Real.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

15 declarations, 0 contain sorry, 0 declared as axioms

Expansions of the real field

This file makes ℝ a structure for the language of rings and for the language of orders. It defines the language Language.exp with a single unary function symbol exp, the language Language.orderedExpField of ordered exponential fields, and the real exponential field (ℝ, +, *, -, 0, 1, ≤, exp) as a structure for that language.

Main declarations

FirstOrder.Language.exp: the language with one unary function symbol exp.
FirstOrder.Language.orderedExpField: the language of ordered exponential fields.

Implementation notes

Mathlib deliberately does not make Ring.compatibleRingOfRing or orderStructure instances, because for a general type R the round trip Ring R → Language.ring.Structure R → Ring R need not commute definitionally. Here they are declared as instances for the single type ℝ. Mathlib has no instance producing a Ring or LE structure from a first-order structure on a concrete type, so no such round trip arises.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

inductive FirstOrder.expFunc The type of function symbols of Language.exp: a single unary symbol exp line 48 source
Source documentation
The type of function symbols of Language.exp: a single unary symbol exp.

no axioms what this means
Statement uses: Nat
```
inductive expFunc : ℕ → Type
  | exp : expFunc 1
  deriving DecidableEq

namespace Language

/-- The language with a single unary function symbol `exp` and no relation symbols. -/
```
def FirstOrder.Language.exp The language with a single unary function symbol exp and no relation symbols line 55 source
Source documentation
The language with a single unary function symbol exp and no relation symbols.

no axioms what this means
Statement uses: FirstOrder.Language
Proof uses: FirstOrder.Language.mk, FirstOrder.expFunc, Nat, Empty
```
protected def exp : Language := ⟨expFunc, fun _ => Empty⟩
…
```
Mentions: FirstOrder.expFunc
abbrev FirstOrder.Language.orderedExpField The language of ordered exponential fields: the language of rings, together with a unary function symbol exp and the order relation ≤ line 60 source
Source documentation
The language of ordered exponential fields: the language of rings, together with a unary function symbol exp and the order relation ≤.

no axioms what this means
Statement uses: FirstOrder.Language
Proof uses: FirstOrder.Language.sum, FirstOrder.Language.ring, FirstOrder.Language.exp, FirstOrder.Language.order
```
protected abbrev orderedExpField : Language :=
…
```
instance FirstOrder.Language.orderedExpField.instIsOrdered The language of ordered exponential fields contains the order symbol ≤ line 64 source
Source documentation
The language of ordered exponential fields contains the order symbol ≤.

no axioms what this means
Statement uses: FirstOrder.Language.IsOrdered, FirstOrder.Language.orderedExpField
Proof uses: FirstOrder.Language.IsOrdered.mk, Sum.inr, FirstOrder.Language.Relations, FirstOrder.Language.ring, OfNat.ofNat, Nat, FirstOrder.Language.sum, FirstOrder.Language.exp, FirstOrder.Language.order, FirstOrder.Language.IsOrdered.leSymb
2 instances, coercions and auxiliary constants left out.
```
instance orderedExpField.instIsOrdered : Language.orderedExpField.IsOrdered :=
…
```
instance FirstOrder.Language.exp.encodableFunctions The function symbol exp is encoded as 0 line 68 source
Source documentation
The function symbol exp is encoded as 0.

standard axioms: Quot.sound, propext what this means
Statement uses: Encodable, Sigma, Nat, FirstOrder.Language.Functions, FirstOrder.Language.exp
Proof uses: Encodable.ofLeftInjection, OfNat.ofNat, Option.some, Sigma.mk, FirstOrder.expFunc.exp
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
instance exp.encodableFunctions : Encodable (Σ n, Language.exp.Functions n) :=
…
```
instance FirstOrder.Language.compatibleRingReal ℝ as a structure for the language of rings, with the usual ring operations line 77 source
Source documentation
ℝ as a structure for the language of rings, with the usual ring operations.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Ring.CompatibleRing, Real
Proof uses: FirstOrder.Ring.compatibleRingOfRing
10 instances, coercions and auxiliary constants left out.
```
instance compatibleRingReal : Ring.CompatibleRing ℝ := Ring.compatibleRingOfRing ℝ
```
instance FirstOrder.Language.orderStructureReal ℝ as a structure for the language of orders, with ≤ interpreted as ≤ line 80 source
Source documentation
ℝ as a structure for the language of orders, with ≤ interpreted as ≤.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Language.Structure, FirstOrder.Language.order, Real
Proof uses: FirstOrder.Language.orderStructure
1 instances, coercions and auxiliary constants left out.
```
instance orderStructureReal : Language.order.Structure ℝ := orderStructure ℝ
```
instance FirstOrder.Language.orderOrderedStructureReal line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Language.OrderedStructure, FirstOrder.Language.order, Real
Proof uses: FirstOrder.Language.OrderedStructure.mk, Fin, OfNat.ofNat, Nat, Iff.rfl, FirstOrder.Language.Structure.RelMap, FirstOrder.Language.IsOrdered.leSymb
7 instances, coercions and auxiliary constants left out.
instance FirstOrder.Language.expStructureReal ℝ as a structure for Language.exp, with exp interpreted as the exponential function line 85 source
Source documentation
ℝ as a structure for Language.exp, with exp interpreted as the exponential function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Language.Structure, FirstOrder.Language.exp, Real
Proof uses: FirstOrder.Language.Structure.mk, Nat, FirstOrder.Language.Functions, FirstOrder.Language.expStructureReal.match_1, Fin, Unit, OfNat.ofNat, Real.exp, FirstOrder.Language.Relations, isEmptyElim
4 instances, coercions and auxiliary constants left out.
```
instance expStructureReal : Language.exp.Structure ℝ where
…
```
theorem FirstOrder.Language.funMap_exp line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fin, OfNat.ofNat, Nat, Real, Eq, FirstOrder.Language.Structure.funMap, FirstOrder.Language.exp, FirstOrder.expFunc.exp, Real.exp
Proof uses: rfl
6 instances, coercions and auxiliary constants left out.
instance FirstOrder.Language.orderedExpFieldOrderedStructureReal In the real exponential field, ≤ is interpreted as ≤ line 95 source
Source documentation
In the real exponential field, ≤ is interpreted as ≤.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Language.OrderedStructure, FirstOrder.Language.orderedExpField, Real, FirstOrder.Language.ring, FirstOrder.Language.sum, FirstOrder.Language.exp, FirstOrder.Language.order
Proof uses: FirstOrder.Language.OrderedStructure.mk, Fin, OfNat.ofNat, Nat, Iff.rfl, FirstOrder.Language.Structure.RelMap, FirstOrder.Language.IsOrdered.leSymb
25 instances, coercions and auxiliary constants left out.
```
instance orderedExpFieldOrderedStructureReal : Language.orderedExpField.OrderedStructure ℝ :=
…
```
example FirstOrder.Language.(example at line 99) line 99 source
Source comment
In the real exponential field, the symbol `exp` is interpreted as the exponential function.
```
example (x : ℝ) :
    Structure.funMap (L := Language.orderedExpField) (Sum.inr (Sum.inl expFunc.exp)) ![x] =
…
```
Mentions: FirstOrder.Language.orderedExpField
example FirstOrder.Language.(example at line 105) line 105 source
Source comment
In the real exponential field, the symbol `+` is interpreted as addition.
```
example (x y : ℝ) :
    Structure.funMap (L := Language.orderedExpField) (Sum.inl Ring.addFunc) ![x, y] = x + y :=
…
```
Mentions: FirstOrder.Language.orderedExpField
example FirstOrder.Language.(example at line 110) line 110 source
Source comment
In the real exponential field, the symbol `≤` is interpreted as the order of `ℝ`.
```
example (x y : ℝ) :
    Structure.RelMap (L := Language.orderedExpField) leSymb ![x, y] ↔ x ≤ y :=
…
```
Mentions: FirstOrder.Language.orderedExpField
example FirstOrder.Language.(example at line 115) line 115 source
Source comment
The Gödel numbering of sentences of `Language.orderedExpField` is found by instance resolution.
```
example : Encodable Language.orderedExpField.Sentence := inferInstance
…
```