Stuff.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

33 declarations, 0 contain sorry, 0 declared as axioms

Random assortments of API lemmas missing in mathlib.

Imports (9)

Imported by 4 files

lemma RingHom.coe_toMonoidHom line 25 source
no axioms what this means
Statement uses: NonAssocSemiring, RingHom, Eq, MonoidHom
Proof uses: rfl
13 instances, coercions and auxiliary constants left out.
def Set.matrixEquiv Set.matrix is equivalent to matrices on the subtype line 30 source
Source documentation
Set.matrix is equivalent to matrices on the subtype.

no axioms what this means
Statement uses: Set, Equiv, Matrix, Set.matrix
Proof uses: Equiv.mk, Subtype.mk, Membership.mem
8 instances, coercions and auxiliary constants left out.
```
def Set.matrixEquiv {m n α : Type*} (S : Set α) :
    S.matrix (m := m) (n := n) ≃ Matrix m n S where
…
```
def Subring.matrixEquiv Subring.matrix is equivalent to matrices on the subtype line 40 source
Source documentation
Subring.matrix is equivalent to matrices on the subtype.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring, Fintype, DecidableEq, Subring, RingEquiv, Subtype, Matrix, Membership.mem, Subring.matrix
Proof uses: Equiv, Set.matrix, AddSubmonoid, Subsemiring.toAddSubmonoid, Subring.toSubsemiring, Set.matrixEquiv, RingEquiv.mk
43 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def Subring.matrixEquiv {n α : Type*} [Ring α] [Fintype n] [DecidableEq n] (S : Subring α) :
    S.matrix (n := n) ≃+* Matrix n n S where
…
```
lemma IsPrincipalIdealRing.exists_isCoprime_and_dvd_pow line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsPrincipalIdealRing, IsDomain, Ne, OfNat.ofNat, Exists, Nat, And, Eq, HMul.hMul, IsCoprime, Dvd.dvd, HPow.hPow
Proof uses: UniqueFactorizationMonoid.induction_on_prime, False, rfl, Eq.refl, IsUnit, Exists.intro, of_eq_true, Eq.trans, True, congr, congrArg, one_mul, eq_self, pow_zero, eq_true, IsUnit.dvd, and_self, Prime, dite, Classical.propDecidable and 26 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
instance ConjAct.continuousConstSMul line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, DivInvMonoid, ContinuousMul, ContinuousConstSMul, ConjAct
Proof uses: ContinuousConstSMul.mk, IsTopologicalGroup.continuous_conj, MulEquiv, ConjAct.ofConjAct
16 instances, coercions and auxiliary constants left out.
lemma ConjAct.isOpen_smul line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, IsTopologicalGroup, Subgroup, IsOpen, ConjAct, HSMul.hSMul, Subgroup.pointwiseMulAction
Proof uses: Iff.mpr, Set.image, Homeomorph, Homeomorph.smul, Homeomorph.isOpen_image
22 instances, coercions and auxiliary constants left out.
lemma Polynomial.Monic.not_irreducible_iff_of_natDegree_eq_two line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, NoZeroDivisors, Polynomial, Polynomial.Monic, Eq, Nat, Polynomial.natDegree, OfNat.ofNat, Iff, Not, Irreducible, Exists, HMul.hMul, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
Proof uses: Subsingleton, Nontrivial, Or, subsingleton_or_nontrivial, Or.inl, of_eq_true, Eq.trans, True, congr, congrArg, False, Subsingleton.elim, not_false_eq_true, funext, eq_self, iff_self, Or.inr, Iff.intro, And, Finsupp and 83 more
148 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma IsPrimitiveRoot.iff_of_prime line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Nat, Nat.Prime, Iff, IsPrimitiveRoot, And, Eq, HPow.hPow, OfNat.ofNat, Ne
Proof uses: Iff.intro, And.intro, IsPrimitiveRoot.pow_eq_one, IsPrimitiveRoot.ne_one, Nat.Prime.one_lt, Iff.mpr, orderOf, IsPrimitiveRoot.iff_orderOf, Or.resolve_left, Nat.Prime.eq_one_or_self_of_dvd, orderOf_dvd_of_pow_eq_one, And.left, of_eq_true, Not, Eq.trans, False, True, congrArg, eq_false, And.right and 1 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Matrix.det_algebraMap line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, CommRing, Algebra, DecidableEq, Fintype, Eq, Matrix.det, RingHom, Matrix, Algebra.algebraMap, HPow.hPow, Nat, Fintype.card
Proof uses: Eq.trans, Matrix.diagonal, Pi.constRingHom, Finset.prod, Finset.univ, Matrix.det_diagonal, of_eq_true, True, congrFun', congrArg, Finset.card, funext, RingHom.id, RingHomCompTriple.comp_apply, Finset.prod_const, eq_self
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Set.injOn_image_iff line 99 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Set.InjOn, Iff, Set.image, Function.comp
Proof uses: Eq.mpr, Membership.mem, Eq, id, congrFun', congrArg, Eq.trans, forall_congr, And, Exists, implies_congr, Eq.refl
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Algebra.norm_one_tmul Finiteness not necessary? line 106 source
Source comment
Finiteness not necessary?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Ring, Algebra, Module.Free, Module.Finite, Eq, MonoidHom, TensorProduct, Algebra.norm, TensorProduct.tmul, OfNat.ofNat, RingHom, Algebra.algebraMap
Proof uses: Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim, Subsingleton, of_eq_true, Nontrivial, RingHom.domain_nontrivial, Sigma, Module.Basis, Nonempty, Module.Free.exists_basis, Fintype, nonempty_fintype, Module.Finite.finite_basis, Nonempty.intro, Eq.mpr, Matrix.det, Classical.propDecidable, AlgHom, Matrix, Algebra.leftMulMatrix, id and 29 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma Algebra.norm_one_tmul
    (R A : Type*) {B : Type*} [CommRing R] [CommRing A] [Ring B] [Algebra R A]
    [Algebra R B] (x : B) [Module.Free R B] [Module.Finite R B] :
    Algebra.norm A ((1 : A) ⊗ₜ[R] x) = algebraMap R A (Algebra.norm R x) := by
…
```
lemma Algebra.norm_eq_det line 121 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Fintype, DecidableEq, Matrix, Eq, MonoidHom, Algebra.norm, HPow.hPow, Nat, Matrix.det, Fintype.card
Proof uses: Eq.mpr, Prod, AlgHom, Algebra.leftMulMatrix, Matrix.stdBasis, id, congrArg, Algebra.norm_eq_matrix_det, Equiv, Matrix.comp, Matrix.map, RingHom, Algebra.algebraMap, Matrix.ext, of_eq_true, Eq.trans, ite, Prod.snd, Prod.fst, OfNat.ofNat and 55 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma vecCons_one_one line 133 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: One, Eq, Fin, Nat.succ, OfNat.ofNat, Nat, Matrix.vecCons, Matrix.vecEmpty
Proof uses: funext, List, List.Mem, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, List.cons, List.Mem.head, Fin.mk, Nat.le_of_lt, Nat.le_refl, List.nil, Eq.refl, eq_of_heq, List.Mem.tail, False.elim, noConfusion_of_Nat and 2 more
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma WithZero.lt_ofAdd_one line 137 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WithZero, Multiplicative, Int, Iff, LT.lt, Equiv, Multiplicative.ofAdd, OfNat.ofNat, LE.le
Proof uses: WithZero.recZeroCoe, of_eq_true, Eq.trans, True, congr, congrArg, iff_self, Multiplicative.toAdd, HAdd.hAdd, zero_add
51 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma WithZero.mul_ofAdd_one_le_one_iff line 145 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WithZero, Multiplicative, Int, Iff, LE.le, HMul.hMul, Equiv, Multiplicative.ofAdd, OfNat.ofNat, LT.lt
Proof uses: WithZero.recZeroCoe, of_eq_true, Eq.trans, True, congr, congrArg, congrFun', MulZeroClass.zero_mul, iff_self, Multiplicative.toAdd
79 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma MeasureSpace.comap_symm line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, Equiv, Eq, MeasurableSpace.comap, Equiv.symm, MeasurableSpace.map
Proof uses: MeasurableSpace.ext, Set, of_eq_true, Iff, MeasurableSet, Eq.trans, Set.preimage, True, congr, congrArg, Exists, And, funext, Set.image, Equiv.image_symm_eq_preimage, iff_self
10 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
instance (instance at line 159) line 159 source
instance (instance at line 166) line 166 source
instance (instance at line 169) line 169 source
instance (instance at line 172) line 172 source
theorem MeasurableEquiv.restrict_preimage line 177 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableEquiv, MeasureTheory.Measure, Set, Eq, MeasureTheory.Measure.restrict, Set.preimage, MeasureTheory.Measure.map, MeasurableEquiv.symm
Proof uses: Eq.mpr, id, congrArg, MeasurableEquiv.restrict_map, Function.comp, MeasureTheory.Measure.map_map, MeasurableEquiv.measurable, of_eq_true, Eq.trans, True, congrFun', MeasurableEquiv.symm_comp_self, MeasureTheory.Measure.map_id, eq_self
6 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def ContinuousAddEquiv.smul smul as a ContinuousAddEquiv line 185 source
Source documentation
smul as a ContinuousAddEquiv.

standard axioms: propext what this means
Statement uses: Group, TopologicalSpace, AddMonoid, DistribMulAction, ContinuousConstSMul, ContinuousAddEquiv
Proof uses: Homeomorph, Homeomorph.smul, ContinuousAddEquiv.mk, AddEquiv.mk, Homeomorph.toEquiv, Homeomorph.continuous_toFun, Homeomorph.continuous_invFun
22 instances, coercions and auxiliary constants left out.
```
def ContinuousAddEquiv.smul {G α : Type*} [Group G] [TopologicalSpace α] [AddMonoid α]
    [DistribMulAction G α]
    [ContinuousConstSMul G α] (g : G) : α ≃ₜ+ α where
…
```
lemma ContinuousAddEquiv.smul_inv line 192 source
standard axioms: Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, AddMonoid, DistribMulAction, ContinuousConstSMul, Eq, ContinuousAddEquiv, ContinuousAddEquiv.smul, Inv.inv, ContinuousAddEquiv.symm
Proof uses: ContinuousAddEquiv.ext, of_eq_true, Eq.trans, HSMul.hSMul, True, congr, congrArg, ContinuousAddEquiv.smul_apply, ContinuousAddEquiv.smul_symm_apply, eq_self
36 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def MulOpposite.opMeasurableEquiv MulOpposite.op as a MeasurableEquiv line 198 source
Source documentation
MulOpposite.op as a MeasurableEquiv.

standard axioms: Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasurableEquiv, MulOpposite
Proof uses: Equiv, MulOpposite.opEquiv, MeasurableEquiv.mk, measurable_mul_op, measurable_mul_unop
2 instances, coercions and auxiliary constants left out.
```
def MulOpposite.opMeasurableEquiv {M : Type*} [MeasurableSpace M] : M ≃ᵐ Mᵐᵒᵖ where
…
```
def Units.opContinuousMulEquiv Units.opEquiv (taking x to x) as a ContinuousMulEquiv line 205 source
Source documentation
Units.opEquiv (taking x to x) as a ContinuousMulEquiv.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Monoid, TopologicalSpace, ContinuousMulEquiv, Units, MulOpposite
Proof uses: MulEquiv, Units.opEquiv, ContinuousMulEquiv.mk
12 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
def Units.opContinuousMulEquiv {M : Type*} [Monoid M] [TopologicalSpace M] :
    Mᵐᵒᵖˣ ≃ₜ* Mˣᵐᵒᵖ where
…
```
lemma Units.coe_opContinuousMulEquiv_symm_apply line 216 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Monoid, TopologicalSpace, MulOpposite, Units, Eq, ContinuousMulEquiv, ContinuousMulEquiv.symm, Units.opContinuousMulEquiv, MulOpposite.op, MulOpposite.unop
Proof uses: rfl
18 instances, coercions and auxiliary constants left out.
lemma Units.coe_opContinuousMulEquiv_symm_apply_inv line 221 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Monoid, TopologicalSpace, MulOpposite, Units, Eq, Inv.inv, ContinuousMulEquiv, ContinuousMulEquiv.symm, Units.opContinuousMulEquiv, MulOpposite.op, MulOpposite.unop
Proof uses: rfl
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
instance (instance at line 225) line 225 source
def MulDistribMulAction.toContinuousMulEquiv smul as a ContinuousMulEquiv line 235 source
Source documentation
smul as a ContinuousMulEquiv.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Monoid, MulDistribMulAction, TopologicalSpace, ContinuousConstSMul, ContinuousMulEquiv
Proof uses: MulEquiv, MulDistribMulAction.toMulEquiv, ContinuousMulEquiv.mk
20 instances, coercions and auxiliary constants left out.
```
def MulDistribMulAction.toContinuousMulEquiv
    {G : Type*} (x : G) (M : Type*) [Group G] [Monoid M] [MulDistribMulAction G M]
    [TopologicalSpace M] [ContinuousConstSMul G M] : M ≃ₜ* M where
…
```
lemma Subgroup.FiniteIndex.of_compactSpace line 244 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, SeparatelyContinuousMul, CompactSpace, Subgroup, IsOpen, Subgroup.FiniteIndex
Proof uses: Iff.mpr, Finite, HasQuotient.Quotient, Subgroup.finiteIndex_iff_finite_quotient, Subgroup.quotient_finite_of_isOpen
15 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
instance (instance at line 249) line 249 source
lemma Subgroup.IsFiniteRelIndex.of_isCompact line 261 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, TopologicalSpace, SeparatelyContinuousMul, Subgroup, IsOpen, IsCompact, Subgroup.IsFiniteRelIndex
Proof uses: CompactSpace, Subtype, Membership.mem, Iff.mp, Set, isCompact_iff_compactSpace, Iff.mpr, Subgroup.FiniteIndex, Subgroup.subgroupOf, Subgroup.isFiniteRelIndex_iff_finiteIndex, Subgroup.FiniteIndex.of_compactSpace, IsOpen.preimage, continuous_subtype_val
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
instance (instance at line 268) line 268 source