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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 | import Mathlib
import OAI.Probability.BinarySweep.Processes.IndependentHistory
import OAI.Probability.BinarySweep.Representations.IrrepSupport
import OAI.Probability.BinarySweep.FiniteLaws.SumDite
namespace OAI
noncomputable section
open scoped BigOperators
namespace BinaryCoordinateSweeps
attribute [local instance] Classical.propDecidable
instance oneAxisOutsideEmpty : IsEmpty {i : Fin 1 // i ≠ 0} :=
⟨fun i => i.property (Subsingleton.elim _ _)⟩
noncomputable def oneAxisSlotEquiv (bits : Fin 1 → ℕ) : GridSlot bits ≃ Slot (bits 0) :=
Equiv.piUnique _
noncomputable def oneAxisChoiceEquiv (bits : Fin 1 → ℕ) :
GridChoices bits ≃ Equiv.Perm (Slot (bits 0)) :=
(Equiv.piUnique _).trans (Equiv.funUnique _ _)
noncomputable def oneAxisSweepEquiv (bits : Fin 1 → ℕ) :
GridChoices bits ≃ Equiv.Perm (GridSlot bits) :=
(oneAxisChoiceEquiv bits).trans ((oneAxisSlotEquiv bits).symm.permCongr)
lemma oneAxisSweepEquiv_apply (bits : Fin 1 → ℕ) (g : GridChoices bits) :
oneAxisSweepEquiv bits g = gridSweep bits g := by
apply Equiv.ext
intro x
funext i
have hi : i = 0 := Subsingleton.elim _ _
subst i
simp only [gridSweep, List.ofFn_succ, List.ofFn_zero, List.reverse_cons,
List.reverse_nil, List.nil_append, List.prod_cons, List.prod_nil, mul_one]
change _ = gridLayer bits g 0 x 0
simp only [oneAxisSweepEquiv, oneAxisChoiceEquiv, oneAxisSlotEquiv, Equiv.trans_apply,
Equiv.permCongr_apply, Equiv.piUnique_apply,
Equiv.funUnique_apply, gridLayer, Equiv.piSplitAt_apply, Equiv.prodCongrLeft_apply,
Equiv.piSplitAt_symm_apply, dite_true, eq_self]
simp only [Equiv.symm_symm, Equiv.piUnique_symm_apply, Equiv.piUnique_apply]
have aux (y : GridOutside bits default) (i : Fin 1) :
uniqueElim (α := fun j => Slot (bits j)) ((g default y) (x default)) i =
(g i (fun j => x j.val)) (x i) := by
obtain rfl := Unique.eq_default i
rw [uniqueElim_default]
apply congrArg (fun y : GridOutside bits default => g default y (x default))
funext j
exact (j.property (Subsingleton.elim _ _)).elim
exact aux _ 0
lemma oneAxis_pathEvent_iff {h : ℕ} {bits : Fin 1 → ℕ} (H : PathFamily bits h)
(g : GridChoices bits) : pathEvent H g ↔
∀ k, gridSweep bits g (H.position 0 k) = H.position (Fin.last 1) k := by
constructor
· exact fun hg k => gridSweep_path H g hg k
· intro hg j k
have hj : j = 0 := Subsingleton.elim _ _
subst j
simpa [gridSweep, List.ofFn_succ] using hg k
def assignmentResidualEquiv {X I : Type*} [Fintype X] [Fintype I]
(A B : I ↪ X) (c : {x : X // x ∉ Set.range A} ≃ {x : X // x ∉ Set.range B}) :
{σ : Equiv.Perm X // Assigns A B σ} ≃ Equiv.Perm {x : X // x ∉ Set.range A} := by
classical
let eA := Equiv.ofInjective A A.injective
let eB := Equiv.ofInjective B B.injective
let e₀ : Set.range A ≃ Set.range B := eA.symm.trans eB
let e : {σ : Equiv.Perm X // Assigns A B σ} ≃
{σ : X ≃ X // ∀ x : Set.range A, σ x = e₀ x} :=
Equiv.subtypeEquiv (Equiv.refl _) (by
intro σ
constructor
· intro hs x
obtain ⟨i, hi⟩ := x.property
have hx : x = eA i := Subtype.ext hi.symm
rw [hx]
simpa [e₀, eA, eB] using hs i
· intro hs i
simpa [e₀, eA, eB] using hs (eA i))
exact (e.trans (Equiv.Set.compl e₀)).trans (Equiv.equivCongr (Equiv.refl _) c.symm)
lemma assignmentResidualEquiv_apply {X I : Type*} [Fintype X] [Fintype I]
(A B : I ↪ X) (c : {x : X // x ∉ Set.range A} ≃ {x : X // x ∉ Set.range B})
(σ : {σ : Equiv.Perm X // Assigns A B σ}) (x : {x : X // x ∉ Set.range A}) :
(c (assignmentResidualEquiv A B c σ x)).val = σ.val x := by
simp [assignmentResidualEquiv, Equiv.equivCongr, Equiv.Set.compl]
rfl
def oneAxisRemainingEquiv {h : ℕ} {bits : Fin 1 → ℕ} (H : PathFamily bits h) :
{g : GridChoices bits // pathEvent H g} ≃ Equiv.Perm (FreeSlot H 0) := by
let A : Fin h ↪ GridSlot bits := ⟨H.position 0, H.disjoint 0⟩
let B : Fin h ↪ GridSlot bits := ⟨H.position (Fin.last 1), H.disjoint _⟩
exact (Equiv.subtypeEquiv (oneAxisSweepEquiv bits) (by
intro g
simpa only [Assigns, oneAxisSweepEquiv_apply, A, B, Function.Embedding.coeFn_mk] using oneAxis_pathEvent_iff H g)).trans
(assignmentResidualEquiv A B (freeIdentification H))
lemma oneAxisRemainingEquiv_apply {h : ℕ} {bits : Fin 1 → ℕ} (H : PathFamily bits h)
(g : {g : GridChoices bits // pathEvent H g}) :
oneAxisRemainingEquiv H g = remainingPerm H g.val g.property := by
apply Equiv.ext
intro x
apply (freeIdentification H).injective
apply Subtype.ext
have he := assignmentResidualEquiv_apply
(⟨H.position 0, H.disjoint 0⟩ : Fin h ↪ GridSlot bits)
(⟨H.position (Fin.last 1), H.disjoint _⟩ : Fin h ↪ GridSlot bits)
(freeIdentification H)
((Equiv.subtypeEquiv (oneAxisSweepEquiv bits) (by
intro g
simpa only [Assigns, oneAxisSweepEquiv_apply, Function.Embedding.coeFn_mk] using oneAxis_pathEvent_iff H g)) g) x
simpa only [oneAxisRemainingEquiv, Equiv.trans_apply, Equiv.subtypeEquiv_apply,
oneAxisSweepEquiv_apply, remainingPerm, Equiv.apply_symm_apply, remainingBijection,
Equiv.coe_fn_mk] using he
lemma oneAxis_gridWeight (bits : Fin 1 → ℕ) (z : ℝ) (g : GridChoices bits) :
gridWeight bits z g = lineLaw (bits 0) z (oneAxisChoiceEquiv bits g) := by
simp [gridWeight, oneAxisChoiceEquiv, Equiv.piUnique, Equiv.funUnique]
lemma oneAxis_gridWeight_zero (bits : Fin 1 → ℕ) (g : GridChoices bits) :
gridWeight bits 0 g = (Fintype.card (Equiv.Perm (GridSlot bits)) : ℝ)⁻¹ := by
rw [oneAxis_gridWeight]
simp only [lineLaw, sub_zero, one_mul, zero_mul, add_zero, uniformLaw]
congr 2
exact Fintype.card_congr (oneAxisSlotEquiv bits).symm.permCongr
lemma oneAxis_conditionalOperator_zero {h D : ℕ} {bits : Fin 1 → ℕ}
(H : PathFamily bits h)
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D))
[ρ.IsIrreducible] (hD : 1 < D) : conditionalOperator H 0 ρ = 0 := by
have hdim : 1 < Module.finrank ℂ (RepSpace D) := by simpa using hD
have hsum := Irrep.sum_eq_zero_of_finrank_gt_one ρ hdim
have hperm : (∑ g : {g : GridChoices bits // pathEvent H g},
ρ (remainingPerm H g.val g.property)) = 0 := by
calc
_ = ∑ g : {g : GridChoices bits // pathEvent H g}, ρ (oneAxisRemainingEquiv H g) := by
simp only [oneAxisRemainingEquiv_apply]
_ = 0 := ((oneAxisRemainingEquiv H).sum_comp (fun σ => ρ σ)).trans hsum
unfold conditionalOperator
have he : (∑ g : GridChoices bits, if hg : pathEvent H g then
(gridWeight bits 0 g : ℂ) • ρ (remainingPerm H g hg) else 0) = 0 := by
rw [sum_dite_zero]
simp only [oneAxis_gridWeight_zero, ← Finset.smul_sum]
apply smul_eq_zero.mpr
right
convert hperm using 1
rw [he, smul_zero]
rfl
end BinaryCoordinateSweeps
end
end OAI
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