mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k m : Fin c.length
⊢ Fin (update c.partSize k (c.partSize k + 1) m) → Fin (n + 1)

Tactic:
by_cases h : m = k

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k m : Fin c.length
h : m = k
⊢ Fin (update c.partSize k (c.partSize k + 1) m) → Fin (n + 1)

case neg
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k m : Fin c.length
h : ¬m = k
⊢ Fin (update c.partSize k (c.partSize k + 1) m) → Fin (n + 1)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
x : Fin n
i : Fin c.length
j : Fin (c.partSize i)
hij : c.emb i j = x.succ
A : c.length = c.length - 1 + 1
⊢ ∃ m y, (c.emb (Fin.cast ⋯ m.succ) y).pred ⋯ = x

Tactic:
have i_ne : i ≠ 0 := by
      intro h
      have : succ x ∈ range (c.emb i) := by rw [← hij]; apply mem_range_self
      rw [h, hc, mem_singleton_iff] at this
      exact Fin.succ_ne_zero _ this

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
x : Fin n
i : Fin c.length
j : Fin (c.partSize i)
hij : c.emb i j = x.succ
A : c.length = c.length - 1 + 1
i_ne : i ≠ 0
⊢ ∃ m y, (c.emb (Fin.cast ⋯ m.succ) y).pred ⋯ = x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
a b : Fin (update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) (c.index 0))
hab : a < b
⊢ a.succ < b.succ

Tactic:
simpa using hab

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
h : ContDiffAt 𝕜 n f x
⊢ ContinuousAt f x

Tactic:
simpa [continuousWithinAt_univ] using h.continuousWithinAt

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

Goal state before:
pp: e : ℂ → ℂ
e' : ℂ
z : ℝ
h : HasStrictDerivAt e e' ↑z
⊢ HasStrictDerivAt (fun x ↦ (e ↑x).re) e'.re z

Tactic:
by
  have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasStrictFDerivAt
  have B :
    HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
      (ofRealCLM z) :=
    h.hasStrictFDerivAt.restrictScalars ℝ
  have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasStrictFDerivAt
  simpa using (C.comp z (B.comp z A)).hasStrictDerivAt

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

Goal state before:
pp: case inr
𝔸 : Type u_2
inst✝³ : Ring 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : T2Space 𝔸
x y : 𝔸
h : Commute x y
val✝ : Algebra ℚ 𝔸
⊢ Commute x (exp y)

Tactic:
rw [exp_eq_tsum ℚ]
    exact Commute.tsum_right x fun n => (h.pow_right n).smul_right _

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

Goal state before:
pp: 𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x a b : 𝔸
h : SemiconjBy x a b
⊢ ∀ (i : ℕ), SemiconjBy x ((↑i !)⁻¹ • a ^ i) ((↑i !)⁻¹ • b ^ i)

Tactic:
exact fun _ ↦ h.pow_right _ |>.smul_right _

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
m k : ℕ
⊢ iteratedDeriv k (fun x ↦ x ^ m) x = ↑(m.descFactorial k) * x ^ (m - k)

Tactic:
by
  simpa using iteratedDerivWithin_pow (Set.mem_univ x) uniqueDiffOn_univ m k

Goal state after:
Module: Mathlib.Topology.Order.Rolle

Goal state before:
pp: case neg
X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).Nonempty
c : X
cmem : c ∈ Icc a b
cle : ∀ x ∈ Icc a b, f c ≤ f x
C : X
Cmem : C ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

Tactic:
· refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩
      exacts [fun h => by rw [h], fun h => by rw [h, hfI]]

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Rolle

Goal state before:
pp: case pos
f : ℝ → ℝ
a b l : ℝ
hab : a < b
hfa : Tendsto f (𝓝[>] a) (𝓝 l)
hfb : Tendsto f (𝓝[<] b) (𝓝 l)
h : ∀ x ∈ Ioo a b, DifferentiableAt ℝ f x
⊢ ∃ c ∈ Ioo a b, deriv f c = 0

Tactic:
exact exists_hasDerivAt_eq_zero' hab hfa hfb fun x hx => (h x hx).hasDerivAt

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Rolle

Goal state before:
pp: case pos
f : ℝ → ℝ
a b l : ℝ
hab : a < b
hfa : Tendsto f (𝓝[>] a) (𝓝 l)
hfb : Tendsto f (𝓝[<] b) (𝓝 l)
h : ∀ x ∈ Ioo a b, DifferentiableAt ℝ f x
⊢ ∃ c ∈ Ioo a b, deriv f c = 0

pp: case neg
f : ℝ → ℝ
a b l : ℝ
hab : a < b
hfa : Tendsto f (𝓝[>] a) (𝓝 l)
hfb : Tendsto f (𝓝[<] b) (𝓝 l)
h : ∃ x ∈ Ioo a b, ¬DifferentiableAt ℝ f x
⊢ ∃ c ∈ Ioo a b, deriv f c = 0

Tactic:
· exact exists_hasDerivAt_eq_zero' hab hfa hfb fun x hx => (h x hx).hasDerivAt

Goal state after:
case neg
f : ℝ → ℝ
a b l : ℝ
hab : a < b
hfa : Tendsto f (𝓝[>] a) (𝓝 l)
hfb : Tendsto f (𝓝[<] b) (𝓝 l)
h : ∃ x ∈ Ioo a b, ¬DifferentiableAt ℝ f x
⊢ ∃ c ∈ Ioo a b, deriv f c = 0
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
⊢ -a < -b

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : -a < x ∧ x < -b
⊢ b < -x

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : -a < x ∧ x < -b
⊢ -x < a

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : -a < x ∧ x < -b
h₁ : -x ∈ Ioo b a
h₂ : deriv f (-x) ≤ -1
⊢ deriv f (-x) < 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
⊢ r' - s > 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁✝ C₂✝ : ConvexCone R M
x : M
C₁ C₂ : ConvexCone R M
⊢ ∀ ⦃x : M⦄, x ∈ ↑C₁ + ↑C₂ → ∀ ⦃y : M⦄, y ∈ ↑C₁ + ↑C₂ → x + y ∈ ↑C₁ + ↑C₂

Tactic:
rintro _ ⟨x₁, hx₁, x₂, hx₂, rfl⟩ y ⟨y₁, hy₁, y₂, hy₂, rfl⟩

Goal state after:
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁✝ C₂✝ : ConvexCone R M
x : M
C₁ C₂ : ConvexCone R M
x₁ : M
hx₁ : x₁ ∈ ↑C₁
x₂ : M
hx₂ : x₂ ∈ ↑C₂
y₁ : M
hy₁ : y₁ ∈ ↑C₁
y₂ : M
hy₂ : y₂ ∈ ↑C₂
⊢ (fun x1 x2 ↦ x1 + x2) x₁ x₂ + (fun x1 x2 ↦ x1 + x2) y₁ y₂ ∈ ↑C₁ + ↑C₂
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
y : ℝ
hy : y ∈ Set.Ioo (-1) 1
this : HasDerivAt F (∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)) y
⊢ F' y = ∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)

Tactic:
calc
      -y ^ n / (1 - y) = ∑ i ∈ Finset.range n, y ^ i + -1 / (1 - y) := by
        simp [field, geom_sum_eq hy.2.ne, sub_ne_zero.2 hy.2.ne, sub_ne_zero.2 hy.2.ne']
        ring
      _ = ∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y) := by
        congr with i
        rw [Nat.cast_succ, mul_div_cancel_left₀ _ (Nat.cast_add_one_pos i).ne']

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

Goal state before:
pp: a : ℝ
h : 0 < a
⊢ 0 < 2 * a + 1

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
h : ∀ (i : ℝ), a < i → x ∈ i • s
ε : ℝ
hε : 0 < ε
⊢ 0 ≤ a + ε / 2

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
h : ∀ (i : ℝ), a < i → x ∈ i • s
ε : ℝ
hε : 0 < ε
⊢ a < a + ε / 2

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
hs : Convex ℝ s
absorbs : Absorbent ℝ s
ε : ℝ
hε : 0 < ε
a : ℝ
ha : 0 < a
x : E
hx : x ∈ s
ha' : a < gauge s ((fun x ↦ a • x) x) + ε / 2
b : ℝ
hb : 0 < b
y : E
hy : y ∈ s
hb' : b < gauge s ((fun x ↦ b • x) y) + ε / 2
⊢ a + b < gauge s (a • x) + gauge s (b • y) + ε

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case refine_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
y : ℝ
hx : y • x₀ ∈ f.domain
⊢ (RingHom.id ℝ) y • 1 ≤ gauge s ↑⟨y • x₀, hx⟩

Tactic:
simp only [mul_one, smul_eq_mul]

Goal state after:
case refine_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
s : Set E
hs₀ : 0 ∈ s
hs₁ : Convex ℝ s
hs₂ : IsOpen s
x₀ : E
hx₀ : x₀ ∉ s
f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯
y : ℝ
hx : y • x₀ ∈ f.domain
⊢ (RingHom.id ℝ) y ≤ gauge s (y • x₀)
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case inr.inr.refine_1.refine_2
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
f : StrongDual ℝ E
hf₁ : f x₀ = 1
hf₂ : ∀ x ∈ C, f x < 1
forall_le : ∀ a ∈ s, ∀ b ∈ t, f a ≤ f b
⊢ f ≠ 0

Tactic:
rintro rfl
      simp at hf₁

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
hs : s.Nonempty
_ht : t.Nonempty
U V : Set E
hU : IsOpen U
hV : IsOpen V
hU₁ : Convex ℝ U
hV₁ : Convex ℝ V
sU : s ⊆ U
tV : t ⊆ V
disj' : Disjoint U V
f : StrongDual ℝ E
u : ℝ
hf₁ : ∀ a ∈ U, f a < u
hf₂ : ∀ b ∈ V, u < f b
x : E
hx₁ : x ∈ s
hx₂ : IsMaxOn (⇑f) s x
this : f x < u
⊢ (f x + u) / 2 < u

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
r s ε x : ℝ
hx : x ∈ B f K r s ε
L : F
LK : L ∈ K
hL₁ : x ∈ A f L r ε
hL₂ : x ∈ A f L s ε
⊢ B f K r s ε ∈ 𝓝[>] x

Tactic:
filter_upwards [A_mem_nhdsGT hL₁, A_mem_nhdsGT hL₂] with y hy₁ hy₂

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
r s ε x : ℝ
hx : x ∈ B f K r s ε
L : F
LK : L ∈ K
hL₁ : x ∈ A f L r ε
hL₂ : x ∈ A f L s ε
y : ℝ
hy₁ : y ∈ A f L r ε
hy₂ : y ∈ A f L s ε
⊢ y ∈ B f K r s ε
Module: Mathlib.Analysis.Normed.Operator.CompleteCodomain

Goal state before:
pp: case intro
𝕜 : Type u_1
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
ι : Type u_4
inst✝⁴ : Finite ι
M : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (M i)
inst✝² : (i : ι) → NormedSpace 𝕜 (M i)
inst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i)
inst✝ : CompleteSpace (ContinuousMultilinearMap 𝕜 M F)
m : (i : ι) → M i
hm : ∀ (i : ι), m i ≠ 0
f : ℕ → F
hf : CauchySeq f
φ : (i : ι) → StrongDual 𝕜 (M i)
hφ : ∀ (i : ι), (φ i) (m i) = 1
val✝ : Fintype ι
g : ℕ → ContinuousMultilinearMap 𝕜 M F :=
  fun n ↦
    (compContinuousLinearMapL φ) (((smulRightL 𝕜 (fun x ↦ 𝕜) F) (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι 𝕜)) (f n))
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
have : CauchySeq g := by
    refine (ContinuousLinearMap.lipschitz _).cauchySeq_comp ?_
    exact (ContinuousLinearMap.lipschitz _).cauchySeq_comp hf

Goal state after:
case intro
𝕜 : Type u_1
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
ι : Type u_4
inst✝⁴ : Finite ι
M : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (M i)
inst✝² : (i : ι) → NormedSpace 𝕜 (M i)
inst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i)
inst✝ : CompleteSpace (ContinuousMultilinearMap 𝕜 M F)
m : (i : ι) → M i
hm : ∀ (i : ι), m i ≠ 0
f : ℕ → F
hf : CauchySeq f
φ : (i : ι) → StrongDual 𝕜 (M i)
hφ : ∀ (i : ι), (φ i) (m i) = 1
val✝ : Fintype ι
g : ℕ → ContinuousMultilinearMap 𝕜 M F :=
  fun n ↦
    (compContinuousLinearMapL φ) (((smulRightL 𝕜 (fun x ↦ 𝕜) F) (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι 𝕜)) (f n))
this : CauchySeq g
⊢ ∃ a, Tendsto f atTop (𝓝 a)
Module: Mathlib.Topology.ContinuousMap.ContinuousMapZero

Goal state before:
pp: X : Type u_1
Y : Type u_2
R : Type u_3
inst✝⁵ : Zero X
inst✝⁴ : Zero Y
inst✝³ : Zero R
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace R
g : C(Y, R)₀
⊢ Continuous g.comp

Tactic:
rw [ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff]
  exact g.toContinuousMap.continuous_postcomp |>.comp <|
    ContinuousMapZero.isEmbedding_toContinuousMap.continuous

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b c : ℝ
μ : Measure ℝ
inst✝ : PseudoMetrizableSpace ε
h : b ∈ [[a, c]]
⊢ IntervalIntegrable f μ a c ↔ IntervalIntegrable f μ a b ∧ IntervalIntegrable f μ b c

Tactic:
simp only [intervalIntegrable_iff, ← integrableOn_union, uIoc_union_uIoc h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
μ : Measure ℝ
h : IntervalIntegrable f μ a b
hab : ¬a ≤ b
⊢ b ≤ a

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
⊢ 0 < (s * r - b) / 3

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
⊢ b + 2 * ((s * r - b) / 3) < s * r

Tactic:
by linarith

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ IntervalIntegrable f volume 0 t

Tactic:
rw [IntervalIntegrable.iff_comp_neg ht]

Goal state after:
case inl
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ IntervalIntegrable (fun x ↦ f (-x)) volume (-0) (-t)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
c d : ℝ
⊢ c⁻¹ • ∫ (x : ℝ) in a..b, f (d - x / c) = ∫ (x : ℝ) in d - b / c..d - a / c, f x

Tactic:
by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_div]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : LocallyCompactSpace E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E → F
r s : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
L : E →L[𝕜] F
this✝ : ProperSpace E
a : α
x : E
r' : ℝ
hr :
  ∀ (y : E),
    dist y (a, x).2 < r' → ∀ (z : E), dist z (a, x).2 < r' → ‖f (a, x).1 z - f (a, x).1 y - (L z - L y)‖ < s * r
Irr' : r / 2 < r'
Ir'r : r' ≤ r
t : ℝ
hrt : r / 2 < t
htr' : t < r'
t' : ℝ
hrt' : r / 2 < t'
ht't : t' < t
b : ℝ
b_lt : b < s * r
hb : ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t, ‖f a z - f a y - (L z - L y)‖ ≤ b
ε : ℝ
εpos : 0 < ε
hε : b + 2 * ε < s * r
u : Set α
u_open : IsOpen[inst✝] u
au : a ∈ u
hu : ∀ (p : α × E), p.1 ∈ u → p.2 ∈ closedBall x t → dist (Function.uncurry f p) (Function.uncurry f (a, p.2)) < ε
this : u ×ˢ ball x (t - t') ∈ 𝓝 (a, x)
a' : α
x' : E
ha'x' : a' ∈ u ∧ dist x' x < t - t'
y : E
hy : dist y (a', x').2 < t'
z : E
hz : dist z (a', x').2 < t'
dyx : dist y x ≤ t
dzx : dist z x ≤ t
⊢ ε + ε + b < s * r

Tactic:
by linarith

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in d..b, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ

Tactic:
rw [integral_interval_sub_interval_comm hab hcd hac, integral_symm b d, integral_symm a c,
    sub_neg_eq_add, sub_eq_neg_add]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntervalIntegrable f μ a b
hb : IntegrableOn f (Ioi b) μ
⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ) in Ioi a, f x ∂μ

Tactic:
rw [integral_interval_add_Ioi _ hb]

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntervalIntegrable f μ a b
hb : IntegrableOn f (Ioi b) μ
⊢ IntegrableOn f (Ioi a) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntervalIntegrable f μ a b
hb : IntegrableOn f (Ioi b) μ
⊢ IntegrableOn f (Ioi a) μ

Tactic:
by_cases! h : a ≤ b

Goal state after:
case pos
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntervalIntegrable f μ a b
hb : IntegrableOn f (Ioi b) μ
h : a ≤ b
⊢ IntegrableOn f (Ioi a) μ

case neg
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ha : IntervalIntegrable f μ a b
hb : IntegrableOn f (Ioi b) μ
h : b < a
⊢ IntegrableOn f (Ioi a) μ