analysis.analytic.isolated_zeros ⟷ Mathlib.Analysis.Analytic.IsolatedZeros

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -5,7 +5,7 @@ Authors: Vincent Beffara
 -/
 import Analysis.Analytic.Basic
 import Analysis.Calculus.Dslope
-import Analysis.Calculus.FderivAnalytic
+import Analysis.Calculus.FDeriv.Analytic
 import Analysis.Calculus.FormalMultilinearSeries
 import Analysis.Analytic.Uniqueness
 
@@ -89,7 +89,7 @@ theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
   refine' hp.mono fun x hx => _
   by_cases h : x = 0
   · convert hasSum_single 0 _ <;> intros <;> simp [*]
-  · have hxx : ∀ n : ℕ, x⁻¹ * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, pow_succ']
+  · have hxx : ∀ n : ℕ, x⁻¹ * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, pow_succ]
     suffices HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀)) by
       simpa [dslope, slope, h, smul_smul, hxx] using this
     · simpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x⁻¹

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -125,7 +125,7 @@ theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
     simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk
   obtain ⟨s, hs1, hs2⟩ := HasSum.exists_hasSum_smul_of_apply_eq_zero hx2 this
   convert hs1.symm
-  simp only [coeff_iterate_fslope] at hx1 
+  simp only [coeff_iterate_fslope] at hx1
   exact hx1.unique hs2
 #align has_fpower_series_at.eq_pow_order_mul_iterate_dslope HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope
 -/
@@ -143,7 +143,7 @@ theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ
 
 #print HasFPowerSeriesAt.locally_zero_iff /-
 theorem locally_zero_iff (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
-  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp )⟩
+  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp)⟩
 #align has_fpower_series_at.locally_zero_iff HasFPowerSeriesAt.locally_zero_iff
 -/
 
@@ -160,7 +160,7 @@ theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
   by
   rcases hf with ⟨p, hp⟩
   by_cases h : p = 0
-  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
+  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
   · exact Or.inr (hp.locally_ne_zero h)
 #align analytic_at.eventually_eq_zero_or_eventually_ne_zero AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 -/
-import Mathbin.Analysis.Analytic.Basic
-import Mathbin.Analysis.Calculus.Dslope
-import Mathbin.Analysis.Calculus.FderivAnalytic
-import Mathbin.Analysis.Calculus.FormalMultilinearSeries
-import Mathbin.Analysis.Analytic.Uniqueness
+import Analysis.Analytic.Basic
+import Analysis.Calculus.Dslope
+import Analysis.Calculus.FderivAnalytic
+import Analysis.Calculus.FormalMultilinearSeries
+import Analysis.Analytic.Uniqueness
 
 #align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
-
-! This file was ported from Lean 3 source module analysis.analytic.isolated_zeros
-! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Analytic.Basic
 import Mathbin.Analysis.Calculus.Dslope
@@ -14,6 +9,8 @@ import Mathbin.Analysis.Calculus.FderivAnalytic
 import Mathbin.Analysis.Calculus.FormalMultilinearSeries
 import Mathbin.Analysis.Analytic.Uniqueness
 
+#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
+
 /-!
 # Principle of isolated zeros
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -49,13 +49,16 @@ namespace HasSum
 
 variable {a : ℕ → E}
 
+#print HasSum.hasSum_at_zero /-
 theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by
   convert hasSum_single 0 fun b h => _ <;>
     first
     | simp [Nat.pos_of_ne_zero h]
     | simp
 #align has_sum.has_sum_at_zero HasSum.hasSum_at_zero
+-/
 
+#print HasSum.exists_hasSum_smul_of_apply_eq_zero /-
 theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
     (ha : ∀ k < n, a k = 0) : ∃ t : E, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t :=
   by
@@ -73,11 +76,13 @@ theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m
     · field_simp [pow_add, smul_smul]
     · simp only [inv_pow]
 #align has_sum.exists_has_sum_smul_of_apply_eq_zero HasSum.exists_hasSum_smul_of_apply_eq_zero
+-/
 
 end HasSum
 
 namespace HasFPowerSeriesAt
 
+#print HasFPowerSeriesAt.has_fpower_series_dslope_fslope /-
 theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
     HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ :=
   by
@@ -92,7 +97,9 @@ theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
       simpa [dslope, slope, h, smul_smul, hxx] using this
     · simpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x⁻¹
 #align has_fpower_series_at.has_fpower_series_dslope_fslope HasFPowerSeriesAt.has_fpower_series_dslope_fslope
+-/
 
+#print HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope /-
 theorem has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : HasFPowerSeriesAt f p z₀) :
     HasFPowerSeriesAt ((swap dslope z₀^[n]) f) ((fslope^[n]) p) z₀ :=
   by
@@ -100,14 +107,18 @@ theorem has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : HasFPowerSeriesA
   · exact hp
   · simpa using ih (has_fpower_series_dslope_fslope hp)
 #align has_fpower_series_at.has_fpower_series_iterate_dslope_fslope HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope
+-/
 
+#print HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero /-
 theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) :
     (swap dslope z₀^[p.order]) f z₀ ≠ 0 :=
   by
   rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
   simpa [coeff_eq_zero] using apply_order_ne_zero h
 #align has_fpower_series_at.iterate_dslope_fslope_ne_zero HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero
+-/
 
+#print HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope /-
 theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
     ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀^[p.order]) f z :=
   by
@@ -120,7 +131,9 @@ theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
   simp only [coeff_iterate_fslope] at hx1 
   exact hx1.unique hs2
 #align has_fpower_series_at.eq_pow_order_mul_iterate_dslope HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope
+-/
 
+#print HasFPowerSeriesAt.locally_ne_zero /-
 theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 :=
   by
   rw [eventually_nhdsWithin_iff]
@@ -129,15 +142,19 @@ theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ
   filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3
   simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3)
 #align has_fpower_series_at.locally_ne_zero HasFPowerSeriesAt.locally_ne_zero
+-/
 
+#print HasFPowerSeriesAt.locally_zero_iff /-
 theorem locally_zero_iff (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
   ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp )⟩
 #align has_fpower_series_at.locally_zero_iff HasFPowerSeriesAt.locally_zero_iff
+-/
 
 end HasFPowerSeriesAt
 
 namespace AnalyticAt
 
+#print AnalyticAt.eventually_eq_zero_or_eventually_ne_zero /-
 /-- The *principle of isolated zeros* for an analytic function, local version: if a function is
 analytic at `z₀`, then either it is identically zero in a neighborhood of `z₀`, or it does not
 vanish in a punctured neighborhood of `z₀`. -/
@@ -149,22 +166,29 @@ theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
   · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
   · exact Or.inr (hp.locally_ne_zero h)
 #align analytic_at.eventually_eq_zero_or_eventually_ne_zero AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
+-/
 
+#print AnalyticAt.eventually_eq_or_eventually_ne /-
 theorem eventually_eq_or_eventually_ne (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
     (∀ᶠ z in 𝓝 z₀, f z = g z) ∨ ∀ᶠ z in 𝓝[≠] z₀, f z ≠ g z := by
   simpa [sub_eq_zero] using (hf.sub hg).eventually_eq_zero_or_eventually_ne_zero
 #align analytic_at.eventually_eq_or_eventually_ne AnalyticAt.eventually_eq_or_eventually_ne
+-/
 
+#print AnalyticAt.frequently_zero_iff_eventually_zero /-
 theorem frequently_zero_iff_eventually_zero {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w) :
     (∃ᶠ z in 𝓝[≠] w, f z = 0) ↔ ∀ᶠ z in 𝓝 w, f z = 0 :=
   ⟨hf.eventually_eq_zero_or_eventually_ne_zero.resolve_right, fun h =>
     (h.filter_mono nhdsWithin_le_nhds).Frequently⟩
 #align analytic_at.frequently_zero_iff_eventually_zero AnalyticAt.frequently_zero_iff_eventually_zero
+-/
 
+#print AnalyticAt.frequently_eq_iff_eventually_eq /-
 theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
     (∃ᶠ z in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ z in 𝓝 z₀, f z = g z := by
   simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)
 #align analytic_at.frequently_eq_iff_eventually_eq AnalyticAt.frequently_eq_iff_eventually_eq
+-/
 
 end AnalyticAt
 
@@ -172,6 +196,7 @@ namespace AnalyticOn
 
 variable {U : Set 𝕜}
 
+#print AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero /-
 /-- The *principle of isolated zeros* for an analytic function, global version: if a function is
 analytic on a connected set `U` and vanishes in arbitrary neighborhoods of a point `z₀ ∈ U`, then
 it is identically zero in `U`.
@@ -182,13 +207,17 @@ theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOn 𝕜 f
   hf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀
     ((hf z₀ h₀).frequently_zero_iff_eventually_zero.1 hfw)
 #align analytic_on.eq_on_zero_of_preconnected_of_frequently_eq_zero AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero
+-/
 
+#print AnalyticOn.eqOn_zero_of_preconnected_of_mem_closure /-
 theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U)
     (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) : EqOn f 0 U :=
   hf.eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀
     (mem_closure_ne_iff_frequently_within.mp hfz₀)
 #align analytic_on.eq_on_zero_of_preconnected_of_mem_closure AnalyticOn.eqOn_zero_of_preconnected_of_mem_closure
+-/
 
+#print AnalyticOn.eqOn_of_preconnected_of_frequently_eq /-
 /-- The *identity principle* for analytic functions, global version: if two functions are
 analytic on a connected set `U` and coincide at points which accumulate to a point `z₀ ∈ U`, then
 they coincide globally in `U`.
@@ -202,13 +231,17 @@ theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOn 𝕜 f U) (hg : A
   simpa [sub_eq_zero] using fun z hz =>
     (hf.sub hg).eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀ hfg' hz
 #align analytic_on.eq_on_of_preconnected_of_frequently_eq AnalyticOn.eqOn_of_preconnected_of_frequently_eq
+-/
 
+#print AnalyticOn.eqOn_of_preconnected_of_mem_closure /-
 theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) :
     EqOn f g U :=
   hf.eqOn_of_preconnected_of_frequently_eq hg hU h₀ (mem_closure_ne_iff_frequently_within.mp hfg)
 #align analytic_on.eq_on_of_preconnected_of_mem_closure AnalyticOn.eqOn_of_preconnected_of_mem_closure
+-/
 
+#print AnalyticOn.eq_of_frequently_eq /-
 /-- The *identity principle* for analytic functions, global version: if two functions on a normed
 field `𝕜` are analytic everywhere and coincide at points which accumulate to a point `z₀`, then
 they coincide globally.
@@ -219,6 +252,7 @@ theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f univ)
   funext fun x =>
     eqOn_of_preconnected_of_frequently_eq hf hg isPreconnected_univ (mem_univ z₀) hfg (mem_univ x)
 #align analytic_on.eq_of_frequently_eq AnalyticOn.eq_of_frequently_eq
+-/
 
 end AnalyticOn
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 
 ! This file was ported from Lean 3 source module analysis.analytic.isolated_zeros
-! leanprover-community/mathlib commit a3209ddf94136d36e5e5c624b10b2a347cc9d090
+! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.Analysis.Analytic.Uniqueness
 /-!
 # Principle of isolated zeros
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves the fact that the zeros of a non-constant analytic function of one variable are
 isolated. It also introduces a little bit of API in the `has_fpower_series_at` namespace that is
 useful in this setup.

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -62,7 +62,7 @@ theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m
   · have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using has_sum_at_zero a)
     exact ⟨a n, by simp [h, hn, this], by simpa [h] using has_sum_at_zero fun m => a (m + n)⟩
   · refine' ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], _⟩
-    have h1 : (∑ i in Finset.range n, z ^ i • a i) = 0 :=
+    have h1 : ∑ i in Finset.range n, z ^ i • a i = 0 :=
       Finset.sum_eq_zero fun k hk => by simp [ha k (finset.mem_range.mp hk)]
     have h2 : HasSum (fun m => z ^ (m + n) • a (m + n)) s := by
       simpa [h1] using (hasSum_nat_add_iff' n).mpr hs

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 
 ! This file was ported from Lean 3 source module analysis.analytic.isolated_zeros
-! leanprover-community/mathlib commit 32253a1a1071173b33dc7d6a218cf722c6feb514
+! leanprover-community/mathlib commit a3209ddf94136d36e5e5c624b10b2a347cc9d090
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -206,5 +206,16 @@ theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : Ana
   hf.eqOn_of_preconnected_of_frequently_eq hg hU h₀ (mem_closure_ne_iff_frequently_within.mp hfg)
 #align analytic_on.eq_on_of_preconnected_of_mem_closure AnalyticOn.eqOn_of_preconnected_of_mem_closure
 
+/-- The *identity principle* for analytic functions, global version: if two functions on a normed
+field `𝕜` are analytic everywhere and coincide at points which accumulate to a point `z₀`, then
+they coincide globally.
+For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
+see `eq_of_eventually_eq`. -/
+theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f univ) (hg : AnalyticOn 𝕜 g univ)
+    (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g :=
+  funext fun x =>
+    eqOn_of_preconnected_of_frequently_eq hf hg isPreconnected_univ (mem_univ z₀) hfg (mem_univ x)
+#align analytic_on.eq_of_frequently_eq AnalyticOn.eq_of_frequently_eq
+
 end AnalyticOn
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -109,7 +109,7 @@ theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
     ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀^[p.order]) f z :=
   by
   have hq := has_fpower_series_at_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp)
-  filter_upwards [hq, has_fpower_series_at_iff'.mp hp]with x hx1 hx2
+  filter_upwards [hq, has_fpower_series_at_iff'.mp hp] with x hx1 hx2
   have : ∀ k < p.order, p.coeff k = 0 := fun k hk => by
     simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk
   obtain ⟨s, hs1, hs2⟩ := HasSum.exists_hasSum_smul_of_apply_eq_zero hx2 this
@@ -123,7 +123,7 @@ theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ
   rw [eventually_nhdsWithin_iff]
   have h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).ContinuousAt
   have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)
-  filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3]with z e1 e2 e3
+  filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3
   simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3)
 #align has_fpower_series_at.locally_ne_zero HasFPowerSeriesAt.locally_ne_zero
 
@@ -181,7 +181,7 @@ theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOn 𝕜 f
 #align analytic_on.eq_on_zero_of_preconnected_of_frequently_eq_zero AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero
 
 theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U)
-    (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({ z | f z = 0 } \ {z₀})) : EqOn f 0 U :=
+    (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) : EqOn f 0 U :=
   hf.eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀
     (mem_closure_ne_iff_frequently_within.mp hfz₀)
 #align analytic_on.eq_on_zero_of_preconnected_of_mem_closure AnalyticOn.eqOn_zero_of_preconnected_of_mem_closure
@@ -201,7 +201,7 @@ theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOn 𝕜 f U) (hg : A
 #align analytic_on.eq_on_of_preconnected_of_frequently_eq AnalyticOn.eqOn_of_preconnected_of_frequently_eq
 
 theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
-    (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({ z | f z = g z } \ {z₀})) :
+    (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) :
     EqOn f g U :=
   hf.eqOn_of_preconnected_of_frequently_eq hg hU h₀ (mem_closure_ne_iff_frequently_within.mp hfg)
 #align analytic_on.eq_on_of_preconnected_of_mem_closure AnalyticOn.eqOn_of_preconnected_of_mem_closure

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -73,14 +73,14 @@ theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m
 
 end HasSum
 
-namespace HasFpowerSeriesAt
+namespace HasFPowerSeriesAt
 
-theorem hasFpowerSeriesDslopeFslope (hp : HasFpowerSeriesAt f p z₀) :
-    HasFpowerSeriesAt (dslope f z₀) p.fslope z₀ :=
+theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
+    HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ :=
   by
   have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
   have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
-  simp only [hasFpowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
+  simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
   refine' hp.mono fun x hx => _
   by_cases h : x = 0
   · convert hasSum_single 0 _ <;> intros <;> simp [*]
@@ -88,24 +88,24 @@ theorem hasFpowerSeriesDslopeFslope (hp : HasFpowerSeriesAt f p z₀) :
     suffices HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀)) by
       simpa [dslope, slope, h, smul_smul, hxx] using this
     · simpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x⁻¹
-#align has_fpower_series_at.has_fpower_series_dslope_fslope HasFpowerSeriesAt.hasFpowerSeriesDslopeFslope
+#align has_fpower_series_at.has_fpower_series_dslope_fslope HasFPowerSeriesAt.has_fpower_series_dslope_fslope
 
-theorem hasFpowerSeriesIterateDslopeFslope (n : ℕ) (hp : HasFpowerSeriesAt f p z₀) :
-    HasFpowerSeriesAt ((swap dslope z₀^[n]) f) ((fslope^[n]) p) z₀ :=
+theorem has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : HasFPowerSeriesAt f p z₀) :
+    HasFPowerSeriesAt ((swap dslope z₀^[n]) f) ((fslope^[n]) p) z₀ :=
   by
   induction' n with n ih generalizing f p
   · exact hp
   · simpa using ih (has_fpower_series_dslope_fslope hp)
-#align has_fpower_series_at.has_fpower_series_iterate_dslope_fslope HasFpowerSeriesAt.hasFpowerSeriesIterateDslopeFslope
+#align has_fpower_series_at.has_fpower_series_iterate_dslope_fslope HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope
 
-theorem iterate_dslope_fslope_ne_zero (hp : HasFpowerSeriesAt f p z₀) (h : p ≠ 0) :
+theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) :
     (swap dslope z₀^[p.order]) f z₀ ≠ 0 :=
   by
   rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
   simpa [coeff_eq_zero] using apply_order_ne_zero h
-#align has_fpower_series_at.iterate_dslope_fslope_ne_zero HasFpowerSeriesAt.iterate_dslope_fslope_ne_zero
+#align has_fpower_series_at.iterate_dslope_fslope_ne_zero HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero
 
-theorem eq_pow_order_mul_iterate_dslope (hp : HasFpowerSeriesAt f p z₀) :
+theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
     ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀^[p.order]) f z :=
   by
   have hq := has_fpower_series_at_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp)
@@ -116,22 +116,22 @@ theorem eq_pow_order_mul_iterate_dslope (hp : HasFpowerSeriesAt f p z₀) :
   convert hs1.symm
   simp only [coeff_iterate_fslope] at hx1 
   exact hx1.unique hs2
-#align has_fpower_series_at.eq_pow_order_mul_iterate_dslope HasFpowerSeriesAt.eq_pow_order_mul_iterate_dslope
+#align has_fpower_series_at.eq_pow_order_mul_iterate_dslope HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope
 
-theorem locally_ne_zero (hp : HasFpowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 :=
+theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 :=
   by
   rw [eventually_nhdsWithin_iff]
   have h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).ContinuousAt
   have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)
   filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3]with z e1 e2 e3
   simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3)
-#align has_fpower_series_at.locally_ne_zero HasFpowerSeriesAt.locally_ne_zero
+#align has_fpower_series_at.locally_ne_zero HasFPowerSeriesAt.locally_ne_zero
 
-theorem locally_zero_iff (hp : HasFpowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
+theorem locally_zero_iff (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
   ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp )⟩
-#align has_fpower_series_at.locally_zero_iff HasFpowerSeriesAt.locally_zero_iff
+#align has_fpower_series_at.locally_zero_iff HasFPowerSeriesAt.locally_zero_iff
 
-end HasFpowerSeriesAt
+end HasFPowerSeriesAt
 
 namespace AnalyticAt
 
@@ -143,7 +143,7 @@ theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
   by
   rcases hf with ⟨p, hp⟩
   by_cases h : p = 0
-  · exact Or.inl (HasFpowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
+  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
   · exact Or.inr (hp.locally_ne_zero h)
 #align analytic_at.eventually_eq_zero_or_eventually_ne_zero AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -47,7 +47,10 @@ namespace HasSum
 variable {a : ℕ → E}
 
 theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by
-  convert hasSum_single 0 fun b h => _ <;> first |simp [Nat.pos_of_ne_zero h]|simp
+  convert hasSum_single 0 fun b h => _ <;>
+    first
+    | simp [Nat.pos_of_ne_zero h]
+    | simp
 #align has_sum.has_sum_at_zero HasSum.hasSum_at_zero
 
 theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
@@ -58,7 +61,7 @@ theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m
   by_cases h : z = 0
   · have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using has_sum_at_zero a)
     exact ⟨a n, by simp [h, hn, this], by simpa [h] using has_sum_at_zero fun m => a (m + n)⟩
-  · refine' ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul] , _⟩
+  · refine' ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], _⟩
     have h1 : (∑ i in Finset.range n, z ^ i • a i) = 0 :=
       Finset.sum_eq_zero fun k hk => by simp [ha k (finset.mem_range.mp hk)]
     have h2 : HasSum (fun m => z ^ (m + n) • a (m + n)) s := by
@@ -77,7 +80,7 @@ theorem hasFpowerSeriesDslopeFslope (hp : HasFpowerSeriesAt f p z₀) :
   by
   have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
   have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
-  simp only [hasFpowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp⊢
+  simp only [hasFpowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
   refine' hp.mono fun x hx => _
   by_cases h : x = 0
   · convert hasSum_single 0 _ <;> intros <;> simp [*]
@@ -111,7 +114,7 @@ theorem eq_pow_order_mul_iterate_dslope (hp : HasFpowerSeriesAt f p z₀) :
     simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk
   obtain ⟨s, hs1, hs2⟩ := HasSum.exists_hasSum_smul_of_apply_eq_zero hx2 this
   convert hs1.symm
-  simp only [coeff_iterate_fslope] at hx1
+  simp only [coeff_iterate_fslope] at hx1 
   exact hx1.unique hs2
 #align has_fpower_series_at.eq_pow_order_mul_iterate_dslope HasFpowerSeriesAt.eq_pow_order_mul_iterate_dslope
 
@@ -125,7 +128,7 @@ theorem locally_ne_zero (hp : HasFpowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ
 #align has_fpower_series_at.locally_ne_zero HasFpowerSeriesAt.locally_ne_zero
 
 theorem locally_zero_iff (hp : HasFpowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
-  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp)⟩
+  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp )⟩
 #align has_fpower_series_at.locally_zero_iff HasFpowerSeriesAt.locally_zero_iff
 
 end HasFpowerSeriesAt
@@ -140,7 +143,7 @@ theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
   by
   rcases hf with ⟨p, hp⟩
   by_cases h : p = 0
-  · exact Or.inl (HasFpowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
+  · exact Or.inl (HasFpowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
   · exact Or.inr (hp.locally_ne_zero h)
 #align analytic_at.eventually_eq_zero_or_eventually_ne_zero AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -32,11 +32,11 @@ useful in this setup.
 -/
 
 
-open Classical
+open scoped Classical
 
 open Filter Function Nat FormalMultilinearSeries Emetric Set
 
-open Topology BigOperators
+open scoped Topology BigOperators
 
 variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E]
   [NormedSpace 𝕜 E] {s : E} {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {n : ℕ} {z z₀ : 𝕜}

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -169,7 +169,7 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
       hfz' hz, smul_right_inj <| zpow_ne_zero _ <| sub_ne_zero.mpr hz] at hfz
     exact hfz hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this
-  rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
+  · rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
   conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_natCast]
   exact (((analyticAt_id _ _).sub analyticAt_const).pow _).smul hg_an
 

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -74,7 +74,7 @@ theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
   refine hp.mono fun x hx => ?_
   by_cases h : x = 0
   · convert hasSum_single (α := E) 0 _ <;> intros <;> simp [*]
-  · have hxx : ∀ n : ℕ, x⁻¹ * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, _root_.pow_succ']
+  · have hxx : ∀ n : ℕ, x⁻¹ * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, _root_.pow_succ]
     suffices HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀)) by
       simpa [dslope, slope, h, smul_smul, hxx] using this
     · simpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x⁻¹

Lemma names around cancellation of multiplication and division are a mess.

This PR renames a handful of them according to the following table (each big row contains the multiplicative statement, then the three rows contain the GroupWithZero lemma name, the Group lemma, the AddGroup lemma name).

| Statement | New name | Old name | |

@@ -165,7 +165,7 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
     apply Filter.Eventually.frequently
     rw [eventually_nhdsWithin_iff] at hg_eq hj_eq ⊢
     filter_upwards [hg_eq, hj_eq] with z hfz hfz' hz
-    rw [← add_sub_cancel' n m, add_sub_assoc, zpow_add₀ <| sub_ne_zero.mpr hz, mul_smul,
+    rw [← add_sub_cancel_left n m, add_sub_assoc, zpow_add₀ <| sub_ne_zero.mpr hz, mul_smul,
       hfz' hz, smul_right_inj <| zpow_ne_zero _ <| sub_ne_zero.mpr hz] at hfz
     exact hfz hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this

... and add a deprecated alias for the old name. This is mostly just me discovering the power of F2

@@ -170,7 +170,7 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
     exact hfz hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this
   rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
-  conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_coe_nat]
+  conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_natCast]
   exact (((analyticAt_id _ _).sub analyticAt_const).pow _).smul hg_an
 
 /-- For a function `f` on `𝕜`, and `z₀ ∈ 𝕜`, there exists at most one `n` such that on a
@@ -180,7 +180,7 @@ lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
     (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
     (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
     m = n := by
-  simp_rw [← zpow_coe_nat] at hm hn
+  simp_rw [← zpow_natCast] at hm hn
   exact Int.ofNat_inj.mp <| unique_eventuallyEq_zpow_smul_nonzero
     (let ⟨g, h₁, h₂, h₃⟩ := hm; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)
     (let ⟨g, h₁, h₂, h₃⟩ := hn; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)

Previously these were syntactically identical to the corresponding zpow_coe_nat and coe_nat_zsmul lemmas, now they are about OfNat.ofNat.

Unfortunately, almost every call site uses the ofNat name to refer to Nat.cast, so the downstream proofs had to be adjusted too.

@@ -170,7 +170,7 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
     exact hfz hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this
   rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
-  conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_ofNat]
+  conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_coe_nat]
   exact (((analyticAt_id _ _).sub analyticAt_const).pow _).smul hg_an
 
 /-- For a function `f` on `𝕜`, and `z₀ ∈ 𝕜`, there exists at most one `n` such that on a
@@ -180,7 +180,7 @@ lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
     (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
     (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
     m = n := by
-  simp_rw [← zpow_ofNat] at hm hn
+  simp_rw [← zpow_coe_nat] at hm hn
   exact Int.ofNat_inj.mp <| unique_eventuallyEq_zpow_smul_nonzero
     (let ⟨g, h₁, h₂, h₃⟩ := hm; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)
     (let ⟨g, h₁, h₂, h₃⟩ := hn; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -161,8 +161,8 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
   let ⟨g, hg_an, _, hg_eq⟩ := hm
   let ⟨j, hj_an, hj_ne, hj_eq⟩ := hn
   contrapose! hj_ne
-  have : ∃ᶠ z in 𝓝[≠] z₀, j z = (z - z₀) ^ (m - n) • g z
-  · apply Filter.Eventually.frequently
+  have : ∃ᶠ z in 𝓝[≠] z₀, j z = (z - z₀) ^ (m - n) • g z := by
+    apply Filter.Eventually.frequently
     rw [eventually_nhdsWithin_iff] at hg_eq hj_eq ⊢
     filter_upwards [hg_eq, hj_eq] with z hfz hfz' hz
     rw [← add_sub_cancel' n m, add_sub_assoc, zpow_add₀ <| sub_ne_zero.mpr hz, mul_smul,

This PR adds "order of vanishing" for meromorphic functions, and some basic API for functions MeromorphicOn a set.

@@ -148,11 +148,13 @@ theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : Anal
   simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)
 #align analytic_at.frequently_eq_iff_eventually_eq AnalyticAt.frequently_eq_iff_eventually_eq
 
-/-- There exists at most one `n` such that locally around `z₀` we have `f z = (z - z₀) ^ n • g z`,
-with `g` analytic and nonvanishing at `z₀`. -/
-lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
-    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
-    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
+/-- For a function `f` on `𝕜`, and `z₀ ∈ 𝕜`, there exists at most one `n` such that on a punctured
+neighbourhood of `z₀` we have `f z = (z - z₀) ^ n • g z`, with `g` analytic and nonvanishing at
+`z₀`. We formulate this with `n : ℤ`, and deduce the case `n : ℕ` later, for applications to
+meromorphic functions. -/
+lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
+    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z)
+    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z) :
     m = n := by
   wlog h_le : n ≤ m generalizing m n
   · exact ((this hn hm) (not_le.mp h_le).le).symm
@@ -160,14 +162,28 @@ lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
   let ⟨j, hj_an, hj_ne, hj_eq⟩ := hn
   contrapose! hj_ne
   have : ∃ᶠ z in 𝓝[≠] z₀, j z = (z - z₀) ^ (m - n) • g z
-  · refine (eventually_nhdsWithin_iff.mpr ?_).frequently
+  · apply Filter.Eventually.frequently
+    rw [eventually_nhdsWithin_iff] at hg_eq hj_eq ⊢
     filter_upwards [hg_eq, hj_eq] with z hfz hfz' hz
-    rwa [← Nat.add_sub_cancel' h_le, pow_add, mul_smul, hfz', smul_right_inj] at hfz
-    exact pow_ne_zero _ <| sub_ne_zero.mpr hz
+    rw [← add_sub_cancel' n m, add_sub_assoc, zpow_add₀ <| sub_ne_zero.mpr hz, mul_smul,
+      hfz' hz, smul_right_inj <| zpow_ne_zero _ <| sub_ne_zero.mpr hz] at hfz
+    exact hfz hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this
-  rw [EventuallyEq.eq_of_nhds this, sub_self, zero_pow, zero_smul]
-  · apply Nat.sub_ne_zero_of_lt (h_le.lt_of_ne' hj_ne)
-  · exact (((analyticAt_id 𝕜 _).sub analyticAt_const).pow _).smul hg_an
+  rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
+  conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_ofNat]
+  exact (((analyticAt_id _ _).sub analyticAt_const).pow _).smul hg_an
+
+/-- For a function `f` on `𝕜`, and `z₀ ∈ 𝕜`, there exists at most one `n` such that on a
+neighbourhood of `z₀` we have `f z = (z - z₀) ^ n • g z`, with `g` analytic and nonvanishing at
+`z₀`. -/
+lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
+    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
+    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
+    m = n := by
+  simp_rw [← zpow_ofNat] at hm hn
+  exact Int.ofNat_inj.mp <| unique_eventuallyEq_zpow_smul_nonzero
+    (let ⟨g, h₁, h₂, h₃⟩ := hm; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)
+    (let ⟨g, h₁, h₂, h₃⟩ := hn; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)
 
 /-- If `f` is analytic at `z₀`, then exactly one of the following two possibilities occurs: either
 `f` vanishes identically near `z₀`, or locally around `z₀` it has the form `z ↦ (z - z₀) ^ n • g z`

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -36,18 +36,13 @@ open scoped Topology BigOperators
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
   [NormedSpace 𝕜 E] {s : E} {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {n : ℕ} {z z₀ : 𝕜}
---  {y : Fin n → 𝕜} -- Porting note: This is used nowhere and creates problem since it is sometimes
--- automatically included as a hypothesis
 
 namespace HasSum
 
 variable {a : ℕ → E}
 
 theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by
-  convert hasSum_single (α := E) 0 fun b h => _ <;>
-    first
-    | simp [Nat.pos_of_ne_zero h]
-    | simp
+  convert hasSum_single (α := E) 0 fun b h ↦ _ <;> simp [*]
 #align has_sum.has_sum_at_zero HasSum.hasSum_at_zero
 
 theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
@@ -56,7 +51,7 @@ theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m
   · simpa
   by_cases h : z = 0
   · have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using hasSum_at_zero a)
-    exact ⟨a n, by simp [h, hn, this], by simpa [h] using hasSum_at_zero fun m => a (m + n)⟩
+    exact ⟨a n, by simp [h, hn.ne', this], by simpa [h] using hasSum_at_zero fun m => a (m + n)⟩
   · refine ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], ?_⟩
     have h1 : ∑ i in Finset.range n, z ^ i • a i = 0 :=
       Finset.sum_eq_zero fun k hk => by simp [ha k (Finset.mem_range.mp hk)]
@@ -171,7 +166,7 @@ lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
     exact pow_ne_zero _ <| sub_ne_zero.mpr hz
   rw [frequently_eq_iff_eventually_eq hj_an] at this
   rw [EventuallyEq.eq_of_nhds this, sub_self, zero_pow, zero_smul]
-  · apply Nat.zero_lt_sub_of_lt (Nat.lt_of_le_of_ne h_le hj_ne.symm)
+  · apply Nat.sub_ne_zero_of_lt (h_le.lt_of_ne' hj_ne)
   · exact (((analyticAt_id 𝕜 _).sub analyticAt_const).pow _).smul hg_an
 
 /-- If `f` is analytic at `z₀`, then exactly one of the following two possibilities occurs: either

This patch defines MeromorphicAt, for functions on nontrivially normed fields, and prove it is preserved by sums and products (routine) and inverses (much less trivial). Along the way, we define "order of vanishing" for a function analytic at a point (as an ENat).

@@ -3,10 +3,9 @@ Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
 -/
-import Mathlib.Analysis.Analytic.Basic
+import Mathlib.Analysis.Analytic.Constructions
 import Mathlib.Analysis.Calculus.Dslope
 import Mathlib.Analysis.Calculus.FDeriv.Analytic
-import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 import Mathlib.Analysis.Analytic.Uniqueness
 
 #align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
@@ -154,6 +153,75 @@ theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : Anal
   simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)
 #align analytic_at.frequently_eq_iff_eventually_eq AnalyticAt.frequently_eq_iff_eventually_eq
 
+/-- There exists at most one `n` such that locally around `z₀` we have `f z = (z - z₀) ^ n • g z`,
+with `g` analytic and nonvanishing at `z₀`. -/
+lemma unique_eventuallyEq_pow_smul_nonzero {m n : ℕ}
+    (hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ m • g z)
+    (hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) :
+    m = n := by
+  wlog h_le : n ≤ m generalizing m n
+  · exact ((this hn hm) (not_le.mp h_le).le).symm
+  let ⟨g, hg_an, _, hg_eq⟩ := hm
+  let ⟨j, hj_an, hj_ne, hj_eq⟩ := hn
+  contrapose! hj_ne
+  have : ∃ᶠ z in 𝓝[≠] z₀, j z = (z - z₀) ^ (m - n) • g z
+  · refine (eventually_nhdsWithin_iff.mpr ?_).frequently
+    filter_upwards [hg_eq, hj_eq] with z hfz hfz' hz
+    rwa [← Nat.add_sub_cancel' h_le, pow_add, mul_smul, hfz', smul_right_inj] at hfz
+    exact pow_ne_zero _ <| sub_ne_zero.mpr hz
+  rw [frequently_eq_iff_eventually_eq hj_an] at this
+  rw [EventuallyEq.eq_of_nhds this, sub_self, zero_pow, zero_smul]
+  · apply Nat.zero_lt_sub_of_lt (Nat.lt_of_le_of_ne h_le hj_ne.symm)
+  · exact (((analyticAt_id 𝕜 _).sub analyticAt_const).pow _).smul hg_an
+
+/-- If `f` is analytic at `z₀`, then exactly one of the following two possibilities occurs: either
+`f` vanishes identically near `z₀`, or locally around `z₀` it has the form `z ↦ (z - z₀) ^ n • g z`
+for some `n` and some `g` which is analytic and non-vanishing at `z₀`. -/
+theorem exists_eventuallyEq_pow_smul_nonzero_iff (hf : AnalyticAt 𝕜 f z₀) :
+    (∃ (n : ℕ), ∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧
+    ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ↔ (¬∀ᶠ z in 𝓝 z₀, f z = 0) := by
+  constructor
+  · rintro ⟨n, g, hg_an, hg_ne, hg_eq⟩
+    contrapose! hg_ne
+    apply EventuallyEq.eq_of_nhds
+    rw [EventuallyEq, ← AnalyticAt.frequently_eq_iff_eventually_eq hg_an analyticAt_const]
+    refine (eventually_nhdsWithin_iff.mpr ?_).frequently
+    filter_upwards [hg_eq, hg_ne] with z hf_eq hf0 hz
+    rwa [hf0, eq_comm, smul_eq_zero_iff_right] at hf_eq
+    exact pow_ne_zero _ (sub_ne_zero.mpr hz)
+  · intro hf_ne
+    rcases hf with ⟨p, hp⟩
+    exact ⟨p.order, _, ⟨_, hp.has_fpower_series_iterate_dslope_fslope p.order⟩,
+      hp.iterate_dslope_fslope_ne_zero (hf_ne.imp hp.locally_zero_iff.mpr),
+      hp.eq_pow_order_mul_iterate_dslope⟩
+
+/-- The order of vanishing of `f` at `z₀`, as an element of `ℕ∞`.
+
+This is defined to be `∞` if `f` is identically 0 on a neighbourhood of `z₀`, and otherwise the
+unique `n` such that `f z = (z - z₀) ^ n • g z` with `g` analytic and non-vanishing at `z₀`. See
+`AnalyticAt.order_eq_top_iff` and `AnalyticAt.order_eq_nat_iff` for these equivalences. -/
+noncomputable def order (hf : AnalyticAt 𝕜 f z₀) : ENat :=
+  if h : ∀ᶠ z in 𝓝 z₀, f z = 0 then ⊤
+  else ↑(hf.exists_eventuallyEq_pow_smul_nonzero_iff.mpr h).choose
+
+lemma order_eq_top_iff (hf : AnalyticAt 𝕜 f z₀) : hf.order = ⊤ ↔ ∀ᶠ z in 𝓝 z₀, f z = 0 := by
+  unfold order
+  split_ifs with h
+  · rwa [eq_self, true_iff]
+  · simpa only [ne_eq, ENat.coe_ne_top, false_iff] using h
+
+lemma order_eq_nat_iff (hf : AnalyticAt 𝕜 f z₀) (n : ℕ) : hf.order = ↑n ↔
+    ∃ (g : 𝕜 → E), AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ n • g z := by
+  unfold order
+  split_ifs with h
+  · simp only [ENat.top_ne_coe, false_iff]
+    contrapose! h
+    rw [← hf.exists_eventuallyEq_pow_smul_nonzero_iff]
+    exact ⟨n, h⟩
+  · rw [← hf.exists_eventuallyEq_pow_smul_nonzero_iff] at h
+    refine ⟨fun hn ↦ (WithTop.coe_inj.mp hn : h.choose = n) ▸ h.choose_spec, fun h' ↦ ?_⟩
+    rw [unique_eventuallyEq_pow_smul_nonzero h.choose_spec h']
+
 end AnalyticAt
 
 namespace AnalyticOn

Purely cosmetic PR.

@@ -120,7 +120,7 @@ theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ
 #align has_fpower_series_at.locally_ne_zero HasFPowerSeriesAt.locally_ne_zero
 
 theorem locally_zero_iff (hp : HasFPowerSeriesAt f p z₀) : (∀ᶠ z in 𝓝 z₀, f z = 0) ↔ p = 0 :=
-  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp )⟩
+  ⟨fun hf => hp.eq_zero_of_eventually hf, fun h => eventually_eq_zero (by rwa [h] at hp)⟩
 #align has_fpower_series_at.locally_zero_iff HasFPowerSeriesAt.locally_zero_iff
 
 end HasFPowerSeriesAt
@@ -134,7 +134,7 @@ theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
     (∀ᶠ z in 𝓝 z₀, f z = 0) ∨ ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 := by
   rcases hf with ⟨p, hp⟩
   by_cases h : p = 0
-  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))
+  · exact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp))
   · exact Or.inr (hp.locally_ne_zero h)
 #align analytic_at.eventually_eq_zero_or_eventually_ne_zero AnalyticAt.eventually_eq_zero_or_eventually_ne_zero
 

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -35,7 +35,7 @@ open Filter Function Nat FormalMultilinearSeries EMetric Set
 
 open scoped Topology BigOperators
 
-variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
   [NormedSpace 𝕜 E] {s : E} {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {n : ℕ} {z z₀ : 𝕜}
 --  {y : Fin n → 𝕜} -- Porting note: This is used nowhere and creates problem since it is sometimes
 -- automatically included as a hypothesis

Move some files to Analysis/Calculus/FDeriv

@@ -5,7 +5,7 @@ Authors: Vincent Beffara
 -/
 import Mathlib.Analysis.Analytic.Basic
 import Mathlib.Analysis.Calculus.Dslope
-import Mathlib.Analysis.Calculus.FDerivAnalytic
+import Mathlib.Analysis.Calculus.FDeriv.Analytic
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 import Mathlib.Analysis.Analytic.Uniqueness
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Vincent Beffara. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vincent Beffara
-
-! This file was ported from Lean 3 source module analysis.analytic.isolated_zeros
-! leanprover-community/mathlib commit a3209ddf94136d36e5e5c624b10b2a347cc9d090
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Analytic.Basic
 import Mathlib.Analysis.Calculus.Dslope
@@ -14,6 +9,8 @@ import Mathlib.Analysis.Calculus.FDerivAnalytic
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 import Mathlib.Analysis.Analytic.Uniqueness
 
+#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
+
 /-!
 # Principle of isolated zeros
 

@@ -41,7 +41,7 @@ open scoped Topology BigOperators
 variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E]
   [NormedSpace 𝕜 E] {s : E} {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {n : ℕ} {z z₀ : 𝕜}
 --  {y : Fin n → 𝕜} -- Porting note: This is used nowhere and creates problem since it is sometimes
--- automatically included as an hypothesis
+-- automatically included as a hypothesis
 
 namespace HasSum
 

@@ -90,20 +90,20 @@ theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
 #align has_fpower_series_at.has_fpower_series_dslope_fslope HasFPowerSeriesAt.has_fpower_series_dslope_fslope
 
 theorem has_fpower_series_iterate_dslope_fslope (n : ℕ) (hp : HasFPowerSeriesAt f p z₀) :
-    HasFPowerSeriesAt ((swap dslope z₀^[n]) f) ((fslope^[n]) p) z₀ := by
+    HasFPowerSeriesAt ((swap dslope z₀)^[n] f) (fslope^[n] p) z₀ := by
   induction' n with n ih generalizing f p
   · exact hp
   · simpa using ih (has_fpower_series_dslope_fslope hp)
 #align has_fpower_series_at.has_fpower_series_iterate_dslope_fslope HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope
 
 theorem iterate_dslope_fslope_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) :
-    (swap dslope z₀^[p.order]) f z₀ ≠ 0 := by
+    (swap dslope z₀)^[p.order] f z₀ ≠ 0 := by
   rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
   simpa [coeff_eq_zero] using apply_order_ne_zero h
 #align has_fpower_series_at.iterate_dslope_fslope_ne_zero HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero
 
 theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
-    ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀^[p.order]) f z := by
+    ∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z := by
   have hq := hasFPowerSeriesAt_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp)
   filter_upwards [hq, hasFPowerSeriesAt_iff'.mp hp] with x hx1 hx2
   have : ∀ k < p.order, p.coeff k = 0 := fun k hk => by

@@ -167,7 +167,7 @@ variable {U : Set 𝕜}
 analytic on a connected set `U` and vanishes in arbitrary neighborhoods of a point `z₀ ∈ U`, then
 it is identically zero in `U`.
 For higher-dimensional versions requiring that the function vanishes in a neighborhood of `z₀`,
-see `eq_on_zero_of_preconnected_of_eventually_eq_zero`. -/
+see `AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero`. -/
 theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOn 𝕜 f U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfw : ∃ᶠ z in 𝓝[≠] z₀, f z = 0) : EqOn f 0 U :=
   hf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀
@@ -184,7 +184,7 @@ theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU
 analytic on a connected set `U` and coincide at points which accumulate to a point `z₀ ∈ U`, then
 they coincide globally in `U`.
 For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
-see `eq_on_of_preconnected_of_eventually_eq`. -/
+see `AnalyticOn.eqOn_of_preconnected_of_eventuallyEq`. -/
 theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : EqOn f g U := by
   have hfg' : ∃ᶠ z in 𝓝[≠] z₀, (f - g) z = 0 :=
@@ -203,7 +203,7 @@ theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : Ana
 field `𝕜` are analytic everywhere and coincide at points which accumulate to a point `z₀`, then
 they coincide globally.
 For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
-see `eq_of_eventually_eq`. -/
+see `AnalyticOn.eq_of_eventuallyEq`. -/
 theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f univ) (hg : AnalyticOn 𝕜 g univ)
     (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g :=
   funext fun x =>