openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /TerminationLaw.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorEntropy | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Entropy | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.IndependentConditioning | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionTerminationLaw | |
| open MatrixMultiplication.Foundation CompletionLabels CompletionColorLaws CompletionLaws | |
| open scoped BigOperators | |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq | |
| attribute [local instance 11000] instDecidableEqFin | |
| universe u | |
| variable {X Y Z : Type u} [Fintype X] [Fintype Y] [Fintype Z] | |
| theorem retained_mass_one (S : FlaggedTensor X Y Z) (first second : Color) | |
| (pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second)) | |
| (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : | |
| eventMass (colorMix S first second pf ps a ha ha') | |
| (fun w => leafColor S w = first ∨ leafColor S w = second) = 1 := by | |
| rw [eventMass, sum_subtype_indicator] | |
| calc | |
| _ = ∑ w, (colorMix S first second pf ps a ha ha').mass w := by | |
| apply Finset.sum_congr rfl | |
| intro w _ | |
| split_ifs with h | |
| · rfl | |
| · exact (colorMix_mass_zero S first second pf ps a ha ha' w | |
| (fun h' => h (Or.inl h')) (fun h' => h (Or.inr h'))).symm | |
| _ = _ := (colorMix S first second pf ps a ha ha').total | |
| def retainedEquiv (S : FlaggedTensor X Y Z) (first second : Color) : | |
| {w : Leaf S // leafColor S w = first ∨ leafColor S w = second} ≃ | |
| RetainedLeaf S first second := Equiv.refl _ | |
| theorem retained_sum (S : FlaggedTensor X Y Z) (first second : Color) (f : Leaf S → ℝ) : | |
| (∑ w : RetainedLeaf S first second, f w.val) = | |
| ∑ w : {w : Leaf S // leafColor S w = first ∨ leafColor S w = second}, f w.val := by | |
| exact ((retainedEquiv S first second).sum_comp | |
| (fun w : RetainedLeaf S first second => f w.val)).symm | |
| def law (S : FlaggedTensor X Y Z) (first second : Color) | |
| (pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second)) | |
| (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : | |
| FiniteLaw (RetainedLeaf S first second) where | |
| mass w := (colorMix S first second pf ps a ha ha').mass w.val | |
| nonneg w := (colorMix S first second pf ps a ha ha').nonneg w.val | |
| total := by | |
| rw [retained_sum] | |
| exact retained_mass_one S first second pf ps a ha ha' | |
| theorem law_mass (S : FlaggedTensor X Y Z) (first second : Color) | |
| (pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second)) | |
| (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) (w : RetainedLeaf S first second) : | |
| (law S first second pf ps a ha ha').mass w = | |
| (colorMix S first second pf ps a ha ha').mass w.val := rfl | |
| theorem law_map_mass {U : Type*} [Fintype U] | |
| (S : FlaggedTensor X Y Z) (first second : Color) | |
| (pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second)) | |
| (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) (f : Leaf S → U) (u : U) : | |
| ((law S first second pf ps a ha ha').map (fun w => f w.val)).mass u = | |
| ((colorMix S first second pf ps a ha ha').map f).mass u := by | |
| rw [FiniteLaw.map_mass] | |
| simp only [law_mass] | |
| rw [retained_sum S first second | |
| (fun w => if f w = u then (colorMix S first second pf ps a ha ha').mass w else 0)] | |
| rw [sum_subtype_indicator (fun w : Leaf S => | |
| leafColor S w = first ∨ leafColor S w = second) | |
| (fun w => if f w = u then (colorMix S first second pf ps a ha ha').mass w else 0)] | |
| rw [FiniteLaw.map_mass] | |
| apply Finset.sum_congr rfl | |
| intro w _ | |
| by_cases h : leafColor S w = first ∨ leafColor S w = second | |
| · simp [h] | |
| · have hz := colorMix_mass_zero S first second pf ps a ha ha' w | |
| (fun h' => h (Or.inl h')) (fun h' => h (Or.inr h')) | |
| simp [h, hz] | |
| theorem law_entropy (S : FlaggedTensor X Y Z) (first second : Color) | |
| (hne : first ≠ second) | |
| (pf : FiniteLaw (ColorSlice S first)) (ps : FiniteLaw (ColorSlice S second)) | |
| (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : | |
| finiteEntropy (law S first second pf ps a ha ha').mass = | |
| entropyTerm a + entropyTerm (1 - a) + | |
| a * finiteEntropy pf.mass + (1 - a) * finiteEntropy ps.mass := by | |
| have hmap : ((law S first second pf ps a ha ha').map Subtype.val).mass = | |
| (colorMix S first second pf ps a ha ha').mass := by | |
| funext w | |
| exact (law_map_mass S first second pf ps a ha ha' id w).trans | |
| ((colorMix S first second pf ps a ha ha').map_mass_apply id Function.injective_id w) | |
| rw [← (law S first second pf ps a ha ha').map_entropy_of_injective | |
| Subtype.val Subtype.val_injective, hmap] | |
| exact colorMix_entropy S first second hne pf ps a ha ha' | |
| end MatrixMultiplication.CompletionTerminationLaw | |
| end | |
| end OAI | |