Poincare.lean`

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

12 declarations, 9 contain sorry, 0 declared as axioms

The Poincaré Conjecture

References:

The formalisations in this file are based on the ones written by Junyan Xu in Mathlib4.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def PoincareConjecture.ConjectureFor The predicate that the generalized Poincaré conjecture holds in dimension line 45 source
Source documentation
The predicate that the generalized Poincaré conjecture holds in dimension $n$, i.e. that any $n$-dimensional manifold that is homotopy equivalent to the sphere is in fact homeomorphic to the sphere.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: TopologicalSpace, T2Space, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, ContinuousMap.HomotopyEquiv, HAdd.hAdd, Metric.sphere, Membership.mem, Set, Nonempty, Homeomorph
29 instances, coercions and auxiliary constants left out.
```
def ConjectureFor (n : ℕ) : Prop :=
…
```
theorem PoincareConjecture.poincare_conjecture The Millennium Problem, solved by Grigori Perelman in 2003: the Poincaré Conjecture holds sorry line 52 source
Source documentation
The Millennium Problem, solved by Grigori Perelman in 2003: the Poincaré Conjecture holds.

depends on sorry what this means
Statement uses: PoincareConjecture.ConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture : ConjectureFor 3 := by
…
```
Mentions: PoincareConjecture.ConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.dimension_two The Generalized Poincaré Conjecture holds for surfaces sorry line 59 source
Source documentation
The Generalized Poincaré Conjecture holds for surfaces.

depends on sorry what this means
Statement uses: PoincareConjecture.ConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.dimension_two : ConjectureFor 2 := by
…
```
Mentions: PoincareConjecture.ConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.dimension_ge_five The Generalized Poincaré Conjecture holds for dimensions at least 5 sorry line 66 source
Source documentation
The Generalized Poincaré Conjecture holds for dimensions at least 5.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, PoincareConjecture.ConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.dimension_ge_five (n : ℕ) (hn : 5 ≤ n) : ConjectureFor n := by
…
```
Mentions: PoincareConjecture.ConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.dimension_four The Generalized Poincaré Conjecture holds in dimension 4 sorry line 73 source
Source documentation
The Generalized Poincaré Conjecture holds in dimension 4.

depends on sorry what this means
Statement uses: PoincareConjecture.ConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.dimension_four : ConjectureFor 4 := by
…
```
Mentions: PoincareConjecture.ConjectureFor
def PoincareConjecture.SmoothConjectureFor The predicate that the smooth Poincaré conjecture holds in dimension line 81 source
Source documentation
The predicate that the smooth Poincaré conjecture holds in dimension $n$, i.e. that any smooth $n$-dimensional manifold that is homotopy equivalent to the sphere is in fact diffeomorphic to the sphere. As in ConjectureFor, the manifold must be Hausdorff: ChartedSpace and IsManifold do not imply this, and in every positive dimension there is a non-Hausdorff smooth manifold that is homotopy equivalent to the sphere.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: TopologicalSpace, T2Space, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, IsManifold, modelWithCornersSelf, ENat, Top.top, ContinuousMap.HomotopyEquiv, HAdd.hAdd, Metric.sphere, Membership.mem, Set, Nonempty, Diffeomorph
48 instances, coercions and auxiliary constants left out.
```
def SmoothConjectureFor (n : ℕ) : Prop :=
…
```
theorem PoincareConjecture.poincare_conjecture.variants.smooth_for_three A reformulation of the Millennium Problem in terms of smooth 3-folds sorry line 87 source
Source documentation
A reformulation of the Millennium Problem in terms of smooth 3-folds.

depends on sorry what this means
Statement uses: PoincareConjecture.SmoothConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.smooth_for_three : SmoothConjectureFor 3 := by
…
```
Mentions: PoincareConjecture.SmoothConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.smooth_implication The smooth formulation of the Millennium Problem implies the general case sorry line 93 source
Source documentation
The smooth formulation of the Millennium Problem implies the general case. This follows from the fact that every topological 3-fold admits a smooth structure [mo296171].

depends on sorry what this means
Statement uses: PoincareConjecture.SmoothConjectureFor, OfNat.ofNat, Nat, PoincareConjecture.ConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.smooth_implication (H : SmoothConjectureFor 3) :
    ConjectureFor 3 := by
…
```
Mentions: PoincareConjecture.SmoothConjectureFor, PoincareConjecture.ConjectureFor
def PoincareConjecture.SmoothTrueValues The values at which the smooth version of the conjecture is known to hold line 98 source
Source documentation
The values at which the smooth version of the conjecture is known to hold.

no axioms what this means
Statement uses: Set, Nat
Proof uses: Insert.insert, OfNat.ofNat, Singleton.singleton
3 instances, coercions and auxiliary constants left out.
```
def SmoothTrueValues : Set ℕ := {1, 2, 3, 5, 6, 12, 56, 61}
…
```
theorem PoincareConjecture.poincare_conjecture.variants.smooth_known_cases The smooth version of the Poincaré conjecture is known to hold in dimensions $1, 2, 3, 5, 6, 12, 56, 61$ sorry line 103 source
Source documentation
The smooth version of the Poincaré conjecture is known to hold in dimensions $1, 2, 3, 5, 6, 12, 56, 61$. See [Wang2017].

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, PoincareConjecture.SmoothTrueValues, PoincareConjecture.SmoothConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.smooth_known_cases (n : ℕ) (hn : n ∈ SmoothTrueValues) :
    SmoothConjectureFor n := by
…
```
Mentions: PoincareConjecture.SmoothTrueValues, PoincareConjecture.SmoothConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.smooth_dimension_four The four dimensional case of the smooth version of the conjecture is still open sorry line 110 source
Source documentation
The four dimensional case of the smooth version of the conjecture is still open. See [Wang2017].

depends on sorry what this means
Statement uses: PoincareConjecture.SmoothConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.smooth_dimension_four : SmoothConjectureFor 4 := by
…
```
Mentions: PoincareConjecture.SmoothConjectureFor
theorem PoincareConjecture.poincare_conjecture.variants.smooth_other_cases It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$ sorry line 116 source
Source documentation
It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Not, Membership.mem, Set, PoincareConjecture.SmoothTrueValues, PoincareConjecture.SmoothConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem poincare_conjecture.variants.smooth_other_cases (n : ℕ) (hn : n > 4)
    (hn' : n ∉ SmoothTrueValues) : ¬ SmoothConjectureFor n := by
…
```
Mentions: PoincareConjecture.SmoothTrueValues, PoincareConjecture.SmoothConjectureFor