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Remove unnecessary hypotheses #191

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May 14, 2024
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Remove unnecessary hypotheses #191

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16 changes: 8 additions & 8 deletions PFR/MoreRuzsaDist.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -32,11 +32,11 @@ variable {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} {V : Type uV}
Let $X,Y$ be random variables. For any function $f, g$ on the range of $X$, we have $I[f(X) : Y] \leq I[X:Y]$.
-/
lemma mutual_comp_le (μ : Measure Ω) [IsProbabilityMeasure μ] (hX : Measurable X)
(hY : Measurable Y) (f : S → U) (hf : Measurable f) [FiniteRange X] [FiniteRange Y] :
(hY : Measurable Y) (f : S → U) [FiniteRange X] [FiniteRange Y] :
I[f ∘ X : Y ; μ] ≤ I[X : Y ; μ] := by
rw [mutualInfo_comm (Measurable.comp hf hX) hY, mutualInfo_comm hX hY,
mutualInfo_eq_entropy_sub_condEntropy hY (Measurable.comp hf hX),
mutualInfo_eq_entropy_sub_condEntropy hY hX]
have h_meas : Measurable (f ∘ X) := Measurable.comp (measurable_discrete f) hX
rw [mutualInfo_comm h_meas hY, mutualInfo_comm hX hY,
mutualInfo_eq_entropy_sub_condEntropy hY h_meas, mutualInfo_eq_entropy_sub_condEntropy hY hX]
gcongr
exact condEntropy_comp_ge μ hX hY f

Expand All @@ -45,13 +45,13 @@ lemma mutual_comp_le (μ : Measure Ω) [IsProbabilityMeasure μ] (hX : Measurabl
have $\bbI[f(X) : g(Y)] \leq \bbI[X : Y]$.
-/
lemma mutual_comp_comp_le (μ : Measure Ω) [IsProbabilityMeasure μ] (hX : Measurable X)
(hY : Measurable Y) (f : S → U) (g : T → V) (hf : Measurable f) (hg : Measurable g) [
FiniteRange X] [FiniteRange Y]:
(hY : Measurable Y) (f : S → U) (g : T → V) (hg : Measurable g)
[FiniteRange X] [FiniteRange Y] :
I[f ∘ X : g ∘ Y ; μ] ≤ I[X : Y ; μ] :=
calc
_ ≤ I[X : g ∘ Y ; μ] := mutual_comp_le μ hX (Measurable.comp hg hY) f hf
_ ≤ I[X : g ∘ Y ; μ] := mutual_comp_le μ hX (Measurable.comp hg hY) f
_ = I[g ∘ Y : X ; μ] := mutualInfo_comm hX (Measurable.comp hg hY) μ
_ ≤ I[Y : X ; μ] := mutual_comp_le μ hY hX g hg
_ ≤ I[Y : X ; μ] := mutual_comp_le μ hY hX g
_ = I[X : Y ; μ] := mutualInfo_comm hY hX μ

/--
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