mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet

Tactic:
by_cases he : Disjoint (Iio d) (Ioi c)

Goal state after:
case pos
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : Disjoint (Iio d) (Ioi c)
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet

case neg
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : ¬Disjoint (Iio d) (Ioi c)
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e : Trivialization F proj
x : Z
B' : Type u_5
inst✝ : TopologicalSpace B'
h : B ≃ₜ B'
x✝ : Z
⊢ x✝ ∈ (e.transHomeomorph (h.prodCongr (Homeomorph.refl F))).source ↔ x✝ ∈ ⇑h ∘ proj ⁻¹' ⇑h.symm ⁻¹' e.baseSet

Tactic:
simp [e.mem_source]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
⊢ ∀ (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C

Tactic:
C

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
⊢ (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
b : A
⊢ ‖a.toProd.2 b‖₊ ^ 2 = ‖a.toProd.2 b * star (a.toProd.2 b)‖₊

Tactic:
by
        simpa only [← sq] using CStarRing.nnnorm_self_mul_star.symm

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
I : Type u_3
inst✝¹ : TopologicalSpace I
inst✝ : DiscreteTopology I
x : X
t : Trivialization I f
hx : x ∈ t.baseSet
xi : ↑t.baseSet × I
⊢ t.invFun (↑xi.1, xi.2) ∈ f ⁻¹' t.baseSet

Tactic:
rw [Set.mem_preimage, ← t.mem_source]

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
I : Type u_3
inst✝¹ : TopologicalSpace I
inst✝ : DiscreteTopology I
x : X
t : Trivialization I f
hx : x ∈ t.baseSet
xi : ↑t.baseSet × I
⊢ t.invFun (↑xi.1, xi.2) ∈ t.source
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
g : G
⊢ (hf.fiberEquivGroup e) e' = g ↔ ↑e' = g • ↑e

Tactic:
rw [fiberEquivGroup, Equiv.symm_apply_eq, Equiv.ofBijective_apply, Subtype.mk.injEq]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x : ℝ
h₁ : -π < x
h₂ : x ≤ π
⊢ (↑(exp x)).arg = x

Tactic:
by
  rw [coe_exp, exp_mul_I, arg_cos_add_sin_mul_I ⟨h₁, h₂⟩]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case h
x✝ : ℝ
⊢ Circle.exp (-x✝) = (Circle.exp x✝)⁻¹

Tactic:
Circle.exp_neg

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
⊢ spectrum ℂ u ⊆ slitPlane ↔ ‖u - 1‖ < 2

Tactic:
by
  simp [subset_slitPlane_iff_of_subset_sphere (spectrum.subset_circle_of_unitary hu),
    norm_sub_one_lt_two_iff hu]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
t : Set (Fin (n + 1))
ht : t ≠ Set.univ
hssubset : t ⊂ Set.univ
i : Fin (n + 1)
hi : i ∈ Set.univ ∧ i ∉ t
w : Fin (n + 1) → k := centroidWeights k univ
h : (affineCombination k univ s.points) w ∈ affineSpan k (s.points '' t)
wdef : w = centroidWeights k univ
hw : ∑ i, w i = 1
h1 : w i = 0
h2 : w i = 1 / (↑n + 1)
⊢ False

Tactic:
simp only [h2, one_div, inv_eq_zero] at h1

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
t : Set (Fin (n + 1))
ht : t ≠ Set.univ
hssubset : t ⊂ Set.univ
i : Fin (n + 1)
hi : i ∈ Set.univ ∧ i ∉ t
w : Fin (n + 1) → k := centroidWeights k univ
h : (affineCombination k univ s.points) w ∈ affineSpan k (s.points '' t)
wdef : w = centroidWeights k univ
hw : ∑ i, w i = 1
h2 : w i = 1 / (↑n + 1)
h1 : ↑n + 1 = 0
⊢ False
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
⊢ (∀ (a : H →L[𝕜] H), a ∘SL ↑((↑f).trans (↑g).symm) = ↑((↑f).trans (↑g).symm) ∘SL a) ↔ ∃ α, ∀ (x : H), f x = (α • g) x

Tactic:
← ContinuousLinearMap.mul_def,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
⊢ (∀ (a : H →L[𝕜] H), a * ↑((↑f).trans (↑g).symm) = ↑((↑f).trans (↑g).symm) * a) ↔ ∃ α, ∀ (x : H), f x = (α • g) x
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
n : ℕ
hc : #Finset.univ = n + 1
i₁ : ι
⊢ finrank k ↥(Submodule.span k (Set.range fun i ↦ p ↑i -ᵥ p i₁)) = Fintype.card { x // x ≠ i₁ } ↔
    finrank k ↥(Submodule.span k (Set.range fun i ↦ p ↑i -ᵥ p i₁)) = n

Tactic:
rw [Fintype.subtype_card]

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
n : ℕ
hc : #Finset.univ = n + 1
i₁ : ι
⊢ finrank k ↥(Submodule.span k (Set.range fun i ↦ p ↑i -ᵥ p i₁)) = #{x | x ≠ i₁} ↔
    finrank k ↥(Submodule.span k (Set.range fun i ↦ p ↑i -ᵥ p i₁)) = n
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
v₀ : V
hv : ∀ v ∈ vectorSpan k s, ∃ r, r • v₀ = v
⊢ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀

Tactic:
use v₀

Goal state after:
case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
v₀ : V
hv : ∀ v ∈ vectorSpan k s, ∃ r, r • v₀ = v
⊢ ∀ p ∈ s, ∃ r, p = r • v₀ +ᵥ p₀
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace ℝ X
inst✝ : HasContDiffBump E
n : ℕ∞
c g : X → E
f : (x : X) → ContDiffBump (c x)
hc : ContDiff ℝ (↑n) c
hr : ContDiff ℝ ↑n fun x ↦ (f x).rIn
hR : ContDiff ℝ ↑n fun x ↦ (f x).rOut
hg : ContDiff ℝ (↑n) g
⊢ ContDiff ℝ ↑n fun x ↦ ↑(f x) (g x)

Tactic:
contDiff_iff_contDiffAt

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace ℝ X
inst✝ : HasContDiffBump E
n : ℕ∞
c g : X → E
f : (x : X) → ContDiffBump (c x)
hc : ∀ (x : X), ContDiffAt ℝ (↑n) c x
hr : ∀ (x : X), ContDiffAt ℝ (↑n) (fun x ↦ (f x).rIn) x
hR : ∀ (x : X), ContDiffAt ℝ (↑n) (fun x ↦ (f x).rOut) x
hg : ∀ (x : X), ContDiffAt ℝ (↑n) g x
⊢ ∀ (x : X), ContDiffAt ℝ (↑n) (fun x ↦ ↑(f x) (g x)) x
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₁, p₂, p₃}

Tactic:
by
  refine (collinear_insert_insert_insert_left_of_mem_affineSpan_pair h₁ h₂ h₃).subset ?_
  gcongr; simp

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E 0)
h : ‖f‖ = 0
i : α
⊢ {i | ¬↑f i = 0} = ∅

Tactic:
simpa [lp.norm_eq_card_dsupport f] using! h

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
hp✝ : p ≠ 0
c : 𝕜
f : ↥(lp E p)
hp : 0 < p.toReal
inst : NNNorm ↥(lp E p)
coe_nnnorm : ∀ (f : ↥(lp E p)), ↑‖f‖₊ = ‖f‖
⊢ ‖c • f‖₊ ^ p.toReal ≤ ‖c‖₊ ^ p.toReal * ‖f‖₊ ^ p.toReal

Tactic:
have hLHS := lp.hasSum_norm hp (c • f)

Goal state after:
case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
hp✝ : p ≠ 0
c : 𝕜
f : ↥(lp E p)
hp : 0 < p.toReal
inst : NNNorm ↥(lp E p)
coe_nnnorm : ∀ (f : ↥(lp E p)), ↑‖f‖₊ = ‖f‖
hLHS : HasSum (fun i ↦ ‖↑(c • f) i‖ ^ p.toReal) (‖c • f‖ ^ p.toReal)
⊢ ‖c • f‖₊ ^ p.toReal ≤ ‖c‖₊ ^ p.toReal * ‖f‖₊ ^ p.toReal
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case top
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
i : α
x : E i
this : Nonempty α
hp : 0 < ∞
⊢ ‖lp.single ∞ i x‖ = ‖x‖

Tactic:
| top =>

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
t : Set H
⊢ DifferentiableOn 𝕜 f t ↔ ∀ (i : ι), DifferentiableOn 𝕜 (fun x ↦ (f x).ofLp i) t

Tactic:
have := Fintype.ofFinite ι
  rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_differentiableOn_iff, differentiableOn_pi]
  rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 ↑f)
i : ι
hi : ‖F i‖ ≤ C
hC : 0 ≤ C
hp : p < ∞
⊢ ‖f‖ ≤ C

Tactic:
· have : 0 < p := zero_lt_one.trans_le _i.elim
    have hp' : 0 < p.toReal := ENNReal.toReal_pos this.ne' hp.ne
    apply norm_le_of_forall_sum_le hp' hC
    exact sum_rpow_le_of_tendsto hp.ne hCF hf

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
_i : Fact (1 ≤ p)
F : ℕ → ↥(lp E p)
hF : CauchySeq F
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) atTop (𝓝 ↑f)
ε : ℝ
hε : 0 < ε
hε' : {p_1 | ‖p_1.1 - p_1.2‖ < ε} ∈ uniformity ↥(lp E p)
n : ℕ
hn : ∀ᶠ (l : ℕ) in atTop, ‖(fun f ↦ F n - f) (F l)‖ < ε
a : α
⊢ Tendsto (fun i ↦ id (fun i ↦ ↑((fun f ↦ F n - f) (F i))) i a) atTop (𝓝 (↑(F n - f) a))

Tactic:
exact (hf.apply_nhds a).const_sub (F n a)

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case mp
α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
⊢ (∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑K * dist x y) →
    ∀ (i : ι), ∀ x ∈ s, ∀ y ∈ s, dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y

Tactic:
intro hfl i x hx y hy

Goal state after:
case mp
α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
hfl : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑K * dist x y
i : ι
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y
Module: Mathlib.Analysis.InnerProductSpace.Calculus

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : NormedSpace ℝ E
f g : ℝ → E
f' g' : E
x : ℝ
⊢ HasDerivAt f f' x → HasDerivAt g g' x → HasDerivAt (fun t ↦ ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x

Tactic:
simpa only [← hasDerivWithinAt_univ] using HasDerivWithinAt.inner 𝕜

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case refine_3
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
⊢ Tendsto (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)) (l.curry (𝓝 x)) (𝓝 0)

Tactic:
refine Tendsto.mono_left ?_ curry_le_prod

Goal state after:
case refine_3
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
⊢ Tendsto (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)) (l ×ˢ 𝓝 x) (𝓝 0)
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
⊢ ContinuousOn (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
by_cases! H : ∀ p ∈ s, ∀ x, g p x = 0
  · apply (continuousOn_const (c := 0)).congr
    rintro ⟨p, x⟩ ⟨hp, -⟩
    apply integral_eq_zero_of_ae (Eventually.of_forall (fun y ↦ ?_))
    simp [H p hp _]
  have : LocallyCompactSpace G := by
    rcases H with ⟨p, hp, x, hx⟩
    have A : support (g p) ⊆ k := support_subset_iff'.2 (fun y hy ↦ hgs p y hp hy)
    have B : Continuous (g p) := by
      refine hg.comp_continuous (.prodMk_right _) fun x => ?_
      simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp
    rcases eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_addGroup hk A B with H | H
    · simp [H] at hx
    · exact H
  /- Since `G` is locally compact, one may thicken `k` a little bit into a larger compact set
  `(-k) + t`, outside of which all functions that appear in the convolution vanish. Then we can
  apply a continuity statement for integrals depending on a parameter, with respect to
  locally integrable functions and compactly supported continuous functions. -/
  rintro ⟨q₀, x₀⟩ ⟨hq₀, -⟩
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := (-k) +ᵥ t
  have k'_comp : IsCompact k' := IsCompact.vadd_set hk.neg t_comp
  let g' : (P × G) → G → E' := fun p x ↦ g p.1 (p.2 - x)
  let s' : Set (P × G) := s ×ˢ t
  have A : ContinuousOn g'.uncurry (s' ×ˢ univ) := by
    have : g'.uncurry = g.uncurry ∘ (fun w ↦ (w.1.1, w.1.2 - w.2)) := by ext y; rfl
    rw [this]
    refine hg.comp (by fun_prop) ?_
    simp +contextual [s', MapsTo]
  have B : ContinuousOn (fun a ↦ ∫ x, L (f x) (g' a x) ∂μ) s' := by
    apply continuousOn_integral_bilinear_of_locally_integrable_of_compact_support L k'_comp A _
      (hf.integrableOn_isCompact k'_comp)
    rintro ⟨p, x⟩ y ⟨hp, hx⟩ hy
    apply hgs p _ hp
    contrapose hy
    exact ⟨y - x, by simpa using hy, x, hx, by simp⟩
  apply ContinuousWithinAt.mono_of_mem_nhdsWithin (B (q₀, x₀) ⟨hq₀, mem_of_mem_nhds ht⟩)
  exact mem_nhdsWithin_prod_iff.2 ⟨s, self_mem_nhdsWithin, t, nhdsWithin_le_nhds ht, Subset.rfl⟩

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case pos
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
hg : ∀ x ∈ ball x₀ R, g x = g x₀
t : G
ht : t ∈ support f
h2t : ‖t‖ < R
⊢ (L (f t)) (g (x₀ - t)) = (L (f t)) (g x₀)

Tactic:
specialize hg (x₀ - t)

Goal state after:
case pos
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
t : G
hg : x₀ - t ∈ ball x₀ R → g (x₀ - t) = g x₀
ht : t ∈ support f
h2t : ‖t‖ < R
⊢ (L (f t)) (g (x₀ - t)) = (L (f t)) (g x₀)
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (𝓝 x).HasBasis (fun r ↦ 0 < r) (closedBall x)

Tactic:
toEuclidean.toHomeomorph.nhds_eq_comap x

Goal state after:
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (Filter.comap (⇑toEuclidean.toHomeomorph) (𝓝 (toEuclidean.toHomeomorph x))).HasBasis (fun r ↦ 0 < r) (closedBall x)
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (𝓝 x).HasBasis (fun r ↦ 0 < r) (ball x)

Tactic:
toEuclidean.toHomeomorph.nhds_eq_comap x

Goal state after:
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (Filter.comap (⇑toEuclidean.toHomeomorph) (𝓝 (toEuclidean.toHomeomorph x))).HasBasis (fun r ↦ 0 < r) (ball x)
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
K : PositiveCompacts G := Classical.arbitrary (PositiveCompacts G)
this : haar (interior ↑K ∩ univ) = 0
⊢ False

Tactic:
inter_univ

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
K : PositiveCompacts G := Classical.arbitrary (PositiveCompacts G)
this : haar (interior ↑K) = 0
⊢ False