Constructions.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

Constructions

Material destined for Mathlib.

Imports (2)

Imported by 4 files

theorem DenseRange.codRestrict_comp line 19 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, DenseRange, Continuous, Set.range, Membership.mem, Set, Function.comp, Set.codRestrict
Proof uses: DenseRange.comp, Function.Surjective.denseRange, Set.mem_range_self, Set.codRestrict_range_surjective, Continuous.codRestrict
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Continuous.piSemialgHomPi line 24 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, RingHom, Semiring, Algebra, TopologicalSpace, SemialgHom, Continuous, Pi.semialgHomPi
Proof uses: id, continuous_pi, Continuous.comp', continuous_apply
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma continuous_of_discreteTopology_snd A map on X × D with D discrete is continuous as soon as all its slices x ↦ g (x, d) are continuous line 38 source
Source documentation
A map on X × D with D discrete is continuous as soon as all its slices x ↦ g (x, d) are continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: TopologicalSpace, DiscreteTopology, Prod, Continuous, Prod.mk
Proof uses: Eq.mpr, ContinuousAt, id, Eq, Filter.Tendsto, nhds, SProd.sprod, Filter, congrArg, nhds_prod_eq, Pure.pure, nhds_discrete, Filter.map, Filter.prod_pure, Iff.mp, Function.comp, Filter.tendsto_map'_iff, Continuous.tendsto
6 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma continuous_of_discreteTopology_snd {X D Y : Type*} [TopologicalSpace X]
    [TopologicalSpace D] [DiscreteTopology D] [TopologicalSpace Y] {g : X × D → Y}
    (hg : ∀ d, Continuous fun x ↦ g (x, d)) : Continuous g := by
…
```