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import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexTypeCounting
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
namespace MatrixMultiplication.LeafVolumes
open MatrixMultiplication.Foundation
open scoped BigOperators
variable {F U : Type*} [CommSemiring F] [DecidableEq U]
def zeroXMatching : Tensor F Unit U U := fun _ y z => if y = z then 1 else 0
def zeroYMatching : Tensor F U Unit U := fun x _ z => if x = z then 1 else 0
def zeroZMatching : Tensor F U U Unit := fun x y _ => if x = y then 1 else 0
theorem zeroXMatching_matrixCoefficients :
Tensor.pullback (fun _ : Unit × Unit => ())
(fun y : Unit × U => y.2) (fun z : U × Unit => z.1)
(zeroXMatching (F := F) (U := U)) =
Tensor.matrixCoefficients Unit Unit U := by
funext x y z
simp [Tensor.pullback, zeroXMatching, Tensor.matrixCoefficients]
theorem zeroYMatching_matrixCoefficients :
Tensor.pullback (fun x : U × Unit => x.1)
(fun _ : Unit × Unit => ()) (fun z : Unit × U => z.2)
(zeroYMatching (F := F) (U := U)) =
Tensor.matrixCoefficients U Unit Unit := by
funext x y z
simp [Tensor.pullback, zeroYMatching, Tensor.matrixCoefficients, eq_comm]
theorem zeroZMatching_matrixCoefficients :
Tensor.pullback (fun x : Unit × U => x.2)
(fun y : U × Unit => y.1) (fun _ : Unit × Unit => ())
(zeroZMatching (F := F) (U := U)) =
Tensor.matrixCoefficients Unit U Unit := by
funext x y z
simp [Tensor.pullback, zeroZMatching, Tensor.matrixCoefficients]
def complementaryMatching {V : Type*} [DecidableEq V] (e : U ≃ V) :
Tensor F Unit U V := fun _ y z => if e y = z then 1 else 0
theorem complementaryMatching_pullback {V : Type*} [DecidableEq V] (e : U ≃ V) :
Tensor.pullback id id e (complementaryMatching (F := F) e) =
zeroXMatching (F := F) (U := U) := by
funext x y z
simp [Tensor.pullback, complementaryMatching, zeroXMatching]
theorem zeroXMatching_subtype (p : U → Prop) [DecidablePred p] :
Tensor.pullback id (Subtype.val : {u // p u} → U) Subtype.val
(zeroXMatching (F := F) (U := U)) =
zeroXMatching (F := F) (U := {u // p u}) := by
funext x y z
simp [Tensor.pullback, zeroXMatching, Subtype.ext_iff]
theorem matchingPower_leaf {X Y Z : Type*} [DecidableEq Y]
(T : Tensor F X Y Z) (x₀ : X) (e : Y ≃ Z)
(hmatch : ∀ y z, T x₀ y (e z) = if y = z then 1 else 0)
(n : ℕ) (p : (Fin n → Y) → Prop) [DecidablePred p] :
Tensor.pullback (fun _ : Unit => fun _ : Fin n => x₀)
(Subtype.val : {w // p w} → (Fin n → Y))
(fun w : {w // p w} => fun i => e (w.val i)) (Tensor.power T n) =
zeroXMatching (F := F) (U := {w // p w}) := by
funext x y z
simp only [Tensor.pullback, Tensor.power, hmatch, zeroXMatching,
Fintype.prod_boole, ← funext_iff, Subtype.ext_iff]
theorem matchingPower_matrixCoefficients {X Y Z : Type*} [DecidableEq Y]
(T : Tensor F X Y Z) (x₀ : X) (e : Y ≃ Z)
(hmatch : ∀ y z, T x₀ y (e z) = if y = z then 1 else 0)
(n : ℕ) (p : (Fin n → Y) → Prop) [DecidablePred p] :
Tensor.pullback (fun _ : Unit × Unit => fun _ : Fin n => x₀)
(fun y : Unit × {w // p w} => y.2.val)
(fun z : {w // p w} × Unit => fun i => e (z.1.val i))
(Tensor.power T n) =
Tensor.matrixCoefficients Unit Unit {w // p w} := by
have h := matchingPower_leaf T x₀ e hmatch n p
have hp := congrArg (Tensor.pullback (fun _ : Unit × Unit => ())
(fun y : Unit × {w // p w} => y.2)
(fun z : {w // p w} × Unit => z.1)) h
rw [zeroXMatching_matrixCoefficients] at hp
unfold Tensor.pullback at hp ⊢
exact hp
theorem rankAtMost_matchingPower_leaf {X Y Z : Type*} [DecidableEq Y]
(T : Tensor F X Y Z) (x₀ : X) (e : Y ≃ Z)
(hmatch : ∀ y z, T x₀ y (e z) = if y = z then 1 else 0)
(n : ℕ) (p : (Fin n → Y) → Prop) [DecidablePred p]
{r : ℕ} (hT : Tensor.RankAtMost (Tensor.power T n) r) :
Tensor.RankAtMost (Tensor.matrixCoefficients (K := F) Unit Unit {w // p w}) r := by
rw [← matchingPower_matrixCoefficients T x₀ e hmatch n p]
exact hT.pullback _ _ _
section ExactLabeledWords
variable {A : Type*} [Fintype A] [DecidableEq A]
abbrev LabeledExactWords (counts b : A → ℕ) :=
Σ w : ExactWords counts, ∀ i : Fin (∑ a, counts a), Fin (b (w.val i))
theorem labelings_card {I : Type*} [Fintype I] [DecidableEq I]
(w : I → A) (b : A → ℕ) :
Fintype.card (∀ i, Fin (b (w i))) =
∏ a, b a ^ wordPopulation w a := by
classical
simp only [Fintype.card_pi, Fintype.card_fin]
calc
(∏ i, b (w i)) = ∏ s : Σ a, {i // w i = a}, b s.1 := by
rw [← (Equiv.sigmaFiberEquiv w).prod_comp (fun i => b (w i))]
apply Finset.prod_congr rfl
intro s _
exact congrArg b s.2.property
_ = ∏ a, b a ^ wordPopulation w a := by
simp [Fintype.prod_sigma, wordPopulation]
theorem labeledExactWords_card (counts b : A → ℕ) :
Fintype.card (LabeledExactWords counts b) =
Nat.multinomial Finset.univ counts * ∏ a, b a ^ counts a := by
classical
rw [Fintype.card_sigma]
have hcard : ∀ w : ExactWords counts,
Fintype.card (∀ i : Fin (∑ a, counts a), Fin (b (w.val i))) =
∏ a, b a ^ counts a := by
intro w
rw [labelings_card]
simp only [w.property]
simp only [hcard, Finset.sum_const, Finset.card_univ, smul_eq_mul,
exactWords_card]
theorem labeledExactWords_card_pos (counts b : A → ℕ)
(hb : ∀ a, counts a ≠ 0 → 0 < b a) :
0 < Fintype.card (LabeledExactWords counts b) := by
classical
rw [labeledExactWords_card]
apply Nat.mul_pos (Nat.multinomial_pos Finset.univ counts)
apply Finset.prod_pos
intro a _
by_cases h : counts a = 0
· simp [h]
· exact pow_pos (hb a h) _
end ExactLabeledWords
structure Dimensions where
rows : ℕ
inner : ℕ
cols : ℕ
deriving DecidableEq
def Dimensions.volume (d : Dimensions) : ℕ := d.rows * d.inner * d.cols
def Dimensions.product (d e : Dimensions) : Dimensions :=
⟨d.rows * e.rows, d.inner * e.inner, d.cols * e.cols⟩
theorem Dimensions.volume_product (d e : Dimensions) :
(d.product e).volume = d.volume * e.volume := by
simp only [Dimensions.product, Dimensions.volume]
ring
def historyDimensions {C : Type*} [Fintype C] (d : C → Dimensions) : Dimensions :=
⟨∏ c, (d c).rows, ∏ c, (d c).inner, ∏ c, (d c).cols⟩
theorem historyDimensions_volume {C : Type*} [Fintype C] (d : C → Dimensions) :
(historyDimensions d).volume = ∏ c, (d c).volume := by
simp [historyDimensions, Dimensions.volume, Finset.prod_mul_distrib]
end MatrixMultiplication.LeafVolumes
end OAI
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