| 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 |
|
|
| private abbrev AllowedLeaf (S : FlaggedTensor X Y Z) (center output : Color) := |
| {a : Leaf S |
|
|
| 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 |
| (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 |
| 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 |
| 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 |
| 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 |
|
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| end OAI |
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|
