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cdec422 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalCW
import Mathlib.Tactic.FinCases
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointCanonicalInitial
open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW
open InheritedMasks CWStrands
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H F : Type*} [Fintype H] [DecidableEq H] [CommRing F]
(counts : H → Shape → ℕ)
abbrev Word := Fin 8 → Fin 7
abbrev RawWords := ∀ h, Positions counts h → Word
abbrev CanonicalWords := ∀ c : ClassKey (H := H), ClassPositions counts c → Word
def wholeEquiv (e : Target counts) : RawWords counts ≃ CanonicalWords counts where
toFun w := fun c j => w c.1 ((positionEquiv counts e c.1).symm ⟨c.2, j⟩)
invFun w := fun h i => w (h, (e h).val i) (positionEquiv counts e h i).2
left_inv w := by
funext h i
change w h ((positionEquiv counts e h).symm (positionEquiv counts e h i)) = w h i
simp
right_inv w := by
funext c j
exact congrArg (fun p : Σ u, Fin (counts c.1 u) => w (c.1, p.1) p.2)
((positionEquiv counts e c.1).apply_symm_apply ⟨c.2, j⟩)
def rawSource : Tensor F (RawWords counts) (RawWords counts) (RawWords counts) :=
classProduct (fun _ : H => strand (Fin 8))
def coarseWord (w : RawWords counts) : Position counts → Fin 17 :=
fun i => CWCoarseIndices.coarseIndex (by simp) (w i.1 i.2)
def ideal (e : Target counts) : Tensor F (RawWords counts) (RawWords counts)
(RawWords counts) :=
JointIdealBranches.ideal (rawSource counts) (coarseWord counts) (coarseWord counts)
(coarseWord counts) (fun _ _ => True) (fun _ _ => True) (fun _ _ => True)
(triple counts e)
def canonical : Tensor F (CanonicalWords counts) (CanonicalWords counts)
(CanonicalWords counts) :=
classProduct (fun c : ClassKey (H := H) => shapeTensor (Fin 8) (shapeNat c.2))
omit [DecidableEq H] in
theorem canonical_pack (e : Target counts) (x y z : RawWords counts) :
canonical counts (wholeEquiv counts e x) (wholeEquiv counts e y)
(wholeEquiv counts e z) =
∏ h, ∏ i, shapeTensor (F := F) (Fin 8) (shapeNat ((e h).val i))
(x h i) (y h i) (z h i) := by
unfold canonical classProduct
rw [Fintype.prod_prod_type]
apply Finset.prod_congr rfl
intro h _
rw [← Fintype.prod_sigma']
apply Fintype.prod_equiv (positionEquiv counts e h).symm
intro i
simp only [wholeEquiv, Equiv.coe_fn_mk, positionEquiv_symm_shape]
omit [DecidableEq H] in
theorem ideal_eq_canonical (e : Target counts) (x y z : RawWords counts) :
ideal (F := F) counts e x y z = canonical counts (wholeEquiv counts e x)
(wholeEquiv counts e y) (wholeEquiv counts e z) := by
rw [canonical_pack]
simp only [ideal, JointIdealBranches.ideal, ExactRecovery.delete, and_true,
rawSource, classProduct, shapeTensor, Fintype.prod_ite_zero, forall_and]
congr 1
simp [coarseWord, triple, shapeNat, shapeSide, CWCoarseIndices.coarseIndex,
funext_iff, Fin.ext_iff, Sigma.forall]
def coordinateMatrix (e : Target counts) : CanonicalWords counts → RawWords counts → F :=
fun x v => if v = (wholeEquiv counts e).symm x then 1 else 0
theorem ideal_restrict (e : Target counts) :
Tensor.restrict (coordinateMatrix (F := F) counts e) (coordinateMatrix counts e)
(coordinateMatrix counts e) (ideal counts e) = canonical counts := by
funext x y z
simp only [Tensor.restrict, coordinateMatrix]
simp only [ite_mul, mul_ite, zero_mul, mul_zero, one_mul, Finset.sum_ite_eq',
Finset.mem_univ, ↓reduceIte]
rw [ideal_eq_canonical]
simp
omit [DecidableEq H] in
theorem coarse_support (x y z : RawWords counts)
(hT : rawSource (F := F) counts x y z ≠ 0) (i : Position counts) :
(coarseWord counts x i).val + (coarseWord counts y i).val +
(coarseWord counts z i).val = 16 := by
have hh : strand (F := F) (Fin 8) (x i.1 i.2) (y i.1 i.2) (z i.1 i.2) ≠ 0 := by
intro hz
apply hT
unfold rawSource classProduct
apply Finset.prod_eq_zero (Finset.mem_univ i.1)
exact Finset.prod_eq_zero (Finset.mem_univ i.2) hz
simpa [coarseWord] using
CWCoarseIndices.strand_coarse_support (F := F) (by simp)
(x i.1 i.2) (y i.1 i.2) (z i.1 i.2) hh
def sideMask (s : Fin 3) (w : RawWords counts) : Prop :=
∀ h a, wordPopulation (fun i => coarseWord counts w ⟨h, i⟩) a =
wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a
def maskedSource : Tensor F (RawWords counts) (RawWords counts) (RawWords counts) :=
ExactRecovery.delete (rawSource counts) (sideMask counts 0) (sideMask counts 1)
(sideMask counts 2)
omit [Fintype H] [DecidableEq H] in
theorem sideMask_of_target (s : Fin 3) (e : Target counts) (w : RawWords counts)
(he : coarseWord counts w =
fun i => sideWord counts s (triple counts e) i.1 i.2) :
sideMask counts s w := by
intro h a
rw [he]
exact marginal_population_eq counts e (canonicalTarget counts) h s a
theorem maskedSource_support (x y z : RawWords counts)
(hT : maskedSource (F := F) counts x y z ≠ 0) :
(coarseWord counts x, coarseWord counts y, coarseWord counts z) ∈
ambientSet counts (fun _ => 16) := by
have hm : sideMask counts 0 x ∧ sideMask counts 1 y ∧ sideMask counts 2 z := by
by_contra hn
simp [maskedSource, ExactRecovery.delete, hn] at hT
have hr : rawSource (F := F) counts x y z ≠ 0 := by
simpa [maskedSource, ExactRecovery.delete, hm] using hT
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ _, ?_, ?_⟩
· intro i
exact coarse_support counts x y z hr i
· intro h s a
fin_cases s
· refine Eq.trans ?_ (hm.1 h a)
apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a)
funext i
simp [sideWord, shapeSide]
· refine Eq.trans ?_ (hm.2.1 h a)
apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a)
funext i
simp [sideWord, shapeSide]
· refine Eq.trans ?_ (hm.2.2 h a)
apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a)
funext i
simp [sideWord, shapeSide]
omit [DecidableEq H] in
theorem ideal_support_masks (e : Target counts) (x y z : RawWords counts)
(hT : ideal (F := F) counts e x y z ≠ 0) :
sideMask counts 0 x ∧ sideMask counts 1 y ∧ sideMask counts 2 z := by
have hc : coarseWord counts x = (triple counts e).1 ∧
coarseWord counts y = (triple counts e).2.1 ∧
coarseWord counts z = (triple counts e).2.2 := by
by_contra hn
simp [ideal, JointIdealBranches.ideal, ExactRecovery.delete, hn] at hT
constructor
· apply sideMask_of_target counts 0 e x
simpa [sideWord, shapeSide] using hc.1
constructor
· apply sideMask_of_target counts 1 e y
simpa [sideWord, shapeSide] using hc.2.1
· apply sideMask_of_target counts 2 e z
simpa [sideWord, shapeSide] using hc.2.2
omit [DecidableEq H] in
theorem delete_ideal_eq (e : Target counts) :
ExactRecovery.delete (ideal (F := F) counts e) (sideMask counts 0)
(sideMask counts 1) (sideMask counts 2) = ideal counts e := by
funext x y z
by_cases h : ideal (F := F) counts e x y z = 0
· rw [ExactRecovery.delete, h]
simp
· rw [ExactRecovery.delete, ite_eq_left (ideal_support_masks counts e x y z h)]
end MatrixMultiplication.JointCanonicalInitial
end
end OAI
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