Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Analysis.Conductivity.Variational.MatrixFiniteLaminate
import OAI.Analysis.Conductivity.Variational.QuantitativeLocalTwoFieldSplit

namespace OAI

noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Matrix Set Filter Topology
open scoped Matrix.Norms.Elementwise

def conjugateDiagonal (Q : Mat3) (α : DiagonalTriple) : Symmetric3 :=
  ⟨Q * Matrix.diagonal α * Qᵀ, by
    rw [Matrix.isHermitian_iff_isSymm]
    unfold Matrix.IsSymm
    rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_transpose,
      Matrix.diagonal_transpose, Matrix.mul_assoc]⟩

lemma conjugateDiagonal_continuous (Q : Mat3) : Continuous (conjugateDiagonal Q) := by
  apply Continuous.subtype_mk
  exact continuous_const.matrix_mul (continuous_pi (fun i => continuous_pi (fun j => by
    by_cases hij : i = j
    · subst j; simpa only [Matrix.diagonal_apply_eq] using continuous_apply i
    · simpa only [Matrix.diagonal_apply_ne _ hij] using
        (continuous_const : Continuous (fun _ : DiagonalTriple => (0:ℝ)))))) |>.matrix_mul continuous_const

lemma conjugateDiagonal_spectral {Q : Mat3} (hQ : Q ∈ orthogonalFrames)
    {α : DiagonalTriple} {T : Set DiagonalTriple} (hα : α ∈ T) :
    conjugateDiagonal Q α ∈ spectralExtension T := ⟨Q, hQ, α, hα, rfl⟩

lemma stretched_preliminary (i : Fin 3) {A α : DiagonalTriple}
    (hq : preliminaryQ i A α ≠ 0) (hA : A i ≠ 0) :
    stretchedDiagonal i (preliminaryQ i A α) (preliminaryB i A α) = α := by
  ext j
  by_cases hj : j = i
  · subst j
    simp [stretchedDiagonal, preliminaryB, preliminaryQ, hA]
  · simp [stretchedDiagonal, preliminaryB, hj, hq]

lemma continuousAt_stretchedDiagonal {i : Fin 3} {q : ℝ → ℝ} {D : ℝ → DiagonalTriple}
    {z : ℝ} (hq : ContinuousAt q z) (hD : ContinuousAt D z) (hn : q z ≠ 0) :
    ContinuousAt (fun t => stretchedDiagonal i (q t) (D t)) z := by
  apply continuousAt_pi.mpr; intro j
  have hj := continuousAt_pi.mp hD j
  by_cases hji : j = i
  · change ContinuousAt (fun y => if j = i then q y * D y j else D y j / q y) z
    simp only [ite_eq_left hji]
    exact hq.mul hj
  · change ContinuousAt (fun y => if j = i then q y * D y j else D y j / q y) z
    simp only [ite_eq_right hji]
    exact hj.div hq hn

lemma contDiffAt_stretchedDiagonal {i : Fin 3} {q : ℝ → ℝ} {D : ℝ → DiagonalTriple}
    {z : ℝ} {k : WithTop ℕ∞} (hq : ContDiffAt ℝ k q z) (hD : ContDiffAt ℝ k D z)
    (hn : q z ≠ 0) : ContDiffAt ℝ k (fun t => stretchedDiagonal i (q t) (D t)) z := by
  apply contDiffAt_pi.mpr; intro j
  have hj := contDiffAt_pi.mp hD j
  by_cases hji : j = i
  · change ContDiffAt ℝ k (fun y => if j = i then q y * D y j else D y j / q y) z
    simp only [ite_eq_left hji]
    exact hq.mul hj
  · change ContDiffAt ℝ k (fun y => if j = i then q y * D y j else D y j / q y) z
    simp only [ite_eq_right hji]
    exact hj.div hq hn

lemma conjugateDiagonal_contDiff (Q : Mat3) :
    ContDiff ℝ (↑(⊤ : ℕ∞)) (fun α : DiagonalTriple => (conjugateDiagonal Q α).val) := by
  rw [contDiff_iff_contDiffAt]; intro α
  apply contDiffAt_matrix_mul _ contDiffAt_const
  apply contDiffAt_matrix_mul contDiffAt_const
  apply contDiffAt_pi.mpr; intro i
  apply contDiffAt_pi.mpr; intro j
  by_cases hij : i = j
  · subst j; simpa only [Matrix.diagonal_apply_eq, id_eq] using
      (contDiffAt_pi.mp (contDiffAt_id (x := α)) i)
  · simpa only [Matrix.diagonal_apply_ne _ hij] using
      (contDiffAt_const : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun _ : DiagonalTriple => (0:ℝ)) α)

theorem spectral_join_plan {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple}
    (hT : ∀ α ∈ T, IsFiniteLaminate a b α)
    (i : Fin 3) {α β : DiagonalTriple} (hα : α ∈ T) (hβ : β ∈ T)
    {θ : ℝ} (hθ : 0 < θ) (hθ1 : θ < 1)
    {Q : Mat3} (hQ : Q ∈ orthogonalFrames) :
    let A := conjugateDiagonal Q (coordinateJoin i θ α β)
    ∃ (d : Coord3) (L : Coord3 →L[ℝ] ℝ) (B : Mat3) (η c m : ℝ)
      (C : ℝ → Symmetric3),
      L d = 1 ∧ B.IsSymm ∧ (∀ v, L (B *ᵥ v) = 0) ∧
      0 < η ∧ η < 1 ∧ 0 < m ∧
      (∀ z ∈ Icc (-η) (1-η), m ≤ 1+c*z) ∧
      (∀ z ∈ Icc (-η) (1-η), ContinuousAt C z) ∧
      (∀ z ∈ Icc (-η) (1-η), ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun t => (C t).val) z) ∧
      (∀ z ∈ Icc (-η) (1-η), 0 < (splitJacobian d L c z).det) ∧
      (∀ z ∈ Icc (-η) (1-η), (C z).val =
        (splitJacobian d L c z).det⁻¹ •
          (splitJacobian d L c z * (A.val + z • B) * (splitJacobian d L c z)ᵀ)) ∧
      (∀ z ∈ Icc (-η) (1-η), C z ∈ matrixFiniteLaminate a b) ∧
      C (-η) ∈ spectralExtension T ∧ C (1-η) ∈ spectralExtension T := by
  let A := coordinateJoin i θ α β
  let P := preliminaryB i A α
  let N := preliminaryB i A β
  let qp := preliminaryQ i A α
  let qm := preliminaryQ i A β
  let η := θ/qp
  have hαi : 0 < α i := ha.trans ((hT α hα).bounds ha i).1
  have hβi : 0 < β i := ha.trans ((hT β hβ).bounds ha i).1
  have hf : 0 < qp ∧ 0 < qm ∧ 0 < η ∧ 0 < (1-θ)/qm ∧
    η+(1-θ)/qm=1 ∧ η*qp+((1-θ)/qm)*qm=1 := by
    simpa [qp, qm, η, A, preliminaryQ, coordinateJoin] using
      preliminary_fractions hαi hβi hθ hθ1
  have hη1 : η < 1 := by linarith [hf.2.2.2.1, hf.2.2.2.2.1]
  let c := qp-qm
  let q : ℝ → ℝ := fun z => 1+c*z
  let D : ℝ → DiagonalTriple := fun z => A + z • (P-N)
  let C : ℝ → Symmetric3 := fun z => conjugateDiagonal Q (stretchedDiagonal i (q z) (D z))
  have hs (z : ℝ) :
      q z = (z+η)*qp+(1-(z+η))*qm ∧
      D z = (z+η) • P + (1-(z+η)) • N := by
    have hh := preliminary_segment i hαi hβi hθ hθ1 z
    refine ⟨?_, hh.1⟩
    change 1 + (qp-qm)*z = _
    have heq : 1+z*(qp-qm)=(z+η)*qp+(1-(z+η))*qm := hh.2.1
    linarith only [heq]
  have hq (z : ℝ) (hz : z ∈ Icc (-η) (1-η)) : min qp qm ≤ q z := by
    rw [(hs z).1]
    have ht : 0 ≤ z+η := by linarith [hz.1]
    have ht' : 0 ≤ 1-(z+η) := by linarith [hz.2]
    have h1 := mul_le_mul_of_nonneg_left (min_le_left qp qm) ht
    have h2 := mul_le_mul_of_nonneg_left (min_le_right qp qm) ht'
    nlinarith
  have hm : 0 < min qp qm := lt_min hf.1 hf.2.1
  have hqpos (z : ℝ) (hz : z ∈ Icc (-η) (1-η)) : 0 < q z := hm.trans_le (hq z hz)
  have hP : P i = N i := by simp [P, N, preliminaryB]
  refine ⟨rotatedDirection Q i, rotatedPhase Q i, Q * Matrix.diagonal (P-N) * Qᵀ,
    η, c, min qp qm, C, rotated_normalization hQ i, ?_,
    rotated_increment_annihilates hQ i hP, hf.2.2.1, hη1, hm, hq, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact Matrix.isHermitian_iff_isSymm.mp (conjugateDiagonal Q (P-N)).property
  · intro z hz
    exact (conjugateDiagonal_continuous Q).continuousAt.comp
      (continuousAt_stretchedDiagonal (continuous_const.add
        (continuous_const.mul continuous_id)).continuousAt
        (continuous_const.add (continuous_id.smul continuous_const)).continuousAt (hqpos z hz).ne')
  · intro z hz
    exact (conjugateDiagonal_contDiff Q).contDiffAt.comp z
      (contDiffAt_stretchedDiagonal (contDiffAt_const.add
        (contDiffAt_const.mul contDiffAt_id))
        (contDiffAt_const.add (contDiffAt_id.smul contDiffAt_const)) (hqpos z hz).ne')
  · intro z hz
    rw [splitJacobian_det _ _ (rotated_normalization hQ i)]
    exact hqpos z hz
  · intro z hz
    rw [splitJacobian_rotated hQ]
    change Q * Matrix.diagonal (stretchedDiagonal i (q z) (D z)) * Qᵀ = _
    have he : (conjugateDiagonal Q A).val + z • (Q * Matrix.diagonal (P-N) * Qᵀ) =
        Q * Matrix.diagonal (D z) * Qᵀ := by
      change Q * Matrix.diagonal A * Qᵀ + z • (Q * Matrix.diagonal (P-N) * Qᵀ) =
        Q * Matrix.diagonal (A + z • (P-N)) * Qᵀ
      have hd : Matrix.diagonal (A+z • (P-N)) =
          Matrix.diagonal A + z • Matrix.diagonal (P-N) := by
        ext j k
        by_cases hjk : j = k
        · subst k; simp
        · simp [hjk]
      rw [hd, Matrix.mul_add, Matrix.add_mul, Matrix.mul_smul, Matrix.smul_mul]
    rw [he, conjugated_push hQ, normalStretch_push i (hqpos z hz).ne']
  · intro z hz
    have hz' : z ∈ Icc (-η) ((1-θ)/qm) := by
      constructor; exact hz.1; linarith [hz.2, hf.2.2.2.2.1]
    obtain ⟨_, w, hw, he⟩ := affine_synchronized_path i hαi hβi hθ hθ1 z hz'
    have hd : stretchedDiagonal i (q z) (D z) = coordinateJoin i w α β := by
      simpa only [q, c, mul_comm] using he
    change conjugateDiagonal Q _ ∈ spectralExtension _
    rw [hd]
    exact conjugateDiagonal_spectral hQ (.join i w hw.1 hw.2 (hT α hα) (hT β hβ))
  · have he := hs (-η)
    have eq : q (-η) = qm := by simpa using he.1
    have ed : D (-η) = N := by simpa using he.2
    change conjugateDiagonal Q (stretchedDiagonal i (q (-η)) (D (-η))) ∈ _
    rw [eq, ed, stretched_preliminary i hf.2.1.ne']
    · exact conjugateDiagonal_spectral hQ hβ
    · simpa [A, coordinateJoin] using (harmonic_parent_pos hαi hβi hθ hθ1).ne'
  · have he := hs (1-η)
    have eq : q (1-η) = qp := by simpa using he.1
    have ed : D (1-η) = P := by simpa using he.2
    change conjugateDiagonal Q (stretchedDiagonal i (q (1-η)) (D (1-η))) ∈ _
    rw [eq, ed, stretched_preliminary i hf.1.ne']
    · exact conjugateDiagonal_spectral hQ hα
    · simpa [A, coordinateJoin] using (harmonic_parent_pos hαi hβi hθ hθ1).ne'
end ScalarConductivity
end
end OAI

	import Mathlib
	import OAI.Analysis.Conductivity.Variational.MatrixFiniteLaminate
	import OAI.Analysis.Conductivity.Variational.QuantitativeLocalTwoFieldSplit

	namespace OAI

	noncomputable section
	open MeasureTheory
	open scoped ENNReal
	open Matrix Filter Topology
	open Set MeasureTheory Filter Topology
	open scoped BigOperators
	open Set MeasureTheory Filter Topology
	open scoped Manifold
	open Set Filter
	open scoped Topology
	open Set Filter MeasureTheory
	open scoped Topology Manifold ENNReal
	open Set
	namespace ScalarConductivity
	open Matrix Set Filter Topology
	open scoped Matrix.Norms.Elementwise

	def conjugateDiagonal (Q : Mat3) (α : DiagonalTriple) : Symmetric3 :=
	⟨Q * Matrix.diagonal α * Qᵀ, by
	rw [Matrix.isHermitian_iff_isSymm]
	unfold Matrix.IsSymm
	rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_transpose,
	Matrix.diagonal_transpose, Matrix.mul_assoc]⟩

	lemma conjugateDiagonal_continuous (Q : Mat3) : Continuous (conjugateDiagonal Q) := by
	apply Continuous.subtype_mk
	exact continuous_const.matrix_mul (continuous_pi (fun i => continuous_pi (fun j => by
	by_cases hij : i = j
	· subst j; simpa only [Matrix.diagonal_apply_eq] using continuous_apply i
	· simpa only [Matrix.diagonal_apply_ne _ hij] using
	(continuous_const : Continuous (fun _ : DiagonalTriple => (0:ℝ)))))) \|>.matrix_mul continuous_const

	lemma conjugateDiagonal_spectral {Q : Mat3} (hQ : Q ∈ orthogonalFrames)
	{α : DiagonalTriple} {T : Set DiagonalTriple} (hα : α ∈ T) :
	conjugateDiagonal Q α ∈ spectralExtension T := ⟨Q, hQ, α, hα, rfl⟩

	lemma stretched_preliminary (i : Fin 3) {A α : DiagonalTriple}
	(hq : preliminaryQ i A α ≠ 0) (hA : A i ≠ 0) :
	stretchedDiagonal i (preliminaryQ i A α) (preliminaryB i A α) = α := by
	ext j
	by_cases hj : j = i
	· subst j
	simp [stretchedDiagonal, preliminaryB, preliminaryQ, hA]
	· simp [stretchedDiagonal, preliminaryB, hj, hq]

	lemma continuousAt_stretchedDiagonal {i : Fin 3} {q : ℝ → ℝ} {D : ℝ → DiagonalTriple}
	{z : ℝ} (hq : ContinuousAt q z) (hD : ContinuousAt D z) (hn : q z ≠ 0) :
	ContinuousAt (fun t => stretchedDiagonal i (q t) (D t)) z := by
	apply continuousAt_pi.mpr; intro j
	have hj := continuousAt_pi.mp hD j
	by_cases hji : j = i
	· change ContinuousAt (fun y => if j = i then q y * D y j else D y j / q y) z
	simp only [ite_eq_left hji]
	exact hq.mul hj
	· change ContinuousAt (fun y => if j = i then q y * D y j else D y j / q y) z
	simp only [ite_eq_right hji]
	exact hj.div hq hn

	lemma contDiffAt_stretchedDiagonal {i : Fin 3} {q : ℝ → ℝ} {D : ℝ → DiagonalTriple}
	{z : ℝ} {k : WithTop ℕ∞} (hq : ContDiffAt ℝ k q z) (hD : ContDiffAt ℝ k D z)
	(hn : q z ≠ 0) : ContDiffAt ℝ k (fun t => stretchedDiagonal i (q t) (D t)) z := by
	apply contDiffAt_pi.mpr; intro j
	have hj := contDiffAt_pi.mp hD j
	by_cases hji : j = i
	· change ContDiffAt ℝ k (fun y => if j = i then q y * D y j else D y j / q y) z
	simp only [ite_eq_left hji]
	exact hq.mul hj
	· change ContDiffAt ℝ k (fun y => if j = i then q y * D y j else D y j / q y) z
	simp only [ite_eq_right hji]
	exact hj.div hq hn

	lemma conjugateDiagonal_contDiff (Q : Mat3) :
	ContDiff ℝ (↑(⊤ : ℕ∞)) (fun α : DiagonalTriple => (conjugateDiagonal Q α).val) := by
	rw [contDiff_iff_contDiffAt]; intro α
	apply contDiffAt_matrix_mul _ contDiffAt_const
	apply contDiffAt_matrix_mul contDiffAt_const
	apply contDiffAt_pi.mpr; intro i
	apply contDiffAt_pi.mpr; intro j
	by_cases hij : i = j
	· subst j; simpa only [Matrix.diagonal_apply_eq, id_eq] using
	(contDiffAt_pi.mp (contDiffAt_id (x := α)) i)
	· simpa only [Matrix.diagonal_apply_ne _ hij] using
	(contDiffAt_const : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun _ : DiagonalTriple => (0:ℝ)) α)

	theorem spectral_join_plan {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple}
	(hT : ∀ α ∈ T, IsFiniteLaminate a b α)
	(i : Fin 3) {α β : DiagonalTriple} (hα : α ∈ T) (hβ : β ∈ T)
	{θ : ℝ} (hθ : 0 < θ) (hθ1 : θ < 1)
	{Q : Mat3} (hQ : Q ∈ orthogonalFrames) :
	let A := conjugateDiagonal Q (coordinateJoin i θ α β)
	∃ (d : Coord3) (L : Coord3 →L[ℝ] ℝ) (B : Mat3) (η c m : ℝ)
	(C : ℝ → Symmetric3),
	L d = 1 ∧ B.IsSymm ∧ (∀ v, L (B *ᵥ v) = 0) ∧
	0 < η ∧ η < 1 ∧ 0 < m ∧
	(∀ z ∈ Icc (-η) (1-η), m ≤ 1+c*z) ∧
	(∀ z ∈ Icc (-η) (1-η), ContinuousAt C z) ∧
	(∀ z ∈ Icc (-η) (1-η), ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun t => (C t).val) z) ∧
	(∀ z ∈ Icc (-η) (1-η), 0 < (splitJacobian d L c z).det) ∧
	(∀ z ∈ Icc (-η) (1-η), (C z).val =
	(splitJacobian d L c z).det⁻¹ •
	(splitJacobian d L c z * (A.val + z • B) * (splitJacobian d L c z)ᵀ)) ∧
	(∀ z ∈ Icc (-η) (1-η), C z ∈ matrixFiniteLaminate a b) ∧
	C (-η) ∈ spectralExtension T ∧ C (1-η) ∈ spectralExtension T := by
	let A := coordinateJoin i θ α β
	let P := preliminaryB i A α
	let N := preliminaryB i A β
	let qp := preliminaryQ i A α
	let qm := preliminaryQ i A β
	let η := θ/qp
	have hαi : 0 < α i := ha.trans ((hT α hα).bounds ha i).1
	have hβi : 0 < β i := ha.trans ((hT β hβ).bounds ha i).1
	have hf : 0 < qp ∧ 0 < qm ∧ 0 < η ∧ 0 < (1-θ)/qm ∧
	η+(1-θ)/qm=1 ∧ ηqp+((1-θ)/qm)qm=1 := by
	simpa [qp, qm, η, A, preliminaryQ, coordinateJoin] using
	preliminary_fractions hαi hβi hθ hθ1
	have hη1 : η < 1 := by linarith [hf.2.2.2.1, hf.2.2.2.2.1]
	let c := qp-qm
	let q : ℝ → ℝ := fun z => 1+c*z
	let D : ℝ → DiagonalTriple := fun z => A + z • (P-N)
	let C : ℝ → Symmetric3 := fun z => conjugateDiagonal Q (stretchedDiagonal i (q z) (D z))
	have hs (z : ℝ) :
	q z = (z+η)qp+(1-(z+η))qm ∧
	D z = (z+η) • P + (1-(z+η)) • N := by
	have hh := preliminary_segment i hαi hβi hθ hθ1 z
	refine ⟨?_, hh.1⟩
	change 1 + (qp-qm)*z = _
	have heq : 1+z(qp-qm)=(z+η)qp+(1-(z+η))*qm := hh.2.1
	linarith only [heq]
	have hq (z : ℝ) (hz : z ∈ Icc (-η) (1-η)) : min qp qm ≤ q z := by
	rw [(hs z).1]
	have ht : 0 ≤ z+η := by linarith [hz.1]
	have ht' : 0 ≤ 1-(z+η) := by linarith [hz.2]
	have h1 := mul_le_mul_of_nonneg_left (min_le_left qp qm) ht
	have h2 := mul_le_mul_of_nonneg_left (min_le_right qp qm) ht'
	nlinarith
	have hm : 0 < min qp qm := lt_min hf.1 hf.2.1
	have hqpos (z : ℝ) (hz : z ∈ Icc (-η) (1-η)) : 0 < q z := hm.trans_le (hq z hz)
	have hP : P i = N i := by simp [P, N, preliminaryB]
	refine ⟨rotatedDirection Q i, rotatedPhase Q i, Q * Matrix.diagonal (P-N) * Qᵀ,
	η, c, min qp qm, C, rotated_normalization hQ i, ?_,
	rotated_increment_annihilates hQ i hP, hf.2.2.1, hη1, hm, hq, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
	· exact Matrix.isHermitian_iff_isSymm.mp (conjugateDiagonal Q (P-N)).property
	· intro z hz
	exact (conjugateDiagonal_continuous Q).continuousAt.comp
	(continuousAt_stretchedDiagonal (continuous_const.add
	(continuous_const.mul continuous_id)).continuousAt
	(continuous_const.add (continuous_id.smul continuous_const)).continuousAt (hqpos z hz).ne')
	· intro z hz
	exact (conjugateDiagonal_contDiff Q).contDiffAt.comp z
	(contDiffAt_stretchedDiagonal (contDiffAt_const.add
	(contDiffAt_const.mul contDiffAt_id))
	(contDiffAt_const.add (contDiffAt_id.smul contDiffAt_const)) (hqpos z hz).ne')
	· intro z hz
	rw [splitJacobian_det _ _ (rotated_normalization hQ i)]
	exact hqpos z hz
	· intro z hz
	rw [splitJacobian_rotated hQ]
	change Q * Matrix.diagonal (stretchedDiagonal i (q z) (D z)) * Qᵀ = _
	have he : (conjugateDiagonal Q A).val + z • (Q * Matrix.diagonal (P-N) * Qᵀ) =
	Q * Matrix.diagonal (D z) * Qᵀ := by
	change Q * Matrix.diagonal A * Qᵀ + z • (Q * Matrix.diagonal (P-N) * Qᵀ) =
	Q * Matrix.diagonal (A + z • (P-N)) * Qᵀ
	have hd : Matrix.diagonal (A+z • (P-N)) =
	Matrix.diagonal A + z • Matrix.diagonal (P-N) := by
	ext j k
	by_cases hjk : j = k
	· subst k; simp
	· simp [hjk]
	rw [hd, Matrix.mul_add, Matrix.add_mul, Matrix.mul_smul, Matrix.smul_mul]
	rw [he, conjugated_push hQ, normalStretch_push i (hqpos z hz).ne']
	· intro z hz
	have hz' : z ∈ Icc (-η) ((1-θ)/qm) := by
	constructor; exact hz.1; linarith [hz.2, hf.2.2.2.2.1]
	obtain ⟨_, w, hw, he⟩ := affine_synchronized_path i hαi hβi hθ hθ1 z hz'
	have hd : stretchedDiagonal i (q z) (D z) = coordinateJoin i w α β := by
	simpa only [q, c, mul_comm] using he
	change conjugateDiagonal Q _ ∈ spectralExtension _
	rw [hd]
	exact conjugateDiagonal_spectral hQ (.join i w hw.1 hw.2 (hT α hα) (hT β hβ))
	· have he := hs (-η)
	have eq : q (-η) = qm := by simpa using he.1
	have ed : D (-η) = N := by simpa using he.2
	change conjugateDiagonal Q (stretchedDiagonal i (q (-η)) (D (-η))) ∈ _
	rw [eq, ed, stretched_preliminary i hf.2.1.ne']
	· exact conjugateDiagonal_spectral hQ hβ
	· simpa [A, coordinateJoin] using (harmonic_parent_pos hαi hβi hθ hθ1).ne'
	· have he := hs (1-η)
	have eq : q (1-η) = qp := by simpa using he.1
	have ed : D (1-η) = P := by simpa using he.2
	change conjugateDiagonal Q (stretchedDiagonal i (q (1-η)) (D (1-η))) ∈ _
	rw [eq, ed, stretched_preliminary i hf.1.ne']
	· exact conjugateDiagonal_spectral hQ hα
	· simpa [A, coordinateJoin] using (harmonic_parent_pos hαi hβi hθ hθ1).ne'

	end ScalarConductivity

	end

	end OAI