum (fun (_ : α) => 0) = 0
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@[simp] axiom ENNReal.sum_dirac : ∀ α [DecidableEq α] (f : α → ENNReal) y,
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ENNReal.sum (fun x => if x = y then f x else 0) = f y
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@[instance] axiom ENNReal.le : LE ENNReal
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axiom ENNReal.le_refl : ∀ (x : ENNReal), x ≤ x
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axiom ENNReal.le_trans : ∀ {x y z: ENNReal}, x ≤ y → y ≤ z → x ≤ z
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axiom ENNReal.le_antisymm : ∀ {x y : ENNReal}, x ≤ y → y ≤ x → x = y
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section
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set_option linter.unusedVariables false
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axiom ENNReal.sum_mono : ∀ {α} (s₁ s₂ : α → ENNReal) (h : ∀ x, s₁ x ≤ s₂ x),
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ENNReal.sum s₁ ≤ ENNReal.sum s₂
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axiom ENNReal.sup_mono : ∀ {α} (s₁ s₂ : α → ENNReal) (h : ∀ x, s₁ x ≤ s₂ x),
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ENNReal.sup s₁ ≤ ENNReal.sup s₂
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axiom ENNReal.mul_mono : ∀ (a b c Distr : ENNReal) (h₁ : a ≤ c) (h₂ : b ≤ Distr),
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a * b ≤ c * Distr
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axiom ENNReal.le_sup : ∀ {α} (a : ENNReal) (s : α → ENNReal) (i : α) (h : a ≤ s i),
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a ≤ ENNReal.sup s
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axiom ENNReal.sup_le : ∀ {α} (a : ENNReal) (s : α → ENNReal) (h : ∀ (i : α), s i ≤ a),
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ENNReal.sup s ≤ a
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end
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/-- Distributions (not normalized, which is curcial, else we don't have ⊥.) -/
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def Distr (α : Type) : Type := α → ENNReal
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noncomputable def Distr.join : Distr (Distr α) → Distr α := fun dd x =>
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ENNReal.sum (fun Distr => Distr x * dd Distr )
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noncomputable instance : Functor Distr where
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map f Distr := fun x => ENNReal.sum (fun y => open Classical in if f y = x then Distr y else 0)
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noncomputable instance : Pure Distr where
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pure x := fun y => open Classical in if x = y then .one else 0
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noncomputable instance : Bind Distr where
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bind Distr f := fun x => ENNReal.sum (fun y => Distr y * f y x)
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open Lean.Order
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noncomputable instance : PartialOrder (Distr α) where
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rel d1 d2 := ∀ x, d1 x ≤ d2 x
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rel_refl _ := ENNReal.le_refl _
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rel_trans h1 h2 _ := ENNReal.le_trans (h1 _) (h2 _)
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rel_antisymm h1 h2 := funext (fun _ => ENNReal.le_antisymm (h1 _) (h2 _))
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noncomputable instance : CCPO (Distr α) where
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has_csup := by
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intro c hchain
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exists fun x => ENNReal.sup fun (Distr : Subtype c) => Distr.val x
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intro d₁
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constructor
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next =>
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intro h d₂ hd₂ x
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apply ENNReal.le_trans ?_ (h x)
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apply ENNReal.le_sup (i := ⟨d₂, hd₂⟩)
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apply ENNReal.le_refl
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next =>
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intro h x
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apply ENNReal.sup_le
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intro Distr
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apply h Distr.1 Distr.2 x
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noncomputable instance : MonoBind Distr where
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bind_mono_left := by
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intro α β d₁ d₂ f h₁₂ y
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unfold bind instBindDistr
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dsimp
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apply ENNReal.sum_mono
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intro x
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apply ENNReal.mul_mono
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· apply h₁₂
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· apply ENNReal.le_refl
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bind_mono_right := by
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intro α β Distr f₁ f₂ h₁₂ y
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apply ENNReal.sum_mono
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intro x
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apply ENNReal.mul_mono
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· apply ENNReal.le_refl
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· apply h₁₂
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noncomputable def coin : Distr Bool := fun _ => .one_half
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noncomputable def geom : Distr Nat := do
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let head ← coin
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if head then
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return 0
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else
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let n ← geom
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return (n + 1)
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partial_fixpoint
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/--
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info: geom.eq_1 :
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geom = do
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let head ← coin
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if head = true then pure 0
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else do
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let n ← geom
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pure (n + 1)
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-/
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#guard_msgs in
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#check geom.eq_1
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-- And we can can do proofs about this
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theorem geom_0 : geom 0 = .one_half := by
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rw [geom]; simp [bind, coin, pure]
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theorem geom_succ : geom (n+1) = .one_half * geom n := by
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conv => lhs; rw [geom]
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simp [bind, coin, pure, apply_ite]
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