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dd14c26 | 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 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.AllSplitBound
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.TensorEnergy
open Function
open scoped Classical BigOperators
def zeroExtend {I J : Type*} (e : I ↪ J) (f : I → ℝ) : J → ℝ :=
Function.extend e f (fun _ => 0)
@[simp] lemma zeroExtend_apply {I J : Type*} (e : I ↪ J) (f : I → ℝ) (i : I) :
zeroExtend e f (e i)=f i := e.injective.extend_apply f _ i
lemma zeroExtend_out {I J : Type*} (e : I ↪ J) (f : I → ℝ) {j : J} (hj : j∉Set.range e) :
zeroExtend e f j=0 := Function.extend_apply' _ _ _ hj
lemma zeroExtend_sum {I J : Type*} [Fintype I] [Fintype J] (e : I ↪ J) (f : I → ℝ) :
(∑j,zeroExtend e f j)=∑i,f i :=
(Fintype.sum_of_injective e e.injective f (zeroExtend e f)
(fun _ hj => zeroExtend_out e f hj) (fun i => (zeroExtend_apply e f i).symm)).symm
lemma zeroExtend_sq {I J : Type*} (e : I ↪ J) (f : I → ℝ) :
(fun j => (zeroExtend e f j)^2)=zeroExtend e (fun i => (f i)^2) := by
funext j
by_cases hj : j∈Set.range e
· obtain ⟨i,rfl⟩ := hj; simp
· rw [zeroExtend_out e f hj,zeroExtend_out e _ hj]; norm_num
lemma zeroExtend_mul {I J : Type*} (e : I ↪ J) (f : I → ℝ) (g : J → ℝ) :
(fun j => zeroExtend e f j*g j)=zeroExtend e (fun i => f i*g (e i)) := by
funext j
by_cases hj : j∈Set.range e
· obtain ⟨i,rfl⟩ := hj; simp
· rw [zeroExtend_out e f hj,zeroExtend_out e _ hj]; simp
lemma sum_restrict_le {I J : Type*} [Fintype I] [Fintype J] (e : I ↪ J) (f : J → ℝ)
(hf : ∀j,0≤f j) : (∑i,f (e i))≤∑j,f j := by
rw [←Finset.sum_image (f:=f) (g:=(e : I → J)) (by intro i _ j _ h; exact e.injective h)]
exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun j _ _ => hf j)
lemma Bound.zeroExtend {I O I' O' : Type*} [Fintype I] [Fintype O] [Fintype I'] [Fintype O']
{A : O → I → ℝ} {C : ℝ} (hA : Bound A C) (hC : 0≤C)
(eI : I ↪ I') (eO : O ↪ O') :
Bound (fun o i => TensorEnergy.zeroExtend eO (fun o => TensorEnergy.zeroExtend eI (A o) i) o) C := by
intro v
have he : (fun o => ∑i,TensorEnergy.zeroExtend eO (fun o => TensorEnergy.zeroExtend eI (A o) i) o*v i)=
TensorEnergy.zeroExtend eO (fun o => ∑i,A o i*v (eI i)) := by
funext o
by_cases ho : o∈Set.range eO
· obtain ⟨o,rfl⟩ := ho
simp only [zeroExtend_apply]
rw [zeroExtend_mul,zeroExtend_sum]
· simp only [zeroExtend_out eO _ ho,zero_mul,Finset.sum_const_zero]
change (∑o,((fun o => ∑i,TensorEnergy.zeroExtend eO (fun o => TensorEnergy.zeroExtend eI (A o) i) o*v i) o)^2)≤_
rw [he,zeroExtend_sq,zeroExtend_sum]
exact (hA (fun i => v (eI i))).trans
(mul_le_mul_of_nonneg_left (sum_restrict_le eI (fun i => (v i)^2) (fun _ => sq_nonneg _)) hC)
end LogConcaveSampling.TensorEnergy
end
end
section
noncomputable section
namespace LogConcaveSampling.TensorEnergy
open Function
open scoped Classical BigOperators
def slotEmbedding {S : Type*} {d D : ℕ} (e : S → Fin d ↪ Fin D) :
(S → Fin d) ↪ (S → Fin D) where
toFun c s := e s (c s)
inj' := by intro a b h; funext s; exact (e s).injective (congrFun h s)
@[simp] lemma slotEmbedding_apply {S : Type*} {d D : ℕ} (e : S → Fin d ↪ Fin D)
(c : S → Fin d) (s : S) : slotEmbedding e c s=e s (c s) := rfl
lemma AllSplitBound.zeroExtend {S : Type} [Fintype S] {d D : ℕ}
{T : (S → Fin d) → ℝ} {M : ℝ} (hT : AllSplitBound T M) (ι : S → Fin d ↪ Fin D) :
AllSplitBound (TensorEnergy.zeroExtend (slotEmbedding ι) T) M := by
intro I O _ _ _ _ _ _ e
let ei := slotEmbedding (fun i => ι (e.symm (.inl i)))
let eo := slotEmbedding (fun o => ι (e.symm (.inr o)))
have he (i : I → Fin d) (o : O → Fin d) :
(fun s => Sum.elim (ei i) (eo o) (e s))=
slotEmbedding ι (fun s => Sum.elim i o (e s)) := by
funext s
rcases h : e s with a|b <;>
simp only [h,Sum.elim_inl,Sum.elim_inr,ei,eo,slotEmbedding_apply,e.symm_apply_eq.mpr h.symm]
have hrow (o : O → Fin D) (i : I → Fin D) :
TensorEnergy.zeroExtend (slotEmbedding ι) T (fun s => Sum.elim i o (e s))=
TensorEnergy.zeroExtend eo (fun o => TensorEnergy.zeroExtend ei (fun i => T (fun s => Sum.elim i o (e s))) i) o := by
by_cases hi : i∈Set.range ei
· obtain ⟨a,rfl⟩ := hi
by_cases ho : o∈Set.range eo
· obtain ⟨b,rfl⟩ := ho
rw [he,zeroExtend_apply,zeroExtend_apply,zeroExtend_apply]
· rw [zeroExtend_out eo _ ho,zeroExtend_out]
rintro ⟨c,hc⟩
apply ho
refine ⟨fun b => c (e.symm (.inr b)),?_⟩
funext b
have hh := congrFun hc (e.symm (.inr b))
change ι (e.symm (.inr b)) (c (e.symm (.inr b)))=o b
simpa only [slotEmbedding_apply,Equiv.apply_symm_apply,Sum.elim_inr] using hh
· rw [zeroExtend_out]
· by_cases ho : o∈Set.range eo
· obtain ⟨b,rfl⟩ := ho
rw [zeroExtend_apply,zeroExtend_out ei _ hi]
· rw [zeroExtend_out eo _ ho]
· rintro ⟨c,hc⟩
apply hi
refine ⟨fun a => c (e.symm (.inl a)),?_⟩
funext a
have hh := congrFun hc (e.symm (.inl a))
change ι (e.symm (.inl a)) (c (e.symm (.inl a)))=i a
simpa only [slotEmbedding_apply,Equiv.apply_symm_apply,Sum.elim_inl] using hh
have hb := (hT I O e).zeroExtend (sq_nonneg M) ei eo
convert hb using 1
ext o i
exact hrow o i
end LogConcaveSampling.TensorEnergy
end
end
end
end OAI
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