| import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Entropy |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.ColorEntropy |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.EntropyAlgebra |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualLawEntropy |
|
|
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws |
| open CompletionProductLaws 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] |
|
|
| def allCenter (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) |
| (w : Fin m → Leaf S) : Prop := ∀ i, leafColor S (w i) = center |
|
|
| theorem iid_allowed (S : FlaggedTensor X Y Z) (center output : Color) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) (w : Fin m → Leaf S) |
| (hw : (iidColorLaw S center output pc po m a ha ha').mass w ≠ 0) (i : Fin m) : |
| leafColor S (w i) = center ∨ leafColor S (w i) = output := by |
| change (∏ j : Fin m, (colorMix S center output pc po a ha ha').mass (w j)) ≠ 0 at hw |
| have hn : (colorMix S center output pc po a ha ha').mass (w i) ≠ 0 := |
| (Finset.prod_ne_zero_iff.mp hw) i (Finset.mem_univ i) |
| by_contra h |
| exact hn (colorMix_mass_zero S center output pc po a ha ha' (w i) |
| (fun h' => h (Or.inl h')) (fun h' => h (Or.inr h'))) |
|
|
| omit [Fintype X] [Fintype Y] [Fintype Z] in |
| theorem not_event_iff_allCenter (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) (m : ℕ) (w : Fin m → Leaf S) |
| (allowed : ∀ i, leafColor S (w i) = center ∨ leafColor S (w i) = output) : |
| ¬ completionEvent S center output m w ↔ allCenter S center m w := by |
| constructor |
| · intro h i |
| rcases allowed i with hi | hi |
| · exact hi |
| · exfalso |
| apply h |
| refine ⟨?_, allowed⟩ |
| simpa only [raisedFlag, ite_eq_right hne] using Exists.intro i hi |
| · intro h he |
| have hx : ∃ i, leafColor S (w i) = output := by |
| simpa only [raisedFlag, ite_eq_right hne] using he.1 |
| obtain ⟨i, hi⟩ := hx |
| exact hne ((h i).symm.trans hi) |
|
|
| theorem event_sum_congr {A : Type*} [Fintype A] (P Q : A → Prop) (f : A → ℝ) |
| (agree : ∀ a, f a ≠ 0 → (P a ↔ Q a)) : |
| (∑ a : {a // P a}, f a.val) = ∑ a : {a // Q a}, f a.val := by |
| rw [sum_subtype_indicator P f, sum_subtype_indicator Q f] |
| apply Finset.sum_congr rfl |
| intro a _ |
| by_cases ha : f a = 0 |
| · simp [ha] |
| · simp only [agree a ha] |
|
|
| theorem removed_event_mass_eq (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : |
| eventMass (iidColorLaw S center output pc po m a ha ha') |
| (fun w => ¬ completionEvent S center output m w) = |
| eventMass (iidColorLaw S center output pc po m a ha ha') (allCenter S center m) := by |
| apply event_sum_congr |
| intro w hw |
| exact not_event_iff_allCenter S center output hne m w |
| (iid_allowed S center output pc po m a ha ha' w hw) |
|
|
| theorem removed_event_entropy_eq (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : |
| (∑ w : {w // ¬ completionEvent S center output m w}, |
| entropyTerm ((iidColorLaw S center output pc po m a ha ha').mass w.val)) = |
| ∑ w : {w // allCenter S center m w}, |
| entropyTerm ((iidColorLaw S center output pc po m a ha ha').mass w.val) := by |
| apply event_sum_congr (fun w => ¬ completionEvent S center output m w) |
| (allCenter S center m) |
| (fun w => entropyTerm ((iidColorLaw S center output pc po m a ha ha').mass w)) |
| intro w hw |
| have hm : (iidColorLaw S center output pc po m a ha ha').mass w ≠ 0 := by |
| intro hm |
| simp [hm] at hw |
| exact not_event_iff_allCenter S center output hne m w |
| (iid_allowed S center output pc po m a ha ha' w hm) |
|
|
| def centerWordEquiv (S : FlaggedTensor X Y Z) (center : Color) (m : ℕ) : |
| {w : Fin m → Leaf S // allCenter S center m w} ≃ (Fin m → ColorSlice S center) where |
| toFun w i := ⟨w.val i, w.property i⟩ |
| invFun w := ⟨fun i => (w i).val, fun i => (w i).property⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| theorem center_word_mass (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) |
| (w : {w : Fin m → Leaf S // allCenter S center m w}) : |
| (iidColorLaw S center output pc po m a ha ha').mass w.val = |
| a ^ m * (independentProduct (fun _ : Fin m => pc)).mass (centerWordEquiv S center m w) := by |
| change (∏ i, (colorMix S center output pc po a ha ha').mass (w.val i)) = |
| a ^ m * ∏ i, pc.mass ⟨w.val i, w.property i⟩ |
| have hslot (i : Fin m) : (colorMix S center output pc po a ha ha').mass (w.val i) = |
| a * pc.mass ⟨w.val i, w.property i⟩ := |
| colorMix_center_mass S center output hne pc po a ha ha' ⟨w.val i, w.property i⟩ |
| simp_rw [hslot] |
| rw [Finset.prod_mul_distrib] |
| simp |
|
|
| theorem allCenter_mass (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : |
| eventMass (iidColorLaw S center output pc po m a ha ha') (allCenter S center m) = a ^ m := by |
| unfold eventMass |
| simp_rw [center_word_mass S center output hne pc po m a ha ha'] |
| rw [← Finset.mul_sum, (centerWordEquiv S center m).sum_comp, |
| (independentProduct (fun _ : Fin m => pc)).total, mul_one] |
|
|
| theorem allCenter_entropy (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (a : ℝ) (ha : 0 ≤ a) (ha' : a ≤ 1) : |
| (∑ w : {w // allCenter S center m w}, |
| entropyTerm ((iidColorLaw S center output pc po m a ha ha').mass w.val)) = |
| entropyTerm (a ^ m) + a ^ m * ((m : ℝ) * finiteEntropy pc.mass) := by |
| simp_rw [center_word_mass S center output hne pc po m a ha ha'] |
| rw [(centerWordEquiv S center m).sum_comp |
| (fun w => entropyTerm (a ^ m * (independentProduct (fun _ : Fin m => pc)).mass w))] |
| change finiteEntropy (fun w => a ^ m * |
| (independentProduct (fun _ : Fin m => pc)).mass w) = _ |
| rw [entropy_scaled_law, independentProduct_entropy] |
| simp |
|
|
| theorem conditionalCompletionLaw_entropy (S : FlaggedTensor X Y Z) (center output : Color) |
| (hne : center ≠ output) |
| (pc : FiniteLaw (ColorSlice S center)) (po : FiniteLaw (ColorSlice S output)) |
| (m : ℕ) (hm : 0 < m) (a : ℝ) (ha : 0 ≤ a) (ha' : a < 1) : |
| finiteEntropy (conditionalCompletionLaw S center output pc po m hm a ha ha').mass = |
| CompletionEntropyAlgebra.conditionalMixEntropy a m |
| (finiteEntropy pc.mass) (finiteEntropy po.mass) := by |
| rw [conditionalCompletionLaw, transport_entropy] |
| have hmass : eventMass (iidColorLaw S center output pc po m a ha ha'.le) |
| (fun w => ¬ completionEvent S center output m w) = a ^ m := |
| (removed_event_mass_eq S center output hne pc po m a ha ha'.le).trans |
| (allCenter_mass S center output hne pc po m a ha ha'.le) |
| have hremoved : |
| (∑ w : {w // ¬ completionEvent S center output m w}, |
| entropyTerm ((iidColorLaw S center output pc po m a ha ha'.le).mass w.val)) = |
| entropyTerm (a ^ m) + a ^ m * ((m : ℝ) * finiteEntropy pc.mass) := |
| (removed_event_entropy_eq S center output hne pc po m a ha ha'.le).trans |
| (allCenter_entropy S center output hne pc po m a ha ha'.le) |
| rw [condition_entropy_of_removed_event _ _ _ (a ^ m) |
| ((m : ℝ) * finiteEntropy pc.mass) hmass hremoved] |
| rw [iidColorLaw_entropy S center output hne pc po m a ha ha'.le] |
| unfold CompletionEntropyAlgebra.conditionalMixEntropy CompletionEntropyAlgebra.binaryEntropy |
| congr 1 |
| ring |
|
|
| end MatrixMultiplication.DualLawEntropy |
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| end |
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| end OAI |
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