openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /PartialResolventCalculus.lean
| import Mathlib.Analysis.InnerProductSpace.LinearPMap | |
| import Mathlib.Analysis.InnerProductSpace.Adjoint | |
| import Mathlib.Analysis.InnerProductSpace.Positive | |
| import Mathlib.Tactic.Linarith | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.PartialResolvent | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] | |
| variable (H : E →ₗ.[ℂ] E) (R : E →L[ℂ] E) | |
| variable (hD : ∀ x : E, R x ∈ H.domain) | |
| variable (hHR : ∀ x : E, H ⟨R x, hD x⟩ = x - R x) | |
| variable (hRH : ∀ x : H.domain, R ((x : E) + H x) = x) | |
| include hD hHR | |
| theorem injective : Function.Injective R := by | |
| apply (LinearMap.ker_eq_bot).mp | |
| apply LinearMap.ker_eq_bot'.mpr | |
| intro x hx | |
| change R x = 0 at hx | |
| have hz : (⟨R x, hD x⟩ : H.domain) = 0 := Subtype.ext hx | |
| have heq := hHR x | |
| rw [hz, H.map_zero, hx, sub_zero] at heq | |
| exact heq.symm | |
| include hRH | |
| omit hHR in | |
| theorem domain_eq_range : H.domain = R.range := by | |
| ext x | |
| constructor | |
| · intro hx | |
| exact ⟨x + H ⟨x, hx⟩, hRH ⟨x, hx⟩⟩ | |
| · rintro ⟨y, rfl⟩ | |
| exact hD y | |
| theorem range_one_sub : (1 - R).range = LinearMap.range H.toFun := by | |
| ext y | |
| constructor | |
| · rintro ⟨x, rfl⟩ | |
| exact ⟨⟨R x, hD x⟩, hHR x⟩ | |
| · rintro ⟨x, rfl⟩ | |
| refine ⟨(x : E) + H x, ?_⟩ | |
| change (x : E) + H x - R ((x : E) + H x) = H x | |
| rw [hRH] | |
| exact add_sub_cancel_left _ _ | |
| omit hRH in | |
| theorem injective_one_sub (hH : Function.Injective H) : | |
| Function.Injective (1 - R : E →L[ℂ] E) := by | |
| apply (LinearMap.ker_eq_bot).mp | |
| apply LinearMap.ker_eq_bot'.mpr | |
| intro x hx | |
| have hzero : H ⟨R x, hD x⟩ = H 0 := by | |
| rw [H.map_zero, hHR] | |
| exact hx | |
| have hRx : R x = 0 := congrArg Subtype.val (hH hzero) | |
| exact injective H R hD hHR (by simpa only [map_zero] using hRx) | |
| omit hRH in | |
| theorem one_sub_mem_inverse_domain (x : E) : | |
| (1 - R) x ∈ H.inverse.domain := by | |
| rw [LinearPMap.inverse_domain] | |
| exact ⟨⟨R x, hD x⟩, hHR x⟩ | |
| omit hRH in | |
| theorem inverse_one_sub (hH : Function.Injective H) (x : E) : | |
| H.inverse ⟨(1 - R) x, one_sub_mem_inverse_domain H R hD hHR x⟩ = R x := by | |
| apply LinearPMap.inverse_apply_eq (LinearPMap.ker_eq_bot.mpr hH) | |
| (x := ⟨R x, hD x⟩) | |
| exact hHR x | |
| theorem inverse_domain_eq_range : H.inverse.domain = (1 - R).range := by | |
| rw [LinearPMap.inverse_domain, range_one_sub H R hD hHR hRH] | |
| omit hD hHR in | |
| theorem one_sub_apply_add_inverse (hH : Function.Injective H) | |
| (y : H.inverse.domain) : | |
| (1 - R) ((y : E) + H.inverse y) = y := by | |
| have hy : (y : E) ∈ LinearMap.range H.toFun := by | |
| rw [← LinearPMap.inverse_domain] | |
| exact y.property | |
| obtain ⟨x, hx⟩ := hy | |
| have hinv : H.inverse y = (x : E) := | |
| LinearPMap.inverse_apply_eq (LinearPMap.ker_eq_bot.mpr hH) hx | |
| rw [hinv, ← hx] | |
| change H x + (x : E) - R (H x + (x : E)) = H x | |
| rw [add_comm (H x) (x : E), hRH] | |
| exact add_sub_cancel_left _ _ | |
| omit hRH in | |
| theorem isSymmetric (hH : H.IsFormalAdjoint H) : R.toLinearMap.IsSymmetric := by | |
| intro x y | |
| have heq := hH ⟨R x, hD x⟩ ⟨R y, hD y⟩ | |
| rw [hHR, hHR] at heq | |
| simp only [inner_sub_left, inner_sub_right] at heq | |
| exact (sub_left_inj.mp heq).symm | |
| omit hRH in | |
| theorem norm_sq_le_re_inner | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) (x : E) : | |
| ‖R x‖ ^ 2 ≤ (inner ℂ (R x) x).re := by | |
| have hp := hpos ⟨R x, hD x⟩ | |
| rw [hHR] at hp | |
| change 0 ≤ (inner ℂ (R x) (x - R x)).re at hp | |
| rw [inner_sub_right, Complex.sub_re] at hp | |
| rw [show (inner ℂ (R x) (R x)).re = ‖R x‖ ^ 2 from | |
| inner_self_eq_norm_sq (𝕜 := ℂ) (R x)] at hp | |
| exact sub_nonneg.mp hp | |
| omit hRH in | |
| theorem one_sub_norm_sq_le_re_inner | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) (x : E) : | |
| ‖(1 - R) x‖ ^ 2 ≤ (inner ℂ ((1 - R) x) x).re := by | |
| have hp := hpos ⟨R x, hD x⟩ | |
| rw [hHR] at hp | |
| have hx : R x = x - (1 - R) x := by | |
| change R x = x - (x - R x) | |
| rw [sub_sub_cancel] | |
| change 0 ≤ (inner ℂ (R x) ((1 - R) x)).re at hp | |
| rw [hx, inner_sub_left, Complex.sub_re] at hp | |
| rw [show (inner ℂ ((1 - R) x) ((1 - R) x)).re = ‖(1 - R) x‖ ^ 2 from | |
| inner_self_eq_norm_sq (𝕜 := ℂ) ((1 - R) x)] at hp | |
| rw [show (inner ℂ x ((1 - R) x)).re = (inner ℂ ((1 - R) x) x).re from | |
| inner_re_symm (𝕜 := ℂ) x ((1 - R) x)] at hp | |
| exact sub_nonneg.mp hp | |
| omit hRH in | |
| theorem isPositive (hH : H.IsFormalAdjoint H) | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) : | |
| R.IsPositive := by | |
| refine ⟨isSymmetric H R hD hHR hH, fun x => ?_⟩ | |
| exact (sq_nonneg ‖R x‖).trans (norm_sq_le_re_inner H R hD hHR hpos x) | |
| omit hRH in | |
| theorem one_sub_isPositive (hH : H.IsFormalAdjoint H) | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) : | |
| (1 - R : E →L[ℂ] E).IsPositive := by | |
| refine ⟨?_, fun x => ?_⟩ | |
| · intro x y | |
| change inner ℂ (x - R x) y = inner ℂ x (y - R y) | |
| rw [inner_sub_left, inner_sub_right] | |
| have hs := isSymmetric H R hD hHR hH x y | |
| change inner ℂ (R x) y = inner ℂ x (R y) at hs | |
| rw [hs] | |
| · exact (sq_nonneg ‖(1 - R) x‖).trans | |
| (one_sub_norm_sq_le_re_inner H R hD hHR hpos x) | |
| omit hD hHR hRH in | |
| private theorem norm_le_one_of_quadratic (T : E →L[ℂ] E) | |
| (hT : ∀ x, ‖T x‖ ^ 2 ≤ (inner ℂ (T x) x).re) : ‖T‖ ≤ 1 := by | |
| apply T.opNorm_le_bound (by norm_num) | |
| intro x | |
| rw [one_mul] | |
| have hinner := re_inner_le_norm (𝕜 := ℂ) (T x) x | |
| change (inner ℂ (T x) x).re ≤ ‖T x‖ * ‖x‖ at hinner | |
| have hquad := hT x | |
| have hnorm := norm_nonneg (T x) | |
| by_contra h | |
| have hstrict : ‖x‖ < ‖T x‖ := lt_of_not_ge h | |
| nlinarith [norm_nonneg x] | |
| omit hRH in | |
| theorem norm_le_one | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) : ‖R‖ ≤ 1 := | |
| norm_le_one_of_quadratic R (norm_sq_le_re_inner H R hD hHR hpos) | |
| omit hRH in | |
| theorem one_sub_norm_le_one | |
| (hpos : ∀ x : H.domain, 0 ≤ (inner ℂ (x : E) (H x)).re) : | |
| ‖(1 - R : E →L[ℂ] E)‖ ≤ 1 := | |
| norm_le_one_of_quadratic (1 - R) (one_sub_norm_sq_le_re_inner H R hD hHR hpos) | |
| variable [CompleteSpace E] | |
| omit hRH in | |
| theorem isSelfAdjoint (hH : H.IsFormalAdjoint H) : IsSelfAdjoint R := | |
| ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr | |
| (isSymmetric H R hD hHR hH) | |
| omit hRH in | |
| theorem one_sub_isSelfAdjoint (hH : H.IsFormalAdjoint H) : | |
| IsSelfAdjoint (1 - R : E →L[ℂ] E) := | |
| by | |
| change star (1 - R) = 1 - R | |
| rw [star_sub, star_one, isSelfAdjoint H R hD hHR hH] | |
| end Invariant233.PartialResolvent | |
| end | |
| end OAI | |