Prove2Me

Canonical compact reflection with phase and cone estimates

Proved
BianchiIX.exists_compact_reflection_estimates

by trppppp · Nov 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

For real variables p,s,a,b,cp,s,a,b,c, set

K=a2+b2+c2−2(ab+bc+ca),S=p2+s2,C=S+K.K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.

The shear components and wall rates are

P=Kp−2a2+b2+c2+ab−2bc+ac,P=Kp-2a^2+b^2+c^2+ab-2bc+ac, Q=Ks+3(−b2+c2+ab−ac),Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), r1=−S+2p,r2=−S−p+3s,r3=−S−p−3s.r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.

These functions define the normalized vacuum Bianchi IX vector field (P,Q,r1a,r2b,r3c)(P,Q,r_1a,r_2b,r_3c). For y=(m,u,v,w)∈R4y=(m,u,v,w)\in\mathbb R^4, define

K(u,v,w)=u2+v2+w2−2(uv+vw+wu),κ=u(v+w−u),K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), Fm=−v2+w2+u(v−w)−mκ,Fu=12(1−m2−(v−w)2+u2)u,F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u, Fv=(m−1−κ)v,Fw=(−m−1−κ)w,F=(Fm,Fu,Fv,Fw),F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),

and E(m,v)=(m−1)2+v2E(m,v)=(m-1)^2+v^2.

For x=(p,s,a,b,c)x=(p,s,a,b,c), put

ψ(p,z)=3 z1+p,Π(x)=(ψ(p,s),ψ(p,a),ψ(p,b),ψ(p,c)).\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).

For y=(m,u,v,w)y=(m,u,v,w), set D(y)=3+m2+K(u,v,w)D(y)=3+m^2+K(u,v,w) and

ΞA(y)=(A−1,Am3,Au3,Av3,Aw3),Ξ(y)=Ξ6/D(y)(y).\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).

Write N(x)=(P,Q,r1a,r2b,r3c)N(x)=(P,Q,r_1a,r_2b,r_3c) for the normalized vector field and C(x)=C(p,s,a,b,c)\mathcal C(x)=C(p,s,a,b,c) for its constraint.

The normalized and projective state spaces are R5\mathbb R^5 and R4\mathbb R^4 with their finite-product norms. The maps Π,Ξ,D,F\Pi,\Xi,D,F act on these state spaces.

Define the core radius, angle, and clock factor by

r(m,v)=(m−1)2+v2,θ(m,v)=arsinh⁡1−mv,d(r,u,w)=1−(u−w)2r2.r(m,v)=\sqrt{(m-1)^2+v^2},\qquad \theta(m,v)=\operatorname{arsinh}\frac{1-m}{v},\qquad d(r,u,w)=1-\frac{(u-w)^2}{r^2}.

For y=(m,u,v,w)y=(m,u,v,w), write ry=r(m,v)r_y=r(m,v), θy=θ(m,v)\theta_y=\theta(m,v), and dy=d(ry,u,w)d_y=d(r_y,u,w). All quotients use the total convention x/0=0x/0=0.

The core state space is R3\mathbb R^3 with coordinates Y=(r,u,p)Y=(r,u,p). Define

η(m,u,v,w)=(r(m,v),u,vw),ιz(r,u,p)=(1−rtanh⁡z,u,rcosh⁡z,pr/cosh⁡z).\eta(m,u,v,w)=(r(m,v),u,vw),\qquad \iota_z(r,u,p)=\left(1-r\tanh z,u,\frac r{\cosh z}, \frac p{r/\cosh z}\right).

For fixed (z,Y)(z,Y) put v=r/cosh⁡zv=r/\cosh z, w=p/vw=p/v, d=d(r,u,w)d=d(r,u,w), and define

V(z,Y)=(−κ(u,v,w)−tanh⁡z (u−w)2/rd,u[−r/2+tanh⁡z+w/cosh⁡z+(u2−w2)/(2r)]d,−2p(1+κ(u,v,w))rd).\mathcal V(z,Y)=\left( \frac{-\kappa(u,v,w)-\tanh z\,(u-w)^2/r}{d},\quad \frac{u[-r/2+\tanh z+w/\cosh z+(u^2-w^2)/(2r)]}{d},\quad \frac{-2p(1+\kappa(u,v,w))}{rd}\right).

For real kk, put εk=1/(10000k)\varepsilon_k=1/(10000k) and define

Kk={(m,u,v,w):∣E(m,v)−k2∣≤1/8,v≥1,0≤u≤εk,0≤vw≤εk}.\mathcal K_k=\{(m,u,v,w): |E(m,v)-k^2|\leq1/8,\quad v\geq1, \quad0\leq u\leq\varepsilon_k,\quad0\leq vw\leq\varepsilon_k\}.

Write X=R5X=\mathbb R^5, Y=R4Y=\mathbb R^4, and let αx,t0\alpha_{x,t_0} be the maximal solution of α′=N(α)\alpha'=N(\alpha) with initial value xx at t0t_0 and domain Imax⁡(x,t0)I_{\max}(x,t_0). Put

Jx,t0(t)=dxαx,t0(t),Ψt0(y,t)=Π(αΞ(y),t0(t)),J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)), Jy,t0(t)=dΠαΞ(y),t0(t)∘JΞ(y),t0(t)∘dΞy.\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.

Write G(y)=6F(y)/D(y)G(y)=6F(y)/D(y), λ(y)=6rydy/D(y)\lambda(y)=6r_yd_y/D(y), and Sk(R,U,P)=(R,kU,P)S_k(R,U,P)=(R,kU,P). Define

Pyξ=dηyξ−dθyξ V(θy,η(y)),Wy,t0,w(t)=PΨt0(y,t)(Jy,t0(t)w).\mathcal P_y\xi=d\eta_y\xi-d\theta_y\xi\,\mathcal V(\theta_y,\eta(y)), \qquad \mathcal W_{y,t_0,w}(t)= \mathcal P_{\Psi_{t_0}(y,t)}(\mathcal J_{y,t_0}(t)w).

The entrance parametrization and radial exit factor are

e(k,u,w)=(1+k2−1,u,1,w),e(k,u,w)=(1+\sqrt{k^2-1},u,1,w), fy=V(θy,η(y))0,cy=−tanh⁡θy,hy=cy+fy/ry,ρy=cy/hy.f_y=\mathcal V(\theta_y,\eta(y))_0,\qquad c_y=-\tanh\theta_y, \qquad h_y=c_y+f_y/r_y,\qquad \rho_y=c_y/h_y.

For projective initial data yy, put

Ay,t0(t)=1+(αΞ(y),t0(t))0.A_{y,t_0}(t)=1+(\alpha_{\Xi(y),t_0}(t))_0.

Coordinates of three-vectors are indexed by 0,1,20,1,2; projective coordinates are indexed by m,u,v,wm,u,v,w.

For Y∈R3Y\in\mathbb R^3, put xY=Ξ(e(Y))x_Y=\Xi(e(Y)), βY(t)=Ψ0(e(Y),t)\beta_Y(t)=\Psi_0(e(Y),t), and

HY={t>0:t∈Imax⁡(xY,0), (βY(t))v=1},T(Y)=inf⁡HY,F(Y)=η(βY(T(Y))).H_Y=\{t>0:t\in I_{\max}(x_Y,0),\ (\beta_Y(t))_v=1\}, \qquad T(Y)=\inf H_Y,\qquad F(Y)=\eta(\beta_Y(T(Y))).

The infimum uses the total real-order convention. Write Valid⁡(Y)\operatorname{Valid}(Y) when T(Y)T(Y) is the least element of HYH_Y, every t∈[0,T(Y)]t\in[0,T(Y)] lies in Imax⁡(xY,0)I_{\max}(x_Y,0) with Ae(Y),0(t)>0A_{e(Y),0}(t)>0, the active wall is strictly above one for 0<t<T(Y)0<t<T(Y), and

dgαxY,0(T(Y))N(αxY,0(T(Y)))<0,g(x)=(Π(x))v.dg_{\alpha_{x_Y,0}(T(Y))}N(\alpha_{x_Y,0}(T(Y)))<0, \qquad g(x)=(\Pi(x))_v.

All vector and operator norms below use the finite-product supremum norm. For real variables p,s,a,b,cp,s,a,b,c, set

K=a2+b2+c2−2(ab+bc+ca),S=p2+s2,C=S+K.K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.

The shear components and wall rates are

P=Kp−2a2+b2+c2+ab−2bc+ac,P=Kp-2a^2+b^2+c^2+ab-2bc+ac, Q=Ks+3(−b2+c2+ab−ac),Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), r1=−S+2p,r2=−S−p+3s,r3=−S−p−3s.r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.

These functions define the normalized vacuum Bianchi IX vector field (P,Q,r1a,r2b,r3c)(P,Q,r_1a,r_2b,r_3c). For y=(m,u,v,w)∈R4y=(m,u,v,w)\in\mathbb R^4, define

K(u,v,w)=u2+v2+w2−2(uv+vw+wu),κ=u(v+w−u),K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), Fm=−v2+w2+u(v−w)−mκ,Fu=12(1−m2−(v−w)2+u2)u,F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u, Fv=(m−1−κ)v,Fw=(−m−1−κ)w,F=(Fm,Fu,Fv,Fw),F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),

and E(m,v)=(m−1)2+v2E(m,v)=(m-1)^2+v^2.

For x=(p,s,a,b,c)x=(p,s,a,b,c), put

ψ(p,z)=3 z1+p,Π(x)=(ψ(p,s),ψ(p,a),ψ(p,b),ψ(p,c)).\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).

For y=(m,u,v,w)y=(m,u,v,w), set D(y)=3+m2+K(u,v,w)D(y)=3+m^2+K(u,v,w) and

ΞA(y)=(A−1,Am3,Au3,Av3,Aw3),Ξ(y)=Ξ6/D(y)(y).\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).

Write N(x)=(P,Q,r1a,r2b,r3c)N(x)=(P,Q,r_1a,r_2b,r_3c) for the normalized vector field and C(x)=C(p,s,a,b,c)\mathcal C(x)=C(p,s,a,b,c) for its constraint.

The normalized and projective state spaces are R5\mathbb R^5 and R4\mathbb R^4 with their finite-product norms. The maps Π,Ξ,D,F\Pi,\Xi,D,F act on these state spaces.

The projective vector field in normalized time is

G(y)=6D(y)F(y).G(y)=\frac6{D(y)}F(y).

Write X=R5X=\mathbb R^5, Y=R4Y=\mathbb R^4, and let αx,t0\alpha_{x,t_0} be the maximal solution of α′=N(α)\alpha'=N(\alpha) with initial value xx at t0t_0 and domain Imax⁡(x,t0)I_{\max}(x,t_0). Put

Jx,t0(t)=dxαx,t0(t),Ψt0(y,t)=Π(αΞ(y),t0(t)),J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)), Jy,t0(t)=dΠαΞ(y),t0(t)∘JΞ(y),t0(t)∘dΞy.\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.

Put c=u(v+w−u)c=u(v+w-u), s(m,u,v,w)=m−1−cs(m,u,v,w)=m-1-c, b(m,u,v,w)=(1−m2−(v−w)2+u2)/2b(m,u,v,w)=(1-m^2-(v-w)^2+u^2)/2, D(m,u,v,w)=3+m2+K(u,v,w)D(m,u,v,w)=3+m^2+K(u,v,w), and r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2}.

Define

Cμ={(m,u,v,w):∣−m−μ∣≤1/2, 0≤u≤1/10000, 0≤v,w≤1},\mathcal C_\mu=\{(m,u,v,w):|-m-\mu|\leq1/2, \ 0\leq u\leq1/10000,\ 0\leq v,w\leq1\}, ω−(y)=s(−ym,yu,yv,yw),H(y)=F(y)/ω−(y).\omega_-(y)=s(-y_m,y_u,y_v,y_w),\qquad H(y)=F(y)/\omega_-(y).

For a normalized trajectory write

β(t)=Π(αx,t0(t)),A(t)=1+(αx,t0(t))0,I=Imax⁡(x,t0).\beta(t)=\Pi(\alpha_{x,t_0}(t)),\quad A(t)=1+(\alpha_{x,t_0}(t))_0, \quad I=I_{\max}(x,t_0).

For p=(μ,u,w)∈R3p=(\mu,u,w)\in\mathbb R^3, set e(p)=(−μ,u,1,w)e(p)=(-\mu,u,1,w), xp=Ξ(e(p))x_p=\Xi(e(p)), and

T−(p)=inf⁡{t>0:t∈Imax⁡(xp,0), (Ψ0(e(p),t))w=1},P−(p)=Ψ0(e(p),T−(p)).T_-(p)=\inf\{t>0:t\in I_{\max}(x_p,0),\ (\Psi_0(e(p),t))_w=1\}, \qquad P_-(p)=\Psi_0(e(p),T_-(p)).

Let V−(p)V_-(p) mean that this infimum is a least element, the full interval [0,T−(p)][0,T_-(p)] lies in the maximal domain with positive AA, the growing wall is below one before the endpoint, and its normalized-time derivative is strictly positive at the endpoint. The infimum has its total real-order meaning.

Write e+(R,u,w)=(1+R2−1,u,1,w)e_+(R,u,w)=(1+\sqrt{R^2-1},u,1,w), e−(R,u,w)=(1−R2−1,u,1,w)e_-(R,u,w)=(1-\sqrt{R^2-1},u,1,w), p−(R,u,w)=(R2−1−1,u,w)p_-(R,u,w)=(\sqrt{R^2-1}-1,u,w), and R(m,u,v,w)=(−m,u,w,v)\mathcal R(m,u,v,w)=(-m,u,w,v). The physical connection coordinates are

C(p)=((−P−(p)0−1)2+1,P−(p)1,P−(p)2).\mathscr C(p)=\left(\sqrt{(-P_-(p)_0-1)^2+1},P_-(p)_1,P_-(p)_2\right).

Below TT denotes the canonical core return time from e+e_+ and F(Y)=η(Ψ0(e+(Y),T(Y)))F(Y)=\eta(\Psi_0(e_+(Y),T(Y))) its radial and wall coordinates. Let R(Y)=C(p−(Y))\mathscr R(Y)=\mathscr C(p_-(Y)), tr(Y)=T−(p−(Y))t_r(Y)=T_-(p_-(Y)), G=R∘FG=\mathscr R\circ F, and tG(Y)=T(Y)+tr(F(Y))t_G(Y)=T(Y)+t_r(F(Y)). A cone direction at YY has d0=1d_0=1, ∣d1∣≤Y1|d_1|\leq\sqrt{Y_1}, ∣d2∣≤Y2|d_2|\leq\sqrt{Y_2}. For a map HH with positive output walls, its positive cone property means dH(Y)ddH(Y)d has positive radial component and ∣(dH(Y)d)i∣≤H(Y)i(dH(Y)d)0|(dH(Y)d)_i|\leq\sqrt{H(Y)_i}(dH(Y)d)_0 for i=1,2i=1,2.

The reflection conclusions with constant CC are: tr(Y)t_r(Y) is the least positive time at which (Ψ0(e−(Y),t))3=1(\Psi_0(e_-(Y),t))_3=1; e+(R(Y))=R(Ψ0(e−(Y),tr(Y)))e_+(\mathscr R(Y))=\mathcal R(\Psi_0(e_-(Y),t_r(Y))); tr,Rt_r,\mathscr R are C2C^2 at YY; and

∣R(Y)0−(Y0−2)∣≤C(Y1+Y2)/Y0,0<R(Y)i≤Yi(i=1,2).|\mathscr R(Y)_0-(Y_0-2)|\leq C(Y_1+Y_2)/Y_0, \qquad0<\mathscr R(Y)_i\leq Y_i\quad(i=1,2).

Every cone direction vv gives d=dR(Y)vd=d\mathscr R(Y)v with

d0≥1/2,∣d0−1∣≤CY1+Y2/Y0,∣di∣≤R(Y)i d0(i=1,2).d_0\geq1/2,\quad |d_0-1|\leq C\sqrt{Y_1+Y_2}/Y_0, \qquad |d_i|\leq\sqrt{\mathscr R(Y)_i}\,d_0\quad(i=1,2).

There are ε,C>0\varepsilon,C>0 such that these conclusions hold whenever 31/8≤Y0≤631/8\leq Y_0\leq6 and 0<Y1,Y2≤ε0<Y_1,Y_2\leq\varepsilon.

Preamble
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveRegularity
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_ProjectiveReflection
import Definitions.Def_BianchiIX_Dynamics_ConnectionExistence
import Definitions.Def_BianchiIX_Dynamics_ConnectionEvents
import Definitions.Def_BianchiIX_Dynamics_CoreEntryDifferential
import Definitions.Def_BianchiIX_Dynamics_CoreTail
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_EntranceReturn
import Definitions.Def_BianchiIX_Dynamics_CompactConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_ConnectionPhysicalReturn
import Definitions.Def_BianchiIX_Dynamics_CoreConnectionChart
import Definitions.Def_BianchiIX_EntrancePhaseResponseData
import Definitions.Def_BianchiIX_EraReturnData
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Tactic.Module
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Normed.Operator.Basic
import Mathlib.LinearAlgebra.Matrix.Notation
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
import Mathlib.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Tactic.Positivity
import Mathlib.Analysis.SpecialFunctions.Pow.Real

noncomputable section
open BianchiIX
open Set
Formal statement
theorem BianchiIX.exists_compact_reflection_estimates :
    ∃ ε > (0 : ℝ), ∃ C > (0 : ℝ), ∀ Y : CoreAngleState,
      (31 / 8 ≤ Y 0 ∧ Y 0 ≤ 6) → (0 < Y 1 ∧ Y 1 ≤ ε) → (0 < Y 2 ∧ Y 2 ≤ ε) →
      IsLeast {t : ℝ | 0 < t ∧ projectiveOriginalFlow (coreTailEntry Y) 0 t 3 = 1}
        (reflectionReturnTime Y) ∧
      coreRadiusEntry (reflectionReturnState Y) =
        projectiveReflection (projectiveOriginalFlow (coreTailEntry Y) 0 (reflectionReturnTime Y)) ∧
      ContDiffAt ℝ 2 reflectionReturnTime Y ∧ ContDiffAt ℝ 2 reflectionReturnState Y ∧
      |reflectionReturnState Y 0 - (Y 0 - 2)| ≤ C * (Y 1 + Y 2) / Y 0 ∧
      (∀ i : Fin 2, 0 < reflectionReturnState Y i.succ ∧ reflectionReturnState Y i.succ ≤ Y i.succ) ∧
      ∀ v : CoreAngleState, v 0 = 1 → |v 1| ≤ Real.sqrt (Y 1) → |v 2| ≤ Real.sqrt (Y 2) →
        let d := fderiv ℝ reflectionReturnState Y v
        1 / 2 ≤ d 0 ∧ |d 0 - 1| ≤ C * Real.sqrt (Y 1 + Y 2) / Y 0 ∧
          ∀ i : Fin 2, |d i.succ| ≤ Real.sqrt (reflectionReturnState Y i.succ) * d 0 := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/CompactReflectionEstimates.lean#L30-L137

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