`PFR/SecondEstimate.lean`

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Project sources: Pinned project documentation, Polynomial Freiman-Ruzsa proof paper, arXiv v2

3 declarations, 0 contain sorry, 0 declared as axioms

Second estimate

The second estimate on tau-minimizers.

Assumptions:

$X_1, X_2$ are tau-minimizers
$X_1, X_2, \tilde X_1, \tilde X_2$ be independent random variables, with $X_1,\tilde X_1$ copies of $X_1$ and $X_2,\tilde X_2$ copies of $X_2$.
$d[X_1;X_2] = k$
$I_1 := I_1 [X_1+X_2 : \tilde X_1 + X_2 | X_1+X_2+\tilde X_1+\tilde X_2]$
$I_2 := I[X_1+X_2 : X_1 + \tilde X_1 | X_1+X_2+\tilde X_1+\tilde X_2]$

Main results

second_estimate : $$ I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$$

Imports (2)

Imported by 2 files

lemma rdist_of_sums_ge' $$ d[X_1+\tilde X_1; X_2+\tilde X_2] \geq k - \frac{\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$ line 62 source
Source documentation
$$ d[X_1+\tilde X_1; X_2+\tilde X_2] \geq k - \frac{\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finite, MeasurableSpace, MeasurableSingletonClass, Module, ZMod, OfNat.ofNat, Nat, MeasurableAdd₂, MeasureTheory.MeasureSpace, MeasureTheory.IsProbabilityMeasure, MeasureTheory.MeasureSpace.volume, refPackage, Measurable, ProbabilityTheory.IdentDistrib, ProbabilityTheory.iIndepFun, Fin, Nat.succ, Matrix.vecCons, Matrix.vecEmpty and 9 more
Proof uses: LE.le.ge, LE.le.trans, refPackage.X₀₁, refPackage.X₀₂, Eq.mpr, LE.le, id, Eq, congrArg, sub_sub, propext, sub_le_sub_iff_left, Eq.symm, mul_add, mul_div_assoc, mul_le_mul_of_nonneg_left, ProbabilityTheory.entropy, condRuzsaDist_diff_le', refPackage.hmeas1, ProbabilityTheory.iIndepFun.indepFun and 88 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
lemma rdist_of_sums_ge' : d[X₁ + X₁' # X₂ + X₂'] ≥ k - p.η * (d[X₁ # X₁] + d[X₂ # X₂]) / 2 := by
…
```
lemma second_estimate_aux line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finite, MeasurableSpace, MeasurableSingletonClass, Module, ZMod, OfNat.ofNat, Nat, MeasurableAdd₂, MeasureTheory.MeasureSpace, MeasureTheory.IsProbabilityMeasure, MeasureTheory.MeasureSpace.volume, refPackage, Measurable, ProbabilityTheory.IdentDistrib, ProbabilityTheory.iIndepFun, Fin, Nat.succ, Matrix.vecCons, Matrix.vecEmpty and 10 more
Proof uses: ProbabilityTheory.IndepFun, ProbabilityTheory.iIndepFun.indepFun, Ne, of_decide_eq_true, Eq, id, Bool, Decidable.decide, Bool.true, Eq.refl, ProbabilityTheory.iIndepFun.indepFun_add_add, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons and 137 more
164 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
lemma second_estimate $$ I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$$ line 103 source
Source documentation
$$ I_2 \leq 2 \eta k + \frac{2 \eta (2 \eta k - I_1)}{1 - \eta}.$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Finite, MeasurableSpace, MeasurableSingletonClass, Module, ZMod, OfNat.ofNat, Nat, MeasurableAdd₂, MeasureTheory.MeasureSpace, MeasureTheory.IsProbabilityMeasure, MeasureTheory.MeasureSpace.volume, refPackage, Measurable, ProbabilityTheory.IdentDistrib, ProbabilityTheory.iIndepFun, Fin, Nat.succ, Matrix.vecCons, Matrix.vecEmpty and 10 more
Proof uses: ProbabilityTheory.IndepFun, ProbabilityTheory.iIndepFun.indepFun, Ne, of_decide_eq_true, Eq, id, Bool, Decidable.decide, Bool.true, Eq.refl, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons, List.Mem.head and 133 more
156 instances, coercions and auxiliary constants left out.

Tactic trace: 37 steps
Open to load the checked before-and-after goal states.
```
lemma second_estimate : I₂ ≤ 2 * p.η * k + (2 * p.η * (2 * p.η * k - I₁)) / (1 - p.η) := by
…
```