EquationalTheories_677_255.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

7 declarations, 2 contain sorry, 0 declared as axioms

Equational Theories

Reference: Equational Theories project site

Imports (1)

FormalConjecturesUtil

class EquationalTheories_677_255.Magma line 30 source
no axioms what this means
abbrev EquationalTheories_677_255.Equation255 line 35 source
no axioms what this means
Statement uses: EquationalTheories_677_255.Magma
Proof uses: Eq, EquationalTheories_677_255.Magma.op
abbrev EquationalTheories_677_255.Equation677 line 37 source
no axioms what this means
Statement uses: EquationalTheories_677_255.Magma
Proof uses: Eq, EquationalTheories_677_255.Magma.op
theorem EquationalTheories_677_255.Equation255_not_implies_Equation677 Equation 255 does not imply Equation 677 line 41 source
Source documentation
Equation 255 does not imply Equation 677.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, EquationalTheories_677_255.Magma, And, EquationalTheories_677_255.Equation255, Not, EquationalTheories_677_255.Equation677
Proof uses: Exists.intro, Fin, OfNat.ofNat, Nat, EquationalTheories_677_255.Magma.mk, Matrix.vecCons, Matrix.vecEmpty, And.intro, List, List.Mem, Eq, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete, EquationalTheories_677_255.Magma.op, List.cons, List.Mem.head and 15 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem Equation255_not_implies_Equation677 :
    ∃ (G : Type) (_ : Magma G), Equation255 G ∧ ¬ Equation677 G :=
…
```
Mentions: EquationalTheories_677_255.Magma, EquationalTheories_677_255.Equation255, EquationalTheories_677_255.Equation677
theorem EquationalTheories_677_255.Equation677_not_implies_Equation255 Equation 677 does not imply Equation 255 sorry line 48 source
Source documentation
Equation 677 does not imply Equation 255.

depends on sorry what this means
Statement uses: Exists, EquationalTheories_677_255.Magma, And, EquationalTheories_677_255.Equation677, Not, EquationalTheories_677_255.Equation255
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Equation677_not_implies_Equation255 :
    ∃ (G : Type) (_ : Magma G), Equation677 G ∧ ¬ Equation255 G := by
…
```
Mentions: EquationalTheories_677_255.Magma, EquationalTheories_677_255.Equation677, EquationalTheories_677_255.Equation255
theorem EquationalTheories_677_255.Finite.Equation255_not_implies_Equation677 Note that this is a stronger form of Equation255_not_implies_Equation677 line 54 source
Source documentation
Note that this is a stronger form of Equation255_not_implies_Equation677.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, EquationalTheories_677_255.Magma, And, Finite, EquationalTheories_677_255.Equation255, Not, EquationalTheories_677_255.Equation677
Proof uses: Exists.intro, Fin, OfNat.ofNat, Nat, EquationalTheories_677_255.Magma.mk, Matrix.vecCons, Matrix.vecEmpty, And.intro, Finite.intro, Fintype.card, Fintype.equivFin, List, List.Mem, Eq, List.finRange, HEq, Membership.mem, Finset, Fintype.elems, Fintype.complete and 18 more
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem Finite.Equation255_not_implies_Equation677 :
    ∃ (G : Type) (_ : Magma G), Finite G ∧ Equation255 G ∧ ¬ Equation677 G :=
…
```
Mentions: EquationalTheories_677_255.Magma, EquationalTheories_677_255.Equation255, EquationalTheories_677_255.Equation677
theorem EquationalTheories_677_255.Finite.Equation677_implies_Equation255 Equational Theories, Problem 8.1 sorry line 67 source
Source documentation
Equational Theories, Problem 8.1. Does Equation 677 imply Equation 255 in every finite magma? The project tentatively conjectures that the answer is no; a false answer is equivalent to the existence of a finite countermodel satisfying Equation 677 but not Equation 255.
Discussion thread: https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255

depends on sorry what this means
Statement uses: Iff, True, EquationalTheories_677_255.Magma, Finite, EquationalTheories_677_255.Equation677, EquationalTheories_677_255.Equation255
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem Finite.Equation677_implies_Equation255 :
    answer(sorry) ↔ ∀ (G : Type) (_ : Magma G), Finite G → Equation677 G → Equation255 G := by
…
```
Mentions: EquationalTheories_677_255.Magma, EquationalTheories_677_255.Equation677, EquationalTheories_677_255.Equation255