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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 | import Mathlib
import OAI.Probability.BinarySweep.FiniteLaws.SumDite
import OAI.Probability.BinarySweep.Trajectories.PathExtension
import OAI.Probability.BinarySweep.Conditional.PlacementBlocks
namespace OAI
noncomputable section
section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Irrep
open Representation
variable {A B X Y : Type*} [Fintype A] [Fintype B] [Fintype X]
def complementMap (f : X ≃ Y) (x : B ↪ X) (y : B ↪ Y)
(hf : x.trans f.toEmbedding=y) : PlacementComplement x ≃ PlacementComplement y where
toFun u := ⟨f u.val,by
rintro ⟨b,hb⟩
apply u.property
refine ⟨b,f.injective ?_⟩
exact (congrArg (fun y : B ↪ Y => y b) hf).trans hb⟩
invFun u := ⟨f.symm u.val,by
rintro ⟨b,hb⟩
apply u.property
refine ⟨b,?_⟩
exact (congrArg (fun y : B ↪ Y => y b) hf).symm.trans
((congrArg f hb).trans (f.apply_symm_apply _))⟩
left_inv u := Subtype.ext (f.symm_apply_apply _)
right_inv u := Subtype.ext (f.apply_symm_apply _)
omit [Fintype B] [Fintype X] in
lemma complementMap_val (f : X ≃ Y) (x : B ↪ X) (y : B ↪ Y)
(hf : x.trans f.toEmbedding=y) (u : PlacementComplement x) :
(complementMap f x y hf u).val=f u.val := rfl
omit [Fintype B] [Fintype X] in
lemma placement_inverse_event (a : Equiv.Perm X) (x y : B ↪ X)
(he : x.trans a.toEmbedding=y) : y.trans a.symm.toEmbedding=x := by
ext b
change a.symm (y b)=x b
rw [← he]
exact a.symm_apply_apply _
omit [Fintype A] [Fintype B] [Fintype X] in
lemma blockPerm_inl (e : A ⊕ B ≃ X) (a : Equiv.Perm A) (u : A) :
blockPerm e a (e (Sum.inl u))=e (Sum.inl (a u)) := by
change e (Equiv.sumCongr a (1:Equiv.Perm B) (e.symm (e (Sum.inl u))))=_
rw [e.symm_apply_apply]
rfl
omit [Fintype X] in
lemma placement_subgroup_complement (e : A ⊕ B ≃ X) (a : Equiv.Perm X)
(x y : B ↪ X) (he : x.trans a.toEmbedding=y) :
((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹)
(placement_cosets_bijective e)).symm ((placementSection e y)⁻¹*a)).1 =
((placementComplementEquiv e x).trans (complementMap a x y he)).trans
(placementComplementEquiv e y).symm := by
have hh := placementCoset_subgroup e a y
rw [placement_inverse_event a x y he] at hh
ext u
apply (placementComplementEquiv e y).injective
apply Subtype.ext
have hu := congrArg (placementSection e y) (Equiv.congr_fun hh (e (Sum.inl u)))
simp only [blockPerm_inl,Equiv.Perm.mul_apply,Equiv.Perm.inv_def,
Equiv.apply_symm_apply] at hu
simp only [Equiv.trans_apply, Equiv.apply_symm_apply]
change placementSection e y (e (Sum.inl _)) = a (placementSection e x (e (Sum.inl u)))
simpa only [Equiv.Perm.inv_def] using hu
end BinaryCoordinateSweeps.Irrep
end
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
attribute [local instance] Classical.propDecidable
variable {b h h' : ℕ} {bits : Fin b → ℕ}
variable {H : PathFamily bits h} {H' : PathFamily bits h'}
def PathExtension.restrictChoice (E : PathExtension H H')
(g : ConditionalChoices H') : ConditionalChoices H :=
⟨g.val,E.pathEvent_mono g.property⟩
lemma PathExtension.choiceWeight (E : PathExtension H H') (z : ℝ)
(hpos : conditionalNormalizer H' z ≠ 0) (g : ConditionalChoices H') :
conditionalChoiceWeight H z (E.restrictChoice g) =
(conditionalNormalizer H' z / conditionalNormalizer H z) *
conditionalChoiceWeight H' z g := by
unfold conditionalChoiceWeight PathExtension.restrictChoice
simp only
field_simp
lemma PathExtension.sum_restrict {V : Type*} [AddCommMonoid V]
(E : PathExtension H H') (f : ConditionalChoices H' → V) :
(∑g : ConditionalChoices H,
if hg : pathEvent H' g.val then f ⟨g.val,hg⟩ else 0) =
∑g : ConditionalChoices H',f g := by
have hL := sum_dite_zero (pathEvent H)
(fun (g : GridChoices bits) (_ : pathEvent H g) =>
if hg' : pathEvent H' g then f ⟨g,hg'⟩ else 0)
have hR := sum_dite_zero (pathEvent H') (fun g hg => f ⟨g,hg⟩)
refine Eq.trans ?_ (Eq.trans hL.symm (Eq.trans ?_ (Eq.trans hR ?_)))
· apply Finset.sum_congr
· ext g; simp only [Finset.mem_univ]
intro g _
split_ifs <;> rfl
· apply Finset.sum_congr rfl
intro g _
by_cases hg : pathEvent H' g
· simp only [hg,E.pathEvent_mono hg,dite_eq_left]
· simp [hg]
· apply Finset.sum_congr
· ext g; simp only [Finset.mem_univ]
intro g _
rfl
lemma PathExtension.conditional_sum {V : Type*} [AddCommGroup V] [Module ℝ V]
(E : PathExtension H H') (z : ℝ) (hpos : conditionalNormalizer H' z ≠ 0)
(f : ConditionalChoices H' → V) :
(∑g : ConditionalChoices H,
if hg : pathEvent H' g.val then conditionalChoiceWeight H z g • f ⟨g.val,hg⟩ else 0) =
(conditionalNormalizer H' z / conditionalNormalizer H z) •
∑g : ConditionalChoices H', conditionalChoiceWeight H' z g • f g := by
rw [Finset.smul_sum]
simp_rw [← mul_smul]
rw [← E.sum_restrict (fun g =>
((conditionalNormalizer H' z / conditionalNormalizer H z) *
conditionalChoiceWeight H' z g) • f g)]
apply Finset.sum_congr rfl
intro g _
split_ifs with hg
· rw [← E.choiceWeight z hpos ⟨g.val,hg⟩]
rfl
· rfl
end BinaryCoordinateSweeps
end
end OAI
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