mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«0».«2»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨0, ⋯⟩ < (fun i ↦ i) ⟨2, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
case «1».«2».«0».«2»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩))
hy : True
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«1».«0»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨1, ⋯⟩ < (fun i ↦ i) ⟨0, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«1».«1»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨1, ⋯⟩ < (fun i ↦ i) ⟨1, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«1».«2»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨1, ⋯⟩ < (fun i ↦ i) ⟨2, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
case «1».«2».«1».«2»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩))
hy : True
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«2».«0»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨2, ⋯⟩ < (fun i ↦ i) ⟨0, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«2».«1»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨2, ⋯⟩ < (fun i ↦ i) ⟨1, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case «1».«2».«2».«2»
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
x✝ :
  dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
    dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
h₁ : dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2)
h₂ : dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2)
h₃ : dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
hx : True
hy : (fun i ↦ i) ⟨2, ⋯⟩ < (fun i ↦ i) ⟨2, ⋯⟩
hxy :
  dist (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) =
    dist (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩))
⊢ (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩ ∧ (fun i ↦ i) ⟨2, ⋯⟩ = (fun i ↦ i) ⟨2, ⋯⟩

Tactic:
simp +decide only at hy

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
hsub : q w - q u = (C u.1 - C w.1) * X + C w.2 - C u.2
hdvd : q u ∣ q w ^ n - (q w - q u) ^ n
⊢ ‖φ x u‖ * M ^ (n - 1) ≤ M ^ n + ‖φ x u - φ x w‖ ^ n

Tactic:
have H' : ((q w - q u) ^ n).natDegree < 2 * n := by rw [hsub]; compute_degree; grind

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
hsub : q w - q u = (C u.1 - C w.1) * X + C w.2 - C u.2
hdvd : q u ∣ q w ^ n - (q w - q u) ^ n
H' : ((q w - q u) ^ n).natDegree < 2 * n
⊢ ‖φ x u‖ * M ^ (n - 1) ≤ M ^ n + ‖φ x u - φ x w‖ ^ n
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
⊢ limsup u f + liminf v f ≤ limsup (u + v) f

Tactic:
refine add_le_of_forall_lt fun a a_u b b_v ↦ (le_limsup_iff h h').2 fun c c_ab ↦ ?_

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a < limsup u f
b : α
b_v : b < liminf v f
c : α
c_ab : c < a + b
⊢ ∃ᶠ (a : ι) in f, c < (u + v) a
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
⊢ f x ≤ f 1 * seminormFromBounded' f x

Tactic:
by_cases h1 : f 1 = 0
  · specialize f_mul x 1
    rw [mul_one, h1, mul_zero] at f_mul
    have hx0 : f x = 0 := f_mul.antisymm (f_nonneg _)
    rw [hx0, h1, zero_mul]
  · rw [mul_comm, ← div_le_iff₀ (lt_of_le_of_ne' (f_nonneg _) h1)]
    conv_lhs => rw [← mul_one x]
    exact le_ciSup (seminormFromBounded_bddAbove_range f_nonneg f_mul x) (1 : R)

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
⊢ f (x * y * z) / f z ≤ seminormFromBounded' f x * seminormFromBounded' f y

Tactic:
← div_le_iff₀ (lt_of_le_of_ne' (seminormFromBounded_nonneg f_nonneg f_mul _) hy)

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
⊢ f (x * y * z) / f z / seminormFromBounded' f y ≤ seminormFromBounded' f x
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun t ↦ -(↑(t % m1) / ↑t)) atTop (𝓝 0)

Tactic:
rw [← neg_zero]

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun t ↦ -(↑(t % m1) / ↑t)) atTop (𝓝 (-0))
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
hμn : μ (x ^ n) = 0
⊢ μ (x ^ n) ^ (1 / ↑n) = 0

Tactic:
hμn,

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
hμn : μ (x ^ n) = 0
⊢ 0 ^ (1 / ↑n) = 0
Module: Mathlib.Analysis.Normed.Field.Krasner

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Normal K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
z : L := x - y
this✝¹ : FiniteDimensional K ↥K⟮y⟯
this✝ : IsUltrametricDist L
y' : ↥K⟮y⟯ := AdjoinSimple.gen K y
zsep : IsSeparable (↥K⟮y⟯) z
this : z ∈ K⟮y⟯ ↔ z ∈ ⊥
hz : z ∉ ⊥
⊢ False

Tactic:
obtain ⟨z', hne, h1⟩ := (notMem_iff_exists_ne_and_isConjRoot zsep
        (minpoly_sub_algebraMap_splits y' (IsIntegral.minpoly_splits_tower_top
          xsep.isIntegral sp))).mp hz

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Normal K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
z : L := x - y
this✝¹ : FiniteDimensional K ↥K⟮y⟯
this✝ : IsUltrametricDist L
y' : ↥K⟮y⟯ := AdjoinSimple.gen K y
zsep : IsSeparable (↥K⟮y⟯) z
this : z ∈ K⟮y⟯ ↔ z ∈ ⊥
hz : z ∉ ⊥
z' : L
hne : z ≠ z'
h1 : IsConjRoot (↥K⟮y⟯) z z'
⊢ False
Module: Mathlib.Analysis.Normed.Group.CocompactMap

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : ProperSpace E
f : E → F
h : ∀ (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖
s : Set F
⊢ s ∈ cocompact F → f ⁻¹' s ∈ cocompact E

Tactic:
hs

Goal state after:
E : Type u_2
F : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : ProperSpace E
f : E → F
h : ∀ (ε : ℝ), ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖
s : Set F
hs : s ∈ cocompact F
⊢ f ⁻¹' s ∈ cocompact E
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
h : μ (x ^ (↑m1 * (n / ↑m1))) ^ (1 / ↑n) ≤ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n)
hL0' : 0 < L + ε / 2
h1 : (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n) < (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n))
h2 : μ (x ^ (n % ↑m1)) ^ (1 / ↑n) ≤ (μ x ^ (n % ↑m1)) ^ (1 / ↑n)
h3 : (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) ≤ L + ε
h4 : 0 < μ (x ^ (n % ↑m1)) ^ (1 / ↑n)
⊢ μ (x ^ (↑m1 * (n / ↑m1) + n % ↑m1)) ^ (1 / ↑n) < L + ε

Tactic:
have h5 : 0 < (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / (n : ℝ))) :=
      mul_pos hL0' (rpow_pos_of_pos hL0' _)

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
h : μ (x ^ (↑m1 * (n / ↑m1))) ^ (1 / ↑n) ≤ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n)
hL0' : 0 < L + ε / 2
h1 : (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n) < (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n))
h2 : μ (x ^ (n % ↑m1)) ^ (1 / ↑n) ≤ (μ x ^ (n % ↑m1)) ^ (1 / ↑n)
h3 : (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) ≤ L + ε
h4 : 0 < μ (x ^ (n % ↑m1)) ^ (1 / ↑n)
h5 : 0 < (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n))
⊢ μ (x ^ (↑m1 * (n / ↑m1) + n % ↑m1)) ^ (1 / ↑n) < L + ε
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: case neg
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : ∀ ⦃x : F⦄, x ∈ f.rootSet F → x = a
⊢ 1 ≤ ‖a - a'‖

Tactic:
rw [isConjRoot_iff_mem_minpoly_rootSet ⟨f, fmon, fa0⟩,
          ← minpoly.eq_of_irreducible_of_monic firr fa0 fmon] at conj

Goal state after:
case neg
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : a' ∈ f.rootSet f.SplittingField
ne : a ≠ a'
hS : ∀ ⦃x : F⦄, x ∈ f.rootSet F → x = a
⊢ 1 ≤ ‖a - a'‖
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
⊢ f x ≤ spectralValue p

Tactic:
by
  by_cases hx0 : f x = 0
  · rw [hx0]; exact spectralValue_nonneg p
  · by_contra h_ge
    have hn_lt (n : ℕ) (hn : n < p.natDegree) : ‖p.coeff n‖ < f x ^ (p.natDegree - n) := by
      have hexp : (‖p.coeff n‖ ^ (1 / (p.natDegree - n : ℝ))) ^ (p.natDegree - n) =
          ‖p.coeff n‖ := by
        rw [← rpow_natCast, ← rpow_mul (norm_nonneg _), mul_comm, rpow_mul (norm_nonneg _),
          rpow_natCast, ← cast_sub (le_of_lt hn), one_div,
          pow_rpow_inv_natCast (norm_nonneg _) (_root_.ne_of_gt (tsub_pos_of_lt hn))]
      have h_base : ‖p.coeff n‖ ^ (1 / (p.natDegree - n : ℝ)) < f x := by
        rw [spectralValue, iSup, not_le, Set.Finite.csSup_lt_iff (spectralValueTerms_finite_range p)
          (Set.range_nonempty (spectralValueTerms p))] at h_ge
        have h_rg : ‖p.coeff n‖ ^ (1 / (p.natDegree - n : ℝ)) ∈
          Set.range (spectralValueTerms p) := by use n; simp only [spectralValueTerms, if_pos hn]
        exact h_ge (‖p.coeff n‖₊ ^ (1 / (p.natDegree - n : ℝ))) h_rg
      rw [← hexp, ← rpow_natCast, ← rpow_natCast]
      gcongr
      exact cast_pos.mpr (tsub_pos_of_lt hn)
    have h_deg : 0 < p.natDegree := natDegree_pos_of_monic_of_aeval_eq_zero hp hx
    have h_lt : f ((Finset.range p.natDegree).sum fun i : ℕ ↦ p.coeff i • x ^ i) <
        f (x ^ p.natDegree) := by
      have hn' (n : ℕ) (hn : n < p.natDegree) : f (p.coeff n • x ^ n) < f (x ^ p.natDegree) := by
        by_cases hn0 : n = 0
        · rw [hn0, pow_zero, map_smul_eq_mul, hf_pm _ (succ_le_iff.mpr h_deg),
            ← Nat.sub_zero p.natDegree, ← hn0]
          exact (mul_le_of_le_one_right (norm_nonneg _) hf_pm.map_one_le_one).trans_lt (hn_lt n hn)
        · have : p.natDegree = p.natDegree - n + n := by rw [Nat.sub_add_cancel (le_of_lt hn)]
          rw [map_smul_eq_mul, hf_pm _ (succ_le_iff.mp (pos_iff_ne_zero.mpr hn0)),
            hf_pm _ (succ_le_iff.mpr h_deg), this, pow_add]
          gcongr
          exact hn_lt n hn
      set g := fun i : ℕ ↦ p.coeff i • x ^ i
      obtain ⟨m, hm_in, hm⟩ : ∃ (m : ℕ) (_ : 0 < p.natDegree → m < p.natDegree),
          f ((Finset.range p.natDegree).sum g) ≤ f (g m) := by
        obtain ⟨m, hm, h⟩ := IsNonarchimedean.finset_image_add (map_zero _) (apply_nonneg _) hf_na g
          (Finset.range p.natDegree)
        rw [Finset.nonempty_range_iff, ← zero_lt_iff, Finset.mem_range] at hm
        exact ⟨m, hm, h⟩
      exact lt_of_le_of_lt hm (hn' m (hm_in h_deg))
    have h0 : f 0 ≠ 0 := by
      have h_eq : f 0 = f (x ^ p.natDegree) := by
        rw [← hx, aeval_eq_sum_range, Finset.sum_range_succ, add_comm, hp.coeff_natDegree,
          one_smul, ← max_eq_left_of_lt h_lt]
        exact IsNonarchimedean.add_eq_max_of_ne hf_na (ne_of_gt h_lt)
      exact h_eq ▸ ne_of_gt (lt_of_le_of_lt (apply_nonneg _ _) h_lt)
    exact h0 (map_zero _)

Goal state after:
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝³ : SeminormedAddCommGroup G
H : Type u_2
inst✝² : SeminormedAddCommGroup H
inst✝¹ : T0Space H
inst✝ : CompleteSpace H
f : NormedAddGroupHom G H
g : NormedAddGroupHom (Completion G) H
hg : ∀ (v : G), f v = g ↑v
v : Completion G
⊢ f.extension v = g v

Tactic:
rw [NormedAddGroupHom.extension_coe_to_fun,
    Completion.extension_unique f.uniformContinuous g.uniformContinuous fun a => hg a]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: V W : SemiNormedGrp
f g✝ : V ⟶ W
this : ∀ (c : Fork f g✝), (NormedAddGroupHom.compHom (Hom.hom (f - g✝))) (Hom.hom c.ι) = 0
c : Fork f g✝
g : c.pt ⟶ (fork f g✝).pt
h : g ≫ (fork f g✝).ι = c.ι
⊢ g = ofHom (NormedAddGroupHom.ker.lift (Hom.hom c.ι) (Hom.hom (f - g✝)) ⋯)

Tactic:
by ext x; dsimp; simp_rw [← h]; rfl

Goal state after: