| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| namespace BinaryCoordinateSweeps | |
| variable {I : Type*} [Fintype I] {Ω G : I → Type*} | |
| [∀i,Fintype (Ω i)] [∀i,Fintype (G i)] | |
| def finitePush {X Y : Type*} [Fintype X] (w : X → ℂ) (f : X → Y) (y : Y) : ℂ := | |
| ∑x, if f x=y then w x else 0 | |
| omit [∀ i, Fintype (G i)] in | |
| lemma finitePush_pi (w : ∀i, Ω i → ℂ) (f : ∀i, Ω i → G i) (g : ∀i, G i) : | |
| (∏i, finitePush (w i) (f i) (g i)) = | |
| finitePush (fun x : ∀i, Ω i => ∏i,w i (x i)) (fun x i => f i (x i)) g := by | |
| unfold finitePush | |
| rw [Fintype.prod_sum] | |
| apply Finset.sum_congr rfl | |
| intro x _ | |
| rw [Fintype.prod_ite_zero] | |
| congr 1 | |
| exact propext funext_iff.symm | |
| lemma finitePush_expectation {X Y V : Type*} [Fintype X] [Fintype Y] | |
| [AddCommMonoid V] [Module ℂ V] (w : X → ℂ) (f : X → Y) (F : Y → V) : | |
| (∑y, finitePush w f y • F y)=∑x,w x • F (f x) := by | |
| classical | |
| unfold finitePush | |
| simp only [Finset.sum_smul] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro x _ | |
| simp only [ite_smul,zero_smul] | |
| simp | |
| lemma independentPush_expectation {V : Type*} [AddCommMonoid V] [Module ℂ V] | |
| (w : ∀i, Ω i → ℂ) (f : ∀i, Ω i → G i) (F : (∀i,G i) → V) : | |
| (∑g : ∀i,G i, (∏i,finitePush (w i) (f i) (g i)) • F g)= | |
| ∑x : ∀i,Ω i, (∏i,w i (x i)) • F (fun i=>f i (x i)) := by | |
| simp_rw [finitePush_pi] | |
| exact finitePush_expectation _ _ _ | |
| end BinaryCoordinateSweeps | |
| end | |
| end OAI | |