feat: port Analysis.SpecialFunctions.Trigonometric.ArctanDeriv (#4764)

Author: Parcly-Taxel

Mathlib.lean

@@ -655,6 +655,7 @@ import Mathlib.Analysis.SpecialFunctions.Pow.Real
 import Mathlib.Analysis.SpecialFunctions.Sqrt
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean

@@ -0,0 +1,232 @@
+/-
+Copyright (c) 2018 Chris Hughes. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
+
+! This file was ported from Lean 3 source module analysis.special_functions.trigonometric.arctan_deriv
+! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv
+
+/-!
+# Derivatives of the `tan` and `arctan` functions.
+
+Continuity and derivatives of the tangent and arctangent functions.
+-/
+
+
+noncomputable section
+
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+
+namespace Real
+
+open Set Filter
+
+open scoped Topology Real
+
+theorem hasStrictDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x := by
+  exact_mod_cast (Complex.hasStrictDerivAt_tan (by exact_mod_cast h)).real_of_complex
+#align real.has_strict_deriv_at_tan Real.hasStrictDerivAt_tan
+
+theorem hasDerivAt_tan {x : ℝ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x := by
+  exact_mod_cast (Complex.hasDerivAt_tan (by exact_mod_cast h)).real_of_complex
+#align real.has_deriv_at_tan Real.hasDerivAt_tan
+
+theorem tendsto_abs_tan_of_cos_eq_zero {x : ℝ} (hx : cos x = 0) :
+    Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop := by
+  have hx : Complex.cos x = 0 := by exact_mod_cast hx
+  simp only [← Complex.abs_ofReal, Complex.ofReal_tan]
+  refine' (Complex.tendsto_abs_tan_of_cos_eq_zero hx).comp _
+  refine' Tendsto.inf Complex.continuous_ofReal.continuousAt _
+  exact tendsto_principal_principal.2 fun y => mt Complex.ofReal_inj.1
+#align real.tendsto_abs_tan_of_cos_eq_zero Real.tendsto_abs_tan_of_cos_eq_zero
+
+theorem tendsto_abs_tan_atTop (k : ℤ) :
+    Tendsto (fun x => abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2)) atTop :=
+  tendsto_abs_tan_of_cos_eq_zero <| cos_eq_zero_iff.2 ⟨k, rfl⟩
+#align real.tendsto_abs_tan_at_top Real.tendsto_abs_tan_atTop
+
+theorem continuousAt_tan {x : ℝ} : ContinuousAt tan x ↔ cos x ≠ 0 := by
+  refine' ⟨fun hc h₀ => _, fun h => (hasDerivAt_tan h).continuousAt⟩
+  exact not_tendsto_nhds_of_tendsto_atTop (tendsto_abs_tan_of_cos_eq_zero h₀) _
+    (hc.norm.tendsto.mono_left inf_le_left)
+#align real.continuous_at_tan Real.continuousAt_tan
+
+theorem differentiableAt_tan {x : ℝ} : DifferentiableAt ℝ tan x ↔ cos x ≠ 0 :=
+  ⟨fun h => continuousAt_tan.1 h.continuousAt, fun h => (hasDerivAt_tan h).differentiableAt⟩
+#align real.differentiable_at_tan Real.differentiableAt_tan
+
+@[simp]
+theorem deriv_tan (x : ℝ) : deriv tan x = 1 / cos x ^ 2 :=
+  if h : cos x = 0 then by
+    have : ¬DifferentiableAt ℝ tan x := mt differentiableAt_tan.1 (Classical.not_not.2 h)
+    simp [deriv_zero_of_not_differentiableAt this, h, sq]
+  else (hasDerivAt_tan h).deriv
+#align real.deriv_tan Real.deriv_tan
+
+@[simp]
+theorem contDiffAt_tan {n x} : ContDiffAt ℝ n tan x ↔ cos x ≠ 0 :=
+  ⟨fun h => continuousAt_tan.1 h.continuousAt, fun h =>
+    (Complex.contDiffAt_tan.2 <| by exact_mod_cast h).real_of_complex⟩
+#align real.cont_diff_at_tan Real.contDiffAt_tan
+
+theorem hasDerivAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
+    HasDerivAt tan (1 / cos x ^ 2) x :=
+  hasDerivAt_tan (cos_pos_of_mem_Ioo h).ne'
+#align real.has_deriv_at_tan_of_mem_Ioo Real.hasDerivAt_tan_of_mem_Ioo
+
+theorem differentiableAt_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π / 2) : ℝ) (π / 2)) :
+    DifferentiableAt ℝ tan x :=
+  (hasDerivAt_tan_of_mem_Ioo h).differentiableAt
+#align real.differentiable_at_tan_of_mem_Ioo Real.differentiableAt_tan_of_mem_Ioo
+
+theorem hasStrictDerivAt_arctan (x : ℝ) : HasStrictDerivAt arctan (1 / (1 + x ^ 2)) x := by
+  have A : cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne'
+  simpa [cos_sq_arctan] using
+    tanLocalHomeomorph.hasStrictDerivAt_symm trivial (by simpa) (hasStrictDerivAt_tan A)
+#align real.has_strict_deriv_at_arctan Real.hasStrictDerivAt_arctan
+
+theorem hasDerivAt_arctan (x : ℝ) : HasDerivAt arctan (1 / (1 + x ^ 2)) x :=
+  (hasStrictDerivAt_arctan x).hasDerivAt
+#align real.has_deriv_at_arctan Real.hasDerivAt_arctan
+
+theorem differentiableAt_arctan (x : ℝ) : DifferentiableAt ℝ arctan x :=
+  (hasDerivAt_arctan x).differentiableAt
+#align real.differentiable_at_arctan Real.differentiableAt_arctan
+
+theorem differentiable_arctan : Differentiable ℝ arctan :=
+  differentiableAt_arctan
+#align real.differentiable_arctan Real.differentiable_arctan
+
+@[simp]
+theorem deriv_arctan : deriv arctan = fun (x : ℝ) => 1 / (1 + x ^ 2) :=
+  funext fun x => (hasDerivAt_arctan x).deriv
+#align real.deriv_arctan Real.deriv_arctan
+
+theorem contDiff_arctan {n : ℕ∞} : ContDiff ℝ n arctan :=
+  contDiff_iff_contDiffAt.2 fun x =>
+    have : cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne'
+    tanLocalHomeomorph.contDiffAt_symm_deriv (by simpa) trivial (hasDerivAt_tan this)
+      (contDiffAt_tan.2 this)
+#align real.cont_diff_arctan Real.contDiff_arctan
+
+end Real
+
+section
+
+/-!
+### Lemmas for derivatives of the composition of `Real.arctan` with a differentiable function
+
+In this section we register lemmas for the derivatives of the composition of `Real.arctan` with a
+differentiable function, for standalone use and use with `simp`. -/
+
+
+open Real
+
+section deriv
+
+variable {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ}
+
+theorem HasStrictDerivAt.arctan (hf : HasStrictDerivAt f f' x) :
+    HasStrictDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x :=
+  (Real.hasStrictDerivAt_arctan (f x)).comp x hf
+#align has_strict_deriv_at.arctan HasStrictDerivAt.arctan
+
+theorem HasDerivAt.arctan (hf : HasDerivAt f f' x) :
+    HasDerivAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') x :=
+  (Real.hasDerivAt_arctan (f x)).comp x hf
+#align has_deriv_at.arctan HasDerivAt.arctan
+
+theorem HasDerivWithinAt.arctan (hf : HasDerivWithinAt f f' s x) :
+    HasDerivWithinAt (fun x => arctan (f x)) (1 / (1 + f x ^ 2) * f') s x :=
+  (Real.hasDerivAt_arctan (f x)).comp_hasDerivWithinAt x hf
+#align has_deriv_within_at.arctan HasDerivWithinAt.arctan
+
+theorem derivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
+    derivWithin (fun x => arctan (f x)) s x = 1 / (1 + f x ^ 2) * derivWithin f s x :=
+  hf.hasDerivWithinAt.arctan.derivWithin hxs
+#align deriv_within_arctan derivWithin_arctan
+
+@[simp]
+theorem deriv_arctan (hc : DifferentiableAt ℝ f x) :
+    deriv (fun x => arctan (f x)) x = 1 / (1 + f x ^ 2) * deriv f x :=
+  hc.hasDerivAt.arctan.deriv
+#align deriv_arctan deriv_arctan
+
+end deriv
+
+section fderiv
+
+variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E}
+  {s : Set E} {n : ℕ∞}
+
+theorem HasStrictFDerivAt.arctan (hf : HasStrictFDerivAt f f' x) :
+    HasStrictFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x :=
+  (hasStrictDerivAt_arctan (f x)).comp_hasStrictFDerivAt x hf
+#align has_strict_fderiv_at.arctan HasStrictFDerivAt.arctan
+
+theorem HasFDerivAt.arctan (hf : HasFDerivAt f f' x) :
+    HasFDerivAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') x :=
+  (hasDerivAt_arctan (f x)).comp_hasFDerivAt x hf
+#align has_fderiv_at.arctan HasFDerivAt.arctan
+
+theorem HasFDerivWithinAt.arctan (hf : HasFDerivWithinAt f f' s x) :
+    HasFDerivWithinAt (fun x => arctan (f x)) ((1 / (1 + f x ^ 2)) • f') s x :=
+  (hasDerivAt_arctan (f x)).comp_hasFDerivWithinAt x hf
+#align has_fderiv_within_at.arctan HasFDerivWithinAt.arctan
+
+theorem fderivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
+    fderivWithin ℝ (fun x => arctan (f x)) s x = (1 / (1 + f x ^ 2)) • fderivWithin ℝ f s x :=
+  hf.hasFDerivWithinAt.arctan.fderivWithin hxs
+#align fderiv_within_arctan fderivWithin_arctan
+
+@[simp]
+theorem fderiv_arctan (hc : DifferentiableAt ℝ f x) :
+    fderiv ℝ (fun x => arctan (f x)) x = (1 / (1 + f x ^ 2)) • fderiv ℝ f x :=
+  hc.hasFDerivAt.arctan.fderiv
+#align fderiv_arctan fderiv_arctan
+
+theorem DifferentiableWithinAt.arctan (hf : DifferentiableWithinAt ℝ f s x) :
+    DifferentiableWithinAt ℝ (fun x => Real.arctan (f x)) s x :=
+  hf.hasFDerivWithinAt.arctan.differentiableWithinAt
+#align differentiable_within_at.arctan DifferentiableWithinAt.arctan
+
+@[simp]
+theorem DifferentiableAt.arctan (hc : DifferentiableAt ℝ f x) :
+    DifferentiableAt ℝ (fun x => arctan (f x)) x :=
+  hc.hasFDerivAt.arctan.differentiableAt
+#align differentiable_at.arctan DifferentiableAt.arctan
+
+theorem DifferentiableOn.arctan (hc : DifferentiableOn ℝ f s) :
+    DifferentiableOn ℝ (fun x => arctan (f x)) s := fun x h => (hc x h).arctan
+#align differentiable_on.arctan DifferentiableOn.arctan
+
+@[simp]
+theorem Differentiable.arctan (hc : Differentiable ℝ f) : Differentiable ℝ fun x => arctan (f x) :=
+  fun x => (hc x).arctan
+#align differentiable.arctan Differentiable.arctan
+
+theorem ContDiffAt.arctan (h : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => arctan (f x)) x :=
+  contDiff_arctan.contDiffAt.comp x h
+#align cont_diff_at.arctan ContDiffAt.arctan
+
+theorem ContDiff.arctan (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => arctan (f x) :=
+  contDiff_arctan.comp h
+#align cont_diff.arctan ContDiff.arctan
+
+theorem ContDiffWithinAt.arctan (h : ContDiffWithinAt ℝ n f s x) :
+    ContDiffWithinAt ℝ n (fun x => arctan (f x)) s x :=
+  contDiff_arctan.comp_contDiffWithinAt h
+#align cont_diff_within_at.arctan ContDiffWithinAt.arctan
+
+theorem ContDiffOn.arctan (h : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => arctan (f x)) s :=
+  contDiff_arctan.comp_contDiffOn h
+#align cont_diff_on.arctan ContDiffOn.arctan
+
+end fderiv
+
+end