NatClassEquation.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 8 curated AI-written annotations and 1204 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, Pinned FLT blueprint introduction

7 declarations, 0 contain sorry, 0 declared as axioms

A class-equation-style counting lemma for conjugates of subgroups

This file proves Dickson.natClassEquation: if H₁, …, Hᵣ are subgroups of a finite group G which pairwise intersect trivially (along with all their conjugates), then counting the nonidentity elements lying in conjugates of the Hᵢ gives the inequality ∑ i, (|Hᵢ| - 1) * |G : N_G(Hᵢ)| ≤ |G| - 1.

This is the integer form of the counting argument underlying the partition equation used in the tame case of Dickson's classification of finite subgroups of PGL₂(𝔽̄_p).

Imports (3)

Imported by 2 files

lemma Dickson.card_conjugates_eq_normalizer_index line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subgroup, Eq, Nat, Fintype.card, Set.ofPred, Exists, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Membership.mem, Set, Subgroup.index, Subgroup.normalizer
Proof uses: Eq.mpr, Nat.card, HasQuotient.Quotient, id, congrArg, Subgroup.index_eq_card, Eq.symm, Nat.card_eq_fintype_card, Nat.card_congr, Equiv.symm, Equiv.ofBijective, Subtype.mk, Quotient.out, QuotientGroup.leftRel, Exists.intro, rfl, And.intro, Function.Injective, Function.Surjective, QuotientGroup.out_eq' and 40 more
87 instances, coercions and auxiliary constants left out.

Tactic trace: 51 steps
Open to load the checked before-and-after goal states.
lemma Dickson.conjClass_unique_index line 112 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Nat, Fin, Subgroup, Exists, Eq, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Ne, Min.min, Bot.bot, OfNat.ofNat, Membership.mem
Proof uses: Decidable.byContradiction, Not, Iff.mp, Iff, Subgroup.ext_iff, Exists.intro, rfl, And.intro, Set, Submonoid, Subgroup.toSubmonoid, HMul.hMul, Inv.inv, Subgroup.ext, Eq.mpr, And, id, congrFun', congrArg, Iff.intro and 23 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
def Dickson.conjClassElements The finset of non-identity elements of G' lying in some conjugate of the subgroup H line 156 source
Source documentation
The finset of non-identity elements of G' lying in some conjugate of the subgroup H.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subgroup, Finset
Proof uses: Finset.filter, And, Ne, OfNat.ofNat, Exists, Membership.mem, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Eq, Finset.univ
19 instances, coercions and auxiliary constants left out.
```
def conjClassElements {G' : Type*} [Group G'] [Fintype G']
    (H : Subgroup G') : Finset G' :=
…
```
def Dickson.conjNonidentity The finset of non-identity elements of the conjugate subgroup H.map (MulAut.conj g) line 163 source
Source documentation
The finset of non-identity elements of the conjugate subgroup H.map (MulAut.conj g).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subgroup, Finset
Proof uses: Finset.filter, And, Ne, OfNat.ofNat, Membership.mem, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Eq, Finset.univ
18 instances, coercions and auxiliary constants left out.
```
def conjNonidentity {G' : Type*} [Group G'] [Fintype G']
    (H : Subgroup G') (g : G') : Finset G' :=
…
```
lemma Dickson.card_conjNonidentity line 169 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subgroup, Eq, Nat, Finset.card, Dickson.conjNonidentity, HSub.hSub, Nat.card, Subtype, Membership.mem, OfNat.ofNat
Proof uses: Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Finset, Finset.erase, Set.toFinset, Subsemigroup.carrier, Submonoid.toSubsemigroup, Subgroup.toSubmonoid, Set, Finset.ext, Eq.mpr, Iff, And, Not, id, congr, congrArg and 24 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Dickson.card_conjClassElements line 188 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Subgroup, GE.ge, Nat, Nat.card, Subtype, Membership.mem, OfNat.ofNat, Exists, Eq, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Ne, Min.min, Bot.bot, Finset.card and 5 more
Proof uses: Fintype.card, Set.ofPred, Set, Dickson.card_conjugates_eq_normalizer_index, Finset, Finset.filter, Finset.univ, Eq.mpr, Finset.biUnion, And, id, congrArg, Finset.ext, Iff, congr, Eq.trans, True, congrFun', true_and, funext and 33 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
lemma Dickson.natClassEquation line 230 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Fintype, Nat, Fin, Subgroup, GE.ge, Nat.card, Subtype, Membership.mem, OfNat.ofNat, Exists, Eq, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, Ne, Min.min, Bot.bot and 7 more
Proof uses: Finset.card, Finset.filter, Dickson.conjClassElements, Eq.mpr, Finset.biUnion, id, congrArg, Eq.symm, Finset.card_biUnion, Set, Finset, Function.onFun, Disjoint, Not, Eq.trans, forall_congr, implies_congr, True, congrFun', true_and and 30 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.