chore: rename lemmas containing "of_open" to match the naming convention (#8229)

Mostly, this means replacing "of_open" by "of_isOpen".
A few lemmas names were misleading and are corrected differently. [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Naming.20convention/near/402702125).

Author: grunweg

Mathlib/AlgebraicGeometry/OpenImmersion.lean

@@ -267,7 +267,7 @@ theorem affineBasisCover_is_basis (X : Scheme) :
     TopologicalSpace.IsTopologicalBasis
       {x : Set X |
         ∃ a : X.affineBasisCover.J, x = Set.range (X.affineBasisCover.map a).1.base} := by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨a, rfl⟩
     exact IsOpenImmersion.open_range (X.affineBasisCover.map a)
   · rintro a U haU hU

Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean

@@ -808,7 +808,7 @@ theorem basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basic
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) := by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact isOpen_basicOpen
   · rintro p U hp ⟨s, hs⟩

Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean

@@ -453,7 +453,7 @@ theorem basicOpen_eq_union_of_projection (f : A) :
 theorem isTopologicalBasis_basic_opens :
     TopologicalSpace.IsTopologicalBasis
       (Set.range fun r : A => (basicOpen 𝒜 r : Set (ProjectiveSpectrum 𝒜))) := by
-  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+  apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
   · rintro _ ⟨r, rfl⟩
     exact isOpen_basicOpen 𝒜
   · rintro p U hp ⟨s, hs⟩

Mathlib/Analysis/Calculus/ContDiff/Basic.lean

@@ -1833,7 +1833,7 @@ theorem LocalHomeomorph.contDiffAt_symm [CompleteSpace E] (f : LocalHomeomorph E
       obtain ⟨t, htu, ht, htf⟩ := mem_nhds_iff.mp (Filter.inter_mem hu h_nhds)
       use f.target ∩ f.symm ⁻¹' t
       refine' ⟨IsOpen.mem_nhds _ _, _⟩
-      · exact f.preimage_open_of_open_symm ht
+      · exact f.isOpen_inter_preimage_symm ht
       · exact mem_inter ha (mem_preimage.mpr htf)
       intro x hx
       obtain ⟨hxu, e, he⟩ := htu hx.2
@@ -1937,11 +1937,11 @@ theorem contDiffOn_succ_iff_derivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s₂)
 
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (formulated with `deriv`) is `C^n`. -/
-theorem contDiffOn_succ_iff_deriv_of_open {n : ℕ} (hs : IsOpen s₂) :
+theorem contDiffOn_succ_iff_deriv_of_isOpen {n : ℕ} (hs : IsOpen s₂) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 n (deriv f₂) s₂ := by
   rw [contDiffOn_succ_iff_derivWithin hs.uniqueDiffOn]
-  exact Iff.rfl.and (contDiffOn_congr fun _ => derivWithin_of_open hs)
-#align cont_diff_on_succ_iff_deriv_of_open contDiffOn_succ_iff_deriv_of_open
+  exact Iff.rfl.and (contDiffOn_congr fun _ => derivWithin_of_isOpen hs)
+#align cont_diff_on_succ_iff_deriv_of_open contDiffOn_succ_iff_deriv_of_isOpen
 
 /-- A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
 there, and its derivative (formulated with `derivWithin`) is `C^∞`. -/
@@ -1961,11 +1961,11 @@ theorem contDiffOn_top_iff_derivWithin (hs : UniqueDiffOn 𝕜 s₂) :
 
 /-- A function is `C^∞` on an open domain if and only if it is differentiable
 there, and its derivative (formulated with `deriv`) is `C^∞`. -/
-theorem contDiffOn_top_iff_deriv_of_open (hs : IsOpen s₂) :
+theorem contDiffOn_top_iff_deriv_of_isOpen (hs : IsOpen s₂) :
     ContDiffOn 𝕜 ∞ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ∞ (deriv f₂) s₂ := by
   rw [contDiffOn_top_iff_derivWithin hs.uniqueDiffOn]
-  exact Iff.rfl.and <| contDiffOn_congr fun _ => derivWithin_of_open hs
-#align cont_diff_on_top_iff_deriv_of_open contDiffOn_top_iff_deriv_of_open
+  exact Iff.rfl.and <| contDiffOn_congr fun _ => derivWithin_of_isOpen hs
+#align cont_diff_on_top_iff_deriv_of_open contDiffOn_top_iff_deriv_of_isOpen
 
 protected theorem ContDiffOn.derivWithin (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂)
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (derivWithin f₂ s₂) s₂ := by
@@ -1978,26 +1978,26 @@ protected theorem ContDiffOn.derivWithin (hf : ContDiffOn 𝕜 n f₂ s₂) (hs
     exact ((contDiffOn_succ_iff_derivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.deriv_within ContDiffOn.derivWithin
 
-theorem ContDiffOn.deriv_of_open (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hmn : m + 1 ≤ n) :
+theorem ContDiffOn.deriv_of_isOpen (hf : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (deriv f₂) s₂ :=
-  (hf.derivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (derivWithin_of_open hs hx).symm
-#align cont_diff_on.deriv_of_open ContDiffOn.deriv_of_open
+  (hf.derivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (derivWithin_of_isOpen hs hx).symm
+#align cont_diff_on.deriv_of_open ContDiffOn.deriv_of_isOpen
 
 theorem ContDiffOn.continuousOn_derivWithin (h : ContDiffOn 𝕜 n f₂ s₂) (hs : UniqueDiffOn 𝕜 s₂)
     (hn : 1 ≤ n) : ContinuousOn (derivWithin f₂ s₂) s₂ :=
   ((contDiffOn_succ_iff_derivWithin hs).1 (h.of_le hn)).2.continuousOn
 #align cont_diff_on.continuous_on_deriv_within ContDiffOn.continuousOn_derivWithin
 
-theorem ContDiffOn.continuousOn_deriv_of_open (h : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂)
+theorem ContDiffOn.continuousOn_deriv_of_isOpen (h : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂)
     (hn : 1 ≤ n) : ContinuousOn (deriv f₂) s₂ :=
-  ((contDiffOn_succ_iff_deriv_of_open hs).1 (h.of_le hn)).2.continuousOn
-#align cont_diff_on.continuous_on_deriv_of_open ContDiffOn.continuousOn_deriv_of_open
+  ((contDiffOn_succ_iff_deriv_of_isOpen hs).1 (h.of_le hn)).2.continuousOn
+#align cont_diff_on.continuous_on_deriv_of_open ContDiffOn.continuousOn_deriv_of_isOpen
 
 /-- A function is `C^(n + 1)` if and only if it is differentiable,
   and its derivative (formulated in terms of `deriv`) is `C^n`. -/
 theorem contDiff_succ_iff_deriv {n : ℕ} :
     ContDiff 𝕜 (n + 1 : ℕ) f₂ ↔ Differentiable 𝕜 f₂ ∧ ContDiff 𝕜 n (deriv f₂) := by
-  simp only [← contDiffOn_univ, contDiffOn_succ_iff_deriv_of_open, isOpen_univ,
+  simp only [← contDiffOn_univ, contDiffOn_succ_iff_deriv_of_isOpen, isOpen_univ,
     differentiableOn_univ]
 #align cont_diff_succ_iff_deriv contDiff_succ_iff_deriv
 

Mathlib/Analysis/Calculus/ContDiff/Defs.lean

@@ -1156,12 +1156,12 @@ theorem contDiffOn_succ_iff_has_fderiv_within {n : ℕ} (hs : UniqueDiffOn 𝕜
 
 /-- A function is `C^(n + 1)` on an open domain if and only if it is
 differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/
-theorem contDiffOn_succ_iff_fderiv_of_open {n : ℕ} (hs : IsOpen s) :
+theorem contDiffOn_succ_iff_fderiv_of_isOpen {n : ℕ} (hs : IsOpen s) :
     ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
       DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn]
   exact Iff.rfl.and (contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx)
-#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_open
+#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_isOpen
 
 /-- A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
 there, and its derivative (expressed with `fderivWithin`) is `C^∞`. -/
@@ -1182,11 +1182,11 @@ theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
 
 /-- A function is `C^∞` on an open domain if and only if it is differentiable there, and its
 derivative (expressed with `fderiv`) is `C^∞`. -/
-theorem contDiffOn_top_iff_fderiv_of_open (hs : IsOpen s) :
+theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :
     ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s := by
   rw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]
   exact Iff.rfl.and <| contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx
-#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_open
+#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_isOpen
 
 protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s := by
@@ -1199,20 +1199,20 @@ protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : Uni
     exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2
 #align cont_diff_on.fderiv_within ContDiffOn.fderivWithin
 
-theorem ContDiffOn.fderiv_of_open (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
+theorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
     ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s :=
   (hf.fderivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (fderivWithin_of_isOpen hs hx).symm
-#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_open
+#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_isOpen
 
 theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderivWithin 𝕜 f s x) s :=
   ((contDiffOn_succ_iff_fderivWithin hs).1 (h.of_le hn)).2.continuousOn
 #align cont_diff_on.continuous_on_fderiv_within ContDiffOn.continuousOn_fderivWithin
 
-theorem ContDiffOn.continuousOn_fderiv_of_open (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
+theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s :=
-  ((contDiffOn_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.continuousOn
-#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_open
+  ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 (h.of_le hn)).2.continuousOn
+#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_isOpen
 
 /-! ### Functions with a Taylor series on the whole space -/
 

Mathlib/Analysis/Calculus/Deriv/Basic.lean

@@ -531,9 +531,9 @@ theorem derivWithin_inter (ht : t ∈ 𝓝 x) : derivWithin f (s ∩ t) x = deri
 theorem derivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : derivWithin f s x = deriv f x := by
   simp only [derivWithin, deriv, fderivWithin_of_mem_nhds h]
 
-theorem derivWithin_of_open (hs : IsOpen s) (hx : x ∈ s) : derivWithin f s x = deriv f x :=
+theorem derivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) : derivWithin f s x = deriv f x :=
   derivWithin_of_mem_nhds (hs.mem_nhds hx)
-#align deriv_within_of_open derivWithin_of_open
+#align deriv_within_of_open derivWithin_of_isOpen
 
 theorem deriv_mem_iff {f : 𝕜 → F} {s : Set F} {x : 𝕜} :
     deriv f x ∈ s ↔

Mathlib/Analysis/Calculus/LineDeriv/Basic.lean

@@ -278,7 +278,7 @@ theorem lineDerivWithin_of_mem_nhds (h : s ∈ 𝓝 x) :
   apply (Continuous.continuousAt _).preimage_mem_nhds (by simpa using h)
   continuity
 
-theorem lineDerivWithin_of_open (hs : IsOpen s) (hx : x ∈ s) :
+theorem lineDerivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) :
     lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v :=
   lineDerivWithin_of_mem_nhds (hs.mem_nhds hx)
 

Mathlib/Analysis/Complex/AbsMax.lean

@@ -240,7 +240,7 @@ theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
       using isOpen_setOf_mem_nhds_and_isMaxOn_norm hd
   have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, hm⟩
   set W := U ∩ {z | ‖f z‖ ≠ ‖f c‖}
-  have hWo : IsOpen W := hd.continuousOn.norm.preimage_open_of_open ho isOpen_ne
+  have hWo : IsOpen W := hd.continuousOn.norm.isOpen_inter_preimage ho isOpen_ne
   have hdVW : Disjoint V W := disjoint_left.mpr fun x hxV hxW => hxW.2 (hV x hxV)
   have hUVW : U ⊆ V ∪ W := fun x hx =>
     (eq_or_ne ‖f x‖ ‖f c‖).imp (fun h => ⟨hx, fun y hy => (hm hy).out.trans_eq h.symm⟩)

Mathlib/Analysis/Convex/Gauge.lean

@@ -359,24 +359,24 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
   exact (gauge_le_of_mem hr₀.le hxr).trans_lt hr₁
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
-theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
+theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     { x | gauge s x < 1 } = s := by
   refine' (gauge_lt_one_subset_self hs₁ ‹_› <| absorbent_nhds_zero <| hs₂.mem_nhds hs₀).antisymm _
   convert interior_subset_gauge_lt_one s
   exact hs₂.interior_eq.symm
-#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
+#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_isOpen
 
 -- porting note: droped unneeded assumptions
-theorem gauge_lt_one_of_mem_of_open (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
+theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
     gauge s x < 1 :=
   interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
-#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ
+#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_isOpenₓ
 
 -- porting note: droped unneeded assumptions
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
     gauge s x < ε := by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
-  have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₂ this
+  have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one]
     at h_gauge_lt
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
@@ -480,14 +480,14 @@ def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Abso
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
-theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
+theorem gaugeSeminorm_lt_one_of_isOpen (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
-  gauge_lt_one_of_mem_of_open hs hx
-#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
+  gauge_lt_one_of_mem_of_isOpen hs hx
+#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_isOpen
 
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s := by
   rw [Seminorm.ball_zero_eq]
-  exact gauge_lt_one_eq_self_of_open hs₁ hs₂.zero_mem hs
+  exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
 
 end IsROrC

Mathlib/Analysis/Convex/Strict.lean

@@ -109,12 +109,13 @@ protected theorem StrictConvex.convex (hs : StrictConvex 𝕜 s) : Convex 𝕜 s
 #align strict_convex.convex StrictConvex.convex
 
 /-- An open convex set is strictly convex. -/
-protected theorem Convex.strictConvex_of_open (h : IsOpen s) (hs : Convex 𝕜 s) : StrictConvex 𝕜 s :=
+protected theorem Convex.strictConvex_of_isOpen (h : IsOpen s) (hs : Convex 𝕜 s) :
+    StrictConvex 𝕜 s :=
   fun _ hx _ hy _ _ _ ha hb hab => h.interior_eq.symm ▸ hs hx hy ha.le hb.le hab
-#align convex.strict_convex_of_open Convex.strictConvex_of_open
+#align convex.strict_convex_of_open Convex.strictConvex_of_isOpen
 
 theorem IsOpen.strictConvex_iff (h : IsOpen s) : StrictConvex 𝕜 s ↔ Convex 𝕜 s :=
-  ⟨StrictConvex.convex, Convex.strictConvex_of_open h⟩
+  ⟨StrictConvex.convex, Convex.strictConvex_of_isOpen h⟩
 #align is_open.strict_convex_iff IsOpen.strictConvex_iff
 
 theorem strictConvex_singleton (c : E) : StrictConvex 𝕜 ({c} : Set E) :=