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92416a2 | 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 99 100 101 102 103 104 | import OAI.Dynamics.ConditionalShuffle.InstrumentBound
namespace OAI
noncomputable section
namespace Revealed.Instrument
open scoped Classical
open Thorp Thorp.Fourier
variable {E S G Ω : Type} [fintype_S : Fintype S] [fintype_G : Fintype G] [group_G : Group G] [fintype_Ω : Fintype Ω] [nonempty_Ω : Nonempty Ω]
variable (I : Data E S G Ω)
lemma base_snoc (e : E) (n : ℕ) (p : Fin n → S) (s : S) :
base I e (n+1) (Fin.snoc p s) = I.next (base I e n p) s := by
let retained_fintype_S := fintype_S
let retained_fintype_G := fintype_G
let retained_group_G := group_G
let retained_fintype_Ω := fintype_Ω
let retained_nonempty_Ω := nonempty_Ω
simp only [base, Fin.init_snoc, Fin.snoc_last]
lemma weights_snoc (e : E) (n : ℕ) (p : Fin n → S) (s : S) :
weights I e (n+1) (Fin.snoc p s) = weights I e n p * probability I (base I e n p) s := by
let retained_fintype_S := fintype_S
let retained_group_G := group_G
let retained_nonempty_Ω := nonempty_Ω
simp only [weights, Fin.init_snoc, Fin.snoc_last]
lemma kernels_snoc (e : E) (n : ℕ) (p : Fin n → S) (s : S) :
kernels I e (n+1) (Fin.snoc p s) = Fin.snoc (kernels I e n p) (kernel I (base I e n p) s) := by
let retained_fintype_S := fintype_S
let retained_group_G := group_G
let retained_nonempty_Ω := nonempty_Ω
simp only [kernels, Fin.init_snoc, Fin.snoc_last]
lemma base_append (e : E) (n m : ℕ) (p : Fin n → S) (q : Fin m → S) :
base I e (n+m) (Fin.append p q) = base I (base I e n p) m q := by
induction m with
| zero =>
have hz : Fin.append p q = p := by funext i; exact Fin.append_left p q i
rw [hz]
rfl
| succ m ih =>
obtain ⟨⟨s,q⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => S)).surjective q
erw [Fin.append_snoc, base_snoc, base_snoc, ih]
lemma weights_append (e : E) (n m : ℕ) (p : Fin n → S) (q : Fin m → S) :
weights I e (n+m) (Fin.append p q) = weights I e n p * weights I (base I e n p) m q := by
induction m with
| zero =>
have hz : Fin.append p q = p := by funext i; exact Fin.append_left p q i
rw [hz, weights, mul_one]
rfl
| succ m ih =>
obtain ⟨⟨s,q⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => S)).surjective q
erw [Fin.append_snoc, weights_snoc, weights_snoc, ih, base_append, mul_assoc]
lemma kernels_append (e : E) (n m : ℕ) (p : Fin n → S) (q : Fin m → S) :
kernels I e (n+m) (Fin.append p q) =
Fin.append (kernels I e n p) (kernels I (base I e n p) m q) := by
induction m with
| zero =>
have hp : Fin.append p q = p := by funext i; exact Fin.append_left p q i
rw [hp]
funext i
exact (Fin.append_left _ _ i).symm
| succ m ih =>
obtain ⟨⟨s,q⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => S)).surjective q
erw [Fin.append_snoc, kernels_snoc, kernels_snoc, ih, base_append, Fin.append_snoc]
lemma realConv_assoc (μ ν η : G → ℝ) : realConv (realConv μ ν) η = realConv μ (realConv ν η) := by
funext z
simp only [realConv, Finset.sum_mul, Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a _
have h := Equiv.sum_comp (Equiv.mulLeft a) (fun b => μ a * ν (a⁻¹*b) * η (b⁻¹*z))
change (∑ b, μ a * ν (a⁻¹*(a*b)) * η ((a*b)⁻¹*z)) = _ at h
simpa only [inv_mul_cancel_left, mul_inv_rev, mul_assoc] using h.symm
lemma realConv_dirac_one (μ : G → ℝ) : realConv (fun g => if g = 1 then 1 else 0) μ = μ := by
funext g
simp [realConv]
lemma realProduct_snoc (n : ℕ) (μ : Fin n → G → ℝ) (ν : G → ℝ) :
realProduct (n+1) (Fin.snoc μ ν) = realConv ν (realProduct n μ) := by
simp only [realProduct, Fin.init_snoc, Fin.snoc_last]
lemma realProduct_append (n m : ℕ) (μ : Fin n → G → ℝ) (ν : Fin m → G → ℝ) :
realProduct (n+m) (Fin.append μ ν) = realConv (realProduct m ν) (realProduct n μ) := by
induction m with
| zero =>
have hz : Fin.append μ ν = μ := by funext i; exact Fin.append_left μ ν i
rw [hz, realProduct, realConv_dirac_one]
rfl
| succ m ih =>
obtain ⟨⟨η,ν⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => G → ℝ)).surjective ν
erw [Fin.append_snoc, realProduct_snoc, realProduct_snoc, ih, realConv_assoc]
end Revealed.Instrument
end
end OAI
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