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 981 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 Carleson blueprint

3 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.Order.ConditionallyCompleteLattice.Indexed

Imported by 3 files

theorem ciSup_le_ciSup line 9 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nonempty, ConditionallyCompleteLattice, Exists, LE.le, BddAbove, Set.range, iSup
Proof uses: ciSup_le, le_ciSup_of_le
13 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem ciSup_eq_ciSup currently unused line 18 source
Source comment
currently unused

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ConditionallyCompleteLinearOrder, Exists, LE.le, Eq, iSup
Proof uses: dite, Nonempty, Classical.propDecidable, BddAbove, Set.range, le_antisymm, ciSup_le_ciSup, Not, False.elim, Membership.mem, Set, False, Eq.mp, congrArg, propext, bddAbove_def, And, LT.lt, not_bddAbove_iff, And.left and 15 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 47 steps
Open to load the checked before-and-after goal states.
```
theorem ciSup_eq_ciSup {α : Type*} {ι ι' : Sort*} [ConditionallyCompleteLinearOrder α]
  {f : ι → α} {g : ι' → α} (h₀ : ∀ i, ∃ j, f i ≤ g j) (h₁ : ∀ j, ∃ i, g j ≤ f i) :
    ⨆ i, f i = ⨆ j, g j := by
…
```
Mentions: g
theorem WithTop.iSup_coe_eq_top' line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ConditionallyCompleteLinearOrderBot, NoTopOrder, Eq, WithTop, iSup, Top.top
Proof uses: Eq.mpr, Not, BddAbove, Set.range, id, congrArg, propext, WithTop.iSup_coe_eq_top, of_eq_true, Eq.trans, False, True, Set.univ, Set, Set.range_id', not_false_eq_true
23 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.