Tournamenton.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 0 contain sorry, 0 declared as axioms

Imports (7)

Imported by 1 file

FormalConjecturesForMathlib.lean

structure LimitObjects.Tournamenton A tournamenton is a measurable function W : [0, 1] × [0, 1] → [0, 1] satisfying the tournament condition W(x, y) + W(y, x) = 1 line 36 source
Source documentation
A tournamenton is a measurable function W : [0, 1] × [0, 1] → [0, 1] satisfying the tournament condition W(x, y) + W(y, x) = 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
```
structure Tournamenton where
…
```
instance LimitObjects.(instance at line 41) line 41 source
theorem LimitObjects.Tournamenton.ext Extensionality principle for Tournamenton line 47 source
Source documentation
Extensionality principle for Tournamenton.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LimitObjects.Tournamenton, Prod, LimitObjects.UnitInterval, Eq, LimitObjects.Tournamenton.toFun
Proof uses: Measurable, Real, Membership.mem, Set, Set.Icc, OfNat.ofNat, HAdd.hAdd, Prod.mk, LimitObjects.Tournamenton.mk, Eq.rec, Eq.refl, funext, Eq.symm
14 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem ext {W₁ W₂ : Tournamenton} (h : ∀ p, W₁ p = W₂ p) : W₁ = W₂ := by
…
```
Mentions: LimitObjects.Tournamenton
def LimitObjects.digraphEdgeFinset The directed edge finset of a finite digraph line 55 source
Source documentation
The directed edge finset of a finite digraph.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Digraph, DecidableRel, Digraph.Adj, Finset, Prod
Proof uses: Finset.filter, Prod.fst, Prod.snd, Finset.univ
1 instances, coercions and auxiliary constants left out.
```
def digraphEdgeFinset (D : Digraph V) [DecidableRel D.Adj] : Finset (V × V) :=
…
```
def LimitObjects.tournamentonHomIntegrand The integrand for the homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ given a configuration x : V → [0, 1] line 62 source
Source documentation
The integrand for the homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ given a configuration x : V → [0, 1].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Digraph, DecidableRel, Digraph.Adj, LimitObjects.Tournamenton, LimitObjects.UnitInterval, Real
Proof uses: Finset.prod, Prod, LimitObjects.digraphEdgeFinset, Membership.mem, Set, Set.Icc, OfNat.ofNat, LimitObjects.Tournamenton.toFun, Prod.mk, Prod.fst, Prod.snd
8 instances, coercions and auxiliary constants left out.
```
def tournamentonHomIntegrand (D : Digraph V) [DecidableRel D.Adj]
    (W : Tournamenton) (x : V → UnitInterval) : ℝ :=
…
```
Mentions: LimitObjects.Tournamenton, LimitObjects.UnitInterval
def LimitObjects.tournamentonHomDensity The homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ on $[0, 1]$ equipped with the Lebesgue measure volume line 70 source
Source documentation
The homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ on $[0, 1]$ equipped with the Lebesgue measure volume.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Digraph, DecidableRel, Digraph.Adj, LimitObjects.Tournamenton, Real
Proof uses: MeasureTheory.integral, LimitObjects.UnitInterval, MeasureTheory.Measure.pi, MeasureTheory.MeasureSpace.volume, LimitObjects.tournamentonHomIntegrand
8 instances, coercions and auxiliary constants left out.
```
noncomputable def tournamentonHomDensity (D : Digraph V) [DecidableRel D.Adj]
    (W : Tournamenton) : ℝ :=
…
```
Mentions: LimitObjects.Tournamenton