mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2.refine_6
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
⊢ (p ∘ fun ta ↦ if ta.1 ≤ t n then g' ta else if f ta ∈ (q e).target then ↑(q e).symm (f ta) else g ta) = ⇑f

Tactic:
ext ta; rw [Function.comp_apply]; split_ifs with _ hv
    · exact congr($g'_lifts ta)
    · rw [hpq e, (q e).right_inv hv]
    · exact congr($g_lifts ta)

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
n : ℤ
hn : ↑n = meromorphicOrderAt f x
⊢ ↑n = meromorphicOrderAt (-f) x

Tactic:
rw [eq_comm, meromorphicOrderAt_eq_int_iff (by fun_prop)] at *

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
n : ℤ
hn : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (-f) z = (z - x) ^ n • g z
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case coe.coe
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁶ : NormedRing R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : Module R E
inst✝³ : IsBoundedSMul R E
inst✝² : Module.IsTorsionFree R E
x : 𝕜
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : IsScalarTower 𝕜 R E
f : 𝕜 → R
g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : meromorphicOrderAt f x = ↑m
n : ℤ
h₂g : meromorphicOrderAt g x = ↑n
⊢ meromorphicOrderAt (f • g) x = ↑m + ↑n

Tactic:
← WithTop.coe_add,

Goal state after:
case coe.coe
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁶ : NormedRing R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : Module R E
inst✝³ : IsBoundedSMul R E
inst✝² : Module.IsTorsionFree R E
x : 𝕜
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : IsScalarTower 𝕜 R E
f : 𝕜 → R
g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : meromorphicOrderAt f x = ↑m
n : ℤ
h₂g : meromorphicOrderAt g x = ↑n
⊢ meromorphicOrderAt (f • g) x = ↑(m + n)
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
w : ℂ
hw : w ∈ ball 0 R
x✝¹ : ℂ
x✝ : x✝¹ ∈ ball 0 R
⊢ x✝¹ ∈ univ

Tactic:
trivial

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : SpecialLinearGroup n R
hn : IsEmpty n
⊢ toGL g ∈ Subgroup.center (GL n R) ↔ g ∈ Subgroup.center (SpecialLinearGroup n R)

Tactic:
then

Goal state after:
n : Type u_1
R : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : SpecialLinearGroup n R
hn : IsEmpty n
⊢ toGL g ∈ Subgroup.center (GL n R) ↔ g ∈ Subgroup.center (SpecialLinearGroup n R)
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖map 1‖⁻¹ * (‖x‖ * ‖map 1‖) = ‖x‖

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case mp.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
hU : IsConnected U
h₂f : ∃ u, meromorphicOrderAt f ↑u ≠ ⊤
this : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = ∅
u : ↑U
⊢ meromorphicOrderAt f ↑u ≠ ⊤

Tactic:
have : u ∉ (∅ : Set U) := by exact fun a => a

Goal state after:
case mp.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
hU : IsConnected U
h₂f : ∃ u, meromorphicOrderAt f ↑u ≠ ⊤
this✝ : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = ∅
u : ↑U
this : u ∉ ∅
⊢ meromorphicOrderAt f ↑u ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
hU✝ : IsConnected U
this : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = univ
u : ↑U
hU : meromorphicOrderAt f ↑u ≠ ⊤
⊢ u ∈ univ

Tactic:
trivial

Goal state after:
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
z : ℂ
hz₁ : z.re = 1
hz₂ : f (↑l + z * (↑u - ↑l)) = e
⊢ (↑l + z * (↑u - ↑l)).re = u ∧ f (↑l + z * (↑u - ↑l)) = e

Tactic:
simp only [add_re, ofReal_re, mul_re, hz₁, sub_re, one_mul, sub_im, ofReal_im, sub_self,
        mul_zero, sub_zero, add_sub_cancel, hz₂, and_self]

Goal state after:
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε ↦ ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0) (𝓝 ‖interpStrip f z‖)

Tactic:
interpStrip_eq_of_mem_verticalStrip _ _ hz

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ Tendsto (fun ε ↦ ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (𝓝[>] 0)
    (𝓝 ‖↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z‖)
Module: Mathlib.Analysis.Distribution.DerivNotation

Goal state before:
pp: V : Type u_11
E : Type u_12
inst✝ : LineDeriv V E E
f : E
n : ℕ
IH : ∀ (m : Fin (n + 1) → V), ∂^{m} f = ∂^{init m} (∂_{m (last n)} f)
m : Fin (n + 1 + 1) → V
hmzero : init m 0 = m 0
hmtail : tail m (last n) = m (last n.succ)
⊢ ∂_{m 0} (∂^{init (tail m)} (∂_{tail m (last n)} f)) = ∂^{init m} (∂_{m (last (n + 1))} f)

Tactic:
by
        rw [hmtail, iteratedLineDerivOp_succ_left, hmzero, tail_init_eq_init_tail]

Goal state after:
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: case hab
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)
I₂ : E := I • ∫ (y : ℝ) in c.im..w.im, f (↑w.re + ↑y * I)
I₃ : E := ∫ (x : ℝ) in c.re..z.re, f (↑x + ↑c.im * I)
I₄ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑z.re + ↑y * I)
I₅ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₆ : E := I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
I₇ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑c.im * I)
I₈ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑w.re + ↑y * I)
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ IntervalIntegrable (fun y ↦ f (↑w.re + ↑y * I)) volume c.im z.im

Tactic:
apply integrableVert

Goal state after:
case hab.hb₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ ↑w.re + ↑c.im * I ∈ ball c r

case hab.hb₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ ↑w.re + ↑z.im * I ∈ ball c r
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: case hbc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ∫ (x : ℝ) in c.re..w.re, f (↑x + ↑c.im * I)
I₂ : E := I • ∫ (y : ℝ) in c.im..w.im, f (↑w.re + ↑y * I)
I₃ : E := ∫ (x : ℝ) in c.re..z.re, f (↑x + ↑c.im * I)
I₄ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑z.re + ↑y * I)
I₅ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₆ : E := I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
I₇ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑c.im * I)
I₈ : E := I • ∫ (y : ℝ) in c.im..z.im, f (↑w.re + ↑y * I)
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ IntervalIntegrable (fun y ↦ f (↑w.re + ↑y * I)) volume z.im w.im

Tactic:
apply integrableVert

Goal state after:
case hbc.hb₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ ↑w.re + ↑z.im * I ∈ ball c r

case hbc.hb₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ ↑w.re + ↑w.im * I ∈ ball c r
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
h : R ≠ 0
⊢ (2 * ↑π)⁻¹ * 1 = I⁻¹ * (2 * ↑π)⁻¹ * I * 1

Tactic:
field

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
s : Set ℝ
c : ℂ
hf : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ({z | ‖z - c‖ ∈ s}.domRestrict f)
hs : ∀ r ∈ s, 0 ≤ r
this : ∀ (x : ↑s × ℝ), circleMap c (↑x.1) x.2 ∈ {z | ‖z - c‖ ∈ s}
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry fun x θ ↦ f (circleMap c (↑x) θ))

Tactic:
apply hf.comp (f := (fun x ↦ ⟨circleMap c x.1 x.2, this x⟩))

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
s : Set ℝ
c : ℂ
hf : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ({z | ‖z - c‖ ∈ s}.domRestrict f)
hs : ∀ r ∈ s, 0 ≤ r
this : ∀ (x : ↑s × ℝ), circleMap c (↑x.1) x.2 ∈ {z | ‖z - c‖ ∈ s}
⊢ Continuous fun x ↦ ⟨circleMap c (↑x.1) x.2, ⋯⟩
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ w ∈ ball c (e + R)

Tactic:
grind [mem_ball]

Goal state after:
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
z : ℂ
hz✝ : z ∈ sphere 0 R
hz : z ≠ 0
⊢ z⁻¹ * (↑q * z / (↑q * z - W) * 1) = ↑q / (↑q * z - W) * 1

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y

Tactic:
by_cases! h : log x ≤ 0

Goal state after:
case pos
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : log x ≤ 0
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y

case neg
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : 0 < log x
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case neg
r : ℝ
hr : ¬r = 0
⊢ 2 * sin r = 2 * r * (sin r / r)

Tactic:
field

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x = 2 - 2 * cos x

Tactic:
by
      rw [sin_sq_add_cos_sq]
      norm_num

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case h
a c : ℂ
R : ℝ
z z₁ z₂ : ℂ
hR : ¬R = 0
hz : ‖z‖ = 1
hz₁ : ‖z₁‖ = 1 ∧ z₁ + (↑R)⁻¹ * (c - a) = 0
hz₂ : ‖z₂‖ = 1 ∧ z₂ + (↑R)⁻¹ * (c - a) = 0
⊢ -((↑R)⁻¹ * (c - a)) = z₂

Tactic:
add_eq_zero_iff_eq_neg.1 hz₂.2

Goal state after:
case h
a c : ℂ
R : ℝ
z z₁ z₂ : ℂ
hR : ¬R = 0
hz : ‖z‖ = 1
hz₁ : ‖z₁‖ = 1 ∧ z₁ + (↑R)⁻¹ * (c - a) = 0
hz₂ : ‖z₂‖ = 1 ∧ z₂ + (↑R)⁻¹ * (c - a) = 0
⊢ -((↑R)⁻¹ * (c - a)) = -((↑R)⁻¹ * (c - a))
Module: Mathlib.Analysis.Complex.Polynomial.UnitTrinomial

Goal state before:
pp: p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : (map (algebraMap ℤ ℂ) q).IsRoot z
⊢ False

Tactic:
IsRoot,

Goal state after:
p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : eval z (map (algebraMap ℤ ℂ) q) = 0
⊢ False
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f c
z : ℂ
hz : c ≤ z
⊢ f c ≤ f z

Tactic:
have h' (n : ℕ) : 0 ≤ iteratedDeriv n (f · - f c) c := by
    cases n with
    | zero => simp only [iteratedDeriv_zero, sub_self, le_refl]
    | succ n =>
      specialize h (n + 1) n.succ_ne_zero
      rw [iteratedDeriv_succ'] at h ⊢
      rwa [funext fun x ↦ deriv_sub_const (f := f) (x := x) (f c)]

Goal state after:
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f c
z : ℂ
hz : c ≤ z
h' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (fun x ↦ f x - f c) c
⊢ f c ≤ f z
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
this :
  ∀ {U : Set ℂ},
    IsOpen U → IsSimplyConnected U → U ≠ univ → 0 ∉ U → ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
hU₀ : ¬0 ∉ U
a : ℂ
ha : a ∉ U
⊢ -a +ᵥ U ≠ univ

Tactic:
simp [hU]

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case hw
w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := ⋯
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
⊢ w ∈ ball 0 (r n)

Tactic:
simp only [mem_ball, dist_zero_right, lt_sub_iff_add_lt, r]
      field_simp
      calc ‖w‖ * (n + 2) + (R - ‖w‖) = ‖w‖ * (n + 1) + R := by ring
        _ < R * (n + 1) + R := by gcongr
        _ = R * (n + 2) := by ring

Goal state after:
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ 0 < ‖f z - x‖

Tactic:
· simpa [sub_eq_zero] using hfx z hz

Goal state after:
Module: Mathlib.Order.Interval.Set.IsoIoo

Goal state before:
pp: k : Type u_1
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsStrictOrderedRing k
x : k
H : 0 < 1 + |x|
⊢ |x / (1 + |x|)| = |x| / (1 + |x|)

Tactic:
abs_div,

Goal state after:
k : Type u_1
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsStrictOrderedRing k
x : k
H : 0 < 1 + |x|
⊢ |x| / |1 + |x|| = |x| / (1 + |x|)
Module: Mathlib.Analysis.RCLike.Sqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
a : 𝕜
⊢ sqrt a = if h : im I = 1 then (complexRingEquiv h).symm ((complexRingEquiv h) a).sqrt else ↑√(re a)

Tactic:
by
  rw [sqrt, eq_comm]
  split_ifs with h
  · simp
  have : (I : 𝕜) = 0 := by grind [I_eq_zero_or_im_I_eq_one]
  simp_all only [Complex.sqrt, im_eq_zero this, map_apply, add_zero, re_to_complex, im_to_complex,
    mul_zero, algebraMap.coe_inj, Complex.cpow_inv_two_re]
  by_cases! ha' : 0 ≤ re a
  · simp [abs_of_nonneg ha', ← two_mul]
  simp [abs_of_nonpos ha'.le, Real.sqrt_eq_zero', ha'.le]

Goal state after:
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case pos
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
⊢ ↑((divisor f (closedBall c |r|)) u) * Real.log (R / r) ≤
    ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹)

Tactic:
by_cases h2 : u ∈ closedBall c |r|
        · --In the smaller ball: the divisors agree and we bound the log factor
          simp only [(h₁f.mono (closedBall_subset_closedBall r_lt_R.le)), h2,
            AnalyticOnNhd.divisor_apply, h₁f, h1]
          by_cases! h3 : u = c --Need to use the divisor is 0 at c rather than comparing the logs
          · rw [h3, (h₁f c (by simp)).analyticOrderAt_eq_zero.mpr h₂f]
            simp
          simp +singlePass only [← log_abs]
          gcongr 2
          · simp
          · have : ‖c - u‖ ≠ 0 := by simpa [sub_eq_zero] using h3.symm
            simpa [field, abs_div, r_pos.trans r_lt_R, dist_eq_norm'] using h2
        · --In the larger ball but not the smaller so LHS is 0 and RHS nonnegative
          simp only [h2, not_false_eq_true, Function.locallyFinsuppWithin.apply_eq_zero_of_notMem,
            Int.cast_zero, zero_mul]
          refine mul_nonneg (mod_cast h₁f.divisor_nonneg ..) ?_
          apply log_abs _ ▸ log_nonneg
          simp only [mem_closedBall, dist_eq_norm', not_le] at h1 h2
          have : ‖c - u‖ ≠ 0 := (r_pos.trans h2).ne'
          simpa [field]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case «0».«0»
K : Type u_1
inst✝¹ : Field K
m : Matrix (Fin 2) (Fin 2) K
inst✝ : NeZero 2
⊢ (m 0 0 - (m 0 0 + m 1 1) / 2) * (m 0 0 - (m 0 0 + m 1 1) / 2) + m 0 1 * m 1 0 =
    ((m 0 0 + m 1 1) * (m 0 0 + m 1 1) - 2 * 2 * (m 0 0 * m 1 1 - m 0 1 * m 1 0)) / (2 * 2)

Tactic:
field

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case «0».«1»
K : Type u_1
inst✝¹ : Field K
m : Matrix (Fin 2) (Fin 2) K
inst✝ : NeZero 2
⊢ (m 0 0 - (m 0 0 + m 1 1) / 2) * m 0 1 + m 0 1 * (m 1 1 - (m 0 0 + m 1 1) / 2) = 0

Tactic:
field

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case «1».«0»
K : Type u_1
inst✝¹ : Field K
m : Matrix (Fin 2) (Fin 2) K
inst✝ : NeZero 2
⊢ m 1 0 * (m 0 0 - (m 0 0 + m 1 1) / 2) + (m 1 1 - (m 0 0 + m 1 1) / 2) * m 1 0 = 0

Tactic:
field

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case «1».«1»
K : Type u_1
inst✝¹ : Field K
m : Matrix (Fin 2) (Fin 2) K
inst✝ : NeZero 2
⊢ m 1 0 * m 0 1 + (m 1 1 - (m 0 0 + m 1 1) / 2) * (m 1 1 - (m 0 0 + m 1 1) / 2) =
    ((m 0 0 + m 1 1) * (m 0 0 + m 1 1) - 2 * 2 * (m 0 0 * m 1 1 - m 0 1 * m 1 0)) / (2 * 2)

Tactic:
field

Goal state after: