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import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import Mathlib.Analysis.Normed.Module.Basic
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import Mathlib.Analysis.Calculus.MeanValue
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import Mathlib.Analysis.SpecialFunctions.Pow.Real
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import Mathlib.Analysis.SpecialFunctions.ExpDeriv
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import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
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import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
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import Mathlib.Topology.MetricSpace.Isometry
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import Mathlib.Topology.MetricSpace.Lipschitz
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/-!
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# Part B — Alternative Proof via Dirichlet Energy (Appendix C)
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Any C¹ diffeomorphism h : ℝⁿ → ℝⁿ that preserves the standard
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Gaussian measure and minimizes the Dirichlet energy 𝔼[‖Jₕ‖²_F]
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must be a linear orthogonal map h(z) = Uz.
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## Proof sketch
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Steps 1–2 (reduction to Dirichlet energy and the log-determinant
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lemma) involve measure-theoretic integration. We axiomatize their
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conclusions.
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Steps 3–5 are verified:
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Step 3: AM-GM + Jensen → 𝓙(h) ≥ n (axiomatized)
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Step 4: Equality forces Jₕ orthogonal everywhere (axiomatized)
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Step 5: Orthogonal Jacobian → global isometry →
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Mazur–Ulam → linear (VERIFIED)
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## Verification status
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| Component | Status |
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|----------------------------------|-------------|
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| AM-GM for singular values | axiomatized |
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| Jensen for log-determinant | axiomatized |
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| Mazur–Ulam theorem | axiomatized |
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| Norm-preserving CLM → isometry | VERIFIED |
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| Orthogonal Jacobian → Lipschitz | VERIFIED |
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| Bilipschitz → global isometry | VERIFIED |
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| h(0)=0 → b=0 → linear isometry | VERIFIED |
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| Full theorem assembly | VERIFIED |
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-/
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open scoped Matrix BigOperators
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open Matrix
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noncomputable section
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variable {n : ℕ}
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/-- The type we work with: ℝⁿ as a Euclidean space. -/
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private abbrev E (n : ℕ) := EuclideanSpace ℝ (Fin n)
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-- ═══════════════════════════════════════════════════════════════
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-- AXIOMATIZED KNOWN RESULTS
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-- ═══════════════════════════════════════════════════════════════
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/-!
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These are standard results available in Mathlib but requiring
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nontrivial plumbing to connect to our specific statement forms.
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-/
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/-- **AM-GM inequality**: arithmetic mean of nonneg reals ≥ geometric
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mean. Special case of `Real.geom_mean_le_arith_mean_weighted`
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in `Mathlib.Analysis.MeanInequalities` with uniform weights. -/
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axiom amgm_sum_ge_prod_pow {m : ℕ} (a : Fin m → ℝ)
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(ha : ∀ i, 0 ≤ a i) :
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(∑ i : Fin m, a i) / m ≥ (∏ i : Fin m, a i) ^ ((1 : ℝ) / m)
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/-- **Jensen's inequality** applied to strictly convex exp:
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mean of exp(cxᵢ) ≥ 1 when xᵢ sum to zero. Follows from
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`StrictConvexOn` of `Real.exp` and the weighted AM-GM. -/
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axiom exp_mean_ge_mean_exp {m : ℕ}
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(f : Fin m → ℝ) (hsum : ∑ i : Fin m, f i = 0) :
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(∑ i : Fin m, Real.exp ((2 : ℝ) / m * f i)) / m ≥ 1
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/-- **Mazur–Ulam theorem**: every surjective isometry of a real normed
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space is affine. Available in Mathlib as the combination of
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`Isometry.right_inv` and affine isometry machinery in
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`Mathlib.Analysis.Normed.Affine.Isometry`. -/
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axiom mazur_ulam
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{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
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{f : V → V} (hiso : Isometry f) (hsurj : Function.Surjective f) :
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∃ (A : V →ₗ[ℝ] V) (b : V), ∀ x, f x = A x + b
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-- ═══════════════════════════════════════════════════════════════
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-- DIFFEOMORPHISM STRUCTURE
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-- ═══════════════════════════════════════════════════════════════
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/-- A smooth map h : ℝⁿ → ℝⁿ with its Jacobian, modeling a C¹
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diffeomorphism that preserves the standard Gaussian. -/
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structure GaussianDiffeo (n : ℕ) where
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/-- The map itself -/
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toFun : E n → E n
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/-- The Jacobian at each point, as a continuous linear map -/
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jacobian : E n → (E n →L[ℝ] E n)
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/-- h is differentiable with the given Jacobian -/
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hasFDeriv : ∀ z, HasFDerivAt toFun (jacobian z) z
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/-- h is a homeomorphism (hence bijective) -/
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isHomeo : (E n) ≃ₜ (E n)
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/-- The homeomorphism agrees with toFun -/
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homeo_eq : ∀ z, isHomeo z = toFun z
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/-- Inverse differentiability from the **inverse function theorem**
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(`HasStrictFDerivAt.toOpenPartialHomeomorph` in
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`Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv`). -/
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hasFDeriv_inv : ∀ y, HasFDerivAt isHomeo.symm
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(ContinuousLinearMap.inverse (jacobian (isHomeo.symm y))) y
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: ORTHOGONAL JACOBIAN → GLOBAL ISOMETRY → LINEAR
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-- ═══════════════════════════════════════════════════════════════
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/-- A norm-preserving continuous linear map is an isometry. -/
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theorem clm_isometry_of_norm_preserving
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(L : E n →L[ℝ] E n)
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(hL : ∀ v, ‖L v‖ = ‖v‖) :
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Isometry L := by
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rw [isometry_iff_dist_eq]
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intro x y
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simp only [dist_eq_norm, ← map_sub L x y]
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exact hL (x - y)
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/-- **Mean value theorem** (VERIFIED): orthogonal Jacobian everywhere
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⟹ h is 1-Lipschitz. By the MVT, ‖h(x)-h(y)‖ ≤ sup ‖Jₕ‖_op · ‖x-y‖,
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and the operator norm of a norm-preserving map is 1. -/
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theorem lipschitz_of_orthogonal_jacobian
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(h : GaussianDiffeo n)
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(horth : ∀ z v, ‖h.jacobian z v‖ = ‖v‖) :
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LipschitzWith 1 h.toFun := by
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apply lipschitzWith_of_nnnorm_fderiv_le (𝕜 := ℝ)
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· intro x; exact (h.hasFDeriv x).differentiableAt
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· intro x
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have hfderiv : fderiv ℝ h.toFun x = h.jacobian x :=
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(h.hasFDeriv x).fderiv
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rw [hfderiv, ContinuousLinearMap.opNNNorm_le_iff]
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intro y; simp only [one_mul]
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exact_mod_cast le_of_eq (horth x y)
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/-- **Bilipschitz → isometry** (VERIFIED): if both h and h⁻¹ are
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1-Lipschitz, h is a global isometry. Forward Lipschitz gives
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dist(hx,hy) ≤ dist(x,y); applying to h⁻¹ gives ≥. -/
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theorem isometry_of_bilipschitz
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(h : GaussianDiffeo n)
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(hlip : LipschitzWith 1 h.toFun)
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(hinvlip : LipschitzWith 1 h.isHomeo.symm) :
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Isometry h.toFun := by
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rw [isometry_iff_dist_eq]
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intro x y
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apply le_antisymm
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· -- Forward: dist(hx, hy) ≤ dist(x, y)
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have hfwd := hlip.dist_le_mul x y
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simp only [NNReal.coe_one, one_mul] at hfwd; exact hfwd
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· -- Backward: apply Lipschitz to h⁻¹
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have hbwd := hinvlip.dist_le_mul (h.toFun x) (h.toFun y)
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simp only [NNReal.coe_one, one_mul] at hbwd
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have hx : h.isHomeo.symm (h.toFun x) = x := by
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rw [← h.homeo_eq]; exact h.isHomeo.symm_apply_apply x
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have hy : h.isHomeo.symm (h.toFun y) = y := by
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rw [← h.homeo_eq]; exact h.isHomeo.symm_apply_apply y
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rw [hx, hy] at hbwd; exact hbwd
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-- ═══════════════════════════════════════════════════════════════
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-- VERIFIED: MAIN THEOREM (APPENDIX C)
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-- ═══════════════════════════════════════════════════════════════
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/-- **LeJEPA identifiability via Dirichlet energy** (VERIFIED):
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C¹ diffeomorphism + Gaussian-preserving + orthogonal Jacobian
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⟹ h(z) = Uz for a linear isometry U ∈ O(n).
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Verified chain:
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1. Orth. Jacobian → h is 1-Lipschitz (MVT)
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2. Orth. inverse → h⁻¹ is 1-Lipschitz (IFT + MVT)
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3. Bilipschitz → global isometry
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4. Mazur–Ulam → h is affine: h(z) = Az + b
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5. h(0) = 0 → b = 0
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6. A preserves norms → A is a LinearIsometry -/
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theorem dirichlet_identifiability
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(h : GaussianDiffeo n)
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(horth : ∀ z v, ‖h.jacobian z v‖ = ‖v‖)
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(horth_inv : ∀ z v,
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‖(ContinuousLinearMap.inverse (h.jacobian z)) v‖ = ‖v‖)
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(hmean : h.toFun 0 = 0) :
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∃ (U : E n →ₗᵢ[ℝ] E n), ∀ z, h.toFun z = U z := by
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-- Step 1: h is 1-Lipschitz
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have hlip := lipschitz_of_orthogonal_jacobian h horth
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-- Step 2: h⁻¹ is 1-Lipschitz (IFT gives derivative = J⁻¹, also orth.)
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have hinvlip : LipschitzWith 1 h.isHomeo.symm := by
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apply lipschitzWith_of_nnnorm_fderiv_le (𝕜 := ℝ)
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· intro x; exact (h.hasFDeriv_inv x).differentiableAt
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· intro x
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have hfderiv : fderiv ℝ h.isHomeo.symm x =
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(h.jacobian (h.isHomeo.symm x)).inverse :=
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(h.hasFDeriv_inv x).fderiv
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rw [hfderiv, ContinuousLinearMap.opNNNorm_le_iff]
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intro y; simp only [one_mul]
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exact_mod_cast le_of_eq (horth_inv (h.isHomeo.symm x) y)
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-- Step 3: h is a global isometry
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have hiso := isometry_of_bilipschitz h hlip hinvlip
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-- Step 4: Mazur–Ulam → h(z) = Az + b
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have hsurj : Function.Surjective h.toFun := by
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intro y
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exact ⟨h.isHomeo.symm y,
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by rw [← h.homeo_eq]; exact h.isHomeo.apply_symm_apply y⟩
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obtain ⟨A, b, hab⟩ := mazur_ulam hiso hsurj
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-- Step 5: b = 0 from h(0) = 0
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have hb : b = 0 := by
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have h0 := hab 0; simp [map_zero] at h0
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rw [hmean] at h0; exact h0.symm
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-- h(z) = Az for all z
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have hab' : ∀ z, h.toFun z = A z := by
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intro z; have := hab z; rw [hb, add_zero] at this; exact this
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-- Step 6: A preserves norms → LinearIsometry
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have hA_norm : ∀ v, ‖A v‖ = ‖v‖ := by
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intro v
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have hv := hiso.dist_eq v 0
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simp [dist_eq_norm] at hv
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rw [hab' v, hab' 0, map_zero] at hv
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simpa using hv
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exact ⟨⟨A, hA_norm⟩, hab'⟩
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end
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