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 10 curated AI-written annotations and 7190 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, Formal Conjectures paper, arXiv v1

9 declarations, 0 contain sorry, 0 declared as axioms

Right ideals

This file gives the basic API for right ideals.

Imports (2)

Imported by 2 files

theorem RightIdeal.ext line 38 source
standard axioms: Quot.sound, propext what this means
Statement uses: Semiring, RightIdeal, Iff, Membership.mem, MulOpposite, Eq
Proof uses: Submodule.ext
10 instances, coercions and auxiliary constants left out.
theorem RightIdeal.mul_mem_right A right ideal is closed under multiplication on the right line 42 source
Source documentation
A right ideal is closed under multiplication on the right.

standard axioms: Quot.sound, propext what this means
Statement uses: Semiring, RightIdeal, Membership.mem, MulOpposite, HMul.hMul
Proof uses: Submodule.smul_mem, MulOpposite.op
13 instances, coercions and auxiliary constants left out.
```
theorem mul_mem_right (I : RightIdeal R) {a : R} (b : R) (ha : a ∈ I) : a * b ∈ I :=
…
```
Mentions: RightIdeal
def RightIdeal.toIdeal A right ideal in a commutative semiring, regarded as an ideal line 52 source
Source documentation
A right ideal in a commutative semiring, regarded as an ideal.

standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, RightIdeal, Ideal
Proof uses: Submodule.mk, AddSubmonoid.mk, AddSubsemigroup.mk, MulOpposite
15 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def toIdeal (I : RightIdeal R) : Ideal R where
…
```
Mentions: RightIdeal
theorem RightIdeal.mem_toIdeal line 62 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, RightIdeal, Iff, Membership.mem, Ideal, RightIdeal.toIdeal, MulOpposite
Proof uses: Iff.rfl
12 instances, coercions and auxiliary constants left out.
def RightIdeal._root_.Ideal.toRightIdeal An ideal in a commutative semiring, regarded as a right ideal line 66 source
Source documentation
An ideal in a commutative semiring, regarded as a right ideal.
```
def _root_.Ideal.toRightIdeal (I : Ideal R) : RightIdeal R where
…
```
Mentions: RightIdeal
theorem RightIdeal._root_.Ideal.mem_toRightIdeal line 76 source
theorem RightIdeal.toIdeal_toRightIdeal line 81 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, RightIdeal, Eq, Ideal.toRightIdeal, RightIdeal.toIdeal
Proof uses: rfl
2 instances, coercions and auxiliary constants left out.
theorem RightIdeal._root_.Ideal.toRightIdeal_toIdeal line 85 source
def RightIdeal.orderIsoIdeal Over a commutative semiring, right ideals and ideals are order isomorphic line 89 source
Source documentation
Over a commutative semiring, right ideals and ideals are order isomorphic.

standard axioms: Quot.sound, propext what this means
Statement uses: CommSemiring, OrderIso, RightIdeal, Ideal, MulOpposite
Proof uses: RelIso.mk, LE.le, Equiv.mk, RightIdeal.toIdeal, Ideal.toRightIdeal, RightIdeal.toIdeal_toRightIdeal, Ideal.toRightIdeal_toIdeal
17 instances, coercions and auxiliary constants left out.
```
def orderIsoIdeal (R : Type u) [CommSemiring R] : RightIdeal R ≃o Ideal R where
…
```
Mentions: RightIdeal