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import OAI.LinearAlgebra.MatrixFields.Extraction.ComplexFiniteSeparation
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
open scoped BigOperators
namespace MatrixMultiplication.Foundation
namespace Tensor
variable {K S : Type*} [CommSemiring K] [DecidableEq S]
variable {X Y Z : S → Type*}
def dependentDirectSum (T : ∀ s, Tensor K (X s) (Y s) (Z s)) :
Tensor K (Σ s, X s) (Σ s, Y s) (Σ s, Z s) :=
fun x y z =>
if hxy : y.1 = x.1 then
if hxz : z.1 = x.1 then T x.1 x.2 (hxy ▸ y.2) (hxz ▸ z.2)
else 0
else 0
@[simp] theorem dependentDirectSum_matching
(T : ∀ s, Tensor K (X s) (Y s) (Z s))
(s : S) (x : X s) (y : Y s) (z : Z s) :
dependentDirectSum T ⟨s, x⟩ ⟨s, y⟩ ⟨s, z⟩ = T s x y z := by
simp [dependentDirectSum]
theorem dependentDirectSum_zero_of_left_ne
(T : ∀ s, Tensor K (X s) (Y s) (Z s))
(x : Σ s, X s) (y : Σ s, Y s) (z : Σ s, Z s)
(h : x.1 ≠ y.1) : dependentDirectSum T x y z = 0 := by
simp [dependentDirectSum, Ne.symm h]
def directSumCoordinateEquiv {A : Type*} (e : ∀ s, X s ≃ A) :
(Σ s, X s) ≃ S × A where
toFun x := (x.1, e x.1 x.2)
invFun x := ⟨x.1, (e x.1).symm x.2⟩
left_inv := by
rintro ⟨s, x⟩
simp
right_inv := by
rintro ⟨s, x⟩
simp
theorem dependentDirectSum_reindex {A B C : Type*}
(T : ∀ s, Tensor K (X s) (Y s) (Z s))
(ex : ∀ s, X s ≃ A) (ey : ∀ s, Y s ≃ B) (ez : ∀ s, Z s ≃ C) :
pullback (directSumCoordinateEquiv ex).symm
(directSumCoordinateEquiv ey).symm (directSumCoordinateEquiv ez).symm
(dependentDirectSum T) =
directSum (fun s => pullback (ex s).symm (ey s).symm (ez s).symm (T s)) := by
funext x y z
rcases x with ⟨s, x⟩
rcases y with ⟨t, y⟩
rcases z with ⟨v, z⟩
by_cases ht : t = s
· subst t
by_cases hv : v = s
· subst v
simp [pullback, directSumCoordinateEquiv, dependentDirectSum, directSum]
· simp [pullback, directSumCoordinateEquiv, dependentDirectSum, directSum,
hv, Ne.symm hv]
· simp [pullback, directSumCoordinateEquiv, dependentDirectSum, directSum,
ht, Ne.symm ht]
end Tensor
variable {X Y Z S U : Type*} [DecidableEq S] [DecidableEq U]
def labelFiberTensor (lx : X → S) (ly : Y → S) (T : Tensor ℂ X Y Z) (s : S) :
Tensor ℂ {x // lx x = s} {y // ly y = s} Z :=
fun x y z => T x.1 y.1 z
def fiberCoordinateEquiv (lx : X → S) :
(Σ s, {x // lx x = s} × U) ≃ X × U where
toFun x := (x.2.1.1, x.2.2)
invFun x := ⟨lx x.1, (⟨x.1, rfl⟩, x.2)⟩
left_inv := by
rintro ⟨s, ⟨⟨x, hx⟩, u⟩⟩
cases hx
rfl
right_inv := by
rintro ⟨x, u⟩
rfl
def tagCoordinateEquiv : (Σ _ : S, Z × Unit) ≃ Z × S where
toFun z := (z.2.1, z.1)
invFun z := ⟨z.2, (z.1, ())⟩
left_inv := by
rintro ⟨s, z, u⟩
cases u
rfl
right_inv := by
rintro ⟨z, s⟩
rfl
theorem separatedTensor_eq_fiber_directSum
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.pullback (fiberCoordinateEquiv (U := U) lx)
(fiberCoordinateEquiv (U := U) ly) (tagCoordinateEquiv (Z := Z) (S := S))
(separatedTensor lx T) =
Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U)) := by
classical
funext x y z
rcases x with ⟨s, x, u⟩
rcases y with ⟨t, y, v⟩
rcases z with ⟨w, z, q⟩
by_cases ht : t = s
· subst t
by_cases hw : w = s
· subst w
rw [Tensor.dependentDirectSum_matching]
simp [Tensor.pullback, fiberCoordinateEquiv, tagCoordinateEquiv,
separatedTensor, Tensor.product, labelFiberTensor, Tensor.dotPairing,
x.property, mul_ite]
· simp [Tensor.pullback, fiberCoordinateEquiv, tagCoordinateEquiv,
separatedTensor, Tensor.dependentDirectSum, x.property, hw, Ne.symm hw]
· have hzero : T x.1 y.1 z = 0 := by
by_contra hzero
have hxy := hlabels x.1 y.1 z hzero
exact ht (y.property.symm.trans (hxy.symm.trans x.property))
simp [Tensor.pullback, fiberCoordinateEquiv, tagCoordinateEquiv,
separatedTensor, Tensor.dependentDirectSum, ht, hzero]
theorem fiber_directSum_eq_separatedTensor
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.pullback (fiberCoordinateEquiv (U := U) lx).symm
(fiberCoordinateEquiv (U := U) ly).symm
(tagCoordinateEquiv (Z := Z) (S := S)).symm
(Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U))) =
separatedTensor lx T := by
rw [← separatedTensor_eq_fiber_directSum T lx ly hlabels]
funext x y z
simp [Tensor.pullback]
theorem separatedTensor_fiber_directSum_rankAtMost
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(h : Tensor.RankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.RankAtMost (Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U))) r := by
rw [← separatedTensor_eq_fiber_directSum T lx ly hlabels]
exact h.pullback _ _ _
theorem separatedTensor_fiber_directSum_rankAtMost_iff
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.RankAtMost (separatedTensor (U := U) lx T) r ↔
Tensor.RankAtMost (Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U))) r := by
constructor
· exact separatedTensor_fiber_directSum_rankAtMost T lx ly hlabels
· intro h
rw [← fiber_directSum_eq_separatedTensor T lx ly hlabels]
exact h.pullback _ _ _
theorem separatedTensor_fiber_directSum_borderRankAtMost
[Fintype X] [Fintype Y] [Fintype Z] [Fintype S] [Fintype U]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(h : Tensor.BorderRankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U))) r := by
classical
rw [← separatedTensor_eq_fiber_directSum T lx ly hlabels]
rw [Tensor.pullback_eq_restrict]
exact h.restrict _ _ _
theorem separatedTensor_uniform_directSum_rankAtMost {A B : Type*}
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(ex : ∀ s, {x // lx x = s} ≃ A) (ey : ∀ s, {y // ly y = s} ≃ B)
{r : ℕ} (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(h : Tensor.RankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.RankAtMost (Tensor.directSum (fun s => Tensor.pullback
(Equiv.prodCongr (ex s) (Equiv.refl U)).symm
(Equiv.prodCongr (ey s) (Equiv.refl U)).symm
(Equiv.refl (Z × Unit)).symm
(Tensor.product (labelFiberTensor lx ly T s) (Tensor.dotPairing U)))) r := by
have hf := separatedTensor_fiber_directSum_rankAtMost T lx ly hlabels h
have hp := hf.pullback
(Tensor.directSumCoordinateEquiv
(fun s => Equiv.prodCongr (ex s) (Equiv.refl U))).symm
(Tensor.directSumCoordinateEquiv
(fun s => Equiv.prodCongr (ey s) (Equiv.refl U))).symm
(Tensor.directSumCoordinateEquiv (fun _ : S => Equiv.refl (Z × Unit))).symm
rw [Tensor.dependentDirectSum_reindex] at hp
exact hp
theorem separatedTensor_localModels_identity {A B C : Type*}
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(fx : S → A → X) (fy : S → B → Y) (fz : S → C → Z)
(hfx : ∀ s a, lx (fx s a) = s) (hfy : ∀ s b, ly (fy s b) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.pullback
(fun x : S × (A × U) => (fx x.1 x.2.1, x.2.2))
(fun y : S × (B × U) => (fy y.1 y.2.1, y.2.2))
(fun z : S × (C × Unit) => (fz z.1 z.2.1, z.1))
(separatedTensor lx T) =
Tensor.directSum (fun s => Tensor.product
(Tensor.pullback (fx s) (fy s) (fz s) T) (Tensor.dotPairing U)) := by
funext x y z
rcases x with ⟨s, a, u⟩
rcases y with ⟨t, b, v⟩
rcases z with ⟨w, c, q⟩
by_cases ht : t = s
· subst t
by_cases hw : w = s
· subst w
simp [Tensor.pullback, separatedTensor, Tensor.directSum,
Tensor.product, Tensor.dotPairing, hfx, mul_ite]
· simp [Tensor.pullback, separatedTensor, Tensor.directSum,
hfx, Ne.symm hw]
· have hzero : T (fx s a) (fy t b) (fz w c) = 0 := by
by_contra hzero
have hxy := hlabels (fx s a) (fy t b) (fz w c) hzero
exact ht ((hfy t b).symm.trans (hxy.symm.trans (hfx s a)))
simp [Tensor.pullback, separatedTensor, Tensor.directSum,
Ne.symm ht, hzero]
theorem separatedTensor_localModels_rankAtMost {A B C : Type*}
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(fx : S → A → X) (fy : S → B → Y) (fz : S → C → Z)
(hfx : ∀ s a, lx (fx s a) = s) (hfy : ∀ s b, ly (fy s b) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
{r : ℕ} (h : Tensor.RankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.RankAtMost (Tensor.directSum (fun s => Tensor.product
(Tensor.pullback (fx s) (fy s) (fz s) T) (Tensor.dotPairing U))) r := by
rw [← separatedTensor_localModels_identity T lx ly fx fy fz hfx hfy hlabels]
exact h.pullback _ _ _
theorem separatedTensor_localModels_borderRankAtMost {A B C : Type*}
[Fintype X] [Fintype Y] [Fintype Z] [Fintype S] [Fintype U]
[Fintype A] [Fintype B] [Fintype C]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(fx : S → A → X) (fy : S → B → Y) (fz : S → C → Z)
(hfx : ∀ s a, lx (fx s a) = s) (hfy : ∀ s b, ly (fy s b) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
{r : ℕ} (h : Tensor.BorderRankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.BorderRankAtMost (Tensor.directSum (fun s => Tensor.product
(Tensor.pullback (fx s) (fy s) (fz s) T) (Tensor.dotPairing U))) r := by
classical
rw [← separatedTensor_localModels_identity T lx ly fx fy fz hfx hfy hlabels]
rw [Tensor.pullback_eq_restrict]
exact h.restrict _ _ _
theorem separatedTensor_scalar_identity
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(sx : S → X) (sy : S → Y) (sz : S → Z)
(hsx : ∀ s, lx (sx s) = s) (hsy : ∀ s, ly (sy s) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(hscalar : ∀ s, T (sx s) (sy s) (sz s) = 1) :
Tensor.pullback
(fun x : S × U => (sx x.1, x.2))
(fun y : S × U => (sy y.1, y.2))
(fun z : S × Unit => (sz z.1, z.1)) (separatedTensor lx T) =
Tensor.directSum (fun _ : S => Tensor.dotPairing U) := by
have hmodels := separatedTensor_localModels_identity (U := U) T lx ly
(fun s (_ : Unit) => sx s) (fun s (_ : Unit) => sy s) (fun s (_ : Unit) => sz s)
(fun s _ => hsx s) (fun s _ => hsy s) hlabels
have h := congrArg (Tensor.pullback
(fun x : S × U => (x.1, ((), x.2)))
(fun y : S × U => (y.1, ((), y.2)))
(fun z : S × Unit => (z.1, ((), z.2)))) hmodels
funext x y z
have hpoint := congrFun (congrFun (congrFun h x) y) z
simpa only [Tensor.pullback, Tensor.directSum, Tensor.product, hscalar, one_mul]
using hpoint
theorem separatedTensor_scalar_rankAtMost
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(sx : S → X) (sy : S → Y) (sz : S → Z)
(hsx : ∀ s, lx (sx s) = s) (hsy : ∀ s, ly (sy s) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(hscalar : ∀ s, T (sx s) (sy s) (sz s) = 1)
{r : ℕ} (h : Tensor.RankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.RankAtMost (Tensor.directSum (fun _ : S => Tensor.dotPairing (K := ℂ) U)) r := by
rw [← separatedTensor_scalar_identity T lx ly sx sy sz hsx hsy hlabels hscalar]
exact h.pullback _ _ _
theorem separatedTensor_scalar_borderRankAtMost
[Fintype X] [Fintype Y] [Fintype Z] [Fintype S] [Fintype U]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(sx : S → X) (sy : S → Y) (sz : S → Z)
(hsx : ∀ s, lx (sx s) = s) (hsy : ∀ s, ly (sy s) = s)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y)
(hscalar : ∀ s, T (sx s) (sy s) (sz s) = 1)
{r : ℕ} (h : Tensor.BorderRankAtMost (separatedTensor (U := U) lx T) r) :
Tensor.BorderRankAtMost (Tensor.directSum (fun _ : S => Tensor.dotPairing U)) r := by
classical
rw [← separatedTensor_scalar_identity T lx ly sx sy sz hsx hsy hlabels hscalar]
rw [Tensor.pullback_eq_restrict]
exact h.restrict _ _ _
end MatrixMultiplication.Foundation
end MatrixAllFields
end OAI
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