mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: case refine_1
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π * |θ|) ^ 2 ≤ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2

pp: case refine_2
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ √((Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2) ≤ |θ|

Tactic:
· rw [mul_pow, ← abs_pow, abs_sq]
      calc
        _ = 2 * (1 - (1 - 2 / π ^ 2 * θ ^ 2)) := by ring
        _ ≤ 2 * (1 - θ.cos) := by
            gcongr; exact Real.cos_le_one_sub_mul_cos_sq <| abs_le.2 <| Ioc_subset_Icc_self hθ
        _ = _ := by linear_combination -θ.cos_sq_add_sin_sq

Goal state after:
case refine_2
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ √((Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2) ≤ |θ|
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: case mp
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2 → ⟪x, y⟫ = 0

Tactic:
intro h

Goal state after:
case mp
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2
⊢ ⟪x, y⟫ = 0
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: case mpr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ⟪x, y⟫ = 0 → ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2

Tactic:
intro h

Goal state after:
case mpr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2
Module: Mathlib.Analysis.Complex.CoveringMap

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
n : ℤ
hn : ↑n ≠ 0
surj : Function.Surjective fun x ↦ x ^ n
⊢ IsQuotientCoveringMap (fun x ↦ x ^ n) ↥(zpowGroupHom n).ker

Tactic:
by
  obtain ⟨n, rfl | rfl⟩ := n.eq_nat_or_neg
  · exact isQuotientCoveringMap_npow n (by aesop) (by simpa using surj)
  rw [show (zpowGroupHom (α := 𝕜ˣ) (-n)).ker = (powMonoidHom n).ker by ext; simp]
  convert (isQuotientCoveringMap_npow n (by aesop) _).homeomorph_comp (.inv 𝕜ˣ)
  · ext; simp
  convert! inv_involutive.surjective.comp surj; simp

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
this : U ∈ 𝓝[≠] x
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
filter_upwards [this, h₁ x hx] with a h₁a h₂a

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
this : U ∈ 𝓝[≠] x
a : 𝕜
h₁a : a ∈ U
h₂a : a ∈ (U \ {x | f₁ x = f₂ x})ᶜ
⊢ f₁ a = f₂ a
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicNFAt f x
⊢ f x ≠ 0 → meromorphicOrderAt f x = 0

Tactic:
intro h

Goal state after:
case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicNFAt f x
h : f x ≠ 0
⊢ meromorphicOrderAt f x = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hf : MeromorphicNFAt f x
hg : MeromorphicNFAt g x
⊢ f =ᶠ[𝓝[≠] x] g → f =ᶠ[𝓝 x] g

Tactic:
intro h

Goal state after:
case mp
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hf : MeromorphicNFAt f x
hg : MeromorphicNFAt g x
h : f =ᶠ[𝓝[≠] x] g
⊢ f =ᶠ[𝓝 x] g
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ (divisor (f ∘ fun x ↦ x + c) U) (x - c) = (divisor f (U + {c})) x

Tactic:
rw [divisor_apply, divisor_apply]

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ (meromorphicOrderAt (f ∘ fun x ↦ x + c) (x - c)).untop₀ = (meromorphicOrderAt f x).untop₀

case neg.hf
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ MeromorphicOn f (U + {c})

case neg.hz
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ x ∈ U + {c}

case neg.hf
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ MeromorphicOn (f ∘ fun x ↦ x + c) U

case neg.hz
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : ¬x ∉ U + {c}
⊢ x - c ∈ U
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
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
h : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤
⊢ analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose ⋯ • f z) x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
obtain ⟨m, hm⟩ := ENat.ne_top_iff_exists.mp h

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
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
h : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤
m : ℕ
hm : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x
⊢ analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose ⋯ • f z) x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp
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
h₁f : MeromorphicNFOn f U
⊢ 0 ≤ MeromorphicOn.divisor f U → AnalyticOnNhd 𝕜 f U

Tactic:
intro h

Goal state after:
case mp
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
h₁f : MeromorphicNFOn f U
h : 0 ≤ MeromorphicOn.divisor f U
⊢ AnalyticOnNhd 𝕜 f U
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
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
h₁f : MeromorphicNFOn f U
⊢ AnalyticOnNhd 𝕜 f U → 0 ≤ MeromorphicOn.divisor f U

Tactic:
intro h

Goal state after:
case mpr
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
h₁f : MeromorphicNFOn f U
h : AnalyticOnNhd 𝕜 f U
⊢ 0 ≤ MeromorphicOn.divisor f U
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case mp
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
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f x ≠ ⊤ → ∀ᶠ (x : 𝕜) in 𝓝[≠] x, f x ≠ 0

Tactic:
intro h

Goal state after:
case mp
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
hf : MeromorphicAt f x
h : meromorphicOrderAt f x ≠ ⊤
⊢ ∀ᶠ (x : 𝕜) in 𝓝[≠] x, f x ≠ 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
⊢ MeromorphicNFOn (g • f) U → MeromorphicNFOn f U

Tactic:
intro h

Goal state after:
case mp
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn (g • f) U
⊢ MeromorphicNFOn f U
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
⊢ MeromorphicNFOn f U → MeromorphicNFOn (g • f) U

Tactic:
intro h

Goal state after:
case mpr
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn f U
⊢ MeromorphicNFOn (g • f) U
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp
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
⊢ toMeromorphicNFOn f U = f → MeromorphicNFOn f U

Tactic:
intro h

Goal state after:
case mp
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
h : toMeromorphicNFOn f U = f
⊢ MeromorphicNFOn f U
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
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
⊢ MeromorphicNFOn f U → toMeromorphicNFOn f U = f

Tactic:
intro h

Goal state after:
case mpr
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
h : MeromorphicNFOn f U
⊢ toMeromorphicNFOn f U = f
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u : 𝕜
⊢ (u ∈ Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) → u ∈ support d

Tactic:
intro h

Goal state after:
case mp
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u : 𝕜
h : u ∈ Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u
⊢ u ∈ support d
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u : 𝕜
⊢ u ∈ support d → u ∈ Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u

Tactic:
intro h

Goal state after:
case mpr
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u : 𝕜
h : u ∈ support d
⊢ u ∈ Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d 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
x : 𝕜
hf : MeromorphicAt f x
⊢ Tendsto f (𝓝[≠] x) (Bornology.cobounded E) ↔ meromorphicOrderAt f x < 0

Tactic:
rcases lt_or_ge (meromorphicOrderAt f x) 0 with ho | ho

Goal state after:
case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x < 0
⊢ Tendsto f (𝓝[≠] x) (Bornology.cobounded E) ↔ meromorphicOrderAt f x < 0

case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : 0 ≤ meromorphicOrderAt f x
⊢ Tendsto f (𝓝[≠] x) (Bornology.cobounded E) ↔ meromorphicOrderAt f x < 0
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
x : 𝕜
hf : MeromorphicAt f x
⊢ (∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ 0 ≤ meromorphicOrderAt f x

Tactic:
rcases lt_or_ge (meromorphicOrderAt f x) 0 with ho | ho

Goal state after:
case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x < 0
⊢ (∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ 0 ≤ meromorphicOrderAt f x

case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : 0 ≤ meromorphicOrderAt f x
⊢ (∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ 0 ≤ meromorphicOrderAt f x
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
f : 𝕜 → 𝕜
h₁ : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
⊢ (g ^ n) x ≠ 0

Tactic:
by simp_all [zpow_ne_zero]

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

Goal state before:
pp: case coe
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hf : MeromorphicAt f x
n : ℕ
hn : meromorphicOrderAt (f ^ n) x = ↑n * meromorphicOrderAt f x
a✝ : ℤ
⊢ ↑n * ↑a✝ + ↑a✝ = (↑n + 1) * ↑a✝

Tactic:
norm_cast
      simp only [Nat.cast_add, Nat.cast_one]
      ring

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
⊢ (u ∉ support fun u x ↦ ↑(D u) * log ‖x - u‖) ↔ u ∉ D.support

Tactic:
simp only [ne_eq, not_not, Function.mem_support]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
⊢ (fun x ↦ ↑(D u) * log ‖x - u‖) = 0 ↔ D u = 0
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
S : Type u_3
inst✝ : CommRing S
f : R →+* S
u : GL n R
hu : u ∈ Subgroup.center (GL n R)
⊢ u ∈ Subgroup.comap (map f) (Subgroup.center (GL n S))

Tactic:
simp only [GeneralLinearGroup.center_eq_range_scalar, MonoidHom.mem_range,
    Subgroup.mem_comap] at hu ⊢

Goal state after:
R : Type u_1
n : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
S : Type u_3
inst✝ : CommRing S
f : R →+* S
u : GL n R
hu : ∃ x, (scalar n) x = u
⊢ ∃ x, (scalar n) x = (map f) u
Module: Mathlib.Analysis.Complex.Isometry

Goal state before:
pp: a : Circle
⊢ rotation a ≠ conjLIE

Tactic:
intro h

Goal state after:
a : Circle
h : rotation a = conjLIE
⊢ False
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
⊢ ∃ Γ, ContinuousOn Γ (Set.Icc 0 (t n)) ∧ Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n)) ∧ Γ 0 = e

Tactic:
induction n with
  | zero =>
    refine ⟨fun _ ↦ e, continuous_const.continuousOn, fun t ht ↦ ?_, rfl⟩
    rw [t_0, Set.Icc_self, Set.mem_singleton_iff] at ht; subst ht; exact γ_0.symm
  | succ n ih => ?_

Goal state after:
case succ
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
ih : ∃ Γ, ContinuousOn Γ (Set.Icc 0 (t n)) ∧ Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n)) ∧ Γ 0 = e
⊢ ∃ Γ, ContinuousOn Γ (Set.Icc 0 (t (n + 1))) ∧ Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t (n + 1))) ∧ Γ 0 = e
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
⊢ fderiv ℝ f z ≠ 0 →
    (((∃ map, restrictScalars ℝ map = fderiv ℝ f z) ∨ ∃ map, restrictScalars ℝ map = fderiv ℝ f z ∘SL ↑conjCLE) ↔
      DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z))

Tactic:
intro h

Goal state after:
f : ℂ → ℂ
z : ℂ
h : fderiv ℝ f z ≠ 0
⊢ ((∃ map, restrictScalars ℝ map = fderiv ℝ f z) ∨ ∃ map, restrictScalars ℝ map = fderiv ℝ f z ∘SL ↑conjCLE) ↔
    DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
⊢ h w =
    ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f (ball 0 R)) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f (sphere 0 R)) i) •
      meromorphicTrailingCoeffAt f w

Tactic:
let S₀R := sphere (0 : ℂ) R

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
⊢ h w =
    ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f (ball 0 R)) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f (sphere 0 R)) i) •
      meromorphicTrailingCoeffAt f w
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ ↑3 - 1 < rexp 1

Tactic:
by exact_mod_cast exp_one_gt_two

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
⊢ log ‖h w‖ =
    ∑ᶠ (i : ℂ), ↑((divisor f (ball 0 R)) i) * log ‖canonicalFactor R i w‖ -
        ∑ᶠ (i : ℂ), ↑((divisor f (sphere 0 R)) i) * log ‖w - i‖ +
      log ‖meromorphicTrailingCoeffAt f w‖

Tactic:
let S₀R := sphere (0 : ℂ) R

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
⊢ log ‖h w‖ =
    ∑ᶠ (i : ℂ), ↑((divisor f (ball 0 R)) i) * log ‖canonicalFactor R i w‖ -
        ∑ᶠ (i : ℂ), ↑((divisor f (sphere 0 R)) i) * log ‖w - i‖ +
      log ‖meromorphicTrailingCoeffAt f w‖
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case mp
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
⊢ γ ∈ (⋯.monodromyPerm x).ker → γ ∈ (FundamentalGroup.mapOfEq { toFun := p, continuous_toFun := ⋯ } ⋯).range

Tactic:
intro h

Goal state after:
case mp
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
h : γ ∈ (⋯.monodromyPerm x).ker
⊢ γ ∈ (FundamentalGroup.mapOfEq { toFun := p, continuous_toFun := ⋯ } ⋯).range
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case mpr
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
⊢ γ ∈ (FundamentalGroup.mapOfEq { toFun := p, continuous_toFun := ⋯ } ⋯).range → γ ∈ (⋯.monodromyPerm x).ker

Tactic:
intro h

Goal state after:
case mpr
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
h : γ ∈ (FundamentalGroup.mapOfEq { toFun := p, continuous_toFun := ⋯ } ⋯).range
⊢ γ ∈ (⋯.monodromyPerm x).ker
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
⊢ ∃ δ < 0, ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))

Tactic:
refine
      ⟨ε * Real.cos (d * b),
        mul_neg_of_neg_of_pos ε₀
          (Real.cos_pos_of_mem_Ioo <| abs_lt.1 <| (abs_of_pos (mul_pos hd₀ hb)).symm ▸ hb'),
        fun w hw => ?_⟩
    replace hw : |im (aff w)| ≤ d * b := by
      rw [← Real.closedBall_eq_Icc, mem_closedBall, Real.dist_eq] at hw
      rw [im_ofReal_mul, sub_im, mul_I_im, ofReal_re, _root_.abs_mul, abs_of_pos hd₀]
      gcongr
    simpa only [aff, re_ofReal_mul, _root_.abs_mul, abs_of_pos hd₀, sub_re, mul_I_re, ofReal_im,
      zero_mul, neg_zero, sub_zero] using
      norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le ε₀.le hw hb'.le

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

Goal state before:
pp: case mp
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
⊢ (∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e) → ∃ x, x.re = l ∧ f x = e

Tactic:
intro h

Goal state after:
case mp
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
h : ∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e
⊢ ∃ x, x.re = l ∧ f x = e
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case mpr
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
⊢ (∃ x, x.re = l ∧ f x = e) → ∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e

Tactic:
intro h

Goal state after:
case mpr
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
h : ∃ x, x.re = l ∧ f x = e
⊢ ∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e