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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 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Labels
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.BigOperators
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionCardinality
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
attribute [local instance] Classical.propDecidable Classical.decEq
universe uCoord
variable {X Y Z : Type uCoord}
def conditionalCount [Fintype X] [Fintype Y] [Fintype Z]
(S : FlaggedTensor X Y Z) (c : Color) : ℕ :=
Fintype.card {a : Leaf S // leafColor S a = c}
private abbrev AllowedLeaf (S : FlaggedTensor X Y Z) (center output : Color) :=
{a : Leaf S // leafColor S a = center ∨ leafColor S a = output}
private def centerLeafEquiv (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) :
{a : Leaf (complete S center m) //
leafColor (complete S center m) a = center} ≃
(Fin m → {a : Leaf S // leafColor S a = center}) :=
(conditionalLeafEquiv S center center m).trans
((Equiv.subtypeEquivRight (fun w => by simp [raisedFlag])).trans
Equiv.subtypePiEquivPi)
private def noncenterLeafEquiv (S : FlaggedTensor X Y Z) (center output : Color)
(m : ℕ) (h : center ≠ output) :
{a : Leaf (complete S center m) //
leafColor (complete S center m) a = output} ≃
{w : Fin m → AllowedLeaf S center output //
¬ ∀ i, leafColor S (w i).val = center} :=
(conditionalLeafEquiv S center output m).trans
{ toFun := fun w =>
⟨fun i => ⟨w.val i, w.property.2 i⟩, by
intro hall
have hex : ∃ i, leafColor S (w.val i) = output := by
simpa only [raisedFlag, ite_eq_right h] using w.property.1
obtain ⟨i, hi⟩ := hex
exact h ((hall i).symm.trans hi)⟩
invFun := fun w =>
⟨fun i => (w.val i).val, by
constructor
· simp only [raisedFlag, ite_eq_right h]
by_contra hex
apply w.property
intro i
rcases (w.val i).property with hc | ho
· exact hc
· exact False.elim (hex ⟨i, ho⟩)
· exact fun i => (w.val i).property⟩
left_inv := fun _ => Subtype.ext rfl
right_inv := fun _ => Subtype.ext rfl }
private def allCenterEquiv (S : FlaggedTensor X Y Z) (center output : Color)
(m : ℕ) :
{w : Fin m → AllowedLeaf S center output //
∀ i, leafColor S (w i).val = center} ≃
(Fin m → {a : Leaf S // leafColor S a = center}) where
toFun w i := ⟨(w.val i).val, w.property i⟩
invFun w := ⟨fun i => ⟨(w i).val, Or.inl (w i).property⟩,
fun i => (w i).property⟩
left_inv _ := Subtype.ext rfl
right_inv _ := rfl
variable [Fintype X] [Fintype Y] [Fintype Z]
private theorem card_allowedLeaf (S : FlaggedTensor X Y Z) (center output : Color)
(h : center ≠ output) :
Fintype.card (AllowedLeaf S center output) =
conditionalCount S center + conditionalCount S output := by
apply Fintype.card_subtype_or_disjoint
exact Pi.disjoint_iff.mpr fun a => Prop.disjoint_iff.mpr
(fun ha => h (ha.1.symm.trans ha.2))
theorem count_complete_center (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) :
conditionalCount (complete S center m) center = conditionalCount S center ^ m := by
calc
conditionalCount (complete S center m) center =
Fintype.card (Fin m → {a : Leaf S // leafColor S a = center}) :=
Fintype.card_congr (centerLeafEquiv S center m)
_ = conditionalCount S center ^ m := by simp [conditionalCount]
theorem count_complete_noncenter (S : FlaggedTensor X Y Z) (center output : Color)
(m : ℕ) (h : center ≠ output) :
conditionalCount (complete S center m) output =
(conditionalCount S center + conditionalCount S output) ^ m -
conditionalCount S center ^ m := by
have hall : Fintype.card {w : Fin m → AllowedLeaf S center output //
∀ i, leafColor S (w i).val = center} = conditionalCount S center ^ m := by
calc
_ = Fintype.card (Fin m → {a : Leaf S // leafColor S a = center}) :=
Fintype.card_congr (allCenterEquiv S center output m)
_ = _ := by simp [conditionalCount]
calc
conditionalCount (complete S center m) output =
Fintype.card {w : Fin m → AllowedLeaf S center output //
¬ ∀ i, leafColor S (w i).val = center} :=
Fintype.card_congr (noncenterLeafEquiv S center output m h)
_ = Fintype.card (Fin m → AllowedLeaf S center output) -
Fintype.card {w : Fin m → AllowedLeaf S center output //
∀ i, leafColor S (w i).val = center} := Fintype.card_subtype_compl _
_ = (conditionalCount S center + conditionalCount S output) ^ m -
conditionalCount S center ^ m := by
rw [hall, Fintype.card_fun, Fintype.card_fin, card_allowedLeaf S center output h]
theorem count_complete (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) (c : Color) :
conditionalCount (complete S center m) c =
if c = center then conditionalCount S center ^ m
else (conditionalCount S center + conditionalCount S c) ^ m -
conditionalCount S center ^ m := by
by_cases h : c = center
· subst c
simpa using count_complete_center S center m
· simpa [h] using count_complete_noncenter S center c m (Ne.symm h)
end MatrixMultiplication.CompletionCardinality
end
end OAI
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