RuzsaDist.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 9 curated AI-written annotations and 485 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

18 declarations, 0 contain sorry, 0 declared as axioms

Ruzsa distance between kernels

Notations

dk[κ ; μ # η ; ν] =

Imports (1)

PFR.ForMathlib.Entropy.Kernel.Group

Imported by 2 files

def ProbabilityTheory.Kernel.rdistm line 26 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, MeasureTheory.Measure, Real
Proof uses: HSub.hSub, ProbabilityTheory.measureEntropy, MeasureTheory.Measure.map, Prod, Prod.fst, Prod.snd, MeasureTheory.Measure.prod, HDiv.hDiv, OfNat.ofNat
12 instances, coercions and auxiliary constants left out.
def ProbabilityTheory.Kernel.rdist line 31 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, ProbabilityTheory.Kernel, MeasureTheory.Measure, Real
Proof uses: MeasureTheory.integral, Prod, MeasureTheory.Measure.prod, ProbabilityTheory.Kernel.rdistm, Prod.fst, Prod.snd
8 instances, coercions and auxiliary constants left out.
lemma ProbabilityTheory.Kernel.rdist_zero_right line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, ProbabilityTheory.Kernel, MeasureTheory.Measure, Eq, Real, ProbabilityTheory.Kernel.rdist, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.integral, Prod, MeasureTheory.Measure.prod, ProbabilityTheory.Kernel.rdistm, Prod.fst, Prod.snd, MeasureTheory.Measure.prod_zero, MeasureTheory.integral_zero_measure, eq_self
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_zero_left line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, ProbabilityTheory.Kernel, MeasureTheory.Measure, Eq, Real, ProbabilityTheory.Kernel.rdist, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.integral, Prod, MeasureTheory.Measure.prod, ProbabilityTheory.Kernel.rdistm, Prod.fst, Prod.snd, MeasureTheory.Measure.zero_prod, MeasureTheory.integral_zero_measure, eq_self
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_eq line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, HSub.hSub, MeasureTheory.integral, Prod, MeasureTheory.Measure.prod, ProbabilityTheory.measureEntropy, MeasureTheory.Measure.map, Prod.fst, Prod.snd, HDiv.hDiv and 3 more
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.rdistm, id, congrArg, ProbabilityTheory.Kernel.rdist.eq_1, MeasureTheory.integral_sub, ProbabilityTheory.integrable_of_finiteSupport, HMul.hMul, Inv.inv, congr, funext, div_eq_mul_inv, congrFun', MeasureTheory.integral_mul_const, NormedAddCommGroup, MeasureTheory.SFinite, NormedSpace, MeasureTheory.IsFiniteMeasure, MeasureTheory.integral_prod, of_eq_true and 9 more
92 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_eq' line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, HSub.hSub, ProbabilityTheory.Kernel.entropy, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, Prod.fst and 5 more
Proof uses: Eq.mpr, MeasureTheory.integral, ProbabilityTheory.measureEntropy, MeasureTheory.Measure.map, id, congrArg, ProbabilityTheory.Kernel.rdist_eq, HSub, Eq.rec, Eq.refl, NormedAddCommGroup, NormedSpace, funext, ProbabilityTheory.Kernel.map_apply, Measurable.fun_sub, Measurable.fst, measurable_id', Measurable.snd, ProbabilityTheory.Kernel.prod_apply, ProbabilityTheory.Kernel.prodMkLeft_apply and 1 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_symm line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist
Proof uses: Eq.mpr, HSub.hSub, ProbabilityTheory.Kernel.entropy, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, Prod.fst, Prod.snd, MeasureTheory.Measure.prod, HDiv.hDiv, OfNat.ofNat, Nat, id, congrArg, ProbabilityTheory.Kernel.rdist_eq', HAdd.hAdd, sub_sub, add_comm and 21 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_zero_kernel_right line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, OfNat.ofNat, HDiv.hDiv, Neg.neg, ProbabilityTheory.Kernel.entropy, Nat
Proof uses: Eq.mpr, HSub.hSub, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, Prod.fst, Prod.snd, MeasureTheory.Measure.prod, id, congrArg, ProbabilityTheory.Kernel.rdist_eq', congrFun', Eq.trans, congr, ProbabilityTheory.Kernel.prodMkLeft_zero, ProbabilityTheory.Kernel.entropy_zero_kernel, zero_div, sub_zero and 75 more
95 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_zero_kernel_left line 101 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, OfNat.ofNat, HDiv.hDiv, Neg.neg, ProbabilityTheory.Kernel.entropy, Nat
Proof uses: Eq.mpr, id, congrArg, ProbabilityTheory.Kernel.rdist_symm, ProbabilityTheory.Kernel.rdist_zero_kernel_right, Eq.refl
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_dirac_zero_right line 107 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, ProbabilityTheory.Kernel.const, MeasureTheory.Measure.dirac, OfNat.ofNat, HDiv.hDiv, ProbabilityTheory.Kernel.entropy, Nat
Proof uses: Eq.mpr, HSub.hSub, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, Prod.fst, Prod.snd, MeasureTheory.Measure.prod, id, congrArg, ProbabilityTheory.Kernel.rdist_eq', congrFun', Eq.trans, HMul.hMul, ENNReal.toReal, Set, ENNReal, Set.univ and 93 more
112 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_dirac_zero_left line 132 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsFiniteKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.rdist, ProbabilityTheory.Kernel.const, MeasureTheory.Measure.dirac, OfNat.ofNat, HDiv.hDiv, ProbabilityTheory.Kernel.entropy, Nat
Proof uses: Eq.mpr, id, congrArg, ProbabilityTheory.Kernel.rdist_symm, ProbabilityTheory.Kernel.rdist_dirac_zero_right, Eq.refl
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.ruzsa_triangle_aux line 138 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, MeasurableSingletonClass, Countable, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.IsMarkovKernel, Eq, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, HSub.hSub, Prod.snd, Prod.fst, ProbabilityTheory.Kernel.snd
Proof uses: Function.comp, Prod.swap, funext, of_eq_true, eq_self, Eq.mpr, id, congrArg, Eq.symm, ProbabilityTheory.Kernel.map_map, measurable_swap, Measurable.of_discrete, ProbabilityTheory.Kernel.map_prod_swap, ProbabilityTheory.Kernel.ext, MeasureTheory.Measure.ext, MeasureTheory.Measure, Set, MeasurableSet, ENNReal, Set.preimage and 12 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.abs_sub_entropy_le_rdist line 149 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, ProbabilityTheory.Kernel.AEFiniteKernelSupport, LE.le, Real, abs, HSub.hSub, ProbabilityTheory.Kernel.entropy, HMul.hMul, OfNat.ofNat, Nat, ProbabilityTheory.Kernel.rdist
Proof uses: Max.max, Prod, ProbabilityTheory.Kernel.prodMkRight, MeasureTheory.Measure.prod, ProbabilityTheory.Kernel.prodMkLeft, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, Prod.fst, Prod.snd, ProbabilityTheory.Kernel.max_entropy_le_entropy_sub_prod, ProbabilityTheory.Kernel.AEFiniteKernelSupport.prodMkRight, ProbabilityTheory.Kernel.AEFiniteKernelSupport.prodMkLeft, Eq.mpr, HDiv.hDiv, id, Eq, congrArg, ProbabilityTheory.Kernel.rdist_eq', And, Neg.neg and 87 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_nonneg line 163 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, ProbabilityTheory.Kernel.AEFiniteKernelSupport, LE.le, Real, OfNat.ofNat, ProbabilityTheory.Kernel.rdist
Proof uses: HMul.hMul, Nat, Trans.trans, abs, HSub.hSub, ProbabilityTheory.Kernel.entropy, abs_nonneg, ProbabilityTheory.Kernel.abs_sub_entropy_le_rdist, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, HPow.hPow, Nat.rawCast, Int.rawCast, inferInstance and 48 more
72 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.ent_of_diff_le The improved entropic Ruzsa triangle inequality line 176 source
Source documentation
The improved entropic Ruzsa triangle inequality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, Countable, MeasurableSingletonClass, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, ProbabilityTheory.Kernel.FiniteKernelSupport, LE.le, Real, ProbabilityTheory.Kernel.entropy, ProbabilityTheory.Kernel.map, HSub.hSub, Prod.fst, Prod.snd, HAdd.hAdd, ProbabilityTheory.Kernel.prod and 2 more
Proof uses: ProbabilityTheory.Kernel.FiniteKernelSupport.prod, Prod.mk, ProbabilityTheory.Kernel.AEFiniteKernelSupport, ProbabilityTheory.Kernel.deleteMiddle, ProbabilityTheory.Kernel.entropy_triple_add_entropy_le, Eq, Function.comp, funext, of_eq_true, eq_self, Eq.mpr, id, congrArg, Eq.symm, ProbabilityTheory.Kernel.map_map, Measurable.fst, measurable_id', Measurable.fun_sub, Measurable.snd, ProbabilityTheory.Kernel.fst_eq and 95 more
90 instances, coercions and auxiliary constants left out.

Tactic trace: 86 steps
Open to load the checked before-and-after goal states.
```
lemma ent_of_diff_le (κ : Kernel T (G × G)) (η : Kernel T G) [IsMarkovKernel κ] [IsMarkovKernel η]
    (μ : Measure T) [IsProbabilityMeasure μ] [FiniteSupport μ]
    (hκ : FiniteKernelSupport κ) (hη : FiniteKernelSupport η) :
    Hk[map κ (fun p : G × G ↦ p.1 - p.2), μ]
      ≤ Hk[map ((fst κ) ×ₖ η) (fun p : G × G ↦ p.1 - p.2), μ]
        + Hk[map (η ×ₖ (snd κ)) (fun p : G × G ↦ p.1 - p.2), μ]
        - Hk[η, μ] := by
…
```
Mentions: ProbabilityTheory.FiniteSupport, ProbabilityTheory.Kernel.FiniteKernelSupport
lemma ProbabilityTheory.Kernel.rdist_triangle_aux1 line 279 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, MeasurableSingletonClass, Countable, ProbabilityTheory.Kernel, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.entropy, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, HSub.hSub, Prod.fst, Prod.snd and 1 more
Proof uses: Filter.Eventually, Membership.mem, Set, Finset, SProd.sprod, MeasureTheory.Measure.support, MeasureTheory.ae, MeasureTheory.Measure.prod_of_full_measure_finset, ProbabilityTheory.measure_compl_support, ENNReal, Compl.compl, OfNat.ofNat, id, Eq.mpr, MeasureTheory.integral, ProbabilityTheory.measureEntropy, MeasureTheory.Measure.restrict, congrArg, NormedAddCommGroup, NormedSpace and 40 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_triangle_aux2 line 310 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, MeasurableSingletonClass, Countable, ProbabilityTheory.Kernel, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, Eq, Real, ProbabilityTheory.Kernel.entropy, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkLeft, ProbabilityTheory.Kernel.prodMkRight, HSub.hSub, Prod.fst, Prod.snd and 1 more
Proof uses: Filter.Eventually, Membership.mem, Set, Finset, SProd.sprod, MeasureTheory.Measure.support, MeasureTheory.ae, MeasureTheory.Measure.prod_of_full_measure_finset, ProbabilityTheory.measure_compl_support, ENNReal, Compl.compl, OfNat.ofNat, id, Eq.mpr, MeasureTheory.integral, ProbabilityTheory.measureEntropy, MeasureTheory.Measure.restrict, congrFun', congrArg, NormedAddCommGroup and 44 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.rdist_triangle line 350 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, AddCommGroup, MeasurableSingletonClass, Countable, ProbabilityTheory.Kernel, ProbabilityTheory.IsMarkovKernel, MeasureTheory.Measure, MeasureTheory.IsProbabilityMeasure, ProbabilityTheory.FiniteSupport, ProbabilityTheory.Kernel.FiniteKernelSupport, LE.le, Real, ProbabilityTheory.Kernel.rdist, HAdd.hAdd
Proof uses: Eq.mpr, HSub.hSub, ProbabilityTheory.Kernel.entropy, Prod, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.prod, ProbabilityTheory.Kernel.prodMkRight, ProbabilityTheory.Kernel.prodMkLeft, Prod.fst, Prod.snd, MeasureTheory.Measure.prod, HDiv.hDiv, OfNat.ofNat, Nat, id, Eq, congrArg, ProbabilityTheory.Kernel.rdist_eq', ProbabilityTheory.Kernel.fst, ProbabilityTheory.Kernel.snd and 75 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.