| import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorLaws |
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| namespace OAI |
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| /-! Readable tensor completion and its finite arithmetic realization. -/ |
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| noncomputable section |
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| universe uCoord |
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| namespace MatrixMultiplication.CompletionColorLaws |
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| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels |
| open CompletionLaws CompletionProductLaws |
| open scoped BigOperators |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
| attribute [local instance 11000] instDecidableEqFin |
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| variable {X Y Z : Type uCoord} [Fintype X] [Fintype Y] [Fintype Z] |
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| def centerWordEquiv (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) : |
| (Fin m → ColorSlice S center) ≃ |
| {w : Fin m → Leaf S |
| toFun w := ⟨fun i => (w i).val, by |
| constructor |
| · simpa [raisedFlag] using (fun i => (w i).property) |
| · exact fun i => Or.inl (w i).property⟩ |
| invFun w := fun i => ⟨w.val i, by |
| have hall : ∀ j, leafColor S (w.val j) = center := by |
| simpa [raisedFlag] using w.property.1 |
| exact hall i⟩ |
| left_inv w := by |
| funext i |
| apply Subtype.ext |
| rfl |
| right_inv w := by |
| apply Subtype.ext |
| rfl |
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| def centerCompletionEquiv (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) : |
| (Fin m → ColorSlice S center) ≃ ColorSlice (complete S center m) center := |
| (centerWordEquiv S center m).trans (conditionalLeafEquiv S center center m).symm |
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| omit [Fintype X] [Fintype Y] [Fintype Z] in |
| theorem centerCompletionEquiv_slots (S : FlaggedTensor X Y Z) (center : Color) |
| (m : ℕ) (w : Fin m → ColorSlice S center) : |
| slot S center m ((centerCompletionEquiv S center m) w).val = |
| fun i => (w i).val := by |
| funext i |
| apply Subtype.ext |
| rfl |
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| def centerCompletionLaw (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) : |
| FiniteLaw (ColorSlice (complete S center m) center) := |
| transport (centerCompletionEquiv S center m) |
| (independentProduct (fun _ : Fin m => p)) |
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| theorem centerCompletionLaw_mass (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) (w : Fin m → ColorSlice S center) : |
| (centerCompletionLaw S center p m).mass ((centerCompletionEquiv S center m) w) = |
| ∏ i, p.mass (w i) := |
| transport_mass (centerCompletionEquiv S center m) _ w |
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| theorem centerCompletionLaw_entropy (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) : |
| finiteEntropy (centerCompletionLaw S center p m).mass = |
| (m : ℝ) * finiteEntropy p.mass := by |
| rw [centerCompletionLaw, transport_entropy, independentProduct_entropy] |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
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| theorem centerCompletionLaw_map_mass {B : Type*} [Fintype B] |
| (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) (f : Leaf S → B) (y : Fin m → B) : |
| ((centerCompletionLaw S center p m).map |
| (fun a i => f (slot S center m a.val i))).mass y = |
| ∏ i, (p.map (fun a => f a.val)).mass (y i) := by |
| rw [FiniteLaw.map_mass] |
| calc |
| (∑ a, if (fun i => f (slot S center m a.val i)) = y then |
| (centerCompletionLaw S center p m).mass a else 0) = |
| ∑ w : Fin m → ColorSlice S center, |
| if (fun i => f (w i).val) = y then ∏ i, p.mass (w i) else 0 := by |
| rw [← (centerCompletionEquiv S center m).sum_comp |
| (fun a => if (fun i => f (slot S center m a.val i)) = y then |
| (centerCompletionLaw S center p m).mass a else 0)] |
| simp only [centerCompletionEquiv_slots, centerCompletionLaw_mass] |
| _ = _ := by |
| simpa only [FiniteLaw.map_mass, independentProduct_mass] using |
| independentProduct_map_mass (fun _ : Fin m => p) |
| (fun _ a => f a.val) y |
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| theorem completionEvent_same_zero_mass (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) : |
| eventMass (iidColorLaw S center center p p m 0 (by norm_num) (by norm_num)) |
| (completionEvent S center center m) = 1 := by |
| unfold eventMass |
| rw [← (centerWordEquiv S center m).sum_comp |
| (fun w => (iidColorLaw S center center p p m 0 (by norm_num) (by norm_num)).mass w.val)] |
| calc |
| _ = ∑ w : Fin m → ColorSlice S center, |
| (independentProduct (fun _ : Fin m => p)).mass w := by |
| apply Finset.sum_congr rfl |
| intro w _ |
| change (∏ i, (colorMix S center center p p 0 (by norm_num) (by norm_num)).mass |
| (w i).val) = ∏ i, p.mass (w i) |
| apply Finset.prod_congr rfl |
| intro i _ |
| rw [colorMix_mass] |
| simp only [zero_mul, sub_zero, one_mul, zero_add] |
| exact p.map_mass_apply Subtype.val Subtype.val_injective (w i) |
| _ = 1 := (independentProduct (fun _ : Fin m => p)).total |
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| theorem conditionalCompletionLaw_same_zero_mass |
| (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) (hm : 0 < m) |
| (w : Fin m → ColorSlice S center) : |
| (conditionalCompletionLaw S center center p p m hm 0 (by norm_num) (by norm_num)).mass |
| ((centerCompletionEquiv S center m) w) = ∏ i, p.mass (w i) := by |
| rw [conditionalCompletionLaw_mass, completionEvent_same_zero_mass, div_one, |
| centerCompletionEquiv_slots] |
| change (∏ i, (colorMix S center center p p 0 (by norm_num) (by norm_num)).mass |
| (w i).val) = ∏ i, p.mass (w i) |
| apply Finset.prod_congr rfl |
| intro i _ |
| rw [colorMix_mass] |
| simp only [zero_mul, sub_zero, one_mul, zero_add] |
| exact p.map_mass_apply Subtype.val Subtype.val_injective (w i) |
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| theorem conditionalCompletionLaw_same_zero_entropy |
| (S : FlaggedTensor X Y Z) (center : Color) |
| (p : FiniteLaw (ColorSlice S center)) (m : ℕ) (hm : 0 < m) : |
| finiteEntropy |
| (conditionalCompletionLaw S center center p p m hm 0 (by norm_num) (by norm_num)).mass = |
| (m : ℝ) * finiteEntropy p.mass := by |
| have hmass : |
| (conditionalCompletionLaw S center center p p m hm 0 (by norm_num) (by norm_num)).mass = |
| (centerCompletionLaw S center p m).mass := by |
| funext a |
| obtain ⟨w, rfl⟩ := (centerCompletionEquiv S center m).surjective a |
| rw [conditionalCompletionLaw_same_zero_mass, centerCompletionLaw_mass] |
| rw [hmass, centerCompletionLaw_entropy] |
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| end MatrixMultiplication.CompletionColorLaws |
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| end |
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| end OAI |
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