Hom.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

8 declarations, 0 contain sorry, 0 declared as axioms

Algebra structures on equivariant homs

Imports (2)

Imported by 2 files

instance (instance at line 17) line 17 source
instance (instance at line 22) line 22 source
def MulActionHom.compLeftAlgHom Composing MulActionHom on the left as an AlgHom line 35 source
Source documentation
Composing MulActionHom on the left as an AlgHom.

standard axioms: Quot.sound, propext what this means
Statement uses: Monoid, MulAction, Semiring, MulSemiringAction, CommSemiring, Algebra, SMulCommClass, MulActionHom, id, AlgHom
Proof uses: AlgHom.mk, RingHom.mk, MonoidHom.mk, OneHom.mk, MulActionHom.comp
44 instances, coercions and auxiliary constants left out.
```
def MulActionHom.compLeftAlgHom (G R Y : Type*) {X X' : Type*}
    [Monoid G] [MulAction G X] [MulAction G X'] [Semiring Y] [MulSemiringAction G Y]
    [CommSemiring R] [Algebra R Y] [SMulCommClass G R Y] (f : X →[G] X') :
    (X' →[G] Y) →ₐ[R] (X →[G] Y) where
…
```
def MulActionHom.toFunAlgHom The coercion from MulActionHom to functions as an AlgHom line 47 source
Source documentation
The coercion from MulActionHom to functions as an AlgHom.

standard axioms: Quot.sound, propext what this means
Statement uses: Monoid, MulAction, Semiring, MulSemiringAction, CommSemiring, Algebra, SMulCommClass, AlgHom, MulActionHom, id
Proof uses: AlgHom.mk, RingHom.mk, MonoidHom.mk, OneHom.mk
49 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
def MulActionHom.toFunAlgHom
    {G H X Y : Type*} [Monoid G] [MulAction G X] [Semiring Y] [MulSemiringAction G Y]
    [CommSemiring H] [Algebra H Y] [SMulCommClass G H Y] :
    (X →[G] Y) →ₐ[H] (X → Y) where
…
```
instance (instance at line 58) line 58 source
def MulActionHom.evalAlgHom Evaluating a MulActionHom is an AlgHom, bundled as a MulActionHom line 65 source
Source documentation
Evaluating a MulActionHom is an AlgHom, bundled as a MulActionHom.

standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, Semiring, Algebra, Monoid, MulAction, MulSemiringAction, SMulCommClass, MulActionHom, id, AlgHom
Proof uses: MulActionHom.mk, AlgHom.comp, Pi.evalAlgHom, MulActionHom.toFunAlgHom
39 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def MulActionHom.evalAlgHom (G R X Y : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y]
    [Monoid G] [MulAction G X] [MulSemiringAction G Y] [SMulCommClass G R Y] :
    X →[G] ((X →[G] Y) →ₐ[R] Y) where
…
```
lemma MulActionHom.evalAlgHom_apply line 72 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, Semiring, Algebra, Monoid, MulAction, MulSemiringAction, SMulCommClass, MulActionHom, id, Eq, AlgHom, MulActionHom.evalAlgHom
Proof uses: rfl
42 instances, coercions and auxiliary constants left out.
def AlgHom.evalMulActionHom Evaluating a AlgHom is an MulActionHom, bundled as a AlgHom line 77 source
Source documentation
Evaluating a AlgHom is an MulActionHom, bundled as a AlgHom.

standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, Semiring, Algebra, Monoid, MulSemiringAction, SMulCommClass, AlgHom, MulActionHom, id
Proof uses: AlgHom.mk, RingHom.mk, MonoidHom.mk, OneHom.mk, MulActionHom.mk
48 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
def AlgHom.evalMulActionHom (G R A Y : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y]
    [Monoid G] [Semiring A] [Algebra R A] [MulSemiringAction G Y] [SMulCommClass G R Y] :
    A →ₐ[R] ((A →ₐ[R] Y) →[G] Y) where
…
```