Lean statement

Area-controlled end replacement and Schwarzschild equality examples

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorems selected for verification: OAI.CKSSourceExterior.area_controlled_end_replacement OAI.CKSMain.schwarzschild_equality_examples

Lean statement file: CKSBondiPenrose.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 260 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem area_controlled_end_replacement    (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N))    (hK : ContMDiff OAI.CKSGeometricCuts.I3 (OAI.CKSGeometricCuts.I3.prod 𝓘(ℝ,OAI.CKSLorentz.SpatialBilinear)) ∞ (OAI.CKSMetricGluing.innerSection OAI.CKSGeometricCuts.I3 K))    (hKsym : ∀ x v w, K x v w = K x w v)    (hOrient : OAI.CKSSourceExterior.Orientable (N := N))    (hScompact : IsCompact (OAI.CKSGeometricCuts.I3.boundary N)) (hSnonempty : (OAI.CKSGeometricCuts.I3.boundary N).Nonempty)    (hcomplete : OAI.CKSReplacementCompleteness.IsComplete OAI.CKSGeometricCuts.I3 g.toContinuousRiemannianMetric)    (hDEC : OAI.CKSSpatialManifold.PhysicalDEC OAI.CKSGeometricCuts.I3 g.inner K)    (d : OAI.CKSSourceExterior.CKSData g K)    (ht : ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖ < (OAI.CKSLorentz.bondiCharge d.massAspect).1) :    ∃ (d' : OAI.CKSSourceExterior.CKSData g K) (R₀ : ℝ) (ε : ℝ → ℝ),      OAI.CKSLorentz.bondiCharge d'.massAspect =        (Real.sqrt ((OAI.CKSLorentz.bondiCharge d.massAspect).1^2 - ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖^2),0) ∧      d'.chart.radius < R₀ ∧ 12 ≤ R₀ ∧ Tendsto ε atTop (𝓝 0) ∧      OAI.CKSFullCutArea.minEnclosingArea g < ⊤ ∧      ∀ R ≥ R₀, (0 ≤ ε R ∧ ε R < 1) ∧        ∃ (gR : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (kR : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N))          (G k : OAI.CKSLorentz.SpatialTensor) (η : ℝ),          ContMDiff OAI.CKSGeometricCuts.I3 (OAI.CKSGeometricCuts.I3.prod 𝓘(ℝ,OAI.CKSLorentz.SpatialBilinear)) ∞ (OAI.CKSMetricGluing.innerSection OAI.CKSGeometricCuts.I3 kR) ∧          (∀ x v w, kR x v w = kR x w v) ∧          (∀ x, x ∉ d'.chart.domain ∨ ‖d'.chart.coordinate x‖ ≤ R →            gR.inner x = g.inner x ∧ kR x = K x) ∧          OAI.CKSSpatialManifold.PhysicalDEC OAI.CKSGeometricCuts.I3 gR.inner kR ∧          (∀ x v, (1-ε R) * g.inner x v v ≤ gR.inner x v v) ∧          OAI.CKSReplacementCompleteness.IsComplete OAI.CKSGeometricCuts.I3 gR.toContinuousRiemannianMetric ∧          OAI.CKSSourceExterior.constraintsIntegrable gR kR ∧          (∀ x ∈ d'.chart.domain,            gR.inner x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 d'.chart.coordinate G x ∧            kR x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 d'.chart.coordinate k x) ∧          (∀ i j : Fin 3, OAI.CKSADM.Symbol 1            (fun x => OAI.CKSLorentz.spatialCartesian G x i j - (if i=j then 1 else 0))) ∧          (∀ x : OAI.CKSLorentz.E, 2*R^2 < ‖x‖ → k x = 0) ∧          Tendsto (OAI.CKSLorentz.spatialADMEnergy G) atTop            (𝓝 (Real.sqrt ((OAI.CKSLorentz.bondiCharge d.massAspect).1^2 - ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖^2) + η)) ∧          Tendsto (OAI.CKSLorentz.spatialADMMomentum G k) atTop (𝓝 0) ∧          0 ≤ η ∧ η ≤ 2 * R ^ (-(1/2 : ℝ)) ∧          OAI.CKSFullCutArea.minEnclosingArea gR < ⊤ ∧          (1-ε R) * OAI.CKSSourceExterior.minimumEnclosingArea g ≤ OAI.CKSSourceExterior.minimumEnclosingArea gR ∧          ∀ D : OAI.CKSGeometricCuts.OuterDomain N,            OAI.CKSFullCutArea.area g D < ⊤ ∧ OAI.CKSFullCutArea.area gR D < ⊤ ∧            (1-ε R) * OAI.CKSSourceExterior.cutArea g D ≤ OAI.CKSSourceExterior.cutArea gR D := by  sorry⋮theorem schwarzschild_equality_examples (m : ℝ) (hm : 0 < m) :    SchwarzschildExample.{v} m hm := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

instance OAI.CKSSpatialManifold.real_id_isometric line 72

local instance real_id_isometric : RingHomIsometric (RingHom.id ℝ) := inferInstance

instance OAI.CKSMetricGluing.real_continuousAdd line 86

local instance real_continuousAdd : ContinuousAdd ℝ := inferInstance

instance OAI.CKSMetricGluing.real_continuousConstSMul line 88

local instance real_continuousConstSMul : ContinuousConstSMul ℝ ℝ := inferInstance

instance OAI.CKSMetricGluing.real_smulCommClass line 90

local instance real_smulCommClass :    @SMulCommClass ℝ ℝ ℝ Algebra.toSMul      (@instSMulOfMul ℝ (@Distrib.toMul ℝ        (@instDistribOfSemiring ℝ (@CommSemiring.toSemiring ℝ Real.instCommSemiring)))) :=  inferInstance

abbreviation OAI.CKSMetricGluing.InnerField line 110

abbrev InnerField :=  letI := OAI.CKSMetricGluing.real_smulCommClass  letI := OAI.CKSMetricGluing.real_continuousConstSMul  ∀ x : M, TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ

abbreviation OAI.CKSMetricGluing.SmoothMetric line 115

abbrev SmoothMetric := ContMDiffRiemannianMetric I ∞ E (fun x : M => TangentSpace I x)

definition OAI.CKSMetricGluing.innerSection line 117

def innerSection (q : OAI.CKSMetricGluing.InnerField I (M := M)) :    M → TotalSpace (E →L[ℝ] E →L[ℝ] ℝ)      (fun x : M => TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ) :=  fun x => ⟨x,q x⟩

abbreviation OAI.CKSAngularGeometry.I line 131
```
abbrev I := Fin 2
```

definition OAI.CKSCalculus.D line 143

def D (e : E) (f : E → ℝ) (x : E) : ℝ := fderiv ℝ f x e

abbreviation OAI.CKSRealizedRound.Point line 234
```
abbrev Point := Fin 3 → ℝ
```
abbreviation OAI.CKSMixedGeometry.A line 298
```
abbrev A := Fin 2
```

abbreviation OAI.CKSMixedGeometry.Angle line 335

abbrev Angle := OAI.CKSMixedGeometry.A → ℝ

abbreviation OAI.CKSInducedSphere.Ix line 352
```
abbrev Ix := Fin 3
```
abbreviation OAI.CKSSphericalHarmonics.Ambient line 362
```
abbrev Ambient := EuclideanSpace ℝ (Fin 3)
```

abbreviation OAI.CKSSphericalHarmonics.Sphere line 364

abbrev Sphere := ↥(Metric.sphere (0 : OAI.CKSSphericalHarmonics.Ambient) 1)

definition OAI.CKSSphericalHarmonics.SmoothSphere line 389

def SmoothSphere (f : OAI.CKSSphericalHarmonics.Sphere → ℝ) : Prop := ContMDiff (𝓡 2) 𝓘(ℝ, ℝ) ∞ f

instance OAI.CKSLorentz.two_atLeastTwo line 399

local instance two_atLeastTwo : Nat.AtLeastTwo 2 := inferInstance

abbreviation OAI.CKSInducedSphere.E line 410

abbrev E := EuclideanSpace ℝ OAI.CKSInducedSphere.Ix

definition OAI.CKSInducedSphere.e line 423

def e (i : OAI.CKSInducedSphere.Ix) : OAI.CKSInducedSphere.E := WithLp.toLp 2 (Pi.single i 1)

abbreviation OAI.CKSLorentz.V line 438
```
abbrev V := ℝ × OAI.CKSLorentz.E
```

structure OAI.CKSLorentz.SourceCKSComponents line 479

structure SourceCKSComponents (j qrad : ℕ) (V : Set OAI.CKSMixedGeometry.Point)    (A : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Point → ℝ) (mr : OAI.CKSMixedGeometry.Angle → ℝ) (mg : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Angle → ℝ) : Prop where  radial : OAI.CKSMixedGeometry.SourceRemainderOn j qrad V (fun y => A 0 0 y-OAI.CKSLorentz.ordinaryLeadingComponent 5 mr y)  mixed : ∀ i k, (i=0 ∧ k≠0) ∨ (i≠0 ∧ k=0) → OAI.CKSMixedGeometry.SourceRemainderOn j 3 V (A i k)  angular : ∀ i k, i≠0 → k≠0 → OAI.CKSMixedGeometry.SourceRemainderOn j 2 V    (fun y => A i k y-OAI.CKSLorentz.ordinaryLeadingComponent 1 (mg i k) y)

definition OAI.CKSLorentz.angularRadialTail line 494

def angularRadialTail (R : ℝ) (W : Set OAI.CKSMixedGeometry.Angle) : Set OAI.CKSMixedGeometry.Point :=  letI := OAI.CKSLorentz.pointDimension_neZero  {y | R < y 0 ∧ OAI.CKSMixedGeometry.angularProjection y ∈ W}

structure OAI.CKSLorentz.SmoothAngularPatch line 507

structure SmoothAngularPatch where  chart : OpenPartialHomeomorph OAI.CKSLorentz.Sphere OAI.CKSMixedGeometry.Angle  param_smooth : ContDiffOn ℝ ∞ (fun θ => (chart.symm θ : OAI.CKSLorentz.E)) chart.target  extension : OAI.CKSLorentz.E → OAI.CKSMixedGeometry.Angle  domain : Set OAI.CKSLorentz.E  isOpen_domain : IsOpen domain  extension_smooth : ContDiffOn ℝ ∞ extension domain  source_domain : ∀ n ∈ chart.source, (n:OAI.CKSLorentz.E) ∈ domain  extension_eq : ∀ n ∈ chart.source, extension n = chart n

definition OAI.CKSLorentz.SmoothAngularPatch.param line 526

def param (P : OAI.CKSLorentz.SmoothAngularPatch) (θ : OAI.CKSMixedGeometry.Angle) : OAI.CKSLorentz.E := P.chart.symm θ

75 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: area, asymptotically, boundary, compact, condition, connected, controlled, data and five more.

Statements in Area-controlled end replacement and the Bondi Penrose inequality in the CKS class come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Area-controlled end replacement and the Bondi Penrose inequality in the CKS class

Theorem (Area-controlled end replacement)

Let (\Omega,g,K) be a smooth connected orientable three-dimensional exterior, complete including its nonempty compact smooth boundary, with exactly one end of class (eq:cks). Suppose DEC holds and E_B>|P_B|. There is a balanced chart such that, for every sufficiently large R, smooth data (G_R,k_R) on the same exterior have the following properties:

1.
They agree with (g,K) on the compact side of r=R and satisfy DEC.
2.
For numbers 0\leq\varepsilon_R<1 with \varepsilon_R\to0, G_R\geq(1-\varepsilon_R)g everywhere on \Omega.
3.
The end is asymptotically flat, with G_R-\delta=O_j(r^{-1}) for every fixed finite j, and k_R=0 sufficiently far out. The constraints are integrable and its charges satisfy P_R=0, E_R=m_B+\eta_R, where 0\leq\eta_R\leq2R^{-1/2}.

The new exterior is complete and A_{\min,G_R}(S)\geq(1-\varepsilon_R)A_{\min,g}(S). No condition on the boundary expansion is needed.

Section 1, Replacing the end and retaining the area Read it in the paper TeX source, line 128

Show the TeX

Let $(\Omega,g,K)$ be a smooth connected orientable three-dimensional
exterior, complete including its nonempty compact smooth boundary,
with exactly one end of class \eqref{eq:cks}. Suppose DEC holds and
$E_B>|P_B|$. There is a balanced chart such that, for every sufficiently
large $R$, smooth data $(G_R,k_R)$ on the same exterior have the following
properties:
\begin{enumerate}
\item They agree with $(g,K)$ on the compact side of $r=R$ and satisfy DEC.
\item For numbers $0\leq\varepsilon_R<1$ with $\varepsilon_R\to0$,
      $G_R\geq(1-\varepsilon_R)g$ everywhere on $\Omega$.
\item The end is asymptotically flat, with $G_R-\delta=O_j(r^{-1})$
      for every fixed finite $j$, and $k_R=0$ sufficiently far out.
      The constraints are integrable and its charges satisfy
      $P_R=0$, $E_R=m_B+\eta_R$, where $0\leq\eta_R\leq2R^{-1/2}$.
\end{enumerate}
The new exterior is complete and
$A_{\min,G_R}(S)\geq(1-\varepsilon_R)A_{\min,g}(S)$.
No condition on the boundary expansion is needed.

Other possible paper matches, closest wording first

Corollary 1.2 of Area-controlled end replacement and the Bondi Penrose inequality in the CKS class. Words of the Lean statement, its names split into words, that also appear in the theorem: cks, bondi, replacement, schwarzschild, regular, horizon and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: disconnected and inequality.
Theorem 6.1 of Area-controlled end replacement and the Bondi Penrose inequality in the CKS class. Words of the Lean statement, its names split into words, that also appear in the theorem: cartesian, dec, across, complement, three, constraints and 26 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: disconnected and including.
Theorem 2.2 of The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors. Words of the Lean statement, its names split into words, that also appear in the theorem: nondegenerate, nonzero, time, angular, graphs, maps and 25 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: penrose and inequality.
Proposition 6.3 of The Penrose inequality for maximal asymptotically hyperbolic initial data. Words of the Lean statement, its names split into words, that also appear in the theorem: projection, ordinary, tangent, direction, plane, mean and 25 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: included.
Proposition 4.1 of A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data. Words of the Lean statement, its names split into words, that also appear in the theorem: mixed, square, root, jet, remainder, riemannian and 22 more.
Theorem 16.1 of Boundary graph deformations for the spacetime Penrose inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: minus, covariant, volume, euclidean, ball, chart and 28 more. Found in the paper’s title, which its main theorem states: graph and spacetime. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: disconnected and included.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 260’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds a supporting or a different result stated Their term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: The two challenge theorems are CKSSourceExterior.area_controlled_end_replacement (a 3-dimensional Cha-Khuri-Sakovich hyperboloidal end can be replaced by asymptotically flat ends keeping the compact interior, completeness and DEC, with Tendsto (spatialADMEnergy G) ... (√(m² - |p|²) + η) and ε R → 0 area loss) and schwarzschild_equality_examples (m) (hm : 0 < m) (the Schwarzschild data satisfy MainHypotheses, bondiMass d = m, minimumEnclosingArea g = 16π m²). Neither states the Penrose inequality: the summary's headline (invariant ADM mass ≥ √(min enclosing area/16π) for asymptotically flat data in every dimension n >= 3 under DEC/weak future trapping, with equality rigidity and charged bounds) is absent; docs: 'The paper's general Bondi-Penrose inequality ... remains unformalized' and the conditional inequality 'is not selected'. The challenge deals with the dimension-3 Bondi/CKS class only.
Definitions the group asks readers to compare with the paper: Entire geometric-analysis stack is hand-built inside one 3.6k-line challenge file (MetricJet, TensorJet, PhysicalDEC, CKSData, minimumEnclosingArea, bondiCharge, spatialADMEnergy, ...) on top of Mathlib manifolds; none is a Mathlib notion and the file has dozens of auxiliary definitions.
Carlos Toledo’s cert-machine
Finds a supporting or a different result stated Their term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: The entire headline. No Penrose inequality of any kind is stated (neither the headline's ADM / asymptotically-flat enclosing-area inequality in any dimension n >= 3, nor even the CKS paper's Bondi-mass inequality), no equality rigidity, no charged dyonic three-dimensional upper-area bound, no higher-dimensional purely electric extension. What is stated is a three-dimensional CKS-class end-replacement construction (from one of the family's 13 manuscripts) and Schwarzschild sharpness examples for an inequality that is itself not stated. The union neither implies the headline nor is implied by it.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The Comparator also includes canonical Schwarzschild equality examples at every positive mass, with $m_{\mathrm{Bondi}}=m$ and enclosing area $16\pi m^2$. The paper's general Bondi–Penrose inequality for possibly disconnected weakly trapped boundaries remains unformalized.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

3,596 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 2,369 to its own declarations, 1,709 to Mathlib’s documentation. 312 declared names also link to the same declaration in OpenAI’s library. 38 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlib/-! Area-controlled end replacement and Schwarzschild equality examples. -/namespace OAI.CKSADMend OAI.CKSADMnamespace OAI.CKSAngularGeometryend OAI.CKSAngularGeometrynamespace OAI.CKSAngularSliceend OAI.CKSAngularSlicenamespace OAI.CKSBoundarySurfaceend OAI.CKSBoundarySurfacenamespace OAI.CKSCalculusend OAI.CKSCalculusnamespace OAI.CKSCartesianOuterend OAI.CKSCartesianOuternamespace OAI.CKSEmbeddingDerivativeend OAI.CKSEmbeddingDerivativenamespace OAI.CKSFullCutAreaend OAI.CKSFullCutAreanamespace OAI.CKSGeometricCutsend OAI.CKSGeometricCutsnamespace OAI.CKSInducedAreaend OAI.CKSInducedAreanamespace OAI.CKSInducedSphereend OAI.CKSInducedSpherenamespace OAI.CKSIntrinsicConstraintsend OAI.CKSIntrinsicConstraintsnamespace OAI.CKSIntrinsicGeometryend OAI.CKSIntrinsicGeometrynamespace OAI.CKSIntrinsicVolumeend OAI.CKSIntrinsicVolumenamespace OAI.CKSLocalBendingend OAI.CKSLocalBendingnamespace OAI.CKSLorentzend OAI.CKSLorentznamespace OAI.CKSLorentz.SmoothAngularPatchend OAI.CKSLorentz.SmoothAngularPatchnamespace OAI.CKSMainend OAI.CKSMainnamespace OAI.CKSMain.InteriorSurfaceend OAI.CKSMain.InteriorSurfacenamespace OAI.CKSMetricGluingend OAI.CKSMetricGluingnamespace OAI.CKSMixedGeometryend OAI.CKSMixedGeometrynamespace OAI.CKSMixedGeometry.FiniteLogDecayend OAI.CKSMixedGeometry.FiniteLogDecaynamespace OAI.CKSRealizedRoundend OAI.CKSRealizedRoundnamespace OAI.CKSReplacementCompletenessend OAI.CKSReplacementCompletenessnamespace OAI.CKSRoundend OAI.CKSRoundnamespace OAI.CKSRound.MetricJetend OAI.CKSRound.MetricJetnamespace OAI.CKSSchwarzschildend OAI.CKSSchwarzschildnamespace OAI.CKSSourceExteriorend OAI.CKSSourceExteriornamespace OAI.CKSSpatialManifoldend OAI.CKSSpatialManifoldnamespace OAI.CKSSphericalHarmonicsend OAI.CKSSphericalHarmonicsnamespace OAI.CKSSurfaceVolumeend OAI.CKSSurfaceVolumenoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSpatialManifoldlocal instance real_id_isometric : RingHomIsometric (RingHom.id ℝ) := inferInstanceend OAI.CKSSpatialManifoldendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMetricGluingopen Bundle Manifold Set Bornologyopen scoped Bundle Manifold ContDiffvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]local instance real_continuousAdd : ContinuousAdd ℝ := inferInstancelocal instance real_continuousConstSMul : ContinuousConstSMul ℝ ℝ := inferInstancelocal instance real_smulCommClass :    @SMulCommClass ℝ ℝ ℝ Algebra.toSMul      (@instSMulOfMul ℝ (@Distrib.toMul ℝ        (@instDistribOfSemiring ℝ (@CommSemiring.toSemiring ℝ Real.instCommSemiring)))) :=  inferInstanceend OAI.CKSMetricGluingendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMetricGluingopen Bundle Manifold Set Bornologyopen scoped Bundle Manifold ContDiffvariable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_2} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] real_continuousAddabbrev InnerField :=  letI := OAI.CKSMetricGluing.real_smulCommClass  letI := OAI.CKSMetricGluing.real_continuousConstSMul  ∀ x : M, TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝabbrev SmoothMetric := ContMDiffRiemannianMetric I ∞ E (fun x : M => TangentSpace I x)def innerSection (q : OAI.CKSMetricGluing.InnerField I (M := M)) :    M → TotalSpace (E →L[ℝ] E →L[ℝ] ℝ)      (fun x : M => TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ) :=  fun x => ⟨x,q x⟩end OAI.CKSMetricGluingendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSAngularGeometryopen Set Filteropen scoped Topology ContDiff NNReal Matrix.Norms.Elementwiseabbrev I := Fin 2end OAI.CKSAngularGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSCalculusopen Filter Setopen scoped Topology ContDiffvariable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]def D (e : E) (f : E → ℝ) (x : E) : ℝ := fderiv ℝ f x eend OAI.CKSCalculusendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSRoundabbrev Idx := Fin 3abbrev Mat := Matrix OAI.CKSRound.Idx OAI.CKSRound.Idx ℝstructure MetricJet where  inv : OAI.CKSRound.Mat  d : OAI.CKSRound.Idx → OAI.CKSRound.Mat  dd : OAI.CKSRound.Idx → OAI.CKSRound.Idx → OAI.CKSRound.Matend OAI.CKSRoundendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSRound.MetricJetdef gamma (j : OAI.CKSRound.MetricJet) (a i k : OAI.CKSRound.Idx) : ℝ :=  (∑ l, j.inv a l * (j.d i k l + j.d k i l - j.d l i k)) / 2def dInv (j : OAI.CKSRound.MetricJet) (s a b : OAI.CKSRound.Idx) : ℝ :=  - ∑ i, ∑ k, j.inv a i * j.d s i k * j.inv k bdef dGamma (j : OAI.CKSRound.MetricJet) (s a i k : OAI.CKSRound.Idx) : ℝ :=  (∑ l, (j.dInv s a l * (j.d i k l + j.d k i l - j.d l i k) +    j.inv a l * (j.dd s i k l + j.dd s k i l - j.dd s l i k))) / 2def ricci (j : OAI.CKSRound.MetricJet) (i k : OAI.CKSRound.Idx) : ℝ :=  (∑ a, (j.dGamma a a i k - j.dGamma k a i a)) +  ∑ a, ∑ b, (j.gamma a a b * j.gamma b i k - j.gamma a k b * j.gamma b i a)def scalar (j : OAI.CKSRound.MetricJet) : ℝ := ∑ i, ∑ k, j.inv i k * j.ricci i kend OAI.CKSRound.MetricJetendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSRoundstructure TensorJet where  val : OAI.CKSRound.Mat  d : OAI.CKSRound.Idx → OAI.CKSRound.Matend OAI.CKSRoundendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSRound.MetricJetdef trace (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) : ℝ :=  ∑ a, ∑ b, g.inv a b * k.val a bdef normSq (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) : ℝ :=  ∑ a, ∑ b, ∑ c, ∑ d, g.inv a c * g.inv b d * k.val a b * k.val c ddef dTrace (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) (i : OAI.CKSRound.Idx) : ℝ :=  ∑ a, ∑ b, (g.dInv i a b * k.val a b + g.inv a b * k.d i a b)def tensorDivergence (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) (i : OAI.CKSRound.Idx) : ℝ :=  ∑ a, ∑ b, g.inv a b * (k.d a b i -    (∑ c, g.gamma c a b * k.val c i) - (∑ c, g.gamma c a i * k.val b c))def energy (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) : ℝ :=  (g.scalar + g.trace k ^ 2 - g.normSq k) / 2def momentum (g : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) (i : OAI.CKSRound.Idx) : ℝ :=  g.tensorDivergence k i - g.dTrace k iend OAI.CKSRound.MetricJetendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSRealizedRoundopen CKSCalculus CKSRoundopen scoped Topology ContDiffabbrev Point := Fin 3 → ℝdef basis (a : OAI.CKSRound.Idx) : OAI.CKSRealizedRound.Point := Pi.single a 1end OAI.CKSRealizedRoundendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLocalBendingopen CKSRounddef momentumSq (j : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) : ℝ :=  ∑ i, ∑ l, j.inv i l * j.momentum k i * j.momentum k ldef DEC (j : OAI.CKSRound.MetricJet) (k : OAI.CKSRound.TensorJet) : Prop :=  Real.sqrt (OAI.CKSLocalBending.momentumSq j k) ≤ j.energy kend OAI.CKSLocalBendingendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSAngularGeometryopen CKSCalculus Set Filteropen scoped Topology ContDiff NNReal Matrix.Norms.Elementwiseabbrev Point := OAI.CKSAngularGeometry.I → ℝdef basis (a : OAI.CKSAngularGeometry.I) : OAI.CKSAngularGeometry.Point := Pi.single a 1end OAI.CKSAngularGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSAngularGeometryopen CKSCalculus Set Filter Matrixopen scoped Topology ContDiff NNReal Matrix.Norms.Elementwiseabbrev AmbientMat := Matrix (Fin 3) (Fin 3) ℝend OAI.CKSAngularGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSAngularGeometryopen Matrix CKSCalculusopen scoped BigOperatorsabbrev PhysicalPoint := Fin 3 → ℝend OAI.CKSAngularGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMixedGeometryopen CKSCalculus Set Filteropen scoped Topology ContDiff NNReal Matrix.Norms.Elementwiseabbrev I := Fin 3abbrev A := Fin 2abbrev Point := OAI.CKSMixedGeometry.I → ℝdef basis (a : OAI.CKSMixedGeometry.I) : OAI.CKSMixedGeometry.Point := Pi.single a 1end OAI.CKSMixedGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMixedGeometryopen CKSCalculus Set Filteropen scoped Topology ContDiff NNReallocal instance pointDimension_neZero : NeZero (3 : ℕ) := inferInstancedef radialFactor (a : OAI.CKSMixedGeometry.I) (y : OAI.CKSMixedGeometry.Point) : ℝ :=  letI := OAI.CKSMixedGeometry.pointDimension_neZero  if a = 0 then y 0 else 1def scaledD (a : OAI.CKSMixedGeometry.I) (f : OAI.CKSMixedGeometry.Point → ℝ) : OAI.CKSMixedGeometry.Point → ℝ :=  fun y => OAI.CKSMixedGeometry.radialFactor a y * OAI.CKSCalculus.D (OAI.CKSMixedGeometry.basis a) f ydef scaledIter : List OAI.CKSMixedGeometry.I → (OAI.CKSMixedGeometry.Point → ℝ) → OAI.CKSMixedGeometry.Point → ℝ  | [], f => f  | a::l,f => OAI.CKSMixedGeometry.scaledD a (scaledIter l f)end OAI.CKSMixedGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMixedGeometryopen CKSCalculus Set Filteropen scoped Topology ContDiff NNReal Matrix.Norms.Elementwiseabbrev Angle := OAI.CKSMixedGeometry.A → ℝdef angularProjection : OAI.CKSMixedGeometry.Point →L[ℝ] OAI.CKSMixedGeometry.Angle :=  ContinuousLinearMap.pi (fun a => ContinuousLinearMap.proj a.succ)def ScaledComponentBound (f : OAI.CKSMixedGeometry.Point → ℝ) (j q : ℕ) (B : ℝ) (y : OAI.CKSMixedGeometry.Point) : Prop :=  letI := OAI.CKSMixedGeometry.pointDimension_neZero  ∀ l : List OAI.CKSMixedGeometry.I, l.length ≤ j → |scaledIter l f y| ≤ B/(y 0)^qend OAI.CKSMixedGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedSphereopen Finsetabbrev Ix := Fin 3end OAI.CKSInducedSphereendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSphericalHarmonicsopen MvPolynomialabbrev Ambient := EuclideanSpace ℝ (Fin 3)abbrev Sphere := ↥(Metric.sphere (0 : OAI.CKSSphericalHarmonics.Ambient) 1)end OAI.CKSSphericalHarmonicsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSphericalHarmonicsopen MvPolynomialopen MeasureTheoryabbrev surfaceMeasure : Measure OAI.CKSSphericalHarmonics.Sphere := (volume : Measure OAI.CKSSphericalHarmonics.Ambient).toSphereend OAI.CKSSphericalHarmonicsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSphericalHarmonicsopen MvPolynomialopen MeasureTheoryopen scoped Pointwiseopen scoped Manifold ContDiff Topologyopen Setdef SmoothSphere (f : OAI.CKSSphericalHarmonics.Sphere → ℝ) : Prop := ContMDiff (𝓡 2) 𝓘(ℝ, ℝ) ∞ fend OAI.CKSSphericalHarmonicsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzlocal instance two_atLeastTwo : Nat.AtLeastTwo 2 := inferInstanceend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedSphereopen Set Filter Finsetopen scoped Topology ContDiffabbrev E := EuclideanSpace ℝ OAI.CKSInducedSphere.Ixend OAI.CKSInducedSphereendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedSphereopen Set Filter Finsetopen scoped Topology ContDiffattribute [local instance] CKSLorentz.two_atLeastTwodef e (i : OAI.CKSInducedSphere.Ix) : OAI.CKSInducedSphere.E := WithLp.toLp 2 (Pi.single i 1)def pd (i : OAI.CKSInducedSphere.Ix) (F : OAI.CKSInducedSphere.E → ℝ) (x : OAI.CKSInducedSphere.E) : ℝ := fderiv ℝ F x (OAI.CKSInducedSphere.e i)end OAI.CKSInducedSphereendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen scoped InnerProductSpace RealInnerProductSpace ContDiff Manifold Topologyopen Setabbrev E := EuclideanSpace ℝ (Fin 3)abbrev V := ℝ × OAI.CKSLorentz.Eabbrev Sphere := ↥(Metric.sphere (0 : OAI.CKSLorentz.E) 1)end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setopen scoped ContDiff Topologylocal instance pointDimension_neZero : NeZero (3 : ℕ) := inferInstancedef ordinaryLeadingComponent (q : ℕ) (M : OAI.CKSMixedGeometry.Angle → ℝ) (y : OAI.CKSMixedGeometry.Point) : ℝ :=  letI := OAI.CKSLorentz.pointDimension_neZero  1/(y 0)^q * M (OAI.CKSMixedGeometry.angularProjection y)end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMixedGeometryopen Setopen scoped ContDiffstructure SourceRemainderOn (j q : ℕ) (V : Set OAI.CKSMixedGeometry.Point) (f : OAI.CKSMixedGeometry.Point → ℝ) : Prop where  regular : ∀ y ∈ V, ContDiffAt ℝ j f y  bound : ∃ B : ℝ, 0 ≤ B ∧ ∀ y ∈ V, OAI.CKSMixedGeometry.ScaledComponentBound f j q B yend OAI.CKSMixedGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setopen scoped ContDiff Topologystructure SourceCKSComponents (j qrad : ℕ) (V : Set OAI.CKSMixedGeometry.Point)    (A : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Point → ℝ) (mr : OAI.CKSMixedGeometry.Angle → ℝ) (mg : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Angle → ℝ) : Prop where  radial : OAI.CKSMixedGeometry.SourceRemainderOn j qrad V (fun y => A 0 0 y-OAI.CKSLorentz.ordinaryLeadingComponent 5 mr y)  mixed : ∀ i k, (i=0 ∧ k≠0) ∨ (i≠0 ∧ k=0) → OAI.CKSMixedGeometry.SourceRemainderOn j 3 V (A i k)  angular : ∀ i k, i≠0 → k≠0 → OAI.CKSMixedGeometry.SourceRemainderOn j 2 V    (fun y => A i k y-OAI.CKSLorentz.ordinaryLeadingComponent 1 (mg i k) y)end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setdef angularRadialTail (R : ℝ) (W : Set OAI.CKSMixedGeometry.Angle) : Set OAI.CKSMixedGeometry.Point :=  letI := OAI.CKSLorentz.pointDimension_neZero  {y | R < y 0 ∧ OAI.CKSMixedGeometry.angularProjection y ∈ W}end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setopen scoped ContDiff Topologystructure SmoothAngularPatch where  chart : OpenPartialHomeomorph OAI.CKSLorentz.Sphere OAI.CKSMixedGeometry.Angle  param_smooth : ContDiffOn ℝ ∞ (fun θ => (chart.symm θ : OAI.CKSLorentz.E)) chart.target  extension : OAI.CKSLorentz.E → OAI.CKSMixedGeometry.Angle  domain : Set OAI.CKSLorentz.E  isOpen_domain : IsOpen domain  extension_smooth : ContDiffOn ℝ ∞ extension domain  source_domain : ∀ n ∈ chart.source, (n:OAI.CKSLorentz.E) ∈ domain  extension_eq : ∀ n ∈ chart.source, extension n = chart nend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentz.SmoothAngularPatchopen CKSMixedGeometry CKSAngularSlice Setopen scoped ContDiff Topologydef param (P : OAI.CKSLorentz.SmoothAngularPatch) (θ : OAI.CKSMixedGeometry.Angle) : OAI.CKSLorentz.E := P.chart.symm θdef sphereRegion (P : OAI.CKSLorentz.SmoothAngularPatch) (W : Set OAI.CKSMixedGeometry.Angle) : Set OAI.CKSLorentz.Sphere :=  P.chart.source ∩ P.chart ⁻¹' Wend OAI.CKSLorentz.SmoothAngularPatchendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setopen scoped ContDiff Topologystructure CKSTensorPatch where  patch : OAI.CKSLorentz.SmoothAngularPatch  region : Set OAI.CKSMixedGeometry.Angle  open_region : IsOpen region  region_target : region ⊆ patch.chart.target  radius : ℝ  metric : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Point → ℝ  second : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Point → ℝ  mr : OAI.CKSMixedGeometry.Angle → ℝ  mg : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Angle → ℝ  mK : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Angle → ℝ  mr_smooth : ContDiffOn ℝ ∞ mr region  mg_smooth : ∀ i k, i≠0 → k≠0 → ContDiffOn ℝ ∞ (mg i k) region  mK_smooth : ∀ i k, i≠0 → k≠0 → ContDiffOn ℝ ∞ (mK i k) region  metric_CKS : OAI.CKSLorentz.SourceCKSComponents 3 6 (OAI.CKSLorentz.angularRadialTail radius region) metric mr mg  second_CKS : OAI.CKSLorentz.SourceCKSComponents 2 5 (OAI.CKSLorentz.angularRadialTail radius region) second (fun _ => 0) mKend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlicelocal instance real_continuousAdd : ContinuousAdd ℝ := inferInstancelocal instance real_continuousConstSMul : ContinuousConstSMul ℝ ℝ := inferInstancelocal instance real_smulCommClass :    @SMulCommClass ℝ ℝ ℝ Algebra.toSMul      (@instSMulOfMul ℝ (@Distrib.toMul ℝ        (@instDistribOfSemiring ℝ (@CommSemiring.toSemiring ℝ Real.instCommSemiring)))) :=  inferInstanceend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSliceattribute [local instance] real_continuousAddabbrev SpatialBilinear :=  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSLorentz.real_continuousConstSMul  OAI.CKSLorentz.E →L[ℝ] OAI.CKSLorentz.E →L[ℝ] ℝabbrev SpatialTensor := OAI.CKSLorentz.E → OAI.CKSLorentz.SpatialBilinearlocal instance three_neZero : NeZero 3 := inferInstancedef polarParam (P : OAI.CKSLorentz.SmoothAngularPatch) (y : OAI.CKSMixedGeometry.Point) : OAI.CKSLorentz.E :=  letI := OAI.CKSLorentz.three_neZero  y 0 • P.param (OAI.CKSMixedGeometry.angularProjection y)def polarTensorComponents (P : OAI.CKSLorentz.SmoothAngularPatch) (A : OAI.CKSLorentz.SpatialTensor)    (i k : OAI.CKSMixedGeometry.I) (y : OAI.CKSMixedGeometry.Point) : ℝ :=  A (OAI.CKSLorentz.polarParam P y) (fderiv ℝ (OAI.CKSLorentz.polarParam P) y (OAI.CKSMixedGeometry.basis i))    (fderiv ℝ (OAI.CKSLorentz.polarParam P) y (OAI.CKSMixedGeometry.basis k))def CKSTensorPatch.Realizes (Q : OAI.CKSLorentz.CKSTensorPatch) (A B : OAI.CKSLorentz.SpatialTensor) : Prop :=  ∀ y ∈ OAI.CKSLorentz.angularRadialTail Q.radius Q.region, ∀ i k,    Q.metric i k y = OAI.CKSLorentz.polarTensorComponents Q.patch A i k y ∧    Q.second i k y = OAI.CKSLorentz.polarTensorComponents Q.patch B i k yend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice CKSCalculusabbrev AngularMatrix := Matrix (Fin 2) (Fin 2) ℝdef angularMetricTrace (σ A : OAI.CKSLorentz.AngularMatrix) : ℝ := (σ⁻¹ * A).tracedef angularRoundMetric (φ : OAI.CKSMixedGeometry.Angle → OAI.CKSLorentz.E) (θ : OAI.CKSMixedGeometry.Angle) : OAI.CKSLorentz.AngularMatrix :=  letI := OAI.CKSLorentz.two_atLeastTwo  letI : Inner ℝ OAI.CKSLorentz.E :=    @InnerProductSpace.toInner ℝ OAI.CKSLorentz.E _      (PiLp.seminormedAddCommGroup 2 (fun _ : Fin 3 => ℝ))      (PiLp.innerProductSpace (fun _ : Fin 3 => ℝ))  fun a b => inner ℝ (fderiv ℝ φ θ (OAI.CKSAngularGeometry.basis a))    (fderiv ℝ φ θ (OAI.CKSAngularGeometry.basis b))end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen scoped RealInnerProductSpace ContDiff Manifold Topologyopen MeasureTheoryopen CKSSphericalHarmonics (SmoothSphere surfaceMeasure)def nullDirection (n : OAI.CKSLorentz.Sphere) : OAI.CKSLorentz.V := (1,(n:OAI.CKSLorentz.E))def rawCharge (M : OAI.CKSLorentz.Sphere → ℝ) : OAI.CKSLorentz.V := ∫ n : OAI.CKSLorentz.Sphere, M n • OAI.CKSLorentz.nullDirection n ∂OAI.CKSSphericalHarmonics.surfaceMeasuredef bondiCharge (M : OAI.CKSLorentz.Sphere → ℝ) : OAI.CKSLorentz.V := (16*Real.pi)⁻¹ • OAI.CKSLorentz.rawCharge Mend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometrylocal instance real_isTopologicalAddGroup : IsTopologicalAddGroup ℝ := inferInstancelocal instance spatialDual_isTopologicalAddGroup : IsTopologicalAddGroup (OAI.CKSLorentz.E →L[ℝ] ℝ) :=  inferInstancelocal instance spatialDual_smulCommClass :    letI := OAI.CKSLorentz.real_smulCommClass    SMulCommClass ℝ ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) :=  inferInstancelocal instance spatialDual_continuousConstSMul : ContinuousConstSMul ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) :=  inferInstancelocal instance spatialDual_isScalarTower : IsScalarTower ℝ ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) := inferInstancelocal instance spatialDual_continuousSMul : ContinuousSMul ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) := inferInstancedef hyperbolicField (x : OAI.CKSLorentz.E) : OAI.CKSLorentz.SpatialBilinear :=  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_continuousConstSMul  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSLorentz.real_isTopologicalAddGroup  letI := OAI.CKSLorentz.spatialDual_isTopologicalAddGroup  letI := OAI.CKSLorentz.spatialDual_smulCommClass  letI := OAI.CKSLorentz.spatialDual_continuousConstSMul  letI := OAI.CKSLorentz.spatialDual_isScalarTower  letI := OAI.CKSLorentz.spatialDual_continuousSMul  let f : OAI.CKSLorentz.SpatialBilinear := innerSL ℝ  f - (1/(1+‖x‖^2)) • (f x).smulRight (f x)def sourceSpatial (A : OAI.CKSLorentz.SpatialTensor) : OAI.CKSLorentz.SpatialTensor := fun x => OAI.CKSLorentz.hyperbolicField x + A xend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen scoped ContDiffattribute [local instance] CKSSpatialManifold.real_id_isometric real_smulCommClass spatialDual_smulCommClasslocal instance hsgNorm : NormedAddCommGroup OAI.CKSLorentz.SpatialBilinear :=  let dualSpace : NormedSpace ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) :=    @ContinuousLinearMap.toNormedSpace ℝ ℝ OAI.CKSLorentz.E ℝ _ _ _ _ _ _ (RingHom.id ℝ)      OAI.CKSSpatialManifold.real_id_isometric ℝ _ _ OAI.CKSLorentz.real_smulCommClass  @ContinuousLinearMap.toNormedAddCommGroup ℝ ℝ OAI.CKSLorentz.E (OAI.CKSLorentz.E →L[ℝ] ℝ)    _ _ _ _ _ dualSpace (RingHom.id ℝ) OAI.CKSSpatialManifold.real_id_isometricend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen scoped ContDiffattribute [local instance] CKSSpatialManifold.real_id_isometric real_smulCommClass spatialDual_smulCommClassattribute [local instance] OAI.CKSLorentz.hsgNormlocal instance hsgSpace : NormedSpace ℝ OAI.CKSLorentz.SpatialBilinear :=  let dualSpace : NormedSpace ℝ (OAI.CKSLorentz.E →L[ℝ] ℝ) :=    @ContinuousLinearMap.toNormedSpace ℝ ℝ OAI.CKSLorentz.E ℝ _ _ _ _ _ _ (RingHom.id ℝ)      OAI.CKSSpatialManifold.real_id_isometric ℝ _ _ OAI.CKSLorentz.real_smulCommClass  @ContinuousLinearMap.toNormedSpace ℝ ℝ OAI.CKSLorentz.E (OAI.CKSLorentz.E →L[ℝ] ℝ)    _ _ _ _ _ dualSpace (RingHom.id ℝ) OAI.CKSSpatialManifold.real_id_isometric    ℝ _ dualSpace OAI.CKSLorentz.spatialDual_smulCommClassend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSADMopen Finset CKSInducedSphereopen scoped RealInnerProductSpace ContDiffdef normal (x : OAI.CKSInducedSphere.E) (i : OAI.CKSInducedSphere.Ix) : ℝ := x i / ‖x‖end OAI.CKSADMendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSADMopen Set Filter Finset CKSSphericalHarmonics CKSInducedSphereopen scoped Topology ContDiffdef spatialInfinity : Filter OAI.CKSInducedSphere.E := Filter.comap (fun x : OAI.CKSInducedSphere.E => ‖x‖) atTopdef TailRegular (F : OAI.CKSInducedSphere.E → ℝ) : Prop :=  ∀ᶠ x in OAI.CKSADM.spatialInfinity, ContDiffAt ℝ ∞ F xdef Decay (d : ℝ) (F : OAI.CKSInducedSphere.E → ℝ) : Prop :=  ∃ C : ℝ, 0 ≤ C ∧ ∀ᶠ x in OAI.CKSADM.spatialInfinity, |F x| ≤ C*‖x‖^(-d)def SymbolN : ℕ → ℝ → (OAI.CKSInducedSphere.E → ℝ) → Prop  | 0, d, F => OAI.CKSADM.Decay d F  | n+1, d, F => OAI.CKSADM.Decay d F ∧ ∀ j : OAI.CKSInducedSphere.Ix, SymbolN n (d+1) (OAI.CKSInducedSphere.pd j F)def Symbol (d : ℝ) (F : OAI.CKSInducedSphere.E → ℝ) : Prop :=  OAI.CKSADM.TailRegular F ∧ ∀ n : ℕ, OAI.CKSADM.SymbolN n d Fend OAI.CKSADMendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSAngularGeometryopen Matrix CKSCalculus Filteropen scoped BigOperators Topology ContDiff Matrix.Norms.Elementwisedef physicalMetricJet (G : OAI.CKSAngularGeometry.PhysicalPoint → OAI.CKSAngularGeometry.AmbientMat) (x : OAI.CKSAngularGeometry.PhysicalPoint) : OAI.CKSRound.MetricJet where  inv := (G x)⁻¹  d := fun a i j => OAI.CKSCalculus.D (OAI.CKSRealizedRound.basis a) (fun y => G y i j) x  dd := fun a b i j => OAI.CKSCalculus.D (OAI.CKSRealizedRound.basis a) (OAI.CKSCalculus.D (OAI.CKSRealizedRound.basis b) (fun y => G y i j)) xdef physicalTensorJet (K : OAI.CKSAngularGeometry.PhysicalPoint → OAI.CKSAngularGeometry.AmbientMat) (x : OAI.CKSAngularGeometry.PhysicalPoint) : OAI.CKSRound.TensorJet where  val := K x  d := fun a i j => OAI.CKSCalculus.D (OAI.CKSRealizedRound.basis a) (fun y => K y i j) xend OAI.CKSAngularGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSCartesianOuteropen Set Filter Matrix CKSADM CKSInducedSphere CKSSphericalHarmonicsopen CKSAngularGeometry (PhysicalPoint)open scoped Topology ContDiff Matrix.Norms.Elementwiseabbrev toPoint : OAI.CKSInducedSphere.E ≃L[ℝ] OAI.CKSAngularGeometry.PhysicalPoint := EuclideanSpace.equiv (Fin 3) ℝend OAI.CKSCartesianOuterendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice CKSCalculusdef cksChartMassAspect (φ : OAI.CKSMixedGeometry.Angle → OAI.CKSLorentz.E) (mr : OAI.CKSMixedGeometry.Angle → ℝ)    (mg mK : OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.I → OAI.CKSMixedGeometry.Angle → ℝ) (θ : OAI.CKSMixedGeometry.Angle) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  OAI.CKSLorentz.angularMetricTrace (OAI.CKSLorentz.angularRoundMetric φ θ)    (fun a b => mg a.succ b.succ θ + 2*mK a.succ b.succ θ) + 2*mr θend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularGeometrydef spatialCoefficients (A : OAI.CKSLorentz.SpatialTensor) (y : OAI.CKSAngularGeometry.PhysicalPoint) : OAI.CKSAngularGeometry.AmbientMat :=  fun i j => A (OAI.CKSCartesianOuter.toPoint.symm y)    (OAI.CKSCartesianOuter.toPoint.symm (OAI.CKSMixedGeometry.basis i))    (OAI.CKSCartesianOuter.toPoint.symm (OAI.CKSMixedGeometry.basis j))def spatialDEC (A B : OAI.CKSLorentz.SpatialTensor) (x : OAI.CKSLorentz.E) : Prop :=  OAI.CKSLocalBending.DEC (OAI.CKSAngularGeometry.physicalMetricJet (OAI.CKSLorentz.spatialCoefficients A) (OAI.CKSCartesianOuter.toPoint x))    (OAI.CKSAngularGeometry.physicalTensorJet (OAI.CKSLorentz.spatialCoefficients B) (OAI.CKSCartesianOuter.toPoint x))end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularSlice Setdef CKSTensorPatch.RepresentsMassAspect (Q : OAI.CKSLorentz.CKSTensorPatch) (M : OAI.CKSLorentz.Sphere → ℝ) : Prop :=  ∀ n ∈ Q.patch.sphereRegion Q.region,    OAI.CKSLorentz.cksChartMassAspect Q.patch.param Q.mr Q.mg Q.mK (Q.patch.chart n) = M nend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularGeometry CKSInducedSphere CKSSphericalHarmonics MeasureTheorydef spatialCartesian (A : OAI.CKSLorentz.SpatialTensor) (x : OAI.CKSLorentz.E) : OAI.CKSAngularGeometry.AmbientMat :=  OAI.CKSLorentz.spatialCoefficients A (OAI.CKSCartesianOuter.toPoint x)local instance sixteen_atLeastTwo : Nat.AtLeastTwo 16 := inferInstanceend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularGeometry CKSInducedSphere CKSSphericalHarmonics MeasureTheoryattribute [local instance] sixteen_atLeastTwodef spatialADMEnergy (A : OAI.CKSLorentz.SpatialTensor) (r : ℝ) : ℝ :=  r^2/(16*Real.pi) * ∫ n : OAI.CKSLorentz.Sphere,    ∑ i : Fin 3, (∑ j : Fin 3,      (OAI.CKSInducedSphere.pd j (fun y => OAI.CKSLorentz.spatialCartesian A y i j) (r • (n:OAI.CKSLorentz.E)) -       OAI.CKSInducedSphere.pd i (fun y => OAI.CKSLorentz.spatialCartesian A y j j) (r • (n:OAI.CKSLorentz.E)))) *      OAI.CKSADM.normal (r • (n:OAI.CKSLorentz.E)) i ∂OAI.CKSSphericalHarmonics.surfaceMeasuredef spatialADMMomentum (A B : OAI.CKSLorentz.SpatialTensor) (r : ℝ) : Fin 3 → ℝ := fun i =>  r^2/(8*Real.pi) * ∫ n : OAI.CKSLorentz.Sphere,    ∑ j : Fin 3, (OAI.CKSLorentz.spatialCartesian B (r • (n:OAI.CKSLorentz.E)) i j -      (∑ a : Fin 3, ∑ b : Fin 3,        (OAI.CKSLorentz.spatialCartesian A (r • (n:OAI.CKSLorentz.E)))⁻¹ a b * OAI.CKSLorentz.spatialCartesian B (r • (n:OAI.CKSLorentz.E)) a b) *      OAI.CKSLorentz.spatialCartesian A (r • (n:OAI.CKSLorentz.E)) i j) * OAI.CKSADM.normal (r • (n:OAI.CKSLorentz.E)) j ∂OAI.CKSSphericalHarmonics.surfaceMeasureend OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSLorentzopen CKSMixedGeometry CKSAngularGeometry MeasureTheorydef spatialEnergy (A B : OAI.CKSLorentz.SpatialTensor) (x : OAI.CKSLorentz.E) : ℝ :=  (OAI.CKSAngularGeometry.physicalMetricJet (OAI.CKSLorentz.spatialCoefficients A) (OAI.CKSCartesianOuter.toPoint x)).energy    (OAI.CKSAngularGeometry.physicalTensorJet (OAI.CKSLorentz.spatialCoefficients B) (OAI.CKSCartesianOuter.toPoint x))end OAI.CKSLorentzendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSReplacementCompletenessopen Bundle Manifoldopen scoped Bundle Manifold ENNRealvariable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_2} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]abbrev Metric := ContinuousRiemannianMetric E (fun x : M => TangentSpace I x)local instance metric_isContinuousRiemannianBundle (g : OAI.CKSReplacementCompleteness.Metric I (M := M)) :    letI : RiemannianBundle (fun x : M => TangentSpace I x) := ⟨g.toRiemannianMetric⟩    IsContinuousRiemannianBundle E (fun x : M => TangentSpace I x) := inferInstance@[instance_reducible] def metricSpace [T3Space M] (g : OAI.CKSReplacementCompleteness.Metric I (M := M)) : EMetricSpace M :=  letI : RiemannianBundle (fun x : M => TangentSpace I x) := ⟨g.toRiemannianMetric⟩  letI := OAI.CKSReplacementCompleteness.metric_isContinuousRiemannianBundle I g  EMetricSpace.ofRiemannianMetric I Mdef IsComplete [T3Space M] (g : OAI.CKSReplacementCompleteness.Metric I (M := M)) : Prop :=  @CompleteSpace M (OAI.CKSReplacementCompleteness.metricSpace I g).toUniformSpaceend OAI.CKSReplacementCompletenessendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSGeometricCutsopen Set Manifold MeasureTheory Bundle CKSReplacementCompletenessopen scoped ENNReal Manifold Bundle ContDiffabbrev E3 := EuclideanSpace ℝ (Fin 3)local instance halfSpaceDimension_neZero : NeZero (3 : ℕ) := inferInstanceabbrev H3 := @EuclideanHalfSpace 3 OAI.CKSGeometricCuts.halfSpaceDimension_neZeroabbrev I3 := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSGeometricCuts.halfSpaceDimension_neZeroend OAI.CKSGeometricCutsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSGeometricCutsopen Set Manifold MeasureTheory Bundle CKSReplacementCompletenessopen scoped ENNReal Manifold Bundle ContDiffvariable (M : Type u) [TopologicalSpace M] [ChartedSpace H3 M] [IsManifold I3 1 M]structure OuterDomain where  Carrier : Type u  topology : TopologicalSpace Carrier  charts : @ChartedSpace OAI.CKSGeometricCuts.H3 _ Carrier topology  smooth : @IsManifold ℝ _ OAI.CKSGeometricCuts.E3 _ _ OAI.CKSGeometricCuts.H3 _ OAI.CKSGeometricCuts.I3 (∞ : ℕ∞ω) Carrier topology charts  inclusion : Carrier → M  embedding : @IsSmoothEmbedding ℝ _ OAI.CKSGeometricCuts.E3 OAI.CKSGeometricCuts.E3 _ _ _ _ OAI.CKSGeometricCuts.H3 OAI.CKSGeometricCuts.H3 _ _ OAI.CKSGeometricCuts.I3 OAI.CKSGeometricCuts.I3    Carrier M topology charts _ _ ∞ inclusion  closed : IsClosed (range inclusion)  connected : IsConnected (range inclusion)  contains_distant_end : ∃ K : Set M, IsCompact K ∧ Kᶜ ⊆ range inclusion  interior_into : inclusion '' (@ModelWithCorners.interior ℝ _ OAI.CKSGeometricCuts.E3 _ _ OAI.CKSGeometricCuts.H3 _ OAI.CKSGeometricCuts.I3    Carrier topology charts) ⊆ OAI.CKSGeometricCuts.I3.interior M  compact_boundary : IsCompact (inclusion '' (@ModelWithCorners.boundary ℝ _ OAI.CKSGeometricCuts.E3 _ _ OAI.CKSGeometricCuts.H3 _ OAI.CKSGeometricCuts.I3    Carrier topology charts))end OAI.CKSGeometricCutsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSpatialManifoldopen Bundle Manifold Set Bornology Filter CKSLorentz CKSMetricGluingopen scoped Bundle Manifold ContDiff Topologyvariable {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] real_id_isometric CKSMetricGluing.real_continuousAdd CKSMetricGluing.real_continuousConstSMul CKSMetricGluing.real_smulCommClasslocal instance real_id_compTriple :    RingHomCompTriple (RingHom.id ℝ) (RingHom.id ℝ) (RingHom.id ℝ) := inferInstanceend OAI.CKSSpatialManifoldendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSpatialManifoldopen Bundle Manifold Set Bornology Filter CKSLorentz CKSMetricGluingopen scoped Bundle Manifold ContDiff Topologyvariable {H : Type u_1} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_2} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] real_id_isometric CKSMetricGluing.real_continuousAdd CKSMetricGluing.real_continuousConstSMul CKSMetricGluing.real_smulCommClassdef endInner (f : M → OAI.CKSLorentz.E) (A : OAI.CKSLorentz.SpatialTensor) : OAI.CKSMetricGluing.InnerField I (M := M) := fun x => by  letI := OAI.CKSSpatialManifold.real_id_isometric  letI := OAI.CKSSpatialManifold.real_id_compTriple  letI : NormedAddCommGroup (TangentSpace I x) := by unfold TangentSpace; infer_instance  letI : NormedSpace ℝ (TangentSpace I x) := by unfold TangentSpace; infer_instance  let df : TangentSpace I x →L[ℝ] OAI.CKSLorentz.E :=    (NormedSpace.fromTangentSpace (f x)).toContinuousLinearMap.comp (mfderiv I 𝓘(ℝ,OAI.CKSLorentz.E) f x)  exact (A (f x)).bilinearComp df dfend OAI.CKSSpatialManifoldendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSpatialManifoldopen Bundle Manifold Set Filter CKSLorentz CKSMetricGluingopen scoped Bundle Manifold ContDiff Topologyvariable {H : Type u_1} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_2} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] CKSLorentz.hsgNorm CKSLorentz.hsgSpacedef PhysicalDECAt (g : OAI.CKSMetricGluing.InnerField I (M := M)) (K : OAI.CKSMetricGluing.InnerField I (M := M)) (x : M) : Prop :=  ∃ (V : Set M) (f : M → OAI.CKSLorentz.E) (A B : OAI.CKSLorentz.SpatialTensor),    IsOpen V ∧ x ∈ V ∧ ContMDiffOn I 𝓘(ℝ,OAI.CKSLorentz.E) ∞ f V ∧    (∀ y ∈ V, Function.Bijective (mfderiv I 𝓘(ℝ,OAI.CKSLorentz.E) f y)) ∧    (∀ y ∈ V, ContDiffAt ℝ ∞ A (f y) ∧ ContDiffAt ℝ ∞ B (f y)) ∧    (∀ y ∈ V, (∀ v w, A (f y) v w = A (f y) w v) ∧      (∀ v : OAI.CKSLorentz.E, v ≠ 0 → 0 < A (f y) v v) ∧      (∀ v w, B (f y) v w = B (f y) w v)) ∧    (∀ y ∈ V, g y = OAI.CKSSpatialManifold.endInner I f A y ∧ K y = OAI.CKSSpatialManifold.endInner I f B y) ∧    OAI.CKSLorentz.spatialDEC A B (f x)def PhysicalDEC (g : OAI.CKSMetricGluing.InnerField I (M := M)) (K : OAI.CKSMetricGluing.InnerField I (M := M)) : Prop :=  ∀ x : M, OAI.CKSSpatialManifold.PhysicalDECAt I g K xend OAI.CKSSpatialManifoldendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicGeometryopen Bundle Bornology Set MeasureTheory Manifold Filter Metricopen scoped ENNReal ContDiff Topology NNReallocal instance halfSpaceDimension_neZero : NeZero (3 : ℕ) := inferInstanceabbrev H3 := @EuclideanHalfSpace 3 OAI.CKSIntrinsicGeometry.halfSpaceDimension_neZeroabbrev I3 := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSIntrinsicGeometry.halfSpaceDimension_neZeroend OAI.CKSIntrinsicGeometryendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filteropen scoped ContDiff Topologyabbrev E2 := EuclideanSpace ℝ (Fin 2)abbrev E3 := EuclideanSpace ℝ (Fin 3)local instance halfSpaceDimension_neZero : NeZero (3 : ℕ) := inferInstanceabbrev H3 := @EuclideanHalfSpace 3 OAI.CKSBoundarySurface.halfSpaceDimension_neZeroabbrev I2 := 𝓘(ℝ,OAI.CKSBoundarySurface.E2)abbrev I3 := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSBoundarySurface.halfSpaceDimension_neZerolocal instance two_atLeastTwo : Nat.AtLeastTwo 2 := inferInstanceend OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filteropen scoped ContDiff Topologyattribute [local instance] two_atLeastTwodef insertNormalZero (point : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.E3 := WithLp.toLp 2 (Fin.cons 0 point)def liftPlane : OAI.CKSBoundarySurface.E2 →L[ℝ] OAI.CKSBoundarySurface.E3 :=  have proof_insertNormalZero_add_0  (first second : OAI.CKSBoundarySurface.E2) :      OAI.CKSBoundarySurface.insertNormalZero (first + second) = OAI.CKSBoundarySurface.insertNormalZero first + OAI.CKSBoundarySurface.insertNormalZero second := by    ext index    cases index using Fin.cases <;> simp [OAI.CKSBoundarySurface.insertNormalZero]  have proof_insertNormalZero_smul_1  (scalar : ℝ) (point : OAI.CKSBoundarySurface.E2) :      OAI.CKSBoundarySurface.insertNormalZero (scalar • point) = scalar • OAI.CKSBoundarySurface.insertNormalZero point := by    ext index    cases index using Fin.cases <;> simp [OAI.CKSBoundarySurface.insertNormalZero]  have proof_insertNormalZero_continuous_2  : Continuous OAI.CKSBoundarySurface.insertNormalZero := by    apply (PiLp.continuous_toLp 2 (fun _ : Fin 3 => ℝ)).comp    apply continuous_pi    intro index    cases index using Fin.cases with    | zero => exact continuous_const    | succ index => exact PiLp.continuous_apply 2 (fun _ : Fin 2 => ℝ) index  {    toFun := OAI.CKSBoundarySurface.insertNormalZero    map_add' := proof_insertNormalZero_add_0    map_smul' := proof_insertNormalZero_smul_1    cont := proof_insertNormalZero_continuous_2  }def tangentialCoordinates (point : OAI.CKSBoundarySurface.E3) : OAI.CKSBoundarySurface.E2 := WithLp.toLp 2 (fun index => point index.succ)def dropPlane : OAI.CKSBoundarySurface.E3 →L[ℝ] OAI.CKSBoundarySurface.E2 :=  have proof_tangentialCoordinates_add_3  (first second : OAI.CKSBoundarySurface.E3) :      OAI.CKSBoundarySurface.tangentialCoordinates (first + second) =        OAI.CKSBoundarySurface.tangentialCoordinates first + OAI.CKSBoundarySurface.tangentialCoordinates second := by    ext index    simp [OAI.CKSBoundarySurface.tangentialCoordinates]  have proof_tangentialCoordinates_smul_4  (scalar : ℝ) (point : OAI.CKSBoundarySurface.E3) :      OAI.CKSBoundarySurface.tangentialCoordinates (scalar • point) = scalar • OAI.CKSBoundarySurface.tangentialCoordinates point := by    ext index    simp [OAI.CKSBoundarySurface.tangentialCoordinates]  have proof_tangentialCoordinates_continuous_5  : Continuous OAI.CKSBoundarySurface.tangentialCoordinates := by    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp    exact continuous_pi fun index => PiLp.continuous_apply 2 (fun _ : Fin 3 => ℝ) index.succ  {    toFun := OAI.CKSBoundarySurface.tangentialCoordinates    map_add' := proof_tangentialCoordinates_add_3    map_smul' := proof_tangentialCoordinates_smul_4    cont := proof_tangentialCoordinates_continuous_5  }end OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filteropen scoped ContDiff Topologydef liftHalf (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.H3 :=  have proof_liftPlane_zero_6  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.liftPlane z 0 = 0 := rfl  have proof_liftPlane_succ_7  (z : OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSBoundarySurface.liftPlane z i.succ = z i := rfl  ⟨OAI.CKSBoundarySurface.liftPlane z, by simp [*]⟩end OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Manifold Bundle Filteropen scoped ContDiff Topologyopen CKSBoundarySurfaceabbrev Radial := {t : ℝ // 0 ≤ t}abbrev Sphere := Metric.sphere (0 : OAI.CKSBoundarySurface.E3) 1abbrev Exterior := OAI.CKSSchwarzschild.Radial × OAI.CKSSchwarzschild.Sphereend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Manifold Bundle Filteropen scoped ContDiff Topologyopen CKSBoundarySurfaceattribute [local instance] Classical.propDecidablelocal instance real_id_invPair : RingHomInvPair (RingHom.id ℝ) (RingHom.id ℝ) := inferInstancedef join : (ℝ × OAI.CKSBoundarySurface.E2) →L[ℝ] OAI.CKSBoundarySurface.E3 :=  letI := OAI.CKSSchwarzschild.real_id_invPair  letI := OAI.CKSBoundarySurface.two_atLeastTwo  let continuousLinearMap : ((ℝ × OAI.CKSBoundarySurface.E2) →ₗ[ℝ] OAI.CKSBoundarySurface.E3) ≃ₗ[ℝ] ((ℝ × OAI.CKSBoundarySurface.E2) →L[ℝ] OAI.CKSBoundarySurface.E3) :=    LinearMap.toContinuousLinearMap  continuousLinearMap {    toFun := fun z => WithLp.toLp 2 (Fin.cons z.1 z.2)    map_add' := by intro a b; ext i; cases i using Fin.cases <;> simp    map_smul' := by intro a b; ext i; cases i using Fin.cases <;> simp }def halfProduct : OAI.CKSSchwarzschild.Radial × OAI.CKSBoundarySurface.E2 ≃ₜ OAI.CKSBoundarySurface.H3 :=  have proof_join_zero_9  (z : ℝ × OAI.CKSBoundarySurface.E2) : OAI.CKSSchwarzschild.join z 0 = z.1 := rfl  have proof_join_succ_10  (z : ℝ × OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSSchwarzschild.join z i.succ = z.2 i := rfl  have proof_join_drop_8  (z : OAI.CKSBoundarySurface.E3) : OAI.CKSSchwarzschild.join (z 0,OAI.CKSBoundarySurface.dropPlane z) = z := by    ext i; cases i using Fin.cases <;> rfl  have proof_drop_join_11  (z : ℝ × OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.dropPlane (OAI.CKSSchwarzschild.join z) = z.2 := by ext i; rfl  letI := OAI.CKSBoundarySurface.halfSpaceDimension_neZero  { toFun z := ⟨OAI.CKSSchwarzschild.join (z.1.val,z.2),z.1.property⟩    invFun z := (⟨z.val 0,z.property⟩,OAI.CKSBoundarySurface.dropPlane z.val)    left_inv z := by cases z; simp [*]    right_inv z := by apply Subtype.ext; exact proof_join_drop_8 _    continuous_toFun := (OAI.CKSSchwarzschild.join.continuous.comp      ((continuous_subtype_val.comp continuous_fst).prodMk continuous_snd)).subtype_mk _    continuous_invFun := ((by fun_prop : Continuous (fun z : H3 => z.val 0)).subtype_mk _).prodMk      (OAI.CKSBoundarySurface.dropPlane.continuous.comp continuous_subtype_val) }def productChart (s : OAI.CKSSchwarzschild.Sphere) : OpenPartialHomeomorph OAI.CKSSchwarzschild.Exterior OAI.CKSBoundarySurface.H3 :=  ((OpenPartialHomeomorph.refl OAI.CKSSchwarzschild.Radial).prod (chartAt OAI.CKSBoundarySurface.E2 s)).trans OAI.CKSSchwarzschild.halfProduct.toOpenPartialHomeomorphinstance exteriorChartedSpace : ChartedSpace OAI.CKSBoundarySurface.H3 OAI.CKSSchwarzschild.Exterior :=  have proof_join_zero_9  (z : ℝ × OAI.CKSBoundarySurface.E2) : OAI.CKSSchwarzschild.join z 0 = z.1 := rfl  have proof_join_succ_10  (z : ℝ × OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSSchwarzschild.join z i.succ = z.2 i := rfl  have proof_halfProduct_val_12  (z : OAI.CKSSchwarzschild.Radial × OAI.CKSBoundarySurface.E2) : (OAI.CKSSchwarzschild.halfProduct z).val = OAI.CKSSchwarzschild.join (z.1.val,z.2) := rfl  have proof_halfProduct_symm_fst_13  (z : OAI.CKSBoundarySurface.H3) : (OAI.CKSSchwarzschild.halfProduct.symm z).1.val = z.val 0 := rfl  have proof_halfProduct_symm_snd_14  (z : OAI.CKSBoundarySurface.H3) : (OAI.CKSSchwarzschild.halfProduct.symm z).2 = OAI.CKSBoundarySurface.dropPlane z.val := rfl  {    atlas := range OAI.CKSSchwarzschild.productChart    chartAt p := OAI.CKSSchwarzschild.productChart p.2    mem_chart_source p := by simp [*, OAI.CKSSchwarzschild.productChart,mem_chart_source OAI.CKSBoundarySurface.E2 p.2]    chart_mem_atlas p := mem_range_self _  }end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Manifold Bundle Filteropen scoped ContDiff Topologyopen CKSBoundarySurfaceattribute [local instance] halfSpaceDimension_neZeroinstance exteriorIsManifold : IsManifold OAI.CKSBoundarySurface.I3 ∞ OAI.CKSSchwarzschild.Exterior :=  have proof_join_zero_9  (z : ℝ × OAI.CKSBoundarySurface.E2) : OAI.CKSSchwarzschild.join z 0 = z.1 := rfl  have proof_join_succ_10  (z : ℝ × OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSSchwarzschild.join z i.succ = z.2 i := rfl  have proof_halfProduct_val_12  (z : OAI.CKSSchwarzschild.Radial × OAI.CKSBoundarySurface.E2) : (OAI.CKSSchwarzschild.halfProduct z).val = OAI.CKSSchwarzschild.join (z.1.val,z.2) := rfl  have proof_halfProduct_symm_fst_13  (z : OAI.CKSBoundarySurface.H3) : (OAI.CKSSchwarzschild.halfProduct.symm z).1.val = z.val 0 := rfl  have proof_halfProduct_symm_snd_14  (z : OAI.CKSBoundarySurface.H3) : (OAI.CKSSchwarzschild.halfProduct.symm z).2 = OAI.CKSBoundarySurface.dropPlane z.val := rfl  have proof_productChart_target_17  (s : OAI.CKSSchwarzschild.Sphere) :      (OAI.CKSSchwarzschild.productChart s).target = (fun z : OAI.CKSBoundarySurface.H3 => OAI.CKSBoundarySurface.dropPlane z.val) ⁻¹' (chartAt OAI.CKSBoundarySurface.E2 s).target := by    ext p; simp [*, OAI.CKSSchwarzschild.productChart]  have proof_productChart_source_18  (s : OAI.CKSSchwarzschild.Sphere) :      (OAI.CKSSchwarzschild.productChart s).source = Prod.snd ⁻¹' (chartAt OAI.CKSBoundarySurface.E2 s).source := by    ext p; simp [*, OAI.CKSSchwarzschild.productChart]  have proof_productChart_symm_19  (s : OAI.CKSSchwarzschild.Sphere) (z : OAI.CKSBoundarySurface.H3) :      (OAI.CKSSchwarzschild.productChart s).symm z = (⟨z.val 0,z.property⟩,(chartAt OAI.CKSBoundarySurface.E2 s).symm (OAI.CKSBoundarySurface.dropPlane z.val)) := rfl  have proof_transition_source_20  (a b : OAI.CKSSchwarzschild.Sphere) :      ((OAI.CKSSchwarzschild.productChart a).symm ≫ₕ OAI.CKSSchwarzschild.productChart b).source =        (fun z : OAI.CKSBoundarySurface.H3 => OAI.CKSBoundarySurface.dropPlane z.val) ⁻¹' ((chartAt OAI.CKSBoundarySurface.E2 a).symm ≫ₕ chartAt OAI.CKSBoundarySurface.E2 b).source := by    ext z    simp only [OpenPartialHomeomorph.trans_source,OpenPartialHomeomorph.symm_source,      mem_inter_iff,mem_preimage,proof_productChart_target_17,proof_productChart_source_18,proof_productChart_symm_19]  have proof_transition_maps_15  (a b : OAI.CKSSchwarzschild.Sphere) :      MapsTo OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3.symm ⁻¹' ((OAI.CKSSchwarzschild.productChart a).symm ≫ₕ OAI.CKSSchwarzschild.productChart b).source ∩ range OAI.CKSBoundarySurface.I3)        ((extChartAt OAI.CKSBoundarySurface.I2 a).symm ≫ extChartAt OAI.CKSBoundarySurface.I2 b).source := by    intro z hz    have he : (OAI.CKSBoundarySurface.I3.symm z).val = z := OAI.CKSBoundarySurface.I3.right_inv hz.2    have h := hz.1    rw [proof_transition_source_20] at h    change OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3.symm z).val ∈ ((chartAt OAI.CKSBoundarySurface.E2 a).symm ≫ₕ chartAt OAI.CKSBoundarySurface.E2 b).source at h    rw [he] at h    simpa only [OAI.CKSBoundarySurface.I2,extChartAt_model_space_eq_id,mfld_simps] using h  have proof_transition_formula_16  (a b : OAI.CKSSchwarzschild.Sphere) {z : OAI.CKSBoundarySurface.E3} (hz : z ∈ range OAI.CKSBoundarySurface.I3) :      OAI.CKSBoundarySurface.I3 (((OAI.CKSSchwarzschild.productChart a).symm ≫ₕ OAI.CKSSchwarzschild.productChart b) (OAI.CKSBoundarySurface.I3.symm z)) =        OAI.CKSSchwarzschild.join (z 0,((extChartAt OAI.CKSBoundarySurface.I2 a).symm ≫ extChartAt OAI.CKSBoundarySurface.I2 b) (OAI.CKSBoundarySurface.dropPlane z)) := by    have he : (OAI.CKSBoundarySurface.I3.symm z).val = z := OAI.CKSBoundarySurface.I3.right_inv hz    change OAI.CKSSchwarzschild.join ((OAI.CKSBoundarySurface.I3.symm z).val 0,      chartAt OAI.CKSBoundarySurface.E2 b ((chartAt OAI.CKSBoundarySurface.E2 a).symm (OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3.symm z).val))) = _    rw [he]    rfl  by    apply isManifold_of_contDiffOn    rintro e e' ⟨a,rfl⟩ ⟨b,rfl⟩    change ContDiffOn ℝ ∞      (fun z => OAI.CKSBoundarySurface.I3 (((OAI.CKSSchwarzschild.productChart a).symm ≫ₕ OAI.CKSSchwarzschild.productChart b) (OAI.CKSBoundarySurface.I3.symm z)))      (OAI.CKSBoundarySurface.I3.symm ⁻¹' ((OAI.CKSSchwarzschild.productChart a).symm ≫ₕ OAI.CKSSchwarzschild.productChart b).source ∩ range OAI.CKSBoundarySurface.I3)    have hs := (contDiffOn_ext_coord_change (I := OAI.CKSBoundarySurface.I2) (n := ∞) b a).comp      OAI.CKSBoundarySurface.dropPlane.contDiff.contDiffOn (proof_transition_maps_15 a b)    have h0 : ContDiff ℝ ∞ (fun z : OAI.CKSBoundarySurface.E3 => z 0) := by fun_prop    apply (OAI.CKSSchwarzschild.join.contDiff.comp_contDiffOn (h0.contDiffOn.prodMk hs)).congr    intro z hz    exact proof_transition_formula_16 a b hz.2end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundleopen scoped ContDiff Topologyopen CKSBoundarySurfacedef height (p : OAI.CKSSchwarzschild.Exterior) : ℝ := p.1.valend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundle Functionopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfacedef radius (m : ℝ) (p : OAI.CKSSchwarzschild.Exterior) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  2*m+OAI.CKSSchwarzschild.height pdef directionAmbient (p : OAI.CKSSchwarzschild.Exterior) : OAI.CKSBoundarySurface.E3 := p.2.valdef position (m : ℝ) (p : OAI.CKSSchwarzschild.Exterior) : OAI.CKSBoundarySurface.E3 := OAI.CKSSchwarzschild.radius m p • OAI.CKSSchwarzschild.directionAmbient pend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filteropen scoped ContDiff Topologydef velocity (m r : ℝ) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo;  -1 + (r+1)*Real.smoothTransition (r-2*m-1)def lapseSquared (m r : ℝ) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  1-2*m/r+(OAI.CKSSchwarzschild.velocity m r)^2def lapse (m r : ℝ) : ℝ := Real.sqrt (OAI.CKSSchwarzschild.lapseSquared m r)end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Manifold Bundle Set Filter CKSLorentz CKSMetricGluing CKSSpatialManifoldopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfacedef euclideanForm : OAI.CKSLorentz.SpatialBilinear := innerSL ℝdef radialUnit (x : OAI.CKSBoundarySurface.E3) : OAI.CKSBoundarySurface.E3 := ‖x‖⁻¹ • xlocal instance real_isScalarTower : IsScalarTower ℝ ℝ ℝ := inferInstancedef radialForm (x : OAI.CKSBoundarySurface.E3) : OAI.CKSLorentz.SpatialBilinear :=  letI := OAI.CKSSpatialManifold.real_id_compTriple  letI := OAI.CKSSpatialManifold.real_id_isometric  letI := OAI.CKSSchwarzschild.real_isScalarTower  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSLorentz.real_isTopologicalAddGroup  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_continuousConstSMul  let innerForm : OAI.CKSLorentz.SpatialBilinear := innerSL ℝ  (ContinuousLinearMap.mul ℝ ℝ).bilinearComp    (innerForm (OAI.CKSSchwarzschild.radialUnit x)) (innerForm (OAI.CKSSchwarzschild.radialUnit x))local instance spatialDual_continuousAdd : ContinuousAdd (OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) := inferInstancedef cartMetric (m : ℝ) (x : OAI.CKSBoundarySurface.E3) : OAI.CKSLorentz.SpatialBilinear :=  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_continuousConstSMul  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSSchwarzschild.spatialDual_continuousAdd  letI := OAI.CKSLorentz.spatialDual_smulCommClass  letI := OAI.CKSLorentz.spatialDual_continuousConstSMul  OAI.CKSSchwarzschild.euclideanForm + ((OAI.CKSSchwarzschild.lapseSquared m ‖x‖)⁻¹-1) • OAI.CKSSchwarzschild.radialForm xdef cartTensor (m : ℝ) (x : OAI.CKSBoundarySurface.E3) : OAI.CKSLorentz.SpatialBilinear :=  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_continuousConstSMul  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSSchwarzschild.spatialDual_continuousAdd  letI := OAI.CKSLorentz.spatialDual_smulCommClass  letI := OAI.CKSLorentz.spatialDual_continuousConstSMul  (OAI.CKSSchwarzschild.velocity m ‖x‖ / ‖x‖) • OAI.CKSSchwarzschild.euclideanForm +    (deriv (OAI.CKSSchwarzschild.velocity m) ‖x‖ / OAI.CKSSchwarzschild.lapseSquared m ‖x‖ - OAI.CKSSchwarzschild.velocity m ‖x‖ / ‖x‖) • OAI.CKSSchwarzschild.radialForm xdef metricInner (m : ℝ) : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := OAI.CKSSchwarzschild.Exterior) := OAI.CKSSpatialManifold.endInner OAI.CKSBoundarySurface.I3 (OAI.CKSSchwarzschild.position m) (OAI.CKSSchwarzschild.cartMetric m)def tensorInner (m : ℝ) : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := OAI.CKSSchwarzschild.Exterior) := OAI.CKSSpatialManifold.endInner OAI.CKSBoundarySurface.I3 (OAI.CKSSchwarzschild.position m) (OAI.CKSSchwarzschild.cartTensor m)end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicConstraintsopen Bundle Manifold Set Filter CKSLorentz CKSMetricGluingopen scoped ContDiff Topologyabbrev E := OAI.CKSLorentz.Elocal instance halfSpaceDimension_neZero : NeZero (3 : ℕ) := inferInstanceabbrev H := @EuclideanHalfSpace 3 OAI.CKSIntrinsicConstraints.halfSpaceDimension_neZeroabbrev I := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSIntrinsicConstraints.halfSpaceDimension_neZeroend OAI.CKSIntrinsicConstraintsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicConstraintsopen Bundle Manifold Set Filter CKSLorentz CKSMetricGluingopen scoped ContDiff Topologyattribute [local instance] halfSpaceDimension_neZeroattribute [local instance] CKSSpatialManifold.real_id_isometric CKSLorentz.real_smulCommClass CKSLorentz.spatialDual_smulCommClassvariable {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]def coordinateMomentumNorm (A B : OAI.CKSLorentz.SpatialTensor) (x : OAI.CKSIntrinsicConstraints.E) : ℝ :=  Real.sqrt (OAI.CKSLocalBending.momentumSq    (OAI.CKSAngularGeometry.physicalMetricJet (OAI.CKSLorentz.spatialCoefficients A) (OAI.CKSCartesianOuter.toPoint x))    (OAI.CKSAngularGeometry.physicalTensorJet (OAI.CKSLorentz.spatialCoefficients B) (OAI.CKSCartesianOuter.toPoint x)))end OAI.CKSIntrinsicConstraintsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicConstraintsopen Bundle Manifold Set Filter CKSLorentz CKSMetricGluingopen scoped ContDiff Topologyattribute [local instance] halfSpaceDimension_neZeroattribute [local instance] CKSSpatialManifold.real_id_isometric CKSLorentz.real_smulCommClass CKSLorentz.spatialDual_smulCommClassvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]structure ConstraintChart (g K : OAI.CKSMetricGluing.InnerField OAI.CKSIntrinsicConstraints.I (M := M)) (x : M) where  domain : Set M  coordinate : M → OAI.CKSIntrinsicConstraints.E  metric : OAI.CKSLorentz.SpatialTensor  tensor : OAI.CKSLorentz.SpatialTensor  isOpen : IsOpen domain  mem : x ∈ domain  smooth : ContMDiffOn OAI.CKSIntrinsicConstraints.I 𝓘(ℝ,OAI.CKSIntrinsicConstraints.E) ∞ coordinate domain  fullRank : ∀ y ∈ domain, Function.Bijective (mfderiv OAI.CKSIntrinsicConstraints.I 𝓘(ℝ,OAI.CKSIntrinsicConstraints.E) coordinate y)  coefficientSmooth :    letI := OAI.CKSLorentz.real_smulCommClass    letI := OAI.CKSLorentz.spatialDual_smulCommClass    ∀ y ∈ domain,      ContDiffAt ℝ ∞ metric (coordinate y) ∧ ContDiffAt ℝ ∞ tensor (coordinate y)  positiveSymmetric : ∀ y ∈ domain,    (∀ v w, metric (coordinate y) v w = metric (coordinate y) w v) ∧    (∀ v : OAI.CKSIntrinsicConstraints.E, v ≠ 0 → 0 < metric (coordinate y) v v) ∧    (∀ v w, tensor (coordinate y) v w = tensor (coordinate y) w v)  represents : ∀ y ∈ domain,    g y = OAI.CKSSpatialManifold.endInner OAI.CKSIntrinsicConstraints.I coordinate metric y ∧    K y = OAI.CKSSpatialManifold.endInner OAI.CKSIntrinsicConstraints.I coordinate tensor ydef ConstraintChart.energy {g K : OAI.CKSMetricGluing.InnerField OAI.CKSIntrinsicConstraints.I (M := M)} {x : M}    (c : OAI.CKSIntrinsicConstraints.ConstraintChart g K x) : ℝ := OAI.CKSLorentz.spatialEnergy c.metric c.tensor (c.coordinate x)def ConstraintChart.momentumNorm {g K : OAI.CKSMetricGluing.InnerField OAI.CKSIntrinsicConstraints.I (M := M)} {x : M}    (c : OAI.CKSIntrinsicConstraints.ConstraintChart g K x) : ℝ := OAI.CKSIntrinsicConstraints.coordinateMomentumNorm c.metric c.tensor (c.coordinate x)end OAI.CKSIntrinsicConstraintsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealattribute [local instance] CKSIntrinsicConstraints.halfSpaceDimension_neZeroabbrev E := EuclideanSpace ℝ (Fin 3)abbrev H := @EuclideanHalfSpace 3 OAI.CKSIntrinsicConstraints.halfSpaceDimension_neZeroabbrev I := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSIntrinsicConstraints.halfSpaceDimension_neZeroabbrev basis (i : Fin 3) : OAI.CKSIntrinsicVolume.E := EuclideanSpace.single i 1local instance model_finiteDimensional : FiniteDimensional ℝ OAI.CKSIntrinsicVolume.E := inferInstanceend OAI.CKSIntrinsicVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealattribute [local instance] CKSIntrinsicConstraints.halfSpaceDimension_neZeroattribute [local instance] model_finiteDimensionallocal instance model_borelSpace : BorelSpace OAI.CKSIntrinsicVolume.E := inferInstanceend OAI.CKSIntrinsicVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealattribute [local instance] CKSIntrinsicConstraints.halfSpaceDimension_neZerovariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]abbrev Metric := ContinuousRiemannianMetric OAI.CKSIntrinsicVolume.E (fun x : M => TangentSpace OAI.CKSIntrinsicVolume.I x)def chartFrame (x : M) (y : OAI.CKSIntrinsicVolume.E) : OAI.CKSIntrinsicVolume.E →L[ℝ] TangentSpace OAI.CKSIntrinsicVolume.I ((extChartAt OAI.CKSIntrinsicVolume.I x).symm y) :=  mfderivWithin 𝓘(ℝ,OAI.CKSIntrinsicVolume.E) OAI.CKSIntrinsicVolume.I (extChartAt OAI.CKSIntrinsicVolume.I x).symm (range OAI.CKSIntrinsicVolume.I) ydef chartMatrix (g : OAI.CKSIntrinsicVolume.Metric (M := M)) (x : M) (y : OAI.CKSIntrinsicVolume.E) : Matrix (Fin 3) (Fin 3) ℝ :=  letI := OAI.CKSMetricGluing.real_continuousAdd  letI := OAI.CKSMetricGluing.real_continuousConstSMul  letI := OAI.CKSMetricGluing.real_smulCommClass  let form : TangentSpace OAI.CKSIntrinsicVolume.I ((extChartAt OAI.CKSIntrinsicVolume.I x).symm y) →L[ℝ]      TangentSpace OAI.CKSIntrinsicVolume.I ((extChartAt OAI.CKSIntrinsicVolume.I x).symm y) →L[ℝ] ℝ :=    g.inner ((extChartAt OAI.CKSIntrinsicVolume.I x).symm y)  fun first second => form (OAI.CKSIntrinsicVolume.chartFrame x y (OAI.CKSIntrinsicVolume.basis first)) (OAI.CKSIntrinsicVolume.chartFrame x y (OAI.CKSIntrinsicVolume.basis second))def chartDensity (g : OAI.CKSIntrinsicVolume.Metric (M := M)) (x : M) (y : OAI.CKSIntrinsicVolume.E) : ℝ :=  Real.sqrt (OAI.CKSIntrinsicVolume.chartMatrix g x y).detend OAI.CKSIntrinsicVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]attribute [local instance] model_finiteDimensional model_borelSpacevariable [MeasurableSpace M] [BorelSpace M]def localVolume (g : OAI.CKSIntrinsicVolume.Metric (M := M)) (x : M) : Measure M :=  Measure.map (extChartAt OAI.CKSIntrinsicVolume.I x).symm    ((volume.restrict (extChartAt OAI.CKSIntrinsicVolume.I x).target).withDensity      (fun y => ENNReal.ofReal (OAI.CKSIntrinsicVolume.chartDensity g x y)))def IsVolume (g : OAI.CKSIntrinsicVolume.Metric (M := M)) (ν : Measure M) : Prop :=  ∀ x : M, ν.restrict (extChartAt OAI.CKSIntrinsicVolume.I x).source = OAI.CKSIntrinsicVolume.localVolume g xend OAI.CKSIntrinsicVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealattribute [local instance] CKSIntrinsicConstraints.halfSpaceDimension_neZerovariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M]  [MeasurableSpace M] [BorelSpace M] [SecondCountableTopology M]variable [IsManifold I 1 M]def riemannianVolume (g : OAI.CKSIntrinsicVolume.Metric (M := M)) : Measure M := Classical.epsilon (fun ν => OAI.CKSIntrinsicVolume.IsVolume g ν)end OAI.CKSIntrinsicVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filteropen scoped ContDiff Topologyattribute [local instance] Classical.propDecidablevariable {M : Type u_2} [TopologicalSpace M] [ChartedSpace H3 M]  [@IsManifold ℝ _ E3 _ _ H3    (@instTopologicalSpaceEuclideanHalfSpace 3 halfSpaceDimension_neZero) I3 ∞ M _ _]abbrev Boundary (M : Type u_2) [TopologicalSpace M] [ChartedSpace OAI.CKSBoundarySurface.H3 M] :=  @ModelWithCorners.boundary ℝ _ OAI.CKSBoundarySurface.E3 _ _ OAI.CKSBoundarySurface.H3    (@instTopologicalSpaceEuclideanHalfSpace 3 OAI.CKSBoundarySurface.halfSpaceDimension_neZero) OAI.CKSBoundarySurface.I3 M _ _end OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filteropen scoped ContDiff Topologyattribute [local instance] Classical.propDecidablevariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H3 M]  [@IsManifold ℝ _ E3 _ _ H3    (@instTopologicalSpaceEuclideanHalfSpace 3 halfSpaceDimension_neZero) I3 ∞ M _ _]def boundaryInverse (x : OAI.CKSBoundarySurface.Boundary M) (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.Boundary M :=  have proof_boundary_chart_zero_22 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSIntrinsicGeometry.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSIntrinsicGeometry.I3 1 M]  (x y : M) (hy : y ∈ (extChartAt OAI.CKSIntrinsicGeometry.I3 x).source)      (hS : y ∈ OAI.CKSIntrinsicGeometry.I3.boundary M) : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 = 0 := by    have hychart : y ∈ (chartAt OAI.CKSIntrinsicGeometry.H3 x).source := by simpa using hy    have hyrange : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ range OAI.CKSIntrinsicGeometry.I3 :=      (extChartAt_target_subset_range x) ((extChartAt OAI.CKSIntrinsicGeometry.I3 x).map_source hy)    have hnonneg : 0 ≤ (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := by      simpa only [OAI.CKSIntrinsicGeometry.I3, range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hyrange    have hnot : ¬OAI.CKSIntrinsicGeometry.I3.IsInteriorPoint y :=      (OAI.CKSIntrinsicGeometry.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mp hS    have hle : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 ≤ 0 := by      by_contra h      have hpos : 0 < (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := lt_of_not_ge h      have hint : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ interior (range OAI.CKSIntrinsicGeometry.I3) := by        simpa only [OAI.CKSIntrinsicGeometry.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hpos      apply hnot      apply (OAI.CKSIntrinsicGeometry.I3.isInteriorPoint_iff_of_mem_atlas one_ne_zero (chart_mem_atlas OAI.CKSIntrinsicGeometry.H3 x) hychart).mpr      exact (chartAt OAI.CKSIntrinsicGeometry.H3 x).mem_interior_extend_target ((chartAt OAI.CKSIntrinsicGeometry.H3 x).map_source hychart) hint    exact le_antisymm hle hnonneg  have proof_liftPlane_zero_6  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.liftPlane z 0 = 0 := rfl  have proof_liftPlane_succ_7  (z : OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSBoundarySurface.liftPlane z i.succ = z i := rfl  have proof_I3_liftHalf_23  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.liftHalf z) = OAI.CKSBoundarySurface.liftPlane z := rfl  have proof_boundary_iff_zero_24 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : M) (hy : y ∈ (chartAt OAI.CKSBoundarySurface.H3 x).source) :      y ∈ OAI.CKSBoundarySurface.I3.boundary M ↔ (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 = 0 := by    constructor    · exact proof_boundary_chart_zero_22 x y (by simp [*])    · intro hz      apply (OAI.CKSBoundarySurface.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mpr      intro hi      have hin := (OAI.CKSBoundarySurface.I3.isInteriorPoint_iff_of_mem_atlas        (by simp [*] : (∞ : ℕ∞ω) ≠ 0) (chart_mem_atlas OAI.CKSBoundarySurface.H3 x) hy).mp hi      have hr := (chartAt OAI.CKSBoundarySurface.H3 x).interior_extend_target_subset_interior_range hin      have hp : 0 < (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 := by        simpa only [OAI.CKSBoundarySurface.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq,          OpenPartialHomeomorph.extend_coe, Function.comp_apply] using hr      rw [hz] at hp      exact (lt_irrefl 0) hp  have proof_chart_inverse_boundary_25 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x).target) :      (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.I3.boundary M := by    apply (proof_boundary_iff_zero_24 x _ ((chartAt OAI.CKSBoundarySurface.H3 x).map_target hz)).mpr    rw [(chartAt OAI.CKSBoundarySurface.H3 x).right_inv hz]    rfl  have proof_boundaryInverse_mem_21 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.Boundary M :=    proof_chart_inverse_boundary_25 x.val hz  if hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target then    ⟨(chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z),proof_boundaryInverse_mem_21 x hz⟩ else xdef boundaryChart (x : OAI.CKSBoundarySurface.Boundary M) : OpenPartialHomeomorph (OAI.CKSBoundarySurface.Boundary M) OAI.CKSBoundarySurface.E2 :=  have proof_lift_drop_30  {z : OAI.CKSBoundarySurface.E3} (hz : z 0 = 0) : OAI.CKSBoundarySurface.liftPlane (OAI.CKSBoundarySurface.dropPlane z) = z := by    ext i    cases i using Fin.cases with    | zero => exact hz.symm    | succ i => rfl  have proof_boundary_chart_zero_22 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSIntrinsicGeometry.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSIntrinsicGeometry.I3 1 M]  (x y : M) (hy : y ∈ (extChartAt OAI.CKSIntrinsicGeometry.I3 x).source)      (hS : y ∈ OAI.CKSIntrinsicGeometry.I3.boundary M) : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 = 0 := by    have hychart : y ∈ (chartAt OAI.CKSIntrinsicGeometry.H3 x).source := by simpa using hy    have hyrange : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ range OAI.CKSIntrinsicGeometry.I3 :=      (extChartAt_target_subset_range x) ((extChartAt OAI.CKSIntrinsicGeometry.I3 x).map_source hy)    have hnonneg : 0 ≤ (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := by      simpa only [OAI.CKSIntrinsicGeometry.I3, range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hyrange    have hnot : ¬OAI.CKSIntrinsicGeometry.I3.IsInteriorPoint y :=      (OAI.CKSIntrinsicGeometry.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mp hS    have hle : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 ≤ 0 := by      by_contra h      have hpos : 0 < (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := lt_of_not_ge h      have hint : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ interior (range OAI.CKSIntrinsicGeometry.I3) := by        simpa only [OAI.CKSIntrinsicGeometry.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hpos      apply hnot      apply (OAI.CKSIntrinsicGeometry.I3.isInteriorPoint_iff_of_mem_atlas one_ne_zero (chart_mem_atlas OAI.CKSIntrinsicGeometry.H3 x) hychart).mpr      exact (chartAt OAI.CKSIntrinsicGeometry.H3 x).mem_interior_extend_target ((chartAt OAI.CKSIntrinsicGeometry.H3 x).map_source hychart) hint    exact le_antisymm hle hnonneg  have proof_liftPlane_zero_6  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.liftPlane z 0 = 0 := rfl  have proof_liftPlane_succ_7  (z : OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSBoundarySurface.liftPlane z i.succ = z i := rfl  have proof_I3_liftHalf_23  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.liftHalf z) = OAI.CKSBoundarySurface.liftPlane z := rfl  have proof_boundary_iff_zero_24 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : M) (hy : y ∈ (chartAt OAI.CKSBoundarySurface.H3 x).source) :      y ∈ OAI.CKSBoundarySurface.I3.boundary M ↔ (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 = 0 := by    constructor    · exact proof_boundary_chart_zero_22 x y (by simp [*])    · intro hz      apply (OAI.CKSBoundarySurface.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mpr      intro hi      have hin := (OAI.CKSBoundarySurface.I3.isInteriorPoint_iff_of_mem_atlas        (by simp [*] : (∞ : ℕ∞ω) ≠ 0) (chart_mem_atlas OAI.CKSBoundarySurface.H3 x) hy).mp hi      have hr := (chartAt OAI.CKSBoundarySurface.H3 x).interior_extend_target_subset_interior_range hin      have hp : 0 < (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 := by        simpa only [OAI.CKSBoundarySurface.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq,          OpenPartialHomeomorph.extend_coe, Function.comp_apply] using hr      rw [hz] at hp      exact (lt_irrefl 0) hp  have proof_liftHalf_drop_chart_26 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : M) (y : OAI.CKSBoundarySurface.Boundary M) (hy : y.val ∈ (chartAt OAI.CKSBoundarySurface.H3 x).source) :      OAI.CKSBoundarySurface.liftHalf (OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y.val))) = chartAt OAI.CKSBoundarySurface.H3 x y.val := by    apply Subtype.ext    exact proof_lift_drop_30 ((proof_boundary_iff_zero_24 x y.val hy).mp y.property)  have proof_chart_inverse_boundary_25 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x).target) :      (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.I3.boundary M := by    apply (proof_boundary_iff_zero_24 x _ ((chartAt OAI.CKSBoundarySurface.H3 x).map_target hz)).mpr    rw [(chartAt OAI.CKSBoundarySurface.H3 x).right_inv hz]    rfl  have proof_boundaryInverse_mem_21 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.Boundary M :=    proof_chart_inverse_boundary_25 x.val hz  have proof_boundaryInverse_val_27 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (OAI.CKSBoundarySurface.boundaryInverse x z).val = (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) := by    simp only [OAI.CKSBoundarySurface.boundaryInverse,dite_eq_left hz]  have proof_dropPlane_apply_31  (z : OAI.CKSBoundarySurface.E3) (i : Fin 2) : OAI.CKSBoundarySurface.dropPlane z i = z i.succ := rfl  have proof_drop_lift_28  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.liftPlane z) = z := by ext i; rfl  have proof_liftHalf_continuous_29  : Continuous OAI.CKSBoundarySurface.liftHalf :=    OAI.CKSBoundarySurface.liftPlane.continuous.subtype_mk (fun _ => by simp [*])  {    toFun y := OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x.val y.val))    invFun := OAI.CKSBoundarySurface.boundaryInverse x    source := Subtype.val ⁻¹' (chartAt OAI.CKSBoundarySurface.H3 x.val).source    target := OAI.CKSBoundarySurface.liftHalf ⁻¹' (chartAt OAI.CKSBoundarySurface.H3 x.val).target    map_source' := by      intro y hy      change OAI.CKSBoundarySurface.liftHalf (OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x.val y.val))) ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target      rw [proof_liftHalf_drop_chart_26 x.val y hy]      exact (chartAt OAI.CKSBoundarySurface.H3 x.val).map_source hy    map_target' := by      intro z hz      change (OAI.CKSBoundarySurface.boundaryInverse x z).val ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).source      rw [proof_boundaryInverse_val_27 x hz]      exact (chartAt OAI.CKSBoundarySurface.H3 x.val).map_target hz    left_inv' := by      intro y hy      have h := proof_liftHalf_drop_chart_26 x.val y hy      apply Subtype.ext      rw [proof_boundaryInverse_val_27 x (by rw [h]; exact (chartAt OAI.CKSBoundarySurface.H3 x.val).map_source hy),h]      exact (chartAt OAI.CKSBoundarySurface.H3 x.val).left_inv hy    right_inv' := by      intro z hz      rw [proof_boundaryInverse_val_27 x hz,(chartAt OAI.CKSBoundarySurface.H3 x.val).right_inv hz,proof_I3_liftHalf_23,proof_drop_lift_28]    open_source := (chartAt OAI.CKSBoundarySurface.H3 x.val).open_source.preimage continuous_subtype_val    open_target := (chartAt OAI.CKSBoundarySurface.H3 x.val).open_target.preimage proof_liftHalf_continuous_29    continuousOn_toFun := OAI.CKSBoundarySurface.dropPlane.continuous.comp_continuousOn      (OAI.CKSBoundarySurface.I3.continuous.comp_continuousOn ((chartAt OAI.CKSBoundarySurface.H3 x.val).continuousOn.comp        continuous_subtype_val.continuousOn (fun _ h => h)))    continuousOn_invFun := by      apply Topology.IsInducing.subtypeVal.continuousOn_iff.mpr      apply ((chartAt OAI.CKSBoundarySurface.H3 x.val).continuousOn_symm.comp        proof_liftHalf_continuous_29.continuousOn (fun _ h => h)).congr      intro z hz      exact proof_boundaryInverse_val_27 x hz  }instance boundaryChartedSpace : ChartedSpace OAI.CKSBoundarySurface.E2 (OAI.CKSBoundarySurface.Boundary M) where  atlas := range OAI.CKSBoundarySurface.boundaryChart  chartAt := OAI.CKSBoundarySurface.boundaryChart  mem_chart_source x := mem_chart_source OAI.CKSBoundarySurface.H3 x.val  chart_mem_atlas x := mem_range_self xend OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSBoundarySurfaceopen Set Manifold Bundle Filter Functionopen scoped ContDiff Topologyvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H3 M] [IsManifold I3 ∞ M]instance boundaryIsManifold : IsManifold OAI.CKSBoundarySurface.I2 ∞ (OAI.CKSBoundarySurface.Boundary M) :=  have proof_lift_mem_ext_target_34 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  [IsManifold OAI.CKSBoundarySurface.I3 ∞ M] (x : M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x).target) :      OAI.CKSBoundarySurface.liftPlane z ∈ (extChartAt OAI.CKSBoundarySurface.I3 x).target := by    change OAI.CKSBoundarySurface.liftPlane z ∈ ((chartAt OAI.CKSBoundarySurface.H3 x).extend OAI.CKSBoundarySurface.I3).target    rw [OpenPartialHomeomorph.extend_target']    exact ⟨OAI.CKSBoundarySurface.liftHalf z,hz,rfl⟩  have proof_ext_symm_lift_35 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  [IsManifold OAI.CKSBoundarySurface.I3 ∞ M] (x : M) (z : OAI.CKSBoundarySurface.E2) :      (extChartAt OAI.CKSBoundarySurface.I3 x).symm (OAI.CKSBoundarySurface.liftPlane z) = (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.liftHalf z) := by    change (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.I3.symm (OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.liftHalf z))) = _    rw [OAI.CKSBoundarySurface.I3.left_inv]  have proof_boundary_chart_zero_22 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSIntrinsicGeometry.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSIntrinsicGeometry.I3 1 M]  (x y : M) (hy : y ∈ (extChartAt OAI.CKSIntrinsicGeometry.I3 x).source)      (hS : y ∈ OAI.CKSIntrinsicGeometry.I3.boundary M) : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 = 0 := by    have hychart : y ∈ (chartAt OAI.CKSIntrinsicGeometry.H3 x).source := by simpa using hy    have hyrange : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ range OAI.CKSIntrinsicGeometry.I3 :=      (extChartAt_target_subset_range x) ((extChartAt OAI.CKSIntrinsicGeometry.I3 x).map_source hy)    have hnonneg : 0 ≤ (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := by      simpa only [OAI.CKSIntrinsicGeometry.I3, range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hyrange    have hnot : ¬OAI.CKSIntrinsicGeometry.I3.IsInteriorPoint y :=      (OAI.CKSIntrinsicGeometry.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mp hS    have hle : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 ≤ 0 := by      by_contra h      have hpos : 0 < (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := lt_of_not_ge h      have hint : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ interior (range OAI.CKSIntrinsicGeometry.I3) := by        simpa only [OAI.CKSIntrinsicGeometry.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hpos      apply hnot      apply (OAI.CKSIntrinsicGeometry.I3.isInteriorPoint_iff_of_mem_atlas one_ne_zero (chart_mem_atlas OAI.CKSIntrinsicGeometry.H3 x) hychart).mpr      exact (chartAt OAI.CKSIntrinsicGeometry.H3 x).mem_interior_extend_target ((chartAt OAI.CKSIntrinsicGeometry.H3 x).map_source hychart) hint    exact le_antisymm hle hnonneg  have proof_liftPlane_zero_6  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.liftPlane z 0 = 0 := rfl  have proof_liftPlane_succ_7  (z : OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSBoundarySurface.liftPlane z i.succ = z i := rfl  have proof_I3_liftHalf_23  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.liftHalf z) = OAI.CKSBoundarySurface.liftPlane z := rfl  have proof_boundary_iff_zero_24 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : M) (hy : y ∈ (chartAt OAI.CKSBoundarySurface.H3 x).source) :      y ∈ OAI.CKSBoundarySurface.I3.boundary M ↔ (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 = 0 := by    constructor    · exact proof_boundary_chart_zero_22 x y (by simp [*])    · intro hz      apply (OAI.CKSBoundarySurface.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mpr      intro hi      have hin := (OAI.CKSBoundarySurface.I3.isInteriorPoint_iff_of_mem_atlas        (by simp [*] : (∞ : ℕ∞ω) ≠ 0) (chart_mem_atlas OAI.CKSBoundarySurface.H3 x) hy).mp hi      have hr := (chartAt OAI.CKSBoundarySurface.H3 x).interior_extend_target_subset_interior_range hin      have hp : 0 < (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 := by        simpa only [OAI.CKSBoundarySurface.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq,          OpenPartialHomeomorph.extend_coe, Function.comp_apply] using hr      rw [hz] at hp      exact (lt_irrefl 0) hp  have proof_chart_inverse_boundary_25 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x).target) :      (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.I3.boundary M := by    apply (proof_boundary_iff_zero_24 x _ ((chartAt OAI.CKSBoundarySurface.H3 x).map_target hz)).mpr    rw [(chartAt OAI.CKSBoundarySurface.H3 x).right_inv hz]    rfl  have proof_boundaryInverse_mem_21 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.Boundary M :=    proof_chart_inverse_boundary_25 x.val hz  have proof_boundaryInverse_val_27 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (OAI.CKSBoundarySurface.boundaryInverse x z).val = (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) := by    simp only [OAI.CKSBoundarySurface.boundaryInverse,dite_eq_left hz]  have proof_boundaryChart_symm_val_36 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2} (hz : z ∈ (OAI.CKSBoundarySurface.boundaryChart x).target) :      ((OAI.CKSBoundarySurface.boundaryChart x).symm z).val = (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) :=    proof_boundaryInverse_val_27 x hz  have proof_transition_lift_maps_32 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : OAI.CKSBoundarySurface.Boundary M) :      MapsTo OAI.CKSBoundarySurface.liftPlane ((OAI.CKSBoundarySurface.boundaryChart x).symm ≫ₕ OAI.CKSBoundarySurface.boundaryChart y).source        ((extChartAt OAI.CKSBoundarySurface.I3 x.val).symm ≫ extChartAt OAI.CKSBoundarySurface.I3 y.val).source := by    intro z hz    change z ∈ (OAI.CKSBoundarySurface.boundaryChart x).target ∧ (OAI.CKSBoundarySurface.boundaryChart x).symm z ∈ (OAI.CKSBoundarySurface.boundaryChart y).source at hz    refine ⟨proof_lift_mem_ext_target_34 x.val hz.1, ?_⟩    change (extChartAt OAI.CKSBoundarySurface.I3 x.val).symm (OAI.CKSBoundarySurface.liftPlane z) ∈ (extChartAt OAI.CKSBoundarySurface.I3 y.val).source    rw [proof_ext_symm_lift_35]    have h := hz.2    change ((OAI.CKSBoundarySurface.boundaryChart x).symm z).val ∈ (chartAt OAI.CKSBoundarySurface.H3 y.val).source at h    rw [proof_boundaryChart_symm_val_36 x hz.1] at h    simpa only [extChartAt_source] using h  have proof_transition_formula_33 {M : Type u_1} [instLocal1 : TopologicalSpace.{u_1} M] [instLocal2 : ChartedSpace.{0, u_1} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u_1} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : z ∈ ((OAI.CKSBoundarySurface.boundaryChart x).symm ≫ₕ OAI.CKSBoundarySurface.boundaryChart y).source) :      ((OAI.CKSBoundarySurface.boundaryChart x).symm ≫ₕ OAI.CKSBoundarySurface.boundaryChart y) z =        OAI.CKSBoundarySurface.dropPlane (((extChartAt OAI.CKSBoundarySurface.I3 x.val).symm ≫ extChartAt OAI.CKSBoundarySurface.I3 y.val) (OAI.CKSBoundarySurface.liftPlane z)) := by    change OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 y.val ((OAI.CKSBoundarySurface.boundaryChart x).symm z).val)) = _    rw [proof_boundaryChart_symm_val_36 x hz.1]    change _ = OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 y.val ((extChartAt OAI.CKSBoundarySurface.I3 x.val).symm (OAI.CKSBoundarySurface.liftPlane z))))    rw [proof_ext_symm_lift_35]  by    apply isManifold_of_contDiffOn    rintro e e' ⟨x,rfl⟩ ⟨y,rfl⟩    change ContDiffOn ℝ ∞ (fun z => ((OAI.CKSBoundarySurface.boundaryChart x).symm ≫ₕ OAI.CKSBoundarySurface.boundaryChart y) z)      (id ⁻¹' ((OAI.CKSBoundarySurface.boundaryChart x).symm ≫ₕ OAI.CKSBoundarySurface.boundaryChart y).source ∩ range id)    rw [preimage_id,range_id,inter_univ]    apply (OAI.CKSBoundarySurface.dropPlane.contDiff.comp_contDiffOn ((contDiffOn_ext_coord_change (I := OAI.CKSBoundarySurface.I3) y.val x.val).comp      OAI.CKSBoundarySurface.liftPlane.contDiff.contDiffOn (proof_transition_lift_maps_32 x y))).congr    intro z hz    exact proof_transition_formula_33 x y hzend OAI.CKSBoundarySurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealabbrev E := EuclideanSpace ℝ (Fin 2)abbrev H := OAI.CKSSurfaceVolume.Eabbrev I := 𝓘(ℝ,OAI.CKSSurfaceVolume.E)abbrev basis (i : Fin 2) : OAI.CKSSurfaceVolume.E := EuclideanSpace.single i 1end OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]abbrev Metric := ContinuousRiemannianMetric OAI.CKSSurfaceVolume.E (fun x : M => TangentSpace OAI.CKSSurfaceVolume.I x)end OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealvariable {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]local instance real_continuousAdd : ContinuousAdd ℝ := inferInstancelocal instance real_continuousConstSMul : ContinuousConstSMul ℝ ℝ := inferInstancelocal instance real_smulCommClass :    @SMulCommClass ℝ ℝ ℝ Algebra.toSMul      (@instSMulOfMul ℝ (@Distrib.toMul ℝ        (@instDistribOfSemiring ℝ (@CommSemiring.toSemiring ℝ Real.instCommSemiring)))) :=  inferInstanceend OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]def chartFrame (x : M) (y : OAI.CKSSurfaceVolume.E) : OAI.CKSSurfaceVolume.E →L[ℝ] TangentSpace OAI.CKSSurfaceVolume.I ((extChartAt OAI.CKSSurfaceVolume.I x).symm y) :=  mfderivWithin 𝓘(ℝ,OAI.CKSSurfaceVolume.E) OAI.CKSSurfaceVolume.I (extChartAt OAI.CKSSurfaceVolume.I x).symm (range OAI.CKSSurfaceVolume.I) ydef chartMatrix (g : OAI.CKSSurfaceVolume.Metric (M := M)) (x : M) (y : OAI.CKSSurfaceVolume.E) : Matrix (Fin 2) (Fin 2) ℝ :=  letI := OAI.CKSSurfaceVolume.real_continuousAdd  letI := OAI.CKSSurfaceVolume.real_continuousConstSMul  letI := OAI.CKSSurfaceVolume.real_smulCommClass  let inner : TangentSpace OAI.CKSSurfaceVolume.I ((extChartAt OAI.CKSSurfaceVolume.I x).symm y) →L[ℝ]      TangentSpace OAI.CKSSurfaceVolume.I ((extChartAt OAI.CKSSurfaceVolume.I x).symm y) →L[ℝ] ℝ :=    g.inner ((extChartAt OAI.CKSSurfaceVolume.I x).symm y)  fun i j => inner (OAI.CKSSurfaceVolume.chartFrame x y (OAI.CKSSurfaceVolume.basis i))    (OAI.CKSSurfaceVolume.chartFrame x y (OAI.CKSSurfaceVolume.basis j))def chartDensity (g : OAI.CKSSurfaceVolume.Metric (M := M)) (x : M) (y : OAI.CKSSurfaceVolume.E) : ℝ :=  Real.sqrt (OAI.CKSSurfaceVolume.chartMatrix g x y).detend OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]variable [MeasurableSpace M] [BorelSpace M]def localVolume (g : OAI.CKSSurfaceVolume.Metric (M := M)) (x : M) : Measure M :=  Measure.map (extChartAt OAI.CKSSurfaceVolume.I x).symm    ((volume.restrict (extChartAt OAI.CKSSurfaceVolume.I x).target).withDensity      (fun y => ENNReal.ofReal (OAI.CKSSurfaceVolume.chartDensity g x y)))def IsVolume (g : OAI.CKSSurfaceVolume.Metric (M := M)) (ν : Measure M) : Prop :=  ∀ x : M, ν.restrict (extChartAt OAI.CKSSurfaceVolume.I x).source = OAI.CKSSurfaceVolume.localVolume g xend OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSurfaceVolumeopen Bundle Manifold Set MeasureTheoryopen scoped ContDiff ENNRealdef riemannianVolume {M : Type u_1} [TopologicalSpace M] [ChartedSpace OAI.CKSSurfaceVolume.H M]    [IsManifold OAI.CKSSurfaceVolume.I 1 M] [MeasurableSpace M] [BorelSpace M] [SecondCountableTopology M]    (g : OAI.CKSSurfaceVolume.Metric (M := M)) : Measure M :=  Classical.epsilon (fun ν => OAI.CKSSurfaceVolume.IsVolume g ν)end OAI.CKSSurfaceVolumeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyabbrev E2 := EuclideanSpace ℝ (Fin 2)abbrev I2 := 𝓘(ℝ,OAI.CKSInducedArea.E2)abbrev E3 := EuclideanSpace ℝ (Fin 3)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]local instance real_continuousAdd : ContinuousAdd ℝ := inferInstancelocal instance real_continuousConstSMul : ContinuousConstSMul ℝ ℝ := inferInstancelocal instance real_id_isometric : RingHomIsometric (RingHom.id ℝ) := inferInstancelocal instance real_id_compTriple :    RingHomCompTriple (RingHom.id ℝ) (RingHom.id ℝ) (RingHom.id ℝ) := inferInstancelocal instance real_smulCommClass :    @SMulCommClass ℝ ℝ ℝ Algebra.toSMul      (@instSMulOfMul ℝ (@Distrib.toMul ℝ        (@instDistribOfSemiring ℝ (@CommSemiring.toSemiring ℝ Real.instCommSemiring)))) :=  inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type u_3} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricabbrev Inner2 :=  letI := OAI.CKSInducedArea.real_smulCommClass  ∀ x : S, TangentSpace OAI.CKSInducedArea.I2 x →L[ℝ] TangentSpace OAI.CKSInducedArea.I2 x →L[ℝ] ℝend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type u_1} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type u_2} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricabbrev Smooth3 := ContMDiffRiemannianMetric I3 ∞ OAI.CKSInducedArea.E3 (fun x : N => TangentSpace I3 x)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricabbrev Bil2 :=  letI := OAI.CKSInducedArea.real_smulCommClass  OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E2 →L[ℝ] ℝabbrev Bil3 :=  letI := OAI.CKSInducedArea.real_smulCommClass  OAI.CKSInducedArea.E3 →L[ℝ] OAI.CKSInducedArea.E3 →L[ℝ] ℝend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type u_1} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type u_2} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type u_3} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricdef inducedInner (g : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) (φ : S → N) : OAI.CKSInducedArea.Inner2 (S := S) :=  letI := OAI.CKSInducedArea.real_id_isometric  letI := OAI.CKSInducedArea.real_id_compTriple  fun x => by    let : NormedAddCommGroup (TangentSpace OAI.CKSInducedArea.I2 x) := by unfold TangentSpace; infer_instance    let : NormedSpace ℝ (TangentSpace OAI.CKSInducedArea.I2 x) := by unfold TangentSpace; infer_instance    let : NormedAddCommGroup (TangentSpace I3 (φ x)) := by unfold TangentSpace; infer_instance    let : NormedSpace ℝ (TangentSpace I3 (φ x)) := by unfold TangentSpace; infer_instance    exact (g.inner (φ x)).bilinearComp (mfderiv OAI.CKSInducedArea.I2 I3 φ x) (mfderiv OAI.CKSInducedArea.I2 I3 φ x)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]local instance tangent_manifold_one : IsManifold I 1 M := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_onelocal instance tangent_vectorBundle : VectorBundle ℝ E (fun x : M => TangentSpace I x) :=  inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundlelocal instance scalar_smulCommClass : SMulCommClass ℝ ℝ ℝ := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClasslocal instance scalar_vectorBundle : VectorBundle ℝ ℝ (Bundle.Trivial M ℝ) := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundlelocal instance scalar_continuousSMul : ContinuousSMul ℝ ℝ := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMullocal instance covector_vectorBundle :    VectorBundle ℝ (E →L[ℝ] ℝ) (fun x : M => TangentSpace I x →L[ℝ] ℝ) :=  inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMulattribute [local instance] covector_vectorBundlelocal instance covector_topologicalAddGroup (x : M) :    IsTopologicalAddGroup (TangentSpace I x →L[ℝ] ℝ) := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMulattribute [local instance] covector_vectorBundleattribute [local instance] covector_topologicalAddGrouplocal instance covector_continuousSMul (x : M) :    ContinuousSMul ℝ (TangentSpace I x →L[ℝ] ℝ) := inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMulattribute [local instance] covector_vectorBundleattribute [local instance] covector_topologicalAddGroupattribute [local instance] covector_continuousSMullocal instance scalarAtlas : MemTrivializationAtlas (Bundle.Trivial.trivialization M ℝ) := ⟨rfl⟩end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMulattribute [local instance] covector_vectorBundleattribute [local instance] covector_topologicalAddGroupattribute [local instance] covector_continuousSMulattribute [local instance] OAI.CKSInducedArea.scalarAtlaslocal instance covector_atlas (e : Trivialization E (π E (fun x : M => TangentSpace I x)))    [MemTrivializationAtlas e] :    MemTrivializationAtlas      (e.continuousLinearMap (RingHom.id ℝ) (Bundle.Trivial.trivialization M ℝ)) :=  inferInstanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricvariable {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E]  {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)  {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]attribute [local instance] tangent_manifold_oneattribute [local instance] tangent_vectorBundleattribute [local instance] scalar_smulCommClassattribute [local instance] scalar_vectorBundleattribute [local instance] scalar_continuousSMulattribute [local instance] covector_vectorBundleattribute [local instance] covector_topologicalAddGroupattribute [local instance] covector_continuousSMulattribute [local instance] OAI.CKSInducedArea.scalarAtlasattribute [local instance] covector_atlasdef bilinearTriv (e : Trivialization E (π E (fun x : M => TangentSpace I x)))    [MemTrivializationAtlas e] :    Trivialization (E →L[ℝ] E →L[ℝ] ℝ) (π (E →L[ℝ] E →L[ℝ] ℝ) (fun x : M =>      TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ)) :=  letI := OAI.CKSInducedArea.real_smulCommClass  e.continuousLinearMap (RingHom.id ℝ)    (e.continuousLinearMap (RingHom.id ℝ) (Bundle.Trivial.trivialization M ℝ))instance bilinearTriv_atlas (e : Trivialization E (π E (fun x : M => TangentSpace I x)))    [MemTrivializationAtlas e] : MemTrivializationAtlas (OAI.CKSInducedArea.bilinearTriv I e) := by  unfold OAI.CKSInducedArea.bilinearTriv  infer_instanceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometriclocal instance covector_smulCommClass {Space : Type u_4} [NormedAddCommGroup Space]    [NormedSpace ℝ Space] :    letI := OAI.CKSInducedArea.real_smulCommClass    SMulCommClass ℝ ℝ (Space →L[ℝ] ℝ) := by  infer_instancelocal instance bilNorm2 : NormedAddCommGroup OAI.CKSInducedArea.Bil2 :=  letI := OAI.CKSInducedArea.real_smulCommClass  ContinuousLinearMap.toNormedAddCommGroup (E := OAI.CKSInducedArea.E2) (F := OAI.CKSInducedArea.E2 →L[ℝ] ℝ) (σ₁₂ := RingHom.id ℝ)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2local instance bilSpace2 : NormedSpace ℝ OAI.CKSInducedArea.Bil2 :=  letI := OAI.CKSInducedArea.real_smulCommClass  letI := OAI.CKSInducedArea.covector_smulCommClass (Space := OAI.CKSInducedArea.E2)  ContinuousLinearMap.toNormedSpaceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2local instance bilNorm3 : NormedAddCommGroup OAI.CKSInducedArea.Bil3 :=  letI := OAI.CKSInducedArea.real_smulCommClass  ContinuousLinearMap.toNormedAddCommGroup (E := OAI.CKSInducedArea.E3) (F := OAI.CKSInducedArea.E3 →L[ℝ] ℝ) (σ₁₂ := RingHom.id ℝ)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3local instance bilSpace3 : NormedSpace ℝ OAI.CKSInducedArea.Bil3 :=  letI := OAI.CKSInducedArea.real_smulCommClass  letI := OAI.CKSInducedArea.covector_smulCommClass (Space := OAI.CKSInducedArea.E3)  ContinuousLinearMap.toNormedSpaceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3attribute [local instance] OAI.CKSInducedArea.bilSpace3local instance mixNorm23 : NormedAddCommGroup (OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3 →L[ℝ] ℝ) :=  letI := OAI.CKSInducedArea.real_smulCommClass  ContinuousLinearMap.toNormedAddCommGroup (E := OAI.CKSInducedArea.E2) (F := OAI.CKSInducedArea.E3 →L[ℝ] ℝ) (σ₁₂ := RingHom.id ℝ)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3attribute [local instance] OAI.CKSInducedArea.bilSpace3attribute [local instance] OAI.CKSInducedArea.mixNorm23local instance mixSpace23 : NormedSpace ℝ (OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3 →L[ℝ] ℝ) :=  letI := OAI.CKSInducedArea.real_smulCommClass  letI := OAI.CKSInducedArea.covector_smulCommClass (Space := OAI.CKSInducedArea.E3)  ContinuousLinearMap.toNormedSpaceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3attribute [local instance] OAI.CKSInducedArea.bilSpace3attribute [local instance] OAI.CKSInducedArea.mixNorm23attribute [local instance] OAI.CKSInducedArea.mixSpace23local instance mixNorm32 : NormedAddCommGroup (OAI.CKSInducedArea.E3 →L[ℝ] OAI.CKSInducedArea.E2 →L[ℝ] ℝ) :=  letI := OAI.CKSInducedArea.real_smulCommClass  ContinuousLinearMap.toNormedAddCommGroup (E := OAI.CKSInducedArea.E3) (F := OAI.CKSInducedArea.E2 →L[ℝ] ℝ) (σ₁₂ := RingHom.id ℝ)end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type*} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type*} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3attribute [local instance] OAI.CKSInducedArea.bilSpace3attribute [local instance] OAI.CKSInducedArea.mixNorm23attribute [local instance] OAI.CKSInducedArea.mixSpace23attribute [local instance] OAI.CKSInducedArea.mixNorm32local instance mixSpace32 : NormedSpace ℝ (OAI.CKSInducedArea.E3 →L[ℝ] OAI.CKSInducedArea.E2 →L[ℝ] ℝ) :=  letI := OAI.CKSInducedArea.real_smulCommClass  letI := OAI.CKSInducedArea.covector_smulCommClass (Space := OAI.CKSInducedArea.E2)  ContinuousLinearMap.toNormedSpaceend OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSInducedAreaopen Bundle Manifold Set Bornology Filteropen scoped Bundle Manifold ContDiff Topologyvariable {H3 : Type u_1} [TopologicalSpace H3] {I3 : ModelWithCorners ℝ E3 H3}  {N : Type u_2} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  {S : Type u_3} [TopologicalSpace S] [ChartedSpace E2 S] [IsManifold I2 ∞ S]attribute [local instance] real_continuousAdd real_continuousConstSMul real_smulCommClass real_id_isometricattribute [local instance] OAI.CKSInducedArea.bilNorm2attribute [local instance] OAI.CKSInducedArea.bilSpace2attribute [local instance] OAI.CKSInducedArea.bilNorm3attribute [local instance] OAI.CKSInducedArea.bilSpace3attribute [local instance] OAI.CKSInducedArea.mixNorm23attribute [local instance] OAI.CKSInducedArea.mixSpace23attribute [local instance] OAI.CKSInducedArea.mixNorm32attribute [local instance] OAI.CKSInducedArea.mixSpace32def inducedMetric (g : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) (φ : S → N)    (hφ : ContMDiff OAI.CKSInducedArea.I2 I3 ∞ φ)    (hinj : ∀ x, Function.Injective (mfderiv OAI.CKSInducedArea.I2 I3 φ x)) :    ContMDiffRiemannianMetric OAI.CKSInducedArea.I2 ∞ OAI.CKSInducedArea.E2 (fun x : S => TangentSpace OAI.CKSInducedArea.I2 x) :=  have proof_inducedInner_symm_37 {H3 : Type u_1} [instLocal1 : TopologicalSpace.{u_1} H3] {I3 : ModelWithCorners.{0, 0, u_1} ℝ OAI.CKSInducedArea.E3 H3} {N : Type u_2} [instLocal4 : TopologicalSpace.{u_2} N] [instLocal5 : ChartedSpace.{u_1, u_2} H3 N] [instLocal6 : IsManifold.{0, 0, u_1, u_2} I3 (∞ : ℕ∞ω) N] {S : Type u_3} [instLocal8 : TopologicalSpace.{u_3} S] [instLocal9 : ChartedSpace.{0, u_3} OAI.CKSInducedArea.E2 S]  (metric : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) (immersion : S → N)      (point : S) (left right : TangentSpace OAI.CKSInducedArea.I2 point) :      OAI.CKSInducedArea.inducedInner metric immersion point left right = OAI.CKSInducedArea.inducedInner metric immersion point right left :=    metric.symm (immersion point) _ _  have proof_inducedInner_pos_38 {H3 : Type u_1} [instLocal1 : TopologicalSpace.{u_1} H3] {I3 : ModelWithCorners.{0, 0, u_1} ℝ OAI.CKSInducedArea.E3 H3} {N : Type u_2} [instLocal4 : TopologicalSpace.{u_2} N] [instLocal5 : ChartedSpace.{u_1, u_2} H3 N] [instLocal6 : IsManifold.{0, 0, u_1, u_2} I3 (∞ : ℕ∞ω) N] {S : Type u_3} [instLocal8 : TopologicalSpace.{u_3} S] [instLocal9 : ChartedSpace.{0, u_3} OAI.CKSInducedArea.E2 S]  (metric : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) (immersion : S → N)      (injective : ∀ point, Function.Injective (mfderiv OAI.CKSInducedArea.I2 I3 immersion point))      (point : S) (vector : TangentSpace OAI.CKSInducedArea.I2 point) (nonzero : vector ≠ 0) :      0 < OAI.CKSInducedArea.inducedInner metric immersion point vector vector :=    metric.pos (immersion point) _      (fun zero => nonzero (injective point (zero.trans (map_zero _).symm)))  have proof_positive_bilinear_bounded_41  (B : OAI.CKSInducedArea.Bil2) (hB : ∀ v : OAI.CKSInducedArea.E2, v ≠ 0 → 0 < B v v) :      IsVonNBounded ℝ {v : OAI.CKSInducedArea.E2 | B v v < 1} := by    have hc : Continuous (fun v : OAI.CKSInducedArea.E2 => B v v) := B.continuous.clm_apply continuous_id    obtain ⟨a,ha,hm⟩ := (isCompact_sphere (0 : OAI.CKSInducedArea.E2) 1).exists_isMinOn      (NormedSpace.sphere_nonempty.mpr (by norm_num : (0 : ℝ) ≤ 1)) hc.continuousOn    have haNorm : ‖a‖ = 1 := by simpa [Metric.mem_sphere] using ha    have ha0 : a ≠ 0 := by intro h; simp [h] at haNorm    have hp : 0 < B a a := hB a ha0    have hlow : ∀ v : OAI.CKSInducedArea.E2, B a a * ‖v‖ ^ 2 ≤ B v v := by      intro v      by_cases hv : v = 0      · simp [hv]      have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv      have hy : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by        rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hn), inv_mul_cancel₀ hn.ne']      have h := hm (show (‖v‖⁻¹ : ℝ) • v ∈ Metric.sphere (0 : OAI.CKSInducedArea.E2) 1 by        simpa [Metric.mem_sphere] using hy)      have he : B ((‖v‖⁻¹ : ℝ) • v) ((‖v‖⁻¹ : ℝ) • v) = B v v / ‖v‖^2 := by        simp only [map_smul, smul_apply, smul_eq_mul]        field_simp      change B a a ≤ B ((‖v‖⁻¹ : ℝ) • v) ((‖v‖⁻¹ : ℝ) • v) at h      rw [he] at h      exact (le_div_iff₀ (sq_pos_of_pos hn)).mp h    apply (NormedSpace.isVonNBounded_iff ℝ).mpr    apply (Metric.isBounded_iff_subset_ball (0 : OAI.CKSInducedArea.E2)).mpr    refine ⟨1 + (B a a)⁻¹, ?_⟩    intro v hv    change dist v 0 < 1 + (B a a)⁻¹    rw [dist_zero_right]    have hbv : B v v < 1 := hv    have hl := hlow v    have hinv : 0 < (B a a)⁻¹ := inv_pos.mpr hp    have he : B a a * (B a a)⁻¹ = 1 := mul_inv_cancel₀ hp.ne'    by_contra hh    have hn : 1 + (B a a)⁻¹ ≤ ‖v‖ := le_of_not_gt hh    have hn1 : 1 ≤ ‖v‖ := by linarith    have hn2 : ‖v‖ ≤ ‖v‖^2 := by nlinarith    have hmul := mul_le_mul_of_nonneg_left hn hp.le    have hmul2 := mul_le_mul_of_nonneg_left hn2 hp.le    nlinarith  have proof_inducedInner_bounded_39 {H3 : Type u_1} [instLocal1 : TopologicalSpace.{u_1} H3] {I3 : ModelWithCorners.{0, 0, u_1} ℝ OAI.CKSInducedArea.E3 H3} {N : Type u_2} [instLocal4 : TopologicalSpace.{u_2} N] [instLocal5 : ChartedSpace.{u_1, u_2} H3 N] [instLocal6 : IsManifold.{0, 0, u_1, u_2} I3 (∞ : ℕ∞ω) N] {S : Type u_3} [instLocal8 : TopologicalSpace.{u_3} S] [instLocal9 : ChartedSpace.{0, u_3} OAI.CKSInducedArea.E2 S]  (metric : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) (immersion : S → N)      (injective : ∀ point, Function.Injective (mfderiv OAI.CKSInducedArea.I2 I3 immersion point)) (point : S) :      IsVonNBounded ℝ {vector : TangentSpace OAI.CKSInducedArea.I2 point |        inducedInner metric immersion point vector vector < 1} :=    proof_positive_bilinear_bounded_41 (OAI.CKSInducedArea.inducedInner metric immersion point)      (proof_inducedInner_pos_38 metric immersion injective point)  have proof_bilinearComp_smoothAt_42 {S : Type u_3} [instLocal1 : TopologicalSpace.{u_3} S] [instLocal2 : ChartedSpace.{0, u_3} OAI.CKSInducedArea.E2 S] [instLocal3 : IsManifold.{0, 0, 0, u_3} OAI.CKSInducedArea.I2 (∞ : ℕ∞ω) S]  [IsManifold OAI.CKSInducedArea.I2 ∞ S]      {B : S → OAI.CKSInducedArea.Bil3} {A : S → OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3} {x : S}      (hB : ContMDiffAt OAI.CKSInducedArea.I2 𝓘(ℝ,OAI.CKSInducedArea.Bil3) ∞ B x)      (hA : ContMDiffAt OAI.CKSInducedArea.I2 𝓘(ℝ,OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3) ∞ A x) :      ContMDiffAt OAI.CKSInducedArea.I2 𝓘(ℝ,OAI.CKSInducedArea.Bil2) ∞ (fun y => (B y).bilinearComp (A y) (A y)) x := by    have flip1 : ContDiff ℝ ∞ (fun B : OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3 →L[ℝ] ℝ => B.flip) :=      by        let L : (OAI.CKSInducedArea.E2 →L[ℝ] OAI.CKSInducedArea.E3 →L[ℝ] ℝ) →L[ℝ] (OAI.CKSInducedArea.E3 →L[ℝ] OAI.CKSInducedArea.E2 →L[ℝ] ℝ) :=          (ContinuousLinearMap.flipₗᵢ ℝ OAI.CKSInducedArea.E2 OAI.CKSInducedArea.E3 ℝ).toContinuousLinearEquiv.toContinuousLinearMap        exact L.contDiff    have flip2 : ContDiff ℝ ∞ (fun B : OAI.CKSInducedArea.Bil2 => B.flip) :=      by        let L : OAI.CKSInducedArea.Bil2 →L[ℝ] OAI.CKSInducedArea.Bil2 :=          (ContinuousLinearMap.flipₗᵢ ℝ OAI.CKSInducedArea.E2 OAI.CKSInducedArea.E2 ℝ).toContinuousLinearEquiv.toContinuousLinearMap        exact L.contDiff    exact flip2.comp_contMDiffAt ((flip1.comp_contMDiffAt (hB.clm_comp hA)).clm_comp hA)  have proof_bilinearTriv_pair_43 {E : Type 0} [instLocal1 : NormedAddCommGroup.{0} E] [instLocal2 : NormedSpace.{0, 0} ℝ E] {H : Type 0} [instLocal4 : TopologicalSpace.{0} H] (I : ModelWithCorners.{0, 0, 0} ℝ E H) {M : Type u_3} [instLocal7 : TopologicalSpace.{u_3} M] [instLocal8 : ChartedSpace.{0, u_3} H M] [instLocal9 : IsManifold.{0, 0, 0, u_3} I (∞ : ℕ∞ω) M]  (e : Trivialization E (π E (fun x : M => TangentSpace I x)))      [MemTrivializationAtlas e] {x : M} (hx : x ∈ e.baseSet)      (B : TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ) (v w : E) :      ((OAI.CKSInducedArea.bilinearTriv I e) ⟨x,B⟩).2 v w = B (e.symmL ℝ x v) (e.symmL ℝ x w) := by    let := OAI.CKSInducedArea.scalarAtlas (M := M)    unfold OAI.CKSInducedArea.bilinearTriv    rw [Trivialization.continuousLinearMap_apply]    dsimp only [ContinuousLinearMap.comp_apply]    rw [Trivialization.continuousLinearMapAt_apply_of_mem (R := ℝ)      (e.continuousLinearMap (RingHom.id ℝ) (Trivial.trivialization M ℝ))      (show x ∈ (e.continuousLinearMap (RingHom.id ℝ) (Trivial.trivialization M ℝ)).baseSet by simpa using hx)]    rw [Trivialization.continuousLinearMap_apply]    dsimp only [ContinuousLinearMap.comp_apply]    rw [Trivialization.continuousLinearMapAt_apply_of_mem (R := ℝ)      (Trivial.trivialization M ℝ) (show x ∈ (Trivial.trivialization M ℝ).baseSet from mem_univ x)]    rfl  have proof_bilinearTriv_pair_44 {E : Type 0} [instLocal1 : NormedAddCommGroup.{0} E] [instLocal2 : NormedSpace.{0, 0} ℝ E] {H : Type u_1} [instLocal4 : TopologicalSpace.{u_1} H] (I : ModelWithCorners.{0, 0, u_1} ℝ E H) {M : Type u_2} [instLocal7 : TopologicalSpace.{u_2} M] [instLocal8 : ChartedSpace.{u_1, u_2} H M] [instLocal9 : IsManifold.{0, 0, u_1, u_2} I (∞ : ℕ∞ω) M]  (e : Trivialization E (π E (fun x : M => TangentSpace I x)))      [MemTrivializationAtlas e] {x : M} (hx : x ∈ e.baseSet)      (B : TangentSpace I x →L[ℝ] TangentSpace I x →L[ℝ] ℝ) (v w : E) :      ((OAI.CKSInducedArea.bilinearTriv I e) ⟨x,B⟩).2 v w = B (e.symmL ℝ x v) (e.symmL ℝ x w) := by    let := OAI.CKSInducedArea.scalarAtlas (M := M)    unfold OAI.CKSInducedArea.bilinearTriv    rw [Trivialization.continuousLinearMap_apply]    dsimp only [ContinuousLinearMap.comp_apply]    rw [Trivialization.continuousLinearMapAt_apply_of_mem (R := ℝ)      (e.continuousLinearMap (RingHom.id ℝ) (Trivial.trivialization M ℝ))      (show x ∈ (e.continuousLinearMap (RingHom.id ℝ) (Trivial.trivialization M ℝ)).baseSet by simpa using hx)]    rw [Trivialization.continuousLinearMap_apply]    dsimp only [ContinuousLinearMap.comp_apply]    rw [Trivialization.continuousLinearMapAt_apply_of_mem (R := ℝ)      (Trivial.trivialization M ℝ) (show x ∈ (Trivial.trivialization M ℝ).baseSet from mem_univ x)]    rfl  have proof_inducedInner_smooth_40 {H3 : Type u_1} [instLocal1 : TopologicalSpace.{u_1} H3] {I3 : ModelWithCorners.{0, 0, u_1} ℝ OAI.CKSInducedArea.E3 H3} {N : Type u_2} [instLocal4 : TopologicalSpace.{u_2} N] [instLocal5 : ChartedSpace.{u_1, u_2} H3 N] [instLocal6 : IsManifold.{0, 0, u_1, u_2} I3 (∞ : ℕ∞ω) N] {S : Type u_3} [instLocal8 : TopologicalSpace.{u_3} S] [instLocal9 : ChartedSpace.{0, u_3} OAI.CKSInducedArea.E2 S] [instLocal10 : IsManifold.{0, 0, 0, u_3} OAI.CKSInducedArea.I2 (∞ : ℕ∞ω) S]  (g : OAI.CKSInducedArea.Smooth3 (I3 := I3) (N := N)) {φ : S → N}      (hφ : ContMDiff OAI.CKSInducedArea.I2 I3 ∞ φ) :      ContMDiff OAI.CKSInducedArea.I2 (OAI.CKSInducedArea.I2.prod 𝓘(ℝ,OAI.CKSInducedArea.Bil2)) ∞        (fun x => TotalSpace.mk' OAI.CKSInducedArea.Bil2 x (OAI.CKSInducedArea.inducedInner g φ x)) := by    let := OAI.CKSInducedArea.real_id_isometric    let := OAI.CKSInducedArea.real_smulCommClass    intro x    let e := trivializationAt OAI.CKSInducedArea.E2 (TangentSpace OAI.CKSInducedArea.I2) x    let e' := trivializationAt OAI.CKSInducedArea.E3 (TangentSpace I3) (φ x)    have hx : x ∈ e.baseSet := mem_baseSet_trivializationAt OAI.CKSInducedArea.E2 (TangentSpace OAI.CKSInducedArea.I2) x    have hx' : φ x ∈ e'.baseSet := mem_baseSet_trivializationAt OAI.CKSInducedArea.E3 (TangentSpace I3) (φ x)    apply (contMDiffAt_section (𝕜 := ℝ) (B := S)      (E := fun z : S => TangentSpace OAI.CKSInducedArea.I2 z →L[ℝ] TangentSpace OAI.CKSInducedArea.I2 z →L[ℝ] ℝ)      (IB := OAI.CKSInducedArea.I2) (n := ∞) (F := OAI.CKSInducedArea.Bil2)      (s := OAI.CKSInducedArea.inducedInner g φ) x).mpr    change ContMDiffAt OAI.CKSInducedArea.I2 𝓘(ℝ,OAI.CKSInducedArea.Bil2) ∞      (fun y : S => ((OAI.CKSInducedArea.bilinearTriv OAI.CKSInducedArea.I2 e)        ⟨y, OAI.CKSInducedArea.inducedInner g φ y⟩).2) x    have hB := (contMDiffAt_section (𝕜 := ℝ) (B := N)      (E := fun z : N => TangentSpace I3 z →L[ℝ] TangentSpace I3 z →L[ℝ] ℝ)      (IB := I3) (n := ∞) (F := OAI.CKSInducedArea.Bil3) (s := g.inner) (φ x)).mp (g.contMDiff (φ x))    change ContMDiffAt I3 𝓘(ℝ,OAI.CKSInducedArea.Bil3) ∞      (fun y : N => ((OAI.CKSInducedArea.bilinearTriv I3 e') ⟨y, g.inner y⟩).2) (φ x) at hB    have hD := (hφ x).mfderiv_const (show (∞ : ℕ∞ω) + 1 ≤ ∞ by simp [*])    apply (proof_bilinearComp_smoothAt_42 (hB.comp x (hφ x)) hD).congr_of_eventuallyEq    filter_upwards [e.open_baseSet.mem_nhds hx,      hφ.continuous.continuousAt.preimage_mem_nhds (e'.open_baseSet.mem_nhds hx')] with y hy hy'    ext v w    rw [proof_bilinearTriv_pair_43 OAI.CKSInducedArea.I2 e hy]    rw [ContinuousLinearMap.bilinearComp_apply]    dsimp only [Function.comp_apply]    rw [proof_bilinearTriv_pair_44 I3 e' hy']    dsimp [OAI.CKSInducedArea.inducedInner,inTangentCoordinates,ContinuousLinearMap.inCoordinates]    rw [e'.symmL_continuousLinearMapAt hy', e'.symmL_continuousLinearMapAt hy']    rfl  {    inner := OAI.CKSInducedArea.inducedInner g φ    symm := proof_inducedInner_symm_37 g φ    pos := proof_inducedInner_pos_38 g φ hinj    isVonNBounded := proof_inducedInner_bounded_39 g φ hinj    contMDiff := proof_inducedInner_smooth_40 g hφ  }end OAI.CKSInducedAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSEmbeddingDerivativeopen Set Manifold Bundle Filter Functionopen scoped ContDiff Topologyabbrev E := EuclideanSpace ℝ (Fin 3)local instance halfSpaceDimension_neZero : NeZero (3 : ℕ) := inferInstanceabbrev H := @EuclideanHalfSpace 3 OAI.CKSEmbeddingDerivative.halfSpaceDimension_neZeroabbrev I := @modelWithCornersEuclideanHalfSpace 3 OAI.CKSEmbeddingDerivative.halfSpaceDimension_neZeroend OAI.CKSEmbeddingDerivativeendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSFullCutAreaopen Set Manifold Bundle Functionopen scoped ContDiff Topologyopen CKSGeometricCuts (OuterDomain)open CKSBoundarySurfacevariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N]instance domainTopology (D : OAI.CKSGeometricCuts.OuterDomain N) : TopologicalSpace D.Carrier := D.topologyinstance domainCharts (D : OAI.CKSGeometricCuts.OuterDomain N) : ChartedSpace OAI.CKSBoundarySurface.H3 D.Carrier := D.chartsinstance domainSmooth (D : OAI.CKSGeometricCuts.OuterDomain N) : IsManifold OAI.CKSBoundarySurface.I3 ∞ D.Carrier := D.smoothabbrev Surface (D : OAI.CKSGeometricCuts.OuterDomain N) := OAI.CKSBoundarySurface.Boundary D.Carrierdef cutInclusion (D : OAI.CKSGeometricCuts.OuterDomain N) : OAI.CKSFullCutArea.Surface D → N := D.inclusion ∘ Subtype.valend OAI.CKSFullCutAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSFullCutAreaopen Set Manifold Bundle Functionopen scoped ContDiff Topologyopen CKSGeometricCuts (OuterDomain)open CKSBoundarySurfacevariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N]variable [@IsManifold ℝ _ E3 _ _ H3  (@instTopologicalSpaceEuclideanHalfSpace 3 CKSBoundarySurface.halfSpaceDimension_neZero) I3 ∞ N _ _]instance domainSecondCountable [SecondCountableTopology N] (D : OAI.CKSGeometricCuts.OuterDomain N) :    SecondCountableTopology D.Carrier := D.embedding.isEmbedding.secondCountableTopologyabbrev SmoothMetric :=  letI := OAI.CKSBoundarySurface.halfSpaceDimension_neZero  letI := OAI.CKSInducedArea.tangent_vectorBundle OAI.CKSBoundarySurface.I3 (M := N)  ContMDiffRiemannianMetric OAI.CKSBoundarySurface.I3 ∞ OAI.CKSBoundarySurface.E3 (fun x : N => TangentSpace OAI.CKSBoundarySurface.I3 x)def cutMetric (g : OAI.CKSFullCutArea.SmoothMetric (N := N)) (D : OAI.CKSGeometricCuts.OuterDomain N) :    ContMDiffRiemannianMetric OAI.CKSBoundarySurface.I2 ∞ OAI.CKSBoundarySurface.E2 (fun x : OAI.CKSFullCutArea.Surface D => TangentSpace OAI.CKSBoundarySurface.I2 x) :=  have proof_boundary_chart_zero_49 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSIntrinsicGeometry.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSIntrinsicGeometry.I3 1 M]  (x y : M) (hy : y ∈ (extChartAt OAI.CKSIntrinsicGeometry.I3 x).source)      (hS : y ∈ OAI.CKSIntrinsicGeometry.I3.boundary M) : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 = 0 := by    have hychart : y ∈ (chartAt OAI.CKSIntrinsicGeometry.H3 x).source := by simpa using hy    have hyrange : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ range OAI.CKSIntrinsicGeometry.I3 :=      (extChartAt_target_subset_range x) ((extChartAt OAI.CKSIntrinsicGeometry.I3 x).map_source hy)    have hnonneg : 0 ≤ (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := by      simpa only [OAI.CKSIntrinsicGeometry.I3, range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hyrange    have hnot : ¬OAI.CKSIntrinsicGeometry.I3.IsInteriorPoint y :=      (OAI.CKSIntrinsicGeometry.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mp hS    have hle : (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 ≤ 0 := by      by_contra h      have hpos : 0 < (extChartAt OAI.CKSIntrinsicGeometry.I3 x y) 0 := lt_of_not_ge h      have hint : extChartAt OAI.CKSIntrinsicGeometry.I3 x y ∈ interior (range OAI.CKSIntrinsicGeometry.I3) := by        simpa only [OAI.CKSIntrinsicGeometry.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq] using hpos      apply hnot      apply (OAI.CKSIntrinsicGeometry.I3.isInteriorPoint_iff_of_mem_atlas one_ne_zero (chart_mem_atlas OAI.CKSIntrinsicGeometry.H3 x) hychart).mpr      exact (chartAt OAI.CKSIntrinsicGeometry.H3 x).mem_interior_extend_target ((chartAt OAI.CKSIntrinsicGeometry.H3 x).map_source hychart) hint    exact le_antisymm hle hnonneg  have proof_liftPlane_zero_6  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.liftPlane z 0 = 0 := rfl  have proof_liftPlane_succ_7  (z : OAI.CKSBoundarySurface.E2) (i : Fin 2) : OAI.CKSBoundarySurface.liftPlane z i.succ = z i := rfl  have proof_I3_liftHalf_23  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.liftHalf z) = OAI.CKSBoundarySurface.liftPlane z := rfl  have proof_boundary_iff_zero_50 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x y : M) (hy : y ∈ (chartAt OAI.CKSBoundarySurface.H3 x).source) :      y ∈ OAI.CKSBoundarySurface.I3.boundary M ↔ (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 = 0 := by    constructor    · exact proof_boundary_chart_zero_49 x y (by simp [*])    · intro hz      apply (OAI.CKSBoundarySurface.I3.isBoundaryPoint_iff_not_isInteriorPoint y).mpr      intro hi      have hin := (OAI.CKSBoundarySurface.I3.isInteriorPoint_iff_of_mem_atlas        (by simp [*] : (∞ : ℕ∞ω) ≠ 0) (chart_mem_atlas OAI.CKSBoundarySurface.H3 x) hy).mp hi      have hr := (chartAt OAI.CKSBoundarySurface.H3 x).interior_extend_target_subset_interior_range hin      have hp : 0 < (OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x y)) 0 := by        simpa only [OAI.CKSBoundarySurface.I3, interior_range_modelWithCornersEuclideanHalfSpace, mem_ofPred_eq,          OpenPartialHomeomorph.extend_coe, Function.comp_apply] using hr      rw [hz] at hp      exact (lt_irrefl 0) hp  have proof_chart_inverse_boundary_51 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x).target) :      (chartAt OAI.CKSBoundarySurface.H3 x).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.I3.boundary M := by    apply (proof_boundary_iff_zero_50 x _ ((chartAt OAI.CKSBoundarySurface.H3 x).map_target hz)).mpr    rw [(chartAt OAI.CKSBoundarySurface.H3 x).right_inv hz]    rfl  have proof_boundaryInverse_mem_52 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) ∈ OAI.CKSBoundarySurface.Boundary M :=    proof_chart_inverse_boundary_51 x.val hz  have proof_boundaryInverse_val_53 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : OAI.CKSBoundarySurface.liftHalf z ∈ (chartAt OAI.CKSBoundarySurface.H3 x.val).target) :      (OAI.CKSBoundarySurface.boundaryInverse x z).val = (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) := by    simp only [OAI.CKSBoundarySurface.boundaryInverse,dite_eq_left hz]  have proof_boundaryChart_symm_val_54 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2} (hz : z ∈ (OAI.CKSBoundarySurface.boundaryChart x).target) :      ((OAI.CKSBoundarySurface.boundaryChart x).symm z).val = (chartAt OAI.CKSBoundarySurface.H3 x.val).symm (OAI.CKSBoundarySurface.liftHalf z) :=    proof_boundaryInverse_val_53 x hz  have proof_inclusion_chart_formula_55 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) {z : OAI.CKSBoundarySurface.E2}      (hz : z ∈ (OAI.CKSBoundarySurface.boundaryChart x).target) :      writtenInExtChartAt OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 x (Subtype.val : OAI.CKSBoundarySurface.Boundary M → M) z = OAI.CKSBoundarySurface.liftPlane z := by    change OAI.CKSBoundarySurface.I3 (chartAt OAI.CKSBoundarySurface.H3 x.val (((OAI.CKSBoundarySurface.boundaryChart x).symm z).val)) = _    rw [proof_boundaryChart_symm_val_54 x hz, (chartAt OAI.CKSBoundarySurface.H3 x.val).right_inv hz]    rfl  have proof_inclusion_chart_eventually_56 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) :      writtenInExtChartAt OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 x (Subtype.val : OAI.CKSBoundarySurface.Boundary M → M) =ᶠ[𝓝 (extChartAt OAI.CKSBoundarySurface.I2 x x)] OAI.CKSBoundarySurface.liftPlane := by    have ht : (OAI.CKSBoundarySurface.boundaryChart x).target ∈ 𝓝 (extChartAt OAI.CKSBoundarySurface.I2 x x) :=      (OAI.CKSBoundarySurface.boundaryChart x).open_target.mem_nhds ((OAI.CKSBoundarySurface.boundaryChart x).map_source (mem_chart_source OAI.CKSBoundarySurface.E2 x))    filter_upwards [ht] with z hz    exact proof_inclusion_chart_formula_55 x hz  have proof_inclusion_smooth_57 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  : ContMDiff OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 ∞ (Subtype.val : OAI.CKSBoundarySurface.Boundary M → M) := by    intro x    apply contMDiffAt_iff.mpr    refine ⟨continuous_subtype_val.continuousAt, ?_⟩    exact ((OAI.CKSBoundarySurface.liftPlane.contDiff.contDiffAt).congr_of_eventuallyEq (proof_inclusion_chart_eventually_56 x)).contDiffWithinAt  have proof_cutInclusion_smooth_47 {N : Type u} [instLocal1 : TopologicalSpace.{u} N] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 N] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) N]  [IsManifold OAI.CKSBoundarySurface.I3 ∞ N] (D : OAI.CKSGeometricCuts.OuterDomain N) :      ContMDiff OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 ∞ (OAI.CKSFullCutArea.cutInclusion D) :=    D.embedding.isImmersion.contMDiff.comp proof_inclusion_smooth_57  have proof_extended_chart_derivative_injective_58 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSEmbeddingDerivative.H M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSEmbeddingDerivative.I (∞ : ℕ∞ω) M]  [IsManifold OAI.CKSEmbeddingDerivative.I ∞ M] (e : OpenPartialHomeomorph M OAI.CKSEmbeddingDerivative.H)      (he : e ∈ IsManifold.maximalAtlas OAI.CKSEmbeddingDerivative.I ∞ M) {x : M} (hx : x ∈ e.source) :      Injective (mfderiv OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) (e.extend OAI.CKSEmbeddingDerivative.I) x) := by    have hed : e.MDifferentiable OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I := ⟨      (contMDiffOn_of_mem_maximalAtlas he).mdifferentiableOn (by simp),      (contMDiffOn_symm_of_mem_maximalAtlas he).mdifferentiableOn (by simp)⟩    have hcomp := mfderiv_comp x OAI.CKSEmbeddingDerivative.I.mdifferentiableAt ((hed.1 x hx).mdifferentiableAt (e.open_source.mem_nhds hx))    change mfderiv OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) (e.extend OAI.CKSEmbeddingDerivative.I) x = _ at hcomp    rw [OAI.CKSEmbeddingDerivative.I.hasMFDerivAt.mfderiv] at hcomp    change mfderiv OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) (e.extend OAI.CKSEmbeddingDerivative.I) x = mfderiv OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I e x at hcomp    rw [hcomp]    exact hed.mfderiv_injective hx  have proof_immersion_derivative_injective_59 {M : Type u} {N : Type u} [instLocal2 : TopologicalSpace.{u} M] [instLocal3 : ChartedSpace.{0, u} OAI.CKSEmbeddingDerivative.H M] [instLocal4 : IsManifold.{0, 0, 0, u} OAI.CKSEmbeddingDerivative.I (∞ : ℕ∞ω) M] [instLocal5 : TopologicalSpace.{u} N] [instLocal6 : ChartedSpace.{0, u} OAI.CKSEmbeddingDerivative.H N] [instLocal7 : IsManifold.{0, 0, 0, u} OAI.CKSEmbeddingDerivative.I (∞ : ℕ∞ω) N]  [IsManifold OAI.CKSEmbeddingDerivative.I ∞ N] {f : M → N} {x : M}      (h : IsImmersionAt OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I ∞ f x) : Injective (mfderiv OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I f x) := by    let A : OAI.CKSEmbeddingDerivative.E →L[ℝ] OAI.CKSEmbeddingDerivative.E := (ContinuousLinearMap.fst ℝ OAI.CKSEmbeddingDerivative.E h.complement).comp      h.equiv.symm.toContinuousLinearMap    let L : N → OAI.CKSEmbeddingDerivative.E := A ∘ (h.codChart.extend OAI.CKSEmbeddingDerivative.I)    have hL : ContMDiffAt OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) ∞ L (f x) :=      A.contMDiff.contMDiffAt.comp _        (h.codChart.contMDiffAt_extend h.codChart_mem_maximalAtlas h.mem_codChart_source)    have heq : (L ∘ f) =ᶠ[𝓝 x] h.domChart.extend OAI.CKSEmbeddingDerivative.I := by      filter_upwards [h.domChart.open_source.mem_nhds h.mem_domChart_source] with y hy      have hy' : y ∈ (h.domChart.extend OAI.CKSEmbeddingDerivative.I).source := by simpa only [OpenPartialHomeomorph.extend_source] using hy      have hw := h.writtenInCharts ((h.domChart.extend OAI.CKSEmbeddingDerivative.I).map_source hy')      dsimp only [Function.comp_apply] at hw      rw [(h.domChart.extend OAI.CKSEmbeddingDerivative.I).left_inv hy'] at hw      change A ((h.codChart.extend OAI.CKSEmbeddingDerivative.I) (f y)) = _      rw [hw]      simp only [A,ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe,        ContinuousLinearEquiv.symm_apply_apply ]      rfl    have hdL := hL.mdifferentiableAt (by simp)    have hdf := h.contMDiffAt.mdifferentiableAt (by simp)    have hder := heq.mfderiv_eq (I := OAI.CKSEmbeddingDerivative.I) (I' := 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E))    rw [mfderiv_comp x hdL hdf] at hder    change _ = mfderiv OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) (h.domChart.extend OAI.CKSEmbeddingDerivative.I) x at hder    have hinj := proof_extended_chart_derivative_injective_58 h.domChart h.domChart_mem_maximalAtlas      h.mem_domChart_source    rw [← hder] at hinj    have hi' : Injective ((mfderiv OAI.CKSEmbeddingDerivative.I 𝓘(ℝ,OAI.CKSEmbeddingDerivative.E) L (f x)) ∘ (mfderiv OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I f x)) := hinj    exact hi'.of_comp  have proof_smooth_embedding_derivative_injective_60 {M : Type u} {N : Type u} [instLocal2 : TopologicalSpace.{u} M] [instLocal3 : ChartedSpace.{0, u} OAI.CKSEmbeddingDerivative.H M] [instLocal4 : IsManifold.{0, 0, 0, u} OAI.CKSEmbeddingDerivative.I (∞ : ℕ∞ω) M] [instLocal5 : TopologicalSpace.{u} N] [instLocal6 : ChartedSpace.{0, u} OAI.CKSEmbeddingDerivative.H N] [instLocal7 : IsManifold.{0, 0, 0, u} OAI.CKSEmbeddingDerivative.I (∞ : ℕ∞ω) N]  {f : M → N} (h : IsSmoothEmbedding OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I ∞ f) :      ∀ x, Injective (mfderiv OAI.CKSEmbeddingDerivative.I OAI.CKSEmbeddingDerivative.I f x) := fun x =>    proof_immersion_derivative_injective_59 (h.isImmersion.isImmersionAt x)  have proof_inclusion_hasMFDeriv_61 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) :      HasMFDerivAt OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 (Subtype.val : OAI.CKSBoundarySurface.Boundary M → M) x OAI.CKSBoundarySurface.liftPlane := by    refine ⟨continuous_subtype_val.continuousAt, ?_⟩    exact (OAI.CKSBoundarySurface.liftPlane.hasFDerivAt.congr_of_eventuallyEq (proof_inclusion_chart_eventually_56 x)).hasFDerivWithinAt  have proof_dropPlane_apply_31  (z : OAI.CKSBoundarySurface.E3) (i : Fin 2) : OAI.CKSBoundarySurface.dropPlane z i = z i.succ := rfl  have proof_drop_lift_28  (z : OAI.CKSBoundarySurface.E2) : OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.liftPlane z) = z := by ext i; rfl  have proof_inclusion_mfderiv_injective_62 {M : Type u} [instLocal1 : TopologicalSpace.{u} M] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 M] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) M]  (x : OAI.CKSBoundarySurface.Boundary M) :      Injective (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 (Subtype.val : OAI.CKSBoundarySurface.Boundary M → M) x) := by    rw [(proof_inclusion_hasMFDeriv_61 x).mfderiv]    intro a b h    change OAI.CKSBoundarySurface.liftPlane a = OAI.CKSBoundarySurface.liftPlane b at h    have hh : OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.liftPlane a) = OAI.CKSBoundarySurface.dropPlane (OAI.CKSBoundarySurface.liftPlane b) := congrArg OAI.CKSBoundarySurface.dropPlane h    change (a : OAI.CKSBoundarySurface.E2) = (b : OAI.CKSBoundarySurface.E2)    exact (proof_drop_lift_28 a).symm.trans (hh.trans (proof_drop_lift_28 b))  have proof_cutInclusion_derivative_injective_48 {N : Type u} [instLocal1 : TopologicalSpace.{u} N] [instLocal2 : ChartedSpace.{0, u} OAI.CKSBoundarySurface.H3 N] [instLocal3 : IsManifold.{0, 0, 0, u} OAI.CKSBoundarySurface.I3 (∞ : ℕ∞ω) N]  [IsManifold OAI.CKSBoundarySurface.I3 ∞ N] (D : OAI.CKSGeometricCuts.OuterDomain N) (x : OAI.CKSFullCutArea.Surface D) :      Injective (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 (OAI.CKSFullCutArea.cutInclusion D) x) := by    unfold OAI.CKSFullCutArea.cutInclusion    rw [mfderiv_comp x ((D.embedding.isImmersion.contMDiff x.val).mdifferentiableAt (by simp [*]))      ((proof_inclusion_smooth_57 x).mdifferentiableAt (by simp [*]))]    · change Injective ((mfderiv OAI.CKSBoundarySurface.I3 OAI.CKSBoundarySurface.I3 D.inclusion x.val) ∘        (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 (Subtype.val : OAI.CKSFullCutArea.Surface D → D.Carrier) x))      exact (proof_smooth_embedding_derivative_injective_60 D.embedding x.val).comp        (proof_inclusion_mfderiv_injective_62 x)  @OAI.CKSInducedArea.inducedMetric OAI.CKSBoundarySurface.H3    (@instTopologicalSpaceEuclideanHalfSpace 3 OAI.CKSBoundarySurface.halfSpaceDimension_neZero)    OAI.CKSBoundarySurface.I3 N _ _ _ (OAI.CKSFullCutArea.Surface D) _ _ _ g (OAI.CKSFullCutArea.cutInclusion D) (proof_cutInclusion_smooth_47 D)    (proof_cutInclusion_derivative_injective_48 D)end OAI.CKSFullCutAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSFullCutAreaopen Set Manifold Bundle MeasureTheoryopen scoped ContDiff Topology ENNRealopen CKSGeometricCuts (OuterDomain)open CKSBoundarySurfacevariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N]  [@IsManifold ℝ _ E3 _ _ H3    (@instTopologicalSpaceEuclideanHalfSpace 3 CKSBoundarySurface.halfSpaceDimension_neZero) I3 ∞ N _ _]  [SecondCountableTopology N]def area (g : OAI.CKSFullCutArea.SmoothMetric (N := N)) (D : OAI.CKSGeometricCuts.OuterDomain N) : ℝ≥0∞ :=  letI : MeasurableSpace (OAI.CKSFullCutArea.Surface D) := borel (OAI.CKSFullCutArea.Surface D)  letI : BorelSpace (OAI.CKSFullCutArea.Surface D) := ⟨rfl⟩  OAI.CKSSurfaceVolume.riemannianVolume (OAI.CKSFullCutArea.cutMetric g D).toContinuousRiemannianMetric univdef minEnclosingArea (g : OAI.CKSFullCutArea.SmoothMetric (N := N)) : ℝ≥0∞ :=  ⨅ D : OAI.CKSGeometricCuts.OuterDomain N, OAI.CKSFullCutArea.area g Dend OAI.CKSFullCutAreaendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicConstraintsopen Set Manifold Bundle MeasureTheory CKSLorentz CKSMetricGluingopen scoped ContDiff Topology ENNRealvariable {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]  [MeasurableSpace M] [BorelSpace M] [SecondCountableTopology M]local instance eight_atLeastTwo : Nat.AtLeastTwo 8 := inferInstanceend OAI.CKSIntrinsicConstraintsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSIntrinsicConstraintsopen Set Manifold Bundle MeasureTheory CKSLorentz CKSMetricGluingopen scoped ContDiff Topology ENNRealvariable {M : Type u_1} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]  [MeasurableSpace M] [BorelSpace M] [SecondCountableTopology M]attribute [local instance] eight_atLeastTwodef IntegrableConstraints (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSIntrinsicConstraints.I (M := M)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSIntrinsicConstraints.I (M := M)) : Prop :=  ∃ μ j : M → ℝ, Continuous μ ∧ Continuous j ∧    Integrable μ (OAI.CKSIntrinsicVolume.riemannianVolume g.toContinuousRiemannianMetric) ∧    Integrable j (OAI.CKSIntrinsicVolume.riemannianVolume g.toContinuousRiemannianMetric) ∧    (∀ x, 0 ≤ j x ∧ j x ≤ μ x) ∧    ∀ x (c : OAI.CKSIntrinsicConstraints.ConstraintChart g.inner K x),      8 * Real.pi * μ x = c.energy ∧ 8 * Real.pi * j x = c.momentumNormend OAI.CKSIntrinsicConstraintsendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Set Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifold CKSGeometricCuts CKSSphericalHarmonicsopen scoped ContDiff Topologyvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]local instance id_invPair (Scalar : Type u_1) [Semiring Scalar] :    RingHomInvPair (RingHom.id Scalar) (RingHom.id Scalar) := inferInstancelocal instance source_manifold_one : IsManifold OAI.CKSGeometricCuts.I3 1 N := inferInstanceend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Set Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifold CKSGeometricCuts CKSSphericalHarmonicsopen scoped ContDiff Topologyvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]attribute [local instance] CKSGeometricCuts.halfSpaceDimension_neZero id_invPair source_manifold_onelocal instance source_trivialization_isLinear (point : N) :    (trivializationAt OAI.CKSLorentz.E (TangentSpace OAI.CKSGeometricCuts.I3) point).IsLinear ℝ := inferInstanceend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Set Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifold CKSGeometricCuts CKSSphericalHarmonicsopen scoped ContDiff Topologyvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]attribute [local instance] CKSGeometricCuts.halfSpaceDimension_neZero id_invPair source_manifold_oneattribute [local instance] source_trivialization_isLineardef Orientable : Prop :=  ∃ o : ∀ x : N, Orientation ℝ (TangentSpace OAI.CKSGeometricCuts.I3 x) (Fin 3), ∀ x : N,    let e := trivializationAt OAI.CKSLorentz.E (TangentSpace OAI.CKSGeometricCuts.I3) x    ∃ V : Set N, IsOpen V ∧ ∃ hx : x ∈ V, ∃ hV : V ⊆ e.baseSet,      ∀ y (hy : y ∈ V),        let atPoint : TangentSpace OAI.CKSGeometricCuts.I3 y ≃L[ℝ] OAI.CKSLorentz.E := e.continuousLinearEquivAt ℝ y (hV hy)        let atBase : TangentSpace OAI.CKSGeometricCuts.I3 x ≃L[ℝ] OAI.CKSLorentz.E := e.continuousLinearEquivAt ℝ x (hV hx)        Orientation.map (Fin 3) atPoint.toLinearEquiv (o y) =        Orientation.map (Fin 3) atBase.toLinearEquiv (o x)structure CoordinateEnd where  radius : ℝ  radius_pos : 0 < radius  domain : Set N  isOpen : IsOpen domain  coordinate : N → OAI.CKSLorentz.E  inverse : OAI.CKSLorentz.E → N  smooth : ContMDiffOn OAI.CKSGeometricCuts.I3 𝓘(ℝ,OAI.CKSLorentz.E) ∞ coordinate domain  inverse_smooth : ContMDiffOn 𝓘(ℝ,OAI.CKSLorentz.E) OAI.CKSGeometricCuts.I3 ∞ inverse {y | radius < ‖y‖}  mapsTo : ∀ x ∈ domain, radius < ‖coordinate x‖  left_inverse : ∀ x ∈ domain, inverse (coordinate x) = x  right_inverse : ∀ y, radius < ‖y‖ → inverse y ∈ domain ∧ coordinate (inverse y) = y  closed_far_side : ∀ r, radius < r → IsClosed {x : N | x ∈ domain ∧ r ≤ ‖coordinate x‖}  compact_inner_side : ∀ r, radius < r → IsCompact {x : N | x ∉ domain ∨ ‖coordinate x‖ ≤ r}structure CKSData (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N)) where  chart : OAI.CKSSourceExterior.CoordinateEnd (N := N)  metricPerturbation : OAI.CKSLorentz.SpatialTensor  tensorPerturbation : OAI.CKSLorentz.SpatialTensor  metric_smooth : ∀ y, chart.radius < ‖y‖ → ContDiffAt ℝ ∞ metricPerturbation y  tensor_smooth : ∀ y, chart.radius < ‖y‖ → ContDiffAt ℝ ∞ tensorPerturbation y  represents : ∀ x ∈ chart.domain,    g.inner x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 chart.coordinate (OAI.CKSLorentz.sourceSpatial metricPerturbation) x ∧    K x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 chart.coordinate (OAI.CKSLorentz.sourceSpatial tensorPerturbation) x  massAspect : OAI.CKSLorentz.Sphere → ℝ  aspect_smooth : OAI.CKSSphericalHarmonics.SmoothSphere massAspect  patches : OAI.CKSLorentz.Sphere → OAI.CKSLorentz.CKSTensorPatch  covers : ∀ n, n ∈ (patches n).patch.sphereRegion (patches n).region  realizes : ∀ n, (patches n).Realizes metricPerturbation tensorPerturbation  aspect_eq : ∀ n, (patches n).RepresentsMassAspect massAspectend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundle MeasureTheoryopen scoped ContDiff Topology InnerProductSpace ENNRealopen CKSBoundarySurface CKSFullCutArea CKSSurfaceVolumeinstance exteriorConnected : ConnectedSpace OAI.CKSSchwarzschild.Exterior := by  have : ConnectedSpace OAI.CKSSchwarzschild.Radial := isConnected_iff_connectedSpace.mp isConnected_Ici  have : ConnectedSpace OAI.CKSSchwarzschild.Sphere := isConnected_iff_connectedSpace.mp    (isConnected_sphere (by rw [← Module.finrank_eq_rank]; norm_num : 1 < Module.rank ℝ E3) (0 : OAI.CKSBoundarySurface.E3) (by norm_num : (0 : ℝ) ≤ 1))  infer_instanceend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Set Manifold Bundle MeasureTheory CKSGeometricCuts CKSMetricGluingopen scoped ContDiffvariable {N : Type u_1} [TopologicalSpace N] [ChartedSpace H3 N]  [@IsManifold ℝ _ E3 _ _ H3    (@instTopologicalSpaceEuclideanHalfSpace 3 CKSGeometricCuts.halfSpaceDimension_neZero) I3 ∞ N _ _]  [SecondCountableTopology N]def cutArea (g : @OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.E3 _ _ OAI.CKSGeometricCuts.H3    (@instTopologicalSpaceEuclideanHalfSpace 3 OAI.CKSGeometricCuts.halfSpaceDimension_neZero)    OAI.CKSGeometricCuts.I3 N _ _ _) (D : OAI.CKSGeometricCuts.OuterDomain N) : ℝ :=  (OAI.CKSFullCutArea.area g D).toRealdef minimumEnclosingArea (g : @OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.E3 _ _ OAI.CKSGeometricCuts.H3    (@instTopologicalSpaceEuclideanHalfSpace 3 OAI.CKSGeometricCuts.halfSpaceDimension_neZero)    OAI.CKSGeometricCuts.I3 N _ _ _) : ℝ :=  ⨅ D : OAI.CKSGeometricCuts.OuterDomain N, OAI.CKSSourceExterior.cutArea g Ddef constraintsIntegrable (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N)) : Prop :=  letI : MeasurableSpace N := borel N  letI : BorelSpace N := ⟨rfl⟩  OAI.CKSIntrinsicConstraints.IntegrableConstraints g Kend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Manifold Bundle Set Filteropen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfacevariable {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V]def covariantPair (g :    letI := OAI.CKSLorentz.real_continuousAdd    letI := OAI.CKSLorentz.real_continuousConstSMul    letI := OAI.CKSLorentz.real_smulCommClass    V → V →L[ℝ] V →L[ℝ] ℝ) (N : V → V)    (x a b : V) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  g x (fderiv ℝ N x a) b + 1/2 *    (fderiv ℝ (fun y => g y (N x) b) x a +     fderiv ℝ (fun y => g y a b) x (N x) -     fderiv ℝ (fun y => g y a (N x)) x b)end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundleopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfaceabbrev Spacetime := ℝ × OAI.CKSBoundarySurface.E3abbrev SpacetimeBilinear :=  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSLorentz.real_continuousConstSMul  OAI.CKSSchwarzschild.Spacetime →L[ℝ] OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝdef timeForm : OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ := ContinuousLinearMap.fst ℝ ℝ OAI.CKSBoundarySurface.E3def spacePart : OAI.CKSSchwarzschild.Spacetime →L[ℝ] OAI.CKSBoundarySurface.E3 := ContinuousLinearMap.snd ℝ ℝ OAI.CKSBoundarySurface.E3def radiusForm (x : OAI.CKSBoundarySurface.E3) : OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ := (OAI.CKSSchwarzschild.euclideanForm (OAI.CKSSchwarzschild.radialUnit x)).comp OAI.CKSSchwarzschild.spacePartdef schwarzschildH (m r : ℝ) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  1-2*m/rlocal instance spacetimeDual_continuousAdd : ContinuousAdd (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) := inferInstancelocal instance spacetimeDual_isTopologicalAddGroup : IsTopologicalAddGroup (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) :=  inferInstancelocal instance spacetimeDual_smulCommClass : SMulCommClass ℝ ℝ (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) := inferInstancelocal instance spacetimeDual_continuousConstSMul : ContinuousConstSMul ℝ (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) :=  inferInstancelocal instance spacetimeDual_isScalarTower : IsScalarTower ℝ ℝ (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) := inferInstancelocal instance spacetimeDual_continuousSMul : ContinuousSMul ℝ (OAI.CKSSchwarzschild.Spacetime →L[ℝ] ℝ) := inferInstancedef advancedMetric (m : ℝ) (z : OAI.CKSSchwarzschild.Spacetime) : OAI.CKSSchwarzschild.SpacetimeBilinear :=  letI := OAI.CKSLorentz.real_continuousAdd  letI := OAI.CKSLorentz.real_continuousConstSMul  letI := OAI.CKSLorentz.real_smulCommClass  letI := OAI.CKSLorentz.real_isTopologicalAddGroup  letI := OAI.CKSSchwarzschild.spacetimeDual_continuousAdd  letI := OAI.CKSSchwarzschild.spacetimeDual_isTopologicalAddGroup  letI := OAI.CKSSchwarzschild.spacetimeDual_smulCommClass  letI := OAI.CKSSchwarzschild.spacetimeDual_continuousConstSMul  letI := OAI.CKSSchwarzschild.spacetimeDual_isScalarTower  letI := OAI.CKSSchwarzschild.spacetimeDual_continuousSMul  letI := OAI.CKSSpatialManifold.real_id_compTriple  letI := OAI.CKSSpatialManifold.real_id_isometric  let spatialMetric : OAI.CKSSchwarzschild.SpacetimeBilinear := OAI.CKSSchwarzschild.euclideanForm.bilinearComp OAI.CKSSchwarzschild.spacePart OAI.CKSSchwarzschild.spacePart  (-OAI.CKSSchwarzschild.schwarzschildH m ‖z.2‖) • OAI.CKSSchwarzschild.timeForm.smulRight OAI.CKSSchwarzschild.timeForm +    OAI.CKSSchwarzschild.timeForm.smulRight (OAI.CKSSchwarzschild.radiusForm z.2) + (OAI.CKSSchwarzschild.radiusForm z.2).smulRight OAI.CKSSchwarzschild.timeForm +    spatialMetric -    (OAI.CKSSchwarzschild.radiusForm z.2).smulRight (OAI.CKSSchwarzschild.radiusForm z.2)def advancedSlope (m r : ℝ) : ℝ := (OAI.CKSSchwarzschild.lapse m r*(OAI.CKSSchwarzschild.lapse m r-OAI.CKSSchwarzschild.velocity m r))⁻¹def advancedTime (m r : ℝ) : ℝ :=  letI := OAI.CKSLorentz.two_atLeastTwo  ∫ s in (2*m)..r, OAI.CKSSchwarzschild.advancedSlope m sdef advancedGraph (m : ℝ) (x : OAI.CKSBoundarySurface.E3) : OAI.CKSSchwarzschild.Spacetime := (OAI.CKSSchwarzschild.advancedTime m ‖x‖,x)def futureNormal (m : ℝ) (z : OAI.CKSSchwarzschild.Spacetime) : OAI.CKSSchwarzschild.Spacetime :=  ((OAI.CKSSchwarzschild.lapse m ‖z.2‖-OAI.CKSSchwarzschild.velocity m ‖z.2‖)⁻¹,OAI.CKSSchwarzschild.velocity m ‖z.2‖ • OAI.CKSSchwarzschild.radialUnit z.2)def graphTangent (m : ℝ) (x a : OAI.CKSBoundarySurface.E3) : OAI.CKSSchwarzschild.Spacetime :=  letI := OAI.CKSLorentz.two_atLeastTwo  letI : Inner ℝ OAI.CKSBoundarySurface.E3 :=    @InnerProductSpace.toInner ℝ OAI.CKSBoundarySurface.E3 _      (PiLp.seminormedAddCommGroup 2 (fun _ : Fin 3 => ℝ))      (PiLp.innerProductSpace (fun _ : Fin 3 => ℝ))  (OAI.CKSSchwarzschild.advancedSlope m ‖x‖ * ⟪OAI.CKSSchwarzschild.radialUnit x,a⟫_ℝ,a)end OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundle CKSSpatialManifold CKSMetricGluing CKSLorentzopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfacestructure HorizonRegularGraph (m : ℝ) : Prop where  ambient_smooth : ∀ z : OAI.CKSSchwarzschild.Spacetime, z.2 ≠ 0 → ContDiffAt ℝ ∞ (OAI.CKSSchwarzschild.advancedMetric m) z  ambient_symmetric : ∀ z a b, OAI.CKSSchwarzschild.advancedMetric m z a b = OAI.CKSSchwarzschild.advancedMetric m z b a  ambient_nondegenerate : ∀ z : OAI.CKSSchwarzschild.Spacetime, z.2 ≠ 0 → ∀ a : OAI.CKSSchwarzschild.Spacetime,    (∀ b, OAI.CKSSchwarzschild.advancedMetric m z a b = 0) → a = 0  graph_smooth_across : ∀ x : OAI.CKSBoundarySurface.E3, m < ‖x‖ → ContDiffAt ℝ ∞ (OAI.CKSSchwarzschild.advancedGraph m) x  graph_injective : Function.Injective (OAI.CKSSchwarzschild.advancedGraph m)  graph_derivative : ∀ x : OAI.CKSBoundarySurface.E3, m < ‖x‖ → ∀ a,    fderiv ℝ (OAI.CKSSchwarzschild.advancedGraph m) x a = OAI.CKSSchwarzschild.graphTangent m x a  horizon_time : OAI.CKSSchwarzschild.advancedTime m (2*m) = 0  horizon_slope : OAI.CKSSchwarzschild.advancedSlope m (2*m) = 1/2  induced_metric : ∀ x : OAI.CKSBoundarySurface.E3, 2*m ≤ ‖x‖ → ∀ a b,    OAI.CKSSchwarzschild.advancedMetric m (OAI.CKSSchwarzschild.advancedGraph m x) (OAI.CKSSchwarzschild.graphTangent m x a) (OAI.CKSSchwarzschild.graphTangent m x b) = OAI.CKSSchwarzschild.cartMetric m x a b  normal_unit : ∀ x : OAI.CKSBoundarySurface.E3, 2*m ≤ ‖x‖ →    OAI.CKSSchwarzschild.advancedMetric m (OAI.CKSSchwarzschild.advancedGraph m x) (OAI.CKSSchwarzschild.futureNormal m (OAI.CKSSchwarzschild.advancedGraph m x))      (OAI.CKSSchwarzschild.futureNormal m (OAI.CKSSchwarzschild.advancedGraph m x)) = -1  normal_orthogonal : ∀ x : OAI.CKSBoundarySurface.E3, 2*m ≤ ‖x‖ → ∀ a,    OAI.CKSSchwarzschild.advancedMetric m (OAI.CKSSchwarzschild.advancedGraph m x) (OAI.CKSSchwarzschild.futureNormal m (OAI.CKSSchwarzschild.advancedGraph m x)) (OAI.CKSSchwarzschild.graphTangent m x a) = 0  future_facing : ∀ x : OAI.CKSBoundarySurface.E3, 2*m ≤ ‖x‖ → 0 < (OAI.CKSSchwarzschild.futureNormal m (OAI.CKSSchwarzschild.advancedGraph m x)).1  induced_second_form : ∀ x : OAI.CKSBoundarySurface.E3, 2*m ≤ ‖x‖ → ∀ a b,    OAI.CKSSchwarzschild.covariantPair (OAI.CKSSchwarzschild.advancedMetric m) (OAI.CKSSchwarzschild.futureNormal m) (OAI.CKSSchwarzschild.advancedGraph m x)      (OAI.CKSSchwarzschild.graphTangent m x a) (OAI.CKSSchwarzschild.graphTangent m x b) = OAI.CKSSchwarzschild.cartTensor m x a bend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundleopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfaceattribute [local instance] CKSInducedArea.real_continuousAdd CKSInducedArea.real_smulCommClass CKSInducedArea.real_continuousConstSMul CKSBoundarySurface.two_atLeastTwolocal instance frameCovectorModule : Module ℝ (OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) :=  @ContinuousLinearMap.module ℝ ℝ ℝ _ _ _ OAI.CKSBoundarySurface.E3 _ _ _ ℝ _ _ _ _    OAI.CKSInducedArea.real_smulCommClass OAI.CKSInducedArea.real_continuousConstSMul    (RingHom.id ℝ) OAI.CKSInducedArea.real_continuousAddend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSchwarzschildopen Set Filter Manifold Bundleopen scoped ContDiff Topology InnerProductSpaceopen CKSBoundarySurfaceattribute [local instance] CKSInducedArea.real_continuousAdd CKSInducedArea.real_smulCommClass CKSInducedArea.real_continuousConstSMul CKSBoundarySurface.two_atLeastTwoattribute [local instance] OAI.CKSSchwarzschild.frameCovectorModuledef christoffelPair (g : OAI.CKSBoundarySurface.E3 → OAI.CKSBoundarySurface.E3 →L[ℝ] OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) (x a b c : OAI.CKSBoundarySurface.E3) : ℝ :=  1/2 * (fderiv ℝ (fun z => g z b c) x a +    fderiv ℝ (fun z => g z a c) x b - fderiv ℝ (fun z => g z a b) x c)def firstFrame (g : OAI.CKSBoundarySurface.E3 →L[ℝ] OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) (T : OAI.CKSBoundarySurface.E2 →L[ℝ] OAI.CKSBoundarySurface.E3) : OAI.CKSBoundarySurface.E2 :=  (Real.sqrt (g (T (EuclideanSpace.single 0 1)) (T (EuclideanSpace.single 0 1))))⁻¹ •    EuclideanSpace.single 0 1def secondResidual (g : OAI.CKSBoundarySurface.E3 →L[ℝ] OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) (T : OAI.CKSBoundarySurface.E2 →L[ℝ] OAI.CKSBoundarySurface.E3) : OAI.CKSBoundarySurface.E2 :=  EuclideanSpace.single 1 1 -    (g (T (EuclideanSpace.single 0 1)) (T (EuclideanSpace.single 1 1)) /      g (T (EuclideanSpace.single 0 1)) (T (EuclideanSpace.single 0 1))) • EuclideanSpace.single 0 1def secondFrame (g : OAI.CKSBoundarySurface.E3 →L[ℝ] OAI.CKSBoundarySurface.E3 →L[ℝ] ℝ) (T : OAI.CKSBoundarySurface.E2 →L[ℝ] OAI.CKSBoundarySurface.E3) : OAI.CKSBoundarySurface.E2 :=  (Real.sqrt (g (T (OAI.CKSSchwarzschild.secondResidual g T)) (T (OAI.CKSSchwarzschild.secondResidual g T))))⁻¹ • OAI.CKSSchwarzschild.secondResidual g Tend OAI.CKSSchwarzschildendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Manifold CKSGeometricCutsopen scoped ContDiffvariable {N : Type*} [TopologicalSpace N] [ChartedSpace H3 N]local instance halfSpace_locallyCompact : LocallyCompactSpace OAI.CKSGeometricCuts.H3 := by  change LocallyCompactSpace {x : EuclideanSpace ℝ (Fin 3) // 0 ≤ x 0}  have hc : IsClosed {x : EuclideanSpace ℝ (Fin 3) | 0 ≤ x 0} :=    isClosed_le continuous_const (by fun_prop)  exact hc.locallyCompactSpaceend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Manifold CKSGeometricCutsopen scoped ContDiffvariable {N : Type u_1} [TopologicalSpace N] [ChartedSpace H3 N]local instance manifold_regular [IsManifold OAI.CKSGeometricCuts.I3 ∞ N] [T2Space N] : T3Space N := by  let := OAI.CKSSourceExterior.halfSpace_locallyCompact  let : LocallyCompactSpace N := ChartedSpace.locallyCompactSpace OAI.CKSGeometricCuts.H3 N  infer_instanceend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSSourceExterioropen Set Manifold Bundle Filter CKSLorentz CKSMetricGluing CKSSpatialManifold CKSGeometricCutsopen CKSADM (Symbol)open scoped ContDiff Topologyvariable {N : Type u_1} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]  [T2Space N] [SecondCountableTopology N] [ConnectedSpace N]attribute [local instance] manifold_regulartheorem area_controlled_end_replacement    (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N))    (hK : ContMDiff OAI.CKSGeometricCuts.I3 (OAI.CKSGeometricCuts.I3.prod 𝓘(ℝ,OAI.CKSLorentz.SpatialBilinear)) ∞ (OAI.CKSMetricGluing.innerSection OAI.CKSGeometricCuts.I3 K))    (hKsym : ∀ x v w, K x v w = K x w v)    (hOrient : OAI.CKSSourceExterior.Orientable (N := N))    (hScompact : IsCompact (OAI.CKSGeometricCuts.I3.boundary N)) (hSnonempty : (OAI.CKSGeometricCuts.I3.boundary N).Nonempty)    (hcomplete : OAI.CKSReplacementCompleteness.IsComplete OAI.CKSGeometricCuts.I3 g.toContinuousRiemannianMetric)    (hDEC : OAI.CKSSpatialManifold.PhysicalDEC OAI.CKSGeometricCuts.I3 g.inner K)    (d : OAI.CKSSourceExterior.CKSData g K)    (ht : ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖ < (OAI.CKSLorentz.bondiCharge d.massAspect).1) :    ∃ (d' : OAI.CKSSourceExterior.CKSData g K) (R₀ : ℝ) (ε : ℝ → ℝ),      OAI.CKSLorentz.bondiCharge d'.massAspect =        (Real.sqrt ((OAI.CKSLorentz.bondiCharge d.massAspect).1^2 - ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖^2),0) ∧      d'.chart.radius < R₀ ∧ 12 ≤ R₀ ∧ Tendsto ε atTop (𝓝 0) ∧      OAI.CKSFullCutArea.minEnclosingArea g < ⊤ ∧      ∀ R ≥ R₀, (0 ≤ ε R ∧ ε R < 1) ∧        ∃ (gR : OAI.CKSMetricGluing.SmoothMetric OAI.CKSGeometricCuts.I3 (M := N)) (kR : OAI.CKSMetricGluing.InnerField OAI.CKSGeometricCuts.I3 (M := N))          (G k : OAI.CKSLorentz.SpatialTensor) (η : ℝ),          ContMDiff OAI.CKSGeometricCuts.I3 (OAI.CKSGeometricCuts.I3.prod 𝓘(ℝ,OAI.CKSLorentz.SpatialBilinear)) ∞ (OAI.CKSMetricGluing.innerSection OAI.CKSGeometricCuts.I3 kR) ∧          (∀ x v w, kR x v w = kR x w v) ∧          (∀ x, x ∉ d'.chart.domain ∨ ‖d'.chart.coordinate x‖ ≤ R →            gR.inner x = g.inner x ∧ kR x = K x) ∧          OAI.CKSSpatialManifold.PhysicalDEC OAI.CKSGeometricCuts.I3 gR.inner kR ∧          (∀ x v, (1-ε R) * g.inner x v v ≤ gR.inner x v v) ∧          OAI.CKSReplacementCompleteness.IsComplete OAI.CKSGeometricCuts.I3 gR.toContinuousRiemannianMetric ∧          OAI.CKSSourceExterior.constraintsIntegrable gR kR ∧          (∀ x ∈ d'.chart.domain,            gR.inner x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 d'.chart.coordinate G x ∧            kR x = OAI.CKSSpatialManifold.endInner OAI.CKSGeometricCuts.I3 d'.chart.coordinate k x) ∧          (∀ i j : Fin 3, OAI.CKSADM.Symbol 1            (fun x => OAI.CKSLorentz.spatialCartesian G x i j - (if i=j then 1 else 0))) ∧          (∀ x : OAI.CKSLorentz.E, 2*R^2 < ‖x‖ → k x = 0) ∧          Tendsto (OAI.CKSLorentz.spatialADMEnergy G) atTop            (𝓝 (Real.sqrt ((OAI.CKSLorentz.bondiCharge d.massAspect).1^2 - ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖^2) + η)) ∧          Tendsto (OAI.CKSLorentz.spatialADMMomentum G k) atTop (𝓝 0) ∧          0 ≤ η ∧ η ≤ 2 * R ^ (-(1/2 : ℝ)) ∧          OAI.CKSFullCutArea.minEnclosingArea gR < ⊤ ∧          (1-ε R) * OAI.CKSSourceExterior.minimumEnclosingArea g ≤ OAI.CKSSourceExterior.minimumEnclosingArea gR ∧          ∀ D : OAI.CKSGeometricCuts.OuterDomain N,            OAI.CKSFullCutArea.area g D < ⊤ ∧ OAI.CKSFullCutArea.area gR D < ⊤ ∧            (1-ε R) * OAI.CKSSourceExterior.cutArea g D ≤ OAI.CKSSourceExterior.cutArea gR D := by  sorryend OAI.CKSSourceExteriorendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regulardef secondForm (q : OAI.CKSLorentz.SpatialTensor) (f : OAI.CKSBoundarySurface.E2 → OAI.CKSBoundarySurface.E3) (y : OAI.CKSBoundarySurface.E2) (ν : OAI.CKSBoundarySurface.E3)    (a b : OAI.CKSBoundarySurface.E2) : ℝ :=  -(q (f y) ν (fderiv ℝ (fun z => fderiv ℝ f z b) y a) +    OAI.CKSSchwarzschild.christoffelPair q (f y) (fderiv ℝ f y a) (fderiv ℝ f y b) ν)def meanCurvature (q : OAI.CKSLorentz.SpatialTensor) (f : OAI.CKSBoundarySurface.E2 → OAI.CKSBoundarySurface.E3) (y : OAI.CKSBoundarySurface.E2) (ν : OAI.CKSBoundarySurface.E3) : ℝ :=  let a := OAI.CKSSchwarzschild.firstFrame (q (f y)) (fderiv ℝ f y)  let b := OAI.CKSSchwarzschild.secondFrame (q (f y)) (fderiv ℝ f y)  OAI.CKSMain.secondForm q f y ν a a + OAI.CKSMain.secondForm q f y ν b bdef tangentialTrace (q k : OAI.CKSLorentz.SpatialTensor) (f : OAI.CKSBoundarySurface.E2 → OAI.CKSBoundarySurface.E3) (y : OAI.CKSBoundarySurface.E2) : ℝ :=  let a := fderiv ℝ f y (OAI.CKSSchwarzschild.firstFrame (q (f y)) (fderiv ℝ f y))  let b := fderiv ℝ f y (OAI.CKSSchwarzschild.secondFrame (q (f y)) (fderiv ℝ f y))  k (f y) a a + k (f y) b bend OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]structure BoundaryGeometry (q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)) where  normal : ∀ p : OAI.CKSBoundarySurface.Boundary N, TangentSpace OAI.CKSBoundarySurface.I3 p.val  unit : ∀ p, q.inner p.val (normal p) (normal p) = 1  orthogonal : ∀ p a, q.inner p.val (normal p)    (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 (Subtype.val : OAI.CKSBoundarySurface.Boundary N → N) p a) = 0  inward : ∀ p, 0 < (show OAI.CKSBoundarySurface.E3 from normal p) 0  chart : ∀ p : OAI.CKSBoundarySurface.Boundary N, OAI.CKSIntrinsicConstraints.ConstraintChart q.inner k p.val  normal_smooth : ∀ p, ContMDiffAt OAI.CKSBoundarySurface.I2 𝓘(ℝ,OAI.CKSBoundarySurface.E3) ∞    (fun z : OAI.CKSBoundarySurface.Boundary N => mfderiv OAI.CKSBoundarySurface.I3 𝓘(ℝ,OAI.CKSBoundarySurface.E3) (chart p).coordinate z.val (normal z)) pdef BoundaryGeometry.parameter {q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)} {k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)}    (B : OAI.CKSMain.BoundaryGeometry q k) (p : OAI.CKSBoundarySurface.Boundary N) : OAI.CKSBoundarySurface.E2 → OAI.CKSBoundarySurface.E3 :=  (B.chart p).coordinate ∘ (Subtype.val : OAI.CKSBoundarySurface.Boundary N → N) ∘ (extChartAt OAI.CKSBoundarySurface.I2 p).symmdef BoundaryGeometry.thetaPlus {q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)} {k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)}    (B : OAI.CKSMain.BoundaryGeometry q k) (p : OAI.CKSBoundarySurface.Boundary N) : ℝ :=  let c := B.chart p  let f := B.parameter p  let y := (extChartAt OAI.CKSBoundarySurface.I2 p) p  let ν := mfderiv OAI.CKSBoundarySurface.I3 𝓘(ℝ,OAI.CKSBoundarySurface.E3) c.coordinate p.val (B.normal p)  OAI.CKSMain.meanCurvature c.metric f y ν + OAI.CKSMain.tangentialTrace c.metric c.tensor f ydef MarginalBoundary (q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)) : Prop :=  ∃ B : OAI.CKSMain.BoundaryGeometry q k, ∀ p, B.thetaPlus p = 0def bondiMass {g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)} {K : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)}    (d : OAI.CKSSourceExterior.CKSData g K) : ℝ :=  Real.sqrt ((OAI.CKSLorentz.bondiCharge d.massAspect).1^2 - ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖^2)end OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topology InnerProductSpaceopen CKSSchwarzschildattribute [local instance] manifold_regularattribute [local instance] CKSIntrinsicConstraints.halfSpaceDimension_neZeroattribute [local instance] CKSSpatialManifold.real_id_isometric CKSLorentz.real_smulCommClass CKSLorentz.spatialDual_smulCommClasslocal instance atlas_manifold_one {N : Type u_1} [TopologicalSpace N] [ChartedSpace OAI.CKSBoundarySurface.H3 N]    [IsManifold OAI.CKSBoundarySurface.I3 ∞ N] : IsManifold OAI.CKSBoundarySurface.I3 1 N := inferInstancelocal instance atlas_tangent_vectorBundle {N : Type u_1} [TopologicalSpace N] [ChartedSpace OAI.CKSBoundarySurface.H3 N]    [IsManifold OAI.CKSBoundarySurface.I3 ∞ N] :    letI := OAI.CKSMain.atlas_manifold_one (N := N)    VectorBundle ℝ OAI.CKSBoundarySurface.E3 (fun x : N => TangentSpace OAI.CKSBoundarySurface.I3 x) := by  infer_instanceabbrev CoefficientAtlas {N : Type u_1} [TopologicalSpace N] [ChartedSpace OAI.CKSBoundarySurface.H3 N]    [IsManifold OAI.CKSBoundarySurface.I3 ∞ N] (q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N)) :=  letI := OAI.CKSMain.atlas_manifold_one (N := N)  letI := OAI.CKSMain.atlas_tangent_vectorBundle (N := N)  let metric : ContMDiffRiemannianMetric OAI.CKSBoundarySurface.I3 ∞ OAI.CKSBoundarySurface.E3 (fun x : N => TangentSpace OAI.CKSBoundarySurface.I3 x) := q  ∀ x : N, OAI.CKSIntrinsicConstraints.ConstraintChart metric.inner k xend OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]structure InteriorSurface (q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N))    (A : OAI.CKSMain.CoefficientAtlas q k) (T : Type v) [TopologicalSpace T] [ChartedSpace OAI.CKSBoundarySurface.E2 T]    [IsManifold OAI.CKSBoundarySurface.I2 ∞ T] where  map : T → N  smooth : ContMDiff OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 ∞ map  embedding : Topology.IsEmbedding map  immersion : ∀ p, Function.Injective (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 map p)  interior : ∀ p, map p ∈ OAI.CKSBoundarySurface.I3.interior N  normal : ∀ p, TangentSpace OAI.CKSBoundarySurface.I3 (map p)  normal_unit : ∀ p, q.inner (map p) (normal p) (normal p) = 1  normal_orthogonal : ∀ p a, q.inner (map p) (normal p) (mfderiv OAI.CKSBoundarySurface.I2 OAI.CKSBoundarySurface.I3 map p a) = 0  normal_smooth : ∀ p, ContMDiffAt OAI.CKSBoundarySurface.I2 𝓘(ℝ,OAI.CKSBoundarySurface.E3) ∞    (fun t => mfderiv OAI.CKSBoundarySurface.I3 𝓘(ℝ,OAI.CKSBoundarySurface.E3) (A (map p)).coordinate (map t) (normal t)) pend OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMain.InteriorSurfaceopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]variable {q : SmoothMetric I3 (M := N)} {k : InnerField I3 (M := N)}  {A : CoefficientAtlas q k} {T : Type v} [TopologicalSpace T] [ChartedSpace E2 T]  [IsManifold I2 ∞ T] (D : InteriorSurface q k A T)def parameter (p : T) : OAI.CKSBoundarySurface.E2 → OAI.CKSBoundarySurface.E3 :=  (A (D.map p)).coordinate ∘ D.map ∘ (extChartAt OAI.CKSBoundarySurface.I2 p).symmdef mean (p : T) : ℝ :=  let c := A (D.map p)  OAI.CKSMain.meanCurvature c.metric (D.parameter p) ((extChartAt OAI.CKSBoundarySurface.I2 p) p)    (mfderiv OAI.CKSBoundarySurface.I3 𝓘(ℝ,OAI.CKSBoundarySurface.E3) c.coordinate (D.map p) (D.normal p))def trace (p : T) : ℝ :=  let c := A (D.map p)  OAI.CKSMain.tangentialTrace c.metric c.tensor (D.parameter p) ((extChartAt OAI.CKSBoundarySurface.I2 p) p)def thetaPlus (p : T) : ℝ := D.mean p + D.trace pdef thetaMinus (p : T) : ℝ := D.mean p - D.trace pdef SeparatesAway (c : OAI.CKSSourceExterior.CoordinateEnd (N := N)) : Prop :=  ∃ B : Set N, B = closure (_root_.interior B) ∧ (_root_.interior B).Nonempty ∧    frontier B = range D.map ∧    ∃ r : ℝ, c.radius < r ∧ B ⊆ {x | x ∉ c.domain ∨ ‖c.coordinate x‖ ≤ r}def ComponentwiseApparent : Prop :=  ∀ p : T, (∀ z ∈ connectedComponent p, D.thetaPlus z = 0) ∨    (∀ z ∈ connectedComponent p, D.thetaMinus z = 0)end OAI.CKSMain.InteriorSurfaceendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]def NoAdditionalHorizons (q : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (k : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N))    (c : OAI.CKSSourceExterior.CoordinateEnd (N := N)) (A : OAI.CKSMain.CoefficientAtlas q k) : Prop :=  ∀ (T : Type v) [TopologicalSpace T] [ChartedSpace OAI.CKSBoundarySurface.E2 T] [IsManifold OAI.CKSBoundarySurface.I2 ∞ T]    [T2Space T] [SecondCountableTopology T] [CompactSpace T] [Nonempty T]    (D : OAI.CKSMain.InteriorSurface q k A T), D.SeparatesAway c → ¬ D.ComponentwiseApparentstructure MainHypotheses [T2Space N] [SecondCountableTopology N] [ConnectedSpace N]    (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := N)) (K : OAI.CKSMetricGluing.InnerField OAI.CKSBoundarySurface.I3 (M := N))    (d : OAI.CKSSourceExterior.CKSData g K) (A : OAI.CKSMain.CoefficientAtlas g K) : Prop where  smooth_tensor : ContMDiff OAI.CKSBoundarySurface.I3 (OAI.CKSBoundarySurface.I3.prod 𝓘(ℝ,OAI.CKSLorentz.SpatialBilinear)) ∞ (OAI.CKSMetricGluing.innerSection OAI.CKSBoundarySurface.I3 K)  symmetric_tensor : ∀ x v w, K x v w = K x w v  orientable : OAI.CKSSourceExterior.Orientable (N := N)  boundary_compact : IsCompact (OAI.CKSBoundarySurface.I3.boundary N)  boundary_nonempty : (OAI.CKSBoundarySurface.I3.boundary N).Nonempty  complete : OAI.CKSReplacementCompleteness.IsComplete OAI.CKSBoundarySurface.I3 g.toContinuousRiemannianMetric  dec : OAI.CKSSpatialManifold.PhysicalDEC OAI.CKSBoundarySurface.I3 g.inner K  timelike : ‖(OAI.CKSLorentz.bondiCharge d.massAspect).2‖ < (OAI.CKSLorentz.bondiCharge d.massAspect).1  boundary_connected : ConnectedSpace (OAI.CKSBoundarySurface.Boundary N)  boundary_marginal : OAI.CKSMain.MarginalBoundary g K  positive_area : 0 < OAI.CKSSourceExterior.minimumEnclosingArea g  no_horizons : OAI.CKSMain.NoAdditionalHorizons.{u,v} g K d.chart Aend OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularvariable {N : Type u} [TopologicalSpace N] [ChartedSpace H3 N] [IsManifold I3 ∞ N]attribute [local instance] sixteen_atLeastTwodef SchwarzschildExample (m : ℝ) (_hm : 0 < m) : Prop :=  ∃ (g : OAI.CKSMetricGluing.SmoothMetric OAI.CKSBoundarySurface.I3 (M := OAI.CKSSchwarzschild.Exterior))    (d : OAI.CKSSourceExterior.CKSData g (OAI.CKSSchwarzschild.tensorInner m))    (A : OAI.CKSMain.CoefficientAtlas g (OAI.CKSSchwarzschild.tensorInner m)),    (∀ p, g.inner p = OAI.CKSSchwarzschild.metricInner m p) ∧    OAI.CKSMain.MainHypotheses.{0,v} g (OAI.CKSSchwarzschild.tensorInner m) d A ∧    OAI.CKSSchwarzschild.HorizonRegularGraph m ∧    OAI.CKSMain.bondiMass d = m ∧    OAI.CKSSourceExterior.minimumEnclosingArea g = 16*Real.pi*m^2 ∧    OAI.CKSMain.bondiMass d = Real.sqrt (OAI.CKSSourceExterior.minimumEnclosingArea g / (16*Real.pi))end OAI.CKSMainendnoncomputable sectionuniverse u v w u_1 u_2 u_3 u_4 u_5 u_6namespace OAI.CKSMainopen Set Filter Manifold Bundle CKSLorentz CKSMetricGluing CKSSpatialManifoldopen CKSBoundarySurface CKSIntrinsicConstraints CKSSourceExterioropen scoped ContDiff Topologyattribute [local instance] manifold_regularopen CKSSchwarzschildtheorem schwarzschild_equality_examples (m : ℝ) (hm : 0 < m) :    SchwarzschildExample.{v} m hm := by  sorryend OAI.CKSMainend

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Area-controlled end replacement and the Bondi Penrose inequality in the CKS class

Theorem (Area-controlled end replacement)

1.
They agree with (g,K) on the compact side of r=R and satisfy DEC.
2.
For numbers 0\leq\varepsilon_R<1 with \varepsilon_R\to0, G_R\geq(1-\varepsilon_R)g everywhere on \Omega.
3.
The end is asymptotically flat, with G_R-\delta=O_j(r^{-1}) for every fixed finite j, and k_R=0 sufficiently far out. The constraints are integrable and its charges satisfy P_R=0, E_R=m_B+\eta_R, where 0\leq\eta_R\leq2R^{-1/2}.

The new exterior is complete and A_{\min,G_R}(S)\geq(1-\varepsilon_R)A_{\min,g}(S). No condition on the boundary expansion is needed.

Section 1, Replacing the end and retaining the area Read it in the paper TeX source, line 128

Show the TeX

Let $(\Omega,g,K)$ be a smooth connected orientable three-dimensional
exterior, complete including its nonempty compact smooth boundary,
with exactly one end of class \eqref{eq:cks}. Suppose DEC holds and
$E_B>|P_B|$. There is a balanced chart such that, for every sufficiently
large $R$, smooth data $(G_R,k_R)$ on the same exterior have the following
properties:
\begin{enumerate}
\item They agree with $(g,K)$ on the compact side of $r=R$ and satisfy DEC.
\item For numbers $0\leq\varepsilon_R<1$ with $\varepsilon_R\to0$,
      $G_R\geq(1-\varepsilon_R)g$ everywhere on $\Omega$.
\item The end is asymptotically flat, with $G_R-\delta=O_j(r^{-1})$
      for every fixed finite $j$, and $k_R=0$ sufficiently far out.
      The constraints are integrable and its charges satisfy
      $P_R=0$, $E_R=m_B+\eta_R$, where $0\leq\eta_R\leq2R^{-1/2}$.
\end{enumerate}
The new exterior is complete and
$A_{\min,G_R}(S)\geq(1-\varepsilon_R)A_{\min,g}(S)$.
No condition on the boundary expansion is needed.

Corollary (Sharp Bondi inequality in the CKS class)

Under the hypotheses of Theorem prop:replacement, assume \theta_+(S)\leq0, with normal into \Omega, and A_{\min,g}(S)>0. Then

\sqrt{E_B^2-|P_B|^2}\geq \sqrt{\frac{A_{\min,g}(S)}{16\pi}}.

The boundary may be disconnected. For every m>0 there is a horizon-regular Schwarzschild hyperboloidal exterior attaining equality with m_B=m and A_{\min,g}(S)=16\pi m^2.

Section 1, Replacing the end and retaining the area Read it in the paper TeX source, line 154

Show the TeX

Under the hypotheses of Theorem~\ref{prop:replacement}, assume
$\theta_+(S)\leq0$, with normal into $\Omega$, and
$A_{\min,g}(S)>0$. Then
\begin{equation}
 \sqrt{E_B^2-|P_B|^2}\geq
 \sqrt{\frac{A_{\min,g}(S)}{16\pi}}.
\end{equation}
The boundary may be disconnected. For every $m>0$ there is a
horizon-regular Schwarzschild hyperboloidal exterior attaining equality
with $m_B=m$ and $A_{\min,g}(S)=16\pi m^2$.

Configuration and file facts

File: lean/ComparatorChallenges/CKSBondiPenrose.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 260 link it.
Solution module: OAI.Geometry.Relativity.CKS.ComparatorMain
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 310 local vocabulary declarations
Lines of Lean, without comments or blank lines: 1,857
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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