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import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly
namespace OAI
section
namespace PerfectCompleteness.CutNativeForms
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces
variable {branch : Nat → Nat} {n k t : Nat}
theorem nodeSlots_fill (lower : Nodes branch n)
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot) :
nodeSlots (CutSlotAssembly.fill (Nodes.path lower) outside inside) lower = inside := by
funext s
exact CutSlotAssembly.fill_at_cut (Nodes.path lower) outside inside s
theorem nodeSlots_fill_of_agree (lower : Nodes branch n)
(p : Path branch n (Nodes.height lower))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot)
(hag : ∀ s, p.slotEmbedding s = (Nodes.path lower).slotEmbedding s) :
nodeSlots (CutSlotAssembly.fill p outside inside) lower = inside := by
funext s
change CutSlotAssembly.fill p outside inside ((Nodes.path lower).slotEmbedding s) = inside s
rw [← hag s]
exact CutSlotAssembly.fill_at_cut p outside inside s
section Equality
variable {a b c d : Slots branch k → Fin t → MixedSupport.Slot}
def spaceCast (hab : a = b) : H a ≃ₗ[F2] H b := by
cases hab
exact LinearEquiv.refl F2 (H a)
def squareCast (hab : a = b) : squareSpace (H a) ≃ₗ[F2] squareSpace (H b) := by
cases hab
exact LinearEquiv.refl F2 (squareSpace (H a))
def dualCast (hab : a = b) (z : Module.Dual F2 (squareSpace (H a))) :
Module.Dual F2 (squareSpace (H b)) := by
cases hab
exact z
def formCast (hab : a = b) (F : SourceLocalOddLists.Form a) : SourceLocalOddLists.Form b := by
cases hab
exact F
@[simp] theorem dualCast_apply (hab : a = b)
(z : Module.Dual F2 (squareSpace (H a))) (f : squareSpace (H a)) :
dualCast hab z (squareCast hab f) = z f := by
cases hab
rfl
@[simp] theorem formCast_apply (hab : a = b) (F : SourceLocalOddLists.Form a)
(f g : H a) : formCast hab F (spaceCast hab f) (spaceCast hab g) = F f g := by
cases hab
rfl
theorem squareCast_product (hab : a = b) (f g : H a) :
squareCast hab (OddListExtraction.productElement (H a) f g) =
OddListExtraction.productElement (H b) (spaceCast hab f) (spaceCast hab g) := by
cases hab
rfl
theorem formCast_multiplication (hab : a = b)
(z : Module.Dual F2 (squareSpace (H a))) :
formCast hab (OddListExtraction.multiplicationForm (H a) z) =
OddListExtraction.multiplicationForm (H b) (dualCast hab z) := by
cases hab
rfl
theorem formRank_formCast (hab : a = b) (F : SourceLocalOddLists.Form a) :
OddListExtraction.formRank (H b) (formCast hab F) =
OddListExtraction.formRank (H a) F := by
cases hab
rfl
theorem squareCast_one (hab : a = b)
(ha : (1 : Domain a → F2) ∈ squareSpace (H a))
(hb : (1 : Domain b → F2) ∈ squareSpace (H b)) :
squareCast hab ⟨1, ha⟩ = ⟨1, hb⟩ := by
cases hab
rfl
theorem dualCast_one (hab : a = b) (z : Module.Dual F2 (squareSpace (H a)))
(ha : (1 : Domain a → F2) ∈ squareSpace (H a))
(hb : (1 : Domain b → F2) ∈ squareSpace (H b)) :
dualCast hab z ⟨1, hb⟩ = z ⟨1, ha⟩ := by
cases hab
rfl
def oneElement (slots : Slots branch k → Fin t → MixedSupport.Slot)
(hbranch : ∀ j < k, 0 < branch j) : squareSpace (H slots) :=
⟨(1 : Domain slots → F2), TreeSourceSpaces.one_mem_squareSpace slots hbranch⟩
theorem dualCast_normalized (hab : a = b) (hbranch : ∀ j < k, 0 < branch j)
(z : Module.Dual F2 (squareSpace (H a)))
(hnorm : z (oneElement a hbranch) = 1) :
dualCast hab z (oneElement b hbranch) = 1 := by
cases hab
exact hnorm
def projectionCast (hab : a = b) (hcd : c = d)
(p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) :
∀ s j, MixedSupport.Projection (b s j) (d s j) := by
cases hab
cases hcd
exact p
theorem spaceCast_HPullback (hab : a = b) (hcd : c = d)
(p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) (f : H c) :
spaceCast hab (HPullback p f) =
HPullback (projectionCast hab hcd p) (spaceCast hcd f) := by
cases hab
cases hcd
rfl
theorem formCast_projection_agreement (hab : a = b) (hcd : c = d)
(p : ∀ s j, MixedSupport.Projection (a s j) (c s j))
(F : SourceLocalOddLists.Form a) (G : SourceLocalOddLists.Form c)
(hagree : ∀ f g, G f g = F (HPullback p f) (HPullback p g)) :
∀ f g, formCast hcd G f g = formCast hab F
(HPullback (projectionCast hab hcd p) f)
(HPullback (projectionCast hab hcd p) g) := by
cases hab
cases hcd
exact hagree
theorem dualCast_projection_agreement (hab : a = b) (hcd : c = d)
(p : ∀ s j, MixedSupport.Projection (a s j) (c s j))
(z : Module.Dual F2 (squareSpace (H a))) (G : SourceLocalOddLists.Form c)
(hagree : ∀ f g, G f g = OddListExtraction.multiplicationForm (H a) z
(HPullback p f) (HPullback p g)) :
∀ f g, formCast hcd G f g =
OddListExtraction.multiplicationForm (H b) (dualCast hab z)
(HPullback (projectionCast hab hcd p) f)
(HPullback (projectionCast hab hcd p) g) := by
cases hab
cases hcd
exact hagree
end Equality
section Native
variable (lower : Nodes branch n) (p : Path branch n (Nodes.height lower))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot)
(hag : ∀ s, p.slotEmbedding s = (Nodes.path lower).slotEmbedding s)
def nativeDual
(z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) :
Module.Dual F2 (squareSpace (H inside)) :=
dualCast (nodeSlots_fill_of_agree lower p outside inside hag) z
def nativeForm
(F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) :
SourceLocalOddLists.Form inside :=
formCast (nodeSlots_fill_of_agree lower p outside inside hag) F
theorem nativeForm_multiplication
(z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) :
nativeForm lower p outside inside hag
(OddListExtraction.multiplicationForm
(H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) z) =
OddListExtraction.multiplicationForm (H inside) (nativeDual lower p outside inside hag z) :=
formCast_multiplication (nodeSlots_fill_of_agree lower p outside inside hag) z
theorem nativeForm_rank
(F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) :
OddListExtraction.formRank (H inside) (nativeForm lower p outside inside hag F) =
OddListExtraction.formRank (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) F :=
formRank_formCast (nodeSlots_fill_of_agree lower p outside inside hag) F
theorem nativeDual_normalized (hbranch : ∀ j < Nodes.height lower, 0 < branch j)
(z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower))))
(hnorm : z (oneElement
(nodeSlots (CutSlotAssembly.fill p outside inside) lower) hbranch) = 1) :
nativeDual lower p outside inside hag z (oneElement inside hbranch) = 1 :=
dualCast_normalized (nodeSlots_fill_of_agree lower p outside inside hag) hbranch z hnorm
variable (outside' : Slots branch n → Fin t → MixedSupport.Slot)
(inside' : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot)
def nativeProjection
(projection : ∀ s j, MixedSupport.Projection
(nodeSlots (CutSlotAssembly.fill p outside inside) lower s j)
(nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j)) :
∀ s j, MixedSupport.Projection (inside s j) (inside' s j) :=
projectionCast (nodeSlots_fill_of_agree lower p outside inside hag)
(nodeSlots_fill_of_agree lower p outside' inside' hag) projection
theorem nativeForm_projection_agreement
(projection : ∀ s j, MixedSupport.Projection
(nodeSlots (CutSlotAssembly.fill p outside inside) lower s j)
(nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j))
(F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower))
(G : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside' inside') lower))
(hagree : ∀ f g, G f g = F (HPullback projection f) (HPullback projection g)) :
∀ f g, nativeForm lower p outside' inside' hag G f g =
nativeForm lower p outside inside hag F
(HPullback (nativeProjection lower p outside inside hag outside' inside' projection) f)
(HPullback (nativeProjection lower p outside inside hag outside' inside' projection) g) :=
formCast_projection_agreement (nodeSlots_fill_of_agree lower p outside inside hag)
(nodeSlots_fill_of_agree lower p outside' inside' hag) projection F G hagree
theorem nativeDual_projection_agreement
(projection : ∀ s j, MixedSupport.Projection
(nodeSlots (CutSlotAssembly.fill p outside inside) lower s j)
(nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j))
(z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower))))
(G : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside' inside') lower))
(hagree : ∀ f g, G f g = OddListExtraction.multiplicationForm
(H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) z
(HPullback projection f) (HPullback projection g)) :
∀ f g, nativeForm lower p outside' inside' hag G f g =
OddListExtraction.multiplicationForm (H inside) (nativeDual lower p outside inside hag z)
(HPullback (nativeProjection lower p outside inside hag outside' inside' projection) f)
(HPullback (nativeProjection lower p outside inside hag outside' inside' projection) g) :=
dualCast_projection_agreement (nodeSlots_fill_of_agree lower p outside inside hag)
(nodeSlots_fill_of_agree lower p outside' inside' hag) projection z G hagree
end Native
end
end PerfectCompleteness.CutNativeForms
end
end OAI
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