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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import Mathlib
import OAI.Probability.BinarySweep.MatrixBounds.TraceHolder
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator
namespace BinaryCoordinateSweeps.TraceHolder
section Expansion
variable {P R : Type*} [Fintype P] [DecidableEq P] [Semiring R]
def pathProduct {n : ℕ} (A : Fin n → Matrix P P R) (x : Fin n → P) (b : P) : R :=
(List.ofFn fun j => A j (x j) (Fin.snoc (α := fun _ => P) x b j.succ)).prod
omit [Fintype P] [DecidableEq P] in
lemma snoc_cons_succ {n : ℕ} (a : P) (x : Fin n → P) (b : P) (j : Fin (n+1)) :
Fin.snoc (α := fun _ => P) (Fin.cons a x) b j.succ =
Fin.snoc (α := fun _ => P) x b j := by
refine Fin.lastCases ?_ (fun i => ?_) j
· simp
· rw [Fin.succ_castSucc]
simp only [Fin.snoc_castSucc, Fin.cons_succ]
omit [Fintype P] [DecidableEq P] in
lemma pathProduct_cons {n : ℕ} (A : Fin (n+1) → Matrix P P R)
(a : P) (x : Fin n → P) (b : P) :
pathProduct A (Fin.cons a x) b =
A 0 a (Fin.snoc (α := fun _ => P) x b 0) * pathProduct (fun j => A j.succ) x b := by
simp only [pathProduct, List.ofFn_succ, List.prod_cons, Fin.cons_zero, Fin.cons_succ,
snoc_cons_succ]
lemma prod_apply_sum_path {n : ℕ} (A : Fin (n+1) → Matrix P P R) (a b : P) :
(List.ofFn A).prod a b = ∑ x : Fin n → P, pathProduct A (Fin.cons a x) b := by
induction n generalizing a with
| zero => simp [pathProduct, Fin.snoc]
| succ n ih =>
rw [List.ofFn_succ, List.prod_cons, Matrix.mul_apply]
simp_rw [ih]
rw [← (Fin.consEquiv (fun _ : Fin (n+1) => P)).sum_comp, Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro c _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
change A 0 a c * pathProduct (fun j => A j.succ) (Fin.cons c x) b =
pathProduct A (Fin.cons a (Fin.cons c x)) b
rw [pathProduct_cons A]
congr 2
omit [Fintype P] [DecidableEq P] in
lemma snoc_cycle {n : ℕ} (x : Fin (n+1) → P) (j : Fin (n+1)) :
Fin.snoc (α := fun _ => P) x (x 0) j.succ = x (finRotate (n+1) j) := by
refine Fin.lastCases ?_ (fun i => ?_) j
· simp
· rw [Fin.succ_castSucc, Fin.snoc_castSucc]
simp [finRotate_apply, Fin.coeSucc_eq_succ]
lemma trace_prod_eq_sum_cycles {n : ℕ} (A : Fin (n+1) → Matrix P P R) :
Matrix.trace (List.ofFn A).prod =
∑ x : Fin (n+1) → P,
(List.ofFn fun j => A j (x j) (x (finRotate (n+1) j))).prod := by
change (∑ i : P, (List.ofFn A).prod i i) = _
simp_rw [prod_apply_sum_path]
calc
_ = ∑ y : P × (Fin n → P), pathProduct A (Fin.cons y.1 y.2) y.1 := by
exact (Fintype.sum_prod_type (fun y : P × (Fin n → P) =>
pathProduct A (Fin.cons y.1 y.2) y.1)).symm
_ = ∑ x : Fin (n+1) → P, pathProduct A x (x 0) := by
simpa only [Fin.consEquiv, Equiv.coe_fn_mk, Fin.cons_zero] using
(Fin.consEquiv (fun _ : Fin (n+1) => P)).sum_comp (fun x => pathProduct A x (x 0))
_ = _ := by
apply Finset.sum_congr rfl
intro x _
simp only [pathProduct, snoc_cycle]
end Expansion
variable {P ι : Type*} [Fintype P] [DecidableEq P] [Fintype ι] [DecidableEq ι]
def flattenBlocks (A : Matrix P P (M ι)) : M (P × ι) :=
fun x y => A x.1 y.1 x.2 y.2
omit [DecidableEq P] [DecidableEq ι] in
lemma flattenBlocks_mul (A B : Matrix P P (M ι)) :
flattenBlocks (A * B) = flattenBlocks A * flattenBlocks B := by
ext x y
simp only [flattenBlocks, Matrix.mul_apply, Fintype.sum_prod_type, Matrix.sum_apply]
omit [Fintype P] [Fintype ι] in
lemma flattenBlocks_one : flattenBlocks (1 : Matrix P P (M ι)) = 1 := by
ext ⟨x,i⟩ ⟨y,j⟩
by_cases hxy : x = y <;> by_cases hij : i = j <;>
simp [flattenBlocks, Matrix.one_apply, hxy, hij, Prod.mk.injEq]
def flattenBlocksHom : Matrix P P (M ι) →* M (P × ι) where
toFun := flattenBlocks
map_one' := flattenBlocks_one
map_mul' := flattenBlocks_mul
omit [DecidableEq P] [DecidableEq ι] in
lemma trace_flattenBlocks (A : Matrix P P (M ι)) :
Matrix.trace (flattenBlocks A) = Matrix.trace (Matrix.trace A) := by
change (∑ x : P × ι, A x.1 x.1 x.2 x.2) = ∑ i : ι, (∑ p : P, A p p) i i
simp only [Fintype.sum_prod_type, Matrix.sum_apply]
exact Finset.sum_comm
lemma trace_flatten_prod {n : ℕ} (A : Fin (n+1) → Matrix P P (M ι)) :
Matrix.trace (List.ofFn (fun j => flattenBlocks (A j))).prod =
∑ x : Fin (n+1) → P,
Matrix.trace (List.ofFn fun j => A j (x j) (x (finRotate (n+1) j))).prod := by
have hp : (List.ofFn (fun j => flattenBlocks (A j))).prod =
flattenBlocks (List.ofFn A).prod := by
simpa only [List.map_ofFn, flattenBlocksHom, MonoidHom.coe_mk, OneHom.coe_mk, Function.comp_def] using
(map_list_prod (flattenBlocksHom (P := P) (ι := ι)) (List.ofFn A)).symm
rw [hp, trace_flattenBlocks, trace_prod_eq_sum_cycles, Matrix.trace_sum]
theorem block_trace_holder {q : ℕ} (hq : 0 < q)
(A : Fin (2*q) → Matrix P P (M ι)) :
‖Matrix.trace (List.ofFn (fun j => flattenBlocks (A j))).prod‖ ≤
∑ x : Fin (2*q) → P,
∏ j, matrixMoment q (A j (x j) (x (finRotate (2*q) j))) ^ (((2*q : ℕ) : ℝ)⁻¹) := by
have ht : Matrix.trace (List.ofFn (fun j => flattenBlocks (A j))).prod =
∑ x : Fin (2*q) → P,
Matrix.trace (List.ofFn fun j => A j (x j) (x (finRotate (2*q) j))).prod := by
generalize hm : 2*q = m at A ⊢
cases m with
| zero => omega
| succ n => exact trace_flatten_prod A
rw [ht]
exact (norm_sum_le _ _).trans (Finset.sum_le_sum fun x _ => matrix_trace_holder hq _)
theorem weighted_block_trace_holder {q : ℕ} (hq : 0 < q)
(p : Fin (2*q) → P → P → ℝ) (hp : ∀ j x y, 0 ≤ p j x y)
(F : Fin (2*q) → P → P → M ι) :
‖Matrix.trace (List.ofFn fun j => flattenBlocks
(fun x y => p j x y • F j x y)).prod‖ ≤
∑ x : Fin (2*q) → P,
(∏ j, p j (x j) (x (finRotate (2*q) j))) *
∏ j, matrixMoment q (F j (x j) (x (finRotate (2*q) j))) ^ (((2*q : ℕ) : ℝ)⁻¹) := by
have ht : Matrix.trace (List.ofFn fun j => flattenBlocks
(fun x y => p j x y • F j x y)).prod =
∑ x : Fin (2*q) → P,
Matrix.trace (List.ofFn fun j => p j (x j) (x (finRotate (2*q) j)) •
F j (x j) (x (finRotate (2*q) j))).prod := by
generalize hm : 2*q = m at p F ⊢
cases m with
| zero => omega
| succ n => exact trace_flatten_prod _
rw [ht]
apply (norm_sum_le _ _).trans
apply Finset.sum_le_sum
intro x _
rw [prod_ofFn_smul, Matrix.trace_smul, norm_smul, Real.norm_eq_abs,
abs_of_nonneg (Finset.prod_nonneg (fun j _ => hp j _ _))]
exact mul_le_mul_of_nonneg_left (matrix_trace_holder hq _)
(Finset.prod_nonneg (fun j _ => hp j _ _))
end BinaryCoordinateSweeps.TraceHolder
end
end OAI
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