Tournament.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, 0 contain sorry, 0 declared as axioms

Imports (6)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Digraph.IsTournament A tournament is a directed graph in which every pair of distinct vertices is connected by exactly one directed edge line 35 source
Source documentation
A tournament is a directed graph in which every pair of distinct vertices is connected by exactly one directed edge.

no axioms what this means
Statement uses: Digraph
Proof uses: And, Not, Digraph.Adj, Ne, Iff
```
def IsTournament (G : Digraph V) : Prop :=
…
```
def Digraph.IsOrientation An orientation of an undirected simple graph H is a digraph D whose edges are obtained by directing each edge of H in exactly one direction line 42 source
Source documentation
An orientation of an undirected simple graph H is a digraph D whose edges are obtained by directing each edge of H in exactly one direction.

no axioms what this means
Statement uses: Digraph, SimpleGraph
Proof uses: And, Digraph.Adj, SimpleGraph.Adj, Iff, Not
```
def IsOrientation (D : Digraph V) (H : SimpleGraph V) : Prop :=
…
```
structure Digraph.Hom The type of digraph homomorphisms D →g D' line 52 source
Source documentation
The type of digraph homomorphisms D →g D'. A function f : V → W is a digraph homomorphism if D.Adj u v → D'.Adj (f u) (f v). Note that f is a general mapping and is not required to be injective.

no axioms what this means
Statement uses: Digraph

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
structure Hom (D : Digraph V) (D' : Digraph W) where
…
```
instance Digraph.(instance at line 58) line 58 source
instance Digraph.(instance at line 61) line 61 source
def Digraph.homCount The number of homomorphisms from a digraph D to a digraph D' line 68 source
Source documentation
The number of homomorphisms from a digraph D to a digraph D'.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Digraph, DecidableRel, Digraph.Adj, Nat
Proof uses: Fintype.card, Digraph.Hom
1 instances, coercions and auxiliary constants left out.
```
noncomputable def homCount (D : Digraph V) (D' : Digraph W) [DecidableRel D.Adj] [DecidableRel D'.Adj] : ℕ :=
…
```
def Digraph.homDensity The homomorphism density $t_D(D')$ of a digraph $D$ in a digraph $D'$ line 75 source
Source documentation
The homomorphism density $t_D(D')$ of a digraph $D$ in a digraph $D'$. It is defined as #Hom(D, D') / |W|^{|V|}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Digraph, DecidableRel, Digraph.Adj, Real
Proof uses: HDiv.hDiv, Digraph.homCount, HPow.hPow, Nat, Fintype.card
9 instances, coercions and auxiliary constants left out.
```
noncomputable def homDensity (D : Digraph V) (D' : Digraph W)
    [DecidableRel D.Adj] [DecidableRel D'.Adj] : ℝ :=
…
```