Basic.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

3 declarations, 0 contain sorry, 0 declared as axioms

Basic

Material destined for Mathlib.

Imports (2)

Imported by 2 files

lemma Representation.comp_def line 31 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, AddCommMonoid, Module, Group, Representation, Eq, LinearMap, RingHom.id, LinearMap.comp, HMul.hMul
Proof uses: rfl
28 instances, coercions and auxiliary constants left out.
def Representation.glMapOfBasis Documented declaration: glMapOfBasis line 36 source
Source documentation
A representation ρ : G → GL(V) of G on a finite free R-module V together with a basis 𝓑 of V gives rise to a group homomorphism G → GL_ι(R) via the matrix of ρ g in the basis 𝓑.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, AddCommMonoid, Module, Group, DecidableEq, Fintype, Representation, Module.Basis, MonoidHom, Matrix.GeneralLinearGroup, Matrix
Proof uses: MonoidHom.mk, OneHom.mk, Units.mk, LinearEquiv, RingHom.id, LinearMap, LinearMap.toMatrix, Inv.inv
44 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
noncomputable def glMapOfBasis
  : G →* Matrix.GeneralLinearGroup ι R where
…
```
def Representation.baseChange Base change of a representation along a commutative R-algebra R': extends ρ : G → GL(V) to R' ⊗[R] V line 49 source
Source documentation
Base change of a representation along a commutative R-algebra R': extends ρ : G → GL(V) to R' ⊗[R] V.

standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, AddCommMonoid, Module, Group, Algebra, Representation, TensorProduct
Proof uses: MonoidHom.mk, LinearMap, RingHom.id, OneHom.mk, LinearMap.baseChange
31 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
noncomputable def baseChange (R' : Type*) [CommRing R'] [Algebra R R'] (ρ : Representation R G V)
    : Representation R' G (R' ⊗[R] V) where
…
```