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f61e547 | 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 96 97 98 | import Mathlib
namespace OAI
namespace Problem367.FiniteChain
variable {A : Type*}
def weight (K : A → A → ℝ) : (n : ℕ) → A → (Fin n → A) → ℝ
| 0, _, _ => 1
| n + 1, a, x => K a (x 0) * weight K n (x 0) (Fin.tail x)
def endpoint : (n : ℕ) → A → (Fin n → A) → A
| 0, a, _ => a
| n + 1, _, x => endpoint n (x 0) (Fin.tail x)
@[simp] theorem weight_zero (K : A → A → ℝ) (a : A) (x : Fin 0 → A) :
weight K 0 a x = 1 := rfl
@[simp] theorem endpoint_zero (a : A) (x : Fin 0 → A) :
endpoint 0 a x = a := rfl
@[simp] theorem weight_cons (K : A → A → ℝ) (n : ℕ) (a b : A) (x : Fin n → A) :
weight K (n + 1) a (Fin.cons b x) = K a b * weight K n b x := by
simp [weight]
@[simp] theorem endpoint_cons (n : ℕ) (a b : A) (x : Fin n → A) :
endpoint (n + 1) a (Fin.cons b x) = endpoint n b x := by
simp [endpoint]
theorem weight_eq_prod (K : A → A → ℝ) (n : ℕ) (a : A) (x : Fin n → A) :
weight K n a x = ∏ i : Fin n, K ((Fin.cons a x : Fin (n+1) → A) i.castSucc) (x i) := by
induction n generalizing a with
| zero => simp
| succ n ih =>
rw [weight, Fin.prod_univ_succ, ih]
simp [Fin.tail]
theorem endpoint_eq_last (n : ℕ) (a : A) (x : Fin n → A) :
endpoint n a x = (Fin.cons a x : Fin (n+1) → A) (Fin.last n) := by
induction n generalizing a with
| zero => rfl
| succ n ih =>
rw [endpoint, ih]
simp
theorem weight_nonneg (K : A → A → ℝ) (hK : ∀ a b, 0 ≤ K a b)
(n : ℕ) (a : A) (x : Fin n → A) : 0 ≤ weight K n a x := by
induction n generalizing a with
| zero => simp
| succ n ih => exact mul_nonneg (hK _ _) (ih _ _)
variable [Fintype A]
theorem sum_cons (n : ℕ) (f : (Fin (n+1) → A) → ℝ) :
∑ x, f x = ∑ a, ∑ x : Fin n → A, f (Fin.cons a x) := by
rw [← (Fin.consEquiv (fun _ : Fin (n+1) => A)).sum_comp f]
rw [Fintype.sum_prod_type]
rfl
def expectation (K : A → A → ℝ) (n : ℕ) (f : A → ℝ) (a : A) : ℝ :=
∑ x : Fin n → A, weight K n a x * f (endpoint n a x)
@[simp] theorem expectation_zero (K : A → A → ℝ) (f : A → ℝ) (a : A) :
expectation K 0 f a = f a := by
simp [expectation]
theorem expectation_succ (K : A → A → ℝ) (n : ℕ) (f : A → ℝ) (a : A) :
expectation K (n+1) f a = ∑ b, K a b * expectation K n f b := by
unfold expectation
rw [sum_cons]
simp only [weight_cons, endpoint_cons, mul_assoc, Finset.mul_sum]
theorem expectation_eigenfunction (K : A → A → ℝ) (f : A → ℝ) (r : ℝ)
(hf : ∀ a, ∑ b, K a b * f b = r * f a) (n : ℕ) (a : A) :
expectation K n f a = r^n * f a := by
induction n generalizing a with
| zero => simp
| succ n ih =>
rw [expectation_succ]
simp_rw [ih]
calc
∑ b, K a b * (r ^ n * f b) = r ^ n * ∑ b, K a b * f b := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro b hb
ring
_ = r ^ (n+1) * f a := by rw [hf, pow_succ]; ring
theorem sum_weight (K : A → A → ℝ) (hK : ∀ a, ∑ b, K a b = 1)
(n : ℕ) (a : A) : ∑ x : Fin n → A, weight K n a x = 1 := by
have h := expectation_eigenfunction K (fun _ => 1) 1 (by simpa using hK) n a
simpa [expectation] using h
end Problem367.FiniteChain
end OAI
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