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

Goal state before:
pp: 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
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).Equilateral ↔ s.Equilateral

Tactic:
refine ⟨fun ⟨r, hr⟩ ↦ ⟨r, fun i j hij ↦ ?_⟩, fun ⟨r, hr⟩ ↦ ⟨r, fun i j hij ↦ ?_⟩⟩

Goal state after:
case refine_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
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
x✝ : (s.reindex e).Equilateral
r : ℝ
hr : ∀ (i j : Fin (n + 1)), i ≠ j → dist ((s.reindex e).points i) ((s.reindex e).points j) = r
i j : Fin (m + 1)
hij : i ≠ j
⊢ dist (s.points i) (s.points j) = r

case refine_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
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
x✝ : s.Equilateral
r : ℝ
hr : ∀ (i j : Fin (m + 1)), i ≠ j → dist (s.points i) (s.points j) = r
i j : Fin (n + 1)
hij : i ≠ j
⊢ dist ((s.reindex e).points i) ((s.reindex e).points j) = r
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
n : ℕ
⊢ (↑(n + 1).factorial)⁻¹ • (n + 1) • x.fst ^ (n + 1).pred • x.snd = ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd

Tactic:
← Nat.cast_smul_eq_nsmul 𝕜 (n + 1),

Goal state after:
𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
n : ℕ
⊢ (↑(n + 1).factorial)⁻¹ • ↑(n + 1) • x.fst ^ (n + 1).pred • x.snd = ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! =
    ((z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! + ((-z) ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) / 2 +
      ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! + ((-z) ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)!) /
        2

Tactic:
neg_sq,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! =
    ((z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! + (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) / 2 +
      ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! + (z ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)!) / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    (((-z) ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! - (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) * I / 2 +
      (((-z) ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)! - (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!) *
          I /
        2

Tactic:
neg_sq,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    ((z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! - (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) * I / 2 +
      ((z ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)! - (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!) * I /
        2
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case refine_2.inr.inr.inr.inr.inl
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
i₁ : Fin 3
i j : Fin (2 + 1)
hij : i ≠ j
h₁₂ : i₁ ≠ i
h₁₃ : i₁ ≠ j
h₂₃ : i ≠ j
h₁ : dist (t.points i₁) (t.points i) = dist (t.points i₁) (t.points j)
h₂ : dist (t.points i₁) (t.points i) = dist (t.points i) (t.points j)
⊢ dist (t.points i) (t.points j) = dist (t.points i₁) (t.points i)

Tactic:
rw [h₂, dist_comm]

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

Goal state before:
pp: case refine_2.inr.inr.inr.inr.inr
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
i₁ : Fin 3
i j : Fin (2 + 1)
hij : i ≠ j
h₁₃ : i₁ ≠ i
h₁₂ : i₁ ≠ j
h₂₃ : j ≠ i
h₁ : dist (t.points i₁) (t.points j) = dist (t.points i₁) (t.points i)
h₂ : dist (t.points i₁) (t.points j) = dist (t.points j) (t.points i)
⊢ dist (t.points i) (t.points j) = dist (t.points i₁) (t.points j)

Tactic:
rw [h₂, dist_comm]

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

Goal state before:
pp: q : ℍ
hq : q.re = 0
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • q ^ n) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)

Tactic:
← expSeries_apply_eq

Goal state after:
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n ↦ (expSeries ℝ ℍ n) fun x ↦ q) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n m : ℕ
hm_lt : m < n + 1
hm : p ((x + y) ^ n) ≤ p (x ^ m) * p (y ^ (n - m))
⊢ p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)

Tactic:
by gcongr

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
hlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))
n : ℕ
⊢ f (c * c ^ n * x) * (f c)⁻¹ ^ n = f (c * c ^ n * x) * f c * (f c)⁻¹ * (f c)⁻¹ ^ n

Tactic:
mul_assoc _ (f c),

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
hlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))
n : ℕ
⊢ f (c * c ^ n * x) * (f c)⁻¹ ^ n = f (c * c ^ n * x) * (f c * (f c)⁻¹) * (f c)⁻¹ ^ n
Module: Mathlib.Analysis.Normed.Field.Instances

Goal state before:
pp: F : Type u_1
inst✝ : NormedField F
f : Filter F
hc : Cauchy f
hn : nhds 0 ⊓ f = ⊥
⊢ Cauchy (map (fun x ↦ x⁻¹) f)

Tactic:
obtain ⟨δ, δ_pos, hδ⟩ := (disjoint_nhds_zero ..).mp <| disjoint_iff.mpr hn

Goal state after:
F : Type u_1
inst✝ : NormedField F
f : Filter F
hc : Cauchy f
hn : nhds 0 ⊓ f = ⊥
δ : ℝ
δ_pos : δ > 0
hδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖
⊢ Cauchy (map (fun x ↦ x⁻¹) f)
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‖
⊢ Tendsto f (cocompact E) (cocompact F)

Tactic:
rw [tendsto_def]

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 ∈ cocompact F, f ⁻¹' s ∈ cocompact E
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: 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))) ^ (1 / ↑n) * μ (x ^ (n % ↑m1)) ^ (1 / ↑n) ≤
    (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n) * μ (x ^ (n % ↑m1)) ^ (1 / ↑n)

Tactic:
by gcongr

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

Goal state before:
pp: 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)) ^ (1 / ↑n) * μ (x ^ (n % ↑m1)) ^ (1 / ↑n) <
    (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * μ (x ^ (n % ↑m1)) ^ (1 / ↑n)

Tactic:
by gcongr

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

Goal state before:
pp: 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))
⊢ (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * μ (x ^ (n % ↑m1)) ^ (1 / ↑n) ≤
    (L + ε / 2) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n)

Tactic:
by gcongr

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : μ x < 1
b : ℕ
a✝ : 0 ≤ b
⊢ μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ 1

Tactic:
exact rpow_le_one (apply_nonneg _ _) (le_of_lt hμx)
          (mul_nonneg (cast_nonneg _) (one_div_nonneg.mpr (cast_nonneg _)))

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
⊢ {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a} ⊆
    {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop, n ≤ a}

Tactic:
intro b hb

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
b : ℝ
hb : b ∈ {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a}
⊢ b ∈ {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop, n ≤ a}
Module: Mathlib.Analysis.Normed.Group.ControlledClosure

Goal state before:
pp: G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i ↦ (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n ↦ ∑ k ∈ range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖-v 0 + h‖ < b 0
hv : ∀ n > 0, ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n ↦ ∑ k ∈ range (n + 1), u k
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
this : ‖v 0‖ ≤ ‖h‖ + b 0
⊢ C * ‖v 0‖ ≤ C * (‖h‖ + b 0)

Tactic:
by gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: 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
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
δpos : δ > 0
hε : (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1) > 0
g : K[X]
gmon : g.Monic
gdeg : f.natDegree = g.natDegree
gcoeff : ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1)
b : f.SplittingField
hab✝ :
  ‖a - b‖ < ((↑f.natDegree + 1) * ((δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1))) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
hab : ‖a - b‖ < δ
bbot : b ∈ ⊥
hb : ¬g = 0 ∧ (aeval b) (map (algebraMap K L) g) = 0
⊢ ∃ x, eval x f = 0

Tactic:
have abot : a ∈ (⊥ : IntermediateField L F) := by
    have masp : ((minpoly L a).map (algebraMap L F)).Splits := by
      simpa [minpoly.eq_of_irreducible_of_monic firr fa0 fmon] using
        (Polynomial.SplittingField.splits f)
    simpa [IntermediateField.adjoin_simple_eq_bot_iff.mpr bbot] using
      IsKrasner.krasner (minpoly.irreducible ⟨f, fmon, fa0⟩).separable
        masp ⟨(g.map _), gmon.map _, hb.2⟩ fun a' h1 h2 ↦ lt_of_lt_of_le hab (norm_sub_le a' h1 h2)

Goal state after:
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
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
δpos : δ > 0
hε : (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1) > 0
g : K[X]
gmon : g.Monic
gdeg : f.natDegree = g.natDegree
gcoeff : ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1)
b : f.SplittingField
hab✝ :
  ‖a - b‖ < ((↑f.natDegree + 1) * ((δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1))) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
hab : ‖a - b‖ < δ
bbot : b ∈ ⊥
hb : ¬g = 0 ∧ (aeval b) (map (algebraMap K L) g) = 0
abot : a ∈ ⊥
⊢ ∃ x, eval x f = 0
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
⊢ ∃ b ∈ ↑(toCompl.comp (incl f.ker)).range, ‖hatg - b‖ < ε

Tactic:
obtain ⟨_, ⟨g : G, rfl⟩, hg : ‖hatg - g‖ < δ⟩ :=
    SeminormedAddCommGroup.mem_closure_iff.mp (Completion.isDenseInducing_coe.dense hatg) δ δ_pos

Goal state after:
G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
⊢ ∃ b ∈ ↑(toCompl.comp (incl f.ker)).range, ‖hatg - b‖ < ε
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
mem_ker : g - g' ∈ f.ker
⊢ ‖f‖ * ‖hatg - ↑g‖ ≤ ‖f‖ * δ

Tactic:
by gcongr

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

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
mem_ker : g - g' ∈ f.ker
this : ‖f g‖ ≤ ‖f‖ * δ
⊢ δ + C' * ‖f g‖ ≤ δ + C' * (‖f‖ * δ)

Tactic:
by gcongr

Goal state after:
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
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ↑‖x‖₊
hf_iso : ∀ (σ : Gal(L/K)) (x : L), f x = f (σ x)
x : L
h_sup : ⨆ σ, f (σ x) = f x
⊢ f x = ⨆ σ, f (σ x)

Tactic:
h_sup

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ↑‖x‖₊
hf_iso : ∀ (σ : Gal(L/K)) (x : L), f x = f (σ x)
x : L
h_sup : ⨆ σ, f (σ x) = f x
⊢ f x = f x
Module: Mathlib.Analysis.Normed.Group.SeparationQuotient

Goal state before:
pp: case mp
M : Type u_1
inst✝ : SeminormedAddCommGroup M
h : normedMk = 0
⊢ ∀ (x : M), ‖x‖ = 0

Tactic:
x

Goal state after:
case mp
M : Type u_1
inst✝ : SeminormedAddCommGroup M
h : normedMk = 0
x : M
⊢ ‖x‖ = 0
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Tactic:
have h1 : ∑' (k : (Tᶜ : Set β)), bound k < ε / 3 := by
    calc _ ≤ ‖∑' (k : (Tᶜ : Set β)), bound k‖ := Real.le_norm_self _
         _ = ‖S - ∑ b ∈ T, bound b‖           := congrArg _ ?_
         _ < ε / 3                            := by rwa [dist_eq_norm, norm_sub_rev] at hT
    simpa only [h_sum.sum_add_tsum_compl, eq_sub_iff_add_eq'] using hS.tsum_eq

Goal state after:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
S : ℝ
hS : HasSum bound S
T : Finset β
hT : dist (∑ b ∈ T, bound b) S < ε / 3
h1 : ∑' (k : ↑(↑T)ᶜ), bound ↑k < ε / 3
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
f : ι → ℝ
hf : Summable f
i : ι
hi : -1 ≤ f i
⊢ Complex.log (1 + ↑(f i)) = ↑(log (1 + f i))

Tactic:
ofReal_log,

Goal state after:
ι : Type u_1
f : ι → ℝ
hf : Summable f
i : ι
hi : -1 ≤ f i
⊢ Complex.log (1 + ↑(f i)) = Complex.log ↑(1 + f i)

ι : Type u_1
f : ι → ℝ
hf : Summable f
i : ι
hi : -1 ≤ f i
⊢ 0 ≤ 1 + f i
Module: Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn

Goal state before:
pp: α : Type u_1
ι : Type u_2
K : Set α
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
hf : TendstoUniformlyOn f g p K
hg : BddAbove ((fun x ↦ (g x).re) '' K)
⊢ ∀ a ∈ upperBounds ((fun x ↦ (g x).re) '' K), ∀ x ∈ K, (g x).re ≤ a

Tactic:
by simp [mem_upperBounds]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn

Goal state before:
pp: α : Type u_1
ι : Type u_2
K : Set α
u : ι → ℝ
f : ι → α → ℂ
hu : Summable u
h : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i
i : ι
hi : ∀ x ∈ K, ‖f i x‖ ≤ u i
hi' : u i ≤ 1 / 2
⊢ ∀ x ∈ K, ‖log (1 + f i x)‖ ≤ 3 / 2 * u i

Tactic:
x

Goal state after:
α : Type u_1
ι : Type u_2
K : Set α
u : ι → ℝ
f : ι → α → ℂ
hu : Summable u
h : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i
i : ι
hi : ∀ x ∈ K, ‖f i x‖ ≤ u i
hi' : u i ≤ 1 / 2
x : α
⊢ x ∈ K → ‖log (1 + f i x)‖ ≤ 3 / 2 * u i