Comp.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

30 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

lemma ProbabilityTheory.Kernel.map_map line 21 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Measurable, Eq, ProbabilityTheory.Kernel.map, Function.comp
Proof uses: ProbabilityTheory.Kernel.ext, MeasureTheory.Measure.ext, MeasureTheory.Measure, Set, MeasurableSet, Eq.mpr, ENNReal, MeasureTheory.Measure.map, id, congrArg, ProbabilityTheory.Kernel.map_apply, Measurable.comp, MeasureTheory.Measure.map_map, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.map_swapRight line 26 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.swapRight, Function.comp, Prod.swap
Proof uses: dite, Measurable, Classical.propDecidable, Eq.mpr, id, congrArg, ProbabilityTheory.Kernel.swapRight_eq, ProbabilityTheory.Kernel.map_map, measurable_swap, Eq.refl, Not, Mathlib.Tactic.Contrapose.contrapose₄, Measurable.comp, of_eq_true, Eq.trans, OfNat.ofNat, True, congr, ProbabilityTheory.Kernel.map_of_not_measurable, eq_self
4 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
def ProbabilityTheory.Kernel.deleteMiddle line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod
Proof uses: ProbabilityTheory.Kernel.mapOfMeasurable, Prod.mk, Prod.fst, Prod.snd
3 instances, coercions and auxiliary constants left out.
lemma ProbabilityTheory.Kernel.deleteMiddle_eq line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.deleteMiddle, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.mapOfMeasurable_eq_map, eq_self
3 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
instance ProbabilityTheory.Kernel.(instance at line 46) line 46 source
instance ProbabilityTheory.Kernel.(instance at line 49) line 49 source
lemma ProbabilityTheory.Kernel.fst_deleteMiddle line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.fst, ProbabilityTheory.Kernel.deleteMiddle
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteMiddle_eq, ProbabilityTheory.Kernel.fst_map_prod, Measurable.comp, measurable_snd, ProbabilityTheory.Kernel.fst_eq, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.snd_deleteMiddle line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.snd, ProbabilityTheory.Kernel.deleteMiddle
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteMiddle_eq, ProbabilityTheory.Kernel.snd_map_prod, measurable_fst, ProbabilityTheory.Kernel.snd_eq, Function.comp, ProbabilityTheory.Kernel.map_map, measurable_snd, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.deleteMiddle_zero line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, Eq, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.Kernel.deleteMiddle, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, ProbabilityTheory.Kernel.deleteMiddle_eq, ProbabilityTheory.Kernel.map_zero, eq_self
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.deleteMiddle_map_prod line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Measurable, Eq, Prod, ProbabilityTheory.Kernel.deleteMiddle, ProbabilityTheory.Kernel.map, Prod.mk
Proof uses: dite, Classical.propDecidable, Measurable.fst, Measurable.snd, Eq.mpr, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteMiddle_eq, Function.comp, ProbabilityTheory.Kernel.map_map, Measurable.prodMk, measurable_id', Eq.refl, Not, Mathlib.Tactic.Contrapose.contrapose₄, Measurable.prod, of_eq_true, Eq.trans and 6 more
4 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.deleteMiddle_compProd line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, ProbabilityTheory.IsSFiniteKernel, Prod, Eq, ProbabilityTheory.Kernel.deleteMiddle, ProbabilityTheory.Kernel.compProd, ProbabilityTheory.Kernel.snd
Proof uses: ProbabilityTheory.Kernel.ext, MeasureTheory.Measure.ext, MeasureTheory.Measure, Set, MeasurableSet, Eq.mpr, ENNReal, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteMiddle_eq, Set.preimage, ProbabilityTheory.Kernel.map_apply', Measurable.prodMk, Measurable.fst, measurable_id', Measurable.snd and 10 more
6 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
def ProbabilityTheory.Kernel.deleteRight line 98 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod
Proof uses: ProbabilityTheory.Kernel.mapOfMeasurable, Prod.mk, Prod.fst, Prod.snd
3 instances, coercions and auxiliary constants left out.
lemma ProbabilityTheory.Kernel.deleteRight_eq line 102 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.deleteRight, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.mapOfMeasurable_eq_map, eq_self
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.deleteRight_zero line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, Eq, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.Kernel.deleteRight, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, ProbabilityTheory.Kernel.deleteRight_eq, ProbabilityTheory.Kernel.map_zero, eq_self
6 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
instance ProbabilityTheory.Kernel.(instance at line 108) line 108 source
instance ProbabilityTheory.Kernel.(instance at line 111) line 111 source
lemma ProbabilityTheory.Kernel.fst_deleteRight line 116 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.fst, ProbabilityTheory.Kernel.deleteRight
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteRight_eq, ProbabilityTheory.Kernel.fst_map_prod, Measurable.comp, measurable_fst, measurable_snd, ProbabilityTheory.Kernel.fst_eq, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.snd_deleteRight line 121 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.snd, ProbabilityTheory.Kernel.deleteRight, ProbabilityTheory.Kernel.fst
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteRight_eq, ProbabilityTheory.Kernel.snd_map_prod, measurable_fst, ProbabilityTheory.Kernel.snd_eq, ProbabilityTheory.Kernel.fst_eq, Function.comp, ProbabilityTheory.Kernel.map_map, measurable_snd, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.deleteRight_map_prod line 127 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Measurable, Eq, Prod, ProbabilityTheory.Kernel.deleteRight, ProbabilityTheory.Kernel.map, Prod.mk
Proof uses: dite, Classical.propDecidable, Measurable.fst, Measurable.snd, Eq.mpr, Prod.fst, Prod.snd, id, congrArg, ProbabilityTheory.Kernel.deleteRight_eq, Function.comp, ProbabilityTheory.Kernel.map_map, Measurable.prodMk, measurable_id', Eq.refl, Not, Mathlib.Tactic.Contrapose.contrapose₄, Measurable.prod, of_eq_true, Eq.trans and 6 more
4 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
def ProbabilityTheory.Kernel.reverse line 145 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod
Proof uses: ProbabilityTheory.Kernel.mapOfMeasurable, Prod.mk, Prod.snd, Prod.fst
3 instances, coercions and auxiliary constants left out.
lemma ProbabilityTheory.Kernel.reverse_eq line 150 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.reverse, ProbabilityTheory.Kernel.map, Prod.mk, Prod.snd, Prod.fst
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.mapOfMeasurable_eq_map, eq_self
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.reverse_zero line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, Eq, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.Kernel.reverse, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ProbabilityTheory.Kernel.map, Prod.mk, Prod.snd, Prod.fst, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.map_zero, eq_self
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.reverse_reverse line 156 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.reverse
Proof uses: Function.comp, Prod.mk, Prod.snd, Prod.fst, id, funext, of_eq_true, Eq.trans, True, congrFun', congrArg, Prod.mk.eta, eq_self, Eq.mpr, ProbabilityTheory.Kernel.map, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.map_map, Measurable.prodMk, Measurable.snd, measurable_id' and 3 more
2 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
instance ProbabilityTheory.Kernel.(instance at line 162) line 162 source
instance ProbabilityTheory.Kernel.(instance at line 165) line 165 source
lemma ProbabilityTheory.Kernel.swapRight_deleteMiddle_reverse line 170 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.swapRight, ProbabilityTheory.Kernel.deleteMiddle, ProbabilityTheory.Kernel.reverse
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.swap, id, congrArg, ProbabilityTheory.Kernel.swapRight_eq, Prod.mk, Prod.snd, Prod.fst, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.deleteMiddle_eq, Function.comp, ProbabilityTheory.Kernel.map_map, Measurable.prodMk, Measurable.fst, measurable_id', Measurable.snd, measurable_swap, Measurable.fun_comp, Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.swapRight_snd_reverse line 177 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.swapRight, ProbabilityTheory.Kernel.snd, ProbabilityTheory.Kernel.reverse, ProbabilityTheory.Kernel.deleteRight
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.swap, id, congrArg, ProbabilityTheory.Kernel.swapRight_eq, Prod.mk, Prod.snd, Prod.fst, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.deleteRight_eq, ProbabilityTheory.Kernel.snd_eq, Function.comp, ProbabilityTheory.Kernel.map_map, Measurable.snd, measurable_id', measurable_swap, Measurable.prodMk, Measurable.fst, Measurable.fun_comp and 1 more
2 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.swapRight_deleteRight_reverse line 184 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ProbabilityTheory.Kernel, Prod, Eq, ProbabilityTheory.Kernel.swapRight, ProbabilityTheory.Kernel.deleteRight, ProbabilityTheory.Kernel.reverse, ProbabilityTheory.Kernel.snd
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.swap, id, congrArg, ProbabilityTheory.Kernel.swapRight_eq, Prod.mk, Prod.snd, Prod.fst, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.deleteRight_eq, ProbabilityTheory.Kernel.snd_eq, Function.comp, ProbabilityTheory.Kernel.map_map, Measurable.prodMk, Measurable.fst, measurable_id', Measurable.snd, measurable_swap, Measurable.fun_comp and 1 more
2 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityTheory.Kernel.FiniteKernelSupport.reverse Reversing preserves finite kernel support line 191 source
Source documentation
Reversing preserves finite kernel support

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, ProbabilityTheory.Kernel, Prod, ProbabilityTheory.Kernel.FiniteKernelSupport, ProbabilityTheory.Kernel.reverse
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.snd, Prod.fst, id, Eq, congrArg, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.FiniteKernelSupport.map
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma FiniteKernelSupport.reverse
    [MeasurableSingletonClass α] [MeasurableSingletonClass β] [MeasurableSingletonClass γ]
    {κ : Kernel T (α × β × γ)} (hκ : FiniteKernelSupport κ) :
    FiniteKernelSupport (reverse κ) := by
…
```
Mentions: ProbabilityTheory.Kernel.FiniteKernelSupport
lemma ProbabilityTheory.Kernel.AEFiniteKernelSupport.reverse line 198 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableSingletonClass, ProbabilityTheory.Kernel, Prod, MeasureTheory.Measure, ProbabilityTheory.Kernel.AEFiniteKernelSupport, ProbabilityTheory.Kernel.reverse
Proof uses: Eq.mpr, ProbabilityTheory.Kernel.map, Prod.mk, Prod.snd, Prod.fst, id, Eq, congrArg, ProbabilityTheory.Kernel.reverse_eq, ProbabilityTheory.Kernel.AEFiniteKernelSupport.map
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.