| import Mathlib |
| import OAI.Probability.Perceptron.Variational.CovarianceMatrix |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
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| variable {K L : Type*} [TopologicalSpace K] [TopologicalSpace L] |
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| def compactMapArray (e : K→L) (Q : CompactArray K) : CompactArray L := fun i j => e (Q i j) |
| def compactMapBlock (e : K→L) (n : ℕ) (Q : CompactBlock K n) : CompactBlock L n := fun i j => e (Q i j) |
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| lemma compactMapArray_continuous {e : K→L} (he : Continuous e) : Continuous (compactMapArray e) := by |
| unfold compactMapArray |
| fun_prop |
| lemma compactMapBlock_continuous {e : K→L} (he : Continuous e) (n : ℕ) : |
| Continuous (compactMapBlock e n) := by |
| unfold compactMapBlock |
| fun_prop |
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| def compactDiagonalMapArray (e d : K→L) (Q : CompactArray K) : CompactArray L := |
| fun i j => if i=j then d (Q i j) else e (Q i j) |
| def compactDiagonalMapBlock (e d : K→L) (n : ℕ) (Q : CompactBlock K n) : CompactBlock L n := |
| fun i j => if i=j then d (Q i j) else e (Q i j) |
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| lemma compactDiagonalMapArray_continuous {e d : K→L} (he : Continuous e) (hd : Continuous d) : |
| Continuous (compactDiagonalMapArray e d) := by |
| apply continuous_pi; intro i; apply continuous_pi; intro j |
| unfold compactDiagonalMapArray |
| split_ifs <;> fun_prop |
| lemma compactDiagonalMapBlock_continuous {e d : K→L} (he : Continuous e) (hd : Continuous d) (n : ℕ) : |
| Continuous (compactDiagonalMapBlock e d n) := by |
| apply continuous_pi; intro i; apply continuous_pi; intro j |
| unfold compactDiagonalMapBlock |
| split_ifs <;> fun_prop |
|
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| omit [TopologicalSpace K] [TopologicalSpace L] in |
| lemma compactDiagonalMapBlock_compactBlock (e d : K→L) (n : ℕ) (Q : CompactArray K) : |
| compactBlock n (compactDiagonalMapArray e d Q)=compactDiagonalMapBlock e d n (compactBlock n Q) := by |
| funext i j |
| simp only [compactBlock,compactDiagonalMapArray,compactDiagonalMapBlock,Fin.val_inj] |
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| variable [MeasurableSpace K] [MeasurableSpace L] [BorelSpace K] [BorelSpace L] |
| [SecondCountableTopology K] [SecondCountableTopology L] |
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| def compactMapLaw (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K)) : |
| ProbabilityMeasure (CompactArray L) := |
| ⟨(μ : Measure (CompactArray K)).map (compactMapArray e), |
| (Measure.isProbabilityMeasure_map_iff (compactMapArray_continuous he).measurable.aemeasurable).2 |
| inferInstance⟩ |
|
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| lemma compactMapLaw_integral (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K)) |
| (F : CompactArray L→ℝ) (hF : Measurable F) : |
| (∫ Q, F Q ∂(compactMapLaw e he μ : Measure (CompactArray L)))= |
| ∫ Q, F (compactMapArray e Q) ∂(μ : Measure (CompactArray K)) := |
| integral_map (compactMapArray_continuous he).measurable.aemeasurable hF.aestronglyMeasurable |
|
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| lemma compactMapLaw_gg (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K)) |
| (hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock K n →ᵇ ℝ) (g : K →ᵇ ℝ), |
| compactGGDefect μ n i f g=0) (n : ℕ) (i : Fin n) |
| (f : CompactBlock L n →ᵇ ℝ) (g : L →ᵇ ℝ) : |
| compactGGDefect (compactMapLaw e he μ) n i f g=0 := by |
| let f' := f.compContinuous ⟨compactMapBlock e n,compactMapBlock_continuous he n⟩ |
| let g' := g.compContinuous ⟨e,he⟩ |
| have hmul (a b : ℕ) : |
| (∫ Q, f (compactBlock n Q)*g (Q a b) ∂(compactMapLaw e he μ : Measure (CompactArray L)))= |
| ∫ Q, f' (compactBlock n Q)*g' (Q a b) ∂(μ : Measure (CompactArray K)) := by |
| exact compactMapLaw_integral e he μ _ |
| ((f.measurable.comp (compactBlock_continuous n).measurable).mul (g.measurable.comp (by fun_prop))) |
| have hf : (∫ Q, f (compactBlock n Q) ∂(compactMapLaw e he μ : Measure (CompactArray L)))= |
| ∫ Q, f' (compactBlock n Q) ∂(μ : Measure (CompactArray K)) := by |
| exact compactMapLaw_integral e he μ _ |
| (f.measurable.comp (compactBlock_continuous n).measurable) |
| have hg : (∫ Q, g (Q 0 1) ∂(compactMapLaw e he μ : Measure (CompactArray L)))= |
| ∫ Q, g' (Q 0 1) ∂(μ : Measure (CompactArray K)) := by |
| exact compactMapLaw_integral e he μ _ (by fun_prop) |
| unfold compactGGDefect |
| simp_rw [hmul,hf,hg] |
| exact hGG n i f' g' |
|
|
| lemma compactMapLaw_pair (e : K→L) (he : Continuous e) (μ : ProbabilityMeasure (CompactArray K)) |
| (a b : ℕ) : |
| (compactMapLaw e he μ : Measure (CompactArray L)).map (fun Q => Q a b)= |
| ((μ : Measure (CompactArray K)).map (fun Q => Q a b)).map e := by |
| change ((μ : Measure (CompactArray K)).map (compactMapArray e)).map (fun Q => Q a b)=_ |
| rw [Measure.map_map (by fun_prop) (compactMapArray_continuous he).measurable, |
| Measure.map_map he.measurable (by fun_prop)] |
| rfl |
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| def compactDiagonalMapLaw (e d : K→L) (he : Continuous e) (hd : Continuous d) |
| (μ : ProbabilityMeasure (CompactArray K)) : ProbabilityMeasure (CompactArray L) := |
| ⟨(μ : Measure (CompactArray K)).map (compactDiagonalMapArray e d), |
| (Measure.isProbabilityMeasure_map_iff |
| (compactDiagonalMapArray_continuous he hd).measurable.aemeasurable).2 inferInstance⟩ |
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| lemma compactDiagonalMapLaw_integral (e d : K→L) (he : Continuous e) (hd : Continuous d) |
| (μ : ProbabilityMeasure (CompactArray K)) (F : CompactArray L→ℝ) (hF : Measurable F) : |
| (∫ Q, F Q ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))= |
| ∫ Q, F (compactDiagonalMapArray e d Q) ∂(μ : Measure (CompactArray K)) := |
| integral_map (compactDiagonalMapArray_continuous he hd).measurable.aemeasurable hF.aestronglyMeasurable |
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| lemma compactDiagonalMapLaw_gg (e d : K→L) (he : Continuous e) (hd : Continuous d) |
| (μ : ProbabilityMeasure (CompactArray K)) |
| (hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock K n →ᵇ ℝ) (g : K →ᵇ ℝ), |
| compactGGDefect μ n i f g=0) (n : ℕ) (i : Fin n) |
| (f : CompactBlock L n →ᵇ ℝ) (g : L →ᵇ ℝ) : |
| compactGGDefect (compactDiagonalMapLaw e d he hd μ) n i f g=0 := by |
| let f' := f.compContinuous ⟨compactDiagonalMapBlock e d n,compactDiagonalMapBlock_continuous he hd n⟩ |
| let g' := g.compContinuous ⟨e,he⟩ |
| have hmul (a b : ℕ) (hab : a≠b) : |
| (∫ Q, f (compactBlock n Q)*g (Q a b) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))= |
| ∫ Q, f' (compactBlock n Q)*g' (Q a b) ∂(μ : Measure (CompactArray K)) := by |
| rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => f (compactBlock n Q)*g (Q a b)) |
| ((f.measurable.comp (compactBlock_continuous n).measurable).mul (g.measurable.comp (by fun_prop)))] |
| simp only [compactDiagonalMapBlock_compactBlock,compactDiagonalMapArray,ite_eq_right hab] |
| rfl |
| have hf : (∫ Q, f (compactBlock n Q) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))= |
| ∫ Q, f' (compactBlock n Q) ∂(μ : Measure (CompactArray K)) := by |
| rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => f (compactBlock n Q)) |
| (f.measurable.comp (compactBlock_continuous n).measurable)] |
| simp only [compactDiagonalMapBlock_compactBlock] |
| rfl |
| have hg : (∫ Q, g (Q 0 1) ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))= |
| ∫ Q, g' (Q 0 1) ∂(μ : Measure (CompactArray K)) := by |
| rw [compactDiagonalMapLaw_integral e d he hd μ (fun Q => g (Q 0 1)) (by fun_prop)] |
| rfl |
| unfold compactGGDefect |
| rw [hmul i.val n (Nat.ne_of_lt i.isLt),hf,hg] |
| have hs : (∑ j ∈ Finset.univ.erase i, ∫ Q, f (compactBlock n Q)*g (Q i j) |
| ∂(compactDiagonalMapLaw e d he hd μ : Measure (CompactArray L)))= |
| ∑ j ∈ Finset.univ.erase i, ∫ Q, f' (compactBlock n Q)*g' (Q i j) |
| ∂(μ : Measure (CompactArray K)) := by |
| apply Finset.sum_congr rfl |
| intro j hj |
| apply hmul |
| intro hij |
| exact (Finset.mem_erase.mp hj).1 (Fin.ext hij.symm) |
| rw [hs] |
| exact hGG n i f' g' |
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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